- Email: [email protected]
Towards unified theory of 2d gravity
NUCLEAR Nuclear Physics North-Holland
P H VS I CS B
B380 (1992) 181—240
Towards unified theory of 2d gravity S. Kharchev, A. Marshakov and A. Mironov Department of Theoretical Physics ~, P.N. Lebedec Physical Institute, Leninsky Prospect 53, 117 924 Moscow. Russia
A. Morozov
**
Institute of Theoretical and Experimental Physics, Bol. Cheremushkinskaya St. 25, 117 259 Moscow, Russia
A. Zabrodin Institute of Chemical Physics, Kosygina St. 4, 117334 Moscow, Russia Received 15 November 1991 Accepted for publication 6 February 1992
We introduce a new one-matrix model with arbitrary potential and matrix-valued background field. Its partition function is a r-function of KP hierarchy, subjected to a kind of -~ constraint. Moreover, the partition function behaves smoothly in the limit of infinitely large 1~~ . this partition function becomes a r-function of matrices. IfKPthehierarchy, potentialobeying is equala set to of X K-reduced 71k~ algebra constraints identical to those conjectured for the double-scaling continuum limit of the (K — I) matrix model. In the case of K 2 the statement reduces to an earlier established relation between the Kontsevich model and ordinary 2d quantum gravity. The Kontsevich model with generic potential may be considered as an interpolation between all the models of 2d quantum gravity, with c <1 preserving the property of integrability and the analogue of the string equation.
1. Motivation and results 1.1. INTRODUCTION
Matrix models play an increasingly important role in the theory of quantum gravity, being a reasonably simple realization of Regge calculus, adequate at least in essentially topological theories. Their conventional application to string theory is based on identification of the critical points of the particular “double-scaling” limit [3—6]of hermitean multimatrix models [7] with the c < I minimal conformal field * **
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—
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theories [8] coupled to 2d gravity. Such models are often supposed [9] to exhaust the entire set of consistent bosonic 2-dimensional string models, while bosonic strings beyond two space-time dimensions are presumably unstable [10]. Technical methods, based on the application of matrix models, appeared rather successful not only in the framework of perturbation theory, but they allow one to estimate the entire sums of perturbation series, thus raising the study of string models to a qualitatively new level. On the other hand, matrix models appeared to possess unexpectedly deep internal mathematical structure [91.In fact, if matrix models are used to describe interpolation between different critical points (i.e. particular c < 1 string models), the interpolating flows are mutually commuting, i.e. the entire pattern of flows constitutes an integrable hierarchy (some reduction of the Kadomtsev—Petviashvili (KP) system). This integrable structure indicates that the somewhat artificial description in terms of matrix integrals should possess a more invariant algebro-geometrical description, which could be used for formulation of the as yet unknown dynamical principle of string theory, unifying all available string models into a distinguished and (hopefully) physically important entity. The main obstacles on the way from matrix models to such a general principle are: (i) the sophisticated intermediate step: the double-scaling limit, (ii) the somewhat obscure origin of integrability: it is not a priori obvious why matrix integrals with generic “potentials” are r-functions of (reduced) KP hierarchies, (iii) the non-universal description of different c < I string models: some of them are nicely unified as different critical points of a particular multimatrix model, but there is no nice interpolation between critical points of models with a different number of matrices. The naive way out is to use the ~-matrix model, but it is no longei- described in terms of finite-dimensional matrix integrals, and, more important, reduction to the case of a particular multimatrix model is too much singular. In this paper we propose a kind of resolution of all these three problems. Namely, we argue that there is a new one-matrix model [11] (which we call generalized Kontsevich model (GKM)), which (i) for a specially adjusted potential describes properly the double-scaling limit of any multimatrix model, (ii) has the partition function, which is a KP T-function, properly reducible at the points associated with multimatrix models, and satisfies an additional equation, which reduces to conventional constraints for multimatrix models, (iii) allows continuous deformation of potential, changing it from the form associated with a given multimatrix model to those corresponding to the others. This makes the study of GKM a natural step in the development of string theory. ~‘
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In the remaining part of this section we describe GKM and its relation to multimatrix models more explicitly. The proofs are presented in sects. 2—4. For a more condensed presentation of our results see ref. [11]. 1.2. GENERALIZED KONTSEVICH MODEL
This paper is devoted to the study of the one-matrix model in an external matrix field A, essentially defined by the integral over the N X N matrix X: 57)[A]~fdXexp[Tr~(X)-i-TrAX],
(ii)
with arbitrary “potential” ~©(X) and dX= 117~~ dX11. Such integrals arise during the consideration of various problems [2,12—14],but this time our purpose is to use (1.1) in order to define a unir’ersal matrix model, which could describe interpolation between all the multimatrix models and their critical points, as explained in subsect. 1.1 above. Namely, we present strong arguments that the partition function of such a universal model may be identified with Z~[M], where fexp[_U(M,Y)]
dY
z~’}[M]=—
,
(1.2)
fexp[_LJ2(M, Y)] dY with U(M, Y) =Tr[~(M+ Y) —~z(M) ~Y’(M)Y], —
U2(M, Y)
=
E limt) —~U(M,cY),
(1.3) (1.4)
c—’
(i.e. U 2 contribution to U(M, Y)). 2(M, is the Y After theY) shift of integration variable X=M+Y
(1.5)
the numerator in (1.2) may be rewritten in terms of (1.1): fe~©M~
dY=
~
(1.6)
The denominator in eq. (1.2) is a simple gaussian integral—a peculiar matrix generalization of [~v~”( 1L)} — i /2
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We shall call Z~’[M], defined by eq. (1.2), the partition function of the generalized Kontsevich model (GKM). The reason for this is that for the special choice of potential *: (1.7) ~2(X) =X3/3, eq. (1.2) becomes the partition function of the original Kontsevich model [15]: N(X)
fdYexp[-_(1/3)
=
Tr Y3_TrMY2]
Z~1[M]= fdY exp[—Tr MY2]
fdXexp[_(1/3) =exp [—(2/3)
TrX3+TrM2XI
Tr M3]
.
(1.8)
fdX exp[—Tr MX2] Eq. (1.8) has a natural generalization, which is also a particular case of (1.2): if
~‘~(X) =7K(X)
=XKi/(K+
1)
(1.9)
eq. (1.2) becomes K Z~[MI~exP[_K+lTrM~]fdXexP
x
{fdX exP[_
_Tr~+TrMKX
(1.10)
~Tr{EMaXMbX~}
We shall argue that Z~1[M], defined by (1.10), is identical to the square root of the partition function of the (K 1)-matrix model in the double-scaling limit with —
=
1 —Tr M”,
n
±0
mod K,
(1.11)
playing the role of generalized Kazakov’s time variables [3,16]:
~ *
(1.12)
We do not specify the integration contour in matrix integrals (1.1), (1.2), since all that we are going to discuss does not depend on it. To make contact with the literature, note that for potential = X~1/(K + 1) integrals should be over hermitean or antihermitean matrices if K + I is even or odd respectively.
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where [7] Fj~=
tim
~
double scaling
(1.13)
=
(the term MKMK+l in the exponent should be omitted; the change from discrete time variables {t~}to continuum (T~}is implicit in performing the double-scaling limit, see refs. [16,17]). Since potential IX) can be continuously deformed from one K in eq. (1.9) to another, the GKM (1.2) as a corollary of (1.12) may be considered as continuous interpolation between all the models of 2d gravity with c < 1, described by particular multicritical points of particular multimatrix models (let us remember that for K fixed the pth multicritical point is defined by the condition [I] that all
1,
=
0, except for T1 and TK+,,,
(1.14)
and according to (1.11) this is a constraint on the form of the matrix M). This is the sense in which the GKM (1.2) may be considered as the universal partition function of bosonic string models. We shall not discuss the implications of this fact here, instead we shall concentrate on the arguments in favour of this conclusion, which is entirely based on the identity7~[M]itself (1.12) (and also it motivated is usually by a the KP nice features of the to interpolating function see Z~, below). Our strategy is to try to i--function, subjected a 2’ constraint, understand as much as possible about Z~[M], on the l.h.s. of (1.12), and compare these results with what is conjectured in ref. [1] about ~/I~~on the r.h.s. —
1.3. PROPERTIES OF Z121
In order to motivate our considerations let us recall first what is known about Z(K}[M] in the particular case of K= 2. Expression (1.2) has been derived in ref. [15] as a representation of the generating functional of intersection numbers of the stable cohomology classes on the universal module space, i.e. it is defined to be a partition function of Witten’s 2d topological gravity [18]. In ref. [2] (see also ref. [13,19] for alternative derivations) it was shown that as N —s Z~}considered as a function of time variables (1.11), satisfies the set of Virasoro constraints ~,
n
~ kTk kodd
(1.15)
a aTk± 2fl
+~
—1,
~ a+b= —2n a ,b odd a nd > (I
a+h=2n dTa3Ti, a ,h odd and> I)
aTabTh
+
~
—
.
T5+21
(1.16)
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(Notations are slightly changed as compared to ref. [2]. This derivation is partly reproduced in subsect. 4.2 below.). Sometimes it may be convenient to consider the appearance of the last item on the r.h.s. of (1.16) as a result on an additional shift of time variables, Tii ~
n
—
ii
~n,K±t,
(117
.
however, these T-times are defined in a K-dependent way and do not seem to have too much sense. Constraints (1.15) are exactly the equations (see refs. [1,20,16]) imposed on ~ Moreover, it is very plausible that they possess a unique solution, so that (1.15) in fact implies that (1.18) An alternative solution to the constraints (1.15) is given [1,20] by a galilean-invariant KdV i--function [21], and, relying upon the same belief that the solution is unique, one concludes that Zc~21is a i--function of the KdV hierarchy, subjected to an additional 5f’~ constraint: (1.19)
Z~21=TKdV=T121,
(1.20) The notation i-121 reflects the fact that are TKdV may be considered as a KP i--function, evaluated at specific points of grassmannian (see subsect. 2.5 below for details), corresponding to the 2-reduction of KP hierarchy. Generically, the i--function of K-reduced hierarchy, i-{K}(j~),is the KP i--function at peculiar points of the grassmannian and possesses the following property:
...TKI, 0,
TK±l...T2KI,
0,
~
(1.21)
This means that the entire dependence of generic i-{1’~T} of all variables TflK (i.e. 7~ with n 0 mod K) is exhausted by the simple exponent =
ex~(~anxi~x)
(1.22)
with time-independent parameters aflK. Moreover, this factor is actually absent in the case of a GKM partition function: an K[Z~] 0. =
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The significance of the just reported results concerning K 2 is two-fold. First, they establish the relation (1.7) between two different models of 2d gravity: Witten’s topological gravity [19], represented by Z~.S21[15],and ordinary quantum 2d gravity [4—61,represented by ~ This was the main emphasise of ref. [2]. The second implication, and it is the one of interest for us in this paper, is the possibility to describe the sophisticated double-scaling limit of matrix model in terms of a completely different matrix model (1.2). It is important that Z~©is in fact a “smooth” function of N as N and all the properties of Z~~<) are just the same as at finite N’s in variance with the double-scaling limit for ordinary matrix models, the limit N —s ~ for GKM is non-singular. Also, to avoid misunderstanding, we should emphasize that what we mean by the double-scaling limit here is not the limit, describing a particular fixed point, but the entire pattern of all the flows between various fixed points of a given (K 1)-matrix model, so that the partition function TJ.~’1is indeed a function of all time variables (1.3). =
—~ ~,
—
—
1.4. THE CASE OF ARBITRARY K
In order to generalize the identity (1.15) to the case of arbitrary K, i.e. to prove (1.12), one can try to prove either an analogue of (1.15) or that of (1.19), (1.20) and compare the result with what is expected for ~TJ~ I) This will be our strategy below. Namely, in sect. 2 a simple and promising formalism is developed, based on interpretation of (1.11) as a Miwa transformation, and we use it to prove that any Z~71[M], defined by (1.2) and (1.11) (i.e. with arbitrary 7~(X))is in fact a KP i--function in particular, it satisfies bilinear Hirota difference equations. Moreover, in the grassmannian parametrization of KP i--functions the point of the grassmannian is specified by the choice of potential ~©(X), so that multimatrix models with different K are [due to eq. (1.12)] associated with different points of the grassmannian. (In terms of the universal module space these points are in fact associated with a certain infinite-genus hyperelliptic surface for K 2, and with certain infinite genus abelian coverings of degrees K in the general case. In this sense interpolation with the changing ?(X) is some flow at infinity of the universal module space.) What is specific for a given K is that whenever 7~(X)is a homogeneous polynomial of degree K+ 1, i.e. if it is of the form (1.9), the KP i--function becomes independent of all TK,, i.e. acquires the property (1.21) and may he considered as a i--function i-1”~ of a K-reduced hierarchy. This will also he explained in more detail in sect. 2. Though any Z~71[M] defined by (1.2) is a KP i--function, the inverse statement is not correct. Z~/1[M] is subjected to the infinite set of constraints, essentially implied by the Ward identities for the integral (1.1) [12,2]: —
=
{Tr e(A)[7~’(~)
-~1}~~=°
(1.23)
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with any c(A). These identities result from invariance of (1.1) under any shift of integration variable X—SX+E(A). Eq. (1.23) seems to provide a complete set of differential equations, unambiguously defining ~ (up to a A- and N-independent constant). This statement is the actual basis for all of the arguments, which make use of the uniqueness of solutions to any constraints. As shown in ref. [2] in the particular case of K = 2 the entire set of such identities with all possible E(A) is equivalent to (1.15) and thus implies (1.19) and (1.20). In fact in sect. 2 we follow a somewhat different logic and give an alternative direct derivation of relations
z~j~1=i-~’<}
Z~/~}=T,
(1.24)
for any X) and K, so it remains only to prove the analogue of (1.20). Such a direct proof is presented in sect. 3, moreover it is valid not only for potentials of the form (1.9), but for any ~(X):
a
1
E
Tr ~V”(M)M”~’
n1
1
ai~ ~/~“(~.)
—~“(~.)
a Z1~1=0,
-
~
(1.25)
(j.t 1)7~
(j.t1)
iij—j.t1
aT1
where p., are eigenvalues of the matrix M. For potentials (1.9) ‘(f~lacquires the conventional form of 5~’~[see eq. (1.28) below] and may be easily supplemented by other generators of Virasoro algebra. Although eqs. (1.24) and (1.25) provide a complete description of the partition function of GKM, the study of direct consequences of Ward identities (1.23) also seems interesting. In particular, it should lead to a direct derivation of the analogue of Virasoro (1.15).parametrizes It also merits noting the meaning of 1[M] constraints is that ~V(X) only some that restricted subset of constraints on Z,~/’ the universal module space. Only this subset is associated with matrix models and interpolations between them. This is also clear from the fact that ~©(X) = is naturally parametrized by an infinite vector (~7~} rather than by a matrix Am,,, which would be natural for the entire grassmannian. Remarkably enough the “dimension” of this subset is just the same as that of the space of time variables (the space of potentials is parametrized by a vector {~}, while that of time variables is parametrized by (1~,,}or by eigenvalue vector {p>,} of the matrix M). This clearly suggests that a formalism should exist for such a restricted set of i--functions, which would treat potential and times in a more symmetrical fashion (see also the discussion in subsect. 3.4). *
*
Let us also remark that such symmetry exposes itself in the fact that the degree of potential K and the order of the multicritical p + K (fixed by the choice of times) give the theory of minimal matter with c = I —6(p — K)2/pK coupled to 2d gravity, and, thus, p and K should enter the theory on equal footing.
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Implications of Ward identities (1.23) for Z~J<1[MI with K> 2 are discussed to some extent in sect. 4. This is a straightforward calculation, though much more tedious than in the simplest case of K = 2. We argue that the proper generalization of eq. (1.15) for the case of K 3 is given by the following set of constraints: na—2:
~
(
~
(3k
a — l)T
I~’3k±3n+
3k..
a,b~0, ~
~ a+h=3n
7/3/~3
Z~
0
=
3a+2
n>—2; (3k
—
2)~2k2~i~2fl +
k>t-f-~I,,~,(,
a
~
~
Z,.
~
a+h=3n
=
0,
3a+I
(1.26)
a,b>~0, n>—3.
Here ~ stands for the Knth harmonics of the jth generator of Zamolodchikov’s WK algebra, and the proper notation would be ~j.x-(i){K) or The way to express these quantities through free fields is described in ref. [1]; for example, ~
~
kTklT/
k,/>1
2
a aTk+/+
~
a
a+h=k+3n
aTaaTh
+3~kTk
3fl
k>l
3
a+h+c’=3n +
a aTaaThaTc
~
aT,,bT~,cT,,; a, b, c>0, a, b, c, k, ls~0 mod 3
(1.27)
a±h-1-c= —3n
and
a
1 k
K
a2
1
aTk±Kfl 2Ka+b=KnaTaaTb a, 6
+
1 ~
E
>0
(K—1)(K+1)
aT,,bT 6
+
24K
~
(1.28)
a+h-’—Kn
a, 6 > 1)
(Note that in order to make formulas a bit more compact, we expressed all these operators through ~ rather than 1, variables. These are defined in (1.17), a/aT = a/aT.) The analogue of (1.26) for K> 3 is a more or less obvious generalization of (1.26), involving all the generators ~ ~ 1) , ~ of W~-a1gebra with the “coefficients” made from T’s and (a/aT)’s. The form of these operators is a bit reminiscent of the 7r-operators, introduced in refs. [14,22] as ingredients of Ward identities in discrete multimatrix models.
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Eqs. (1.26) (and their analogues for solution of the simpler set of equations, ~Z~1=0,
K>
3) are obviously resolved by any
k=2,3,...,K;
n~1—k
(1.29)
[eq. (1.20) is the particular case: k = 2, n = — 1]. On the other hand the constraints (1.26) are equivalent to (1.23) for the particular form of potential (1.9), and thus are supposed to possess a single solution. Therefore from this point of view it also looks plausible that (1.26) (and its counterpart for any K) is simply equivalent to (1.29). Unfortunately we do not provide an explicit proof of this equivalence (though we could rely, of course, upon the indirect proof from ref. [1], deducing (1.29) from eqs. (1.19), (1.20), which are proved in subsect. 2.3 without any reference to (1.23)). If one believes in this transition from (1.26) to (1.29), we can finally return to multimatrix models. Namely, in ref. [1] it was suggested that ~/TJ~11 (i) is a i--function of K-reduced hierarchy, (ii) satisfies the constraint n~ —1,
(1.30)
and, as a corollary of (i) and (ii), (iii) satisfies the entire set of eqs. (1.29): ~k~YT’i~~=0
k=2,3
K;
n~l—k.
(1.31)
Now it remains to use the above-mentioned assertion that this system has a unique solution in order to arrive at our identification (1.32) and thus to the central idea of this paper: unit’ersality of GKM (1.2). 1.5. COMMENTS
Before we proceed to actual derivation of all these statements, let us comment briefly on the very status of the constrains (1.29) imposed on ~T’J~’1.In order to honestly derive such constraints on the lines of ref. [161it is first necessary to find out their counterparts (loop equations) in the discrete multimatrix models (these are expressed in terms of the so-called ~ operators in refs. [2,22]), then carefully perform the appropriate reductions (including identification of all the K 1 a priori different potentials), and take the double-scaling limit. The actual form of the ~/constraints is very similar to (1.26) and therefore it may happen that the —
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final constraints would arise just in the form of (1.26) rather than (1.29). This could make the relation (1.26) even more important. After this general description of our reasoning and results it remains to prove that (i) Z,~/~1as defined by (1.2) (the limit N—*~ is smooth) is indeed a KP T-function; (ii) for any iV(X) it is subjected to an additional constraint (1.25); (iii) if IX) = const .X~I, Z~’© in fact becomes a i--function i-~’~ of the K-reduced KP hierarchy; (iv) in this case the entire set of constraints like (1.15) for K = 2 and (1.26) for K = 3 are also imposed on Z~K}. The proof of points (i) and (iii) is given in sect. 2, point (ii) is discussed in sect. 3 and point (iv) in sect. 4. While (i), (ii), (iii) are proved in full generality — for any potential ~©~(X) — the universal form of the constraints (iv) is not completely understood. Moreover, the proof of(iv) for the case of X)=X~/(K+ 1), as presented in sect. 4, is not complete even in the simplest case of K = 3. Also the proof of (iv) is not independent of (iii), since eq. (1.21) is used. The deep understanding of these ft~.constraints and their universal formulation remains the first obvious problem for GKM to be resolved in the future. The second obvious problem is to find out the algebro-geometrical origin of the model (1.2) itself.
2. Partition function of the GKM and KP r-function 2.1. SHORT SUMMARY
The purpose of this section is to prove that (A) The partition function Z~<)[M], if considered as a function of time variables =
—tr M’, n
(2.1)
is a KP i--function for any value of N and any potential ~[X]. (B) As soon as ~[X] is a homogeneous polynomial of degree K+ 1, Z~<1[M] is in fact a i--function of K-reduced KP hierarchy. Moreover, actually, aZ11~/aT,,K= 0. In order to prove these statements, first, in subsect. 2.2 we rewrite the r.h.s. of eq. (1.2) in terms of the determinant formula (2.21) Z~.7~)[Lv!]
det =
‘
i,
j
=
1
N.
(2.2)
We show in subsect. 2.3 that any KP i--function in the Miwa parametrization has
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the same determinant form. Finally, as a check of self-consistency, we prove in subsect. 2.4 that any determinant formula (2.2) with any set of functions (41(p.)} satisfies the Hirota difference any determinant formula (2.2) with any set of functions {41(p.)) satisfies the Hirota difference equation. A brief description of grassmannian (or Universal module space) language is given in subsect. 2.5. The point of the grassmannian corresponding to the i--function (2.2) is defined by the infinite-vector function {~1(p.)}.Remarkably, the results of ref. [23] which concern identification of the points in the grassmannian, associated with multi-matrix models, are immediately reproduced from the explicit expressions for the basis {~1(p.)}.This exhausts our discussion of (A). The main this which distinguishes matrix models from the point of view of solutions to the KP hierarchy is that the set of functions {~1(p.)}in (2.2) is not arbitrary. Moreover, this whole infinite set of functions is expressed in terms of a single potential ~[X] (i.e. instead of arbitrary matrix A1 in 4.(p.) = EA11p.~we have here only a ti’ector ~ or 2~[p.] = E~p.’).This is the origin of and other ~:constraints (which in the context of KP hierarchy may be considered as implications of 2’.~).All these constraints are in fact contained in the Ward identity (1.23). While a detailed discussion of these constraints is postponed to subsect. 3.4, in subsect. 2.6 we discuss another implication of (1.23) arising in the case of polynomial 2’~[X]. Namely, whenever ~[X] is a homogeneous polynomial of degree K + 1, ~±~(p.) in (2.2) can be substituted by p.~1(p.)and in such points of the grassmannian the KP i--function is known to possess the property (1.21) and thus may be considered as a i--function i-~ of K-reduced KP hierarchy. In subsect. 2.7 we prove that the factor (1.22) is actually absent in GKM. Further discussion of (B) will be given below. ~/
2.2. FROM 6KM TO DETERMINANT FORMULA
Numerator of (1.2) We begin from evaluation of the integral (1.1): .9~}[A]mfdXexp(—Tr[?~(X)—TrAXI}.
(2.3)
If eigenvalues of X and A are denoted by {x,} and (A,) respectfully, this integral can be rewritten as
[~
fdx1 exp[ -~(x~)+ A1x1]~(X).
(2.4)
VN stands for the volume of the unitary group U(N) and is unessential in what follows; ~(X) and i.1(A) are Van der Monde determinants, e.g. L1(X) = 11,>1(x~— xi). The transformation, leading from (2.3) to (2.4), is a trick, familiar from the
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study of multi-matrix models in the formalism of orthogonal polynomials [24,7,25], and we do not dwell on this point. For what follows it is very important that the term Tr AX in (2.3) is linear in X (were it instead, say, Tr AX”, the polynomial factor .~iAIX)at the r.h.s. of eq. (2.4) would be substituted by z.12(X)/~i(X°) and our reasoning below would be inapplicable). The r.h.s. of (2.4) is proportional to the determinant of the N x N matrix det(
~i~(A)
11)F,(AJ)
(2.5)
with ~±~(A) mfdxx
exp[—~(x) +Ax]
=
(~~Fi(A).
(2.6)
Note that F1(A) =3’~?Ii~[A].
(2.7)
If we recall that in GKtvI (2.8)
A=~’(M)
and denote the eigenvalues of M by {p.1}, eq. (2.6) acquires the form det ~(p.) =
,
,
,
fl(~ (p.~)-7
(2.9)
(p.1))
i >1
where (2.10) Denominator Proceed now to the denominator of (1.2): I~[M}
fdX e:(M~.
(2.11)
Making use of the U( N) invariance of the Haar measure d X one can easily diagonalize M in (2.11). Of course, this does not imply any integration over angular variables and provides no factors like .i(X). Then to evaluate (2.11) we just have to use the obvious rule of gaussian integration,
fdX ex~(_ ~~1x11x11)
flu_t/2
(2.12)
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(an unessential) constant factor is omitted), and substitute the explicit expression for (/1(M). If the potential is represented as a formal series, ~(X)=
E~/~X”,
(2.13)
0
(and thus is supposed to be analytic in X at X= 0), eq. (1.4) implies that U7(M, X)
=
(n
~
+
1)~+~{ ~
n=0
Tr MUXMhX},
a+h~-n—
I
and
E
~(n+1)7~±i{
U~
a±h=n—I
n=()
/L/L~ =
~
(n+1)~,±~—~- =
‘(p.1)—~’(p.1)
0=0
Coming hack to (1.2), we conclude that
1[M]=exp{Tr[ii~(M)
—M1V’(M)]}
Z
u [det ~
I-I
1(p.1)]
[det cP.(p..)j ~1(M) exp[~V(p. 2 [Liç1]~’ 1) — =
“
— ~
,
(p.1))
fls(p.~) i=t
N
fls(p.~).
=
s(p.)
(~(p.1) ,
j>~
Here s(p.,)
,1N[~(M)1
p.11~’(p.,)],
[~‘(p.)]
1/2
(2.14)
i.e.
exp[7 ~(p.) — p.~’(p.)].
(2.15)
The product of s-factors at the r.h.s. of (2.16) can be absorbed into ~-functions
Z~.f~[M]= [det
cP1(p.1)]/~(M),
(2.16)
where cP,(p.)
=s(p.)cP1(p.).
(2.17)
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195
Kac—Schwarz operator From eqs. (2.6), (2.10) and (2.15) one can deduce that c1(p.) can be derived from the basic function t1 1(p.) by the relation =
[~/.“(p.)]
y + p.)’’ e~~> dy
t/2f(
A4(p.)~1(p.), (2.18)
=
where U(p., y) is defined by eq. (1.3) and AU }(p.) is the first-order differential operator
a
exp[77(p.) —p.~’(,n)J a exp[—~(p.) +p.~’(p.)] 2 hp. =
1/)
a ~‘~“(p.) ap. 1
=
[7.n(p.)~i/
—+p.—
2[~”(p.)}
(2.19)
~.
In the particular case (1.9) 1 k-I
=
k—I
a —
k
kp. dp. 2kp. coincides (up to the scale transformation of p. and A{~}(p.))with the operator which determines the finite dimensional subspace of the grassmannian in ref. [23] (we shall return to this point in subsect. 2.5). We emphasize that the property ~
(p.) =A~( p.)~ 1(p.),
(~+(A)
=
~F1(A))
(2.20)
is exactly the thing which distinguishes partition functions of GKM from expression (2.2) for a generic i--function, =
[det ~1(p.,.)j/i(M).
(2.21)
with arbitrary sets of functions ~1(p.). In sect. 3 we demonstrate that the quantity (2.21) is in fact a KP i--function in Miwa coordinates. Implications of additional constraint (2.20) will be discussed in subsects. 2.6 and 3.4. N-dependence Before we proceed to a discussion of Miwa coordinates, two more things about formula (2.21) deserve mentioning. First, the entire set {c~(p.)}is naturally N-independent and infinite. It is reasonable to require also that ~‘s are linear independent and ~(p.)/~(p.)
=c1p.~(l + 0(1/p.)),
(2.22)
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with p.-independent c• [this is true for functions (2.7)]. Then the set {41(p.)) is identified as projective coordinates of a point of the 1[M] grassmannian, see subsect. for an infinitely large 2.5. TheM.r.h.s. of eq. (2.21) to naturally matrix In order to return the caserepresents of finite Ni-~’ it is enough to require that all eigenvalues of M, except p.~,. p.~’tend to infinity. In this sense the function i-~[M] in (2.21) is independent of N; the entire dependence on N comes from the argument M: N is the quantity of finite eigenvalues of M. As a simple check of consistency, let us additionally carry p.~to infinity in (2.21), then, according to (2.22) . .
det~~ 1(p.J)
~
.
det~~(p.1) (1 .
+
O(1/p.~))
and ~N(M)
(p.~)N_t~~1(M)(1
+
O(1/p.~)),
so that [c~cb1(p.~)](1 + O(1/p.N)).
~
(2.23)
This is the exact statement about the N-dependence of i-N~ Actually in GKM cN4~~(p.) = 1 at p. —~ In what follows we often omit the subscript N. ~.
Multiplicities The second remark concerns the form of the r.h.s. of (2.21), when some eigenvalues of M coincide [see also eq. (2.30) below]. If eigenvalue p., appears with the multiplicity p,, eq. (2.21) looks like det[~1(p.1)A171~~(p.1) ...
r[(p.1,
pj)]
II
i i (p.,
(2.24)
.
— p.1)
i >1
Notation in this formula is not very transparent: it is implied that the matrix in the numerator has rows of the form ~1(p.1), A1~1~1(p.1)
A(~/~1(p.1), ~1(p.2), A~~1~1(p.2) A[~1 ‘~1(p.2), ~~(p.3), A1>~1~1(p.3),...,
where operator A1.~is defined by eq. (2.19). Expression (2.24) will be used in subsect. 2.4 in the proof of the Hirota identity. Note also that if some p., ~, this may be treated as vanishing of the corresponding multiplicity, p, = 0. Such p., obviously do not contribute to (2.24). Thus, the value of N (the finite size of the —‘
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matrix M in GKM) may be interpreted as the number of p.’s which appear with non-vanishing multiplicities. 2.3. KP r-FUNCTION IN THE MIWA PARAMETRIZATION
The Generic KP i--function is a correlator of a special form: i-G{i,}
=
K0I:e~T,t,G~0)
(2.25)
with J(z) =~(z)t/i(z); G=:exp(~’,,,,,t/i,,,t/i,,):
(2.26)
in the theory of free 2-dimensional fermionic fields i/i(z), ~(z) with the action ft/i at/s. Vacuum states are defined by conditions t/’,,IO)=O,
n<0,
i/i,,I0~=0, n>0
(2.27)
dzU2. where tfi(z)= ~t/J,,z” dzl/2, t~(Z) ~ The crucial restriction on the form of the correlator, implied by (2.26) is that the operator :e~T,~J* G is the gaussian exponential (exponent is quadratic in fields), so that the insertion of this operator may be considered as a modification of the i/s — i/i propagator, and Wick’s theorem is usually applicable. Namely, the correlators Cr;((~i},
(A 1})
~
(2.28)
for any relevant G are expressed through the pair correlators of the same form: CG({p.1},
{A1})
=
det(11)CG{p./, A1}.
(2.29)
Operator G in eq. (2.28) can be safely substituted by the entire: e~’~”:G, but we do not need this for our purposes1’~~”: below. Instead with the help of the Miwa in (2.25) through insertions of fermionic transformation we shall express :e operators. Then (2.25) acquires the form of (2.28) and after that the application (2.29) provides a representation for i--function in determinant form, to be compared with eq. (2.21) for the partition function of GKM. There are slightly different forms of Miwa transformation. Usually one writes
(2.30) p., and treats the r.h.s. as an integral fp(p.)/p.” over a Riemann sphere with coordi-
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nate p. and singular function p(p.) =
Unified theory of 2d gra,ity L,p
15(p.
—
p.,). We use instead the transfor-
mation (2.1) (2.31)
T,,=—TrM”=—~~, n
interpreting p.’s as eigenvalues of the matrix M, all coming with unit multiplicities p, = 1 (while p,> 1 in (2.30) may be interpreted as a result of the coincidence of p, eigenvalues). The form (2.30) of the Miwa transformation is preferable from the point of view of invertibility and also is convenient for the proof of Hirota identities (see subsect. 2.4 below). The simplest way to understand what happens to the operator el~T,J~ after the substitution of (2.31), is to use the free-boson representation of the current J(z)=aq,(z). Then
~T,,J,,=L i
~nn.p.,
=E~(p.1)~
and :exp( ~~(p.1))
:=
fl(p.
fl:
1p.1)
i
1:.
(2.32)
e~
In the fermionic representation it is better to start from 1
1
n
p.~
1 p.~
(2.33)
instead of (2.31). Then
fl(Th-~1) :exp(ET~J,,)’=
‘1
1>1
-
-
fl~(~)ç~(p.~). (2.34)
i>i
In order to come back to (2.31) and (2.32) it is necessary to shift all Th’s to infinity. This may be expressed by saying that the left vacuum in (2.25) is substituted by (N~ ~K0(’(~)...~°(~).
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of 2d grality
The i--function (2.25) can be represented in various forms: =
(0 :exp( ~
0)
7~,J,):G
=
~i(M)
fl: e~’~:G JO)
- ‘(N
ii(Th~1) =
lim ~—‘~
=
lim ~
“~
-
-
11(p.,p.1)11(p.1p.1) 1>1 1>, C(;({p.j},
{p.1})
C~({~1},
{p.J)
(OJ fl~(f~1)q~(p.1)GJ0) i
(2.35
and applying Wick’s theorem (2.29) we obtain (~
.
i-~[M] = lim det(/f)
CG(j~1,p.1) -
det ~1(p.1)
(2.36)
=
C,(p.1,p.1)
~(M)
with functions (0 ~
~(~)t/(p.)G
JO).
(2.37)
Thus, we proved that the KP i--function in Miwa coordinates (2.2) has the determinant form (2.1). Below we shall need a more detailed expression of the right-hand side in eq. (2.35), when all points /1> tend to the same value 1/t with fixed t ~ 0. From (2.35) one can obtain i-,~[tJM]
fl(1_tp.,)N fl(p.1-p.1)
det
.
(i-I).
¶a~(0J~(t)~(p.J)GI0)
I>’
=
—
fl
______a~ (0 J:exp[ ~n’(p.7”
(1— tp.1)N
fl(p.1—p.1)
(I— 1)!
— t”)J,,]
:G
JO)
I
‘
I>’
(2.38) In the limit of ~1(p.)
=
t
—~
0 we obtain eq. (2.37) with
ltm~1(p., t) i-’O
=
lima’ (z—1).t-*o .
(0 :exp[ ~n~(p.” — t’)J,,} l—tp.
:G
JO) (2.39)
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Eqs. (2.38) and (2.39) will be used in sect. 3.2 in one of our derivations of the constraint. One more remark is in order now. From the form of the operator expansion for i).~x)i/i(p.) it is clear that the function as given by (2.39), behaves as + 0(1/p.)) when p. —~ in accordance with (2.37), and all the functions are independent. This last statement should be more or less clear from the fact that the set {41(p.)} contains all the information about fermionic propagator ~‘..
4(p.),
~,
C~(j~,p.),
which is exactly the same as the information in operator G, which should be of the form (2.26), and defines the actual fermionic action. To make this mutual independence of qYs even more transparent we prove in subsect. 2.4 that Hirota identities are satisfied by the quantity (2.36) with any set of functions (~1(p.)}.The possibility to choose {~1(p.))in any way is important for us to argue without going into any more details, that the partition function of GKM, which was expressed in determinant from (2.36) with the specific choice of ‘/, is indeed a KP i--function. =
~,,
2.4. HIROTA EQUATIONS FOR r-FUNCTION IN MIWA COORDINATES
The usual form of bilinear Hirota equations, which are the defining equations of KP i--functions, is ~
where D7 are Hirota symbols, D (D7, ~DT,,...) and 3”, are Schur polynomials. Note that these equations are in fact more than KP equations themselves, which 2 log i-/aT 2) and all describe evolution of the functions u,(T~) (u(1.,) u2(T~) a 1 other u,(T~)can be determined from the relations
~~logi-=(L~)
1and
1
with
see eqs. (3.35) and (3.36) rather than
Lm3+ Eu~±1a i-
itself~Indeed, any transformation
i-(T,,)—*i-(T,,)exp[H(T 2,T3...)+TiH],
(2.40)
with al-I/aT1 0 and H const does not change u~(I,),i.e. H and H cannot be fixed by the entire of KP equations. They are, however, fixed by Hirota equations (up to linear function H + T1H Eb~T,,,b,, const). This remark is important for the subject of matrix models whenever interpolation between critical points is considered. For example, one can use the representation =
=
=
~
=exp{fTl(Ti
=
—x)u(x, T2...) dx+H(T2, T3...)
+
TIH}
(2.41)
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and at any given multicritical point the functions H, H are inessential. However, they are very important for interpolations, and, in particular, for the relevance of Virasoro and ~Il~-constraints. Our first task in this subsection is to rewrite the Hirota equations in Miwa coordinates (2.1). Since this is a widely known transformation (see, e.g. ref. [26]), we just cite here the answer: Hirota equations in Miwa coordinates state that (p.,, —p.6)i-(p,,, P6, Pc +
(p.6
— p.~)i-(p,,+
1,
+
1)i-(p,,
+
1, p6
Ph, Pc)T(Pa, p6 +
+
1, p~)
1, Pc
+
1)
+(p.c—p.,,)r(p,,,p6+l,p,)i-(p,,+l,p6,p~+l)O,
(2.42)
where the i--function is expressed through the Miwa variables (2.30) according to eq. (2.24) and three arbitrary eigenvalues from the set (p.,) with corresponding multiplicities are chosen. The second task is to prove that i-[M] = ~(MY’ det ~1(p.1) with any {4,(p.)} satisfies eq. (2.42). All we need to derive the desired equations is the well-known Jacobi identity for the determinants. For any (N + 2) x (N + 2) determinant J we denote by J(~’,) the determinant obtained from J with s rows i,,.. .,i, and s columns j,,. . omitted. Then the Jacobi identity reads J~’~ J=J~ 1,3
Let us consider i-N({p.}) set of eigenvalues p.1
p.,~, into
the
p.~+02
-J
3
i-~(p.,,..., p.N)
~ where
J~ 1
L
J~. 3
1
and divide the given (a priori different) “clusters” of sizes p1,. .. PL:
...;p.,,±+,,,+,
Then one should introduce two additional eigenvalues p.N+i, p.N+2 and apply the Jacobi identity for i = N + 1, j = N + 2 to the function i-N+2((p.}, p.N÷~,p.N÷2).Then simple calculations gives the following system of equations: E,p1 =IV.
p.N±l,
=
—
p.N+I
p.N+2)TN({p.})
(i-N+I((p.},
—i-~~1({p.},p.N+l)TN+l((p.},
where
~
p.N+2)TN÷J({p.},
p.N±,)
p.N±2 p.N+2)},
(2.43)
denotes some new i--function which is obtained from the given
i-N+,
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~N±2~ Now let all the eigenvalues in each by a change of the last row: 4N± cluster tend to the values p. 1, . . , p.,~and 1 ~ p.N+2 —~p.6 where p.,, and p.~, belong to different clusters. Then eq. (2.43) acquires the form [in the notations as in eq. (2.42)]: .
(p.o =
—
p.6)i-(p~+ 1,
T(Pa, Ph +
1,
Ph +
~
~
1, Pc)i-( Pa, p1,,
Pc)~(Pa+
1, p6,
Pc) — i-(p,, +
1,
Ph’ Pc)~(Pa,Ph +
1,
Pc),
(2.44) where p~ describes some arbitrary third cluster different from the previous ones. One can multiply this equation by the factor i-(p,,, Ph’ Pc + 1)/i-(p,,, p,,, p~)and write down a couple of other equations obtained by cyclic permutations among indices (a, b, c). The sum of these three equations coincides with eq. (2.42). Another interesting expression can be derived from eq. (2.44), as follows. Let us make the replacement Pa ~ — 1, N —s N — 1 and put p6 = 0 (this means that we have the cluster with the single element p.6 in the corresponding i--functions in eq. (2.44)). Then in the limit p.1, —s p.,, we have ‘r(pa, Ph + 1) —~i-(p,, + 1) (notations of other clusters will be omitted since they do not change) and the equation now takes the form i-N+I(P,, + l)TN1( Pa_l)
=i-N(Pa)
lim ~i~*5L,,
a ~n(Pa, p.,,) ap.,,
n(Pa)
lim iLh*tL*
a ~~i-N(Pa, ap.,,
p.,,).
Simple calculation gives further that lim ~
a ap.,,
a TN(Pa) p,, ap.~ 1
~i-N(Pa,
p.h)
=
and now we obtain 2
PaTN+I(Pa)TN_t(Pa)N(PaY~
a ~(pa) ap.~i-(p,,)
.
(2.45)
This equation holds for every function in the form (2.21). A more concrete expression can be obtained for the GKM partition function 3~~/,~7} defined by eq. (2.9) in terms of A variables [see eq. (2.8)]. For this quantity the following relation holds: La N
=
,~ V
~c~-{~)N
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203
Therefore we have Pa4T~i(Pa +
1)5”)~),(p,,— 1) L a2 6=1
a
aAa A,,
a
L
a
3’)~”1(p,,)— ~—3’~V1(p,,) ~ a
h=1
h
We should remark that in the case of Pi = N, p,, = 0, a ~ I (A, A) this equation reduces to the Toda-chain one. Indeed, /7~/1(p,,)now has the very simple determinant form 1
det a’~~F(A).
=
(N-i)!
It is a determinant of a matrix with specific Toda-chain symmetry of constant entries along the anti-diagonal [27]. From the other point of view, the matrix integral (1.1) reduces in the present case of the diagonal matrix A to a standard discrete model partition function which is well-known to correspond to Toda-chain hierarchy [28,17]. If we return to the original description in the terms of p.’s [see eqs. (2.16) and (2.24)], then our final equation is TN+i(Pa +
i)TNI(
Pa —
1) =i-
=
(n (p.
—
p.
)P*P~)-
I ~
a2
L
2
~
~
([~“(p.,,)I
i-N(P,,)
I/2
exp[~(p.,,)
~ —
p.~~~’(p.,,)])”’ (2.46)
We shall return to eq. (2.46) in subsect. 3.3 in the context of the string equation. 2.5. GRASSMANNIAN DESCRIPTION OF KP r-FUNCTION
In this subsection we present the necessary material about infinite-dimensional grassmannians and their connection to integrable hierarchies. The details and proofs can be found in refs. [29—31]. Definitions. Let us consider the infinite dimensional Hilbert space H realized as the space of Laurent series ~(p.) on the unit circle S’(p. e S’) on the complex plane. This space is naturally represented as the direct sum H = ~ H, where H~(H_) is spanned by the vectors (p.”), n ~ 0 ((p. “}, n > 0), n ~ f. These vectors form the orthogonal basis in H:(p.”, p.”) = The infinite-dimensional grassmannian Gr is the set of subspaces W E H (grassmannian points) which satisfy the two conditions:
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S. Kharchet’ ci al.
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(i) the orthogonal projector pr~: W —s H÷ is a Fredholm operator, i.e. the kernel and co-kernel are finite dimensional spaces; (ii) The orthogonal projector pr_: W H is a compact operator. This definition corresponds to Sato’s grassmannian rather that to the Segal— Wilson one, where q5(p.) should be convergent in some neighborhood of ~. The index of pr~is called the relative dimension of W: —*
md pr~=dim(ker pr~)— dim(coker pr~). The grassmannian is decomposed into the direct sum of the connected components: Gr= ~ Gr,,,
(2.47)
where Gr~consists of W’s with the relative dimension n. For our purposes it is sufficient to deal with Gr0. It is convenient to represent linear operators acting in H as block matrices with respect to the decomposition H H + ~
a=[a
~],
aEGL(~),
(2.48)
where the operators a, b, c, d act as follows: a: H~—sH~,b: H~.—sH~,and so on. Clearly, b and c should be compact operators. For applications to integrable hierarchies the following commutative 1 to subgroup non-zero F c GL(oc) is of importance. It consists of the mapping from S complex numbers and acts on H by multiplications of the Laurent series. There are two natural commutative subgroups F~ and F in F: the elements of F~ (F) can be analytically continued inside (outside) SI. Their elements ~ E F~ and E F can be parametrized through the infinite sets {Tk} and (Tk) (k ~ 1) of independent variables (“times”): y~(p.)= exp( ~ Tkp.k), k>I
y(p.)
=
exp( ~ Tkp._k).
(2.49)
k>I
Let {4~}(n~ 0) be a basis in W. It is called admissible if the operator transforming p.” to pr~(/s,,)in H÷has a well-defined determinant. We can write
=
~ (~±)nkp.~ + ~ (~)fl~p._k, k=1 k=I
where 4i~has a determinant, çb. is compact. In particular, for W always possible to choose
(4~2),,,,,
=
s,,,,,.
(2.50) E
Gr
0 it is In this case we call this basis {~,(c)}
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canonical for W.Generally, the convenient choice for admissible 4~ (to be used below) is a lower-triangle matrix with unit diagonal elements: (4+),,k = 0 if n
p.k_l(~
[cf. eq. (2.22)]. The i--function of KP hierarchy orthogonal projection y~W—s H~: =
+
i-~
O(p.~)),
k~1
(2.51)
is defined as the determinant of the
exp(pr~:y+({T,,})W —s H+),
(2.52)
det(1 +a~bB),
(2.53)
or, more explicitly, =
1 [see eq. (2.50)1. where y, is represented in the block form (2.48), b = (4~Y Given a Riemann surface of finite genus one can associate with it a point in Gr by constructing the admissible basis according to the well-known rules [30]. In this case the formal Laurent series for the basis vectors are in fact convergent. Our situation is different: our basis vectors are asymptotic (formal) series which do not need to converge. This should be interpreted as adequate generalization to infinite genus Riemann surfaces. In this sense we often identify Gr with the universal module space (of line bundles over Riemann surfaces with punctures). Instead of W one can work with the equivalent space yj{Tk}), the i--function being changed in the trivial way:
i-~w({Tk})=exP(~kTkfk)i-w({Tk}).
(2.54)
Let us note that in Miwa variables (2.30) the “evolution factor” y~(p.)(2.49) looks like a product of pole factors: y~(p.)= fl(1—p./p. 1)”. The various reductions of the KP hierarchy can be described in these terms as follows. Let f(p.) be a Laurent series in p.. Those W’s which satisfy the condition f(p.)WcW
(2.55)
corresponds to the well-known Boussinesq,...) give rise1< to solutions of “f-reduction”K-reductions of KP. In (KdV, particular, the choice f(p.) = p. which are in fact associated with Ak_I Lie algebra [32]. In this case the i--function
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does not depend on 7, with n = 0 mod K (modulo trivial exponential factors linear in times like that in eq. (2.54)). An admissible (but non-canonical) basis can be chosen in such a way that n ~ 1.
p.K4~(p.)
=
(2.56)
Vice versa, if we have a basis of the form (2.56), the corresponding point of Gr leads to a solution of the K-reduced KP hierarchy. In subsect. 2.6 we show that the GKM (1.2) with arbitrary polynomial potential ~(x) corresponds to the ~‘(p.) reduction of the KP hierarchy. Note that K-reduction is equivalent to the existence of some admissible basis with the property (2.56). However, other admissible bases (including canonical) in this case satisfy a weaker condition: N
4~±,,(p.) =p.~+ ~a1/, with some constant a,’s. Fermionic formalism The grassmannian approach is in fact equivalent to the fermionic language used in subsect. 2.3. Let us briefly describe the relation between them. in the fermionic interpretation H becomes the one-particle Fock space spanned by i/i,,. The fermionic operator G of the form (2.26) makes linear transformations in H according to i/i,,
-~Gt/i,,G~ ~
H.
The positive (negative) modes of the current J(p.) [eq. (2.26)] are the generators of F~ (FJ. The matrix ~ in eq. (2.26) can be expressed through the canonical basis ~~p.) of the corresponding W as follows: 1 +
G,,,,,
=
dp.dv
(2i-ri)
p.~
~)
~w(p.’
~“’~“~‘w(
2
.
=
~ p.~~(v). k=I
A more invariant expression is
.~w(p.’
~)
=
1 p.—v i-~
(0kp._k
k
p.,
p),
(2.57)
(2.58)
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Unified theory of 2d graiitv
207
We emphasize that the explicit basis t/sk = ~ [see eq. (2.18)] is not canonical, but instead satisfies eq. (2.56). However, eqs. (2.57) and (2.58) are very important for any interpretation in terms of (infinite-genus) Riemann surfaces, in particular, for the identification of GKM with appropriate Liouville-like models of 2d gravity. Explicit formulae for can be easily derived, at least, in the case of 27= 27k. ~
i--functions of reduced hierarchies Let us note that in the fermionic language it is trivial to understand that the K-reduction condition leads to the i--function i-1~<1 almost independent of times 1,K, i.e. the property (1.21). Indeed, eqs. (2.56)—(2.58) imply that the condition of such a reduction can be transformed into the matrix {,~,,,,}: ~
E”~]= 0,
where E is the shift matrix, (E,,,,,) = On the same “infinite-matrix language” the elements of F~ can be parametrized as y~=exp(ETkE’<), ~ exp(~E~),i.e. the currents “k from eq. (2.26) act as E”. Now it is evident that exponent of currents ~k with k = 0 mod K can be pulled to the right vacuum and cancelled, eliminating the dependence of the corresponding times (up to the exponential factor like that in eq. (2.54) — this is due to the freedom to insert the exponential y= exp(ETkE~’)between G and y~). Now let us discuss the freedom in the definition of the y-function from the viewpoint of the Hirota equation. As was pointed out in eq. (2.54) there is a freedom to multiply the T-function by a linear exponential of times. Another possible modification is dealing with other components Gr,, of the grassmannian. It corresponds to the shift of the first basis vector or, equivalently, multiplying it by p.”. All these can be trivially understood in terms of the Hirota equation in the Miwa parametrization (2.42). Indeed, it is invariant with respect to multiplying by the product LI•f(p.,), where f(p.) is an arbitrary Laurent series. If the first term of this series p.°, one can expand f(p.,) —~exp(~~p.~T~) and reproduce the correct factor in eq. (2.54). If the first term behaves as p.”, one should deal with Gr,,. Thus, in Miwa variables we readily describe all full freedom in the definition of the i--function. ~
‘~
and specific points in the grassmannian To conclude this subsection let us note that the problem of correspondence between matrix models and the points of the grassmannian has been already addressed in ref. [23]. That paper examined the implications of the 3’/ constraint in the double-scaling limit of the K-matrix model and concluded that the corresponding point of the grassmannian is defined by the set of K linear independent vectors (A’~ 1),where operator A for 27= X’~~ ‘/(K + 1) essentially coincides with ours [see eqs. (2.19) and (2.20)]. Moreover, from our study of GKM we obtain GKM
208
5. Kharcher ci a!.
/
Unified theory of 2d gratity
straightforwardly the explicit form of all basis vectors for any potentials: /, = 1, as given in eq. (2.18). Note also that in ref. [23] a shift of time variables analogous to ours in eq. (1.17) was introduced. As to ref. [23], we would explain the appearance of the Airy functions (in the case of the K = 2 model) from the string equation in the following way. If we put all Tk = 0 except for T1 the string equation can be satisfied only if the i--function is zero. This means that the relevant point of the grassmannian is singular. Instead, we can put T3 = c 0 keeping all other Tk with k ~ 5 equal to zero. In this case we have
a log ai,
2
Choosing c = — ~, so that and using the relation
T3
=
T3 +
+~!~T~=0. ~ =
0, taking the derivative with respect to
T,
2 log
a
u(T 1)
=
_______
where u(T~)is the KdV potential, we obtain 2u(T1)
=
T1.
According to the ideology of the inverse scattering method in the case of 2-reduction we should consider the Schrodinger equation for this potential: 2iIi +2u~I”p.21I’, a aT~ i.e.
a2 ~i’ (2.59) Solving this equation, we obtain the Airy function ~!‘(p.JT 1)JT,,,,=
2) ~I’,)(p.)Ai(—T~ +p. ~1’ 0(p.)fexp[_x/3+(p.T1)x]
dx.
The wave function ~I’is nothing other than the Baker—Akhiezer function. ~I’~is
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Unified theory of 2d grarity
adjusted from the normalization requirement ~It(p.)e~T~~” general ~I’can be expressed through the i--function as [33]
~I’(p.
J 1,)
=
exp( ~. 1,p.”)
i-(T —p.”/n)
209 —~1
as p.
—~oc~
In
(2.60)
(see subsect. 3.3 below for more details, in eq. (2.59) we have a ~P-function of the form (2.60) with Tk = 0, k ~ 3. The admissible basis for the corresponding point of Gr,, can be constructed by means of the 1It~functionas follows [30]:
a~’iit k-I all
T, =1)
The basis vectors, constructed according to this expression, obviously coincide with those which were obtained in subsect. 2.2 [eq. (2.6)] for the particular case of 7z(X)
=
3.
~X
For generic K the role of eq. (2.59) is played by a Kth order differential equations (L’~I’= p.2w) and the result is the Airy function of level K as defined in eq. (2.6). 2.6. 0KM AND REDUCTIONS OF KP HIERARCHY
We prove here that whenever potential 27fX) in GKM is a homogeneous polynomial of degree K+ 1, i.e. ~V(X)= const..XK ), the partition function Z~[M], which is a KP i--function, in fact can be treated as a i--function i-{~c1of K-reduced KP hierarchy. For this purpose let us return to subsect. 2.2 and note that because of eq. 92.7) F 1(A) satisfies the Ward identity (1.23):
[~‘(a/aA)
—
A] F,(A)
=
0.
(2.61)
If the potential 2~(X) is a polynomial of finite degree K + 1, then (2.6) may be used to express FK+ 1 in the form of a linear combination of F,’s with i ~ K. Namely, if 27(X) = —~2~7X’, VK±,= 1/(K+ 1), eq. (2.6) implies that FK±,= — ~i7/F,+AF1.
(2.62)
Clearly, all the terms on the r.h.s. of eq. (2.62) except for the last one AF, will drop out of the determinant (2.9), i.e. in this case FK+i may be defined as AF~ rather than by eq. (2.62). Obviously in the same way any FK+,fl may be substituted by AF,,,, while any F0K+,,, can be substituted by A”I~0.In other words, .9~,[A] is
210
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Unified theory of 2dgraiity
given by eq. (2.9) with the first K functions F
1 ... FK defined by eq. (2.6), while other elements of the basis are given by the recurrent relation
A”F,~.
(2.63)
Note that this is true when 2Y(X) is any potential of degree K + 1, not obligatory homogeneous. However, we should recall that A = 27~’(p.)and p. rather than A is the proper parameter to deal with the grassmannian picture. Therefore eq. (2.63) implies the appearance of KP reduction, associated with the function ~27”(p.).If 27 is further restricted to be 77(X)=X~’/(K+ 1), A = p.K and (2.63) acquires the form F
‘-~A”F ,,,
4
p. ?~KF a,,
which is characteristic of conventional K-reduction. As we already explained, K-reduction implies the relation (1.21) with some
factor of the form (1.22). However, even this factor3 is(K absent in the of GKM = 2) this wascase exhaustively (1.10). the [15], particular caseproperties of ~7(X) of = const . X provedFor in ref. using the symplectic structure on (the model of) the universal module space. Since interpretation of entire GKM in such terms is not yet available, the analog of such a proof for generic K is still lacking. 2.7. ON T,,K INDEPENDENCE OF Z{K}
Reformulation of the problem. First of all, let us choose our variables (p.,) in such a way that the only non-vanishing times 7, = n - 1Ep.~” are those with n = 0 mod K. To do this it is enough to have only K finite parameters p. 1, . , p.,.~.,which are essentially Kth order roots of unity: 1, E = e2’~’~, j = 1... K. (2.65) p.1 = p. For such a choice of variables the formula (1.21) . .
T~(T 1
=
.. .
TK_l, TK, TK+,
.
. .
exP(~aflK~,K)i-~(T,.
T2KI, T2K, T2K+I
. .
TK_l, 0,
TK+,
. ..
...) T2K_l,
0,
T2K±l ...),
(2.66)
turns into i-~(0...O, TK,O...O, T2K,0...)=exP(~a,,K~,K)i-~(0)
=
exp(~(aflK/n)p.fr~K)i-~~(O).(2.67)
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211
This relation is valid for any i--function i-~ of a K-reduced KP hierarchy. Our purpose is to prove that for the particular example of such r1’~1, namely, for partition function Z~<~of GKM, associated with potential of the form 77(X) = XK+I/(K+ 1), a,,K[Z]—a
log Z~/a1,~=O,
(2.68)
holds or, equivalently, the function in the r.h.s. of eq. (2.67) is independent of p.. In other words, we need to prove that ~Z[p.f]~(
~{K}(p.)
2(,~)
1.
=
(2.69)
The l.h.s. of this relation can be evaluated with the help of eq. (2.16), Z{K}
=
We emphasize that in order to prove 1,,,~independence of Z{K} it is enough to examine the determinant of the K X K matrix in (2.16) with entries
=
et’~(p.) =
(s(A) ~s(AY’) ~
~1(p.)
~
Co
=
~
=
p.’
‘e~’~(p.), (2.70)
1
(2.71) 1’~depend
(of quantities ~, s,thus A [see eqs. (2.15), (2.8i)—(2.20)] and e on course, the formallofthe potential ~V and on K). Straightforward calculation (K = 2) We begin our proof that ~1~(p.) = 1 from direct calculation of this quantity. If we substitute the expansion (2.71) with p.’s defined in (2.65) into (2.16) the result reads ~{K}(p.)
1
K (fle~t)
=
{a}
d
(—a,+i—I)j (i-I)j
et(,J)E
p.
~
,
(2.72)
e
The remaining determinant at the r.h.s. is actually vanishing unless ~a 1
=
0 mod K, so that eq. (2.68) is in fact equivalent to the set of identities K
~ (n e~2). {a,} =
nK
det
(—a,+i—I)j
d~ (o)
(I
~
=
s,,,, for all
n ~ 0
(2.73)
212
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S. Kharcheu eta!.
Unified theory of2dgra,ity
for coefficients e~of series expansions of modified K-level Airy functions tj~(K)(p.)=p.(K_I)/2exp[Kp.K+I/(K+
xJexp[_x~’/(K+ For example, for
K
=
1)]
1) +p.Kx~x~~ dx.
(2.74)
2 identities eq. (2.73) looks like ~
(2.75)
()“e(1)e(2)~6
a+/3=2n
Since for K=2
=
exp(—~x3+xp.2)
~exp(—4p.3)fdx 1
2
=
rfdz exp —z
=
~ei~’fdx’x
=
~fdz(z+p.) 1
— —
—
____
3p.3/2
—
J
F(3k+1/2)
1 ~ 9k(2k)!
=
1(1/2)
1 p.~’
exp(—~x3+xp.2)
exp{_z2_ 6k
+
1
1(3k
3p.3/2}
+
1/2)
1
~ 9k(~~
F(1/2)
6k—I
p.~’
we have e~’~ =
1 9”(2a)!
F(3a+1/2) 1(1/2)
2~ = — 6n+l e~. 6c~
a
—
e~
~
(2.76)
A more explicit form of eq. (2.75) is 36a/3—1 a±fl=2n
a,f3~O
(2a) !(2f3)! F(3a
2 -
~)F(3p
-
~)
=
-4[F(fl]
~,
(2.77)
which is indeed a valid F-function identity. This calculation, though absolutely straightforward, is hard to repeat for generic K. Still it is useful for illustrative purposes and we included it in this section.
S. Kharchet at a!.
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Unified theory of2dgraiity
213
The proof A much easier approach is to prove that p.
a
.~{K}(p.)
=
0.
(2.78)
ap.
Since it is obvious from eq. (2.72) that at least ~1K} 1 and p. ~, this would provide a complete proof on eq. (2.69). in order to prove eq. (2.78) it is enough to act with p.a/ap. upon —~
—~
det(,J)E”e1’1(p.E3)
~{KI(p.)
(2.79)
and then make use of the fact, that p.(a/ap.)e~’kp.e’)can be decomposed into a linear combination of e”kp.c~). e~(p.’). This decomposition, which is, of course, the implication of K-reduction, can be worked out from the relation (2.20), ..
p.’e~’~°(p.) =A{~}p.’e~(p.).
(2.80)
Since I AK
=
Kp.’<’
a K—i —+p.— ap. 2Kp.”
,
(2.81)
(see also ref. [23]), we have a (p.~)e(~)(~EJ)
=
(
K+1—2i 2
— K1p.K~)e(~)(~E1)+ KE1p.Ke+~)(p.c1).
(2.82) This should be supplemented by the condition I) =
e~°
(2.83)
(the equation Ac =p.’
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Unified theory of2dgra,ity
Then (p.~)~(2)(p.)
(p.~)(et1)(p.)e(2)(
=
—p.)
+
—p.)) e~2~(p.)e’°(
[(~— 2p.3)e°~(p.)+ 2p.3e(2)(p.)je(2)( —p.) +e(p.)[_2p.3e(I)(_p.)+(_~+2p.3)e(2)(_p.)]
(—~— 2p.3)e(2)(p.)]e(~(_p.)
+
[2p.3eW(p.)
+
e~21(p.)[(~+ 2p.3)e~>(—p.) — 2p.3e~2~( —p.)],
+
and the r.h.s. is identically zero. If we return to the case of arbitrary K, note that after the substitution of eq. (2.79) into the l.h.s. of eq. (2.78) we obtain a sum of K determinants like eq. (2.79) with e”e~’~(p.e~) in the row i’s substituted by eq. (2.82):
(p.~)E~e(~)(p.E1) =
(K
+
l— 2i’
+ Kp.K+
— Ke1p.
I)EiJe(l)(p.Ei)
leut~ ‘e’1’~~(p.e’).
Next, note that the last item in the r.h.s. coincides with the entry E~’~ ‘~‘e~’~ ‘~(p.~) of the next line of the same determinant, and this can be eliminated. Moreover, the first item on the r.h.s. just implies that the i’th line is multiplied by a factor of ~(K+ 1— 2i’) and the effect of this cancels in the sum over i’: E~,~(K+ 1— 2i’) = 0. Therefore, we conclude that
a p. ___~{K}(p.) dp.
a ap. a ~ ~p.
K
—Kp.~ ~
(2.84)
where 3’~,is just the same determinant 3’, only in the i’th line d~ E’’e~’~(p.E’) is substituted by ‘d~ 1= E” ~‘~‘e”’(p.e’).The sum of such 3’, (for any original 3’) is identically zero as a consequence of the identity ~ ~e’= 0.
/
S. Kharche, eta!.
Unified theory of 2dgraiity
215
Examples: K=2:
fd,, 3’=det~ d2,
3’, =det(
—
d12\ d22
d11
/ d11 3’2=det~ — d-,,
)
d11d22—d12d21
(—) 2 d~,I d,2
=
J
2
(—) d221
3’ +9/2=0.
2~’~3:
K
=
3:
E
=
e
(d,,
d
12 d,7 (13~ d32 3’=det~d2I =
9/,
d11d22d3,
=
det
(
+
Ied~1 d
2, d3,
d,3\ d23 d33 )
d~2d2~d3, + d,3d32d21 2d~, ~ E d27 d,3 d32 d33
3d,~d 2d23d,1
ed, ,d22d33 +
+
e
2d
32d2,—
e
3d, 12d21d33 — E
3d~~d22
—
d~, d13 \ 2d 3d e 22 e 21 ~ d,2 d33 / 3’2=det(ed2’ 2d 3d, = 11d22d33 + E 2d23d3~+ Ed13d32d2, (
d,1
— E3d, 3d32d,
I’
I’
)
2d, =
— d13d,,d22 — d23d32d,
— d,2d2d13
d d
2d
— Ed,2d21d3~—
~
13
31
22
216
S. Kharchei eta!.
9/3
=
d1,
d12
d~3
det d21
d22
d23
~3l
=
3d E 1
—
ed32
1d22d33
+
/
Unified theory of2dgraiity
3d
E
33
Ed,2d23d3,
+
3d E 13d32d2,
3d — E
12d21d33
— Ed13d31d22
2d e 23d32d11, ~l
+9/2+9/3—0.
These two examples are enough to illustrate the general phenomenon: determinant 9/ is an algebraic sum of products d~1.. ~ Let us pick up one of these items. It appears in 9/, with the coefficient e”, where j’ is defined as the second subscript of the element d,1 in this product. The map i’ —*j’ depends on the particular item we choose, but when we sum over all i’ this is the same as summing over all j’, since every subscript appears in the product once and only once. Therefore in the sum E,9/,. every item appears with the universal coefficient = 0, i.e. ~,9/, = 0. This completes our proof of eq. (2.78) and thus of eq. (2.68). .
Remark. To avoid possible confusion, let us note that it is important Airy functions 1’~(p.)are that expanded in series are analytic functions of p., but not of A = p.K (e.g. e with integer powers of p. _(1<+ I) = A1 I/K) This is what makes the proof slightly non-trivial. Otherwise, if one suggested that the entries of the matrix in eq. (2.79) are analytic functions of A, the A-derivative of this determinant would be simply a combination of determinants with coincident rows and thus vanishing. However, non-analyticity in A requires one to be more accurate: AUI( takes different values p.E1 when appearing in different places, and a detailed calculation, as given above,
is necessary.
3. Universal 3’/..
1
constraint and the string equation
3.1. MOTIVATIONS
Since it was proved in sect. 2 that Z{KI[M] is a i--function i-1~ of the K-reduced KP hierarchy, in order to define it completely (i.e. to fix the point of the grassmannian) it is enough to deduce somehow a single additional constraint (3.1)
/
S. Kharche, at a!.
Unified theory of 2d gra city
217
with i
a ~
K
(n+K)1,,±K~-
~>,
0 mod
K
1
a —
+ ~T
1
~ aT,bT,,. 2K~6~
(3.2)
According to arguments of ref. [1] a constraint such as (3.1), when imposed on = i-~, implies the entire set of WK algebra constraints in the form of eq. (1.29). Another reason for the study of 9/~ constraint is that it is much simpler than any other one: 9// is the only operator of interest which does not contain double- and higher-order T-derivatives (to be exact, there is one more such generator: ~ Among other things, this means that it is easier to formulate the unit ersal (i.e. for any potential 77(X)) S~i,constraint, than any other corollary of (universal!) Ward identity (1.23). This section contains a formulation and derivation of the S’i//~ constraint for GKM with arbitrary potential 27~(X),making use of explicit formulas, derived in sect. 2. Namely, we are going to exploit the fact that the functions ct,(p.) in eq. (2.10) are not independent, but are rather related by the action of the B/GA operator: see eq. (2.20). The proof itself is described in subsect. 3.2, and subsect. 3.3 contains additional remarks about the string equation. We emphasize that not only is the 9/— constraint valid for any 7~(X)(with the only restriction that 7’~(x) grows faster than x as x —, ~), but it is just the same for17 any size N of the = i-I.’ I), this conmatrices.is Just like the property of integrability that Z construction behave straint not sensitive to N, and this makes (i.e. the entire smoothly in the continuum limit, as N —*
3.2.
DIRECT EVALUATION
OF 5~’
It is well known [21,28], that the of operator
~1
a
Tr— aAtr
=
Tr
constraint is closely related to the action i
a
(3.3)
—.
77”(M) aMir
Therefore it is natural to examine how this operator acts upon det i13(p..) Z1~[M]= s(p.) ~,(p.)
=
(3.4)
~1(1vI)[Js(p.,), (27~”(p.))/2 exp[27(p.)
=~(A) = (a/aA)’1F(A),
—
p.7~’(p.)],
A
=
(3.5)
218
/
S. Kharchei at a!.
First of all, if
Unified theory of 2d gracity
1~~ is considered as a function of
Z
1
=
In the particular case of
7~<(X)=
7~(X) =
KZIK)
1
az11<1
M?~+K
a1,
Tr
~,
,,~,
,,2Omod
—
~
(3.6)
K
1 =
.
1), eq. (3.6) turns into
X’~~ ‘/(K +
i
(lAtr
alogz1~1
~
—
a
~I1~
~
1
a
i
T-variables,
—
~
a1,bT,, +
a
Z1”~1.
(3.7)
a+6==K a,h>~I
We made use of the definition (3.2) of the 9/{1~ operator and the fact that Z{KI is independent of all TJ,K. Time variables are defined by the Miwa transformation T = ,~—I Tr M”. On the other hand, if we apply (3.3) to explicit formula (3.4), we obtain
a
I
z1~’1
=
aA,~
—Tr M
E +2
7~”(p.,)— 77”(p. 7~”(p..)7~”(p.) p.—p.. 1)
a
+Tr—~-—logdetF(A1), or, in the case of
7~/X) =XK~/(K+
a
1
—Tr M
=
(3.8)
I),
a
+
~
aT,,bTh + Tr—~—log det F,(A1). (3.9)
a+b=K a,6 ~ I
Ir
The combination of (3.7) and (3.9) implies that I
_~9/(~IZ{K}= —
a ~—log
~(!~ +
a Tr M — Tr~—logdet F,(A~). (3.10)
The r.h.s. here is practically independent of K, and this may be used, together with eqs. (3.6) and (3.8) in order to suggest the formula for the universal operator ~ the idea is to preserve the relation (3.10): 1 =
—
a
~—log
Z(~ +
a
Tr M — Tr—~-—logdet F(A1).
(3.11)
S. Kharcliei at a!.
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Unifled theory of
2d grai dy
219
Here
a
I 9/{~’}= ~Tr —l
7~(M)M/~+
,,~,,
1
________—~
— a7~,
a
27~”(p.)— 27~”(p..) ‘
‘
7 ‘(p.,)7 ‘(p.1)
—
(3.12)
—,
aT1
p.,— p.1
and this turns into (3.2) when 77(X)=X’<~/(K+1) (note that the items with i =j are included in the sum on the r.h.s. in eq. (3.12)). So, in order to prove the 9/f~’ constraint, one should prove that the r.h.s. of eq. (3.9) vanishes,
a
—i-log 1 = i-/~ ~,
Z~1
=
Tr M
a
—
Tr~-j-—logdet F,(A,.),
(3.13)
defined by eq. (3.4). The l.h.s. may he represented as the residue
forthe Z~ratio of res~
i-~1(I,)
a
—~--Iogi-~,(1(7~,).
=
(3.14)
However, if expressed through Miwa coordinates, the i--function in the numerator is given by the same formula (3.4) with one extra parameter p., i.e. is in fact equal to i-t~J~. This idea is almost enough to deduce eq. (3.13). Since eq. (3.13) is valid for any value of N, it is reasonable to begin with an illustrative example of N = 1. Then (A = TV1(
~,) =
i-p’ )[p.,]
=
exp[ 7( p.,)
— p.7
~‘( p.~)]
[7
~(
p.
2F( A,),
1)] i-~71(~,+p.”/n)
~
=rY1[p.,, p.] =
exp[7’(p.,) —p.,7~’(p. 1)]exp[7’(p.) —p.7~’(p.)] x
2 [77(p.1)7~(p.)]I/
[F(A)aF(A)/aA—F(A)aF(A,)/aA,]
p. —p.,
exp[7~(p.) — p.7~’(p.)][7~(p.)]
‘~2F(A) i-~
=
p. —p.
}[p.,]
1
[—a log F(A)/aA, -i-a log F(A)/aA].
(3.15)
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The function F(A)
=
fdx exp[—~(x) +Ax] exp[27~(p.) — p.7~’(p.)}[7~”(p.)]
_1/2{1 +
0(7~”/7~”V”)}.(3.16)
p. and for our purposes it If ?7(p.) grows with p.fl as p. ~, then 77”/(7Y”)2 is enough to have n> i, so that in the braces on the r.h.s. we have (1 + o(1/p.)((p. o(p.) 0 ad p. oc). Then numerator on the r.h.s. of eq. (3.15) is 1 + o(l/p.), while the second item in square brackets behaves as a log F(A)/aA p.(1 + o(l/p.)). Combining all this with eq. (3.14), we obtain “,
—*
—*
a ~log
—*
1+o(l/p.) TV1=res~{ p.p. 1
=p.1
[—alog F(Al)/aAi+~(i+o(i/~))I}
—a log F(A,)/aA1.
(3.17)
Thus (3.13) is proved for the particular case of N — 1. The proof is literally the same for any N, only instead of a relatively simple expression in square brackets on the r.h.s. of (3.14) one has the ratio
F(A,) F(A2)
aF(A1)/aA1 aF(A2)/aA2
det
... ...
:
‘
aNF(A1)/aA~ a”F(A1)/aA~’ 6’F(A aN_lF(A2)/aA~~ a 2)/aA~’ :
.
F(AN)
aF(AN)/aAN
...
F(A)
aF(A)/aA
...
(3.18)
:
a~’F(AN)/aA~~aNF(AN)/aA~ aN~F(A)/aAN~ aNF(A)/aAN
over F(A1) det
F(A2)
aF(A1)/aA aF(A2)/aA2
a’F(A,)/aA~~
...
aNF(A2)/aA~~
.
,
F(AN)
...
aF(AN)/aAN
.
..:
,
(3.19)
a~’F(AN)/aA~7’
which is in fact equal to +
o(l/p.)
-
1 [Tr~lo~
det ~(A1)~’ [1+ 0(1/~)I}. (3.20)
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We used here the estimates
a2F/aA2 F
=p.(i+o(i/p.)),
F
{ }
3F/~A 2 F
a
aNF/aA~v
F
=p.N(1+o(1/p.))
which allow us to pick up only the contributions with aNF(A)/aAN and a”1F(A)/aA~’ to eq. (3.18) all others are lower order in p. as p. cc, The Nth derivative in (3.18) is multiplied by the determinant of the N >< N matrix, which is exactly (3.19), while the (N — 1)th derivative is multiplied by —
F(A,)
aF(A)/aA~
—~
...
aN_2F(A,)/aA~~2 avF(A~)/aA~/~
aF(A2)/aA2
...
a”~
aF(AN)/aAN
..:
2F(A
F(A 2) det
2 a”F(A 2)/aA~
2)/aA~’
2F(AN)/aA~2 aNF(AN)/aA~ F(AN)
a~ (3.21)
This is, however, nothing but Tr(a/aAtr) or the logarithm of (3.19), and for this to be true it is absolutely essential that 1~ (a/aA)’~F
1. With the estimate (3.20) we obtain from eq. (3.14):
a ~log
l+o(l/p.) i-~=res~ fl(p._p.)p.{[1+o(l/p.)]
__[Tr~lo~det
~(A1)] ‘[1
a =
~p.1—Tr 7=1
—log det I~.(A,).
(3.22)
tr
This completes the proof of eq. (3.13) and thus of the universal =
0.
9/~[1
constraint: (3.23)
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We proved the crucial equation (3.13) by direct calculation. Now we would like to describe an alternative proof which originates from representing the i--function through the current correlators described in subsect. 2.3. This approach may be useful for evaluation of the higher T-derivatives (see subsect. 3.4 below). Namely, from eqs. (2.36) and (2.38) it is obvious that i-~[t MJ i-(t, {7j) depends on t and i.e. in eq. (2.38) as follows: i-(t, (1,}) = i-(T1 — Nt, T2 — ~Nt2,...),
a
i-~j[tJM]=exp -N~—~n
i-(T)= Lt”3’~,(a)i-(T),
(3.24)
a
,,
where 9/~,(a)are the Schur polynomials and symbol a represents the vector with components (—Na/aT~, ~Na/aT,,...). Therefore from eqs. (2.38), (2.39) and (3.24) one can deduce —
-N~i-(T) =a,r~[t J M] J,=o
=
—N~p., i-(T) .
iirna, det ~
+
t),
where ~,(p. 1, t) are defined by eq. (2.29). From this expression it follows that ~i-(T)
(3.25)
Ep..i-(T)~(T),
where ~(T) denotes some new i--function which is obtained from the given i-(T) by the shift of the last row in the determinant: ~~(p.j) ~N+ 1(p.1). Eq. (3.25) is nothing but eq. (3.13). —~
3.3. UNIVERSAL STRING EQUATION
The string equation, as implied by (3.23), is by definition 1i-{71 I
a ~‘ aT,
=0.
(3.26)
i-I7)
If we substitute the explicit expression (3.11) of the 9/, equation acquires the form ~Tr n:a()
I
d2logr =u.
7Y”(M)M”~
aT 1aT,,
operator the string
(3.27)
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Unified theory of 2d gralitv
223
Here we introduce a conventional notation u a2 log i-/aT,2 and also defined new I’,, and T~ derivatives, so that a log i-/aT,, 0, a log i-/aTI T 1. The derivation of (3.27) involves taking T1 derivatives of the objects 9/{~1
Tr
(3.28)
7/P~(M)M~+I .
In order to get some impression about a~7~,/aT,, let us imagine that 7(M) is some polynomial of degree Q + 1, so that [7/”(M)]~ -.~M~~(1 + t1Q’ + + ...). Then 5’, (Q + fl)TQ÷,,+ t~(Q+ n + l)TQ~,,±I+ t’2(Q + n + 2)T~+,,±2 + .... Therefore whenever Q + n > 1 (i.e. Q > 0, as it is already necessary for the derivation of eq. (3.23))
a/?,,/aT1=o, forn~1. In order to derive eq. (3.27) one also needs the eq. (3.11), 1
1
2
,~
derivative of the second item in
T,
27”(p.,)
~“(p.,)~”(p.1)
(3.29)
—
27~”(
p.1)
30)
p.,—p.1
Under the same assumptions about 7(p.) the second ratio in this sum is a polynomial in p.’s of degree Q — 2. The only contribution to eq. (3.30), which contains pure factors p./I or p.7~,is 7”(p.,) ‘//“(p. 1 1 1) p., + p.1 = T,~i,, —
others are expanded as bilinear series in 1,,1, with m, n > 1. Therefore, the T, derivative of (3.30) (if Q> 2) is just 9/, and is described by the n = 0 term in eq. (3.27). So we derived the universal string equation (3.27) in the form 2 log i-t7~1
a aT,aT,,
If potential 7~(X)=const .XK±I, the variables .9’, return to the familiar relation [34]
a2 log
1 —
(n
~ ,,*() mod
K
+
(3.31)
=u. =
[(n +K)/K]1,+K, and we
i-{’~}
K)l,±K
=
T~1,
u.
(3.32)
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Unified theory of 2d gravity
Note that even if we restore the factor (1.22), describing the maximal possible ‘,K dependence of i-t”1 it drops out of the string equation (3.32) after taking the a/aT, derivative. Our next task is reformulation of (3.31) in terms of pseudo-differential operators and in the form of a bilinear relation. If L
=a+
(3.33)
~
then [33]
a2 log aT 1aT,,
(3.34)
For n = — 1, 0, 1 we have (L ‘L, = 1, (L°) , = 0, L —1 with (3.27). K-reduction is associated with the condition
=
K
u in accordance
do not appear in eq. (3.32)).
~M”~(L”),=u.
Tr~()
=
(3.35)
(L”)=O
(this guarantees that items with n = 0 mod Rewritten in these terms, eq. (3.31) turns into
u2
(3.36)
Recall now1.that the framework of can the bedressing [33] L =[a, KaK The ineigenfunctions of L defined formalism as (Kez~),where x] = I,I denote the operator K acts on e~x(i.e. (Ke~) is a function of z, not an operator]. The Baker—Akhiezer function in these terms is given by [331 l/J(z I 7,) = exp(ET,,zn)(Kez~), while its conjugate 1/f *(z 1;,) = exp(—E1.,z”)[(K’)~ e~’].These definitions are useful for us, since [33]
L” and=Ka”K the brackets
(L”)
=
res~zt~1/J*(z)1/,(z).
(3.37)
We assume that the contour integral over z, implicit in the definition of res,, is around zero. if eq. (3.37) is substituted into eq. (3.36), we obtain M i/f*(z)1/J(z) u= —resTr-~77 (M) z(zI—M)
.
Since ~(zI7,)=exp(E7,z)
i-(1, —z”/n) i-(T)
(3.38)
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225
and 1/J*(zIT)exp(
E7~,z”)
-r(T r(T +z”/n) 0)
the product 1/f *(z)clf(z) 1 + O(1/z) (as z cc) and the contour integral over z, when pulled to infinity, picks up the contributions from eigenvalues of M. Finally we obtain our universal string equation in the form of a bilinear relation for the Baker—Akhiezer functions: —~
+
u
=
0.
(3.39)
Comparison of eqs. (2.46) and (3.39) enables us to derive a useful representation for the u-function: 2
u= ~,
~
a
log—. .iV
(3.40)
Note that eqs. (2.45) and (2.46) which follow from the general formula (2.44), provide the connection between residues of the Baker—Akhiezer function and time derivatives of the i--function without reference to the pseudo-differential operator language. Indeed, it can be easily understood if one substitutes eq. (3.25) into eq. (2.45). 3.4.
DISCUSSION
Earlier in this section we derived the universal 9/~1 constraint (1.25) on the partition function Z{~v}. Its simplest implication, the universal string equation, was considered in subsect. 3.3. The string equation is nothing but the T, derivative of this 9/., constraint. Remarkably enough, it appears to be much simpler than the constraint itself [cf. eqs. (3.27) and (3.12)]. However, the string equation is not the only implication of (1.23). Another one should be the entire tower (1.29) of Virasoro and 7constraints imposed on Zt’~. In this subsection we discuss three different ways to derive these remaining relations for any 77(X) in the GKM. Being technically different, they emphasize different (though related) properties of the partition function, and therefore each of the three deserves careful investigation, which is, however, beyond the scope of this paper. KP hierarchy approach This one is the most popular in the literature (see refs. [1,20]). The idea is that the Jf, constraint, when imposed on the KP i--function, automatically implies all
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other constraints. In other words, 9/, constraint is enough to fix the point in the grassmannian with which the KP i--function is associated and, thus, it fixes this i--function completely. It would be interesting to find the explicit dependence of the point in the grassmannian on the potential 77(X) in GKM. Even more interesting would be any alternative description of the entire subset U,. in the grassmannian, associated with GKM with arbitrary 77(X), and any reasonable parametrization U,., which would be an alternative parametrization of the space of potentials. Since 77(X) can be changed smoothly, U,. should be a kind of a manifold, lying at infinity of the universal module space (if the latter is embedded into Sato’s grassmannian), and should surely possess some amusing properties. Unfortunately, the already existing papers on the subject (of course, they concern conventional (multi) matrix models rather than GKM) either emphasize implications of K-reduction, like ref. [23], or rely upon the formalism of pseudo-differential operators, like refs. [1,201, and thus need to be translated into grassmannian language. A separate question in the framework of this approach is why should the constraint always be associated with 7iç algebra (as they actually are). Technically this is more or less obvious whenever pseudo-differential operators are used. In KP grassmannian language, the crucial ingredient should be the ~,. covariance of KP hierarchy, discovered in refs. [35—37]. The constraints of interest involve operators from some Borel subalgebra ~ of this ~, and the choice of subalgebra depends on the choice of potential. Reductions of GKM to (double-scaling limit of) multimatrix models, i.e. what happens if 77(X) ~ in grassmannian language should be attributed to intersections of U,. and submanifolds Grtt’1 in Sato’s grassmannian, associated with K-reductions. Somehow at these points a closed Zamolodchikov 7f~subalgebra emerges from entire It is certainly interesting to see how this happens and how the Lie algebra structure is broken. It is also unclear whether such a phenomenon takes place only at U,. fl Gr11<~ or everywhere in Gr~. One more important question is what is the relation between the just discussed ~ algebra of refs. [35—37]and other ~, which is presumably relevant in c = 1 models, and is naively generated by operators, which involve finite differences instead of derivatives in the free-field representation. We use this chance to note that the problem of how c = 1 models are included into the framework of GK.M is still open, and the invariant description of the U,. subset in the grassmannian would also be helpful for its resolution. ~,.
Straightforward derit’ation of 9/ and ~ constraints This approach is just the straightforward generalization of what was done in subsect. 3.2 in the derivation of the 9/~) constraint from the knowledge of explicit expression (2.2) for Z17~1.We sketch here several steps of the derivation for the
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Unified theory of 2d gravity
Virasoro case. The idea is to apply the operator Tr ~t”~’(a/ai1ir) the one hand this is equal to a
171, ~ Tr 77”(M)M”~’ —Z a1,
227
to eq. (2.2). On
(3.41)
n~l
on the other hand, it can be explicitly evaluated. Expression (3.41) is very close to main difference is that the sum in eq. (3.41) goes from n = 1 and for small enough n Tr((77’(M))”~’/77”(M)M”~’)contains powers of positite powers of M, which cannot be expressed through T-variahles. For 7—=X”~ this happens for n ~ Km. In order to get rid of these positive powers of M one should use the analogues of eq. (3.13), saying that
9/~/,~}Z17).The
~log a
Tr M’
Z171
+
(3.42)
This is the origin of conventional a2/d7~, dT,, terms in operators 9/~. As to generalization of the shift (1.17), it results in an additional contribution of the form .,
-E~~7~
(3.43)
in the definition of operators. All extra terms cancel, giving rise to the proper universal constraint of the form .9~ 1Z1~= 0, m ~ — 1. ~-constraints can be (at least in principle) derived in the same manner, though actual calculations are increasingly sophisticated. ~
Implications of Ward identities This third approach exploits the fundamental Ward identity (1.23). Technically its main difference from the previous approach is that the operators Tr 77’(a/aAtr), non-linear in a/aAtr, are involved. Conceptually this approach is different, since it does not exploit explicitly any integrable features of the partition function. We shall discuss this method in more detail in sect. 4.
4. From Ward identities to
W
constraints
4.1. GENERAL DISCUSSION
This section is devoted to the derivation (unfortunately, incomplete) of the entire set of Virasoro and W constraints in GKM. The role of such constraints in the study of any matrix model is two-fold. First of all, they can be considered as a
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complete set of differential equations, which specify the partition function of the model as a function of time variables. Then this set of equations implies, among other things, that the solution is a KP i--function (and sometimes a reduced i-(~<1 function). The first application, discussed in detail in sect. 3, is reasonable if one already knows about the integrable structure. Then the partition function is identified with some (reduced) KP i--function (i.e. evaluated at some point of the grassmannian), and the role of the Virasoro constraint is to specify this i--function (i.e. fix the point in the grassmannian). In this second type of circumstances it is enough to have a single constraint, namely 9/ 1Z =corollaries 0; all other constraints, if t”1, are deducible of this one [1,20]. imposed on the KP i--function or iWith identification of Zt7~1 with a i--function in sect. 2 and the proof of the universal 9/~/,} constraint in sect. 3 we exhausted this line of reasoning. The subject of this section is to concentrate instead on another approach and ignore almost all that we already studied about integrable structure of GKM. Then the constraints can be considered as a complete set of differential equations, which specify the partition function of the model as a function of time variables. Among other things, the set of equations implies that the solution is the KP i--function (and sometimes a reduced i-~ function). To be more concrete, we shall investigate direct corollaries of Ward identity (1.23), {Tr E(n)[77’(a/antr)
‘~]}~N[’t1=0.
(4.1)
We shall, however, discuss only specific potentials, 77(X) = const x~~± I, and use explicitly the fact that the partition function is independent of all T,,K. The main problem with the implications of Ward identity (4.1) is that they acquire the form of conventional Virasoro or W-constraints only in the limit of N = cc~The reliable results from our point of view, should, however, be N-independent. But as soon as N is finite, the complete set of independent Ward identities (4.1) is also finite. Remarkably enough they can still be expressed through generators of W-algebras, but the constraints arise as a finite number of vanishing conditions for infinite linear combinations of ill~operators, acting on partition function. As N cc, the number of such conditions tends to infinity, implying that every item in linear combinations vanishes by itself. These items have exactly the form of conventional Virasoro constraints in the particular case of K = 2, while for K = 3 they rather look like eqs. (1.26), and generalization of (1.26) for K> 3 is more less obvious. In any case the honest statement is that any solution to the 7~17constraintsin the conventional form (1.29) does satisfy the equations, which follow from (4.1), for any N (to make N finite one should take all but the first N eigenvalues of matrix M to infinity, see subsect. 2.2). In this sense transition from finite to infinite N is smooth. However, the inverse can be true, i.e. (4.1) can imply (1.29) only as N = cc, Moreover, since (1.26), which is actually implied by (4.1) as N = cc, is not quite identical to (1.29), one needs also to rely upon the (very ,
—~
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Unified theory of 2d gravity
229
plausible) assumption that solutions to (1.26) and (1.29) are unique (up to essential constant factors) and thus coincide. Such an approach has been already applied in ref. [2] to the study of the Kontsevich model itself, i.e. for the case of K=2: 772(X)=X3/3. Our purpose now is to extend this consideration to other potentials 77~(X)=X”~’/(K+ 1). However, while the structure of the answers is very clear obvious from (1.26), — the actual calculations are very tedious. Therefore we restrict our presentation below only to the first non-trivial case of K = 3, which will be considered in subsect. 4.3; in subsect 4.2 we reproduce from ref. [21 the derivation of Virasoro constraint in the case of K = 2. —
4.2. VIRASORO CONSTRAINTS IN KONTSEVICH MODEL
(K
2)
The problem. Our purpose in this section is to prove the identity itr(EP~
—
e~A)9/= ~
~
9/,,Z
tr(e
2)
(4.2)
0A° for
*
9/~21{A) JDX exp( —tr =
X3 +
tr
AX)
c[~] exp(~tr A3/2)Z~(~~),m odd
(4.3)
with
c{~]=det(~®I+I®~)~2
(4.4)
and 1
~
kT~
a
~i
a2 ___
2fl ~
a+h=2n aTaaTh a,b~0odd
aTk±
2 ~ kodd
T,~ +3fl~~)+~fl()
4
1 16
a .
aT2~~3
(4.5)
Below in this section we omit label (2) and do not indicate explicitly the fact that all sums run over only odd times. *
To avoid any comparison note that in the original version of ref. [2] there was a mistake in the signs in the exponential in (4.3) and in the shift of times (1.17) (normalization of times are also different).
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While (4.2)is valid for any size of the matrix cc) we can insist that all the quantities
A, in the limit of infinitely large A
tr(e1,A°2)
(4.6)
(N —s
(e.g. tr A’~’2for and (4.1) imply that
=
A°)become algebraically independent, so that eqs. (4.2) 9/~,Z{T)
=
0,
n> —1.
(4.7)
Method. Note that 9/(A} in eq. (4.3), which we have to differentiate in order to prove (4.2), depends only upon eigenvalues (Ak) of the matrix A. Therefore, it is natural to consider eq. (4.2) at the diagonal point A, 1 = 0, i ~ uj. The only “non-diagonal” piece of (4.2) which survives at this point is proportional to
2Ak ‘I
—
6ki
_____ a aA..aA.. 3’
=
5,, =0.in
.ii
A ‘
6kj
—A 1
for
i±f.
(4.8)
Eq. (4.8) is nothing but a familiar formula for the second-order correction to hamiltonian eigenvalues in ordinary quantum-mechanical perturbation theory. It can be easily derived from the variation of determinant formula: 6 log(det .4) For diagonal
tr~6A — ~tr(~6A~6A)
=
k
....
(4.9)
A,
1 = A,6,1, but, generically, non-diagonal 6A,1, this equation gives
6A~iE5AlJ6A1f.iE ~
+
Ak
~
ii
6A~I6AIIH
A.A 1
~
A.
A1
A,—A,.
which proves (4.8). Proof Now we shall turn directly to the proof of eq. (4.2). Since is assumed to be a function of A, it can be, in fact, treated as a function of eigenvalues A. After that, (4.2) can be rewritten in the following way: e — 2/3 tr C(~)Z{T}
Z
~
[tr
~
2
a
e2~3~
_A}]C(~)
a2z
aT
-e(A)--~’--~ a,b~-.0 TUaT,, aA, “
‘
~~2Z{T)
aT, aAi
(4.10) (4.11)
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I
~
z ,,>,,
Unified theoiy of 2d gravity
a21,
az[
~J~E,,(A)
aT
a
231
aT, alogC
aA~
‘
1aA~,
[,.j
3”2 1I
a
(4.12)
j
aA,j3— —tr aA, A
a7~f2) +
(
I~e
L 1,(A,)
2
a (2~ 3”2 I tr A 13/ /
— ~
—
~
—1
~A 1Ef)(Af)
f2)
2
+
~~(A,)( a ii aA,~aA
tr
A3”2
1,1
I /
(4.14)
3~2a log C aA~
(4.15)
‘2 a traA, A 131 —
I
\
/
a2c
1
+
~ ~,,(A,)
1
aA ,~aA
(4.16)
11j
3”2
with tr
=
~k’~k
A
(4.13)
and
c=
fl(~,~~
2~
(4.17)
1)/
i .1
First, since aT,
-
-
A~°3~2,
(4.18)
It is easy to notice that the term with the second derivatives (4.11) can be immediately written in the desired form:
(4.11)
1 =
—
~ nb—i
I =
-~ ~
~e,,(A,)A~”2
~
a2z
a+b—2n
I
a2z
tr{E 1,(A,)A ~~n_2}
(4.19)
~ a+h=211
~
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For the first-derivative terms with the help of (4.17), alogC
1
aA,
—
—
=
3 2
a
—tr A3”2 aA~
_______ —
1 ~
(4.20)
~,
+
and
~
a2A,,~
~
2T,,
t9T,
a
1
k
aA~
aAk
~
(4.21)
(4.22)
Then for (4.12) we have
az
I
I
~L{
2Z
A’~3~2A’~3~2
(~) (~)b
+ a±b=2(n±1) ~
+V~I
1 _______
—
~i+~1i
azl
1
2E~(A,)A1~l I
1
n—l
~n~0~h11,Ja=~0iJ
=
~
1
i
~
~
[E~A)Ai~~2
I
~
—
j
~ 0~1tr(pA~2){~
I aT
1 I
az
~
~
az _____
2fl÷k]
az =
I
k,31
~‘
—
j.
aT2~~3
(4.23)
k ~ 3,, +
The remaining part contains terms proportional to Z itself which need a bit more care. First, it is easy to notice that two items in (4.13) just cancel each other, so (4.13) gives no contribution to the final result. For (4.14) we have 2 tr A3~2 a tr A3”2 a2Ak E~(A,) a + Ee~(A,) aAk aAIJaAJI (4.24) Using (4.21) and (4.8) it can be transformed into 1
~-vc~~ A,—A3
1
=
~e~(A,)
1 ~
(4.25)
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233
and this cancels (4.15) (transformed with the help of eqs. (4.20) and (4.21)). Thus, the only contribution is (4.16), which gives a2~
2C
1
~
C ~
—
a log
~2C
~ aA,
1aA~, aAk
=
I
C
~
+ ~
2(A,-A1) 1
1 + ~i(~I+~k)j(1_6IJ)
[a2
aA~ log C
+
)2l
aA, C\ j. ía log
(4.26)
Now, using (4.8) and (4.20) we obtain for (4.16): (i
1
— ~E
16. I
2
_________
Ee~(A,)A~/
+ —
I,’
1,(A,)A1
1
1 ~I(~I
1
________
+ ~i)
(~,
+
1
(~+~.) (VA~+V~)
+~ ~EP(AI)AI i,j k
~1
~ e~,(A,)~ 1 2,~jkAj_AjL~j(~j+~k)
~~E
1
+
2+
1 ~I(~i+~k)]J
~(~Ep(A,)A[1A:I+~Ep(A,)A~3/2Ar3/2)
0(A,)A1
+
~(A,)( 4A,(~,+
~
I +
~k))
i*j*k
F 1 I _____
I +
2(A,
-A
1) 2
=
[~j(~~
1 +
+ ~
~k)
-
~
+ ~k)j
(~EP(AI)A~’AT1 + ~
~~(A,)A1
~
1 I
_____
12(A,)AT3/2Ar3/2) i “.3
1
2A~l/2,
i.’.jk ~ (A)A[IA/I/
(4.27)
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234
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Unified theory of 2d gravity
In the last transformation we used the fact that 1 (A,
-
F
1
A1)
1
~
+ V~k) -
1 + ~k)J
~ (4.28) Finally, (4.27) can be rewritten as 2+~ ~~(A,)A~’
EA~2A~”2 j,k
j~Le~(A,)A~
1
~2 ~-n-2I~
=
—6
1F2EA~I/2~+ ~
/~16
n
LI
I
{
(4.29)
j
{
Now taking together (4.19), (4.23) and (4.29) we obtain our main result:
3/2) a2 \1 1 tr e —~ —A 3/2Z(T}= — E tr(e~A~2) C(VA)Z{T} aA Zn>, 1jC(VA) e~r5
exp(—~trA
{
‘)
1 2
a
—
1
a2
I
i
aT
~
kTk
8fl
21I+k
+~
a +h=2n
~
~
+
6
fl.O +
4
a +
I,OT, — 1Z(T)
a =
0. (4.30)
This completes the derivation of (4.2) and, thus, of Virasoro constraints for the case K = 2. 4.3. EXAMPLE OF
K
=3
In the case of generic K the analogue of the derivation, described in subsect. 4.2, should involve the following steps, (1) Represent 9/[A] as t”t[A] =g~[A]Zt1’1(T,,), (4.31)
9/
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235
Unified theory of2dgraiity
with
=
exp[(p.,7’(p.,)
fl [~=~(p..yI/2
g~[A] = ~(A)
~
exp{a[K/(K+
[A1)~’2K
- 77(p.,))]]
I)]A*”‘}].
(4.32)
Parameter a is introduced here for the sake of convenience, in fact, a = 1. (ii) Substitute this .~1[A] into (4.1), which in this particular case of 77K(X) X”~/(K+ 1), looks like
- aKA])~[A]Z{K)(T)
{Tr E(A)[(~) When a/aAtr acts on
=
0.
=
(4.33)
Z(T, 1),
the following rule is applied:
~~tr
~
(4.34)
and the sum at the r.h.s goes over all positive n 0 mod K. As to higher-order derivatives, a’Z/aA’,~,they are defined with the help of relations like (4.8). (iii) When all the A-derivatives in (4.1) act on the exponent in g[A] =g[77’(M)], we get the term, which is equal to 77’(a Tr[M77’(M) — 77(M)1/aA,r) = 77’(M) = A and cancels the A-term in (4.1). The next contribution, when all but one of the A-derivatives act on the exponent, vanishes, Actually this reflects the fact that there are no ~ generators among the final W-constraints. (iv) Perform a shift of variables K —~1.,
—
=
~
(4.35)
I
(this procedure does not change a/aT,, —~a/aT,). (v) After all these substitutions the l.h.s. of eq. (4.32) acquires the form of an infinite series where every item is a product of Tr[~(M)M°] and of a K linear t1’~(i,),In the case = 3 combination of generators of WK algebra, acting on Z this equation looks like + 9~M3”
~ (3k
— 2)~k277~3k
+ a+b=3n a,h +1);n —3
E(3k
— 2)~k277~2k
+
~ a+h=3n a,h+O~n7. —3
/3
a aTIU+I -_a
+
,~,
9 ~
M3’~2~3
Z~31= 0, (4.36)
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/
Unified theory of 2d gravity
(see eqs. (1.27) and (1.28) for explicit expressions for v-operators). This relation is valid for any value of N, and in this sense is an example of identity, which behaves smoothly in the limit N —s oc~ (vi) If N = cc all the quantities Tr e(M)M~with given p but varying ~(M) become independent, and (4.35) may be said to imply (1.26). In what follows we restrict ourselves to an illustrative calculation: we prove that (4.35) holds for N = 1 (i.e. when there are numbers instead of matrices), Even this calculation is long enough. Additional technical details, necessary to deal with matrices rather than ordinary numbers, are exhaustively discussed in subsect. 4.2. Example When (4.31) with used, we obtain (Z a2
O=’~(p.) ~~ 1p. +
K
3, N = I is substituted into (4.32) and the rule (4.33) is
=
31):
Z~
az
1
a
aT0p.
1
27
az
28
p.’1 ST,,
27
1
n+4 1
+
p.
~
1 1 a3z p.3 t,~ 27p.’~~’~ aT,aT~aT~+ n2+12n+39
a2Z
m+fl aTmaTa
n+m+8 18
~
E ~
az
1
a2z
p.m+n
aTmaTn
7
(4.37)
Our purpose now is to compare this expression with the l.h.s, of eq. (4.35). In order to understand the structure of (4.35) let us note that the first item (with 7f(3)) in eq. (4.35) contains the contribution
a3z
1 p.(+/fl+fl
aT 1aT,0aT~
just the same as in eq. (4.36), but only under the restriction I + m + n = 0 mod 3. However, no such restriction is imposed in (4.36), and in order to restore the equivalence (4.35) and which (4.36), add onetheneeds add the terms with 2)z)/aT between to the l.h,s. of (4.35), missingtocontributions
a(7#”’
i p.l±?fl+fl
a3z
aT,aT,~aT,, withl+m+n=1
2
mod3.
If we analyze the terms without a in eq. (4.36), it is important that a also appears in the shift (4.34): nhj31
=
nT,, — 3a6,, 4,
(4.38)
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Unified theory of 2d gravity
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so that at the same time we substitute all t’s by T’s in (4.35), (1.26)—(1.28). Then the term
(K—1)(3K—1)(5K—1)...((2K—1)K—1)28 (2K)K in (4.36) should be compared to the contributions without Z-derivatives to eq. (4.35). These come from the negative harmonics of ir”~ and ~(2) operators. Namely, so modified 7cf’,.~ contains (2T2)3 + 3(TI)2(4T4) = 4/p.°, 7f~P.~contains 3 = 1/p.3, and there is no constant term in ~ As to Virasoro operators, (TI) there is /, ‘2(T,)(2T 3 in 7l7j~ and (K + 1)/24 = 1/9 in ~2) 2) = 1/3p. Also the contribution of interest to a7~v.~/~/aT, is / ‘2(2T 2, and to 2) = 1/3p. a7r~/aT 2/3p.. Also (2T 5, 2—is (5T ~ ‘4(T1) = contributes -~‘2(T~)(2T2)~ = 1/3p. (5T 21 5, (2T 2)~T~3~ 2~ — (2T 2, (T 5)~T,.~ 5) ‘/~= ‘/~= 1/9p. — (TI)’ 1)7’~ 2(2T 4, (4T1/9p. 2~(4T 2)~ 42)and (TI)~2~ = l/9p.. 2(T1) 2) = 1/3p. 4)~.ir~ 4)~ = 1/9p. Putting this all together we obtain the coefficient —
1)(K
—
—
=~-
{[4
+
1]
+
9[_
+
+
+
+
9[~
+
+
+
+
0(a)
=
+
0(a)
in front of Z in (4.35), in accordance with (4.36). in order to reproduce the term [(K 1)(K + 1)(2K + 1)/24K]a’<2Z = ~aZ in (4.36) it is necessary to restore the shift (4.36) of the T 4 variable in (4.35). There are also contributions with T,~ in ~2 term andinin(4.36). (3k — 2)T2k..27#1~3,,, which are responsible for the occurrence of the a This illustrative comparison of (4.35) and (4.36) is enough to recognize the proper structure of (4.35). After it is found out it is easy to ensure that all the remaining contributions to (4.35) and (4.36) also coincide. Of course, in order to check the trace structure of (4.35) our simple example of N = 1 is not enough the analogue of detailed consideration of subsect. 4.2 is required. Such detailed derivation, as well as consideration of the case of K> 3, is beyond the scope of this paper. —
5. Conclusion To conclude, we presented enough evidence that the Generalized Kontsevich model (1.2) interpolates between all the multimatrix models, while preserving the property of integrability and the 9/, constraint. This makes 0KM a very appealing candidate for the role of a theory, which could unify all “stable” (i.e. with c ~ 1) bosonic string models. However, the study of GKM was at most originated here. Let us list a set of problems, which seem interesting for the future development.
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Unified theory of 2d gravity
(1) Though the integrable structure and constraint in GKM have been discussed more or less exhaustively, the situation with generic ~constraint and their implications remains less satisfactory. Of course, they may be derived from the .9/— I constraint, as suggested in ref. [1], but it is still desirable: (i) to complete the derivation of v-constraints from Ward identity (1.23) for N = cc, as suggested in sect. 3, for K> 3; (ii) to prove the equivalence of these -constraints [which look like (1.26)] to conventional constraints (1.29); (iii) to describe the subvariety U,. in the grassmannian (at infinity of the Universal module space), specified by the universal 9/~1”~ constraint; (iv) to take the double-scaling continuum limit of the properly reduced v-constraint in discrete multimatrix models (which look like “[M1] = ~>‘~
I
for the 2-matrix case [TI) in order to derive the constraints (1.31) for \/T’.~~ (as was done in ref. [16] for the one-matrix case); (v) to add the problems discussed in subsect, 3.4 to this list. (2) We tried to argue that a particular (K — 1) matrix model arises from 0KM for a particular choice of potential: 77(X) 77~(X)= const ‘X’~’. A particular (CFT + 2d gravity) model is specified by additional adjustment of the M-matrix in 2/pK). These such a way that all T,, = 0, n ~ 1, p + K (for c = I — 6(p — K) particular points in the space of parameters (77(X), M) should be interpreted as critical points of GKM. The relevant questions are: (i) what is the proper criterion, distinguishing critical points in terms of GKM itself; (ii) are there any other critical points? One can even hope that the answer to the last question is positive and some other critical points may be identified with non-bosonic string models. Let us bear in mind that the stability criterion like c ~ 1 (implying the absence of tachionic instabilities, leading to disintegration of Riemann surfaces through the creation of holes), which in the bosonic case implies the restriction d ~ 2 for the space-time dimension, in the case of N = 2 superstrings turns into a much more promising condition: d ~ 4. (3) A separate class of problems is related to identification of (double scaling limit) c = 1 models in the theory of GK.M. This could lead to a more natural formulation of 0KM, in particular, help to restore the symmetry between arguments 77(X) and M of Z1” 1[M]. Keeping in mind the topological implications of the K = 2 model it is also natural to ask what corresponds to the Penner model [38,39]? (4) The last group of problems concerns the meaning and generalizations of 0KM. First of all, both formulations in terms of topological (like refs, [15,40,38,41,39]) and conformal (in the spirit of refs. [42,43]) are required in order to understand explicitly the connection to Liouville theory. Second, the hope that just GK.M (1.2) is enough to include N = 2 superstrings may be too optimistic, and —
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then one needs to look for more universal models. Third, in any case one should search for a more invariant (algebraico-geometric) interpretation of (1.2): GKJvI is already much simpler than the original multi-matrix models (it is a one-matrix model with a smooth continuum limit), but still it is not simple enough to be appealing as a fundamental theory. The main aesthetic problem is the asymmetry between M and 77(X) and, perhaps, the very relevance of matrix integrals. Fourth, GKM does describe interpolation between sufficiently many string models, but it is not a dynamical interpolation: one may still need to look for an “off-shell” version of GKM, involving a sort of integration over 77(X). This problem can be after all related to the question about critical points. In particular, it may be interesting to find out something similar to the “off-shell” actions of refs. [17,44,45], which reproduce Virasoro constraints in the ordinary 1-matrix models as dynamical equations of motion, for the case of GKM. Possible links between 0KM and double-loop algebras (in the spirit of ref. [46]) also deserve investigation. We are indebted for stimulating discussions to A. Gerasimov, E. Gava, M. Kontsevich, Yu. Makeenko, K. Narain, A. Orlov, S. Pakulyak and I. Zakharevich.
References 11] M. Fukuma, H. Kawai and R. Nakayama, Int. J. Mod. Phys. A6 (1991) 1385; Infinite dimensional grassmannian structure of two-dimensional quantum gravity, preprint UT-572, KEK-TH-272 (1990); Explicit solution for p—q duality in two-dimensional quantum gravity, preprint UT-582. KEK-TH-289 (1991)
[2] A. Marshakov, A. Mironov and A. Morozov, Phys. Lett. B274 (1992) 280 [3] V.A. Kazakov, Mod. Phys. Lett. A4 (1989) 2125 [4] E. Brézin and V.A. Kazakov, Phys. Lett. B236 (1990) 144 [5] M. Douglas and S. Shenker, NucI. Phys. B335 (1990) 635 161 D. Gross and A.A. Migdal, Phys. Rev. Lett. 64 (1990) 127 [7] 5. Chadha, G. Mahoux and M. Mehta, J. Phys. A14 (1981) 579 [8] A.A. Belavin, AM. Polyakov and A.B. Zamolodchikov, Nuci. Phys. B241 (1984) 333 [9] M. Douglas, Phys. Lett. B238 (1990) 176 [10] N. Seiberg, Notes on quantum Liouville theory and quantum gravity, preprint RU-90-29 (1990) [11] 5. Kharchev, A. Marshakov, A. Mironov. A. Morozov and A. Zabrodin, Phys. Lett. B275 (1992)311 1121 D.J. Gross and M.J. Newman, Phys. Lett. B266 (1991) 291 [13] Yu. Makeenko and G. Semenoff, Mod. Phys. Lett. A6 (1991) 3455 114] A. Marshakov, A. Mironov and A. Morozov, From Virasoro constraints in Kontsevich’s model to W-constraints in 2-matrix models, preprint FIAN/TD/05-91, ITEP-M-7/91 (August 1991) [15] ML. Kontsevich, Funk. Anal. & Prilozh. 25 (1991) 50 (in Russian) 116] Yu. Makeenko, A. Marshakov, A. Mironov and A. Morozov. Nucl. Phys. B356 (1991) 574 [17] A. Gerasimov, Yu. Makeenko, A. Marshakov, A. Mironov, A. Morozov and A. Orlov, Mod. Phys. Lett. A6 (1991) 3079 [18] E. Witten, Nuci. Phys. B341) (1990) 281 [19] F. Witten, On the Kontsevich model and other models of two-dimensional gravity, preprint IASSNS-HEP-9l/24 (June 1991) [20] R. Dijkgraaf, H. Verlinde and E. Verlinde, Mud. Phys. B348 (1991) 435 121] A.Yu. Orlov, in Proc. Kiev Int. Workshop (World Scientific. Singapore, 1988) p. 116
240
S. Kharchav at a!.
/
Unified theory of 2d gravity
[221 W-constraints in discrete multi-matrix models, preprint FIAN-ITEP (in preparation) [23] V. Kac and AS. Schwarz, Phys. Lett. B257 (1991) 329 [24] CD. Itzyson and J.-B. Zuber, J. Math. Phys. 21(1980) 411 [25] ML. Mehta, Commun. Math. Phys. 79 (1981) 327 [26] T. Miwa, Proc. Japan Academy 58 (1982) 9 [27] S. Kharchev, A. Marshakov, A. Mironov, A. Orlov and A. Zabrodin, NucI. Phys. B366 (1991) 569 [28] A. Gerasimov, A. Marshakov, A. Mironov, A. Morozov and A. Orlov, Mud. Phys. B357 (1991) 565 [29] N. Sato, RIMS Kokyuroku 439 (1981) 30 [30] G. Segal and G. Wilson, PubI. I.H.E.S. 61(1985) 1 [31] A. Zabrodin, Teor. Mat. Phys., 78 (1989) 234 (in Russian) [32] V. Drinfeld and V. Soko~ov,J. Soy. Math. 30 (1984) 1975 [33] E. Date, M. Jimbo, M. Kashiwara and T. Miwa, in Proc. RIMS Symp. Nonlinear integrable systems — classical theory and quantum theory (RIMS Kyoto, 1983) p. 39 [34] D. Gross and A.A. Migdal, Mud. Phys. B340 (1990) 333 [35] A. Orlov and F. Schulman, Lett. Math. Phys. 12 (1986) 171 [36] P. Grinevich and A. Orlov, Flag spaces in KP theory and Virasoro action on det and Segal—Wilson r-function, preprint Cornell Univ. (September 1989) [37] A. Semikhatov, Int. J. Mod. Phys. A4 (1989) 467 [38] R. Penner, J. Dill. Geom. 27 (1987) 35 [39] J. Distler and C. Vafa, Mod. Phys. Lett. A6 (1991) 259 [40] J. Harer and D. Zagier, Invent. Math. 85 (1986) 457 [41] C. Itzyzson and J.-B. Zuber, Matrix integration and combinatorics of modular groups, preprint SPhT/90-004 (1990) [42] A. Marshakov, A. Mironov and A. Morozov, Phys. Lett. B265 (1991) 99 [43] A. Gerasimov, preprint ITEP (1991) [44] A. Jevicki and T. Yoneya, Mod. Phys. Lett. AS (1990) 1615 [45] P. Ginsparg, M. Goulian, MR. Plesser and J. Zinn-Justin, NucI. Phys. B342 (1990) 539 [46] A. Gerasimov, D. Lebedev and A. Morozov, mt. J. Mod. Phys. A6 (1991) 877
P H VS I CS B
B380 (1992) 181—240
Towards unified theory of 2d gravity S. Kharchev, A. Marshakov and A. Mironov Department of Theoretical Physics ~, P.N. Lebedec Physical Institute, Leninsky Prospect 53, 117 924 Moscow. Russia
A. Morozov
**
Institute of Theoretical and Experimental Physics, Bol. Cheremushkinskaya St. 25, 117 259 Moscow, Russia
A. Zabrodin Institute of Chemical Physics, Kosygina St. 4, 117334 Moscow, Russia Received 15 November 1991 Accepted for publication 6 February 1992
We introduce a new one-matrix model with arbitrary potential and matrix-valued background field. Its partition function is a r-function of KP hierarchy, subjected to a kind of -~ constraint. Moreover, the partition function behaves smoothly in the limit of infinitely large 1~~ . this partition function becomes a r-function of matrices. IfKPthehierarchy, potentialobeying is equala set to of X K-reduced 71k~ algebra constraints identical to those conjectured for the double-scaling continuum limit of the (K — I) matrix model. In the case of K 2 the statement reduces to an earlier established relation between the Kontsevich model and ordinary 2d quantum gravity. The Kontsevich model with generic potential may be considered as an interpolation between all the models of 2d quantum gravity, with c <1 preserving the property of integrability and the analogue of the string equation.
1. Motivation and results 1.1. INTRODUCTION
Matrix models play an increasingly important role in the theory of quantum gravity, being a reasonably simple realization of Regge calculus, adequate at least in essentially topological theories. Their conventional application to string theory is based on identification of the critical points of the particular “double-scaling” limit [3—6]of hermitean multimatrix models [7] with the c < I minimal conformal field * **
E-mail address: [email protected]. E-mail address: [email protected].
0550-3213/92/$O5.OO © 1992
—
Elsevier Science Publishers By. All rights reserved
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Unified theojy of 2il grauitv
theories [8] coupled to 2d gravity. Such models are often supposed [9] to exhaust the entire set of consistent bosonic 2-dimensional string models, while bosonic strings beyond two space-time dimensions are presumably unstable [10]. Technical methods, based on the application of matrix models, appeared rather successful not only in the framework of perturbation theory, but they allow one to estimate the entire sums of perturbation series, thus raising the study of string models to a qualitatively new level. On the other hand, matrix models appeared to possess unexpectedly deep internal mathematical structure [91.In fact, if matrix models are used to describe interpolation between different critical points (i.e. particular c < 1 string models), the interpolating flows are mutually commuting, i.e. the entire pattern of flows constitutes an integrable hierarchy (some reduction of the Kadomtsev—Petviashvili (KP) system). This integrable structure indicates that the somewhat artificial description in terms of matrix integrals should possess a more invariant algebro-geometrical description, which could be used for formulation of the as yet unknown dynamical principle of string theory, unifying all available string models into a distinguished and (hopefully) physically important entity. The main obstacles on the way from matrix models to such a general principle are: (i) the sophisticated intermediate step: the double-scaling limit, (ii) the somewhat obscure origin of integrability: it is not a priori obvious why matrix integrals with generic “potentials” are r-functions of (reduced) KP hierarchies, (iii) the non-universal description of different c < I string models: some of them are nicely unified as different critical points of a particular multimatrix model, but there is no nice interpolation between critical points of models with a different number of matrices. The naive way out is to use the ~-matrix model, but it is no longei- described in terms of finite-dimensional matrix integrals, and, more important, reduction to the case of a particular multimatrix model is too much singular. In this paper we propose a kind of resolution of all these three problems. Namely, we argue that there is a new one-matrix model [11] (which we call generalized Kontsevich model (GKM)), which (i) for a specially adjusted potential describes properly the double-scaling limit of any multimatrix model, (ii) has the partition function, which is a KP T-function, properly reducible at the points associated with multimatrix models, and satisfies an additional equation, which reduces to conventional constraints for multimatrix models, (iii) allows continuous deformation of potential, changing it from the form associated with a given multimatrix model to those corresponding to the others. This makes the study of GKM a natural step in the development of string theory. ~‘
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In the remaining part of this section we describe GKM and its relation to multimatrix models more explicitly. The proofs are presented in sects. 2—4. For a more condensed presentation of our results see ref. [11]. 1.2. GENERALIZED KONTSEVICH MODEL
This paper is devoted to the study of the one-matrix model in an external matrix field A, essentially defined by the integral over the N X N matrix X: 57)[A]~fdXexp[Tr~(X)-i-TrAX],
(ii)
with arbitrary “potential” ~©(X) and dX= 117~~ dX11. Such integrals arise during the consideration of various problems [2,12—14],but this time our purpose is to use (1.1) in order to define a unir’ersal matrix model, which could describe interpolation between all the multimatrix models and their critical points, as explained in subsect. 1.1 above. Namely, we present strong arguments that the partition function of such a universal model may be identified with Z~[M], where fexp[_U(M,Y)]
dY
z~’}[M]=—
,
(1.2)
fexp[_LJ2(M, Y)] dY with U(M, Y) =Tr[~(M+ Y) —~z(M) ~Y’(M)Y], —
U2(M, Y)
=
E limt) —~U(M,cY),
(1.3) (1.4)
c—’
(i.e. U 2 contribution to U(M, Y)). 2(M, is the Y After theY) shift of integration variable X=M+Y
(1.5)
the numerator in (1.2) may be rewritten in terms of (1.1): fe~©M~
dY=
~
(1.6)
The denominator in eq. (1.2) is a simple gaussian integral—a peculiar matrix generalization of [~v~”( 1L)} — i /2
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Unified theory of 2d gracity
We shall call Z~’[M], defined by eq. (1.2), the partition function of the generalized Kontsevich model (GKM). The reason for this is that for the special choice of potential *: (1.7) ~2(X) =X3/3, eq. (1.2) becomes the partition function of the original Kontsevich model [15]: N(X)
fdYexp[-_(1/3)
=
Tr Y3_TrMY2]
Z~1[M]= fdY exp[—Tr MY2]
fdXexp[_(1/3) =exp [—(2/3)
TrX3+TrM2XI
Tr M3]
.
(1.8)
fdX exp[—Tr MX2] Eq. (1.8) has a natural generalization, which is also a particular case of (1.2): if
~‘~(X) =7K(X)
=XKi/(K+
1)
(1.9)
eq. (1.2) becomes K Z~[MI~exP[_K+lTrM~]fdXexP
x
{fdX exP[_
_Tr~+TrMKX
(1.10)
~Tr{EMaXMbX~}
We shall argue that Z~1[M], defined by (1.10), is identical to the square root of the partition function of the (K 1)-matrix model in the double-scaling limit with —
=
1 —Tr M”,
n
±0
mod K,
(1.11)
playing the role of generalized Kazakov’s time variables [3,16]:
~ *
(1.12)
We do not specify the integration contour in matrix integrals (1.1), (1.2), since all that we are going to discuss does not depend on it. To make contact with the literature, note that for potential = X~1/(K + 1) integrals should be over hermitean or antihermitean matrices if K + I is even or odd respectively.
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Unified theory of2dgracity
185
where [7] Fj~=
tim
~
double scaling
(1.13)
=
(the term MKMK+l in the exponent should be omitted; the change from discrete time variables {t~}to continuum (T~}is implicit in performing the double-scaling limit, see refs. [16,17]). Since potential IX) can be continuously deformed from one K in eq. (1.9) to another, the GKM (1.2) as a corollary of (1.12) may be considered as continuous interpolation between all the models of 2d gravity with c < 1, described by particular multicritical points of particular multimatrix models (let us remember that for K fixed the pth multicritical point is defined by the condition [I] that all
1,
=
0, except for T1 and TK+,,,
(1.14)
and according to (1.11) this is a constraint on the form of the matrix M). This is the sense in which the GKM (1.2) may be considered as the universal partition function of bosonic string models. We shall not discuss the implications of this fact here, instead we shall concentrate on the arguments in favour of this conclusion, which is entirely based on the identity7~[M]itself (1.12) (and also it motivated is usually by a the KP nice features of the to interpolating function see Z~, below). Our strategy is to try to i--function, subjected a 2’ constraint, understand as much as possible about Z~[M], on the l.h.s. of (1.12), and compare these results with what is conjectured in ref. [1] about ~/I~~on the r.h.s. —
1.3. PROPERTIES OF Z121
In order to motivate our considerations let us recall first what is known about Z(K}[M] in the particular case of K= 2. Expression (1.2) has been derived in ref. [15] as a representation of the generating functional of intersection numbers of the stable cohomology classes on the universal module space, i.e. it is defined to be a partition function of Witten’s 2d topological gravity [18]. In ref. [2] (see also ref. [13,19] for alternative derivations) it was shown that as N —s Z~}considered as a function of time variables (1.11), satisfies the set of Virasoro constraints ~,
n
~ kTk kodd
(1.15)
a aTk± 2fl
+~
—1,
~ a+b= —2n a ,b odd a nd > (I
a+h=2n dTa3Ti, a ,h odd and> I)
aTabTh
+
~
—
.
T5+21
(1.16)
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(Notations are slightly changed as compared to ref. [2]. This derivation is partly reproduced in subsect. 4.2 below.). Sometimes it may be convenient to consider the appearance of the last item on the r.h.s. of (1.16) as a result on an additional shift of time variables, Tii ~
n
—
ii
~n,K±t,
(117
.
however, these T-times are defined in a K-dependent way and do not seem to have too much sense. Constraints (1.15) are exactly the equations (see refs. [1,20,16]) imposed on ~ Moreover, it is very plausible that they possess a unique solution, so that (1.15) in fact implies that (1.18) An alternative solution to the constraints (1.15) is given [1,20] by a galilean-invariant KdV i--function [21], and, relying upon the same belief that the solution is unique, one concludes that Zc~21is a i--function of the KdV hierarchy, subjected to an additional 5f’~ constraint: (1.19)
Z~21=TKdV=T121,
(1.20) The notation i-121 reflects the fact that are TKdV may be considered as a KP i--function, evaluated at specific points of grassmannian (see subsect. 2.5 below for details), corresponding to the 2-reduction of KP hierarchy. Generically, the i--function of K-reduced hierarchy, i-{K}(j~),is the KP i--function at peculiar points of the grassmannian and possesses the following property:
...TKI, 0,
TK±l...T2KI,
0,
~
(1.21)
This means that the entire dependence of generic i-{1’~T} of all variables TflK (i.e. 7~ with n 0 mod K) is exhausted by the simple exponent =
ex~(~anxi~x)
(1.22)
with time-independent parameters aflK. Moreover, this factor is actually absent in the case of a GKM partition function: an K[Z~] 0. =
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The significance of the just reported results concerning K 2 is two-fold. First, they establish the relation (1.7) between two different models of 2d gravity: Witten’s topological gravity [19], represented by Z~.S21[15],and ordinary quantum 2d gravity [4—61,represented by ~ This was the main emphasise of ref. [2]. The second implication, and it is the one of interest for us in this paper, is the possibility to describe the sophisticated double-scaling limit of matrix model in terms of a completely different matrix model (1.2). It is important that Z~©is in fact a “smooth” function of N as N and all the properties of Z~~<) are just the same as at finite N’s in variance with the double-scaling limit for ordinary matrix models, the limit N —s ~ for GKM is non-singular. Also, to avoid misunderstanding, we should emphasize that what we mean by the double-scaling limit here is not the limit, describing a particular fixed point, but the entire pattern of all the flows between various fixed points of a given (K 1)-matrix model, so that the partition function TJ.~’1is indeed a function of all time variables (1.3). =
—~ ~,
—
—
1.4. THE CASE OF ARBITRARY K
In order to generalize the identity (1.15) to the case of arbitrary K, i.e. to prove (1.12), one can try to prove either an analogue of (1.15) or that of (1.19), (1.20) and compare the result with what is expected for ~TJ~ I) This will be our strategy below. Namely, in sect. 2 a simple and promising formalism is developed, based on interpretation of (1.11) as a Miwa transformation, and we use it to prove that any Z~71[M], defined by (1.2) and (1.11) (i.e. with arbitrary 7~(X))is in fact a KP i--function in particular, it satisfies bilinear Hirota difference equations. Moreover, in the grassmannian parametrization of KP i--functions the point of the grassmannian is specified by the choice of potential ~©(X), so that multimatrix models with different K are [due to eq. (1.12)] associated with different points of the grassmannian. (In terms of the universal module space these points are in fact associated with a certain infinite-genus hyperelliptic surface for K 2, and with certain infinite genus abelian coverings of degrees K in the general case. In this sense interpolation with the changing ?(X) is some flow at infinity of the universal module space.) What is specific for a given K is that whenever 7~(X)is a homogeneous polynomial of degree K+ 1, i.e. if it is of the form (1.9), the KP i--function becomes independent of all TK,, i.e. acquires the property (1.21) and may he considered as a i--function i-1”~ of a K-reduced hierarchy. This will also he explained in more detail in sect. 2. Though any Z~71[M] defined by (1.2) is a KP i--function, the inverse statement is not correct. Z~/1[M] is subjected to the infinite set of constraints, essentially implied by the Ward identities for the integral (1.1) [12,2]: —
=
{Tr e(A)[7~’(~)
-~1}~~=°
(1.23)
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with any c(A). These identities result from invariance of (1.1) under any shift of integration variable X—SX+E(A). Eq. (1.23) seems to provide a complete set of differential equations, unambiguously defining ~ (up to a A- and N-independent constant). This statement is the actual basis for all of the arguments, which make use of the uniqueness of solutions to any constraints. As shown in ref. [2] in the particular case of K = 2 the entire set of such identities with all possible E(A) is equivalent to (1.15) and thus implies (1.19) and (1.20). In fact in sect. 2 we follow a somewhat different logic and give an alternative direct derivation of relations
z~j~1=i-~’<}
Z~/~}=T,
(1.24)
for any X) and K, so it remains only to prove the analogue of (1.20). Such a direct proof is presented in sect. 3, moreover it is valid not only for potentials of the form (1.9), but for any ~(X):
a
1
E
Tr ~V”(M)M”~’
n1
1
ai~ ~/~“(~.)
—~“(~.)
a Z1~1=0,
-
~
(1.25)
(j.t 1)7~
(j.t1)
iij—j.t1
aT1
where p., are eigenvalues of the matrix M. For potentials (1.9) ‘(f~lacquires the conventional form of 5~’~[see eq. (1.28) below] and may be easily supplemented by other generators of Virasoro algebra. Although eqs. (1.24) and (1.25) provide a complete description of the partition function of GKM, the study of direct consequences of Ward identities (1.23) also seems interesting. In particular, it should lead to a direct derivation of the analogue of Virasoro (1.15).parametrizes It also merits noting the meaning of 1[M] constraints is that ~V(X) only some that restricted subset of constraints on Z,~/’ the universal module space. Only this subset is associated with matrix models and interpolations between them. This is also clear from the fact that ~©(X) = is naturally parametrized by an infinite vector (~7~} rather than by a matrix Am,,, which would be natural for the entire grassmannian. Remarkably enough the “dimension” of this subset is just the same as that of the space of time variables (the space of potentials is parametrized by a vector {~}, while that of time variables is parametrized by (1~,,}or by eigenvalue vector {p>,} of the matrix M). This clearly suggests that a formalism should exist for such a restricted set of i--functions, which would treat potential and times in a more symmetrical fashion (see also the discussion in subsect. 3.4). *
*
Let us also remark that such symmetry exposes itself in the fact that the degree of potential K and the order of the multicritical p + K (fixed by the choice of times) give the theory of minimal matter with c = I —6(p — K)2/pK coupled to 2d gravity, and, thus, p and K should enter the theory on equal footing.
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Implications of Ward identities (1.23) for Z~J<1[MI with K> 2 are discussed to some extent in sect. 4. This is a straightforward calculation, though much more tedious than in the simplest case of K = 2. We argue that the proper generalization of eq. (1.15) for the case of K 3 is given by the following set of constraints: na—2:
~
(
~
(3k
a — l)T
I~’3k±3n+
3k..
a,b~0, ~
~ a+h=3n
7/3/~3
Z~
0
=
3a+2
n>—2; (3k
—
2)~2k2~i~2fl +
k>t-f-~I,,~,(,
a
~
~
Z,.
~
a+h=3n
=
0,
3a+I
(1.26)
a,b>~0, n>—3.
Here ~ stands for the Knth harmonics of the jth generator of Zamolodchikov’s WK algebra, and the proper notation would be ~j.x-(i){K) or The way to express these quantities through free fields is described in ref. [1]; for example, ~
~
kTklT/
k,/>1
2
a aTk+/+
~
a
a+h=k+3n
aTaaTh
+3~kTk
3fl
k>l
3
a+h+c’=3n +
a aTaaThaTc
~
aT,,bT~,cT,,; a, b, c>0, a, b, c, k, ls~0 mod 3
(1.27)
a±h-1-c= —3n
and
a
1 k
K
a2
1
aTk±Kfl 2Ka+b=KnaTaaTb a, 6
+
1 ~
E
>0
(K—1)(K+1)
aT,,bT 6
+
24K
~
(1.28)
a+h-’—Kn
a, 6 > 1)
(Note that in order to make formulas a bit more compact, we expressed all these operators through ~ rather than 1, variables. These are defined in (1.17), a/aT = a/aT.) The analogue of (1.26) for K> 3 is a more or less obvious generalization of (1.26), involving all the generators ~ ~ 1) , ~ of W~-a1gebra with the “coefficients” made from T’s and (a/aT)’s. The form of these operators is a bit reminiscent of the 7r-operators, introduced in refs. [14,22] as ingredients of Ward identities in discrete multimatrix models.
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Eqs. (1.26) (and their analogues for solution of the simpler set of equations, ~Z~1=0,
K>
3) are obviously resolved by any
k=2,3,...,K;
n~1—k
(1.29)
[eq. (1.20) is the particular case: k = 2, n = — 1]. On the other hand the constraints (1.26) are equivalent to (1.23) for the particular form of potential (1.9), and thus are supposed to possess a single solution. Therefore from this point of view it also looks plausible that (1.26) (and its counterpart for any K) is simply equivalent to (1.29). Unfortunately we do not provide an explicit proof of this equivalence (though we could rely, of course, upon the indirect proof from ref. [1], deducing (1.29) from eqs. (1.19), (1.20), which are proved in subsect. 2.3 without any reference to (1.23)). If one believes in this transition from (1.26) to (1.29), we can finally return to multimatrix models. Namely, in ref. [1] it was suggested that ~/TJ~11 (i) is a i--function of K-reduced hierarchy, (ii) satisfies the constraint n~ —1,
(1.30)
and, as a corollary of (i) and (ii), (iii) satisfies the entire set of eqs. (1.29): ~k~YT’i~~=0
k=2,3
K;
n~l—k.
(1.31)
Now it remains to use the above-mentioned assertion that this system has a unique solution in order to arrive at our identification (1.32) and thus to the central idea of this paper: unit’ersality of GKM (1.2). 1.5. COMMENTS
Before we proceed to actual derivation of all these statements, let us comment briefly on the very status of the constrains (1.29) imposed on ~T’J~’1.In order to honestly derive such constraints on the lines of ref. [161it is first necessary to find out their counterparts (loop equations) in the discrete multimatrix models (these are expressed in terms of the so-called ~ operators in refs. [2,22]), then carefully perform the appropriate reductions (including identification of all the K 1 a priori different potentials), and take the double-scaling limit. The actual form of the ~/constraints is very similar to (1.26) and therefore it may happen that the —
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final constraints would arise just in the form of (1.26) rather than (1.29). This could make the relation (1.26) even more important. After this general description of our reasoning and results it remains to prove that (i) Z,~/~1as defined by (1.2) (the limit N—*~ is smooth) is indeed a KP T-function; (ii) for any iV(X) it is subjected to an additional constraint (1.25); (iii) if IX) = const .X~I, Z~’© in fact becomes a i--function i-~’~ of the K-reduced KP hierarchy; (iv) in this case the entire set of constraints like (1.15) for K = 2 and (1.26) for K = 3 are also imposed on Z~K}. The proof of points (i) and (iii) is given in sect. 2, point (ii) is discussed in sect. 3 and point (iv) in sect. 4. While (i), (ii), (iii) are proved in full generality — for any potential ~©~(X) — the universal form of the constraints (iv) is not completely understood. Moreover, the proof of(iv) for the case of X)=X~/(K+ 1), as presented in sect. 4, is not complete even in the simplest case of K = 3. Also the proof of (iv) is not independent of (iii), since eq. (1.21) is used. The deep understanding of these ft~.constraints and their universal formulation remains the first obvious problem for GKM to be resolved in the future. The second obvious problem is to find out the algebro-geometrical origin of the model (1.2) itself.
2. Partition function of the GKM and KP r-function 2.1. SHORT SUMMARY
The purpose of this section is to prove that (A) The partition function Z~<)[M], if considered as a function of time variables =
—tr M’, n
(2.1)
is a KP i--function for any value of N and any potential ~[X]. (B) As soon as ~[X] is a homogeneous polynomial of degree K+ 1, Z~<1[M] is in fact a i--function of K-reduced KP hierarchy. Moreover, actually, aZ11~/aT,,K= 0. In order to prove these statements, first, in subsect. 2.2 we rewrite the r.h.s. of eq. (1.2) in terms of the determinant formula (2.21) Z~.7~)[Lv!]
det =
‘
i,
j
=
1
N.
(2.2)
We show in subsect. 2.3 that any KP i--function in the Miwa parametrization has
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the same determinant form. Finally, as a check of self-consistency, we prove in subsect. 2.4 that any determinant formula (2.2) with any set of functions (41(p.)} satisfies the Hirota difference any determinant formula (2.2) with any set of functions {41(p.)) satisfies the Hirota difference equation. A brief description of grassmannian (or Universal module space) language is given in subsect. 2.5. The point of the grassmannian corresponding to the i--function (2.2) is defined by the infinite-vector function {~1(p.)}.Remarkably, the results of ref. [23] which concern identification of the points in the grassmannian, associated with multi-matrix models, are immediately reproduced from the explicit expressions for the basis {~1(p.)}.This exhausts our discussion of (A). The main this which distinguishes matrix models from the point of view of solutions to the KP hierarchy is that the set of functions {~1(p.)}in (2.2) is not arbitrary. Moreover, this whole infinite set of functions is expressed in terms of a single potential ~[X] (i.e. instead of arbitrary matrix A1 in 4.(p.) = EA11p.~we have here only a ti’ector ~ or 2~[p.] = E~p.’).This is the origin of and other ~:constraints (which in the context of KP hierarchy may be considered as implications of 2’.~).All these constraints are in fact contained in the Ward identity (1.23). While a detailed discussion of these constraints is postponed to subsect. 3.4, in subsect. 2.6 we discuss another implication of (1.23) arising in the case of polynomial 2’~[X]. Namely, whenever ~[X] is a homogeneous polynomial of degree K + 1, ~±~(p.) in (2.2) can be substituted by p.~1(p.)and in such points of the grassmannian the KP i--function is known to possess the property (1.21) and thus may be considered as a i--function i-~ of K-reduced KP hierarchy. In subsect. 2.7 we prove that the factor (1.22) is actually absent in GKM. Further discussion of (B) will be given below. ~/
2.2. FROM 6KM TO DETERMINANT FORMULA
Numerator of (1.2) We begin from evaluation of the integral (1.1): .9~}[A]mfdXexp(—Tr[?~(X)—TrAXI}.
(2.3)
If eigenvalues of X and A are denoted by {x,} and (A,) respectfully, this integral can be rewritten as
[~
fdx1 exp[ -~(x~)+ A1x1]~(X).
(2.4)
VN stands for the volume of the unitary group U(N) and is unessential in what follows; ~(X) and i.1(A) are Van der Monde determinants, e.g. L1(X) = 11,>1(x~— xi). The transformation, leading from (2.3) to (2.4), is a trick, familiar from the
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study of multi-matrix models in the formalism of orthogonal polynomials [24,7,25], and we do not dwell on this point. For what follows it is very important that the term Tr AX in (2.3) is linear in X (were it instead, say, Tr AX”, the polynomial factor .~iAIX)at the r.h.s. of eq. (2.4) would be substituted by z.12(X)/~i(X°) and our reasoning below would be inapplicable). The r.h.s. of (2.4) is proportional to the determinant of the N x N matrix det(
~i~(A)
11)F,(AJ)
(2.5)
with ~±~(A) mfdxx
exp[—~(x) +Ax]
=
(~~Fi(A).
(2.6)
Note that F1(A) =3’~?Ii~[A].
(2.7)
If we recall that in GKtvI (2.8)
A=~’(M)
and denote the eigenvalues of M by {p.1}, eq. (2.6) acquires the form det ~(p.) =
,
,
,
fl(~ (p.~)-7
(2.9)
(p.1))
i >1
where (2.10) Denominator Proceed now to the denominator of (1.2): I~[M}
fdX e:(M~.
(2.11)
Making use of the U( N) invariance of the Haar measure d X one can easily diagonalize M in (2.11). Of course, this does not imply any integration over angular variables and provides no factors like .i(X). Then to evaluate (2.11) we just have to use the obvious rule of gaussian integration,
fdX ex~(_ ~~1x11x11)
flu_t/2
(2.12)
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(an unessential) constant factor is omitted), and substitute the explicit expression for (/1(M). If the potential is represented as a formal series, ~(X)=
E~/~X”,
(2.13)
0
(and thus is supposed to be analytic in X at X= 0), eq. (1.4) implies that U7(M, X)
=
(n
~
+
1)~+~{ ~
n=0
Tr MUXMhX},
a+h~-n—
I
and
E
~(n+1)7~±i{
U~
a±h=n—I
n=()
/L/L~ =
~
(n+1)~,±~—~- =
‘(p.1)—~’(p.1)
0=0
Coming hack to (1.2), we conclude that
1[M]=exp{Tr[ii~(M)
—M1V’(M)]}
Z
u [det ~
I-I
1(p.1)]
[det cP.(p..)j ~1(M) exp[~V(p. 2 [Liç1]~’ 1) — =
“
— ~
,
(p.1))
fls(p.~) i=t
N
fls(p.~).
=
s(p.)
(~(p.1) ,
j>~
Here s(p.,)
,1N[~(M)1
p.11~’(p.,)],
[~‘(p.)]
1/2
(2.14)
i.e.
exp[7 ~(p.) — p.~’(p.)].
(2.15)
The product of s-factors at the r.h.s. of (2.16) can be absorbed into ~-functions
Z~.f~[M]= [det
cP1(p.1)]/~(M),
(2.16)
where cP,(p.)
=s(p.)cP1(p.).
(2.17)
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Kac—Schwarz operator From eqs. (2.6), (2.10) and (2.15) one can deduce that c1(p.) can be derived from the basic function t1 1(p.) by the relation =
[~/.“(p.)]
y + p.)’’ e~~> dy
t/2f(
A4(p.)~1(p.), (2.18)
=
where U(p., y) is defined by eq. (1.3) and AU }(p.) is the first-order differential operator
a
exp[77(p.) —p.~’(,n)J a exp[—~(p.) +p.~’(p.)] 2 hp. =
1/)
a ~‘~“(p.) ap. 1
=
[7.n(p.)~i/
—+p.—
2[~”(p.)}
(2.19)
~.
In the particular case (1.9) 1 k-I
=
k—I
a —
k
kp. dp. 2kp. coincides (up to the scale transformation of p. and A{~}(p.))with the operator which determines the finite dimensional subspace of the grassmannian in ref. [23] (we shall return to this point in subsect. 2.5). We emphasize that the property ~
(p.) =A~( p.)~ 1(p.),
(~+(A)
=
~F1(A))
(2.20)
is exactly the thing which distinguishes partition functions of GKM from expression (2.2) for a generic i--function, =
[det ~1(p.,.)j/i(M).
(2.21)
with arbitrary sets of functions ~1(p.). In sect. 3 we demonstrate that the quantity (2.21) is in fact a KP i--function in Miwa coordinates. Implications of additional constraint (2.20) will be discussed in subsects. 2.6 and 3.4. N-dependence Before we proceed to a discussion of Miwa coordinates, two more things about formula (2.21) deserve mentioning. First, the entire set {c~(p.)}is naturally N-independent and infinite. It is reasonable to require also that ~‘s are linear independent and ~(p.)/~(p.)
=c1p.~(l + 0(1/p.)),
(2.22)
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with p.-independent c• [this is true for functions (2.7)]. Then the set {41(p.)) is identified as projective coordinates of a point of the 1[M] grassmannian, see subsect. for an infinitely large 2.5. TheM.r.h.s. of eq. (2.21) to naturally matrix In order to return the caserepresents of finite Ni-~’ it is enough to require that all eigenvalues of M, except p.~,. p.~’tend to infinity. In this sense the function i-~[M] in (2.21) is independent of N; the entire dependence on N comes from the argument M: N is the quantity of finite eigenvalues of M. As a simple check of consistency, let us additionally carry p.~to infinity in (2.21), then, according to (2.22) . .
det~~ 1(p.J)
~
.
det~~(p.1) (1 .
+
O(1/p.~))
and ~N(M)
(p.~)N_t~~1(M)(1
+
O(1/p.~)),
so that [c~cb1(p.~)](1 + O(1/p.N)).
~
(2.23)
This is the exact statement about the N-dependence of i-N~ Actually in GKM cN4~~(p.) = 1 at p. —~ In what follows we often omit the subscript N. ~.
Multiplicities The second remark concerns the form of the r.h.s. of (2.21), when some eigenvalues of M coincide [see also eq. (2.30) below]. If eigenvalue p., appears with the multiplicity p,, eq. (2.21) looks like det[~1(p.1)A171~~(p.1) ...
r[(p.1,
pj)]
II
i i (p.,
(2.24)
.
— p.1)
i >1
Notation in this formula is not very transparent: it is implied that the matrix in the numerator has rows of the form ~1(p.1), A1~1~1(p.1)
A(~/~1(p.1), ~1(p.2), A~~1~1(p.2) A[~1 ‘~1(p.2), ~~(p.3), A1>~1~1(p.3),...,
where operator A1.~is defined by eq. (2.19). Expression (2.24) will be used in subsect. 2.4 in the proof of the Hirota identity. Note also that if some p., ~, this may be treated as vanishing of the corresponding multiplicity, p, = 0. Such p., obviously do not contribute to (2.24). Thus, the value of N (the finite size of the —‘
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matrix M in GKM) may be interpreted as the number of p.’s which appear with non-vanishing multiplicities. 2.3. KP r-FUNCTION IN THE MIWA PARAMETRIZATION
The Generic KP i--function is a correlator of a special form: i-G{i,}
=
K0I:e~T,t,G~0)
(2.25)
with J(z) =~(z)t/i(z); G=:exp(~’,,,,,t/i,,,t/i,,):
(2.26)
in the theory of free 2-dimensional fermionic fields i/i(z), ~(z) with the action ft/i at/s. Vacuum states are defined by conditions t/’,,IO)=O,
n<0,
i/i,,I0~=0, n>0
(2.27)
dzU2. where tfi(z)= ~t/J,,z” dzl/2, t~(Z) ~ The crucial restriction on the form of the correlator, implied by (2.26) is that the operator :e~T,~J* G is the gaussian exponential (exponent is quadratic in fields), so that the insertion of this operator may be considered as a modification of the i/s — i/i propagator, and Wick’s theorem is usually applicable. Namely, the correlators Cr;((~i},
(A 1})
~
(2.28)
for any relevant G are expressed through the pair correlators of the same form: CG({p.1},
{A1})
=
det(11)CG{p./, A1}.
(2.29)
Operator G in eq. (2.28) can be safely substituted by the entire: e~’~”:G, but we do not need this for our purposes1’~~”: below. Instead with the help of the Miwa in (2.25) through insertions of fermionic transformation we shall express :e operators. Then (2.25) acquires the form of (2.28) and after that the application (2.29) provides a representation for i--function in determinant form, to be compared with eq. (2.21) for the partition function of GKM. There are slightly different forms of Miwa transformation. Usually one writes
(2.30) p., and treats the r.h.s. as an integral fp(p.)/p.” over a Riemann sphere with coordi-
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nate p. and singular function p(p.) =
Unified theory of 2d gra,ity L,p
15(p.
—
p.,). We use instead the transfor-
mation (2.1) (2.31)
T,,=—TrM”=—~~, n
interpreting p.’s as eigenvalues of the matrix M, all coming with unit multiplicities p, = 1 (while p,> 1 in (2.30) may be interpreted as a result of the coincidence of p, eigenvalues). The form (2.30) of the Miwa transformation is preferable from the point of view of invertibility and also is convenient for the proof of Hirota identities (see subsect. 2.4 below). The simplest way to understand what happens to the operator el~T,J~ after the substitution of (2.31), is to use the free-boson representation of the current J(z)=aq,(z). Then
~T,,J,,=L i
~nn.p.,
=E~(p.1)~
and :exp( ~~(p.1))
:=
fl(p.
fl:
1p.1)
i
1:.
(2.32)
e~
In the fermionic representation it is better to start from 1
1
n
p.~
1 p.~
(2.33)
instead of (2.31). Then
fl(Th-~1) :exp(ET~J,,)’=
‘1
1>1
-
-
fl~(~)ç~(p.~). (2.34)
i>i
In order to come back to (2.31) and (2.32) it is necessary to shift all Th’s to infinity. This may be expressed by saying that the left vacuum in (2.25) is substituted by (N~ ~K0(’(~)...~°(~).
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of 2d grality
The i--function (2.25) can be represented in various forms: =
(0 :exp( ~
0)
7~,J,):G
=
~i(M)
fl: e~’~:G JO)
- ‘(N
ii(Th~1) =
lim ~—‘~
=
lim ~
“~
-
-
11(p.,p.1)11(p.1p.1) 1>1 1>, C(;({p.j},
{p.1})
C~({~1},
{p.J)
(OJ fl~(f~1)q~(p.1)GJ0) i
(2.35
and applying Wick’s theorem (2.29) we obtain (~
.
i-~[M] = lim det(/f)
CG(j~1,p.1) -
det ~1(p.1)
(2.36)
=
C,(p.1,p.1)
~(M)
with functions (0 ~
~(~)t/(p.)G
JO).
(2.37)
Thus, we proved that the KP i--function in Miwa coordinates (2.2) has the determinant form (2.1). Below we shall need a more detailed expression of the right-hand side in eq. (2.35), when all points /1> tend to the same value 1/t with fixed t ~ 0. From (2.35) one can obtain i-,~[tJM]
fl(1_tp.,)N fl(p.1-p.1)
det
.
(i-I).
¶a~(0J~(t)~(p.J)GI0)
I>’
=
—
fl
______a~ (0 J:exp[ ~n’(p.7”
(1— tp.1)N
fl(p.1—p.1)
(I— 1)!
— t”)J,,]
:G
JO)
I
‘
I>’
(2.38) In the limit of ~1(p.)
=
t
—~
0 we obtain eq. (2.37) with
ltm~1(p., t) i-’O
=
lima’ (z—1).t-*o .
(0 :exp[ ~n~(p.” — t’)J,,} l—tp.
:G
JO) (2.39)
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Eqs. (2.38) and (2.39) will be used in sect. 3.2 in one of our derivations of the constraint. One more remark is in order now. From the form of the operator expansion for i).~x)i/i(p.) it is clear that the function as given by (2.39), behaves as + 0(1/p.)) when p. —~ in accordance with (2.37), and all the functions are independent. This last statement should be more or less clear from the fact that the set {41(p.)} contains all the information about fermionic propagator ~‘..
4(p.),
~,
C~(j~,p.),
which is exactly the same as the information in operator G, which should be of the form (2.26), and defines the actual fermionic action. To make this mutual independence of qYs even more transparent we prove in subsect. 2.4 that Hirota identities are satisfied by the quantity (2.36) with any set of functions (~1(p.)}.The possibility to choose {~1(p.))in any way is important for us to argue without going into any more details, that the partition function of GKM, which was expressed in determinant from (2.36) with the specific choice of ‘/, is indeed a KP i--function. =
~,,
2.4. HIROTA EQUATIONS FOR r-FUNCTION IN MIWA COORDINATES
The usual form of bilinear Hirota equations, which are the defining equations of KP i--functions, is ~
where D7 are Hirota symbols, D (D7, ~DT,,...) and 3”, are Schur polynomials. Note that these equations are in fact more than KP equations themselves, which 2 log i-/aT 2) and all describe evolution of the functions u,(T~) (u(1.,) u2(T~) a 1 other u,(T~)can be determined from the relations
~~logi-=(L~)
1and
1
with
see eqs. (3.35) and (3.36) rather than
Lm3+ Eu~±1a i-
itself~Indeed, any transformation
i-(T,,)—*i-(T,,)exp[H(T 2,T3...)+TiH],
(2.40)
with al-I/aT1 0 and H const does not change u~(I,),i.e. H and H cannot be fixed by the entire of KP equations. They are, however, fixed by Hirota equations (up to linear function H + T1H Eb~T,,,b,, const). This remark is important for the subject of matrix models whenever interpolation between critical points is considered. For example, one can use the representation =
=
=
~
=exp{fTl(Ti
=
—x)u(x, T2...) dx+H(T2, T3...)
+
TIH}
(2.41)
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and at any given multicritical point the functions H, H are inessential. However, they are very important for interpolations, and, in particular, for the relevance of Virasoro and ~Il~-constraints. Our first task in this subsection is to rewrite the Hirota equations in Miwa coordinates (2.1). Since this is a widely known transformation (see, e.g. ref. [26]), we just cite here the answer: Hirota equations in Miwa coordinates state that (p.,, —p.6)i-(p,,, P6, Pc +
(p.6
— p.~)i-(p,,+
1,
+
1)i-(p,,
+
1, p6
Ph, Pc)T(Pa, p6 +
+
1, p~)
1, Pc
+
1)
+(p.c—p.,,)r(p,,,p6+l,p,)i-(p,,+l,p6,p~+l)O,
(2.42)
where the i--function is expressed through the Miwa variables (2.30) according to eq. (2.24) and three arbitrary eigenvalues from the set (p.,) with corresponding multiplicities are chosen. The second task is to prove that i-[M] = ~(MY’ det ~1(p.1) with any {4,(p.)} satisfies eq. (2.42). All we need to derive the desired equations is the well-known Jacobi identity for the determinants. For any (N + 2) x (N + 2) determinant J we denote by J(~’,) the determinant obtained from J with s rows i,,.. .,i, and s columns j,,. . omitted. Then the Jacobi identity reads J~’~ J=J~ 1,3
Let us consider i-N({p.}) set of eigenvalues p.1
p.,~, into
the
p.~+02
-J
3
i-~(p.,,..., p.N)
~ where
J~ 1
L
J~. 3
1
and divide the given (a priori different) “clusters” of sizes p1,. .. PL:
...;p.,,±+,,,+,
Then one should introduce two additional eigenvalues p.N+i, p.N+2 and apply the Jacobi identity for i = N + 1, j = N + 2 to the function i-N+2((p.}, p.N÷~,p.N÷2).Then simple calculations gives the following system of equations: E,p1 =IV.
p.N±l,
=
—
p.N+I
p.N+2)TN({p.})
(i-N+I((p.},
—i-~~1({p.},p.N+l)TN+l((p.},
where
~
p.N+2)TN÷J({p.},
p.N±,)
p.N±2 p.N+2)},
(2.43)
denotes some new i--function which is obtained from the given
i-N+,
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~N±2~ Now let all the eigenvalues in each by a change of the last row: 4N± cluster tend to the values p. 1, . . , p.,~and 1 ~ p.N+2 —~p.6 where p.,, and p.~, belong to different clusters. Then eq. (2.43) acquires the form [in the notations as in eq. (2.42)]: .
(p.o =
—
p.6)i-(p~+ 1,
T(Pa, Ph +
1,
Ph +
~
~
1, Pc)i-( Pa, p1,,
Pc)~(Pa+
1, p6,
Pc) — i-(p,, +
1,
Ph’ Pc)~(Pa,Ph +
1,
Pc),
(2.44) where p~ describes some arbitrary third cluster different from the previous ones. One can multiply this equation by the factor i-(p,,, Ph’ Pc + 1)/i-(p,,, p,,, p~)and write down a couple of other equations obtained by cyclic permutations among indices (a, b, c). The sum of these three equations coincides with eq. (2.42). Another interesting expression can be derived from eq. (2.44), as follows. Let us make the replacement Pa ~ — 1, N —s N — 1 and put p6 = 0 (this means that we have the cluster with the single element p.6 in the corresponding i--functions in eq. (2.44)). Then in the limit p.1, —s p.,, we have ‘r(pa, Ph + 1) —~i-(p,, + 1) (notations of other clusters will be omitted since they do not change) and the equation now takes the form i-N+I(P,, + l)TN1( Pa_l)
=i-N(Pa)
lim ~i~*5L,,
a ~n(Pa, p.,,) ap.,,
n(Pa)
lim iLh*tL*
a ~~i-N(Pa, ap.,,
p.,,).
Simple calculation gives further that lim ~
a ap.,,
a TN(Pa) p,, ap.~ 1
~i-N(Pa,
p.h)
=
and now we obtain 2
PaTN+I(Pa)TN_t(Pa)N(PaY~
a ~(pa) ap.~i-(p,,)
.
(2.45)
This equation holds for every function in the form (2.21). A more concrete expression can be obtained for the GKM partition function 3~~/,~7} defined by eq. (2.9) in terms of A variables [see eq. (2.8)]. For this quantity the following relation holds: La N
=
,~ V
~c~-{~)N
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203
Therefore we have Pa4T~i(Pa +
1)5”)~),(p,,— 1) L a2 6=1
a
aAa A,,
a
L
a
3’)~”1(p,,)— ~—3’~V1(p,,) ~ a
h=1
h
We should remark that in the case of Pi = N, p,, = 0, a ~ I (A, A) this equation reduces to the Toda-chain one. Indeed, /7~/1(p,,)now has the very simple determinant form 1
det a’~~F(A).
=
(N-i)!
It is a determinant of a matrix with specific Toda-chain symmetry of constant entries along the anti-diagonal [27]. From the other point of view, the matrix integral (1.1) reduces in the present case of the diagonal matrix A to a standard discrete model partition function which is well-known to correspond to Toda-chain hierarchy [28,17]. If we return to the original description in the terms of p.’s [see eqs. (2.16) and (2.24)], then our final equation is TN+i(Pa +
i)TNI(
Pa —
1) =i-
=
(n (p.
—
p.
)P*P~)-
I ~
a2
L
2
~
~
([~“(p.,,)I
i-N(P,,)
I/2
exp[~(p.,,)
~ —
p.~~~’(p.,,)])”’ (2.46)
We shall return to eq. (2.46) in subsect. 3.3 in the context of the string equation. 2.5. GRASSMANNIAN DESCRIPTION OF KP r-FUNCTION
In this subsection we present the necessary material about infinite-dimensional grassmannians and their connection to integrable hierarchies. The details and proofs can be found in refs. [29—31]. Definitions. Let us consider the infinite dimensional Hilbert space H realized as the space of Laurent series ~(p.) on the unit circle S’(p. e S’) on the complex plane. This space is naturally represented as the direct sum H = ~ H, where H~(H_) is spanned by the vectors (p.”), n ~ 0 ((p. “}, n > 0), n ~ f. These vectors form the orthogonal basis in H:(p.”, p.”) = The infinite-dimensional grassmannian Gr is the set of subspaces W E H (grassmannian points) which satisfy the two conditions:
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(i) the orthogonal projector pr~: W —s H÷ is a Fredholm operator, i.e. the kernel and co-kernel are finite dimensional spaces; (ii) The orthogonal projector pr_: W H is a compact operator. This definition corresponds to Sato’s grassmannian rather that to the Segal— Wilson one, where q5(p.) should be convergent in some neighborhood of ~. The index of pr~is called the relative dimension of W: —*
md pr~=dim(ker pr~)— dim(coker pr~). The grassmannian is decomposed into the direct sum of the connected components: Gr= ~ Gr,,,
(2.47)
where Gr~consists of W’s with the relative dimension n. For our purposes it is sufficient to deal with Gr0. It is convenient to represent linear operators acting in H as block matrices with respect to the decomposition H H + ~
a=[a
~],
aEGL(~),
(2.48)
where the operators a, b, c, d act as follows: a: H~—sH~,b: H~.—sH~,and so on. Clearly, b and c should be compact operators. For applications to integrable hierarchies the following commutative 1 to subgroup non-zero F c GL(oc) is of importance. It consists of the mapping from S complex numbers and acts on H by multiplications of the Laurent series. There are two natural commutative subgroups F~ and F in F: the elements of F~ (F) can be analytically continued inside (outside) SI. Their elements ~ E F~ and E F can be parametrized through the infinite sets {Tk} and (Tk) (k ~ 1) of independent variables (“times”): y~(p.)= exp( ~ Tkp.k), k>I
y(p.)
=
exp( ~ Tkp._k).
(2.49)
k>I
Let {4~}(n~ 0) be a basis in W. It is called admissible if the operator transforming p.” to pr~(/s,,)in H÷has a well-defined determinant. We can write
=
~ (~±)nkp.~ + ~ (~)fl~p._k, k=1 k=I
where 4i~has a determinant, çb. is compact. In particular, for W always possible to choose
(4~2),,,,,
=
s,,,,,.
(2.50) E
Gr
0 it is In this case we call this basis {~,(c)}
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205
canonical for W.Generally, the convenient choice for admissible 4~ (to be used below) is a lower-triangle matrix with unit diagonal elements: (4+),,k = 0 if n
p.k_l(~
[cf. eq. (2.22)]. The i--function of KP hierarchy orthogonal projection y~W—s H~: =
+
i-~
O(p.~)),
k~1
(2.51)
is defined as the determinant of the
exp(pr~:y+({T,,})W —s H+),
(2.52)
det(1 +a~bB),
(2.53)
or, more explicitly, =
1 [see eq. (2.50)1. where y, is represented in the block form (2.48), b = (4~Y Given a Riemann surface of finite genus one can associate with it a point in Gr by constructing the admissible basis according to the well-known rules [30]. In this case the formal Laurent series for the basis vectors are in fact convergent. Our situation is different: our basis vectors are asymptotic (formal) series which do not need to converge. This should be interpreted as adequate generalization to infinite genus Riemann surfaces. In this sense we often identify Gr with the universal module space (of line bundles over Riemann surfaces with punctures). Instead of W one can work with the equivalent space yj{Tk}), the i--function being changed in the trivial way:
i-~w({Tk})=exP(~kTkfk)i-w({Tk}).
(2.54)
Let us note that in Miwa variables (2.30) the “evolution factor” y~(p.)(2.49) looks like a product of pole factors: y~(p.)= fl(1—p./p. 1)”. The various reductions of the KP hierarchy can be described in these terms as follows. Let f(p.) be a Laurent series in p.. Those W’s which satisfy the condition f(p.)WcW
(2.55)
corresponds to the well-known Boussinesq,...) give rise1< to solutions of “f-reduction”K-reductions of KP. In (KdV, particular, the choice f(p.) = p. which are in fact associated with Ak_I Lie algebra [32]. In this case the i--function
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does not depend on 7, with n = 0 mod K (modulo trivial exponential factors linear in times like that in eq. (2.54)). An admissible (but non-canonical) basis can be chosen in such a way that n ~ 1.
p.K4~(p.)
=
(2.56)
Vice versa, if we have a basis of the form (2.56), the corresponding point of Gr leads to a solution of the K-reduced KP hierarchy. In subsect. 2.6 we show that the GKM (1.2) with arbitrary polynomial potential ~(x) corresponds to the ~‘(p.) reduction of the KP hierarchy. Note that K-reduction is equivalent to the existence of some admissible basis with the property (2.56). However, other admissible bases (including canonical) in this case satisfy a weaker condition: N
4~±,,(p.) =p.~+ ~a1/, with some constant a,’s. Fermionic formalism The grassmannian approach is in fact equivalent to the fermionic language used in subsect. 2.3. Let us briefly describe the relation between them. in the fermionic interpretation H becomes the one-particle Fock space spanned by i/i,,. The fermionic operator G of the form (2.26) makes linear transformations in H according to i/i,,
-~Gt/i,,G~ ~
H.
The positive (negative) modes of the current J(p.) [eq. (2.26)] are the generators of F~ (FJ. The matrix ~ in eq. (2.26) can be expressed through the canonical basis ~~p.) of the corresponding W as follows: 1 +
G,,,,,
=
dp.dv
(2i-ri)
p.~
~)
~w(p.’
~“’~“~‘w(
2
.
=
~ p.~~(v). k=I
A more invariant expression is
.~w(p.’
~)
=
1 p.—v i-~
(0kp._k
k
p.,
p),
(2.57)
(2.58)
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We emphasize that the explicit basis t/sk = ~ [see eq. (2.18)] is not canonical, but instead satisfies eq. (2.56). However, eqs. (2.57) and (2.58) are very important for any interpretation in terms of (infinite-genus) Riemann surfaces, in particular, for the identification of GKM with appropriate Liouville-like models of 2d gravity. Explicit formulae for can be easily derived, at least, in the case of 27= 27k. ~
i--functions of reduced hierarchies Let us note that in the fermionic language it is trivial to understand that the K-reduction condition leads to the i--function i-1~<1 almost independent of times 1,K, i.e. the property (1.21). Indeed, eqs. (2.56)—(2.58) imply that the condition of such a reduction can be transformed into the matrix {,~,,,,}: ~
E”~]= 0,
where E is the shift matrix, (E,,,,,) = On the same “infinite-matrix language” the elements of F~ can be parametrized as y~=exp(ETkE’<), ~ exp(~E~),i.e. the currents “k from eq. (2.26) act as E”. Now it is evident that exponent of currents ~k with k = 0 mod K can be pulled to the right vacuum and cancelled, eliminating the dependence of the corresponding times (up to the exponential factor like that in eq. (2.54) — this is due to the freedom to insert the exponential y= exp(ETkE~’)between G and y~). Now let us discuss the freedom in the definition of the y-function from the viewpoint of the Hirota equation. As was pointed out in eq. (2.54) there is a freedom to multiply the T-function by a linear exponential of times. Another possible modification is dealing with other components Gr,, of the grassmannian. It corresponds to the shift of the first basis vector or, equivalently, multiplying it by p.”. All these can be trivially understood in terms of the Hirota equation in the Miwa parametrization (2.42). Indeed, it is invariant with respect to multiplying by the product LI•f(p.,), where f(p.) is an arbitrary Laurent series. If the first term of this series p.°, one can expand f(p.,) —~exp(~~p.~T~) and reproduce the correct factor in eq. (2.54). If the first term behaves as p.”, one should deal with Gr,,. Thus, in Miwa variables we readily describe all full freedom in the definition of the i--function. ~
‘~
and specific points in the grassmannian To conclude this subsection let us note that the problem of correspondence between matrix models and the points of the grassmannian has been already addressed in ref. [23]. That paper examined the implications of the 3’/ constraint in the double-scaling limit of the K-matrix model and concluded that the corresponding point of the grassmannian is defined by the set of K linear independent vectors (A’~ 1),where operator A for 27= X’~~ ‘/(K + 1) essentially coincides with ours [see eqs. (2.19) and (2.20)]. Moreover, from our study of GKM we obtain GKM
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straightforwardly the explicit form of all basis vectors for any potentials: /, = 1, as given in eq. (2.18). Note also that in ref. [23] a shift of time variables analogous to ours in eq. (1.17) was introduced. As to ref. [23], we would explain the appearance of the Airy functions (in the case of the K = 2 model) from the string equation in the following way. If we put all Tk = 0 except for T1 the string equation can be satisfied only if the i--function is zero. This means that the relevant point of the grassmannian is singular. Instead, we can put T3 = c 0 keeping all other Tk with k ~ 5 equal to zero. In this case we have
a log ai,
2
Choosing c = — ~, so that and using the relation
T3
=
T3 +
+~!~T~=0. ~ =
0, taking the derivative with respect to
T,
2 log
a
u(T 1)
=
_______
where u(T~)is the KdV potential, we obtain 2u(T1)
=
T1.
According to the ideology of the inverse scattering method in the case of 2-reduction we should consider the Schrodinger equation for this potential: 2iIi +2u~I”p.21I’, a aT~ i.e.
a2 ~i’ (2.59) Solving this equation, we obtain the Airy function ~!‘(p.JT 1)JT,,,,=
2) ~I’,)(p.)Ai(—T~ +p. ~1’ 0(p.)fexp[_x/3+(p.T1)x]
dx.
The wave function ~I’is nothing other than the Baker—Akhiezer function. ~I’~is
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adjusted from the normalization requirement ~It(p.)e~T~~” general ~I’can be expressed through the i--function as [33]
~I’(p.
J 1,)
=
exp( ~. 1,p.”)
i-(T —p.”/n)
209 —~1
as p.
—~oc~
In
(2.60)
(see subsect. 3.3 below for more details, in eq. (2.59) we have a ~P-function of the form (2.60) with Tk = 0, k ~ 3. The admissible basis for the corresponding point of Gr,, can be constructed by means of the 1It~functionas follows [30]:
a~’iit k-I all
T, =1)
The basis vectors, constructed according to this expression, obviously coincide with those which were obtained in subsect. 2.2 [eq. (2.6)] for the particular case of 7z(X)
=
3.
~X
For generic K the role of eq. (2.59) is played by a Kth order differential equations (L’~I’= p.2w) and the result is the Airy function of level K as defined in eq. (2.6). 2.6. 0KM AND REDUCTIONS OF KP HIERARCHY
We prove here that whenever potential 27fX) in GKM is a homogeneous polynomial of degree K+ 1, i.e. ~V(X)= const..XK ), the partition function Z~[M], which is a KP i--function, in fact can be treated as a i--function i-{~c1of K-reduced KP hierarchy. For this purpose let us return to subsect. 2.2 and note that because of eq. 92.7) F 1(A) satisfies the Ward identity (1.23):
[~‘(a/aA)
—
A] F,(A)
=
0.
(2.61)
If the potential 2~(X) is a polynomial of finite degree K + 1, then (2.6) may be used to express FK+ 1 in the form of a linear combination of F,’s with i ~ K. Namely, if 27(X) = —~2~7X’, VK±,= 1/(K+ 1), eq. (2.6) implies that FK±,= — ~i7/F,+AF1.
(2.62)
Clearly, all the terms on the r.h.s. of eq. (2.62) except for the last one AF, will drop out of the determinant (2.9), i.e. in this case FK+i may be defined as AF~ rather than by eq. (2.62). Obviously in the same way any FK+,fl may be substituted by AF,,,, while any F0K+,,, can be substituted by A”I~0.In other words, .9~,[A] is
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given by eq. (2.9) with the first K functions F
1 ... FK defined by eq. (2.6), while other elements of the basis are given by the recurrent relation
A”F,~.
(2.63)
Note that this is true when 2Y(X) is any potential of degree K + 1, not obligatory homogeneous. However, we should recall that A = 27~’(p.)and p. rather than A is the proper parameter to deal with the grassmannian picture. Therefore eq. (2.63) implies the appearance of KP reduction, associated with the function ~27”(p.).If 27 is further restricted to be 77(X)=X~’/(K+ 1), A = p.K and (2.63) acquires the form F
‘-~A”F ,,,
4
p. ?~KF a,,
which is characteristic of conventional K-reduction. As we already explained, K-reduction implies the relation (1.21) with some
factor of the form (1.22). However, even this factor3 is(K absent in the of GKM = 2) this wascase exhaustively (1.10). the [15], particular caseproperties of ~7(X) of = const . X provedFor in ref. using the symplectic structure on (the model of) the universal module space. Since interpretation of entire GKM in such terms is not yet available, the analog of such a proof for generic K is still lacking. 2.7. ON T,,K INDEPENDENCE OF Z{K}
Reformulation of the problem. First of all, let us choose our variables (p.,) in such a way that the only non-vanishing times 7, = n - 1Ep.~” are those with n = 0 mod K. To do this it is enough to have only K finite parameters p. 1, . , p.,.~.,which are essentially Kth order roots of unity: 1, E = e2’~’~, j = 1... K. (2.65) p.1 = p. For such a choice of variables the formula (1.21) . .
T~(T 1
=
.. .
TK_l, TK, TK+,
.
. .
exP(~aflK~,K)i-~(T,.
T2KI, T2K, T2K+I
. .
TK_l, 0,
TK+,
. ..
...) T2K_l,
0,
T2K±l ...),
(2.66)
turns into i-~(0...O, TK,O...O, T2K,0...)=exP(~a,,K~,K)i-~(0)
=
exp(~(aflK/n)p.fr~K)i-~~(O).(2.67)
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Unified theory of2dgratity
211
This relation is valid for any i--function i-~ of a K-reduced KP hierarchy. Our purpose is to prove that for the particular example of such r1’~1, namely, for partition function Z~<~of GKM, associated with potential of the form 77(X) = XK+I/(K+ 1), a,,K[Z]—a
log Z~/a1,~=O,
(2.68)
holds or, equivalently, the function in the r.h.s. of eq. (2.67) is independent of p.. In other words, we need to prove that ~Z[p.f]~(
~{K}(p.)
2(,~)
1.
=
(2.69)
The l.h.s. of this relation can be evaluated with the help of eq. (2.16), Z{K}
=
We emphasize that in order to prove 1,,,~independence of Z{K} it is enough to examine the determinant of the K X K matrix in (2.16) with entries
=
et’~(p.) =
(s(A) ~s(AY’) ~
~1(p.)
~
Co
=
~
=
p.’
‘e~’~(p.), (2.70)
1
(2.71) 1’~depend
(of quantities ~, s,thus A [see eqs. (2.15), (2.8i)—(2.20)] and e on course, the formallofthe potential ~V and on K). Straightforward calculation (K = 2) We begin our proof that ~1~(p.) = 1 from direct calculation of this quantity. If we substitute the expansion (2.71) with p.’s defined in (2.65) into (2.16) the result reads ~{K}(p.)
1
K (fle~t)
=
{a}
d
(—a,+i—I)j (i-I)j
et(,J)E
p.
~
,
(2.72)
e
The remaining determinant at the r.h.s. is actually vanishing unless ~a 1
=
0 mod K, so that eq. (2.68) is in fact equivalent to the set of identities K
~ (n e~2). {a,} =
nK
det
(—a,+i—I)j
d~ (o)
(I
~
=
s,,,, for all
n ~ 0
(2.73)
212
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S. Kharcheu eta!.
Unified theory of2dgra,ity
for coefficients e~of series expansions of modified K-level Airy functions tj~(K)(p.)=p.(K_I)/2exp[Kp.K+I/(K+
xJexp[_x~’/(K+ For example, for
K
=
1)]
1) +p.Kx~x~~ dx.
(2.74)
2 identities eq. (2.73) looks like ~
(2.75)
()“e(1)e(2)~6
a+/3=2n
Since for K=2
=
exp(—~x3+xp.2)
~exp(—4p.3)fdx 1
2
=
rfdz exp —z
=
~ei~’fdx’x
=
~fdz(z+p.) 1
— —
—
____
3p.3/2
—
J
F(3k+1/2)
1 ~ 9k(2k)!
=
1(1/2)
1 p.~’
exp(—~x3+xp.2)
exp{_z2_ 6k
+
1
1(3k
3p.3/2}
+
1/2)
1
~ 9k(~~
F(1/2)
6k—I
p.~’
we have e~’~ =
1 9”(2a)!
F(3a+1/2) 1(1/2)
2~ = — 6n+l e~. 6c~
a
—
e~
~
(2.76)
A more explicit form of eq. (2.75) is 36a/3—1 a±fl=2n
a,f3~O
(2a) !(2f3)! F(3a
2 -
~)F(3p
-
~)
=
-4[F(fl]
~,
(2.77)
which is indeed a valid F-function identity. This calculation, though absolutely straightforward, is hard to repeat for generic K. Still it is useful for illustrative purposes and we included it in this section.
S. Kharchet at a!.
/
Unified theory of2dgraiity
213
The proof A much easier approach is to prove that p.
a
.~{K}(p.)
=
0.
(2.78)
ap.
Since it is obvious from eq. (2.72) that at least ~1K} 1 and p. ~, this would provide a complete proof on eq. (2.69). in order to prove eq. (2.78) it is enough to act with p.a/ap. upon —~
—~
det(,J)E”e1’1(p.E3)
~{KI(p.)
(2.79)
and then make use of the fact, that p.(a/ap.)e~’kp.e’)can be decomposed into a linear combination of e”kp.c~). e~(p.’). This decomposition, which is, of course, the implication of K-reduction, can be worked out from the relation (2.20), ..
p.’e~’~°(p.) =A{~}p.’e~(p.).
(2.80)
Since I AK
=
Kp.’<’
a K—i —+p.— ap. 2Kp.”
,
(2.81)
(see also ref. [23]), we have a (p.~)e(~)(~EJ)
=
(
K+1—2i 2
— K1p.K~)e(~)(~E1)+ KE1p.Ke+~)(p.c1).
(2.82) This should be supplemented by the condition I) =
e~°
(2.83)
(the equation Ac =p.’
214
S. Kharcher eta!.
/
Unified theory of2dgra,ity
Then (p.~)~(2)(p.)
(p.~)(et1)(p.)e(2)(
=
—p.)
+
—p.)) e~2~(p.)e’°(
[(~— 2p.3)e°~(p.)+ 2p.3e(2)(p.)je(2)( —p.) +e(p.)[_2p.3e(I)(_p.)+(_~+2p.3)e(2)(_p.)]
(—~— 2p.3)e(2)(p.)]e(~(_p.)
+
[2p.3eW(p.)
+
e~21(p.)[(~+ 2p.3)e~>(—p.) — 2p.3e~2~( —p.)],
+
and the r.h.s. is identically zero. If we return to the case of arbitrary K, note that after the substitution of eq. (2.79) into the l.h.s. of eq. (2.78) we obtain a sum of K determinants like eq. (2.79) with e”e~’~(p.e~) in the row i’s substituted by eq. (2.82):
(p.~)E~e(~)(p.E1) =
(K
+
l— 2i’
+ Kp.K+
— Ke1p.
I)EiJe(l)(p.Ei)
leut~ ‘e’1’~~(p.e’).
Next, note that the last item in the r.h.s. coincides with the entry E~’~ ‘~‘e~’~ ‘~(p.~) of the next line of the same determinant, and this can be eliminated. Moreover, the first item on the r.h.s. just implies that the i’th line is multiplied by a factor of ~(K+ 1— 2i’) and the effect of this cancels in the sum over i’: E~,~(K+ 1— 2i’) = 0. Therefore, we conclude that
a p. ___~{K}(p.) dp.
a ap. a ~ ~p.
K
—Kp.~ ~
(2.84)
where 3’~,is just the same determinant 3’, only in the i’th line d~ E’’e~’~(p.E’) is substituted by ‘d~ 1= E” ~‘~‘e”’(p.e’).The sum of such 3’, (for any original 3’) is identically zero as a consequence of the identity ~ ~e’= 0.
/
S. Kharche, eta!.
Unified theory of 2dgraiity
215
Examples: K=2:
fd,, 3’=det~ d2,
3’, =det(
—
d12\ d22
d11
/ d11 3’2=det~ — d-,,
)
d11d22—d12d21
(—) 2 d~,I d,2
=
J
2
(—) d221
3’ +9/2=0.
2~’~3:
K
=
3:
E
=
e
(d,,
d
12 d,7 (13~ d32 3’=det~d2I =
9/,
d11d22d3,
=
det
(
+
Ied~1 d
2, d3,
d,3\ d23 d33 )
d~2d2~d3, + d,3d32d21 2d~, ~ E d27 d,3 d32 d33
3d,~d 2d23d,1
ed, ,d22d33 +
+
e
2d
32d2,—
e
3d, 12d21d33 — E
3d~~d22
—
d~, d13 \ 2d 3d e 22 e 21 ~ d,2 d33 / 3’2=det(ed2’ 2d 3d, = 11d22d33 + E 2d23d3~+ Ed13d32d2, (
d,1
— E3d, 3d32d,
I’
I’
)
2d, =
— d13d,,d22 — d23d32d,
— d,2d2d13
d d
2d
— Ed,2d21d3~—
~
13
31
22
216
S. Kharchei eta!.
9/3
=
d1,
d12
d~3
det d21
d22
d23
~3l
=
3d E 1
—
ed32
1d22d33
+
/
Unified theory of2dgraiity
3d
E
33
Ed,2d23d3,
+
3d E 13d32d2,
3d — E
12d21d33
— Ed13d31d22
2d e 23d32d11, ~l
+9/2+9/3—0.
These two examples are enough to illustrate the general phenomenon: determinant 9/ is an algebraic sum of products d~1.. ~ Let us pick up one of these items. It appears in 9/, with the coefficient e”, where j’ is defined as the second subscript of the element d,1 in this product. The map i’ —*j’ depends on the particular item we choose, but when we sum over all i’ this is the same as summing over all j’, since every subscript appears in the product once and only once. Therefore in the sum E,9/,. every item appears with the universal coefficient = 0, i.e. ~,9/, = 0. This completes our proof of eq. (2.78) and thus of eq. (2.68). .
Remark. To avoid possible confusion, let us note that it is important Airy functions 1’~(p.)are that expanded in series are analytic functions of p., but not of A = p.K (e.g. e with integer powers of p. _(1<+ I) = A1 I/K) This is what makes the proof slightly non-trivial. Otherwise, if one suggested that the entries of the matrix in eq. (2.79) are analytic functions of A, the A-derivative of this determinant would be simply a combination of determinants with coincident rows and thus vanishing. However, non-analyticity in A requires one to be more accurate: AUI( takes different values p.E1 when appearing in different places, and a detailed calculation, as given above,
is necessary.
3. Universal 3’/..
1
constraint and the string equation
3.1. MOTIVATIONS
Since it was proved in sect. 2 that Z{KI[M] is a i--function i-1~ of the K-reduced KP hierarchy, in order to define it completely (i.e. to fix the point of the grassmannian) it is enough to deduce somehow a single additional constraint (3.1)
/
S. Kharche, at a!.
Unified theory of 2d gra city
217
with i
a ~
K
(n+K)1,,±K~-
~>,
0 mod
K
1
a —
+ ~T
1
~ aT,bT,,. 2K~6~
(3.2)
According to arguments of ref. [1] a constraint such as (3.1), when imposed on = i-~, implies the entire set of WK algebra constraints in the form of eq. (1.29). Another reason for the study of 9/~ constraint is that it is much simpler than any other one: 9// is the only operator of interest which does not contain double- and higher-order T-derivatives (to be exact, there is one more such generator: ~ Among other things, this means that it is easier to formulate the unit ersal (i.e. for any potential 77(X)) S~i,constraint, than any other corollary of (universal!) Ward identity (1.23). This section contains a formulation and derivation of the S’i//~ constraint for GKM with arbitrary potential 27~(X),making use of explicit formulas, derived in sect. 2. Namely, we are going to exploit the fact that the functions ct,(p.) in eq. (2.10) are not independent, but are rather related by the action of the B/GA operator: see eq. (2.20). The proof itself is described in subsect. 3.2, and subsect. 3.3 contains additional remarks about the string equation. We emphasize that not only is the 9/— constraint valid for any 7~(X)(with the only restriction that 7’~(x) grows faster than x as x —, ~), but it is just the same for17 any size N of the = i-I.’ I), this conmatrices.is Just like the property of integrability that Z construction behave straint not sensitive to N, and this makes (i.e. the entire smoothly in the continuum limit, as N —*
3.2.
DIRECT EVALUATION
OF 5~’
It is well known [21,28], that the of operator
~1
a
Tr— aAtr
=
Tr
constraint is closely related to the action i
a
(3.3)
—.
77”(M) aMir
Therefore it is natural to examine how this operator acts upon det i13(p..) Z1~[M]= s(p.) ~,(p.)
=
(3.4)
~1(1vI)[Js(p.,), (27~”(p.))/2 exp[27(p.)
=~(A) = (a/aA)’1F(A),
—
p.7~’(p.)],
A
=
(3.5)
218
/
S. Kharchei at a!.
First of all, if
Unified theory of 2d gracity
1~~ is considered as a function of
Z
1
=
In the particular case of
7~<(X)=
7~(X) =
KZIK)
1
az11<1
M?~+K
a1,
Tr
~,
,,~,
,,2Omod
—
~
(3.6)
K
1 =
.
1), eq. (3.6) turns into
X’~~ ‘/(K +
i
(lAtr
alogz1~1
~
—
a
~I1~
~
1
a
i
T-variables,
—
~
a1,bT,, +
a
Z1”~1.
(3.7)
a+6==K a,h>~I
We made use of the definition (3.2) of the 9/{1~ operator and the fact that Z{KI is independent of all TJ,K. Time variables are defined by the Miwa transformation T = ,~—I Tr M”. On the other hand, if we apply (3.3) to explicit formula (3.4), we obtain
a
I
z1~’1
=
aA,~
—Tr M
E +2
7~”(p.,)— 77”(p. 7~”(p..)7~”(p.) p.—p.. 1)
a
+Tr—~-—logdetF(A1), or, in the case of
7~/X) =XK~/(K+
a
1
—Tr M
=
(3.8)
I),
a
+
~
aT,,bTh + Tr—~—log det F,(A1). (3.9)
a+b=K a,6 ~ I
Ir
The combination of (3.7) and (3.9) implies that I
_~9/(~IZ{K}= —
a ~—log
~(!~ +
a Tr M — Tr~—logdet F,(A~). (3.10)
The r.h.s. here is practically independent of K, and this may be used, together with eqs. (3.6) and (3.8) in order to suggest the formula for the universal operator ~ the idea is to preserve the relation (3.10): 1 =
—
a
~—log
Z(~ +
a
Tr M — Tr—~-—logdet F(A1).
(3.11)
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Unifled theory of
2d grai dy
219
Here
a
I 9/{~’}= ~Tr —l
7~(M)M/~+
,,~,,
1
________—~
— a7~,
a
27~”(p.)— 27~”(p..) ‘
‘
7 ‘(p.,)7 ‘(p.1)
—
(3.12)
—,
aT1
p.,— p.1
and this turns into (3.2) when 77(X)=X’<~/(K+1) (note that the items with i =j are included in the sum on the r.h.s. in eq. (3.12)). So, in order to prove the 9/f~’ constraint, one should prove that the r.h.s. of eq. (3.9) vanishes,
a
—i-log 1 = i-/~ ~,
Z~1
=
Tr M
a
—
Tr~-j-—logdet F,(A,.),
(3.13)
defined by eq. (3.4). The l.h.s. may he represented as the residue
forthe Z~ratio of res~
i-~1(I,)
a
—~--Iogi-~,(1(7~,).
=
(3.14)
However, if expressed through Miwa coordinates, the i--function in the numerator is given by the same formula (3.4) with one extra parameter p., i.e. is in fact equal to i-t~J~. This idea is almost enough to deduce eq. (3.13). Since eq. (3.13) is valid for any value of N, it is reasonable to begin with an illustrative example of N = 1. Then (A = TV1(
~,) =
i-p’ )[p.,]
=
exp[ 7( p.,)
— p.7
~‘( p.~)]
[7
~(
p.
2F( A,),
1)] i-~71(~,+p.”/n)
~
=rY1[p.,, p.] =
exp[7’(p.,) —p.,7~’(p. 1)]exp[7’(p.) —p.7~’(p.)] x
2 [77(p.1)7~(p.)]I/
[F(A)aF(A)/aA—F(A)aF(A,)/aA,]
p. —p.,
exp[7~(p.) — p.7~’(p.)][7~(p.)]
‘~2F(A) i-~
=
p. —p.
}[p.,]
1
[—a log F(A)/aA, -i-a log F(A)/aA].
(3.15)
220
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Unified theory of 2d gratity
The function F(A)
=
fdx exp[—~(x) +Ax] exp[27~(p.) — p.7~’(p.)}[7~”(p.)]
_1/2{1 +
0(7~”/7~”V”)}.(3.16)
p. and for our purposes it If ?7(p.) grows with p.fl as p. ~, then 77”/(7Y”)2 is enough to have n> i, so that in the braces on the r.h.s. we have (1 + o(1/p.)((p. o(p.) 0 ad p. oc). Then numerator on the r.h.s. of eq. (3.15) is 1 + o(l/p.), while the second item in square brackets behaves as a log F(A)/aA p.(1 + o(l/p.)). Combining all this with eq. (3.14), we obtain “,
—*
—*
a ~log
—*
1+o(l/p.) TV1=res~{ p.p. 1
=p.1
[—alog F(Al)/aAi+~(i+o(i/~))I}
—a log F(A,)/aA1.
(3.17)
Thus (3.13) is proved for the particular case of N — 1. The proof is literally the same for any N, only instead of a relatively simple expression in square brackets on the r.h.s. of (3.14) one has the ratio
F(A,) F(A2)
aF(A1)/aA1 aF(A2)/aA2
det
... ...
:
‘
aNF(A1)/aA~ a”F(A1)/aA~’ 6’F(A aN_lF(A2)/aA~~ a 2)/aA~’ :
.
F(AN)
aF(AN)/aAN
...
F(A)
aF(A)/aA
...
(3.18)
:
a~’F(AN)/aA~~aNF(AN)/aA~ aN~F(A)/aAN~ aNF(A)/aAN
over F(A1) det
F(A2)
aF(A1)/aA aF(A2)/aA2
a’F(A,)/aA~~
...
aNF(A2)/aA~~
.
,
F(AN)
...
aF(AN)/aAN
.
..:
,
(3.19)
a~’F(AN)/aA~7’
which is in fact equal to +
o(l/p.)
-
1 [Tr~lo~
det ~(A1)~’ [1+ 0(1/~)I}. (3.20)
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Unified theory of 2d gravity
221
We used here the estimates
a2F/aA2 F
=p.(i+o(i/p.)),
F
{ }
3F/~A 2 F
a
aNF/aA~v
F
=p.N(1+o(1/p.))
which allow us to pick up only the contributions with aNF(A)/aAN and a”1F(A)/aA~’ to eq. (3.18) all others are lower order in p. as p. cc, The Nth derivative in (3.18) is multiplied by the determinant of the N >< N matrix, which is exactly (3.19), while the (N — 1)th derivative is multiplied by —
F(A,)
aF(A)/aA~
—~
...
aN_2F(A,)/aA~~2 avF(A~)/aA~/~
aF(A2)/aA2
...
a”~
aF(AN)/aAN
..:
2F(A
F(A 2) det
2 a”F(A 2)/aA~
2)/aA~’
2F(AN)/aA~2 aNF(AN)/aA~ F(AN)
a~ (3.21)
This is, however, nothing but Tr(a/aAtr) or the logarithm of (3.19), and for this to be true it is absolutely essential that 1~ (a/aA)’~F
1. With the estimate (3.20) we obtain from eq. (3.14):
a ~log
l+o(l/p.) i-~=res~ fl(p._p.)p.{[1+o(l/p.)]
__[Tr~lo~det
~(A1)] ‘[1
a =
~p.1—Tr 7=1
—log det I~.(A,).
(3.22)
tr
This completes the proof of eq. (3.13) and thus of the universal =
0.
9/~[1
constraint: (3.23)
222
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Unified theory of 2d gravity
We proved the crucial equation (3.13) by direct calculation. Now we would like to describe an alternative proof which originates from representing the i--function through the current correlators described in subsect. 2.3. This approach may be useful for evaluation of the higher T-derivatives (see subsect. 3.4 below). Namely, from eqs. (2.36) and (2.38) it is obvious that i-~[t MJ i-(t, {7j) depends on t and i.e. in eq. (2.38) as follows: i-(t, (1,}) = i-(T1 — Nt, T2 — ~Nt2,...),
a
i-~j[tJM]=exp -N~—~n
i-(T)= Lt”3’~,(a)i-(T),
(3.24)
a
,,
where 9/~,(a)are the Schur polynomials and symbol a represents the vector with components (—Na/aT~, ~Na/aT,,...). Therefore from eqs. (2.38), (2.39) and (3.24) one can deduce —
-N~i-(T) =a,r~[t J M] J,=o
=
—N~p., i-(T) .
iirna, det ~
+
t),
where ~,(p. 1, t) are defined by eq. (2.29). From this expression it follows that ~i-(T)
(3.25)
Ep..i-(T)~(T),
where ~(T) denotes some new i--function which is obtained from the given i-(T) by the shift of the last row in the determinant: ~~(p.j) ~N+ 1(p.1). Eq. (3.25) is nothing but eq. (3.13). —~
3.3. UNIVERSAL STRING EQUATION
The string equation, as implied by (3.23), is by definition 1i-{71 I
a ~‘ aT,
=0.
(3.26)
i-I7)
If we substitute the explicit expression (3.11) of the 9/, equation acquires the form ~Tr n:a()
I
d2logr =u.
7Y”(M)M”~
aT 1aT,,
operator the string
(3.27)
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Unified theory of 2d gralitv
223
Here we introduce a conventional notation u a2 log i-/aT,2 and also defined new I’,, and T~ derivatives, so that a log i-/aT,, 0, a log i-/aTI T 1. The derivation of (3.27) involves taking T1 derivatives of the objects 9/{~1
Tr
(3.28)
7/P~(M)M~+I .
In order to get some impression about a~7~,/aT,, let us imagine that 7(M) is some polynomial of degree Q + 1, so that [7/”(M)]~ -.~M~~(1 + t1Q’ + + ...). Then 5’, (Q + fl)TQ÷,,+ t~(Q+ n + l)TQ~,,±I+ t’2(Q + n + 2)T~+,,±2 + .... Therefore whenever Q + n > 1 (i.e. Q > 0, as it is already necessary for the derivation of eq. (3.23))
a/?,,/aT1=o, forn~1. In order to derive eq. (3.27) one also needs the eq. (3.11), 1
1
2
,~
derivative of the second item in
T,
27”(p.,)
~“(p.,)~”(p.1)
(3.29)
—
27~”(
p.1)
30)
p.,—p.1
Under the same assumptions about 7(p.) the second ratio in this sum is a polynomial in p.’s of degree Q — 2. The only contribution to eq. (3.30), which contains pure factors p./I or p.7~,is 7”(p.,) ‘//“(p. 1 1 1) p., + p.1 = T,~i,, —
others are expanded as bilinear series in 1,,1, with m, n > 1. Therefore, the T, derivative of (3.30) (if Q> 2) is just 9/, and is described by the n = 0 term in eq. (3.27). So we derived the universal string equation (3.27) in the form 2 log i-t7~1
a aT,aT,,
If potential 7~(X)=const .XK±I, the variables .9’, return to the familiar relation [34]
a2 log
1 —
(n
~ ,,*() mod
K
+
(3.31)
=u. =
[(n +K)/K]1,+K, and we
i-{’~}
K)l,±K
=
T~1,
u.
(3.32)
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Unified theory of 2d gravity
Note that even if we restore the factor (1.22), describing the maximal possible ‘,K dependence of i-t”1 it drops out of the string equation (3.32) after taking the a/aT, derivative. Our next task is reformulation of (3.31) in terms of pseudo-differential operators and in the form of a bilinear relation. If L
=a+
(3.33)
~
then [33]
a2 log aT 1aT,,
(3.34)
For n = — 1, 0, 1 we have (L ‘L, = 1, (L°) , = 0, L —1 with (3.27). K-reduction is associated with the condition
=
K
u in accordance
do not appear in eq. (3.32)).
~M”~(L”),=u.
Tr~()
=
(3.35)
(L”)=O
(this guarantees that items with n = 0 mod Rewritten in these terms, eq. (3.31) turns into
u2
(3.36)
Recall now1.that the framework of can the bedressing [33] L =[a, KaK The ineigenfunctions of L defined formalism as (Kez~),where x] = I,I denote the operator K acts on e~x(i.e. (Ke~) is a function of z, not an operator]. The Baker—Akhiezer function in these terms is given by [331 l/J(z I 7,) = exp(ET,,zn)(Kez~), while its conjugate 1/f *(z 1;,) = exp(—E1.,z”)[(K’)~ e~’].These definitions are useful for us, since [33]
L” and=Ka”K the brackets
(L”)
=
res~zt~1/J*(z)1/,(z).
(3.37)
We assume that the contour integral over z, implicit in the definition of res,, is around zero. if eq. (3.37) is substituted into eq. (3.36), we obtain M i/f*(z)1/J(z) u= —resTr-~77 (M) z(zI—M)
.
Since ~(zI7,)=exp(E7,z)
i-(1, —z”/n) i-(T)
(3.38)
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225
and 1/J*(zIT)exp(
E7~,z”)
-r(T r(T +z”/n) 0)
the product 1/f *(z)clf(z) 1 + O(1/z) (as z cc) and the contour integral over z, when pulled to infinity, picks up the contributions from eigenvalues of M. Finally we obtain our universal string equation in the form of a bilinear relation for the Baker—Akhiezer functions: —~
+
u
=
0.
(3.39)
Comparison of eqs. (2.46) and (3.39) enables us to derive a useful representation for the u-function: 2
u= ~,
~
a
log—. .iV
(3.40)
Note that eqs. (2.45) and (2.46) which follow from the general formula (2.44), provide the connection between residues of the Baker—Akhiezer function and time derivatives of the i--function without reference to the pseudo-differential operator language. Indeed, it can be easily understood if one substitutes eq. (3.25) into eq. (2.45). 3.4.
DISCUSSION
Earlier in this section we derived the universal 9/~1 constraint (1.25) on the partition function Z{~v}. Its simplest implication, the universal string equation, was considered in subsect. 3.3. The string equation is nothing but the T, derivative of this 9/., constraint. Remarkably enough, it appears to be much simpler than the constraint itself [cf. eqs. (3.27) and (3.12)]. However, the string equation is not the only implication of (1.23). Another one should be the entire tower (1.29) of Virasoro and 7constraints imposed on Zt’~. In this subsection we discuss three different ways to derive these remaining relations for any 77(X) in the GKM. Being technically different, they emphasize different (though related) properties of the partition function, and therefore each of the three deserves careful investigation, which is, however, beyond the scope of this paper. KP hierarchy approach This one is the most popular in the literature (see refs. [1,20]). The idea is that the Jf, constraint, when imposed on the KP i--function, automatically implies all
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other constraints. In other words, 9/, constraint is enough to fix the point in the grassmannian with which the KP i--function is associated and, thus, it fixes this i--function completely. It would be interesting to find the explicit dependence of the point in the grassmannian on the potential 77(X) in GKM. Even more interesting would be any alternative description of the entire subset U,. in the grassmannian, associated with GKM with arbitrary 77(X), and any reasonable parametrization U,., which would be an alternative parametrization of the space of potentials. Since 77(X) can be changed smoothly, U,. should be a kind of a manifold, lying at infinity of the universal module space (if the latter is embedded into Sato’s grassmannian), and should surely possess some amusing properties. Unfortunately, the already existing papers on the subject (of course, they concern conventional (multi) matrix models rather than GKM) either emphasize implications of K-reduction, like ref. [23], or rely upon the formalism of pseudo-differential operators, like refs. [1,201, and thus need to be translated into grassmannian language. A separate question in the framework of this approach is why should the constraint always be associated with 7iç algebra (as they actually are). Technically this is more or less obvious whenever pseudo-differential operators are used. In KP grassmannian language, the crucial ingredient should be the ~,. covariance of KP hierarchy, discovered in refs. [35—37]. The constraints of interest involve operators from some Borel subalgebra ~ of this ~, and the choice of subalgebra depends on the choice of potential. Reductions of GKM to (double-scaling limit of) multimatrix models, i.e. what happens if 77(X) ~ in grassmannian language should be attributed to intersections of U,. and submanifolds Grtt’1 in Sato’s grassmannian, associated with K-reductions. Somehow at these points a closed Zamolodchikov 7f~subalgebra emerges from entire It is certainly interesting to see how this happens and how the Lie algebra structure is broken. It is also unclear whether such a phenomenon takes place only at U,. fl Gr11<~ or everywhere in Gr~. One more important question is what is the relation between the just discussed ~ algebra of refs. [35—37]and other ~, which is presumably relevant in c = 1 models, and is naively generated by operators, which involve finite differences instead of derivatives in the free-field representation. We use this chance to note that the problem of how c = 1 models are included into the framework of GK.M is still open, and the invariant description of the U,. subset in the grassmannian would also be helpful for its resolution. ~,.
Straightforward derit’ation of 9/ and ~ constraints This approach is just the straightforward generalization of what was done in subsect. 3.2 in the derivation of the 9/~) constraint from the knowledge of explicit expression (2.2) for Z17~1.We sketch here several steps of the derivation for the
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Virasoro case. The idea is to apply the operator Tr ~t”~’(a/ai1ir) the one hand this is equal to a
171, ~ Tr 77”(M)M”~’ —Z a1,
227
to eq. (2.2). On
(3.41)
n~l
on the other hand, it can be explicitly evaluated. Expression (3.41) is very close to main difference is that the sum in eq. (3.41) goes from n = 1 and for small enough n Tr((77’(M))”~’/77”(M)M”~’)contains powers of positite powers of M, which cannot be expressed through T-variahles. For 7—=X”~ this happens for n ~ Km. In order to get rid of these positive powers of M one should use the analogues of eq. (3.13), saying that
9/~/,~}Z17).The
~log a
Tr M’
Z171
+
(3.42)
This is the origin of conventional a2/d7~, dT,, terms in operators 9/~. As to generalization of the shift (1.17), it results in an additional contribution of the form .,
-E~~7~
(3.43)
in the definition of operators. All extra terms cancel, giving rise to the proper universal constraint of the form .9~ 1Z1~= 0, m ~ — 1. ~-constraints can be (at least in principle) derived in the same manner, though actual calculations are increasingly sophisticated. ~
Implications of Ward identities This third approach exploits the fundamental Ward identity (1.23). Technically its main difference from the previous approach is that the operators Tr 77’(a/aAtr), non-linear in a/aAtr, are involved. Conceptually this approach is different, since it does not exploit explicitly any integrable features of the partition function. We shall discuss this method in more detail in sect. 4.
4. From Ward identities to
W
constraints
4.1. GENERAL DISCUSSION
This section is devoted to the derivation (unfortunately, incomplete) of the entire set of Virasoro and W constraints in GKM. The role of such constraints in the study of any matrix model is two-fold. First of all, they can be considered as a
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complete set of differential equations, which specify the partition function of the model as a function of time variables. Then this set of equations implies, among other things, that the solution is a KP i--function (and sometimes a reduced i-(~<1 function). The first application, discussed in detail in sect. 3, is reasonable if one already knows about the integrable structure. Then the partition function is identified with some (reduced) KP i--function (i.e. evaluated at some point of the grassmannian), and the role of the Virasoro constraint is to specify this i--function (i.e. fix the point in the grassmannian). In this second type of circumstances it is enough to have a single constraint, namely 9/ 1Z =corollaries 0; all other constraints, if t”1, are deducible of this one [1,20]. imposed on the KP i--function or iWith identification of Zt7~1 with a i--function in sect. 2 and the proof of the universal 9/~/,} constraint in sect. 3 we exhausted this line of reasoning. The subject of this section is to concentrate instead on another approach and ignore almost all that we already studied about integrable structure of GKM. Then the constraints can be considered as a complete set of differential equations, which specify the partition function of the model as a function of time variables. Among other things, the set of equations implies that the solution is the KP i--function (and sometimes a reduced i-~ function). To be more concrete, we shall investigate direct corollaries of Ward identity (1.23), {Tr E(n)[77’(a/antr)
‘~]}~N[’t1=0.
(4.1)
We shall, however, discuss only specific potentials, 77(X) = const x~~± I, and use explicitly the fact that the partition function is independent of all T,,K. The main problem with the implications of Ward identity (4.1) is that they acquire the form of conventional Virasoro or W-constraints only in the limit of N = cc~The reliable results from our point of view, should, however, be N-independent. But as soon as N is finite, the complete set of independent Ward identities (4.1) is also finite. Remarkably enough they can still be expressed through generators of W-algebras, but the constraints arise as a finite number of vanishing conditions for infinite linear combinations of ill~operators, acting on partition function. As N cc, the number of such conditions tends to infinity, implying that every item in linear combinations vanishes by itself. These items have exactly the form of conventional Virasoro constraints in the particular case of K = 2, while for K = 3 they rather look like eqs. (1.26), and generalization of (1.26) for K> 3 is more less obvious. In any case the honest statement is that any solution to the 7~17constraintsin the conventional form (1.29) does satisfy the equations, which follow from (4.1), for any N (to make N finite one should take all but the first N eigenvalues of matrix M to infinity, see subsect. 2.2). In this sense transition from finite to infinite N is smooth. However, the inverse can be true, i.e. (4.1) can imply (1.29) only as N = cc, Moreover, since (1.26), which is actually implied by (4.1) as N = cc, is not quite identical to (1.29), one needs also to rely upon the (very ,
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229
plausible) assumption that solutions to (1.26) and (1.29) are unique (up to essential constant factors) and thus coincide. Such an approach has been already applied in ref. [2] to the study of the Kontsevich model itself, i.e. for the case of K=2: 772(X)=X3/3. Our purpose now is to extend this consideration to other potentials 77~(X)=X”~’/(K+ 1). However, while the structure of the answers is very clear obvious from (1.26), — the actual calculations are very tedious. Therefore we restrict our presentation below only to the first non-trivial case of K = 3, which will be considered in subsect. 4.3; in subsect 4.2 we reproduce from ref. [21 the derivation of Virasoro constraint in the case of K = 2. —
4.2. VIRASORO CONSTRAINTS IN KONTSEVICH MODEL
(K
2)
The problem. Our purpose in this section is to prove the identity itr(EP~
—
e~A)9/= ~
~
9/,,Z
tr(e
2)
(4.2)
0A° for
*
9/~21{A) JDX exp( —tr =
X3 +
tr
AX)
c[~] exp(~tr A3/2)Z~(~~),m odd
(4.3)
with
c{~]=det(~®I+I®~)~2
(4.4)
and 1
~
kT~
a
~i
a2 ___
2fl ~
a+h=2n aTaaTh a,b~0odd
aTk±
2 ~ kodd
T,~ +3fl~~)+~fl()
4
1 16
a .
aT2~~3
(4.5)
Below in this section we omit label (2) and do not indicate explicitly the fact that all sums run over only odd times. *
To avoid any comparison note that in the original version of ref. [2] there was a mistake in the signs in the exponential in (4.3) and in the shift of times (1.17) (normalization of times are also different).
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While (4.2)is valid for any size of the matrix cc) we can insist that all the quantities
A, in the limit of infinitely large A
tr(e1,A°2)
(4.6)
(N —s
(e.g. tr A’~’2for and (4.1) imply that
=
A°)become algebraically independent, so that eqs. (4.2) 9/~,Z{T)
=
0,
n> —1.
(4.7)
Method. Note that 9/(A} in eq. (4.3), which we have to differentiate in order to prove (4.2), depends only upon eigenvalues (Ak) of the matrix A. Therefore, it is natural to consider eq. (4.2) at the diagonal point A, 1 = 0, i ~ uj. The only “non-diagonal” piece of (4.2) which survives at this point is proportional to
2Ak ‘I
—
6ki
_____ a aA..aA.. 3’
=
5,, =0.in
.ii
A ‘
6kj
—A 1
for
i±f.
(4.8)
Eq. (4.8) is nothing but a familiar formula for the second-order correction to hamiltonian eigenvalues in ordinary quantum-mechanical perturbation theory. It can be easily derived from the variation of determinant formula: 6 log(det .4) For diagonal
tr~6A — ~tr(~6A~6A)
=
k
....
(4.9)
A,
1 = A,6,1, but, generically, non-diagonal 6A,1, this equation gives
6A~iE5AlJ6A1f.iE ~
+
Ak
~
ii
6A~I6AIIH
A.A 1
~
A.
A1
A,—A,.
which proves (4.8). Proof Now we shall turn directly to the proof of eq. (4.2). Since is assumed to be a function of A, it can be, in fact, treated as a function of eigenvalues A. After that, (4.2) can be rewritten in the following way: e — 2/3 tr C(~)Z{T}
Z
~
[tr
~
2
a
e2~3~
_A}]C(~)
a2z
aT
-e(A)--~’--~ a,b~-.0 TUaT,, aA, “
‘
~~2Z{T)
aT, aAi
(4.10) (4.11)
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I
~
z ,,>,,
Unified theoiy of 2d gravity
a21,
az[
~J~E,,(A)
aT
a
231
aT, alogC
aA~
‘
1aA~,
[,.j
3”2 1I
a
(4.12)
j
aA,j3— —tr aA, A
a7~f2) +
(
I~e
L 1,(A,)
2
a (2~ 3”2 I tr A 13/ /
— ~
—
~
—1
~A 1Ef)(Af)
f2)
2
+
~~(A,)( a ii aA,~aA
tr
A3”2
1,1
I /
(4.14)
3~2a log C aA~
(4.15)
‘2 a traA, A 131 —
I
\
/
a2c
1
+
~ ~,,(A,)
1
aA ,~aA
(4.16)
11j
3”2
with tr
=
~k’~k
A
(4.13)
and
c=
fl(~,~~
2~
(4.17)
1)/
i .1
First, since aT,
-
-
A~°3~2,
(4.18)
It is easy to notice that the term with the second derivatives (4.11) can be immediately written in the desired form:
(4.11)
1 =
—
~ nb—i
I =
-~ ~
~e,,(A,)A~”2
~
a2z
a+b—2n
I
a2z
tr{E 1,(A,)A ~~n_2}
(4.19)
~ a+h=211
~
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For the first-derivative terms with the help of (4.17), alogC
1
aA,
—
—
=
3 2
a
—tr A3”2 aA~
_______ —
1 ~
(4.20)
~,
+
and
~
a2A,,~
~
2T,,
t9T,
a
1
k
aA~
aAk
~
(4.21)
(4.22)
Then for (4.12) we have
az
I
I
~L{
2Z
A’~3~2A’~3~2
(~) (~)b
+ a±b=2(n±1) ~
+V~I
1 _______
—
~i+~1i
azl
1
2E~(A,)A1~l I
1
n—l
~n~0~h11,Ja=~0iJ
=
~
1
i
~
~
[E~A)Ai~~2
I
~
—
j
~ 0~1tr(pA~2){~
I aT
1 I
az
~
~
az _____
2fl÷k]
az =
I
k,31
~‘
—
j.
aT2~~3
(4.23)
k ~ 3,, +
The remaining part contains terms proportional to Z itself which need a bit more care. First, it is easy to notice that two items in (4.13) just cancel each other, so (4.13) gives no contribution to the final result. For (4.14) we have 2 tr A3~2 a tr A3”2 a2Ak E~(A,) a + Ee~(A,) aAk aAIJaAJI (4.24) Using (4.21) and (4.8) it can be transformed into 1
~-vc~~ A,—A3
1
=
~e~(A,)
1 ~
(4.25)
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233
and this cancels (4.15) (transformed with the help of eqs. (4.20) and (4.21)). Thus, the only contribution is (4.16), which gives a2~
2C
1
~
C ~
—
a log
~2C
~ aA,
1aA~, aAk
=
I
C
~
+ ~
2(A,-A1) 1
1 + ~i(~I+~k)j(1_6IJ)
[a2
aA~ log C
+
)2l
aA, C\ j. ía log
(4.26)
Now, using (4.8) and (4.20) we obtain for (4.16): (i
1
— ~E
16. I
2
_________
Ee~(A,)A~/
+ —
I,’
1,(A,)A1
1
1 ~I(~I
1
________
+ ~i)
(~,
+
1
(~+~.) (VA~+V~)
+~ ~EP(AI)AI i,j k
~1
~ e~,(A,)~ 1 2,~jkAj_AjL~j(~j+~k)
~~E
1
+
2+
1 ~I(~i+~k)]J
~(~Ep(A,)A[1A:I+~Ep(A,)A~3/2Ar3/2)
0(A,)A1
+
~(A,)( 4A,(~,+
~
I +
~k))
i*j*k
F 1 I _____
I +
2(A,
-A
1) 2
=
[~j(~~
1 +
+ ~
~k)
-
~
+ ~k)j
(~EP(AI)A~’AT1 + ~
~~(A,)A1
~
1 I
_____
12(A,)AT3/2Ar3/2) i “.3
1
2A~l/2,
i.’.jk ~ (A)A[IA/I/
(4.27)
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In the last transformation we used the fact that 1 (A,
-
F
1
A1)
1
~
+ V~k) -
1 + ~k)J
~ (4.28) Finally, (4.27) can be rewritten as 2+~ ~~(A,)A~’
EA~2A~”2 j,k
j~Le~(A,)A~
1
~2 ~-n-2I~
=
—6
1F2EA~I/2~+ ~
/~16
n
LI
I
{
(4.29)
j
{
Now taking together (4.19), (4.23) and (4.29) we obtain our main result:
3/2) a2 \1 1 tr e —~ —A 3/2Z(T}= — E tr(e~A~2) C(VA)Z{T} aA Zn>, 1jC(VA) e~r5
exp(—~trA
{
‘)
1 2
a
—
1
a2
I
i
aT
~
kTk
8fl
21I+k
+~
a +h=2n
~
~
+
6
fl.O +
4
a +
I,OT, — 1Z(T)
a =
0. (4.30)
This completes the derivation of (4.2) and, thus, of Virasoro constraints for the case K = 2. 4.3. EXAMPLE OF
K
=3
In the case of generic K the analogue of the derivation, described in subsect. 4.2, should involve the following steps, (1) Represent 9/[A] as t”t[A] =g~[A]Zt1’1(T,,), (4.31)
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with
=
exp[(p.,7’(p.,)
fl [~=~(p..yI/2
g~[A] = ~(A)
~
exp{a[K/(K+
[A1)~’2K
- 77(p.,))]]
I)]A*”‘}].
(4.32)
Parameter a is introduced here for the sake of convenience, in fact, a = 1. (ii) Substitute this .~1[A] into (4.1), which in this particular case of 77K(X) X”~/(K+ 1), looks like
- aKA])~[A]Z{K)(T)
{Tr E(A)[(~) When a/aAtr acts on
=
0.
=
(4.33)
Z(T, 1),
the following rule is applied:
~~tr
~
(4.34)
and the sum at the r.h.s goes over all positive n 0 mod K. As to higher-order derivatives, a’Z/aA’,~,they are defined with the help of relations like (4.8). (iii) When all the A-derivatives in (4.1) act on the exponent in g[A] =g[77’(M)], we get the term, which is equal to 77’(a Tr[M77’(M) — 77(M)1/aA,r) = 77’(M) = A and cancels the A-term in (4.1). The next contribution, when all but one of the A-derivatives act on the exponent, vanishes, Actually this reflects the fact that there are no ~ generators among the final W-constraints. (iv) Perform a shift of variables K —~1.,
—
=
~
(4.35)
I
(this procedure does not change a/aT,, —~a/aT,). (v) After all these substitutions the l.h.s. of eq. (4.32) acquires the form of an infinite series where every item is a product of Tr[~(M)M°] and of a K linear t1’~(i,),In the case = 3 combination of generators of WK algebra, acting on Z this equation looks like + 9~M3”
~ (3k
— 2)~k277~3k
+ a+b=3n a,h +1);n —3
E(3k
— 2)~k277~2k
+
~ a+h=3n a,h+O~n7. —3
/3
a aTIU+I -_a
+
,~,
9 ~
M3’~2~3
Z~31= 0, (4.36)
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(see eqs. (1.27) and (1.28) for explicit expressions for v-operators). This relation is valid for any value of N, and in this sense is an example of identity, which behaves smoothly in the limit N —s oc~ (vi) If N = cc all the quantities Tr e(M)M~with given p but varying ~(M) become independent, and (4.35) may be said to imply (1.26). In what follows we restrict ourselves to an illustrative calculation: we prove that (4.35) holds for N = 1 (i.e. when there are numbers instead of matrices), Even this calculation is long enough. Additional technical details, necessary to deal with matrices rather than ordinary numbers, are exhaustively discussed in subsect. 4.2. Example When (4.31) with used, we obtain (Z a2
O=’~(p.) ~~ 1p. +
K
3, N = I is substituted into (4.32) and the rule (4.33) is
=
31):
Z~
az
1
a
aT0p.
1
27
az
28
p.’1 ST,,
27
1
n+4 1
+
p.
~
1 1 a3z p.3 t,~ 27p.’~~’~ aT,aT~aT~+ n2+12n+39
a2Z
m+fl aTmaTa
n+m+8 18
~
E ~
az
1
a2z
p.m+n
aTmaTn
7
(4.37)
Our purpose now is to compare this expression with the l.h.s, of eq. (4.35). In order to understand the structure of (4.35) let us note that the first item (with 7f(3)) in eq. (4.35) contains the contribution
a3z
1 p.(+/fl+fl
aT 1aT,0aT~
just the same as in eq. (4.36), but only under the restriction I + m + n = 0 mod 3. However, no such restriction is imposed in (4.36), and in order to restore the equivalence (4.35) and which (4.36), add onetheneeds add the terms with 2)z)/aT between to the l.h,s. of (4.35), missingtocontributions
a(7#”’
i p.l±?fl+fl
a3z
aT,aT,~aT,, withl+m+n=1
2
mod3.
If we analyze the terms without a in eq. (4.36), it is important that a also appears in the shift (4.34): nhj31
=
nT,, — 3a6,, 4,
(4.38)
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237
so that at the same time we substitute all t’s by T’s in (4.35), (1.26)—(1.28). Then the term
(K—1)(3K—1)(5K—1)...((2K—1)K—1)28 (2K)K in (4.36) should be compared to the contributions without Z-derivatives to eq. (4.35). These come from the negative harmonics of ir”~ and ~(2) operators. Namely, so modified 7cf’,.~ contains (2T2)3 + 3(TI)2(4T4) = 4/p.°, 7f~P.~contains 3 = 1/p.3, and there is no constant term in ~ As to Virasoro operators, (TI) there is /, ‘2(T,)(2T 3 in 7l7j~ and (K + 1)/24 = 1/9 in ~2) 2) = 1/3p. Also the contribution of interest to a7~v.~/~/aT, is / ‘2(2T 2, and to 2) = 1/3p. a7r~/aT 2/3p.. Also (2T 5, 2—is (5T ~ ‘4(T1) = contributes -~‘2(T~)(2T2)~ = 1/3p. (5T 21 5, (2T 2)~T~3~ 2~ — (2T 2, (T 5)~T,.~ 5) ‘/~= ‘/~= 1/9p. — (TI)’ 1)7’~ 2(2T 4, (4T1/9p. 2~(4T 2)~ 42)and (TI)~2~ = l/9p.. 2(T1) 2) = 1/3p. 4)~.ir~ 4)~ = 1/9p. Putting this all together we obtain the coefficient —
1)(K
—
—
=~-
{[4
+
1]
+
9[_
+
+
+
+
9[~
+
+
+
+
0(a)
=
+
0(a)
in front of Z in (4.35), in accordance with (4.36). in order to reproduce the term [(K 1)(K + 1)(2K + 1)/24K]a’<2Z = ~aZ in (4.36) it is necessary to restore the shift (4.36) of the T 4 variable in (4.35). There are also contributions with T,~ in ~2 term andinin(4.36). (3k — 2)T2k..27#1~3,,, which are responsible for the occurrence of the a This illustrative comparison of (4.35) and (4.36) is enough to recognize the proper structure of (4.35). After it is found out it is easy to ensure that all the remaining contributions to (4.35) and (4.36) also coincide. Of course, in order to check the trace structure of (4.35) our simple example of N = 1 is not enough the analogue of detailed consideration of subsect. 4.2 is required. Such detailed derivation, as well as consideration of the case of K> 3, is beyond the scope of this paper. —
5. Conclusion To conclude, we presented enough evidence that the Generalized Kontsevich model (1.2) interpolates between all the multimatrix models, while preserving the property of integrability and the 9/, constraint. This makes 0KM a very appealing candidate for the role of a theory, which could unify all “stable” (i.e. with c ~ 1) bosonic string models. However, the study of GKM was at most originated here. Let us list a set of problems, which seem interesting for the future development.
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Unified theory of 2d gravity
(1) Though the integrable structure and constraint in GKM have been discussed more or less exhaustively, the situation with generic ~constraint and their implications remains less satisfactory. Of course, they may be derived from the .9/— I constraint, as suggested in ref. [1], but it is still desirable: (i) to complete the derivation of v-constraints from Ward identity (1.23) for N = cc, as suggested in sect. 3, for K> 3; (ii) to prove the equivalence of these -constraints [which look like (1.26)] to conventional constraints (1.29); (iii) to describe the subvariety U,. in the grassmannian (at infinity of the Universal module space), specified by the universal 9/~1”~ constraint; (iv) to take the double-scaling continuum limit of the properly reduced v-constraint in discrete multimatrix models (which look like “[M1] = ~>‘~
I
for the 2-matrix case [TI) in order to derive the constraints (1.31) for \/T’.~~ (as was done in ref. [16] for the one-matrix case); (v) to add the problems discussed in subsect, 3.4 to this list. (2) We tried to argue that a particular (K — 1) matrix model arises from 0KM for a particular choice of potential: 77(X) 77~(X)= const ‘X’~’. A particular (CFT + 2d gravity) model is specified by additional adjustment of the M-matrix in 2/pK). These such a way that all T,, = 0, n ~ 1, p + K (for c = I — 6(p — K) particular points in the space of parameters (77(X), M) should be interpreted as critical points of GKM. The relevant questions are: (i) what is the proper criterion, distinguishing critical points in terms of GKM itself; (ii) are there any other critical points? One can even hope that the answer to the last question is positive and some other critical points may be identified with non-bosonic string models. Let us bear in mind that the stability criterion like c ~ 1 (implying the absence of tachionic instabilities, leading to disintegration of Riemann surfaces through the creation of holes), which in the bosonic case implies the restriction d ~ 2 for the space-time dimension, in the case of N = 2 superstrings turns into a much more promising condition: d ~ 4. (3) A separate class of problems is related to identification of (double scaling limit) c = 1 models in the theory of GK.M. This could lead to a more natural formulation of 0KM, in particular, help to restore the symmetry between arguments 77(X) and M of Z1” 1[M]. Keeping in mind the topological implications of the K = 2 model it is also natural to ask what corresponds to the Penner model [38,39]? (4) The last group of problems concerns the meaning and generalizations of 0KM. First of all, both formulations in terms of topological (like refs, [15,40,38,41,39]) and conformal (in the spirit of refs. [42,43]) are required in order to understand explicitly the connection to Liouville theory. Second, the hope that just GK.M (1.2) is enough to include N = 2 superstrings may be too optimistic, and —
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then one needs to look for more universal models. Third, in any case one should search for a more invariant (algebraico-geometric) interpretation of (1.2): GKJvI is already much simpler than the original multi-matrix models (it is a one-matrix model with a smooth continuum limit), but still it is not simple enough to be appealing as a fundamental theory. The main aesthetic problem is the asymmetry between M and 77(X) and, perhaps, the very relevance of matrix integrals. Fourth, GKM does describe interpolation between sufficiently many string models, but it is not a dynamical interpolation: one may still need to look for an “off-shell” version of GKM, involving a sort of integration over 77(X). This problem can be after all related to the question about critical points. In particular, it may be interesting to find out something similar to the “off-shell” actions of refs. [17,44,45], which reproduce Virasoro constraints in the ordinary 1-matrix models as dynamical equations of motion, for the case of GKM. Possible links between 0KM and double-loop algebras (in the spirit of ref. [46]) also deserve investigation. We are indebted for stimulating discussions to A. Gerasimov, E. Gava, M. Kontsevich, Yu. Makeenko, K. Narain, A. Orlov, S. Pakulyak and I. Zakharevich.
References 11] M. Fukuma, H. Kawai and R. Nakayama, Int. J. Mod. Phys. A6 (1991) 1385; Infinite dimensional grassmannian structure of two-dimensional quantum gravity, preprint UT-572, KEK-TH-272 (1990); Explicit solution for p—q duality in two-dimensional quantum gravity, preprint UT-582. KEK-TH-289 (1991)
[2] A. Marshakov, A. Mironov and A. Morozov, Phys. Lett. B274 (1992) 280 [3] V.A. Kazakov, Mod. Phys. Lett. A4 (1989) 2125 [4] E. Brézin and V.A. Kazakov, Phys. Lett. B236 (1990) 144 [5] M. Douglas and S. Shenker, NucI. Phys. B335 (1990) 635 161 D. Gross and A.A. Migdal, Phys. Rev. Lett. 64 (1990) 127 [7] 5. Chadha, G. Mahoux and M. Mehta, J. Phys. A14 (1981) 579 [8] A.A. Belavin, AM. Polyakov and A.B. Zamolodchikov, Nuci. Phys. B241 (1984) 333 [9] M. Douglas, Phys. Lett. B238 (1990) 176 [10] N. Seiberg, Notes on quantum Liouville theory and quantum gravity, preprint RU-90-29 (1990) [11] 5. Kharchev, A. Marshakov, A. Mironov. A. Morozov and A. Zabrodin, Phys. Lett. B275 (1992)311 1121 D.J. Gross and M.J. Newman, Phys. Lett. B266 (1991) 291 [13] Yu. Makeenko and G. Semenoff, Mod. Phys. Lett. A6 (1991) 3455 114] A. Marshakov, A. Mironov and A. Morozov, From Virasoro constraints in Kontsevich’s model to W-constraints in 2-matrix models, preprint FIAN/TD/05-91, ITEP-M-7/91 (August 1991) [15] ML. Kontsevich, Funk. Anal. & Prilozh. 25 (1991) 50 (in Russian) 116] Yu. Makeenko, A. Marshakov, A. Mironov and A. Morozov. Nucl. Phys. B356 (1991) 574 [17] A. Gerasimov, Yu. Makeenko, A. Marshakov, A. Mironov, A. Morozov and A. Orlov, Mod. Phys. Lett. A6 (1991) 3079 [18] E. Witten, Nuci. Phys. B341) (1990) 281 [19] F. Witten, On the Kontsevich model and other models of two-dimensional gravity, preprint IASSNS-HEP-9l/24 (June 1991) [20] R. Dijkgraaf, H. Verlinde and E. Verlinde, Mud. Phys. B348 (1991) 435 121] A.Yu. Orlov, in Proc. Kiev Int. Workshop (World Scientific. Singapore, 1988) p. 116
240
S. Kharchav at a!.
/
Unified theory of 2d gravity
[221 W-constraints in discrete multi-matrix models, preprint FIAN-ITEP (in preparation) [23] V. Kac and AS. Schwarz, Phys. Lett. B257 (1991) 329 [24] CD. Itzyson and J.-B. Zuber, J. Math. Phys. 21(1980) 411 [25] ML. Mehta, Commun. Math. Phys. 79 (1981) 327 [26] T. Miwa, Proc. Japan Academy 58 (1982) 9 [27] S. Kharchev, A. Marshakov, A. Mironov, A. Orlov and A. Zabrodin, NucI. Phys. B366 (1991) 569 [28] A. Gerasimov, A. Marshakov, A. Mironov, A. Morozov and A. Orlov, Mud. Phys. B357 (1991) 565 [29] N. Sato, RIMS Kokyuroku 439 (1981) 30 [30] G. Segal and G. Wilson, PubI. I.H.E.S. 61(1985) 1 [31] A. Zabrodin, Teor. Mat. Phys., 78 (1989) 234 (in Russian) [32] V. Drinfeld and V. Soko~ov,J. Soy. Math. 30 (1984) 1975 [33] E. Date, M. Jimbo, M. Kashiwara and T. Miwa, in Proc. RIMS Symp. Nonlinear integrable systems — classical theory and quantum theory (RIMS Kyoto, 1983) p. 39 [34] D. Gross and A.A. Migdal, Mud. Phys. B340 (1990) 333 [35] A. Orlov and F. Schulman, Lett. Math. Phys. 12 (1986) 171 [36] P. Grinevich and A. Orlov, Flag spaces in KP theory and Virasoro action on det and Segal—Wilson r-function, preprint Cornell Univ. (September 1989) [37] A. Semikhatov, Int. J. Mod. Phys. A4 (1989) 467 [38] R. Penner, J. Dill. Geom. 27 (1987) 35 [39] J. Distler and C. Vafa, Mod. Phys. Lett. A6 (1991) 259 [40] J. Harer and D. Zagier, Invent. Math. 85 (1986) 457 [41] C. Itzyzson and J.-B. Zuber, Matrix integration and combinatorics of modular groups, preprint SPhT/90-004 (1990) [42] A. Marshakov, A. Mironov and A. Morozov, Phys. Lett. B265 (1991) 99 [43] A. Gerasimov, preprint ITEP (1991) [44] A. Jevicki and T. Yoneya, Mod. Phys. Lett. AS (1990) 1615 [45] P. Ginsparg, M. Goulian, MR. Plesser and J. Zinn-Justin, NucI. Phys. B342 (1990) 539 [46] A. Gerasimov, D. Lebedev and A. Morozov, mt. J. Mod. Phys. A6 (1991) 877