determinants of cauchy-riemann operators as τ-functions

Acta Applicandae Mathematicae 18: 199-223, 1990. 199 © 1990 Kluwer Academic Publishers. Printed in the Netherlands.

Determinants of Cauchy-Riemann Operators

as z-Functions

J O H N P A L M E R Mathematics Department, University o f Arizona, Tucson, AZ 85721, U.S.A.

(Received: 20 May 1989; accepted: 19 December 1989)

Abstract. The z-functions introduced by Sato, Miwa, and Jimbo for the deformation theory associated with the Riemann-Hilbert problem on p1 is shown to be a determinant for a singular Cauchy-Riemann operator whose domain incorporates functions with prescribed branching behavior.The analysis relies heavily on previous work of Malgrange on monodromy preserving deformation theory.

AMS subject classification (1980). 35Q 15.

Kew words. Cauchy-Riemann operators, z-functions.

0. Introduction

The object of this paper is to give an interpretation of the z-functions introduced by Sato, Miwa, and Jimbo for monodromy preserving deformations of the Cauchy- Riemann equations on p1 [10, II] as the determinant of an associated Cauchy- Riemann operator on the spin bundle over pl. The monodromy in the problem is now reflected by prescribed branch discontinuities in the domain of the associated 'Cauchy-Riemann' operator. This point of view is at least implicit in the fifth paper [10, V] on Holonomic quantum fields and it is also implicit in the notion of vertex insertions in conformal field theory [2].

One motivation for our desire to reinterpret the z-function as the determinant of a Cauchy-Riemann operator is that there are natural situations in which one has a 'z- function' but there is no clear analogue of the linear differential equations that are the arena for deformation theory. An example is provided by the correlations of the two- dimensional Ising model on a lattice. These correlation functions are naturally determinants (more precisely Pfaitians) of certain inhomogeneous difference operators on 12(Z 2) [-7] but it is not terribly natural to formulate monodromy preserving deformation theory on the lattice. On the other hand, it was the discovery that a certain continuum scaling limit of the two point Ising correlation could be expressed in terms of Painlev6 transcendents by Wu, McCoy, Trace, and Barouch in [-14] that blossomed into a general theory of monodromy preserving deformations and associated z-functions in the series of papers [10, I-V]. In this theory, the scaled correlations of the Ising model are the prototypical example of a z-function. In

200 JOHN PALMER

studying the scaling limit of the two-dimensional Ising model it is useful to have a picture in which the nature of the T-function does not change as one passes from the lattice to the continuum.

In [4], Malgrange gives a very elegant geometric interpretation for the z-function that arises for monodromy preserving deformations of the Cauchy-Riemann equations as part of his analysis of the deformation equations themselves. Another reason for our interest in reinterpreting the z-function comes from the desire to generalize this work of Malgrange to other settings. It was not immediately clear to us from the work of Sato, Miwa, and Jimbo or Malgrange what the appropriate definition of a ~-function should be for a Riemann surface different from p1. This paper does not answer this question but I believe it does at least give a clear formulation of the problem to be solved.

There is a also a difficulty in generalizing the Malgrange analysis to the Euclidean Dirac equation. In Malgrange the ~-function is understood to define a divisor (in the 'space of branch points') where an auxilliary family of holomorphic line bundles on p1 is holomorphically nontrivial. There are no holomorphic bundles evident in the Euclidean Dirac problem but the notion that the ~-function is the determinant of the Euclidean Dirac operator with a domain incorporating specified branching does carry over. We hope to return to the problem of generalizing the Malgrange analysis to the Euclidean Dirac equation in another place.

This paper is organized in four sections. The first section deals with determinant bundles for Cauchy-Riemann operators on the spin bundle on P1 following the ideas of Quillen [9]. A localization of the determinant bundles is introduced following some ideas in Witten [13] and this is shown to lead to a mapping into the det* bundle on the boundary of the localization as defined by Segal and Wilson [11]. Although our main interest in this paper is in singular Cauchy-Riemann operators, for simplicity we treat only smooth operators in this first section. The reader should be aware that in referring to singular Cauchy-Riemann operators we step outside the bounds of convention. In as much as Cauchy-Riemann operators define complex structures on C °~ vector bundles they are always smooth operators. Still, we believe that the reason for referring to the operators that we introduce as singular Cauchy-Riemann operators will be apparent to the reader as soon as they are introduced. Our strategy in dealing with these singular Cauchy-Riemann operators will be to localize them away from their singularities.

The second section presents an account of a connection in a group of bundle automorphism of the det* bundle that will be used in trivializing the determinant bundle we are interested in. The connection one form of this connection relative to the canonical section is the regularization of the logarithmic derivative of the determinant of a Toeplitz operator introduced by Malgrange [4].

The third section introduces the singular Cauchy-Riemann operators of interest to us and reviews their relation to the Riemann-Hilbert problem. We also begin the translation of the Malgrange analysis into this setting.

The fourth section is largely devoted to connecting the analysis of the appropriate

DETERMINANTS OF CAUCHY-RIEMANN OPERATORS AS z-FUNCTIONS 201

det* bundle with the analysis in Malgrange. One feature of this work that complicates

matters is that we are not able to directly localize to a neighborhood of the branch

points as is done in Malgrange. To do so directly would mean for us the sacrifice of

leaving the smooth Grassmannian. Nevertheless, after the appropriate incantations we do end up back in the neighborhood of the branch points and from that point on we are able to follow Malgrange in detail. Our final result is that the z-function of Sato, Miwa, and Jimbo is the determinant of a singular Cauchy-Riemann operator. This determinant is obtained by comparing the canonical section of the determinant bundle with a local trivialization that arises from a flat connection.

1. Cauchy-Riemann Operators on the Spin Bundle over p1

We are interested in the determinant bundle over the space of Cauchy-Riemann operators on the spin bundle over p1. The reason for considering the spin bundle is

that the Cauchy-Riemann operators on this bundle all have index 0 and this makes for a minimum amount of fuss in the discussion of determinants of such operators.

The fact about the index is a consequence of the Riemann-Roch theorem although for this special case it is not hard to check directly.

We will start with a description of the spin bundle over p1. Choose a positive

number e between 0 and 1 and define:

O, = {zeC:lzl < 1 + e} and D'~ = {z~C:lz[ > 1 - e} u {~}

(we think of W as C w {oo} in the usual fashion). We also write D for the closed unit disk and D' for the complement of the open unit disk in px. Let D e x C" and D', x C"

denote the trivial bundles over De and D'~, respectively, and write eo(P ) for the row vector whose j t h entry is the section (P, ej) of D, × C" and write e~(P) for the row vector whose j th entry is the section (P, ej) of D'~ x C n ({ej} denotes the standard basis for Cn). The spin bundle E ~ over p1 is defined by the transition function z7 t (where z is

the usual local parameter x + iy on D) as follows:

eo~(P) = z(P)- leo(P ), P ~ D e ~ D'~.

Iffo is a map from D e to C n then we define the local section:

eo(P)fo(P ) = ~ eoj(P)fo,;(P) j = l

of E", and we refer to f0 as the local coordinate of this section. Since D, has the natural local parameter z we will think of a local coordinate fo as functions of z without

introducing special notation. In a precisely analogous manner mapsf~o from D', into C n determine local sections eo~f~ ofE" and we will think of the local coordinatef~o as a function of the natural local parameter w = z - ~ without introducing special notation.

Let A (p'q) denote the space of (p, q) forms on p1. A Cauchy-Riemann operator, X, on E" is a first order differential operator which maps smooth sections of E" into smooth sections of E" ® A (°'1) and which has a special local form. Relative to the local

202 JOHN PALMER

parameter z and the trivialization eo the action of X is

Xeo(z)fo(z) = Co(Z)d~'(~z + Ao(z))fo(z),

where ~ : = ~ ( 0 x - iOy) and Ao(z) is a smooth n × n matrix valued function on D. Relative to the local parameter w and the trivialization e~ the action of X is

Xe~(w)f~(w) = eo~(w) dw(0w + A ~(w))f~(w)

for a smooth complex valued n × n matrix valued function A®(w) on D',. It is clear that for X to be well defined we must have

d~Ao(z ) = d#A®(w)

so that Ao and A~ fit together to give a (0, 1) form valued endomorphism of the bundle E n '

Cauchy-Riemann operators are important because the choice of a holomorphic

structure for the bundle E" is the same as the choice of a Cauchy-Riemann operator

on E". A local sec t ionf is holomorphic with respect to the complex structure on E" determined by a Cauchy-Riemann operator, X, if and only if Xf = 0 (a more detailed description of this connection can be found in the book [3]).

Quillen [9] has defined a determinant bundle over the space of Cauchy-Riemann operators on any fixed C ~ vector bundle whose base is a compact Riemann surface.

We are interested in Quillen's determinant bundle for the space of Cauchy-Riemann

operators on E" and in particular to a 'localization' of this bundle to the disks D and

D'. We will now explain what we mean by this. Suppose that X is a Cauchy-Riemann

operator on E". Then X induces a natural 'Cauchy-Riemann operator ' on the disk D

in the following manner. Relative to the local parameter z and the trivialization e o one

may realize X as an operatorfo(z) --+ ( ~ + Ao(z))fo(z ) for some smooth matrix valued function Ao. This differential operator has an infinite-dimensional kernel when it acts

on smooth functions on the disk and so it does not very closely resemble the operator

X. However, there is a natural way to pose boundary conditions for the domain of the

operator, 0z + Ao, so that the resulting operator quite closely reflects the operator X. Let H~(E ") denote the Sobolev space of order s of sections of E". Let HA(D ) denote the subspace of HI(E ") consisting of those sect ionsf which are killed by X in the exterior

of the closed unit disk D (that is Xf(p) = 0 for p e D~). We may identify elements in

Hi(D) with C" valued functions on D via the trivialization eo and we do so henceforth.

Let XD denote the operator obtained by restricting ~, + Ao(z ) to the domain HA(D). Then XD is a natural restriction of X in the sense that the kernel and cokernel of X can be identified with the kernel and cokernel of Xo in a fashion that we will now indicate.

Suppose thatfo e HA(D) and that Xofo = 0. Thenfo is by definition the eo coordinate of a section feH~(E ") that is killed by X in the exterior of the unit disk. Since XD f0 = 0 it follows that Xf = 0. We see then that ker(Xo) is contained in ker(X). Since the other inclusion is obvious we have identified the kernel of X with the kernel

of X o. Suppose now that fo ~ LZ(D) and consider the map

L2(O)~ fo(z) ~ d~eo(z)fo(z ) + Zl(X),


where ~(X) denotes the range of X. It is easy to see that f0 is in the kernel of this map if

and only if there is a section g ~ Hlx(D) (temporarily regarded as an element in HI(E")) such that Xg(z) = d~eo(z)fo(z) over D. But this implies that f0 is in the range of Xo. Thus the displayed map induces an injection from the cokernel of X o to the cokernel of X. We wish to show that this map is surjective. To see this it is enough to show that any sect ionfe L2(E ") differs from one supported on D by an element in the range of X. Let q~ denote a C ~ function on p1 which is 1 on the complement of D and 0 on the complement of D'~ for some positive e. It is always possible to solve Xg = ~bf for g locally defined on D'~ (see [1]). Now multiply g by a smooth cutoff ~ which is identically 1 on D' and which is identically 0 in some neighborhood of the complement of D'~ so that q/g is naturally an element in Hi(E"). Then f - X(~,g) has support in D.

The fiber in Quillen's determinant bundle over X is isomorphic to ker(X)* ® coker(X) (or canonically to C when X is invertible). Since the kernel and cokernel of

XD agree with the kernel and cokernel of X it is not too surprising that we can define a determinant bundle over the family of operators Xo so that the map X ~ Xo is covered by a bundle map on the associated determinant bundles. We next consider one way to do this.

There is an alternative way to understand the fiber in Quillen's determinant bundle that will be useful for us. Each Cauchy-Riemann operator on E" is a Fredholm map from HI(E ") to LZ(E" ® A tO'l)) with index O. Thus one can find an invertible map q from HI(E n) to L2(E n ® A (°'1)) so that q -xX is a compact perturbation of the identity.

In fact one may choose the map q so that q -1X is a trace class (or even finite rank) perturbation of the identity. We will say that q (or more precisely q- 1) is an admissible parametrix for X provided that q - i X is a trace class perturbation of the identity. If q l and q2 are both admissible parametrices for some Cauchy-Riemann operator X then one may verify that q] lql is a trace class perturbation of the identity. The fiber in the determinant bundle over X may now be identified with the set of ordered pairs (q, 2) with q an admissible parametrix for X and 2 e C* subject to the equivalence relation (qt, 21) = (q2, 22) if and only if:

21 = 2 2 det(ql-~q2)

The determinant is well defined [12] and the multiplicative property for determinants may be used to show that this is a well defined equivalence relation. The 'relative' determinant det(q- 1X) makes sense for any admissible parametrix q but, of course, its value depends on q. The determinant bundle is designed so that the map

X ~ (q, det(q-1X)) is a well defined section (it is called the canonical section a). The problem of assigning a determinant to a family, ~ , of Cauchy-Riemann operators translates, in this scheme of things, into the problem of finding a trivialization fi of the determinant bundle over o~. Given such a trivialization one can sensibly define a determinant as follows:

~(x) det~(X) = - -

~(x)

204 JOHN PALMER

By making 'locally uniform' choices for q one may exhibit triviatizations for the determininant bundle with holomorphic transition functions. We will not do this in the simplest fashion here since we are precisely interested in describing one way of uniformly picking such paramatrices for the operators Xa that will mesh well with the Grassmannian formalism on the boundary [13]. Each operator Xo has a finite dimensional kernel and a finite dimensional cokernel with the same dimensions and is consequently a Fredholm operator with index 0. We would like to model the fiber in the determinant bundle over this family of Fredholm maps as we did above using admissible paramatrices. In order to exhibit the holomorphic nature of this bundle we will now describe a locally uniform way to find such paramatrices. It will be useful for us to introduce a slight modification of the Grassmannian first introduced by Segal and Wilson in [11]. Let H1/2($1) denote the Sobolev space of order ½ on the circle. We write H+ for the subspace of H1/z(S1) whose elements have holomorphic continuations into the interior of the unit disk and H_ for the subspace of H1/2(S 1) whose elements have holomorphic extensions into the exterior of the unit disk which vanish at oo. Suppose that f e HI/2(SI) ", w e write

If0" fk ---- ~ f (e iO) e -ikO dO

for the kth (keZ) Fourier coefficient o f f If f and g are both in Hx/2(S 1) then the natural inner product on HX/2(SX) is

( f , g )= ~ (1 + Ikl)A'gk, k = - o o

where x .y = xay 1 + -.. + x,y, is the standard bilinear form on C". Note that the subspaces H+ and H_ are orthogonal with respect to this inner product.

We will say that a closed subspace W of HX/z(S~) is close to H± if the difference of the orthogonal projection on Wand the orthogonal projection on H i is a Schmidt class operator. We will write Gr(H+)(Gr(H_)) for the Grassmannian of all closed subspaces of HI/2(S 1) that are close to H+ (H_).

Let ON(X, D) denote the H~/2(SI) closure of the subspace obtained by restricting solutions f e H~(D,) of Xf(z)= 0 to S ~ (identified with functions in the eo trivialization). It is a result of Segal and Wilson [11] that the subspace so obtained is in the Grassmannian Gr(H+). In fact if we let Gro(H+ ) denote the connected component of Gr(H+) which contains H+ then it is also true that

ON(X, D) ~ Gr0(H +)

We define ON(X, D') in a similar fashion but with one small difference. The subspace ON(X, D') is again obtained by taking the Hxn(s ~) closure of the space of functions on S ~ that arise by restricting solutions f e H~(D',) of Xf(w) = 0 to the circle. However, these sections are identified with functions on the circle via the eo trivialization (not the e~ trivialization that might be suggested by symmetry). The subspaces ON(X, D') are in the Grassmannian, Gr(H_). In fact, as before, ON(X, O') is in the connected


component of the Grassmannian containing H_, that is

8N(X, D') e Gro(H_).

The results we have described for the subspaces ON(X, D) and ON(X, D') are simple consequences of the fact that any holomorphic bundle over an open Riemann surface is holomorphically trivial I-l]. Let A(z) denote a C ~ matrix valued function on De and consider the Cauchy-Riemann operator d~(~z + A(z)) on the trivial bundle De × C". Let {~bl, q~2 . . . . . q~,} denote a holomorphic trivialization for the Cauchy-Riemann operator d~(~ + A(z)). Define an n x n matrix valued function ~b(z) whosej th column is the vector valued function r~j(z). Then tk is a smooth map from D e into Gl(n, C) and

4,-qz)(~z + A(z))q,(z) = ~z,

where ~b is regarded as a matrix valued multiplication operator. It follows from this that ON(X, D) = 4)Is,H+ so that ON(X, D) is in the Grassmannian of smooth subspaces close to H+ I-8]. The fact that the restriction of q~ to the boundary of the disk extends to a continuous map from the disk to GI(n,C) implies directly that

ON(X, D) E Gro(H +). Now choose such a trivialization $ and consider the splitting

H~/2($1) = ckH+ + (kH_ = ON(X, D) + cbH_ where we have written q~ = Sis, to avoid fussy notation. Since the function ~b is smooth and invertible on the circle the associated matrix valued multiplication operator is continuous and invertible on H~/2($1) so that the splitting H1/2(S ~) = ckH+ + ckH_ is also continuous. The

subspace 4~H- is a smooth subspace in Gro(H_) which is transverse to ON(X, D). Let H~(D) denote the Sobolev space of order one on D whose elements have boundary values in q~H_ and write X~)(4~) for the restriction of X to the domain H~(D). We will show that XD(q~) is invertible and we will use this map to construct uniform admissible paramatrices for XD.

We begin by showing that the projection in Ht/2(S ') on the subspace ON(X, D) along the subspace 4)H_ naturally induces a continuous map P , from H~(D) to N(X, D). Each function f~ Ht(D) continuously restricts to a function fin1 ~ Ht/2(S1). We write flsl = q~O+ + q~O-, where o±EH+. The map which takes O+ to its

holomorphic extension to D (which we continue to denote by ~/+) is continuous from HI/2(S1) to H~(D). F o r f ~ Hi(D) we define a continuous map P , : Hi(D)-~ HX(D) by

P4,f(z) = ~(z)9+(z), zeD.

Suppose now that f~Htx(D). Then

Xof = Xo(dpXI - P+)f

for the simple reason that f - Pq~fe H~(D) and XPq, f = O. We will now show that the map (1 - P,): H~(D) ~ H~(D) is Fredholm with index O. Once we know this then the relation between Xo (which also has index O) and Xo(~b) given above will imply that XD(4~) has index 0 as well. But Xo(~b ) was constructed so that it has a trivial null space. It follows that Xo(~b) must be invertible if I - P , has index O.

206 JOHN PALMER

To see that I - P , is Fredholm it is useful to split Hi(D):

Hi(D) = H~o(D) + nar(D),

where HI(D ) is the subspace of HI(D) whose elements vanish on the boundary of D, and Har(D) is the subspace of H I(D) whose elements are harmonic in the interior of D. Evidently

H~:(D) = H~(D) + nar(D) c~ H~(D)

and

HI(D ) = H~(D) + Har(D) n HI(D ).

It is clear that relative to these splittings the map I - P , is the identity on H~(D) and as a map from Har(D)n Hlx(D) to Har(D)n H~(D) is induced by the projection of ON(X, D') on 0 H - along q~H +. It makes good sense to restrict to the boundary since the map:

Har(D) e f -* flsl ~ H1/2(S1)

is a Hilbert space isomorphism. The projection of ON(X, D') on ~bH_ along ~bH+ is a Fredholm map and has index 0 in H1/2(S1) since both ON(X, D') and q~H_ are in

Gro(H_) and ~bH+ is in Gro(H+) [8]. We see then that Xo(c~) is invertible and so to construct a parametrix for

X o = XD((a)(I- P~,) it is enough to construct a parametrix for I - Po. The map I - Po from Har(D) n H~(D) to Har(D) n HI(D ) is determined by its induced action on the boundary. Thus to find a parametrix for I - Po it will suffice to find a parametrix for the projection of ON(X, D') on q~H_ along q~H+ (which we will also denote by I - P , to avoid introducing extra notation). We will do this in a uniform fashion by choosing a local holomorphic trivialization for the bundle of ~b-admissible frames in the Grassmannian Gr(H_) near the point t~N(X, D'). If W is a subspace in

Gr(H_) then a ~b-admissible frame for W is a continuous isomorphism w: q~H_ ~ W with the property that the composition of w with the projection of W on ~bH_ along q~H+ is a trace class perturbation of the identity on ~bH_ [8].

We will now briefly describe one way to locally trivialize the bundle of admissible frames. Suppose that ~, is a smooth map from the circle to Gl(n, C). We write:

(a4,(~b) b~,OP)~ = \ % ( ~ ) d,(O),l

for the matrix of the multiplication operator ~ relative to the splitting H1/2($1) = dpH+ + ~H . The part of the Grassmanian we are interested in can be identified with the smooth based loops in U(n). A based loop is simply one that maps the point 1 on the circle to the identity in U(n). Our subspaces are always in the connected component containing H_+ so we can even restrict our attention to maps into U(n) whose determinants have winding number zero. Suppose now that Oo is a smooth based loop in U(n) whose determinant has winding number zero (this winding


number is also minus the index of the operator d(qJo) l8]). If d,(qJo) is invertible then it is trivial to check that w = ~bod,(qJo)-1 is a ~b-admissible frame for W = q/oH_. It is

also easy to extend this to a local trivialization in a neighborhood of such a map ~k o. Let U(~bo) denote the open set (in the C ® topology) of all smooth based loops qJ in U(n) with the property that d,(qJ~'o 1) is invertible. Define:

w(q,) = ~0[a~(~o 1)d,(qJo)]- 1

for q/• U(qJo), the operator multiplying ~O being regarded as a map from 4~H- to H1/2($1). We claim that w(qJ) is a ~b-admissible frame for ~OH_. To see this observe first that the rules for matrix multiplication imply that d~(~bl~,2) = d~(~kl)d~,(~,2) + c~(qJ0b~(q,2). If we let ~O 1 = ~bqJ o i and ~O 2 = ~b o in this last equality we see that the d~ matrix element for ~O differs from the product of the d~ matrix elements for ~'qJo ~ and qJo by the product ofb~ and c~ matrix elements for smooth loops. The b~ and c~ matrix elements of any smooth loop are Schmidt class maps [8] so that d~(~k) differs from d~(~kq/o ~)d6(qJo) by a trace class map. It is easy to see that this implies w(~O) defined above is a q~-admissible frame for ~kH_. This gives a trivialization of the frames over U(~ko). We would like to see that the open sets U(~ko) with d~(q~o) invertible cover the smooth based loops in U(n) with winding number zero for the determinant. It is a result of Pressley and Segal that the smooth maps on the circle with values in U(n) and invertible d~ matrix elements are dense in the space of all smooth maps into U(n) with winding number zero for the determinant (this is the 'big cell' in the cell decomposition in [8]). Thus if ~ is any smooth map from S 1 to the unitary group then every neighborhood of ~b contains elements ~b o with d~(~Oo) invertible. If this open neighborhood is chosen small enough then d~(q~'o 1) will be invertible and we have

• u(q~o). We are now ready to explain the sense in which we wish to use Xo to 'localize' the

description of the determinant bundle for Cauchy-Riemann operators on E". The operator X o will be useful for us in focussing on variations of the operator X which occur in the exterior of the unit disk. If o~ is a family of Cauchy-Riemann operators on E" which all agree over D then the family of operators Xo for X • ~ all agree as differential operators over D except for the boundary conditions on the unit circle which determine their domains. Thus in discussing operators Xo which arise from X • ~ we may once and for all fix the smooth trivialization ~b for X on D. The only thing that does vary in the family X o for X • ~" is the subspace ON(X, D'). Thus it is natural to topologize the family X o by regarding it as the collection of subspaces ~N(X, D') in the smooth Grassmannian Gro(H_) for X • ~ . Once this is done, one easily recognizes the determinant bundle over the family X o as the pullback of the det* bundle over Gro(H_ ). The calculation of special admissible paramatrices for X o given above shows that we can identify elements in the fiber of the determinant bundle over Xo with pairs (q, 2), where

q = Xo(~ )w- 1

and w is a 4~-admissible frame for ON(X, D'). It is understood in this formula that the map w extends first to a map from Har(D) c~ HI(D ) to Har(D) c~ H~(D) by identifying

208 JOHN PALMER

boundary values with their harmonic extensions and then to a map from HI(D ) to HIe(D) by declaring it to be the identity on Hlo(D). This sort of extension will be understood for q%admissible frames in the rest of this section.

If ql is associated with the q%admissible frame wl and q2 is associated with the ~b- admissible frame w 2 then since Xv(d? ) does not change for the family ~ we find that the equivalence relation for (qx, 21) and (q2, 22) that defines the fiber in the determinant bundle over XD is

21 = 22 det(w~ Iwl).

This is precisely the equivalence relation that identifies pairs (w, 2) (w an admissible frame for W) with elements in the fiber of the det* bundle over the Grassmannian [8],

We are now prepared to state the principal result of this section. Let A denote a fixed C OO map from D, to the n x n complex matrices. Let ~ a denote the family of Cauchy-Riemann operators on E" which agree with d~(~ + A) in the eo trivialization over D (not D~!)

THEOREM. The map which takes X E f f a to the subspace ~?N(X,D') in the Grassmannian Gro(H_ ) lifts to a map from Quillen's determinant bundle over ~a to the det* bundle of Seoal and Wilson over the Grassmannian Gro(H_ ). The lift is an isomorphism on the fibers.

Proof. We begin by showing how to choose parametrices for X ~ ~A that will make the lift obvious. Fix a choice of Riemannian structure on P1 to obtain a distinguished volume element on PI. Fix a choice of Hermitian structure on the bundle E ". The Riemannian structure on pl induces an Hermitian structure on the bundle A ~ of one forms on P~ and combining this with the volume form we have natural inner products on L2(E n) and L2(E n ® An). Using these Hermitian structures we can define an inner product, ( . , • ), on HI(E ~) as follows:

(f , g ) : = (df, dg) + (f , g),

where the first inner product on the right is in L2(E n ® A 1) and the second is in L2(E"). Let Hlx(D) z denote the orthogonal complement in HI(E ~) of Hi(D) relative to this inner product (recall that the elements in Hlx(D) are naturally sections in HI(E~)). Let L2(D) denote the space of sections of L2(E ~ ® A ~°' 1)) which vanish outside the unit disk and let L2(D ') denote the space of sections of LZ(E ~ ® A ~°'1)) which vanish on the unit disk. We have the continuous splittings:

HI(E n) = HA(O) @ "~x(O)',

L2(E ~ ® A t°'l)) = L2(D) @ L2(D').

Since X kills the elements of Hi(D) in the exterior of the unit disk it follows that the matrix of X: Ht(E ") ~ L2(E" @ A t°'t)) relative to the two splittings given above is

upper triangular:


It is clear that X1 can be identified with Xo defined above. Since X is Fredholm it follows that X2 must be Fredholm and since the index of Xo is the same as the index of X it follows that the index of X2 must be zero. Now suppose that feHlx(D) ± and X 2 f = 0. This implies that Xfvanishes in the complement of D and so fe H~x(D). Thus we must h a v e f = 0. Since X 2 does not have a kernel and is Fredholm with index 0 it follows that X 2 must be invertible. Now fix a trivialization ~ for the family ~A over D~ and choose a @admissible frame, w, for 9N(X, D') as above. With the extension of w to Sobolev spaces of order one on the disk understood as above define:

q w,

We have already verified that the map Xo(~)w-1 is an admissible parametrix for Xo = XI and since X2 is invertible it follows that q(w) is an admissible parametrix for X. It is now a simple matter to Check that the map which takes the point (q(w), 2) in the determinant bundle over the point X into the point (w, 2) in the det* bundle over ON(X, D') is well defined. It is clear by construction that the lift is an isomorphism on the fibers. []

Because the lift of the map X ~ dN(X, D') is an isomorphism on the fibers it follows that one can pull back sections of the det* bundle to the determinant bundle over ~'~A. We leave it to the reader to check that the 'canonical' section of [11] in det* pulls back to the 'canonical' section of [9] in det under this map. We will study determinants of families of operators in ~ a by first trivializing the image bundle in det* and then comparing the resulting trivialization with the canonical section in det*. Because the canonical sections of the two bundles are related by pull back this procedure is equivalent to first trivializing in the determinant bundle and then comparing with the canonical section over ~A.

A

2. The det* Bundle and a Connection on LG

In this section we will consider Cauchy-Riemann operators on E n which 'live' on the disk D. We do this to set the stage for an analogy with 'singular' Cauchy-Riemann operators which arise when considering the Riemann-Hilbert problem. Since, in this section, we will deal exclusively with the group Gl(n, C) it is notationally convenient to declare

G = Gl(n, C).

Now let Y(z) denote a smooth map from D~ to G which is holomorphic in the neighborhood D~ n D'~ of S1. Each such map Y gives rise to a Cauchy-Riemann operator on E n as follows. On D~ define the action of X in the eo trivialization by

Xfo(z ) = d~' V(z)- '-~z r(z)fo(z).

On D'~ define X in the e~ trivialization by

Xfoo(w) = d~f~o(w) .

210 JOHN PALMER

Because Y(z) is holomorphic on D~ n D',(~Y(z) = 0) these two definitions mesh to give a globally defined Cauchy-Riemann operator on E". Let ~ denote the family of all such maps Y. For each Ye ~ the potential, Y(z)-l~Y(z), vanishes in a neighborhood of S 1 so that all the variation in the associated family of Cauchy-Riemann operators is confined to the interior of the unit disk. We may thus use the scheme described in the preceeding section to localize the associated Cauchy-Riemann

operators by restricting them to the exterior of the unit disk. Each localization Xo, of a Cauchy-Riemann operator X which arises from an element Ye ~ is given by the differential operator d ~ w with boundary conditions ~N(X,D)= Y-1H+. The restriction of holomorphic sections for d;v~ w to S ~ is just H in the e 0 trivialization. Since d#~w is already trivial in D', the map q~ discussed in the preceeding section may be taken to be the identity. This choice gives q~H+ = H+ as the subspace of H~/2($1) which we choose transverse to ~bH_ = H_. The results of the previous section identify the determinant bundle over the family of operators XD, with the standard det* bundle over the subspaces Y-~H + e Gr(H +). The admissible frames for the subspaces in Gr(H+) are based, in this case, on the standard splitting H+ + H_ of H1/2(81).

Eventually we will be interested in trivializing the determinant bundle over a family of singular Cauchy-Riemann operators that live on the disk D. We will do this by trivializing the det* bundle over the corresponding subspaces in Gr(H+). This procedure has the advantage that the subspaces we consider will be smooth in Gr(H+) even though the corresponding Cauchy-Riemann operators are quite singular. It has the disadvantage that the det* bundle over the Grassmannian is topologically nontrivial. Thus, although the determinant bundle over the space of all Cauchy- Riemann operators on E" is trivializable, the image of ~ in det* is not. In the special

case we are interested in the choice of a local trivialization in the det* bundle depends on a connection in a group of bundle automorphisms acting on det*. The rest of this section is devoted to a description of this connection (which is closely related to one defined by Mickelsson in [5]).

It is well known that the group LG of smooth loops in G acts on the Grassmannian Gr(H+) but that this action does not lift to an action of this group on det*. However, there is a central extension LG of LG which does act on det* covering the action of LG on the base. We are exclusively interested in the connected component of the identity in LG, for which we write LoG , and for simplicity we confine our discussion to a

description of the central extension LoG over LoG (note that the connected

component of the identity in LG consists of those loops whose determinant has winding number 0 on S 1 [-8]).

We recall the construction of LoG that can be found in [11] and [8]. The elements in LoG can be identified with equivalence classes of pairs (g, q) where g e LoG, and q is an admissible parametrix for a(g). Recall that the matrix of the multiplication operator associated with g relative to the H+ + H_ splitting of H1/2(S1) is

(a(g) b(g)~ g = kc (v ) a(o)J

DETERMINANTS OF CAUCHY-RIEMANN OPERATORS AS r-FUNCTIONS 211

and an admissible parametrix for a(g) is an invertible map q: H+ ~ H+ such that

a(g)q- x is a trace class perturbation of the identity.

Using the fact that the b and c matrix elements of any smooth loop are Schmidt

class operators one may easily verify that the set of such pairs is a group with respect to the composition law:

(g~, ql)'(g2, q2) = (gig2, q~q2).

One only needs to check that qlq2 is an admissible parametrix for a(gtg2). The reason

for considering this group of pairs is that this extension of LoG may be seen to act on the bundle of admissible frames over Gr(H +). If w: H + ~ W is an admissible frame for

W~Gr(H+) then one may check that (g, q ) -w:= gwq-1 defines an admissible frame for the subspace gW when q is an admissible parametrix for a(g).

Let w be an admissible frame for W as above and suppose # e C. Recall that the fiber in the det* bundle over We Gro(H ÷) can be thought of as an equivalence class of pairs

(w,/0. We declare the action of (g, q) on (w, p) to be

(g, q)'(w, 1 a) = (gwq-1, #).

The equivalence relation that defines the det* bundle is (w~,/al) = (w2, #2) if and only if

# 1 = ]A2 det(w21w0 -

The group which acts on the det* bundle can thus be identified with the group of

pairs (g, q) subject to the equivalence relation (gl, q0 = (g2, q2) if and only if g~ = g2 and det(q2ql I) = 1.

We let LoG denote the group of pairs (g, q) subject to this equivalence relation and

we observe that the map (g, q) ~ g induces a homomorphism LoG ~ LoG with C* as its kernel.

There is a natural connection on the principal bundle LoG ~ LoG which we will

now describe. To give a connection on LoG it is enough to specify a horizontal subspace of the tangent space to LoG at the identity which is invariant under the natural action of C*. One may then obtain a horizontal distribution by moving this

subspace around LoG using the group action. Suppose that t ~ gt is a smooth curve in LoG which passes through the identity at t = 0. For all values of t sufficiently close to 0 the map a(gt ) will be invertible. Thus the curve t ~ (Or, a(gt)) determines a tangent to LoG at the identity. The set of all such tangents we declare to be the horizontal

subspace in LoG at the identity. Thinking of tangent vectors to LoG at (g, q) as equivalence classes of curves passing

through (g, q) at t = 0 we can identify the elements of this tangent space as equivalence classes of pairs (6g, 6q) where 69 is a smooth n x n matrix valued function on the circle and 6q is a continuous linear map on H + which differs from a(fg) by a trace class map. Two such pairs (6gk, 6qk) for k = 1, 2 are declared equal if and only if 691 = 692 and Tr(q-1(6q2 - 6qO ) = O.

212 JOHN PALMER

One may write this last relation more suggestively as

Tr(q-l(6qt - a(t~g))) = Tr(q-1(6q2 - a(~g)))

which allows one to identify the fiber with C.

The connection one form ~ on LoG associated with a horizontal distribution of subspaces in T(LoG ) assigns to each tangent vector its vertical component. If a is a local section of LoG then the pullback a*ct is the local one form for the connection

relative to the section a. In order to make a comparison between what we do here and the calculations in the work of Malgrange [4] it will be useful for us to determine the connection one form relative to the canonical section a(g):= (g, a(g)) defined over

those 0 ~ LoG with a(g) invertible (it is a globally defined section on the associated line bundle which vanishes over those g for which a(g) is not invertible). Suppose now that

g ~ L o G and a(g) is invertible. Let 6g denote a tangent to LoG at g. We wish to write da(6g) = (6g, a(6g)) as the sum of a horizontal vector (g, a(g)). (6h, a(6h)) and a vertical

vector (0, 6q). Employing the equivalence relation given above we find that we must h a v e 6 h = g 16g and

Tr(a(g) 16q) = Tr(a(g)-la(t~g) - a(g-16g)).

The map (0, 6 q ) ~ Tr(q-16q) canonically identifies the vertical tangents over (g, q) with C. It follows that

a*ct((Sg) = Tr(a(g)- i a(t~g) - a(g - i 6g)).

We make one final observation concerning the group LoG. The map from LoG into Gro(H÷ ) given by g ~ gH+ lifts to a map from LoG into det*. A lift is given by (g, q) ~ (gq- l, 1). It is easy to check that this map is an isomorphism on the fibers of the associated line bundles and that the pull back of the canonical section over det* is the canonical section of LoG described above.

3. Cauchy-Riemann Operators Associated with the Riemann-Hilbert Problem

In this section we will introduce singular Cauchy-Riemann operators that are naturally associated with the Riemann-Hilbert problem. Part of the motivation for

the introduction of these operators comes from Sato, Miwa and Jimbo's work on z- functions for Holonomic quantum fields [10] and part comes from the notion of vertex insertions in string theory and conformal field theory [2].

We begin with a brief review of the classical Riemann-Hilbert problem on p1. Suppose that A(z) is an n × n matrix valued function on pa whose matrix elements are rational functions ofz with poles at {al . . . . . ap}. The fundamental solution Y(z) to the linear differential equation d Y/dz = A(z)Y(z) is, in general, a multivalued function of z with branch points at z = aj for j = 1 . . . . . p. To describe the multivalued nature of such fundamental solutions, choose a base point ao in p1 different from any of the branch points {an . . . . . ap}. We may suppose that ao is not collinear with any of the points a~ and by relabeling if necessary we may insure that the points al , . - -, ap occur in


order as one makes a counterclockwise circuit of ao crossing the line segments joining

a0 to aj. Let ?j denote a simple closed curve based at a0 and encircling aj (but none of

the other branch points)just once in the counterclockwise sense. The fundamental

group for p1 _ {al . . . . . ap} := P~ is generated by the equivalence classes [~j] of the curves ~,j with the single relation

[~q][?2]"" [?p] = identity

among the generators (this identity is the reason for the fussiness in the labeling of the

points a~, . . . , %). Let Pa ~ denote the simply connected covering space of P]. The fundamental group of P~ acts by deck transformations on P~ and one may describe the fundamental solutions Y(z) to Y' = A Y as holomorphic functions on P] which

transform Y([Tj]P) = Y(P)Mf 1 according to a linear representation n([Tj]) = M r of the fundamental group (the monodromy group of the equation).

Riemann's success in analyzing the solutions to the hypergeometric equation in terms of its monodromy representation led him to consider the problem of character- izing linear differential equations in terms of their monodromy representations.

One starts with p invertible matrices M 1 . . . . , Mp satisfying the relation M~.. . Mp = identity so that n[Tj] := Mj determines a representation of the fundamental group of p1. The classical Riemann-Hilbert problem is to determine an

invertible matrix valued hoiomorphic function Y(z) on P~ which transforms on the right according to this representation of the fundamental group.

To establish some measure of uniqueness and to make the connection with linear

equations with rational coefficients one further requires that the isolated singularities of the single valued one form A := d Y Y - 1 at the points aj are all simple poles. This

makes Y a fundamental solution for a linear differential equation on p1 with regular singular points. One consequence of this is that for some choice of a logarithm 2niLj for M r the local behavior of Y near aj is the same as the behavior of the multivalued function (z - a j) -Lj. If one fixes a branch of Y(z) and (z - a j)-/~J in the neighborhood of some point close to aj then the function Y(z)(z - a j) nj will have a single valued analytic continuation into a punctured neighborhood of aj since both Y and

(z - ag) -L~ change by a factor of M / 1 when z makes a counterclockwise circuit of a). We say that Y(z) behaves like (z - aj)-LJ for z near aj if Y(z)(z - aj) L~ has a removable singularity at aj.

In his paper on isomonodromic deformations, Malgrange [4] employs a slight reformulation of the Riemann-Hilbert problem to good effect. It will be useful for us

to explain Malgrange's terminology. We can define a connection VA on the trivial bundle P~ x C" so that A is the connection one form for VA in the standard trivialization. Let ej(P) (j = 1 . . . . . n) denote the standard trivialization ej(P) := (P, eg) of P~ x C". Let e(P) denote the row vector with j th entry ej(P) and le t f (P) denote a column vector of functions on P~. We write e(P)f(P) for the section:

P---* ~ ~(P)ej(P) j = l

214 JOHN PALMER

The connection VA is defined by VAef= e d f + eAf Since ~zY= 0, it follows that the connection VA is integrable (has zero curvature) and is holomorphic (its one form contains only a d z term). When the connection one form A(z) has simple poles at

z = a i we say that the connection VA has logarithmic poles at aj. The version of the

Riemann-Hilbert problem we consider is the problem of constructing an integrable holomorphic connection on the trivial bundle P] × C" with logarithmic poles at z = aj

and prescribed holonomy M f 1 on the curves 7j [4]. The holonomy is determined by

parallel translation of the basis e(ao) at the base point a 0 around the curves ~j. We now wish to describe the construction of a Cauchy-Riemann operator from

data closely connected with the formulation of the Riemann-Hilbert problem.

Roughly speaking we wish to define a 'Cauchy-Riemann' operator on the spin bundle over p1 whose domain consists of sections with prescribed branching (monodromy Ms) at the points a~. To make an explicit determination of the domain of this operator the specification of a logarithm 2giLj for M~ will be important and we will have to define branch cuts to keep the sections single valued (in the case of unitary

monodromy one might work on the simply connected covering space ~x but I do not know what to do in the most general situation).

For simplicity, we will suppose that the branch points {al . . . . , ap} are all contained in the open unit disk D. If necessary one can always Moebius transform ao to 0 and then make a scale transformation to insure that the image of each of the points aj is inside the unit circle. The operator we wish to introduce is the standard Cauchy- Riemann operator on the spin bundle over p1 except that the domain now consists of

multivalued functions with 'monodromy Mj at a S. To be more precise about the domain we will introduce a system of branch cuts Fj emanating from the branch points a i. Each curve Fj is chosen to be a simple piecewise smooth curve in D the initial endpoint of which is the point aj. The final endpoint of each of the curves F i is chosen from among a finite collection of points {s 1 . . . . . s~} which we refer to as sinks, all of which are distinct from the branch points {aa, . . . , a~}. No two of the curves F~

are to intersect except at a sink and if (F j,, . . . , Fir ) represents the counterclockwise disposition of branch cuts arriving at a sink s then we require M jl "" M jr = identity so that the point s is indeed a 'sink' for monodromy. The simplest way to choose such curves Fj is to pick the sink s = 0(= a0) (which is not collinear with any two branch points) and let F~ denote the straight line joining aj to s. The choice of the curves Fj will not be very important in our constructions. In a sense that we hope to explain elsewhere deforming the curves F i and moving the sinks sk about is accomplished by gauge transformations.

In order to simply describe the domain for the Cauchy-Riemann operator that we wish to define we first recall the solution of the Riemann-Hilbert problem on an open Riemann surface. On the open disk D~ it is always possible to solve the Riemann- Hilbert problem with prescribed singularities ( z - a j) -Lj at aj for any choice of logarithms 2giLj for Mj (see [1]). In what follows we suppose that we have made a choice of the logarithms 2niL i. Fixing the branch of a multivalued solution we obtain a single valued function Y(z) on the connected open set D~ - w~= a Fj. It is easy to see


that any two such functions Y1 and I12 differ by a map Yx(z)Y2(z) -1 which is an invertible holomorphic function on the disk D~. Now choose e small enough so that D'~ does not contain any of the points {al . . . . . ap} and let {~k, I - ~} denote a partition of unity subordinate to the covering {D~, D',} of p1. Let ~t' be the set of measurable sections of E". We define a domain ~.,L = J4 by:

~ a j : = {f~J/t ' : Y~OfeH~(D~) and ( 1 - ~O)f~H~(D'~)}

It is not hard to check that the domain ~. ,z does not depend on the partition of unity or on the choice of the function Y (since HI(D~) is invariant under multiplication by smooth maps into Gl(n, C)).

We now define a Cauchy-Riemann operator ~.a. acting on sections f in the domain ~.,L by -Oa.Lf= -O~,L~kf+ 8.,L(1 -- ~)f, where

0.,LOf:= d~ Co(Z) Y(z)- 1~: y(z)O(z)fo(z),

?.,L(1 -- O)f:= d# e~(W)?w(1 - O(w))f~(w).

Since Y~,fe H 1 it makes sense to apply 0~ to the coordinates of this section. Also because Y is holomorphic in a neighborhood of the circle it follows that 8.,L does not depend on the choice of the function ~. Indeed, since different solutions of the Riemann-Hilbert problem on D, differ by invertible holomorphic matrices on the left it follows that the operator 8.,L does not in fact depend on the choice of the solution Y. The operator 8.,L acts on sectionsfe ~.,L, supported in D~, precisely as the differential operator d ~ , at least if one stays away from the curves Fj where the coordinate o f f has discontinuities.

The principal result of this paper is to explain the sense in which one may understand that the z-function for the Riemann-Hilbert problem introduced by Sato, Miwa, and Jimbo is a determinant for the Cauchy-Riemann operator 0.,L. In making this connection we will closely follow the treatment in Malgrange [4] (though with a somewhat different interpretation of the calculations).

One may prove that the map 8..z is Fredholm from ~a,L into L2(D ') • Y-1LZ(D). However, it may happen that the index of ~..z is not zero. If one considers the restriction of 0.,L to the exterior of the unit circle with the natural boundary conditions Y- 1H + then as in the first section it is not hard to see that the index of this operator is the same as the virtual dimension of the subspace Y- ~H+ (the dimension of the kernel of the projection on H+ minus the codimension of the range of the projection in H+). This virtual dimension is, in turn, the winding number of the function z ~ det(Y(z)) on the unit circle [8]. Since d(log[det(Y(z)]) = Tr(dY(z)Y(z)-1) and dY(z) = 8zY(Z)dz in a neighborhood of the circle (because "~.Y(z) = 0 near S 1) it follows that the index in question is:

1 ~s ind(0.,L) = ~ ~ Tr(O~ Y(z) Y(z)- a) dz.

Because the function Y(z) looks like (z - ai) -zj near the point aj one may do this last

2 1 6 J O H N PALMER

integral by residues to get

p

ind(~a,L) = -- ~ Tr(Lj). j--1

To keep things as simple as possible we will hereafter suppose that

p

Tr(Lj) = 0. j = l

This will imply that the index of ~a,L on the spin bundle over p1 is zero (this condition is known as the Fuchs relation). It is clear that in the general case one can adjust the index of ~a,L SO that it is 0 by simply regarding it as an operator on the sections of a different bundle on p1. It is also clear that one can adjust the index on the spin bundle

to any value one desires by making different choices of the logarithms 27riL i. Let us briefly consider what the solution to the Riemann-Hilbert problem on p l

signifies in this setting. The problem will have a solution in p1 precisely when one of the solutions Y(z) in the disk, D, has an invertible holomorphic extension to the exterior of the disk, D'. If Y is a fixed solution in the disk then every solution has the form Y÷ Y for some invertible holomorphic map Y÷. Thus the Riemann-Hilbert problem will have a solution on all of p1 precisely when the restriction of Y to the unit circle has a canonical factorization Y= Y~_IY_ where Y has an invertible holomorphic extension into the exterior of the unit disk. This is not always possible but when YIs, does have such a canonical factorization one may regard the matrix valued function on P~ obtained by matching Y_ in the exterior of the disk with Y÷ Yon the interior of the disk as a global 'gauge transformation' intertwining ~,,L with the

standard Cauchy-Riemann operator on the spin bundle.

We are interested in the behavior of the Cauchy-Riemann operators ~a.z in the neighborhood of points where the Riemann-Hilbert problem has a solution. Suppose then that the classical Riemann-Hilbert problem has a solution at some collection of points {ao.1 . . . . . ao,p} all contained in the open unit disk. Let Yo(z) denote the matrix valued solution to the Riemann-Hilbert problem with singularity (z - aoj)-Lj at ao, j, fixed branch cuts Fj, and normalized so that Yo(oO) = identity.

Let D~ denote the open disk centered at aoj with radius Rj. Suppose that the radius Rj is chosen small enough so that the closures of the disks Dj + are all contained in the open unit disk and no two of the disks Dj + intersect. Choose positive numbers rj and pj

so that 0 < rj < p~ < Rj. Let D~ denote the disk of radius pj about aod and let Cj denote the circle of radius pj

about aod. Choose rj small enough so that when laj - aod[ < rj and z is in the annulus rj < Iz - aoj[ < Rj we have

z - a ~ + a o , jCD[ f o r k # j .

To study the solution of the Riemann-Hilbert problem for the collection of points {aa . . . . . ap} with laj - aod[ < rj we would like to introduce the functions

Y~(z) := Yo(z - aj + aoj) for [z - aoj[ < R i.


These have the right local monodromy at aj and the problem becomes one of altering

this collection of functions by local holomorphic maps so that they fit together globally. This is essentially what we shall do but because the function Yo has fixed branch cuts it does not compare smoothly with its translates on the annuli of interest. We follow Malgrange on this point and compare connections instead. Let Ao:= dYoYo ~ denote the one form derived from the solution Yo (note that in the domain of the local parameter z one has Ao(z) = dzOzYo(z)Yo(z)-1) and write Vo for the associated connection on the trivial bundle over P]. For [a t - a o a l < rj and

[z - aoal < Rt define

At(z, a) = Ao(z - a t + aoj),

where we mean by this the pull back of the form Ao by the translation aj - aoa. Let Vj denote the connection on the trivial bundle over the disk Iz - aoa[ < Rj determined by the one form Aj and observe that by construction the connection V t has a logarithmic pole at aj.

On the annular region r t < Iz - aoal < Rt the connections Vo and V t have the same holonomy (Mr 1) on the circle C t, the connection V t depends holomorphically on the parameter a t and the two connections agree at a t = aoa. Thus there exists a gauge transformation St(z, a) which takes the connection V t into the connection V0 on the

annulus rj < Iz - ao,tl < Rt. That is S t V t S f I = Vo. We fix S t uniquely by requiring that St(z, ao) = identity, it is then analytic in the

parameter a t. To solve the Riemann-Hilbert problem in D e it suffices to factor the restrictions

Stlc~ = S S f where S is the restriction to C t of a fixed holomorphic invertible map S(z) defined for z ~ D~ - u~= 1 Dr, and Sf(z) is holomorphic and invertible for z inside D t. A connection which solves the Riemann-Hilbert problem on D~ is then obtained by

matching up the connection S-1VoS on D e - u T = l D j with the connections S~VtS f -~ on D t (which is possible since they agree in a neighborhood o f the circle c j).

Since St(z, a) depends analytically on the parameter a t it is possible to solve the factorization problem so that S(z, a) also depends analytically on the parameters aj and is normalized so that S(z, a o a ) = identity. We may use this solution to the Riemann-Hilbert problem to define a family of Cauchy-Riemann operators 0a.L for [at - aoal < ft. The only missing ingredient is the selection of branch cuts for each of the points a t. We are not much concerned with the detailed selection of these branch cuts Ft(a) but for simplicity we will require that however these branch cuts are chosen they differ from the branch cuts F t for aoa only inside the circle of radius r t about aoa.

As in the first section one may show that the map which takes ~a.L into its natural restriction to the exterior of the disk, lifts to a map on the determinant bundles. As in the first part of the second section the determinant bundle over the restriction is naturally identified with the det* bundle over the subspaces Y- IH ÷ e Gro(H +) (where Y:= S - I Y o near S ~) which are the natural boundary conditions for the restriction. Thus to define a determinant for the local family of Cauchy-Riemann operators ~a.L

218 J O H N P A L M E R

defined by the choice of branch cuts Fj(a) it will suffice to trivialize the det* bundle over the subspaces Yo IS(., a)H+. It is this problem that we turn to in the final section.

4. A Local Trivialization the det* Bundle

In the preceding section we defined the 'Cauchy-Riemann' operator ~a,L and we mapped the determinant bundle over this family into the det* bundle over the family of subspaces YoXS( ., a)H+ in the Grassmannian Gro(H+) of subspaces of H1/2(S 1) which are close to H+. The restriction of Yo(z) to the unit circle has an invertible holomorphic extension to the exterior of the unit disk in p1. It follows that the

multiplication operator Yo acting on Gro(H+) has a canonical lift Yo to a map on the det* bundle. Using this map we can identify the bundle we are interested in with the det* bundle over the family of subspaces S(', a)H+. One may also easily check that the canonical section on det* is equivariant with respect to the map ~'o so that we may continue to use the canonical section to define a determinant.

Following Malgrange we will study the det* bundle over the subspaces S(., a)H÷ by pulling back to the circles Cj of radius pj about the points ao, j. Write C for the union of the circles Cj and let H(C) denote the Hilbert space direct sum:

p

H(C):~- 2 (~ H1/2(Cj) • j = l

The Hilbert space H(C) splits as a direct sum H+(C) (~ H_(C) of functions which have holomorphic continuations into the interior D i of the circles Cj(H+(C)) and functions

which have holomorphic continuations into the exterior of the union wiD~(H_(C)). It will be useful for us to describe the projections, P_+(C), on these subspaces in terms of the Hardy space decompositions for the individual circles Cj.

For each space H1/2(Cj) let P+ denote the orthogonal projection on those functions in H1/2(C~) that have holomorphic continuations into the interior of the circle Cj. We write P j := I - ,of. Iffj~H1/2(C~) then P~fj has a holomorphic extension to the complement of the interior of Cj and so it makes sense to restrict this holomorphic extension to any of the circles Ck. We denote this restriction by P~,jfj (so that P~j is a map from H1/2(C~) to HI/2(Ck)). We now describe the splitting of the Hilbert space H(C) which is of interest to us. Each vec to r f = (fl . . . . . fp) ~ H(C) can be uniquely split into a vector 9- - (91 . . . . . 9p) each component 9j of which has a holomorphic extension into the interior of the circle Cj and a vector h = ( h , . . . , hp) each component h~ of which is the restriction to the circle Cj of fixed function h holomorphic in the intersection of the exteriors of the circles Cj (that is hj = h[cj) and which vanishes at oo. Then 9 = P+(C)fand h = P_ (C) f an d it is straightforward to check that:

p

9j = Pf f j - Z Pj,kfk, h(z)---- ~ Pj-kfk(z), k#j k= 1

where Pj.j:= P ; is understood in the equation for h.


There is a natural map from Gro(H+(C)) to Gro(H+) which we will now describe. In

fact it will be simpler and it will suffÉce for our purposes to describe this map on the

analytic Grassmannian. Let Gr,o(H+) denote the collection of subspaces in Gr(H+)

which are of the form q~H + for some element tk e LG which extends holomorphically to a neighborhood of S ~ as a map into G.

Now let L~G denote the group of smooth loops from C i into G := Gl(n, C). We write LcG for the direct sum:

p

Cca:-- Z ]=1

The group LcG acts in an obvious fashion on the Hilbert space H(C) and this action induces an action on the Grassmannian Gr(H+(C)). We define Gr,o(H+(C)) as the

collection of subspaces in Gr(H + (C)) which are of the form 4~H +(C) where q~ ~ LcG has a holomorphic extension to a neighborhood of the collection or circles C as a map into G. Suppose now that ckH+(C) is in the analytic Grassmannian Gr~,(H.(C)). Then we may think of $ as a collection ofp functions ~bj (j = 1 . . . . , p) the j th function being holomorphic in a neighborhood of the circle Cj. This collection {~bj} defines a

holomorphic vector bundle on D, obtained by glueing together the trivial bundles over each disk Dj with the trivial bundle in a neighborhood of the complement of

~']= ~ Dj via the transition functions ~bj. The fact that every holomorphic vector bundle on the disk D, is holomorphically trivial implies that one may factor the functions ~bj as tkj = ~0tki +, where in this last equation q~f is a holomorphic map from a

neighborhood of D i into G and q~ is a holomorphic map from a neighborhood of the complement of ~ = 1Di in D~ into G. The equation which relates q~j and q~0 + is understood to hold in a neighborhood of the circle C i. We now define a map T from

Gr,o(H+(C)) to Gr,~(H+) by T4)H+(C):= q)ls,H+. It is not difficult to check that this map is well defined. It is also easy to check that if

<bH+(C) is in Gro(H+(C)) then the image Tc~H+(C) is in Gro(H+). We write T o for the restriction of T to Gro(H+(C)).

We now show that T o lifts to a map To from the det* bundle over Gro(H+(C)) to the det* bundle over Gro(H+). We will do this by exhibiting a canonical isomorphism

between the kernel of P+[~,+ and the kernel of P+(C)l~on.(c) and a canonical isomorphism between the respective cokernels. Suppose to begin that u~H+(C) is such that P +(C)tpu = O.

It follows that tpu has a holomorphic extension into the exterior of the disks Dj (which vanishes at ~) . Since ~0 is holomorphic and invertible between the circles C and S 1 it follows that u has a holomorphic extension from C to S 1. Since u is already

holomorphic inside the disks Dj it follows that Uls, is in H+. Finally since ¢pu extends to a holomorphic function in the exterior of the circle S 1 it follows that P+tpUlsl = 0 and hence that ~0ulsl is in the kernel of the restriction of P+ to the subspace (pH+. Without difficulty one may check that the map which takes ~0u in the kernel of P+(C) restricted to ~oH÷(C) to qml s' in the kernel of P÷ restricted to ~0H+ is an isomorphism.

220 JOHN PALMER

Next suppose that f e H+ and consider the map:

f + P + goH + ~ f lc + P +(C)tpH +(C)

from the cokernel of the restriction of P+ to rpH+ to the cokernel of the restriction of

P+(C) to goH+(C). If f + P+q~H+ is in the kernel of this map thenf l c = P+(C)~og for some geH+(C). N o w f l c has a holomorphic continuation from C to S 1 and since f lc = (l - P_(C))gog and P_(C)gog has a holomorphic continuation from C to S 1 it follows that gog must have a holomorphic continuation from C to S 1. As before, it follows from this that g itself has an analytic continuation from an element of H+(C) to an element of H+. We see from the results above that f e l l + differs from the analytic continuation of g0g to S 1 by a function which has a holomorphic extension into the exterior of the unit circle that vanishes at oo. Thus f = P+(rpg) and it follows that the map on cokernels defined above is injective. Since the index of P ÷ I~n+ and the index of P÷(C)l~on+~c) are both zero and we've seen that the dimensions of the kernels of each of these projections are equal it follows that the cokernels must have the same dimension and hence that our map relating the cokernels must be an isomorphism.

Let V denote a finite dimensional complex vector space. We write 2(V) for the highest exterior power of V. It is not difficult to see that the fiber of the det* bundle over a subspace W is canonically isomorphic to 2(ker P + I w)* ® 2(coker P + I w) if P + I W is not an isomorphism and is canonically isomorphic to C if P + I w is an isomorphism. The isomorphism of kernels and cokernels we have exhibited thus induces an isomorphism on the fibers of the det* bundle. We have:

THEOREM. The map T O lifts to a map To from the det* bundle over Gr,0(H+(C)) to the det* bundle over Gr,o(H+) which is an isomorphism on the fibers. The canonical section over Gr,o(H+) pulls back to the canonical section over Gr(H+(C)) under this map.

Proof It remains only to verify the assertion about the canonical sections. In case P+(C) is an isomorphism when restricted to q~H+(C) the same will be true for P+ restricted to goH 4. Over those subspaces on which P + restricts to an isomorphism the canonical identification of the fiber in det* with C is/~ ~ (P~ 1, p). Thus

To(P+(C) -1, ~):= ( P ~ , #)

The canonical section vanishes over those subspaces on which P+(C) (or P+) is not an isomorphism and is equal to (P+(C)- 1, 1) (or (P+ 1, 1)) over those subspaces on which this projection is an isomorphism. The claim about the pull back of the canonical section is immediate. []

Suppose that S(z) is a holomorphic map from some neighborhood of the closed region between the circles Cj and S 1 into the general linear group Gl(n, C). Then the

map T O intertwines with S in the following fashion SIsIT o = ToSIc. From the preceding section we know that the map S(z, a) is holomorphic for z in a

neighborhood of the closed region between the circle S ~ and the circles Cj with values in Gl(n, C). Using the intertwining property above we see that the subspace S(' , a)lcH+(C) is transformed into S(' , a)H+ under the map To. If we use To to identify


the respective det* bundles then we see that the det* bundle of interest to us is equivalent to the det* bundle over the subspaces S(', a)lcH+(C). Since the canonical section pulls back under this map it is once again appropriate to define determinants

by comparison with the canonical section. In the preceding section we saw that the restriction of S( ", a) to Cj factors as S~Sf - 1

where each Sj is holomorphic in an annulus containing C~ and each Sf is holomorphic and invertible in the interior of Cj. It will be convenient to write

S(a) := Sl(a) @ '" @ Sv(a).

Then since the functions Sf are invertible and holomorphic inside C~ we have:

S( ' , a)lcH +(C) = S(a)H +(C).

There is a map from LcG into Gr(H+(C)) given by LcG ~ S ~ SH+(C) and as in the second section we can define a central extension ~Lc(G) of LcG so that this map lifts to a map from LcG into det*. If we define the matrix

a(S) b(S)') c(S~ d(s)j

of the map S acting on H(C) relative to the splitting H(C) = H+(C) O) H_(C) then the construction of Lc G is precisely analogous to the construction of LG given in Section 2. Indeed the construction of the left invariant connection for LG described in Section 2 carries over as well. We are now prepared to describe the trivialization of the det* bundle which leads to the z-function of Sato, Miwa, and Jimbo. The choice of the element S{a)eLcG allows us to regard the det* bundle over S(a)H+(C) as the LeG bundle over S(a) s LcG. Following Malgrange we now calculate the curvature of the left invariant connection in this last bundle to see that it is zero. This allows us to

trivialize the bundle with a flat section and we shall see (again following Malgrange) that the z-function of [10] is the determinant of ~,,L obtained in this fashion. Let d, denote exterior differentiation with respect to the parameters a = (a 1, a2 . . . . . av). Then the pull back of the connection one form relative to the canonical section is

o*0t = Tr(a(S(a))-la(d.S(a))- a(S(a)-ld, S(a))).

The curvature of the associated connection is the da-exterior derivative of this one form. In order to calculate this simply we make some preliminary modifications in the formula for a*e. Recall that Pf is the orthogonal projection in H1/2(Ci) on those functions which continue analytically into the interior of the circle Cj and H+(Cj) = pfHI/2(Cj). In the formula for datr*ct we wish to replace a(S-1 d.S) by its diagonal part relative to the decomposition

p

~I+(C) = ~ n+(cj~ j = l

Since the trace depends only on this diagonal part the replacement will not effect this trace. We may use the formula for P+(C) given above to calculate this diagonal part.

222 JOHN PALMER

The action of a(S-ldsS ) on H+(C) is found by restricting S - ld . S to H+(C) and then

projecting the result back on H+(C) via the projection P+(C). Since the diagonal part

of the projection P+(C) is given by P~- ~ ' " q) Pp+ we find that the diagonal part, ~ ,

of a(S ld,S) is given by:

d = a(S1 td.S1) G " " @ a(Sp 1 daSp),

where a(gt) = P] otP~-. Now write

M : = a(S), f l := M d M -1 - d a M M -1

and observe that

a*ct = Tr(M 1dam - ~ ) = -Tr(fl) .

Since fl is a trace class map we have Tr(fl ^ fl) = 0 and it follows that

daa*Ct = -Tr (d . f l ) = - T r ( d . f l + fl ^ fl).

Since fl is a 'gauge transformation' of ~ by M the relation between curvature forms:

d . f l + f l ^ f l = M ( d . d + d ^ ~ ) M - 1

is well known and it follows that

daa*a = - T r ( d . d + d ^ d ) .

Returning to the construction of the maps St(a ) in the last section we see that Sj

depends only on the j t h variable a t and it follows that d . ( S j l d . S j ) = 0 and

d ^ sd = 0. Thus the curvature of the connection is zero on the family a ~ S(a).

Let ~(a) denote a flat section over a --* S(a) in L,c(G) with respect to the connection

with one form a. Then we can define a local determinant for Oa,L as follows:

det(~.L ) _ tT(a) ~(a)"

Let • = det(~ad.). Then a = z~. If we pull back the covariant derivative of both sides of this last relation to the space o f ' a ' parameters then we find:

V,a = d , ~ + ~V,~ = d , ~ ,

since ~ is a fiat section. But Vaa = a * e - a and it follows that d, zz -1 = a*~.

At this point we have established the same formula for the logarithmic derivative of

our determinant that Malgrange uses in his Proposition 6.11. We quote his result:

T H E O R E M .

- Tr(Aj A k) 1 d(aj ak) o o d. lo9(det ~.,L) = -~ ~ Tr(AjAk) + ~k - - - - - - dak.

j:[:k a t -- ak j ao,j -- aO,k

In this last formula the matrices At(a ) are the residues of the connection one form for the solution to the Riemann-Hi lber t problem at the point aj close to aoj and


Aj.-° . _ Aj(ao)" That is

d Y @ At(a) _ _ y - 1 = ~

dz j= 1 (z - at)

As explained in Malgrange [4] these matrices are solutions to the Schlesinger equations

daAj = - ~ [A t, Ak] d(aj - ak) k~j a t -- ak

Finally we see then that our det(~a,L) is essentially the z-function of [10] and can be expressed in terms of the solution to the deformation equations (the Schlesinger equations).

References

1. Forster, O.: Lectures on Riemann Surfaces, Springer-Verlag, New York, 1981. 2. Green, M. B., Schwarz, J. H., and Witten, E.: Superstring Theory, Vol, 1, Cambridge University Press,

1987, pp. 401-411. 3. Griffiths, P., and Harris, J.: Principles of Al#ebraic Geometry, Wiley, New York, 1978. 4. Malgrange, B.: Sur les deformations isomonodromiques, in L. B. de Monvel, A. Doudy, and J. L.

Verdier, (eds.), Mathematique et Physique: Seminaire de rEcole Normale Superieure, 1979 1982, Birkhauser, Boston, 1983, pp. 400-426.

5. Mickelsson, J.: Kac Moody groups and the Dirac determinant line bundle, in Topological and Geometrical Methods in Field Theory, World Scientific, Singapore, 1986, pp. 117-131.

6. Miwa, T.: Painlev6 property of monodromy preserving equations and the analyticity of the r function, Publ. R.I.M.S. Kyoto Univ. 17-2 (1981), 703-721.

7. Palmer, J.: Pfattian bundles and the Ising model, Comm. Math. Phys. 120 (1989), 547-574. 8. Pressley, A., and Segal, G.: Loop Groups, Clarendon Press, Oxford, 1986. 9. Quillen, D.: Determinants of Cauchy Riemann operators on a Riemann surface, Funct. Anal. Appl. 19

(1985), 37-41. 10. Sato, M., Miwa, T., and Jimbo, M.: Holonomic quantum fields I-V, Puhl. RIMS, Kyoto Univ., 14 (1978)

223-267; 15 (1979) 201-278; 15 (1979) 577-629; 15 (1979) 871-972; 16 (1980) 531-584. 11. Segal, G. and Wilson, G.: Loop groups and equations of KdV type, Pub. Math. IHES 61 (1985), 5-65. 12. Simon, B.: Notes on infinite determinants of Hilbert space operators, Adv. in Math. 2,4 (1977), 244-273. 13. Witten, E.: Quantum field theory, Grassmannians and algebraic curves, Comm. Math. Phys. 113 (1988),

529-60O. 14. Wu, T. T., McCoy, B. M., Tracy, C. A., and Barouch, E.: Spin-spin correlation functions for the two

dimensional Ising model: Exact theory in the scaling region, Phys. Rev. BI3 (1976), 316 374.

determinants of cauchy-riemann operators as τ-functions

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