- Email: [email protected]
Grassmannians, multiplicative ward identities and theta-function identities
Volume 203, number 3
PHYSICS LETTERSB
31 March 1988
GRASSMANNIANS, MULTIPLICATIVE WARD IDENTITIES AND THETA-FUNCTION IDENTITIES Siddhartha SEN School of Mathematics, Trinity College, Dublin, Ireland
and A.K. RAINA Theoretical Physics Group, TIFR, Bombay 400 005, India
Received 27 November 1987
The symmetrygroup Divo(Y) on a Riemann surface is used to derive multiplicativeWard identities. These are used to determine the propagatorfor a spin-J system and to derive Fay's trisecant identity.
Recently there has been some interest in an infinite grassmannian formulation of string theory [ 1-3 ]. These infinite grassmannians contain the moduli space of all compact Riemann surfaces of every genus and include certain "degenerate limits" e.g. infinite genus surfaces and surfaces with nodes [ 4,5 ]. They also, rather surprisingly represent the space of solutions of certain non-linear differential equations, the so-called KadomtsevPetviashvili hierarchy [4,6 ]. The connection between Riemann surfaces, conformal field theories, infinite grassmannians and the K - P hierarchy has been used to obtain interesting results [ 1,2 ]. In this paper we develop Witten's [ 3 ] idea of multiplicative Ward identities, formulated in the framework of infinite grassmannians and show how by their use we can derive the propagator for a fermionic system of arbitrary conformal spin J on an arbitrary compact Riemann surface without boundary of genus g. We will denote such a surface by Y~g.We then proceed to show how by extending the method to N-point functions leads to Fay's trisecant identity [ 7 ]. Let us start by briefly summarising Witten's approach. We consider a fermionic system described by the action Sac f ~IT)w,
(1)
where ~, ~/are sections o f K 1-2, K 2, respectively, with K the canonical line bundle on Xg. In a local coordinate system involving the complex variable t, Kcan be represented by dt. The I7) is essentially the differential operator --0/0t, where trepresents the complex conjugate of the variable t. The partition function corresponding to (1) is Z(I7)) = f ~ u ~ e x p ( - S ) .
(2)
Witten makes the observation that (2) is invariant under the transformation V---,f~v, ~---,f- ~~ ,
(3)
wherefis a holomorphic function on Eg so that [D,J] =0. Furthermore this transformation being only a coordinate change should keep Z invariant. This simple remark is the basis of the multiplicative Ward identity. For 256
0370-2693/88/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
Volume 203, number 3
PHYSICS LETTERSB
31 March 1988
Zg such a global holomorphic functionfis, of course, a constant. If, however we consider a Riemann surface with boundaries, ~, t h e n f n e e d not be a constant. The boundaries that we will consider will all be diffeomorphic to boundaries S ~ of discs on Z. In order to guarantee Z ¢ 0 suitable boundary conditions on the sections ~, have to be introduced to eliminate possible zero modes of the operator IS). On the boundary circle S ~ we will assume that both ~u and ~ belong to the Hilbert space H of square integrable functions on S ~ (in the sense that if ~u(t) oc ~ ( t ) ~ , t = exp (i0), then ~(t) is a square integrable function). However, the requirement that ~u, be chosen so as to eliminate all but a finite number of zero modes of the IS) operator means that not all subspaces of H are suitable. The suitable subspaces form an infinite grassmannian G r ( H ) . An example of such a subspace is provided by the subspace H+ of H. In terms of a local coordinate t = It[ exp(i0)~ S', H+ is described as the space spanned by the vectors ( 1, t, t 2, ...). Its orthogonal complement, using the natural inner product fo2~ ~,* (0) ~u(0) dO, is H_ which is spanned by (1/t, 1/t 2, ...). It is easy to see that boundary conditions from the subspace H_ lead to zero modes of the operator IS) in the region outside the disc whose boundary is S ~. This is because starting with elements from H_ a convergent power series expansion in 1/t for It I > 1 is obtained [ 3 ]. This leads to a function f ( t ) holomorphic outside the disc which is thus a zero mode off). A formal definition of the infinite grassmannian G r ( H ) is as follows [ 5 ]: D e f i n i t i o n I. G r ( H ) is the set of all closed subspaces W of a separable complex Hilbert space H with H = H + ~ H_, H +, H_ being two infinite dimensional orthogonal closed subspaces such that: (i) The orthogonal projection: p r : W ~ H + is a Fredholm operator (i.e. has finite dimensional kernel and cokernel) and: (ii) The orthogonal projection: pr: W ~ H _ is a Hilbert-Schmidt operator.
The grassmannian G r ( H ) is an interesting space as far as string theory is concerned for the following reasons: (I) Given a Riemann surface Z, a line bundle L which describes a conformal field on Z, then there is a map, called the Krichever map [ 8 ], which associates with this conformal field theory on Z a point W in G r ( H ) [ 1,2 ]. (II) Associated with any W e G r ( H ) there is a semi infinite form in a Fock space F [ 1,2,9]. The correspondence when W = H + is explicitly given by Vo^ v_ ~^ v_2 ^ ..., where v_~ is the vector in Fock space that corresponds to the vector in Fock space that corresponds to the vector t i in H+. (III) The partition function Z may be interpreted as a holomorphic section F of a DET* line bundle over W~ G r ( H ) [ 3-5 ]. A sketch of the reason for this is as follows. For W = H+ we first construct the semi-infinite form Vo ^ v_ ~ A V-2 A .... This form can be interpreted as a DET line bundle over W. The partition function Z is a functional of this object, hence Z is a section F of a DET* line bundle i.e. a bundle, dual to the DET line bundle. (IV) It is possible to show that there is a 1-1 correspondence between the Fock space semi-infinite forms constructed from particular points W~ G r ( H ) and holomorphic sections of the DET* line bundle that we have just described [ 4,5 ]. Thus for a compact Riemann surface Z with one boundary on which boundary conditions are chosen from W o ~ G r ( H ) we rewrite (2) as Z(Dwo)= ~ ~
~exp(-S)
(4)
Wo
to emphasize the important role played by Wo. The transformation (3) should keep Z invariant simply because (3) does not change the action S and represents only a change in the coordinates ~, ~. This invariance means j~Z(Dwo) = Z ( D f w o ) .
(5)
The left-hand side records the fact that Z(Dwo) is not a function of Wo but is rather, as we stated in (III), a holomorpbic section of a DET* line bundle over Wo. Thus findicates the lift of the action of f on Wo to the 257
Volume 203, number 3
PHYSICS LETTERSB
31 March 1988
section of the DET* bundle [ 1,3-5 ]. The right-hand side records the fact that introducingfchanges the subspace Wo to fWo. Eq. (5) presents the simplest form of Witten's multiplicative Ward identity. Witten next proceeds to consider the multiplicative Ward identities that would follow from the fact that meromorphic functions on Zg form a group: the so-called group of Principal Divisors, Divp(Y ) [ 10,11 ]. We find that in order to deal with genus g ~ 0 surfaces the interesting group is not Divp(E ) but the group of divisors of Z of degree zero, Divo (E). It is a simple matter to deal with such groups in the spirit of Witten's approach [ 3 ]. We have the following theorems [ 10]:
Theorem 1. Every divisor of degree zero is the divisor of a unique (up to a multiplicative constant) multivalued function.
Theorem 2. Every divisor of degree ( g - 1 ) is the divisor of a unique (up to a multiplicative constant) theta function [ 7 ]. In order to define a multivalued function f ( z ) , ze X g we recall that the first homology group of Y,g, H ~(£ g, Z) has 2g generators which we label Ai, Bi(i= 1..... g) [ 10]. We then have:
Definition 2. (10) A multivalued function f ( z ) , ze E g is characterized by the fact that when analytically continued round A~f(z) ~ f ( z ) z ( A D . Similarly when analytically continued round B~,f(z) --,f(z)z(B~), where z(Ai), z(Bg) are phase factors called multipliers. Note for multivalued functions z(Ai), z(Bi) are independent ofz. For theta functions they are z-dependent. This is crucial and is responsible for the difference between theorem 1 and 2. Using multivalued functions is natural, in view of theorem 1, for divisors of degree zero. Let us then proceed to study transformations of the type described by (3) but we allowfto be a multivalued meromorphic function in Zg. Let us also, for the moment, suppose that ~u, ~ are both sections of K 1/2 with ~ conjugate to ~u. Let us finally suppose that our multivalued functionfhas one zero at the point W~Eg and one pole at the point Z~52g. Clearly the transformations (3) are no longer well defined. In order to deal with this situation we follow Witten [ 3 ] and cut out discs of radii e surrounding the points z and w (we will ultimately take the limit E~ 0). On the boundaries of these two discs we impose boundary conditions from the subspace H+. We also think of the points z, w as points where the "vertex operators" equal to the identity operator (lz, lw) have been introduced. For such a system the generalized transformations (3) involving multivalued functions f i s now a symmetry. However introducing fwill obviously modify the boundary conditions given by the subspace H + to f H +. Thus near the zero o f f ( u s i n g local coordinates with t = 0 at w), H+ = (1, t, t 2, ...) is transformed to tH+ = (t, t 2, ...) i.e. the vector Vocorresponding to t o is removed. Thus the Fock space vector Vo^ v_ 1^ v_2.., is changed to v_ ~A V-2 A .... Such a change can be brought about, in the limit [tl ~ 0 , by replacing lw by ~ ( W ) . Similarly at the pole o f f again using a local coordinate t with t = 0 corresponding to the point z e £ , H+ = (1, t, t 2.... ) is transformed to (l/t)H+=(1/t, 1, t, t 2, ...). Thus, again using the correspondence between H+ and semi-infinite form VoA V_ ~A V-2 A ... we can see (1/t)H+ represents the semi-infinite form v+ t A VoA V_ ~A V--2..., i.e. an extra vector v+ ~has been introduced. Such a change can, again, be brought about, in the limit t-,0 by replacing the vertex operator (1)z by q/(z). Thus the transformation by the multivalued function f c a n be implemented at the poles and zeros o f f by introducing vertex operators qt (z), q~(w), respectively, and keeping the boundary condition H+ on the boundaries unchanged. These insertions, however, do not have the local transformation properties required i.e. they do not transform as objects of conformal spin zero. To remedy this situation Witten's prescription is to introduce the insertion, at the pole off(t):
~(z)/x/( ( d f - ' /dt)dt)t=z , while at a zero o f f ( t ) the insertion is 258
(6)
Volume 203, number 3
PHYSICS LETTERS B
31 March 1988
~( w)/x/ ( ( df/dt)dt),=w .
(7)
The multiplicative Ward identity now reads
f .@~ ~exp(-S)[~t(z)/x/((df-l/dt)dt),:z] ty(w)/x/((df/dt)dt),=w.
ff 99/~exp(-S)= (Recall ~t(z) = ~ ( z ) x / ~
(8)
to within a phase factor.) We can rewrite this equation as
(tu(z)~(w) ) = f ~ ' ~ exp(-S)p'(z)~(w)=x/((df_l/dt)dt),_ z x/((df/dt)dt),_~K(z, w) f~
~
exp(-S)
-
-
(9) '
where . gTf~qz ~ e x p ( - S ) exp(-S)
K(z, w~ = f~qz ~
(10)
"
We now proceed to construct a suitable f ( t ) and then to determine the object K(z, w), f ( t ) is easily constructed using prime forms [ 7 ]. Thus
f(t) =E(t, w)/E(z, t).
(11 )
To determine the factor K(z, w), global information regarding the sections ~u(z), ~(w) as well as the global properties of the prime forms has to be used. These are all multivalued functions. Thus a continuation of E(z, w) in z round the homology cycle Aj leads to a multiplier 1 while a continuation round a Bj cycle of Z leads to the multiplier exp(giQjj-2gi)fBjwj where wj represents a suitably normalised abelian differential of the first kind with fAjwi~ij, fB~wi=~2ij [ 10]. Finally we impose, as boundary conditions, the requirement that the multiplier for q/(z) be exp(2gio~j) (for the Aj cycle) and be e x p ( - 2 g / ~ j ) for the Bj cycle. Keeping in mind the fact that ~(z), qT(w) are sections o f K 1/2 and thus must both have degree ( g - 1 ) we see that a K(z, w) meeting all these requirements is K(z, w) = const. 0(e + z - w) [ 7 ]. By requiring the residue of (~u(z) ~(w) ) at z = w be one we finally get (assuming 0(e) ¢ 0) (~(z)~7(w)) = O(e+z-w)
O(e)
1
E(z, w) '
(12)
where ei= ~jo~j +fl~ = Y{:l :~ w~- ~ , Ai = Riemann constant and z - w = f ~ w~, which is the standard result [ 12]. Note in the approach we have followed (12) is unique because of theorems 1 and 2. This is the reason why Witten's approach works. We now turn to the case of the fields ~ts, ~ 1 - s _ sections o f K i, K l -:, respectively. In dealing with this system we have to tackle the problem of zero modes. By a simple application of the Riem a n n - R o c h theorem [ 11 ] it is easy to see that ( 2 J - 1 ) ( g - 1 ) holomorphic sections Ms exist (for J > 1 ). These all represent zero modes of the operator I) hence, unless they are eliminated the partition function Z for this system will be zero. A convenient way to eliminate the zero modes is to introduce points Zl.... , zK= ( 2 J - 1) ( g - 1 ) and consider 2(D) = f ~q/~
e x p ( - S ) (I~I i=~ ~u(zi)),
(13)
Because of the fermionic nature of the variables involved the factor l-[ k= ~~(Z~) acts effectively as a projection operator onto the space orthogonal to the space of zero modes of the operator I). Note that 2~(D), because of the fermionic nature of the variables, is invariant modulo phases under the generalized transformations of the type described by eq. (3). Introducing as for the spin-½ case multivalued functionsf(t) with a zero at weZg and a pole at z~Y~ we can write
f(t) =E(t, w)/E(z, t).
(14) 259
Volume 203, n u m b e r 3
PHYSICS LETTERS B
31 March 1988
Again, removing discs of radii e surrounding the points z and WeEg and introducing H+ boundary conditions on the boundaries of these discs leads to vertex insertions (in the limit ~-,0) proportional to ~/J(z) and ~ - J ( w ) , respectively. These insertions, as for the spin-½ case, have to be suitably normalised in order for them to transform as conformal spin-zero objects. This is achieved by noting that factors 1/x/((df -1/dt) dt),=z for ~S(z) and 1/x/((df/dt)dt)t=w for ~ l - J ( w ) leave objects transforming as ~]J-1/2(z), ~l/2-J(w), respectively. Thus the remaining problem is to construct some natural object ZJ- 1/2(t) which fits the bill. To construct ZJ- 1/2(t) we observe that the presence of 1-[~=1~](zi) means that whenever z equals one of the zi of this product the propagator must vanish. These ( 2 J - 1 ) ( g - 1 ) zeros are also the number of zeros an object of the type ZJ- 1/2(t) must have on the basis of topological arguments, i.e. degree Z~- 1/2= ( j _ ½) deg K = ( 2 J - 1 ) ( g - 1 ). Thus we consider
Zj_,/z(t) =
]-I~=lE(t,z~)
[ I-[~-11E(t,
[h(t)]z,_,
(15)
Qt)] 2s-1
The factors in (15 ) are easily explained. The prime forms in the numerator represent the vanishing of x s- 1/2(t) at the points t = z~, i = 1,..., k. The prime forms in the denominator are equal in number to those in the numerator and guarantee that the ratio represents a multivalued meromorphie function on Zg. The apparent poles ofz J- 1/2(t) at t = Q~, ..., Qg_ 1 are cancelled by the zeros of/~(t) which are chosen to be at the same set of points t = Q1, ... • Moreover, h (t) is a half order differential so that [ h (t)] 2J-1 represents a KJ- l/Z-like object. Thus the insertion at the pole off(t) at t=z is
[~'S(z)/x/((df -l /dt)dt),=zl/zS-'/2(z) ,
(16)
while at the zero off(t) at t = w, the insertion is
[~1-J( w)/x/((df/dt)dt) t=~]ff-1/2( w).
(17)
Using these insertions and proceeding as for the spin-½ case, the propagator is found to be ( J > 1 )
O(e+z-w) 1 ,2,-1),g-1, E(z, zi) (~_11E(w, Q,) ~2J-1( h(z) ~2J-1 O(e) E(z, w) ,~__~ E(w,z,) --E--~,-~)] \ff(--~.] '
(~J(z)~l-S(w) ) -
(18)
which is the standard result [ 13 ]. The theta function appears, as in the spin-½ case, by matching phases of multipliers round A,, B~ cycles, etc. In (18)
t~(t)=( ~= w~(t) ~O[~](Olg2))l/Zexp(2zfiot~ i w~),
(19)
to
with e~=g2o%+fl~=
wj-(2J-1)Ai, 0
~
k=l
(01~)¢0,
PO
wi.
(z-w)~= w
Again because of theorems 1 and 2, (18) is unique. With the help of these results it is a simple matter to derive various theta function identities. Let us first consider the spin-½ case. Consider the 2N-point function q/(zi)
j=l
~(wj)
.
(20)
There are two ways of evaluating this object. First by using Wick's theorem and (12). This gives
(~ I --fO(e+zi-wj ) 1 ) i~=l~'l(Zi)Jf=i l ~(Wj) =oet~ ~ g(z:i, w5 .
(21)
Alternatively we could determine (17) by using the multiplicative Ward identity approach directly. In that 260
Volume 203, number 3
PHYSICS LETTERS B
31 March 1988
approach we start with a multivalued m e r o m o r p h i c function f ( t ) with N zeros at t = wb w2, ... and N poles at t=z~, z2 ..., i.e.f(t) = 1-[N=~E(t, wi)/E(zi, t) and proceeding as for the case when N = 1 we get
-]i
(22)
where the 0-function term comes from the requirements. Balance phases coming from the prime forms and the phase coming from the sections ~t(z), ~(w). The fact that the sections q/(z), ~(w) have degree ( g - 1 ). Residue at (E z~= E w~) is one. The result obtained is, again, unique because of theorems 1 and 2. Equating these two expressions gives us the Fay identity [ 7,14 ] -
d .{O(e+z~-wj) et~
0~
1 "~ O(e+ Zizi-E,wi) ]-[~
(23)
Following the same procedure the conformal spin J, 1 - J system leads to the identity
d.FO(e+zi-wj) et L
0-~
1 (2J-l)(g-l)E(zi,dk) (g~=llE(Wj,Ql,)2J-l {~l(Zi))2J-I] E(z,, wj)
k~__~ E(wj, ak)
E(zi, Q,) J
\h-~j)J
O(e+ Y.izi- Y~iw,) I-[i,jE(zi, zj)E(w,, wj) ,2J-l),g-l) E(zi, ak) ;gl-i1E(wi, Qt) ]zJ-, ; h(zi) ~2J_1 = O(e) ]-LaE(z,, wj) k~-_l E(Wj, ak) \]=~ E(z,, a,) J kh-~j)J " (24) As a check on this identity we consider the spatial case k = 1, 2; i = 1, 2 ; j = 1,2; 1= 1. We get
O(e+z~-w,) O(e+z2-w2) E(z~, w2)E(z2, w l ) O(e) O(e) = O(e+zl +z2-wl -w2) E(z,, z2)E(wl, O(e)
O(e+zl-w2) O(e+zz-wj) O(e)
O(e)
E(z,, w,)E(z2, w2)
w2) .
(25)
The complicated factors involving h ( t ) , etc., cancel from both sides, leaving the well-known Fay trisecant identity [ 7 ]. Thus no new theta function identities emerge by looking at the conformal spin J, 1 - J system. From these examples it should be clear that the multiplicative Ward identities are useful objects.
S.S. would like to thank the Theory G r o u p T I F R for their hospitality where this work was started and A.R. would like to thank the School of Mathematics, Trinity College Dublin and the Institute for Advanced Studies in Dublin for their hospitality while this work was being completed.
R e f e r e n c e s
[ 1] L. Alvarez-Gaum6, C. Gomez and G. Reina, Phys. Lett. B 190 (1987) 55; preprint CERN-4775 (1987); C. Vafa, Phys. Lett. B 190 (1987) 47. [2] N. Ishibashi, Y. Matsuo and H. Ooguri, Mod. Phys. Lett. A 2 (1987) 119; S. Saito, Phys. Rev. D 36 (1987) 1819. [3] E. Witten, Princeton University preprint PUPT-1057 (1987). [4] G.B. Segal and G. Wilson, Pub. Math. I.H.E.S., 61 (1985) 5. [5] A. Pressley and G.B. Segal, Loop groups (Oxford U.P., Oxford, 1986). 261
Volume 203, number 3
PHYSICS LETTERS B
31 March 1988
[6] E. Date, M. Jimbo, M. Kashiwara and T, Miwa, J. Phys. Soc. Japan 50 (1981) 3806. [ 7 ] J.D. Fay, Theta functions on Riemann surfaces, Lecture Notes in Mathematics, Vol. 352 (Springer, Berlin, 1973 ); D. Mumford, Tata Lectures in Theta, Vols. I, II (Birkh~iuser, Basel, 1984). [8] I.M. Krichever, Russ. Math. Surv. 32 (1977) 185. [9] V. Kac and A.K. Raina, Lectures on Highest weight representations of infinite dimensional Lie algebras (World Scientific, Singapore, 1987). [ 10] H.M. Farkas and I. Kra, Riemann surfaces (Springer, Berlin, 1980). [ 11 ] O. Forster, Lectures on Riemann surfaces (Springer, Berlin, 1980). [ 12] M.A. Namazie, K.S. Narain and M.H. Sarmadi, Phys. Lett. B 178 (1986) 329; H. Sonoda, Phys. Lett. B 178 (1986) 390. [ 13] M. Dugan and H. Sonoda, Nucl. Phys. B 289 (1987) 227. [ 14] J.B. Bost and P. Nelson, Phys. Rev. Lett. 57 (1986) 795; L. Alvarez-Gaume, J.B. Bost, G. Moore, P. Nelson and C. Vafa, Phys. Lett. B 178 (1986) 41; T. Eguchi and H. Ooguri, Phys. Lett. B 187 (1987) 127.
262
PHYSICS LETTERSB
31 March 1988
GRASSMANNIANS, MULTIPLICATIVE WARD IDENTITIES AND THETA-FUNCTION IDENTITIES Siddhartha SEN School of Mathematics, Trinity College, Dublin, Ireland
and A.K. RAINA Theoretical Physics Group, TIFR, Bombay 400 005, India
Received 27 November 1987
The symmetrygroup Divo(Y) on a Riemann surface is used to derive multiplicativeWard identities. These are used to determine the propagatorfor a spin-J system and to derive Fay's trisecant identity.
Recently there has been some interest in an infinite grassmannian formulation of string theory [ 1-3 ]. These infinite grassmannians contain the moduli space of all compact Riemann surfaces of every genus and include certain "degenerate limits" e.g. infinite genus surfaces and surfaces with nodes [ 4,5 ]. They also, rather surprisingly represent the space of solutions of certain non-linear differential equations, the so-called KadomtsevPetviashvili hierarchy [4,6 ]. The connection between Riemann surfaces, conformal field theories, infinite grassmannians and the K - P hierarchy has been used to obtain interesting results [ 1,2 ]. In this paper we develop Witten's [ 3 ] idea of multiplicative Ward identities, formulated in the framework of infinite grassmannians and show how by their use we can derive the propagator for a fermionic system of arbitrary conformal spin J on an arbitrary compact Riemann surface without boundary of genus g. We will denote such a surface by Y~g.We then proceed to show how by extending the method to N-point functions leads to Fay's trisecant identity [ 7 ]. Let us start by briefly summarising Witten's approach. We consider a fermionic system described by the action Sac f ~IT)w,
(1)
where ~, ~/are sections o f K 1-2, K 2, respectively, with K the canonical line bundle on Xg. In a local coordinate system involving the complex variable t, Kcan be represented by dt. The I7) is essentially the differential operator --0/0t, where trepresents the complex conjugate of the variable t. The partition function corresponding to (1) is Z(I7)) = f ~ u ~ e x p ( - S ) .
(2)
Witten makes the observation that (2) is invariant under the transformation V---,f~v, ~---,f- ~~ ,
(3)
wherefis a holomorphic function on Eg so that [D,J] =0. Furthermore this transformation being only a coordinate change should keep Z invariant. This simple remark is the basis of the multiplicative Ward identity. For 256
0370-2693/88/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
Volume 203, number 3
PHYSICS LETTERSB
31 March 1988
Zg such a global holomorphic functionfis, of course, a constant. If, however we consider a Riemann surface with boundaries, ~, t h e n f n e e d not be a constant. The boundaries that we will consider will all be diffeomorphic to boundaries S ~ of discs on Z. In order to guarantee Z ¢ 0 suitable boundary conditions on the sections ~, have to be introduced to eliminate possible zero modes of the operator IS). On the boundary circle S ~ we will assume that both ~u and ~ belong to the Hilbert space H of square integrable functions on S ~ (in the sense that if ~u(t) oc ~ ( t ) ~ , t = exp (i0), then ~(t) is a square integrable function). However, the requirement that ~u, be chosen so as to eliminate all but a finite number of zero modes of the IS) operator means that not all subspaces of H are suitable. The suitable subspaces form an infinite grassmannian G r ( H ) . An example of such a subspace is provided by the subspace H+ of H. In terms of a local coordinate t = It[ exp(i0)~ S', H+ is described as the space spanned by the vectors ( 1, t, t 2, ...). Its orthogonal complement, using the natural inner product fo2~ ~,* (0) ~u(0) dO, is H_ which is spanned by (1/t, 1/t 2, ...). It is easy to see that boundary conditions from the subspace H_ lead to zero modes of the operator IS) in the region outside the disc whose boundary is S ~. This is because starting with elements from H_ a convergent power series expansion in 1/t for It I > 1 is obtained [ 3 ]. This leads to a function f ( t ) holomorphic outside the disc which is thus a zero mode off). A formal definition of the infinite grassmannian G r ( H ) is as follows [ 5 ]: D e f i n i t i o n I. G r ( H ) is the set of all closed subspaces W of a separable complex Hilbert space H with H = H + ~ H_, H +, H_ being two infinite dimensional orthogonal closed subspaces such that: (i) The orthogonal projection: p r : W ~ H + is a Fredholm operator (i.e. has finite dimensional kernel and cokernel) and: (ii) The orthogonal projection: pr: W ~ H _ is a Hilbert-Schmidt operator.
The grassmannian G r ( H ) is an interesting space as far as string theory is concerned for the following reasons: (I) Given a Riemann surface Z, a line bundle L which describes a conformal field on Z, then there is a map, called the Krichever map [ 8 ], which associates with this conformal field theory on Z a point W in G r ( H ) [ 1,2 ]. (II) Associated with any W e G r ( H ) there is a semi infinite form in a Fock space F [ 1,2,9]. The correspondence when W = H + is explicitly given by Vo^ v_ ~^ v_2 ^ ..., where v_~ is the vector in Fock space that corresponds to the vector in Fock space that corresponds to the vector t i in H+. (III) The partition function Z may be interpreted as a holomorphic section F of a DET* line bundle over W~ G r ( H ) [ 3-5 ]. A sketch of the reason for this is as follows. For W = H+ we first construct the semi-infinite form Vo ^ v_ ~ A V-2 A .... This form can be interpreted as a DET line bundle over W. The partition function Z is a functional of this object, hence Z is a section F of a DET* line bundle i.e. a bundle, dual to the DET line bundle. (IV) It is possible to show that there is a 1-1 correspondence between the Fock space semi-infinite forms constructed from particular points W~ G r ( H ) and holomorphic sections of the DET* line bundle that we have just described [ 4,5 ]. Thus for a compact Riemann surface Z with one boundary on which boundary conditions are chosen from W o ~ G r ( H ) we rewrite (2) as Z(Dwo)= ~ ~
~exp(-S)
(4)
Wo
to emphasize the important role played by Wo. The transformation (3) should keep Z invariant simply because (3) does not change the action S and represents only a change in the coordinates ~, ~. This invariance means j~Z(Dwo) = Z ( D f w o ) .
(5)
The left-hand side records the fact that Z(Dwo) is not a function of Wo but is rather, as we stated in (III), a holomorpbic section of a DET* line bundle over Wo. Thus findicates the lift of the action of f on Wo to the 257
Volume 203, number 3
PHYSICS LETTERSB
31 March 1988
section of the DET* bundle [ 1,3-5 ]. The right-hand side records the fact that introducingfchanges the subspace Wo to fWo. Eq. (5) presents the simplest form of Witten's multiplicative Ward identity. Witten next proceeds to consider the multiplicative Ward identities that would follow from the fact that meromorphic functions on Zg form a group: the so-called group of Principal Divisors, Divp(Y ) [ 10,11 ]. We find that in order to deal with genus g ~ 0 surfaces the interesting group is not Divp(E ) but the group of divisors of Z of degree zero, Divo (E). It is a simple matter to deal with such groups in the spirit of Witten's approach [ 3 ]. We have the following theorems [ 10]:
Theorem 1. Every divisor of degree zero is the divisor of a unique (up to a multiplicative constant) multivalued function.
Theorem 2. Every divisor of degree ( g - 1 ) is the divisor of a unique (up to a multiplicative constant) theta function [ 7 ]. In order to define a multivalued function f ( z ) , ze X g we recall that the first homology group of Y,g, H ~(£ g, Z) has 2g generators which we label Ai, Bi(i= 1..... g) [ 10]. We then have:
Definition 2. (10) A multivalued function f ( z ) , ze E g is characterized by the fact that when analytically continued round A~f(z) ~ f ( z ) z ( A D . Similarly when analytically continued round B~,f(z) --,f(z)z(B~), where z(Ai), z(Bg) are phase factors called multipliers. Note for multivalued functions z(Ai), z(Bi) are independent ofz. For theta functions they are z-dependent. This is crucial and is responsible for the difference between theorem 1 and 2. Using multivalued functions is natural, in view of theorem 1, for divisors of degree zero. Let us then proceed to study transformations of the type described by (3) but we allowfto be a multivalued meromorphic function in Zg. Let us also, for the moment, suppose that ~u, ~ are both sections of K 1/2 with ~ conjugate to ~u. Let us finally suppose that our multivalued functionfhas one zero at the point W~Eg and one pole at the point Z~52g. Clearly the transformations (3) are no longer well defined. In order to deal with this situation we follow Witten [ 3 ] and cut out discs of radii e surrounding the points z and w (we will ultimately take the limit E~ 0). On the boundaries of these two discs we impose boundary conditions from the subspace H+. We also think of the points z, w as points where the "vertex operators" equal to the identity operator (lz, lw) have been introduced. For such a system the generalized transformations (3) involving multivalued functions f i s now a symmetry. However introducing fwill obviously modify the boundary conditions given by the subspace H + to f H +. Thus near the zero o f f ( u s i n g local coordinates with t = 0 at w), H+ = (1, t, t 2, ...) is transformed to tH+ = (t, t 2, ...) i.e. the vector Vocorresponding to t o is removed. Thus the Fock space vector Vo^ v_ 1^ v_2.., is changed to v_ ~A V-2 A .... Such a change can be brought about, in the limit [tl ~ 0 , by replacing lw by ~ ( W ) . Similarly at the pole o f f again using a local coordinate t with t = 0 corresponding to the point z e £ , H+ = (1, t, t 2.... ) is transformed to (l/t)H+=(1/t, 1, t, t 2, ...). Thus, again using the correspondence between H+ and semi-infinite form VoA V_ ~A V-2 A ... we can see (1/t)H+ represents the semi-infinite form v+ t A VoA V_ ~A V--2..., i.e. an extra vector v+ ~has been introduced. Such a change can, again, be brought about, in the limit t-,0 by replacing the vertex operator (1)z by q/(z). Thus the transformation by the multivalued function f c a n be implemented at the poles and zeros o f f by introducing vertex operators qt (z), q~(w), respectively, and keeping the boundary condition H+ on the boundaries unchanged. These insertions, however, do not have the local transformation properties required i.e. they do not transform as objects of conformal spin zero. To remedy this situation Witten's prescription is to introduce the insertion, at the pole off(t):
~(z)/x/( ( d f - ' /dt)dt)t=z , while at a zero o f f ( t ) the insertion is 258
(6)
Volume 203, number 3
PHYSICS LETTERS B
31 March 1988
~( w)/x/ ( ( df/dt)dt),=w .
(7)
The multiplicative Ward identity now reads
f .@~ ~exp(-S)[~t(z)/x/((df-l/dt)dt),:z] ty(w)/x/((df/dt)dt),=w.
ff 99/~exp(-S)= (Recall ~t(z) = ~ ( z ) x / ~
(8)
to within a phase factor.) We can rewrite this equation as
(tu(z)~(w) ) = f ~ ' ~ exp(-S)p'(z)~(w)=x/((df_l/dt)dt),_ z x/((df/dt)dt),_~K(z, w) f~
~
exp(-S)
-
-
(9) '
where . gTf~qz ~ e x p ( - S ) exp(-S)
K(z, w~ = f~qz ~
(10)
"
We now proceed to construct a suitable f ( t ) and then to determine the object K(z, w), f ( t ) is easily constructed using prime forms [ 7 ]. Thus
f(t) =E(t, w)/E(z, t).
(11 )
To determine the factor K(z, w), global information regarding the sections ~u(z), ~(w) as well as the global properties of the prime forms has to be used. These are all multivalued functions. Thus a continuation of E(z, w) in z round the homology cycle Aj leads to a multiplier 1 while a continuation round a Bj cycle of Z leads to the multiplier exp(giQjj-2gi)fBjwj where wj represents a suitably normalised abelian differential of the first kind with fAjwi~ij, fB~wi=~2ij [ 10]. Finally we impose, as boundary conditions, the requirement that the multiplier for q/(z) be exp(2gio~j) (for the Aj cycle) and be e x p ( - 2 g / ~ j ) for the Bj cycle. Keeping in mind the fact that ~(z), qT(w) are sections o f K 1/2 and thus must both have degree ( g - 1 ) we see that a K(z, w) meeting all these requirements is K(z, w) = const. 0(e + z - w) [ 7 ]. By requiring the residue of (~u(z) ~(w) ) at z = w be one we finally get (assuming 0(e) ¢ 0) (~(z)~7(w)) = O(e+z-w)
O(e)
1
E(z, w) '
(12)
where ei= ~jo~j +fl~ = Y{:l :~ w~- ~ , Ai = Riemann constant and z - w = f ~ w~, which is the standard result [ 12]. Note in the approach we have followed (12) is unique because of theorems 1 and 2. This is the reason why Witten's approach works. We now turn to the case of the fields ~ts, ~ 1 - s _ sections o f K i, K l -:, respectively. In dealing with this system we have to tackle the problem of zero modes. By a simple application of the Riem a n n - R o c h theorem [ 11 ] it is easy to see that ( 2 J - 1 ) ( g - 1 ) holomorphic sections Ms exist (for J > 1 ). These all represent zero modes of the operator I) hence, unless they are eliminated the partition function Z for this system will be zero. A convenient way to eliminate the zero modes is to introduce points Zl.... , zK= ( 2 J - 1) ( g - 1 ) and consider 2(D) = f ~q/~
e x p ( - S ) (I~I i=~ ~u(zi)),
(13)
Because of the fermionic nature of the variables involved the factor l-[ k= ~~(Z~) acts effectively as a projection operator onto the space orthogonal to the space of zero modes of the operator I). Note that 2~(D), because of the fermionic nature of the variables, is invariant modulo phases under the generalized transformations of the type described by eq. (3). Introducing as for the spin-½ case multivalued functionsf(t) with a zero at weZg and a pole at z~Y~ we can write
f(t) =E(t, w)/E(z, t).
(14) 259
Volume 203, n u m b e r 3
PHYSICS LETTERS B
31 March 1988
Again, removing discs of radii e surrounding the points z and WeEg and introducing H+ boundary conditions on the boundaries of these discs leads to vertex insertions (in the limit ~-,0) proportional to ~/J(z) and ~ - J ( w ) , respectively. These insertions, as for the spin-½ case, have to be suitably normalised in order for them to transform as conformal spin-zero objects. This is achieved by noting that factors 1/x/((df -1/dt) dt),=z for ~S(z) and 1/x/((df/dt)dt)t=w for ~ l - J ( w ) leave objects transforming as ~]J-1/2(z), ~l/2-J(w), respectively. Thus the remaining problem is to construct some natural object ZJ- 1/2(t) which fits the bill. To construct ZJ- 1/2(t) we observe that the presence of 1-[~=1~](zi) means that whenever z equals one of the zi of this product the propagator must vanish. These ( 2 J - 1 ) ( g - 1 ) zeros are also the number of zeros an object of the type ZJ- 1/2(t) must have on the basis of topological arguments, i.e. degree Z~- 1/2= ( j _ ½) deg K = ( 2 J - 1 ) ( g - 1 ). Thus we consider
Zj_,/z(t) =
]-I~=lE(t,z~)
[ I-[~-11E(t,
[h(t)]z,_,
(15)
Qt)] 2s-1
The factors in (15 ) are easily explained. The prime forms in the numerator represent the vanishing of x s- 1/2(t) at the points t = z~, i = 1,..., k. The prime forms in the denominator are equal in number to those in the numerator and guarantee that the ratio represents a multivalued meromorphie function on Zg. The apparent poles ofz J- 1/2(t) at t = Q~, ..., Qg_ 1 are cancelled by the zeros of/~(t) which are chosen to be at the same set of points t = Q1, ... • Moreover, h (t) is a half order differential so that [ h (t)] 2J-1 represents a KJ- l/Z-like object. Thus the insertion at the pole off(t) at t=z is
[~'S(z)/x/((df -l /dt)dt),=zl/zS-'/2(z) ,
(16)
while at the zero off(t) at t = w, the insertion is
[~1-J( w)/x/((df/dt)dt) t=~]ff-1/2( w).
(17)
Using these insertions and proceeding as for the spin-½ case, the propagator is found to be ( J > 1 )
O(e+z-w) 1 ,2,-1),g-1, E(z, zi) (~_11E(w, Q,) ~2J-1( h(z) ~2J-1 O(e) E(z, w) ,~__~ E(w,z,) --E--~,-~)] \ff(--~.] '
(~J(z)~l-S(w) ) -
(18)
which is the standard result [ 13 ]. The theta function appears, as in the spin-½ case, by matching phases of multipliers round A,, B~ cycles, etc. In (18)
t~(t)=( ~= w~(t) ~O[~](Olg2))l/Zexp(2zfiot~ i w~),
(19)
to
with e~=g2o%+fl~=
wj-(2J-1)Ai, 0
~
k=l
(01~)¢0,
PO
wi.
(z-w)~= w
Again because of theorems 1 and 2, (18) is unique. With the help of these results it is a simple matter to derive various theta function identities. Let us first consider the spin-½ case. Consider the 2N-point function q/(zi)
j=l
~(wj)
.
(20)
There are two ways of evaluating this object. First by using Wick's theorem and (12). This gives
(~ I --fO(e+zi-wj ) 1 ) i~=l~'l(Zi)Jf=i l ~(Wj) =oet~ ~ g(z:i, w5 .
(21)
Alternatively we could determine (17) by using the multiplicative Ward identity approach directly. In that 260
Volume 203, number 3
PHYSICS LETTERS B
31 March 1988
approach we start with a multivalued m e r o m o r p h i c function f ( t ) with N zeros at t = wb w2, ... and N poles at t=z~, z2 ..., i.e.f(t) = 1-[N=~E(t, wi)/E(zi, t) and proceeding as for the case when N = 1 we get
-]i
(22)
where the 0-function term comes from the requirements. Balance phases coming from the prime forms and the phase coming from the sections ~t(z), ~(w). The fact that the sections q/(z), ~(w) have degree ( g - 1 ). Residue at (E z~= E w~) is one. The result obtained is, again, unique because of theorems 1 and 2. Equating these two expressions gives us the Fay identity [ 7,14 ] -
d .{O(e+z~-wj) et~
0~
1 "~ O(e+ Zizi-E,wi) ]-[~
(23)
Following the same procedure the conformal spin J, 1 - J system leads to the identity
d.FO(e+zi-wj) et L
0-~
1 (2J-l)(g-l)E(zi,dk) (g~=llE(Wj,Ql,)2J-l {~l(Zi))2J-I] E(z,, wj)
k~__~ E(wj, ak)
E(zi, Q,) J
\h-~j)J
O(e+ Y.izi- Y~iw,) I-[i,jE(zi, zj)E(w,, wj) ,2J-l),g-l) E(zi, ak) ;gl-i1E(wi, Qt) ]zJ-, ; h(zi) ~2J_1 = O(e) ]-LaE(z,, wj) k~-_l E(Wj, ak) \]=~ E(z,, a,) J kh-~j)J " (24) As a check on this identity we consider the spatial case k = 1, 2; i = 1, 2 ; j = 1,2; 1= 1. We get
O(e+z~-w,) O(e+z2-w2) E(z~, w2)E(z2, w l ) O(e) O(e) = O(e+zl +z2-wl -w2) E(z,, z2)E(wl, O(e)
O(e+zl-w2) O(e+zz-wj) O(e)
O(e)
E(z,, w,)E(z2, w2)
w2) .
(25)
The complicated factors involving h ( t ) , etc., cancel from both sides, leaving the well-known Fay trisecant identity [ 7 ]. Thus no new theta function identities emerge by looking at the conformal spin J, 1 - J system. From these examples it should be clear that the multiplicative Ward identities are useful objects.
S.S. would like to thank the Theory G r o u p T I F R for their hospitality where this work was started and A.R. would like to thank the School of Mathematics, Trinity College Dublin and the Institute for Advanced Studies in Dublin for their hospitality while this work was being completed.
R e f e r e n c e s
[ 1] L. Alvarez-Gaum6, C. Gomez and G. Reina, Phys. Lett. B 190 (1987) 55; preprint CERN-4775 (1987); C. Vafa, Phys. Lett. B 190 (1987) 47. [2] N. Ishibashi, Y. Matsuo and H. Ooguri, Mod. Phys. Lett. A 2 (1987) 119; S. Saito, Phys. Rev. D 36 (1987) 1819. [3] E. Witten, Princeton University preprint PUPT-1057 (1987). [4] G.B. Segal and G. Wilson, Pub. Math. I.H.E.S., 61 (1985) 5. [5] A. Pressley and G.B. Segal, Loop groups (Oxford U.P., Oxford, 1986). 261
Volume 203, number 3
PHYSICS LETTERS B
31 March 1988
[6] E. Date, M. Jimbo, M. Kashiwara and T, Miwa, J. Phys. Soc. Japan 50 (1981) 3806. [ 7 ] J.D. Fay, Theta functions on Riemann surfaces, Lecture Notes in Mathematics, Vol. 352 (Springer, Berlin, 1973 ); D. Mumford, Tata Lectures in Theta, Vols. I, II (Birkh~iuser, Basel, 1984). [8] I.M. Krichever, Russ. Math. Surv. 32 (1977) 185. [9] V. Kac and A.K. Raina, Lectures on Highest weight representations of infinite dimensional Lie algebras (World Scientific, Singapore, 1987). [ 10] H.M. Farkas and I. Kra, Riemann surfaces (Springer, Berlin, 1980). [ 11 ] O. Forster, Lectures on Riemann surfaces (Springer, Berlin, 1980). [ 12] M.A. Namazie, K.S. Narain and M.H. Sarmadi, Phys. Lett. B 178 (1986) 329; H. Sonoda, Phys. Lett. B 178 (1986) 390. [ 13] M. Dugan and H. Sonoda, Nucl. Phys. B 289 (1987) 227. [ 14] J.B. Bost and P. Nelson, Phys. Rev. Lett. 57 (1986) 795; L. Alvarez-Gaume, J.B. Bost, G. Moore, P. Nelson and C. Vafa, Phys. Lett. B 178 (1986) 41; T. Eguchi and H. Ooguri, Phys. Lett. B 187 (1987) 127.
262