- Email: [email protected]
On the equivalence of topological and quantum 2D gravity
Physics Letters B 274 ( 1992 ) 280-288 North-Holland
PHYSICS LETTERS B
On the equivalence of topological and quantum 2D gravity A. Marshakov, A. M i r o n o v Department of Theoretical Physics, P.N. Lebedev Physical Institute, Leninsky prospect 53, SU- l 17 924 Moscow, USSR
and A. M o r o z o v Institute of Theoretical and Experimental Physics, Bol. Cheremushkinskaya street 25, SU- 117 259 Moscow, USSR Received 24 September 1991
We demonstrate the equivalence of Virasoro constraints imposed on the continuum limit of the partition function of the hermitean one-matrix model and the Ward identities of Kontsevich's model. Since the first model describes ordinary d = 2 q u a n t u m gravity, while the second one is supposed to coincide with Witten's topological gravity, the result provides a strong implication that the two models are indeed the same.
I. Introduction
From the early days of matrix models there is a belief that partition functions of two a priori different models of 2D gravity should coincide. The first is the (square root of the) partition function Z,,//~qg{7", } of ordinary Polyakov 2D quantum gravity, described by the continuum limit of the hermitean one-matrix model [ 1 ] (and is known to be a r-function of the KdV hierarchy [2 ] ). The second partition function Z,g of Witten's topological gravity [ 3 ] is a priori defined as a kind of generating functional of intersection indices of divisors on module spaces of Riemann surfaces with punctures. Later it was discovered [4,5 ] that Zx/~qg{T,, } satisfies a set of Virasoro constraints 73 = 0,
5f.=
~_.
(k+½)Tk~k+,+
k >~ ¢~n+ 1,0
Z
a+b=n-a,b>~O
_
I
_
1
2
OTaOT,,+ g~+,.o' T~To + ~.o' ~6 •
(1)
Eq. ( 1 ) can be deduced [6 ] as a continuum limit of Ward identities in a discrete matrix model, associated with the shift of integration variables [ 7 ]
X ~ X + eX p+ I
(2)
in the integral over the N × N hermitean matrix Z[e'{t}= f O X e x p ( - t r V{X}),
V{X}= ~ tkX ~.
(3)
k=O
However, it may be more reasonable to take these Virasoro constraints for a straightforward definition of Zqg{73, E-mail address: [email protected].
280
Elsevier Science Publishers B.V.
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16 January 1992
which does not refer to a sophisticated change of variables {t}-- {7"} [ 6 ] and to the discrete model ( 3 ) at all. As for Zig it was recently represented by Kontsevich [ 8 ] in terms of another matrix model Ztg=
lim
Zttga),
(4)
size X~oo
where Zt(ga) -
1 f DXexp(-tr{MX2+X3}) C[M]
(5)
with
C[M] = f D X exp ( - tr{MX 2} ) = det ( M ® I + I ® M ) - '/2
(6)
and X, M being (anti) hermitean matrices. It is a simple combinatorial result, that as soon as the size of Xgoes to infinity Zt~ga) becomes only dependent on the variables ~t 4
3 2m+l
Tin-
m+½
trM-Zm-l+
~m,l.
(7)
The explicit formulation of the original suggestion in these terms would be
= z./2
vo},
(8)
which in particular implies what is known as Witten's suggestion, i.e., that Ztg{ T} is like x/~qg{ T} a r-function of the KdV-hierarchy. A necessary condition for ( 8 ) to hold is that Ztg{ 7"} defined by ( 4 ) - ( 6 ) satisfies the same Virasoro constraints ( 1 ). This is the statement which we are going to prove in the paper. Moreover, we shall prove the relation involving Z(ga) [see eq. (16) below], i.e., valid forfinite-dimensional matrices X, which implies the entire set o f Virasoro constraints only as the size goes to infinity. This statement is equivalent to (8) modulo (i) the assertions which have been made by Kontsevich in the derivation of ( 4 ) - ( 6 ) from Penner's formalism of fat graphs, and (ii) the so far subtle problem o f uniqueness of solutions to the Virasoro constraints ( 1 ). We know at least about two recent attempts to develop some similar (i.e., exploiting the Virasoro constraints ) approach to the study o f suggestion (8). One is due to Witten [9] ~2. Another one is due to Makeenko and Semenoff [10] who investigated the relation between the tree-level approximation to x/~qg and to Z,g. An explicit relation between ref. [ 10 ] and our reasoning below deserves further investigation.
2. Main results Since the Ward identities are a little obscure in Kontsevich's presentation ( 4 ) - ( 6 ), we begin with a slight reformulation of his model. After a shift o f the integration variable X - , X - ~ M a n d the redefinition M = 3A 2, we obtain ~ We will see the origin of the d,,,l term and the 1/(m+~) factor below. In both these aspects (7) is different from Kontsevich's definition of time variables. Though his choice is more appropriate for the study of the generating functional for intersection indices, (7) is better to establish the correspondence with the one-matrix model. In other words, the generating functional of intersection indices appears to be a z-function only after appropriate rescaling of"time "-variables. ~2 We were informed about this talk by Professor I, Singer, but unfortunately we do not know what exactly has been done. 281
Volume 274, number 3,4
,~{A}-
f
PHYSICS LETTERS B
DXexp(-trX3+trAX)=C[x/A]exp
(2
-~trA
16 January 1992
)
3/2 Z(gd),
(9)
with C [ , f A ] = det ( x / ~ ® l + I ® x f A ) - ,/2
(10)
The functional .~{A} can be considered as a kind of matrix-Fourier transform of the exponential cubic potential tr X 3. It satisfies the obvious Ward identities, associated with a shift of the integration variables X-~ X + ep,
( 11 )
namely tr(q, ~ 02 -]~pA).~{A}=O.
(12)
Here Ep stands for any X-independent matrix, and Atr denotes a transponent matrix. There are as m a n y as N 2 different choices of ep (N being the size of the X matrix ). Though a concrete choice of ev should not be essential, we shall assume it to be a function of A, e.g., % = A p (with integer or half-integer p, again it is not essential). In principle, there is nothing to forbid a consideration of X-dependent ep'S, for example %=X p+l (as implied by ref. [ 7 ] ). This case was investigated to some extent in ref. [ 10 ], but it leads to much more complicated Ward identities than (12) (because the integration measure is no longer invariant, and, what is more important, the cubic form of the potential is not preserved). Clearly, the complete basis ep= X ~'+~is somehow related to the basis %=A p and after all the Ward identities should be essentially the same (indeed, they were interpreted in ref. [ 10] as Virasoro constraints), but this is not so simple to prove ~3 Let us return to our Ward identities ( 12 ). Substitute in ( 12 ) the functional .~- in the form, inspired by (9), .~(A)=C[x/A lexp
(2
)
3/2 Z{rm},
-~trA
(13)
with 1
T i n - m + ½ trA
4
.... I/2..1_ ~__~ ~,,,., .
(14)
Our main statement (to be proved in section 3 below) is that after the substitution of (13) into (12) it turns into tr(epA
,,-2) 5 ( ~ Z = 0 .
(15)
n>~ -- I
The identity
1 { °2
~tr~eP~
)
%Z
-~evA J = Z,~_, ~Ztr(EvA .... 2)
(16)
is valid for any size of the matrix A, but only in the limit of infinitely large A: (i) it is reasonable to substitute Z~gd) in (9) by Zig, which depends only on tr A-q with half-integer q"s; (ii) the T,,'s in ( 14 ) are really independent variables; (iii) all the quantities ~3 In fact, in the actual derivation in ref. [10] some other "master equations" are used (but not derived) instead of these Virasoro constraints. The master equation (eq. (3.1) in ref. [ 10] ) is certainly nothing bul our Ward identity ( 12 ) with (E)0= ~pfipj (p= 1..... N) in the basis, where A is diagonal [compare with (20) below]. 282
Volume 274, number 3,4
PHYSICS LETTERSB
tr (eRA - , - 2 )
16 January 1992 (17)
(e.g., trA p - ' - 2 ) become algebraically independent, so that eq. ( 15 ) implies that Lf, Zig {T} = 0,
n>~-l.
(18)
This concludes our derivation of the Virasoro constraints for Ztg{T) defined as a continuum limit of Kontsevich's partition function. Note that the fact that the operators Lf, in (15) contain only second T-derivatives, is a direct consequence of that only double A-derivatives arise on the LHS and thus of the cubic nature of the potential tr X 3 in Kontsevich's model. It is very interesting (in particular, from the point of view of multimatrix and Potts models ) to study the matrix-Fourier transform of potentials like tr X K+~ for any degree K + 1. Amusingly enough, the corresponding Ward identities being transformed to the form of ( 15 ) contain operators similar to the higherspin operators of the WK-algebra instead of the Virasoro generators LP,. This should he somehow related to the recent suggestion of Gava and Narain [ 11 ], namely that the Ward identities in the continuum limit of the twomatrix model acquire the form of the WK-algebra constraints if one of the potentials is of degree K. For a more close relation between ref. [ 11 ] and the Ward identity ( 12 ) see ref. [ 12 ].
3. The proof of the main result
3. I. An important relation First, let us point out that Y~{A}defined by ( 13 ) which we have to differentiate to prove ( 16 ) depends only upon the eigenvalues {2k} of the matrix A. Therefore, it is natural to consider eq. (16) at the diagonal point A~j=O, iv~j. The only "non-diagonal" piece of (16) which survives at this point is proportional to 022
6ki--Skj
OAijOAjiAm,=o,.~,, - 2i-2j
for
i#j
(19)
Eq. (19) 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 the determinant formula
61og(detA)=trl 6A-½ tr(15A16A)+ .... For the diagonal A~j=2 fl~j,but, generically, non-diagonal 62k
ZW =
_l
8AoSAji
! ~ (1
8Aij, this equation gives
l ~ SAo6Aj, +...,
which proves (19). If applied to a function ~{A} depending only on eigenvalues of A, eq. (19) implies that (02 )
-~
02~{A}] ii ~{A} Amn=Om#n= ~J OAij OAJi Amn Om¢n= ~J
0~102,-0~102j 2i--~J
(20)
(note that the item j = i is not omitted from the sum). The general formula for a matrix derivative of such a functional at the diagonal point is 283
Volume 274, number 3,4 0n+ 1 x
PHYSICS LETTERS B
l
OA Jit
16 January 1992
1~,~/02 t
,4..... o,,,.~,,
J,.:.J, . . . . p. . . . talions (21--J.jl) ""('~t--J'jk)
(21)
[i.jb..jn]
3.2. Simplified example In order to illustrate the idea, we shall begin now with a simpler calculation which is (i) much more transparent than the actual derivation of (16), and (ii) can be also of interest from the point of view of the relation between discrete and continuum matrix models. Namely, let us derive the formula for the action oftr{% 02/0A 2} on a function Z~a){tk}, tk= (1/k) tr(1/Ak), to= - t r { l o g A } [in variance with (14) the integer powers of A are involved]. Then [using (19) ] (02 ) tr ~p 0A~r
~
022
Z ( t ) = k ~. o ~ t r ( e p A - k - l - 2 ) +
OZ ( 0 2 t n ~ _ ePOA2r,]- ~ -~~ ~nZtr(epA-~-2),
~" ~-~tr n~0
(22)
with
~:,= E kt, k~>l
+
52
,+t .... a,b >~0
(23)
Ot. Ott,
Remarkably enough, the operators (23) are exactly the Virasoro operators annihilating the partition function (3) of the discrete matrix model. In these models also the to-dependence of Z ~a) is fixed:
Oto
Z~a)= _NZ~a~ .
This corresponds to extracting the factor exp ( with
Nto) = (det A ) - N from Z (a). This factor should be compared
C(,,/A) = [det ( x / A ® I + I ® , , / A ) ] - ,/2 in the continuum case. Note that while C(M) = [det(M®l+IQM) ] -~/2 is a normalization factor for the gaussian measure d X e x p [ - t r ( M X 2) ], in the discrete case we have (det A ) - ' v = det-~/2MX= det-~/: ( x / M ® x / M ) .
(24)
This represents a normalization factor for a slightly different gaussian measure, that is, d X e x p [ - t r ( x / M
X
x,/Mx)I. The derivation in this subsection is not of immediate use, since (i) we did not find anything like the Ward identity tr el,
Z(t)=0,
(25)
and (ii) in the case of a finite size N of the matrices there is no way to distinguish particular terms in the sum over n in (22) (the quantities tr{ eo A - " - 2 } are algebraically dependent ). Still it is an amusing reformulation o f the discrete Virasoro constraints, especially because it is somewhat generalizable: t r e p a[ n )0"
~Z~d)rt}_ ~ - .>~'l-x lYV~"~"Z'a'{t} tr(epA-X-')
is similarly represented as a sum of harmonics of the (discrete-model) I~K)-operators, 284
(26)
Volume274, number 3,4
I~(~K)=
X
kl,...,kK-I
PHYSICSLETTERSB
16 January 1992
0 OK k~t~...kK_~t,,~_,O t k l + . . . + k K - l + n + ""+ax+...~+ax=NOtal...Ota,"
(27)
The tilde over W stands to emphasize that the relative coefficients in the sum (27) do not coincide with the conventional definition of the W (m-operators of the W-algebra. While ( 16 ) - the continuum-model analogue of (22) - will be explicitly derived in section 3.3 below, the analogue of (27), which hopefully exists as well, deserves more efforts to be worked out.
3.3. Derivationof eq. (16) Now we shall turn directly to the proof of ( 16 ). Since Ep is assumed to be a function of A, it can be, in fact, treated as a function of eigenvalues 2,. After that, (16) can be rewritten in the following way:
exp[(2/3v/3) trA3/2][trEp(~--~SA2-~A)lC(,f-A)exp[-(2/3,ff3)trA3/2] Z{T} C(v)z{T} 02Z
0Ta 0Yb
= 1Za,b>~O ~ 0Ta 0Tb ~ ~v(2~)
02t
(29)
02 i
+f~n~o-ff-~ 'P(2i)OAoOAji+2~i 'p(2') 0k, 02,~t-2~i 'p(2i) 02i\-3---w/~w/3 ~ / t r A +{l ~i ep(2,)- ~'/,p(2i)[o~i (- +)trA3/2]2
[OAjj02OAji( -~ )
+ ~,.sep(2,)
(28)
trA3/2
02,
(30) (31,
]
(32)
+2~ep(2j(_3-~)OtrA3/2Ol°gC •
J
02,
(33)
02C 1 ~ ~p(2,) + C ~,s 0A~j 0AjJ'
(34)
with trA3/2= X,23/2 and C= I-I ( , ~ + ~ )
-1/2 .
(35)
i,j
First, since
OT. _
(36)
__2zn_3/2
02,
it is easy to notice that the term with the second derivatives (29) can be immediately written in the desired
form:
02Z
(29)=
Z n>~--I
"-2 a + b =En - - I
i
OToOT - n>~--I Y"
02Z
tr{ev(2i)/-"-2}
a+b=n--E1OTaOTb"
(37)
For the first-derivative terms with the help of (35), 0 log C 04,
1 -
1
(38)
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16 January 1992
0 trA3/2=}x/~, 02,
(39)
and
02Tn =~
022k
OAoOAj,
aTn
OA~jOAji02k
02Tn + ~
(40)
dr"
Then we have
~ o~o~,,
~p(~') ~ + ~
1 - -Z n~o ~,,~
o=oZ~,(,~,)~7°-2+°,~. ~-''-~ + z ~
1
- - ,,>-~ E E.• ~p(,~,),~-,,_2 k>~a.+,.o E -Z
1
----Z
Z
1
+ a+h=2(n+l)
tr(ep A - ' - 2 )
tl>~ -- [
Z k~> ~n+ 1,0
2
%(2i)2.,,_ i ]
~,,(,~,)
2 )oz
,l-J *-''2 + -~- a ~ . ,
OT,,+k
OZ
(k+½)TkoT,,+~,
(41)
where
k--I~2_}_
1
Tk= k ~ t r A
4
~ - ~ ~k,l •
The remaining part contains terms proportional to Z itself which need a bit more care. First, it is easy to notice that the two items in (31 ) just cancel each other, so (31 ) gives no contribution to the final result. For (32) we have 2 (~ 02 trA3/2 0trA3/2 3,~ %(2,) 022 + ~,,,q,(2~) 02~
022k ]
(42)
OAn OA,,,]"
Using (39) and (19) it can be transformed into
i(
,,,
•
,<-,~,
j = - ~
,
i
(43)
Z ','('~') ,/L, +,,/~, i,.I
and this cancels (33) [using (38), (39) ]. Thus, the only contribution is (34), which gives 1
02C
022k
1(
01ogC
1 02C
1
I
= ~2(~,-~,) ,G,(~+4x;)- ~(,G,+,/~) Now, using (19) and (38) we obtain for (34) 286
).~(l_6,s)+~oro210gc ,{010gc~2]
L ~
*k~7-,; J
(44)
Volume 274, number 3,4
PHYSICSLETTERSB
16 January 1992
1
1
)
f6 Z, ,.(~,);,-:+12,, ,.(~,);:' ./Z,t./Z,+v%) + ( ~ , + ~ ) :
1 +~,~,~(~,~+~
.1_1 2 (P (Z/)'~IT-1 Lj, k
i
t#J
[
l
+1
+ i.,.*Z ev(2') 4,,1.,(.v~+~-)(.~i+~,/~k) 5
, ) - ~,/s,+~
E (}P(2i)/~]-2"~- 2 i
2(2,-2j)
[/•P(2i)2t"-lZJ 1 - ] - l e p ( 2 i ) 2 Z 3 / 2 2 7 3 / 2 ]
(
,
l
( ~ ; + . ~ k ) - .~j(~,+V/~k) JFI
i~j
)] (45)
2 e(R)2CLtT'/22i-'/2. i,Lj~sk
In the last equality we used the fact that
1
1
z -4-' (~+,/z j¢k 2i --2j -a •
j~k
~- ,/z, ~ + ~ ~) (~j.jl_~k)
-
,
~.](~i_,[_~k
;) -['-(j*~,'k).]
(46)
Finally, (45) can be rewritten as l t
i
ep(2,)2;-'~-2 j,k
n
~6~,~_~ 2 ~ . , ;
]
T~v,,,oj.
(47)
'
Now taking together ( 37 ), (41 ) and (47) we obtain our main result:
exp[(2/3v/3)trA3/2][treo(~--~SAz_~A)]C(x/A)exp[_(2/3x/~)trA3/2]Z{T } c(,/5)z{r} 1
=Z
2
n>~-- 1
tr(ep A - " - 2 )
(
l
~
c32
2 (k+~)Tk-~,+ a+b=n-E 1 0T~OTb+~6a.,o+~6a.+,.oTo2 k~n+l,0
)
Z{T}=0.
(48)
a>~O,b>~O
This completes the derivation.
Acknowledgement We are deeply indebted to Professor I. Singer who informed us about the talk by Witten [ 9 ] and encouraged us to complete our study of Ward identities in Kontsevich's model. We benefited a lot from illuminating discussions on the subject with E. Corrigan, A. Gerasimov, M. Kontsevich, Yu. Makeenko, A. Niemi and I. Zakharevich. We are grateful for the hospitality of the Mathematical Science Research Institute, Berkeley, CA and the Research Institute for Theoretical Physics (TFT) at Helsinski University where some pieces of this work have been done. A. Mor. is also very much indebted for the hospitality of the Department of Mathematical Sciences of Durham University. The work of A. Mar. and A. Mir. was partially supported by the Research Institute for Theoretical Physics (TFT) at Helsinki University while that of A. Mor. by the UK Science and Engineering Council through the Visiting Fellowship program and by NORDITA.
287
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PHYSICS LETTERS B
References [ l ] V. Kazakov, Mod. Phys. Lett. A 4 (1989) 2125; E. Brezin and V. Kazakov, Phys. Lett. B 236 (1990) 144; M. Douglas and S. Shenker, Nucl. Phys. B 335 (1990) 635; D. Gross and A. Migdal, Phys. Rev. Lett. 64 (1990) 127. [2] M. Douglas, Phys. Lett. B 238 (1990) 176. [3] E. Witten, Nucl. Phys. B 340 (1990) 281; R. Dijkgraaf and E. Witten, Nucl. Phys. B 342 (1990) 486. [4] M. Fukuma, H. Kawai and R. Nakayama, Intern. J. Mod. Phys. A 6 (1991) 1385. [5] R. Dijkgraaf, E. Verlinde and H. Verlinde, Nucl. Phys. B 356 ( 1991 ) 574. [6 ] Yu. Makeenko et al., Nucl. Phys. B 356 ( 1991 ) 574. [7] J. Ambjorn, J. Jurkiewicz and Yu. Makeenko, Phys. Lett. B 251 (1990) 517; A. Gerasimov et al., Nucl. Phys. B 357 ( 1991 ) 565; A. Mironov and A. Morozov, Phys. Lelt. B 252 (1990) 47. [ 8] M. Kontsevich, Funct. Anal. Appl. (1990), to be published. [9 ] E. Witten, talk NYC Conf. (June 1991 ). [ 10] Yu. Makeenko and G. Semenoff, ITEP/UBC preprint (July 1991 ). [ 11 ] E. Gava and K. Narain, Phys. Lett. B 263 ( 1991 ) 213. [ 12 ] A. Marshakov, A. Mironov and A. Morozov, preprint ITEP-M-5/91, FIAN/TD/05-91.
288
16 January 1992
PHYSICS LETTERS B
On the equivalence of topological and quantum 2D gravity A. Marshakov, A. M i r o n o v Department of Theoretical Physics, P.N. Lebedev Physical Institute, Leninsky prospect 53, SU- l 17 924 Moscow, USSR
and A. M o r o z o v Institute of Theoretical and Experimental Physics, Bol. Cheremushkinskaya street 25, SU- 117 259 Moscow, USSR Received 24 September 1991
We demonstrate the equivalence of Virasoro constraints imposed on the continuum limit of the partition function of the hermitean one-matrix model and the Ward identities of Kontsevich's model. Since the first model describes ordinary d = 2 q u a n t u m gravity, while the second one is supposed to coincide with Witten's topological gravity, the result provides a strong implication that the two models are indeed the same.
I. Introduction
From the early days of matrix models there is a belief that partition functions of two a priori different models of 2D gravity should coincide. The first is the (square root of the) partition function Z,,//~qg{7", } of ordinary Polyakov 2D quantum gravity, described by the continuum limit of the hermitean one-matrix model [ 1 ] (and is known to be a r-function of the KdV hierarchy [2 ] ). The second partition function Z,g of Witten's topological gravity [ 3 ] is a priori defined as a kind of generating functional of intersection indices of divisors on module spaces of Riemann surfaces with punctures. Later it was discovered [4,5 ] that Zx/~qg{T,, } satisfies a set of Virasoro constraints 73 = 0,
5f.=
~_.
(k+½)Tk~k+,+
k >~ ¢~n+ 1,0
Z
a+b=n-a,b>~O
_
I
_
1
2
OTaOT,,+ g~+,.o' T~To + ~.o' ~6 •
(1)
Eq. ( 1 ) can be deduced [6 ] as a continuum limit of Ward identities in a discrete matrix model, associated with the shift of integration variables [ 7 ]
X ~ X + eX p+ I
(2)
in the integral over the N × N hermitean matrix Z[e'{t}= f O X e x p ( - t r V{X}),
V{X}= ~ tkX ~.
(3)
k=O
However, it may be more reasonable to take these Virasoro constraints for a straightforward definition of Zqg{73, E-mail address: [email protected].
280
Elsevier Science Publishers B.V.
Volume 274, number 3,4
PHYSICS LETTERS B
16 January 1992
which does not refer to a sophisticated change of variables {t}-- {7"} [ 6 ] and to the discrete model ( 3 ) at all. As for Zig it was recently represented by Kontsevich [ 8 ] in terms of another matrix model Ztg=
lim
Zttga),
(4)
size X~oo
where Zt(ga) -
1 f DXexp(-tr{MX2+X3}) C[M]
(5)
with
C[M] = f D X exp ( - tr{MX 2} ) = det ( M ® I + I ® M ) - '/2
(6)
and X, M being (anti) hermitean matrices. It is a simple combinatorial result, that as soon as the size of Xgoes to infinity Zt~ga) becomes only dependent on the variables ~t 4
3 2m+l
Tin-
m+½
trM-Zm-l+
~m,l.
(7)
The explicit formulation of the original suggestion in these terms would be
= z./2
vo},
(8)
which in particular implies what is known as Witten's suggestion, i.e., that Ztg{ T} is like x/~qg{ T} a r-function of the KdV-hierarchy. A necessary condition for ( 8 ) to hold is that Ztg{ 7"} defined by ( 4 ) - ( 6 ) satisfies the same Virasoro constraints ( 1 ). This is the statement which we are going to prove in the paper. Moreover, we shall prove the relation involving Z(ga) [see eq. (16) below], i.e., valid forfinite-dimensional matrices X, which implies the entire set o f Virasoro constraints only as the size goes to infinity. This statement is equivalent to (8) modulo (i) the assertions which have been made by Kontsevich in the derivation of ( 4 ) - ( 6 ) from Penner's formalism of fat graphs, and (ii) the so far subtle problem o f uniqueness of solutions to the Virasoro constraints ( 1 ). We know at least about two recent attempts to develop some similar (i.e., exploiting the Virasoro constraints ) approach to the study o f suggestion (8). One is due to Witten [9] ~2. Another one is due to Makeenko and Semenoff [10] who investigated the relation between the tree-level approximation to x/~qg and to Z,g. An explicit relation between ref. [ 10 ] and our reasoning below deserves further investigation.
2. Main results Since the Ward identities are a little obscure in Kontsevich's presentation ( 4 ) - ( 6 ), we begin with a slight reformulation of his model. After a shift o f the integration variable X - , X - ~ M a n d the redefinition M = 3A 2, we obtain ~ We will see the origin of the d,,,l term and the 1/(m+~) factor below. In both these aspects (7) is different from Kontsevich's definition of time variables. Though his choice is more appropriate for the study of the generating functional for intersection indices, (7) is better to establish the correspondence with the one-matrix model. In other words, the generating functional of intersection indices appears to be a z-function only after appropriate rescaling of"time "-variables. ~2 We were informed about this talk by Professor I, Singer, but unfortunately we do not know what exactly has been done. 281
Volume 274, number 3,4
,~{A}-
f
PHYSICS LETTERS B
DXexp(-trX3+trAX)=C[x/A]exp
(2
-~trA
16 January 1992
)
3/2 Z(gd),
(9)
with C [ , f A ] = det ( x / ~ ® l + I ® x f A ) - ,/2
(10)
The functional .~{A} can be considered as a kind of matrix-Fourier transform of the exponential cubic potential tr X 3. It satisfies the obvious Ward identities, associated with a shift of the integration variables X-~ X + ep,
( 11 )
namely tr(q, ~ 02 -]~pA).~{A}=O.
(12)
Here Ep stands for any X-independent matrix, and Atr denotes a transponent matrix. There are as m a n y as N 2 different choices of ep (N being the size of the X matrix ). Though a concrete choice of ev should not be essential, we shall assume it to be a function of A, e.g., % = A p (with integer or half-integer p, again it is not essential). In principle, there is nothing to forbid a consideration of X-dependent ep'S, for example %=X p+l (as implied by ref. [ 7 ] ). This case was investigated to some extent in ref. [ 10 ], but it leads to much more complicated Ward identities than (12) (because the integration measure is no longer invariant, and, what is more important, the cubic form of the potential is not preserved). Clearly, the complete basis ep= X ~'+~is somehow related to the basis %=A p and after all the Ward identities should be essentially the same (indeed, they were interpreted in ref. [ 10] as Virasoro constraints), but this is not so simple to prove ~3 Let us return to our Ward identities ( 12 ). Substitute in ( 12 ) the functional .~- in the form, inspired by (9), .~(A)=C[x/A lexp
(2
)
3/2 Z{rm},
-~trA
(13)
with 1
T i n - m + ½ trA
4
.... I/2..1_ ~__~ ~,,,., .
(14)
Our main statement (to be proved in section 3 below) is that after the substitution of (13) into (12) it turns into tr(epA
,,-2) 5 ( ~ Z = 0 .
(15)
n>~ -- I
The identity
1 { °2
~tr~eP~
)
%Z
-~evA J = Z,~_, ~Ztr(EvA .... 2)
(16)
is valid for any size of the matrix A, but only in the limit of infinitely large A: (i) it is reasonable to substitute Z~gd) in (9) by Zig, which depends only on tr A-q with half-integer q"s; (ii) the T,,'s in ( 14 ) are really independent variables; (iii) all the quantities ~3 In fact, in the actual derivation in ref. [10] some other "master equations" are used (but not derived) instead of these Virasoro constraints. The master equation (eq. (3.1) in ref. [ 10] ) is certainly nothing bul our Ward identity ( 12 ) with (E)0= ~pfipj (p= 1..... N) in the basis, where A is diagonal [compare with (20) below]. 282
Volume 274, number 3,4
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tr (eRA - , - 2 )
16 January 1992 (17)
(e.g., trA p - ' - 2 ) become algebraically independent, so that eq. ( 15 ) implies that Lf, Zig {T} = 0,
n>~-l.
(18)
This concludes our derivation of the Virasoro constraints for Ztg{T) defined as a continuum limit of Kontsevich's partition function. Note that the fact that the operators Lf, in (15) contain only second T-derivatives, is a direct consequence of that only double A-derivatives arise on the LHS and thus of the cubic nature of the potential tr X 3 in Kontsevich's model. It is very interesting (in particular, from the point of view of multimatrix and Potts models ) to study the matrix-Fourier transform of potentials like tr X K+~ for any degree K + 1. Amusingly enough, the corresponding Ward identities being transformed to the form of ( 15 ) contain operators similar to the higherspin operators of the WK-algebra instead of the Virasoro generators LP,. This should he somehow related to the recent suggestion of Gava and Narain [ 11 ], namely that the Ward identities in the continuum limit of the twomatrix model acquire the form of the WK-algebra constraints if one of the potentials is of degree K. For a more close relation between ref. [ 11 ] and the Ward identity ( 12 ) see ref. [ 12 ].
3. The proof of the main result
3. I. An important relation First, let us point out that Y~{A}defined by ( 13 ) which we have to differentiate to prove ( 16 ) depends only upon the eigenvalues {2k} of the matrix A. Therefore, it is natural to consider eq. (16) at the diagonal point A~j=O, iv~j. The only "non-diagonal" piece of (16) which survives at this point is proportional to 022
6ki--Skj
OAijOAjiAm,=o,.~,, - 2i-2j
for
i#j
(19)
Eq. (19) 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 the determinant formula
61og(detA)=trl 6A-½ tr(15A16A)+ .... For the diagonal A~j=2 fl~j,but, generically, non-diagonal 62k
ZW =
_l
8AoSAji
! ~ (1
8Aij, this equation gives
l ~ SAo6Aj, +...,
which proves (19). If applied to a function ~{A} depending only on eigenvalues of A, eq. (19) implies that (02 )
-~
02~{A}] ii ~{A} Amn=Om#n= ~J OAij OAJi Amn Om¢n= ~J
0~102,-0~102j 2i--~J
(20)
(note that the item j = i is not omitted from the sum). The general formula for a matrix derivative of such a functional at the diagonal point is 283
Volume 274, number 3,4 0n+ 1 x
PHYSICS LETTERS B
l
OA Jit
16 January 1992
1~,~/02 t
,4..... o,,,.~,,
J,.:.J, . . . . p. . . . talions (21--J.jl) ""('~t--J'jk)
(21)
[i.jb..jn]
3.2. Simplified example In order to illustrate the idea, we shall begin now with a simpler calculation which is (i) much more transparent than the actual derivation of (16), and (ii) can be also of interest from the point of view of the relation between discrete and continuum matrix models. Namely, let us derive the formula for the action oftr{% 02/0A 2} on a function Z~a){tk}, tk= (1/k) tr(1/Ak), to= - t r { l o g A } [in variance with (14) the integer powers of A are involved]. Then [using (19) ] (02 ) tr ~p 0A~r
~
022
Z ( t ) = k ~. o ~ t r ( e p A - k - l - 2 ) +
OZ ( 0 2 t n ~ _ ePOA2r,]- ~ -~~ ~nZtr(epA-~-2),
~" ~-~tr n~0
(22)
with
~:,= E kt, k~>l
+
52
,+t .... a,b >~0
(23)
Ot. Ott,
Remarkably enough, the operators (23) are exactly the Virasoro operators annihilating the partition function (3) of the discrete matrix model. In these models also the to-dependence of Z ~a) is fixed:
Oto
Z~a)= _NZ~a~ .
This corresponds to extracting the factor exp ( with
Nto) = (det A ) - N from Z (a). This factor should be compared
C(,,/A) = [det ( x / A ® I + I ® , , / A ) ] - ,/2 in the continuum case. Note that while C(M) = [det(M®l+IQM) ] -~/2 is a normalization factor for the gaussian measure d X e x p [ - t r ( M X 2) ], in the discrete case we have (det A ) - ' v = det-~/2MX= det-~/: ( x / M ® x / M ) .
(24)
This represents a normalization factor for a slightly different gaussian measure, that is, d X e x p [ - t r ( x / M
X
x,/Mx)I. The derivation in this subsection is not of immediate use, since (i) we did not find anything like the Ward identity tr el,
Z(t)=0,
(25)
and (ii) in the case of a finite size N of the matrices there is no way to distinguish particular terms in the sum over n in (22) (the quantities tr{ eo A - " - 2 } are algebraically dependent ). Still it is an amusing reformulation o f the discrete Virasoro constraints, especially because it is somewhat generalizable: t r e p a[ n )0"
~Z~d)rt}_ ~ - .>~'l-x lYV~"~"Z'a'{t} tr(epA-X-')
is similarly represented as a sum of harmonics of the (discrete-model) I~K)-operators, 284
(26)
Volume274, number 3,4
I~(~K)=
X
kl,...,kK-I
PHYSICSLETTERSB
16 January 1992
0 OK k~t~...kK_~t,,~_,O t k l + . . . + k K - l + n + ""+ax+...~+ax=NOtal...Ota,"
(27)
The tilde over W stands to emphasize that the relative coefficients in the sum (27) do not coincide with the conventional definition of the W (m-operators of the W-algebra. While ( 16 ) - the continuum-model analogue of (22) - will be explicitly derived in section 3.3 below, the analogue of (27), which hopefully exists as well, deserves more efforts to be worked out.
3.3. Derivationof eq. (16) Now we shall turn directly to the proof of ( 16 ). Since Ep is assumed to be a function of A, it can be, in fact, treated as a function of eigenvalues 2,. After that, (16) can be rewritten in the following way:
exp[(2/3v/3) trA3/2][trEp(~--~SA2-~A)lC(,f-A)exp[-(2/3,ff3)trA3/2] Z{T} C(v)z{T} 02Z
0Ta 0Yb
= 1Za,b>~O ~ 0Ta 0Tb ~ ~v(2~)
02t
(29)
02 i
+f~n~o-ff-~ 'P(2i)OAoOAji+2~i 'p(2') 0k, 02,~t-2~i 'p(2i) 02i\-3---w/~w/3 ~ / t r A +{l ~i ep(2,)- ~'/,p(2i)[o~i (- +)trA3/2]2
[OAjj02OAji( -~ )
+ ~,.sep(2,)
(28)
trA3/2
02,
(30) (31,
]
(32)
+2~ep(2j(_3-~)OtrA3/2Ol°gC •
J
02,
(33)
02C 1 ~ ~p(2,) + C ~,s 0A~j 0AjJ'
(34)
with trA3/2= X,23/2 and C= I-I ( , ~ + ~ )
-1/2 .
(35)
i,j
First, since
OT. _
(36)
__2zn_3/2
02,
it is easy to notice that the term with the second derivatives (29) can be immediately written in the desired
form:
02Z
(29)=
Z n>~--I
"-2 a + b =En - - I
i
OToOT - n>~--I Y"
02Z
tr{ev(2i)/-"-2}
a+b=n--E1OTaOTb"
(37)
For the first-derivative terms with the help of (35), 0 log C 04,
1 -
1
(38)
285
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16 January 1992
0 trA3/2=}x/~, 02,
(39)
and
02Tn =~
022k
OAoOAj,
aTn
OA~jOAji02k
02Tn + ~
(40)
dr"
Then we have
~ o~o~,,
~p(~') ~ + ~
1 - -Z n~o ~,,~
o=oZ~,(,~,)~7°-2+°,~. ~-''-~ + z ~
1
- - ,,>-~ E E.• ~p(,~,),~-,,_2 k>~a.+,.o E -Z
1
----Z
Z
1
+ a+h=2(n+l)
tr(ep A - ' - 2 )
tl>~ -- [
Z k~> ~n+ 1,0
2
%(2i)2.,,_ i ]
~,,(,~,)
2 )oz
,l-J *-''2 + -~- a ~ . ,
OT,,+k
OZ
(k+½)TkoT,,+~,
(41)
where
k--I~2_}_
1
Tk= k ~ t r A
4
~ - ~ ~k,l •
The remaining part contains terms proportional to Z itself which need a bit more care. First, it is easy to notice that the two items in (31 ) just cancel each other, so (31 ) gives no contribution to the final result. For (32) we have 2 (~ 02 trA3/2 0trA3/2 3,~ %(2,) 022 + ~,,,q,(2~) 02~
022k ]
(42)
OAn OA,,,]"
Using (39) and (19) it can be transformed into
i(
,,,
•
,<-,~,
j = - ~
,
i
(43)
Z ','('~') ,/L, +,,/~, i,.I
and this cancels (33) [using (38), (39) ]. Thus, the only contribution is (34), which gives 1
02C
022k
1(
01ogC
1 02C
1
I
= ~2(~,-~,) ,G,(~+4x;)- ~(,G,+,/~) Now, using (19) and (38) we obtain for (34) 286
).~(l_6,s)+~oro210gc ,{010gc~2]
L ~
*k~7-,; J
(44)
Volume 274, number 3,4
PHYSICSLETTERSB
16 January 1992
1
1
)
f6 Z, ,.(~,);,-:+12,, ,.(~,);:' ./Z,t./Z,+v%) + ( ~ , + ~ ) :
1 +~,~,~(~,~+~
.1_1 2 (P (Z/)'~IT-1 Lj, k
i
t#J
[
l
+1
+ i.,.*Z ev(2') 4,,1.,(.v~+~-)(.~i+~,/~k) 5
, ) - ~,/s,+~
E (}P(2i)/~]-2"~- 2 i
2(2,-2j)
[/•P(2i)2t"-lZJ 1 - ] - l e p ( 2 i ) 2 Z 3 / 2 2 7 3 / 2 ]
(
,
l
( ~ ; + . ~ k ) - .~j(~,+V/~k) JFI
i~j
)] (45)
2 e(R)2CLtT'/22i-'/2. i,Lj~sk
In the last equality we used the fact that
1
1
z -4-' (~+,/z j¢k 2i --2j -a •
j~k
~- ,/z, ~ + ~ ~) (~j.jl_~k)
-
,
~.](~i_,[_~k
;) -['-(j*~,'k).]
(46)
Finally, (45) can be rewritten as l t
i
ep(2,)2;-'~-2 j,k
n
~6~,~_~ 2 ~ . , ;
]
T~v,,,oj.
(47)
'
Now taking together ( 37 ), (41 ) and (47) we obtain our main result:
exp[(2/3v/3)trA3/2][treo(~--~SAz_~A)]C(x/A)exp[_(2/3x/~)trA3/2]Z{T } c(,/5)z{r} 1
=Z
2
n>~-- 1
tr(ep A - " - 2 )
(
l
~
c32
2 (k+~)Tk-~,+ a+b=n-E 1 0T~OTb+~6a.,o+~6a.+,.oTo2 k~n+l,0
)
Z{T}=0.
(48)
a>~O,b>~O
This completes the derivation.
Acknowledgement We are deeply indebted to Professor I. Singer who informed us about the talk by Witten [ 9 ] and encouraged us to complete our study of Ward identities in Kontsevich's model. We benefited a lot from illuminating discussions on the subject with E. Corrigan, A. Gerasimov, M. Kontsevich, Yu. Makeenko, A. Niemi and I. Zakharevich. We are grateful for the hospitality of the Mathematical Science Research Institute, Berkeley, CA and the Research Institute for Theoretical Physics (TFT) at Helsinski University where some pieces of this work have been done. A. Mor. is also very much indebted for the hospitality of the Department of Mathematical Sciences of Durham University. The work of A. Mar. and A. Mir. was partially supported by the Research Institute for Theoretical Physics (TFT) at Helsinki University while that of A. Mor. by the UK Science and Engineering Council through the Visiting Fellowship program and by NORDITA.
287
Volume 274, number 3,4
PHYSICS LETTERS B
References [ l ] V. Kazakov, Mod. Phys. Lett. A 4 (1989) 2125; E. Brezin and V. Kazakov, Phys. Lett. B 236 (1990) 144; M. Douglas and S. Shenker, Nucl. Phys. B 335 (1990) 635; D. Gross and A. Migdal, Phys. Rev. Lett. 64 (1990) 127. [2] M. Douglas, Phys. Lett. B 238 (1990) 176. [3] E. Witten, Nucl. Phys. B 340 (1990) 281; R. Dijkgraaf and E. Witten, Nucl. Phys. B 342 (1990) 486. [4] M. Fukuma, H. Kawai and R. Nakayama, Intern. J. Mod. Phys. A 6 (1991) 1385. [5] R. Dijkgraaf, E. Verlinde and H. Verlinde, Nucl. Phys. B 356 ( 1991 ) 574. [6 ] Yu. Makeenko et al., Nucl. Phys. B 356 ( 1991 ) 574. [7] J. Ambjorn, J. Jurkiewicz and Yu. Makeenko, Phys. Lett. B 251 (1990) 517; A. Gerasimov et al., Nucl. Phys. B 357 ( 1991 ) 565; A. Mironov and A. Morozov, Phys. Lelt. B 252 (1990) 47. [ 8] M. Kontsevich, Funct. Anal. Appl. (1990), to be published. [9 ] E. Witten, talk NYC Conf. (June 1991 ). [ 10] Yu. Makeenko and G. Semenoff, ITEP/UBC preprint (July 1991 ). [ 11 ] E. Gava and K. Narain, Phys. Lett. B 263 ( 1991 ) 213. [ 12 ] A. Marshakov, A. Mironov and A. Morozov, preprint ITEP-M-5/91, FIAN/TD/05-91.
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