Higgs fields, non-abelian Cauchy kernels and the Goldman symplectic structure
aa r X i v : . [ m a t h . AG ] F e b Higgs fields, non-abelian Cauchy kernels and the Goldmansymplectic structure
M. Bertola †‡♣ C. Norton ♥ G. Ruzza ♦ . † Department of Mathematics and Statistics, Concordia University1455 de Maisonneuve W., Montr´eal, Qu´ebec, Canada H3G 1M8 ‡ SISSA, International School for Advanced Studies, via Bonomea 265, Trieste,Italy ♣ Centre de recherches math´ematiques, Universit´e de Montr´ealC. P. 6128, succ. centre ville, Montr´eal, Qu´ebec, Canada H3C 3J7 ♥ Department of Mathematics, University of Michigan, Ann Arbor, MI 48109,USA ♦ Institut de recherche en math´ematique et physique, Universit´e catholique de Lou-vain, Chemin du Cyclotron 2, 1348 Louvain-la-Neuve, Belgium
Abstract
We consider the moduli space of vector bundles of rank n and degree ng over a fixed Riemann surface ofgenus g ≥
2. We use the explicit parametrization in terms of the Tyurin data. In the moduli space there is a“non-abelian” Theta divisor, consisting of bundles with h ≥
1. On the complement of this divisor we constructa non-abelian Cauchy kernel explicitly in terms of the Tyurin data. With the additional datum of a non-specialdivisor, we can construct a reference flat holomorphic connection which is also dependent holomorphically onthe moduli of the bundle. This allows us to identify the bundle of Higgs fields, i.e. the cotangent bundle ofthe moduli space, with the affine bundle of holomorphic connections and provide a monodromy map into theGL n character variety. We show that the Goldman symplectic structure on the character variety pulls backalong this map to the complex canonical symplectic structure on the cotangent bundle and hence also on thespace of affine connections. The pull-back of the Liouville one-form to the affine bundle of connections is thenshown to be a logarithmic form with poles along the non-abelian theta divisor and residue given by h . Contents T ∗ b V
126 Flat connections and the Cauchy kernel 137 Equivalence of the (complex) symplectic structure on T ∗ V and the Goldman bracket on R ( C , GL n )
168 Singularities of the Malgrange-Fay form along the non-abelian theta divisor 199 Comments and open problems 21A Proof of Theorem 7.4 22 Marco.Bertola@ { concordia.ca, sissa.it } [email protected] [email protected] Introduction and results
In the study of stable holomorphic vector bundles over a Riemann surface C of genus g > holomorphically . This famous result establishesa one-to-one correspondence between stable holomorphic bundles and projective unitary representations and thusprovides the moduli space with the structure of real smooth manifold; the trace invariants of monodromies arethe smooth functions on the moduli space.Of a similar nature is the even more classical result of Fuchsian uniformization of Riemann surfaces, whichprovides real coordinates on the moduli space of Riemann surfaces of genus g ≥
2; the Schwartzian derivative withthe uniformizing coordinate provides a natural projective connection on the Riemann surface C . However, again,this projective connection, while holomorphic with respect to the conformal structure of the Riemann surface,does not depend holomorphically on the moduli.The space of (holomorphic) projective connections is an affine space modelled on the space of quadraticdifferentials on C ; the latter is identified with the fiber of the cotangent bundle to the Teichm¨uller space T g above C . A projective connection is equivalent to the datum of an “oper”, namely, a second order ordinary differentialequation ( ∂ z + U ( z )) ψ ( z ) = 0 (1.1)for a holomorphic ( − / ψ , where U is the projective connection. The monodromy representation of(1.1) is a function of the chosen projective connection U . The set of holomorphic projective connections is anaffine bundle over T g . The fiber is an affine space modelled over the corresponding fiber of the cotangent bundleand, hence, over the space of holomorphic quadratic differentials. If we choose a reference projective connection U C then we can identify the cotangent space with the space of projective connetctions by the map φ U C + φ . Ifthe reference connection U C depends holomorphically on the moduli, then the above map allows us to map T ∗ T g into the affine bundle of projective connections and then to the P SL character variety by the monodromy map.This was the driving idea in [17, 13, 4].On the P SL ( C ) character variety there is a natural complex–analytic Poisson symplectic structure due toGoldman [15]; the question posed and answered in [17, 4] was precisely how to choose U C in such a way that theabove map from T ∗ T g to the character variety is complex-analytic and establishes a Poisson morphism betweenthe canonical complex symplectic structure on T ∗ T g and the character variety. For this purpose, the Fuchsianprojective connection is not appropriate because it does not depend analytically on the moduli.A parallel problem can be posed in the context of holomorphic vector bundles. The space of holomorphic(flat) connections on a flat holomorphic vector bundle X of rank n is also an affine vector space modelled on thespace of Higgs fields H (End( X ) ⊗ K ), where K denotes the canonical bundle of the curve C . Similarly to thecase of quadratic differentials, a holomorphic Higgs field on a bundle X can be thought of as a cotangent vectorof the holomorphic cotangent bundle of the moduli space of vector bundles. Hence we can view the space, A , ofholomorphic connections as an affinization of the cotangent bundle of the moduli space.With regard to symplectic structures, the broad strokes picture is then quite similar to the situation with opersand projective connections; on one side the P GL n ( C ) character variety has a complex–analytic Poisson symplecticstructure due to Goldman, and on the other side the holomorphic cotangent bundle of the moduli space has anatural complex symplectic structure. The question we pose and answer here is the exact analog of the case ofopers; namely, whether there is a natural choice of reference holomorphic connection for every vector bundle X ,depending holomorphically on the moduli of the bundle, and such that the mapping of the cotangent bundle tothe character variety results in a Poisson morphism (at least on an open dense set). Outline of results.
We consider the moduli space V of vector bundles E of rank n and degree ng such thatthe generic fiber is spanned by the holomorphic sections; these vector bundles admit a convenient and explicitparametrization due to Tyurin. There seems to be not much literature using this parametrization and the originalpapers [22, 23] have not been translated from Russian. We also could not really follow the original paper due tothe language barrier. The notion of Tyurin data was notably used in [8] and works of Krichever [18, 19]; in loc.cit. only the generic case where the Tyurin divisor is simple is considered.For this reason we re-derive a self-contained treatment of the notion in Section 2 including the general case ofarbitrary multiplicity of the Tyurin points. We work mostly in the context of framed vector bundles; namely we2 ∗ V A D R ( C , GL n ) V A = F D +Φ M F D (1.2)Figure 1: Illustration of the main spaces and maps.choose a point ∞ ∈ C and a basis of global holomorphic sections that trivialize the fiber at ∞ .In terms of the Tyurin data we can derive the Riemann–Roch theorem for vector bundles in the same spiritas the classical treatment of line bundles, using an analogue of the Brill–Noether matrix, see Section 3.The central object in our construction is the non–abelian Cauchy kernel for a framed vector bundle E . This is ageneralization of the standard notion of Cauchy kernel on a Riemann surface [9]. This kernel is a matrix, C ∞ ( q, p )whose rows (as functions of p ) are meromorphic sections of the bundle E with pole at q and zero at ∞ while thecolumns (as functions of q ) are meromorphic sections of E ∗ ⊗ K with only poles at p, ∞ . The kernel is determinedby the normalization condition that res q = p C ∞ ( q, p ) is the identity matrix, see Section 4 and Theorem-Definition4.2. Furthermore, it exists if h ( E ) = 0 and has a simple pole as a function of the holomorphic moduli on thenon-abelian Theta divisor, i.e., the locus (Θ) ⊂ V where h ( E ) >
0. We show how to construct explicitly suchkernels in term of the Tyurin data, see (3.2) in the general case and (4.13) for the generic case of simple Tyurinpoints.The Cauchy kernel has the following two primary uses:- it allows to construct explicitly all Higgs fields (see Sec. 5);- it naturally defines an affine connection that depends holomorphically on the moduli of the bundle (Sec. 6).This allows to holomorphically map the cotangent bundle of the moduli space to the character variety. Toconstruct this connection we fix an arbitrary reference line bundle of degree g corresponding to a non-specialdivisor D and tensor E by its dual so as to obtain a bundle of degree zero. Then we define the referenceconnection F D on this resulting bundle by (6.8), which is a holomorphic connection on the bundle.The connection d − F D is holomorphic on the bundle E ∗ ⊗ O ( D ) or can be equivalently viewed as meromorphicon the trivial vector bundle; in this latter perspective F D is simply a matrix of meromorphic differentials on C such that the local monodromy of the connection around each pole is trivial (Thm. 6.3). This induces a map,parametrized by the choice of D , from the moduli space V to the GL n ( C ) character variety of the fundamentalgroup of C R ( C , GL n ) := Hom (cid:0) π ( C , ∞ ) , GL n (cid:1) / P GL n , (1.3)where the action of P GL n is the global conjugation.The choice of the divisor D affects only projectively the representation (Sec. 6.2) which is therefore independentof D in the P GL n character variety: this means that we can construct a canonically defined immersion of themoduli space V in the P GL n character variety; since C ∞ depends holomorphically on the Tyurin data, so doesthe representation. This construction provides an alternative description to the projective-unitary one used inconjunction with Narasimhan–Seshadri’s theorem.In Sec. 6 we then consider A D , the bundle over V whose fiber consists of connections that are holomorphicon E ∗ ⊗ O ( D ): the reference connection F D provides a section of the canonical projection A D → V which isholomorphic away from the non-abelian Theta divisor (Θ) and with a simple pole therein. This section allows usto identify the cotangent bundle T ∗ V (away from the theta divisor) with A D by sending the Higgs field Φ to theconnection d − ( F D + Φ). Composing this identification with the monodromy map, we obtain a map from thecotangent bundle T ∗ V to the character variety R ( C , GL n ). The resulting picture is illustrated in Fig. 1.We consider the symplectic properties in Sec. 7. Consider the tautological holomorphic (Liouville) one-form λ on the (holomorphic) co–tangent bundle of the moduli space: we can pull it back onto the space of connections A D by our choice of reference connection F D . The resulting expression Ξ is the Malgrange–Fay one–form (Definition7.1), written as an integral over the canonical dissection of the curve C . This defines a symplectic structure on A D . On the other hand, we can pull back to A D the (complex) symplectic structure induced by the Goldman3oisson bracket [15] (which is non-degenerate for closed surfaces). Then our result is that these two structurescoincide up to a simple proportionality factor (Thm. 7.5).Finally we analyze the singularity structure of the Malgrange–Fay form Ξ; it is ill–defined on the non-abeliantheta divisor (Θ) and we show that it is a logarithmic form with a simple pole along (Θ) and residue equalto h ( E ) (Thm. 8.1). This is the parallel of a result of Fay (Theorem 2 in [10]). However the form that Fayintroduced in loc. cit. was not identified with the tautological form on the cotangent bundle of the moduli space,and furthermore no connection with the Goldman bracket was identified.As a logarithmic form, the Malgrange–Fay form Ξ has the same singularities, locally, as the differential ofthe logarithm of a determinant; this is identified as the determinant of the inverse to the Brill–Noether–Tyurinmatrix used in the construction of the non-abelian Cauchy kernel. Illustrative example: line bundles.
Given a generic line bundle E (equivalently, a non-special divisor T ) ofdegree g and a point ∞ ∈ C there is a unique Cauchy kernel C ∞ ( q, p ) with poles at T , q and a zero at ∞ in thevariable p and poles at T , p, ∞ in the variable q . It is a differential in q and a function with respect to p . Thenormalization is determined by res q = p C ∞ ( q, p ) = 1.In this case the reference connection F D can be written simply in terms of Riemann Theta functions as follows: F D = d p ln Θ( A ( p − ∞ ) − f )Θ( A ( p − D ) − K ) + ~ω ( p ) · ∇ ln Θ( f ) , where A denotes the Abel map, K the vector of Riemann constants, f = A ( T − ∞ ) + K and ~ω the row vectorof the g holomorphic α –normalized differentials and ∇ denotes the gradient operator. Inspection shows that F D has a simple pole at every point f of the Jacobian where Θ = 0. These correspond precisely to line bundles T of degree g such that the corresponding line bundle E has h ( E ) ≥
1. The connection F D is a holomorphicconnection on O ( T − D ) and is single–valued on the Picard variety (i.e. with respect to T or, equivalently, f inthe Jacobian).The fiber of A D above E can be identified with meromorphic differentials of the form A = φ + F D where φ (aholomorphic differential) represents the Higgs field. In this case the character variety consists of the exponentialsof the periods of A (note that the integer residues at the points of T , D do not contribute to the monodromyrepresentation). This gives a (scalar) representation of the fundamental group with monodromy factors χ ( γ ) = exp I γ A . A direct computation of the Malgrange form (Def. 7.1) in this simple case gives Ξ = − c t · δ f where c is the vectorof coefficients of φ = P c ℓ ω ℓ . Here and in the rest of the paper δ denotes the exterior derivative operator on V Inthis scalar case the Goldman Poisson bracket is simply { ln χ ( γ ) , ln χ ( e γ ) } = γ ◦ e γ, denoting by ◦ the intersection number of the curves. This is a non-degenerate Poisson bracket and correspondsto the symplectic form g X j =1 δ ln χ ( α j ) ∧ δ ln χ ( β j ) = g X j =1 δ I α j A ∧ δ I β j A = − π i δ f ∧ δ c = − π i δ Ξ . (1.4)In order to pull back the corresponding symplectic form on A D and see the type of singularity along the thetadivisor we need to choose a locally smooth section of the map A D → V : we choose A = F D + ~ω ( p )( c −∇ ln Θ( f )).Then the Malgrange form becomes Ξ = δ ln Θ( f ) − c t · δ f , which shows its logarithmic nature on the Theta divisor.The fact that the residue of Ξ on (Θ) is h ( E ) is simply a restatement of the classical Riemann’s singularity theorem.This example illustrates the gist of the results of our paper. Common notations • C : a curve of genus g ≥
2, with a choice of point ∞ ∈ C . K : the canonical bundle of C • V : the moduli spaces of vector bundles, E , of rank n and degree ng such that the fiber over ∞ is spannedby global holomorphic sections (as is the generic fiber).4 b V : like the above but also with n “framing” sections linearly independent at ∞ . • (Θ) , d (Θ): the non-abelian theta divisors where h ( E ) > V , b V , respectively. • V := V \ (Θ), b V := b V \ d (Θ). • T the Tyurin divisor of a vector bundle see Sec. 2. We use T for the Tyurin space (2.11). • T : the Brill–Noether–Tyurin matrix (3.2). The Theta divisor is the locus det T = 0. • A D E : the affine space of holomorphic connections on E ∗ ⊗ O ( D ). Then A D is the corresponding bundleover V , whose fiber at E is A D E . See Sec. 6. • ω j , j = 1 , . . . , g a Torelli marked basis of holomorphic differentials; ω z,w ( p ) the third kind normalizeddifferential with residue +1 at z and − w . Acknowledgements.
The authors wish to thank Prof. Indranil Biswas for valuable suggestions. This materialis based upon work supported by the National Science Foundation under Grant No. 1440140, while the firstauthor was in residence at the Mathematical Sciences Research Institute in Berkeley, California, during the fallsemester of 2019.C.N. was supported from the AMS-Simons Travel Grants, which are administered by the American Mathe-matical Society with support from the Simons Foundation.G.R. acknowledges support from European Union’s H2020 research and innovation programme under theMarie Sk lowdoska–Curie grant No. 778010
IPaDEGAN and from the Fonds de la Recherche Scientifique-FNRSunder EOS project O013018F.
Throughout this paper, let C be a fixed compact Riemann surface of genus g ≥
2. For any vector bundle E ofrank n and degree ng on C , the Riemann-Roch theorem h ( E ) = h ( E ) + deg E − n ( g −
1) = h ( E ) + n, h i ( E ) := dim H i ( E ) , (2.1)implies that E has at least n holomorphic sections. We shall see in Section 3 that generically, we have h ( E ) = n ,namely E has exactly n holomorphic sections. The degree ng case is particularly interesting, as the n sectionsprovide a convenient parametrization of (an open part in) the moduli space of vector bundles on C , as we nowexplain expanding on the works of Tyurin and Krichever [22, 23, 19]. Let us fix a point ∞ ∈ C , and let V be the set of isomorphism classes of vector bundles E over C such thatrank E = n , deg E = ng , and the locus of points p ∈ C such that the fiber E p is spanned by global holomorphicsections is a dense set in C containing ∞ . One may cover a more general moduli space letting the point ∞ vary,but for our purposes it is convenient to fix it once for all.A (global) framing of E ∈ V is a collection S = ( s , . . . , s n ) of global holomorphic sections s i ∈ H ( E ), suchthat s ( ∞ ) , . . . , s n ( ∞ ) are linearly independent. Let b V the set of isomorphism classes of pairs ( E , S ) with E ∈ V and S = ( s , . . . , s n ) a framing of E ; we require isomorphisms to commute with the framing sections s i .For any ( E , S ) ∈ b V let us introduce the Tyurin divisor T of points of C where s ∧ · · · ∧ s n = 0. By ourassumptions on the degree of E , T = P t n t t is a positive divisor of degree P t n t = ng on C ; moreover, ∞ 6∈ T .We also fix a conformal D (for example by Fuchsian uniformization) such that T ⊂ D ⊂ C \ {∞} . Let usdenote z the coordinate on D , writing z ∈ D both for the point and the coordinate representing it.Given ( E , S ) ∈ b V we can trivialize the bundle E over C \ T , where the sections s i span the fiber and we identifythem with the canonical basis e i of C n , and D , which is contractible. Throughout this paper, all vectors are meantas column vectors unless otherwise explicitly stated. Denote g i ( z ) the vectors representing the trivialization of s i on D and set G ( z ) = (cid:2) g ( z ) · · · g n ( z ) (cid:3) t , z ∈ D . (2.2)5he matrix G ( z ) is a holomorphic function of z ∈ D and det G vanishes precisely at T . The matrix G serves therole of transition function for E ; namely, we can identify a local holomorphic section of E over the open U ⊆ C with a vector meromorphic function s = s ( p ), p ∈ U with poles at T ∩ U only such that s t ( z ) G ( z ) is analytic at z ∈ T ∩ U. (2.3)For instance, the framing global sections s i are identified with the constant functions e i .Let R be the ring of holomorphic functions of the variable z in a domain containing the closure D . Lemma 2.1
Let G ( z ) ∈ Mat n ( R ) be such that det G ( z ) ∈ R does not vanish identically. There exist uniquematrices H ∈ GL n ( R ) and P ∈ Mat n ( C [ z ]) such that G ( z ) = P ( z ) H ( z ) (2.4) with P (termed polynomial normal form of G ) a polynomial matrix of the form P ( z ) = p ( z ) 0 0 · · · f ( z ) p ( z ) 0 · · · f ( z ) f ( z ) p ( z ) · · · ... ... ... . . . ...f n ( z ) f n ( z ) f n ( z ) . . . p n ( z ) (2.5) where p j ( z ) are monic polynomials of degree d j with all their zeros in D , and the entries f jk ( z ) are polynomialsof degree ≤ d j − . Moreover, two matrices G , G ∈ Mat n ( R ) have the same polynomial normal form if and onlyif G = G H for some H ∈ GL n ( R ) . Proof.
This is a minor elaboration on a similar statement in [14, Pag 137]. Consider the first row of G andassume it is not identically zero. We find an element whose divisor of zeroes in D is of minimal degree and bring itto the (1 ,
1) entry by a column permutation. Now G , ( z ) = p ( z ) a ( z ) with a ( z ) ∈ GL ( R ) and p ( z ) a polynomialwith all its zeros in D . So we can divide the first column by a ( z ) and assume G , is a polynomial. By adding a R -multiple of the first column to the remaining columns, we can assume that the whole first row of G is made ofpolynomials with deg G ,j ≤ deg G , −
1. Suppose G ,j are not all zero ( j = 2 , . . . , n ). We select the one whosedivisor in D is of lowest degree and then bring it to the first entry by a column permutation and repeat. At eachiteration the degree of G , decreases by at least one and the process stops when either G , is a constant or allthe other entries on the first rows are zero.We then repeat the process on the submatrix (2 ..n ; 2 ..n ) and so on. At the end of this first pass the matrix G has been transformed to a lower triangular matrix with polynomials on the diagonal whose roots are all in D . G ≃ p ( z ) 0 . . . ⋆ p ( z ) . . . ... . . .⋆ ⋆ . . . p n ( z ) , deg p j = d j . (2.6)The elements on the lower triangular part are some analytic functions, and ≃ denotes identity up to rightmultiplication by GL n ( R ). Again, by column operations only, we can reduce the ( j, k ) entry to a polynomial ofdegree ≤ d j −
1. This proves existence of
P, H as in the statement.To prove the uniqueness of
P, H it is enough to show that H = is the only solution of the equation P = e P H with P, e P ∈ Mat n ( C [ z ]) of the form (4.3) and H ∈ GL n ( R ). To see it, assume P = e P H and call h ab the entries of H . It is clear that H is also lower-triangular and hence for the diagonal entries we have p i = e p i h ii , which is onlypossible if h ii = 1 and p i = e p i . Then e f a +1 ,a = f a +1 ,a + h a +1 ,a p a +1 , and since deg( f a +1 ,a − e f a +1 ,a ) < deg p a +1 wemust have h a +1 ,a = 0. We proceed to the next sub-diagonal entries, and we have e f a +2 ,a = f a +2 ,a + h a +2 ,a p a +2 ,and since deg( f a +2 ,a − e f a +2 ,a ) < deg p a +2 we must have h a +2 ,a = 0. Repeating this argument iteratively we provethat H = .For the last statement, suppose G i = P i H i for some matrices P i , H i ( i = 1 ,
2) of the form in the statement.If P = P then clearly G = G H for H = H − H ∈ GL n ( R ). Conversely, if G = G H and G = P H , then G = P H H − , and G , G have the same normal form P . (cid:4) Now, let G ∈ Mat n ( R ) be given in (2.2): we call the coefficients of the polynomials of P in (2.5) the moduli of the framed bundle ( E , S ) ∈ b V .To see that the moduli provide local coordinates on b V , consider the following construction of representative ofthe equivalence class of framed vector bundles. 6 emark 2.2 (Construction of a framed bundle from its moduli.) Suppose we have fixed a coordinate disk D ⊂ C as above, and within it we fix a polynomial normal form P ∈ Mat n ( C [ z ]) as in (2.5) of Lemma 2.1, suchthat the divisor T := (det P ) has degree ng ; restricting P : D \ T → GL n ( C ) we obtain a transition functiondefining a vector bundle E , trivialized over D and C \ T . Namely, as above, local holomorphic sections of thebundle E just constructed are vector meromorphic functions s = s ( p ) of p ∈ U with poles at T ∩ U only such that s t ( z ) P ( z ) is analytic at z ∈ T ∩ U. (2.7)We also canonically associate the framing S = ( s , . . . , s n ) with s i equal to the constant vector e i . It followsthat if P ∈ Mat n ( C [ z ]) is the polynomial normal of the framed vector bundle ( E , S ), then the new vector bundleproduced by this construction is isomorphic to E . Indeed from the construction of Lemma 2.1 we have G = P H with H analytic and invertible in D , so that the transition functions G, P define isomorphic bundles. △ Denoting d j = deg p j as in Lemma 2.1, we have P nj =1 d j = ng . By (2.5) there are P nj =1 jd j ≤ n g moduli.Therefore, the set M D of moduli (for a fixed coordinate disk D ) is stratified as M D = G M ( d ,...,d n ) D , (2.8)where the strata are labeled by ( d , . . . , d n ) with d j ≥ , P nj =1 d j = ng ; the moduli provide a complex manifoldstructure of dimension P nj =1 jd j to M ( d ,...,d n ) D . The open stratum M (0 ,..., ,ng ) D , of dimension n g , is described inthe following example. Example 2.3
According to Lemma 2.1, in the generic case ( d = 0 , . . . , d n = ng ) the matrix G has the form G ( z ) = · · · · · · ... ... . . . ... ... · · · h ( z ) h ( z ) · · · h n − ( z ) p ( z ) H ( z ) (2.9) where H ( z ) ∈ GL n ( R ) , p ( z ) is a monic polynomial of degree ng with zeroes in D and h j ( z ) are polynomials in C [ z ] / h p ( z ) i , i.ee., arbitrary polynomials of degree ≤ ng − . The number of parameters of the polynomials ( h j ) n − j =1 is ( n − ng , thus providing, along with the coefficients of p , a total number of n g parameters. In geometric termsthis shows that E is generically an extension of a rank n − trivial subbundle by the line bundle det E ; → O n − → E → det E → . For different coordinate disks D ⊂ C this construction provides an atlas of complex charts of dimension n g on the open part in b V where the normal form is generic in the sense of this example. Remark 2.4
Lemma 2.1 also shows that the transition function can always be put in lower-triangular form.Hence there is a complete flag E = { } ⊂ E ⊂ · · · ⊂ E n = E of sub-bundles of E , with E r of rank r (seee.g. [16]). Clearly deg E r = P ri =1 d i ; in particular (semi-)stability implies P rj =1 d j < rg ( P rj =1 d j ≤ rg ) for all r ∈ { , . . . , n } . △ Let us denote, as above, R the ring of functions of z holomorphic in a domain containing D ; let also ~ R = C n ⊗ R ,thought of as row-vectors. Moreover, for a (possibly vector-valued) meromorphic function f = f ( z ) of z ∈ D wedenote C − [ f ]( z ) := X p ∈ ( f ) − res z ′ = z ( p ) f ( z ′ )d z ′ z − z ′ , (2.10)where the sum extends over the poles of f in D , the Cauchy operator extracting the polar part of f in D . Definition 2.5
The
Tyurin space associated with the framed bundle ( E , S ) ∈ b V is the complex vector space T := C − h ~ R P − i , (2.11) where P is the polynomial normal form introduced above. v t ( z ) , . . . , v tng ( z ) ∈ T defined by v td + ··· + d j − + k ( z ) = C − [ z k e tj F − ( z )] , k = 0 , . . . , d j − , j = 1 , . . . , n, (2.12)where d j is the degree of the diagonal entries p j ( z ) in the polynomial normal form P , as in (2.5) (we remind that P nj =1 d j = ng ). The proof of the following lemma is elementary and hence omitted. Lemma 2.6
The vectors in (2.12) form a basis of T ; in particular dim T = ng . Remark 2.7 (Relationship with Tyurin vectors)
If the Tyurin divisor T associated with ( E , S ) ∈ b V isdenoted T = P j n j t j , then T is a direct sum of spaces of Laurent series in z − z ( t j ) of dimensions n j ; moreover C − [ zT ] ⊆ T , and in particular each of the “localized” subspaces is invariant under multiplication by z − z ( t j ) andprojection to the polar tail. If in addition, the Tyurin divisor is simple (i.e. n j = 1) we have T = ng M j =1 C v tj , v j ( z ) := h j z − z ( t j ) (2.13)for some nonzero vectors h , . . . , h ng ∈ C n ( Tyurin vectors [22, 23, 18, 8]). △ Remark 2.8 (Cohomological interpretation of the Tyurin space)
Given ( E , S ) ∈ b V , we can define a shortexact sequence of sheaves on C → O n → E → T → . (2.14)Here O n is the sheaf of C n -valued holomorphic functions on C , and E is identified with the sheaf of its holomorphicsections; the first map is defined, for any open U ⊆ C , by( f , . . . , f n ) ∈ O n ( U ) f s + · · · + f n s n ∈ E ( U ) . (2.15)The quotient sheaf T is a skyscraper sheaf supported at the Tyurin divisor T . The Tyurin space T introducedabove in (2.11) is isomorphic to the space H ( T ) of global sections of T ; indeed, H ( T ) can be regarded as thespace of polar tails of rational functions of z ∈ D , such that right-multiplied by P ( z ) they become analytic on D . △ Let (Θ) ⊂ V the set of isomorphism classes of vector bundles E with h ( E ) > n . Let us note that, in view ofthe Riemann-Roch theorem and of the Serre duality, this condition is equivalent to h ( E ) = h ( E ∗ ⊗ K ) > d (Θ) ⊂ b V consist of framed bundles ( E , S ) with h ( E ) > n . Let us also denote V = V \ (Θ) and b V = b V \ d (Θ). We shall refer to (Θ) and d (Θ) as non-abelian theta divisors ; the goal of this section is to provideexplicit holomorphic equations for d (Θ).Let T be the Tyurin space (2.11) associated with ( E , S ) ∈ b V . Denote K the canonical bundle on C andintroduce a bilinear pairing between T and H ( K n ) as follows; v t ⊗ ν (cid:10) v t , ν (cid:11) = X t ∈ T res t v t ν . (3.1)Hereafter we agree that elements of H ( K n ) are (column) vector holomorphic differentials on C .Let us fix a basis ω , . . . , ω g of holomorphic differentials on C and let e j be the standard basis of C n . Withrespect to the basis v tj ( z ) (2.12) of T , and the basis ω i ⊗ e j of H ( K n ), the pairing (3.1) is represented by the Brill-N¨other-Tyurin matrix T = X t ∈ T res z = z ( t ) v t ( z ) ... v tng ( z ) ⊗ [ ω ( z ) , . . . , ω g ( z )] ∈ Mat ng × ng ( C ) , (3.2)where ⊗ denotes the Kronecker product. Note that T depends holomorphically on the moduli of ( E , S ).We explicitly identify the left and right null-spaces of this pairing. To this end, given ( E , S ) ∈ b V we considerthe bundle E ∗ ⊗ K trivialized similarly as explained above for E ; namely we identify its local holomorphic sectionsover the open U ⊆ C with vector holomorphic differentials η = η ( p ) of p ∈ U satisfying P − ( z ) η ( z ) analytic at z ∈ T ∩ U, (3.3)with P ( z ) the polynomial normal form. 8 roposition 3.1 [1] The left null-space of (3.1) is identified with the space H ( E ⊗O ( −∞ )) of global holomorphicsections of E vanishing at ∞ by the map r ∈ H ( E ( −∞ )) C − [ r ( z )] ∈ T. (3.4) In particular h ( E ) = n + corank T . (3.5) [2] The right null-space of the pairing (3.1) is identified with H ( E ∗ ⊗ K ) . In particular h ( E ∗ ⊗ K ) = h ( E ) = corank T . (3.6) Proof. [1]
The section r ∈ H ( E ( −∞ )) by the above discussion is a vector meromorphic function r = r ( p )of p ∈ C with r ( ∞ ) = 0 and poles at T only, such that r t P is analytic at T . For all vectors of holomorphicdifferentials ν ∈ H ( K n ) we have h C − [ r ( z )] , ν i = X t ∈ T res t r t ν = 0 , (3.7)as r t ν is a globally defined meromorphic differential on C with poles at T only.Conversely, given v t ( z ) ∈ T and fixing a symplectic basis { a , b , . . . , a g , b g } of H ( C , Z ), there exists a uniquerow vector meromorphic differential Ω t on C satisfying( Ω ) ≥ − T , Ω ( z ) − d v t ( z ) analytic in D , I a j Ω = 0 , j = 1 , . . . , g. (3.8)Define a row vector function r by r ( p ) = R p ∞ Ω ; r ( p ) is a single-valued vector meromorphic function of p ∈ C if and only if Ω has vanishing b -periods too. It follows from the Riemann bilinear relations for second kindabelian differentials that this is the case if and only if v t ( z ) ∈ T is in the left null-space of h , i . In such case, r ∈ H ( E ( −∞ )) and we have explicitly inverted (3.4). [2] It follows from the fact that P − η is regular at T if and only if η is in the right-kernel of h , i . (cid:4) Incidentally, the Riemann-Roch formula (2.1) also follows by comparing (3.5) and (3.6).
Corollary 3.2
The locus d (Θ) is a divisor in b V , defined by local holomorphic equations det T = 0 . The fact that det T is not identically zero on b V follows by considering the bundle E = O ( D ) ⊕ n for a non-specialdegree g divisor D on C ; taking D special shows instead that d (Θ) is non-empty. Example 3.3
In the case of line bundles, n = 1 , and distinct Tyurin points t , . . . , t g , T reduces to the Brill–N¨other matrix T = [ ω i ( t j )] gi,j =1 and the condition det T = 0 defines the locus of degree g special divisors. Remark 3.4 (Cohomological interpretation of the pairing)
The short exact sequence (2.14) induces thelong exact sequence in cohomology0 → H ( O n ) = C n → H ( E ) → H ( T ) → H ( O n ) → H ( E ) → H ( T ) = 0. By Serre duality and theidentification of H ( T ) with the Tyurin space T (Rem. 2.8), we rearrange this exact sequence as0 → H ( E ( −∞ )) → T φ −→ H ( K n ) ∗ → H ( E ∗ ⊗ K ) ∗ → , (3.10)where we identify H ( E ) / C n ≃ H ( E ( −∞ )). It follows that the map φ induces a pairing h , i : T ⊗ H ( K n ) → C . (3.11)By exactness of (3.10) the left null-space is isomorphic to H ( E ( −∞ )) and right null-space is isomorphic to H ( E ∗ ⊗ K ), as shown explicitly in Prop. 3.1. △ Remark 3.5 (Non-abelian theta divisor and semi-stability) If E ∈ V then E is necessarily semi-stable.For, suppose E is not semi-stable and let F be a sub-bundle of rank r = 0 , n such that deg F = rg + k for some k ≥
1. On one side, the Riemann-Roch Theorem applied to F implies h ( F ) = h ( F ) + deg F − r ( g − ≥ r + k ;on the other side, since E is generically spanned, we must have at least another n − r sections of E , and so h ( E ) ≥ n + k . △ We thank Prof. Indranil Biswas for pointing this out. Non-abelian Cauchy kernel
We bow introduce the main tool in the subsequent analysis, namely the matrix Cauchy kernel. The constructionis explicit in terms of the moduli of the bundle.
Proposition 4.1
Let ( E , S ) ∈ b V . Fix a point p ∈ C \ ( T + ∞ ) . There exist exactly n linearly independent globalholomorphic sections of the bundle E ∗ ⊗ K ( p + ∞ ) . Proof.
A global holomorphic section of E ∗ ⊗ K ( p + ∞ ) is a vector meromorphic differential ν such that ν hassimple poles at p, ∞ only and P − ν is analytic at T . Let us fix a third kind abelian differential ω p, ∞ on C withsimple poles at p, ∞ only; ν must be in the form ν = ω p, ∞ e + b ν , where b ν is a vector of holomorphic differentialson C and e ∈ C n is any constant vector. The condition that P − ν does not have poles at T translates into thecondition X t ∈ T res z = z ( t ) v t ( z ) [ ω p, ∞ ( z ) e + b ν ( z )] = 0 , ∀ v t ( z ) ∈ T. (4.1)The condition uniquely determines b ν due to the assumption E 6∈ (Θ). Indeed, recalling the basis(2.12) of T , fixinga basis ω , . . . , ω g of H ( K ), and writing b ν = P gk =1 b k ω k for the unknown constant vectors b , . . . , b g ∈ C n , thecondition (4.1) is equivalent to T b ... b g + X t ∈ T res z = z ( t ) ω p, ∞ ( z ) v t ( z ) ...ω p, ∞ ( z ) v ng ( z ) e = 0 , (4.2)where T is given in (3.2), for the same choice of ω , . . . , ω g . This has a unique solution b , . . . , b g , as det T = 0since E 6∈ (Θ). The conclusion follows as e is arbitrary in C n . (cid:4) Theorem–Definition 4.2 (Non-abelian Cauchy kernel)
Fix a framed vector bundle ( E , S ) ∈ b V . Thereexists a unique n × n (matrix) Cauchy kernel C ∞ ( q, p ) such that- it is a meromorphic differential in the variable q with its divisor satisfying ( C ∞ ( q, p )) q ≥ − p − ∞ ;- it is a meromorphic function in the variable p with its divisor satisfying ( C ∞ ( q, p )) p ≥ ∞ − q − T ,and such that the following properties hold:1. the expression P − ( q ) C ∞ ( q, p ) is regular for q ∈ D , and2. it satisfies the normalization condition res q = p C ∞ ( q, p ) = n × n = − res q = ∞ C ∞ ( q, p ) . (4.3) Proof.
According to Proposition 4.1, let η ℓ ( q ; p, ∞ ) be the unique sections of E ∗ ⊗ K (in q ) with simple poles at q = p, ∞ only, normalized byres q = p η ℓ ( q ; p, ∞ ) = e ℓ = − res q = ∞ η ℓ ( q ; p, ∞ ) , ℓ = 1 , . . . , n. (4.4)Then the matrix C ∞ ( q, p ) := (cid:2) η ( q ; p, ∞ ) · · · η n ( q ; p, ∞ ) (cid:3) (4.5)has the desired properties. The uniqueness follows from Prop. 4.1. (cid:4) We can write the Cauchy kernel more explicitly in terms of the basis v tj ( z ) given in (2.12), of a fixed basis ~ω = ( ω , . . . , ω g ) (understood as a row-vector) of holomorphic differentials on C , and of the matrix T given in(3.2). Indeed from (4.2) and (4.5) we infer that the Cauchy kernel can be written as C ∞ ( q, p ) = ω p, ∞ ( q ) n × n − ( ~ω ( q ) ⊗ n × n ) · T − · X t ∈ T res z = z ( t ) v t ( z ) ... v tng ( z ) ω p, ∞ ( z ) (4.6)10here ⊗ denotes the Kronecker product, and ω p + ,p − ( q ) denotes the third–kind differential with simple poles at p ± and residue ±
1, respectively, normalized to have vanishing a –cycles. In more transparent terms, the entriesof C ∞ ( q, p ) can be expressed as( C ∞ ( q, p )) i,j = det T P t ∈ T res z = z ( t ) v t ( z ) e i ω p, ∞ ( z ) ... v tng ( z ) e i ω p, ∞ ( z ) ~ω ( q ) ⊗ e tj δ ij ω p, ∞ ( q ) det T . (4.7)From the expression (4.7) it follows that C ∞ ( p, q ) is a meromorphic function of the moduli with pole at d (Θ) only.It follows from the uniqueness of the Cauchy kernel, that (4.7) is single-valued in p . This can also be checkeddirectly from the monodromy properties of normalized third kind differentials ω p + α ℓ , ∞ ( q ) = ω p, ∞ ( q ) , ω p + β ℓ , ∞ ( q ) = ω p, ∞ ( q ) + ω ℓ ( q ) (4.8)with respect to analytic continuations along a symplectic basis of generators α , β , . . . , α g , β g for π ( C , ∞ ) forwhich H α ℓ ω k = δ ℓk ; from this we see that the determinant of the matrix in the numerator of (4.7) is single-valued.Furthermore, (4.7) shows that the Cauchy kernel has poles as a function of p as p → T . To see this, for p ∈ D we have X t ∈ T res z = z ( t ) ω p, ∞ ( z ) v tj ( z ) = − res z = z ( p ) ω p, ∞ ( z ) v tj ( z ) = − v tj ( z ( p )) , j = 1 , . . . , ng, (4.9)which diverges as p → T . However, (4.9) shows that the singular part of C ∞ ( q, p ) as p → T belongs to theTyurin space T (row-wise); then C ∞ ( q, p ) P ( p ) is regular as p → T . Corollary 4.3
The expression P − ( q ) C ∞ ( q, p ) P ( p ) is regular for q, p ∈ D away from the diagonal locus q = p . Example 4.4
In the case of line bundles, n = 1 , C ∞ ( q, p ) reduces to the known (scalar) Cauchy kernel [9]; e.g.when the Tyurin points t , . . . , t g are simple, we have C ∞ ( q, p ) = det ω ( t ) · · · ω g ( t ) ω p, ∞ ( t ) ... . . . ... ...ω ( t g ) · · · ω g ( t g ) ω p, ∞ ( t g ) ω ( q ) · · · ω g ( q ) ω p, ∞ ( q ) det ω ( t ) · · · ω g ( t ) ... . . . ...ω ( t g ) · · · ω g ( t g ) . (4.10) Example 4.5 (Cauchy kernel in the case of simple Tyurin points)
If all Tyurin points t , . . . , t ng are sim-ple, let h j be the Tyurin vectors as in (2.13). Define H = [ h , . . . , h ng ] ∈ Mat n,ng ( C ) (the normalization of theTyurin vectors are chosen arbitrarily, and the final formulæ will be independent of this choice), and, for anydifferential, ω define ω ( T ) := diag( ω ( t ) , . . . , ω ( t ng )) . (4.11)Then the Brill–Noether-Tyurin matrix is T = (cid:2) ω ( T ) H t | · · · | ω g ( T ) H t (cid:3) (4.12)and the Cauchy kernel is given explicitly in terms of the normalized Abelian differentials (or any other basis ofholomorphic differentials) of the first and third kind as the following expression C ij ( q, p ) = det ω ( T ) H t · · · ω g ( T ) H t ω p, ∞ ( T ) H t e j ω ( q ) e ti · · · ω g ( q ) e ti ω p, ∞ ( q ) δ ij det T . (4.13)11 Framed Higgs fields and T ∗ b V E ∈ V , a framed Higgs field
Φ is an element Φ ∈ H (End( E ) ⊗ K ( ∞ )), namely a meromorphic section ofEnd( E ) ⊗ K with only a simple pole at ∞ . Given a framing S of E , inducing the polynomial normal form P , aframed Higgs field is trivialized as a matrix meromorphic differential Φ with poles at T + ∞ only, such that P − ( z )Φ( z ) P ( z ) is analytic on z ∈ D . (5.1)The next lemma, whose proof we omit, provides an equivalent characterization of framed Higgs fields, and isa generalization of a result of [18]. Lemma 5.1
A meromorphic matrix differential Φ on C with (Φ) ≥ − T − ∞ defines a framed Higgs field if andonly if C − [Φ] ∈ T (row-wise) and C − [ T Φ] ⊆ T . Here T is the Tyurin space (2.11), C − is the Cauchy operator(2.10), and T Φ denotes the set of elements v t Φ for all v t ∈ T . The Cauchy kernel can be used to construct explicitly Higgs fields in terms of local data near the Tyurinpoints as we now explain.
Proposition 5.2
Given an arbitrary collection ( φ t ) t ∈ T of germs of matrix valued holomorphic differentials, thenthe expression Φ( q ) := X t ∈ T res p = t C ∞ ( q, p ) P ( p ) φ t ( p ) P − ( p ) (5.2) defines a Higgs field. Viceversa, given any Higgs field Φ , then the expression at the right hand side of (5.2) with φ t ( p ) equal to the germ of P − ( p )Φ( p ) P ( p ) at p = t recovers Φ . The proof is a simple verification based on the properties of the Cauchy kernel in Thm-Def. 4.2 and it is omitted.It is well known that the Higgs fields are generically identifiable with the cotangent vectors to the modulispace of vector bundles; we need to make this identification explicit in our setting. To this end, introduce thefollowing pairing between a framed Higgs field Φ ∈ H (End( E ) ⊗ K ( ∞ )) and a deformation ∂ of the transitionfunction G : (Φ , ∂ ) = X t ∈ T res t tr (cid:0) Φ ∂GG − (cid:1) . (5.3)Note that this expression is invariant under G GH with H analytic and invertible in D . Therefore we mayreplace G in (5.3) by the normal form P . Proposition 5.3
Let
E ∈ b V : then the pairing (5.3) is nondegenerate. Proof.
We need to show that (5.3) vanishes for all framed Higgs fields Φ ∈ H (End( E ) ⊗ K ( ∞ )) if and only ifthe deformation is trivial. The only nontrivial part is to show that if (Φ , ∂ ) = 0 for all framed Higgs fields thenthe deformation is trivial. Let us compute explicitly the pairing when the Higgs field is expressed via (5.2).Let ( φ t ) t ∈ T be a collection of germs of matrix valued holomorphic differentials at t ∈ T , and let Φ be theHiggs field defined in (5.2). Then (Φ , ∂ ) = X t ∈ T res t tr (cid:0) φ t P − ∂P (cid:1) . (5.4)This is seen by inserting the expression (5.2) in (5.3) (with G replaced by P ) and then evaluate the residuesusing the properties of the Cauchy kernel. Since now the collection ( φ t ) of holomorphic matrix germs at t ∈ T is completely arbitrary, we must have ∂P = 0. (cid:4) From framed to unframed vector bundles.
There is an action of P GL n ( C ) on b V defined by a change ofglobal framing; it is induced by the GL n ( C )-action of multiplication of the transition function G on the left by agiven constant matrix C ∈ GL n ( C ).The orbits of this P GL n ( C )-action on b V are in one-to-one correspondence with points of V ; V = b V / P GL n ( C ) . (5.5)Therefore the cotangent space T ∗E V is identified with the subspace of framed Higgs fields Φ annihilating ofthe infinitesimal version ∂G = AG , A ∈ Mat n ( C ), of the deformation; using (5.3) we get(Φ , ∂ ) = X t ∈ T res t tr (cid:0) Φ ∂GG − (cid:1) = X t ∈ T res t tr (Φ A ) = − res ∞ tr (Φ A ) = 0 . (5.6)Since the matrix A ∈ Mat n ( C ) in (5.6) is arbitrary we immediately get the following well-known result.12 roposition 5.4 The cotangent space T ∗E V is identified with the space H (End( E ) ⊗ K ) of holomorphic Higgsfields.
The previous sections have described the holomorphic moduli on V (and b V ) and the identification of the cotangentbundle in terms of the explicitly realized Higgs fields in terms of local holomorphic germs (Prop. 5.2) togetherwith the tautological one form on the cotangent bundle (5.3).The main tool is the Cauchy kernel (Thm-Def. 4.2) of the framed bundle which allows us to define a holomor-phic map from the cotangent bundle T ∗ V to the GL n ( C ) character variety. This hinges on the definition of the“Fay” differential as the regular term in the expansion of C ∞ along the diagonal and subsequently by Theorem6.3. Let us fix a non-special positive divisor of degree g , D = P p n p p (i.e. h ( K ( − D )) = 0) with ∞ 6∈ D .For an arbitrary E ∈ V , let A D E be the affine space of holomorphic connections on E ∗ ⊗ O ( D ); this is an affinespace modelled on H ( E ∗ ⊗ E ⊗ K ) which is isomorphic to T ∗E V by Prop. 5.4.Let A D be the affine bundle over V , whose fiber at E ∈ V is A D E ; equivalently, this is the set of pairs ( E , ∇ )of E ∈ V and ∇ ∈ A D E .The final goal of this section is to provide a reference connection ∇ D in the affine bundle A D → V to beidentified with the zero section of T ∗ V → V . We also remark that if E ∈ (Θ) the space of connections A D E isstill an affine space modelled over H ( E ∗ ⊗ E ⊗ K ), but it has larger dimension and hence A D extends over thewhole V only as a sheaf and not a bundle.It is convenient to consider an arbitrary framing S of E ∈ V ; this is unique up to the P GL n action describedabove (5.5). The framing S of E allows us to regard a section of E ∗ ⊗ O ( D ) as a vector valued meromorphicfunction ψ on C such that ( ψ ) ≥ − D , (6.1)and such that P − ( z ) ψ ( z ) is analytic at T , (6.2)where P ( z ) is the polynomial normal form in D as in the previous sections.Denote by A D ( E , S ) the space of matrix-valued differentials A with poles only at T + D and such that P − A P − P − d P is analytic at T , (6.3) A + n p d z p z p is analytic at D , (6.4)for any local coordinate z p near p ∈ D = P p n p p , z p ( p ) = 0 . We have an identification A D E ≃ A D ( E , S ) , where theconnection ∇ ∈ A D E acts on the section ψ in (6.1)–(6.2) as ∇ ψ = d ψ − A ψ . The space A D ( E , S ) is an affine spaceover the space of Higgs fields Φ ∈ H (End( E ) ⊗ K ); indeed if Φ is a matrix of differentials with (Φ) ≥ − T and P − Φ P analytic at T , then A + Φ satisfies (6.3)–(6.4) whenever A does. Theorem 6.1
For any A ∈ A D ( E , S ) the differential equation dΨ = A Ψ for the matrix-valued matrix Ψ hasapparent singularities at T and monodromy-free singularities at D . More precisely we have the following.1. For any solution Ψ , in D we have the local factorization Ψ =
P H with H analytic at T .2. For any p ∈ D of multiplicity n p and any solution Ψ , we have the local factorization Ψ = z − n p p H , with H locally analytic near p , for any local coordinate near p , z p ( p ) = 0 . If a point p of D belongs to T as well, then the requirement for A is simply: P − A P − P − d P + n p d z p z p = is analytic at p. roof. For the first statement, define H := P − Ψ. Then d H = A + H where A + := P − A P − P − d P . By(6.2) A + is analytic at T and so H is analytic at T as well. The proof of the second statement is completelyanalogous. (cid:4) In geometric terms, the holomorphic connection ∇ ∈ A D E , written ∇ = d − A for an arbitrary framing S and A ∈ A D ( E , S ) , equips the bundle E ∗ ⊗ O ( D ) with a flat structure whose flat sections are the columns of a matrixsolution of dΨ = A Ψ. The monodromy of this equation computes the holonomy of the flat bundle, a fact whichwill be exploited below in Sec. 6.2.
The Fay-differential.
For any ( E , S ) ∈ b V , let C ∞ be the associated Cauchy kernel introduced in Sec. 4. For z, w local coordinates of two points in the same coordinate chart, define F ( w ) by the formula F ( w ) := lim z → w (cid:18) C ∞ ( w, z ) − d ww − z (cid:19) . (6.5)One can check that the matrix F ( w ) transforms as follows under a change of local coordinate ζ = ζ ( w ); F ( ζ ) = d ζ d w F ( w ) + (cid:18) d ζ d w (cid:19) . (6.6)Because of our initial assumptions on D , there is a unique half-differential h D (up to scalar) with divisor properties( h D ) ≥ D − ∞ . (6.7)and with U(1) multiplier system (see [9]). Definition 6.2
We define the
Fay-differential as F D ( p ) := F ( p ) − d ln h D ( p ) . (6.8)From (6.6) it follows that F D is a matrix of differential forms with poles at T + D , while the pole at ∞ of F is canceled by the pole of d ln h D ; indeed, near ∞ the behaviour of C ∞ is given in a local coordinate z with z ( ∞ ) = 0 by C ∞ ( w, z )d w = w − z − w + Q ∞ ( w ) + O( z ) (6.9)and hence F D ( z ) = O(1)d z near ∞ . Theorem 6.3 [1]
The Fay-differential F D belongs to A D ( E , S ) . [2] Under the P GL n action of change of framing we have F D C F D C − . Therefore the Fay-differential providesa well-defined connection ∇ D = d − F D ∈ A D E . [3] Let e D be another nonspecial divisor of degree g . Then F e D = F D + Ω D , e D , (6.10) where Ω D , e D = d ln (cid:0) h D /h e D (cid:1) is the unique third–kind differential with purely imaginary periods and with simplepoles at the points p ∈ D with residue n p and with simple poles at the points q ∈ e D with residue − n q (where n p , n q are the multiplicities of the points in the corresponding divisors). Proof. [1]
By Corollary 4.3 the expression P − ( w ) C ∞ ( w, z ) P ( z ) is regular in D except along the diagonal w = z .Here w, z denote, by a slight abuse of notation, both the point and the coordinates. Near w = z we have P − ( w ) C ∞ ( w, z ) P ( z ) = d ww − z − P − ( w ) P ′ ( w )d w + P − ( w ) F ( w ) P ( w ) + O( z − w ) . (6.11)On the other hand P − ( w ) C ∞ ( w, z ) P ( z ) = d ww − z + e A ( w ) + O( z − w ) , (6.12)where e A ( w ) is analytic for w ∈ D , because P − ( w ) C ∞ ( w, z ) P ( z ) has no singularities other than the pole on thediagonal. By comparison of (6.11) and (6.12) we deduce that − P − ( w ) P ′ ( w )d w + P − ( w ) F ( w ) P ( w ) = e A ( w ) is analytic at T , (6.13)14nd so − P − ( w ) P ′ ( w )d w + P − ( w ) F D ( w ) P ( w ) = e A ( w ) − d ln h D is analytic at T (6.14)and (6.3) is established.To verify the condition (6.4) it suffices to note that the multiplicity n p of p in D coincides with the order ofvanishing of h D . [2] It follows from the easily established fact that the non-abelian Cauchy kernel transforms as C ∞ ( q, p ) C C ∞ ( q, p ) C − under the change of framing. [3] This statement follows immediately from the definition (6.8). (cid:4)
Corollary 6.4
Identifying the fibers T ∗E V ≃ H (End( E ) ⊗ K ) of the bundle T ∗ V as in Prop. 5.4, the map I : A D → T ∗ V ( E , ∇ ) ( E , ∇ D − ∇ ) (6.15) is an isomorphism of bundles over V . For any ( E , S ) ∈ b V and any A ∈ A D ( E , S ) , a matrix of flat sections Ψ is uniquely defined bydΨ = A Ψ , Ψ( ∞ ) = . (6.16)According to Thm. 6.1 the matrix of flat sections Ψ extends to a meromorphic matrix function on the universalcover of C , with poles only at D . Under analytic continuation along γ ∈ π ( C , ∞ ), Ψ undergoes a transformationΨ( γ · p ) = Ψ( p ) M − γ , M γ ∈ GL n ( C ) . (6.17)Note that the homotopy class of γ is in π ( C ) and not π ( C \ D ) because Ψ has trivial local monodromy at D .Therefore we obtain the map c M E : A D ( E , S ) → Hom( π ( C , ∞ ) , GL n ( C )) . (6.18)The group P GL n ( C ) acts on Hom ( π ( C , ∞ ) , GL n ( C )) via conjugation, and the quotient under this action is calledthe character variety which we denote by R ( C , GL n ). Proposition 6.5
The map (6.18) for all
E ∈ V factors through a well defined map M : A D → R ( C , GL n ) , (6.19) independent of the framing of E . Proof.
We note that under the change of framing the matrices A, Ψ transform as follows A C A C − , Ψ C Ψ C − , (6.20)the second of which enforces the normalization condition at ∞ in (6.16). This implies that the monodromymatrices transform as M γ CM γ C − . Therefore the map A D ( E , S ) → R ( C , GL n ) obtained by composing c M E in(6.18) with the quotient projection Hom ( π ( C , ∞ ) , GL n ( C )) → R ( C , GL n ), factors through a well defined map M as in the statement. (cid:4) Note that the monodromy map M in (6.19) is holomorphic; this is in contrast with the standard Donaldsonmap [7] which is only real-analytic (since it maps into the U n -character variety). Remark 6.6 If D , e D are non-special divisors of degree g , then the corresponding solutions Ψ , e Ψ of the equation(6.16) are related by, see (6.10), e Ψ( p ) = c h D ( p ) h e D ( p ) Ψ( p ) , c = lim p →∞ h e D ( p ) h D ( p ) ∈ C ∗ . (6.21)Hence the monodromy map M in (6.19) depends on the choice of D only up to a scalar unitary representationof π ( C , ∞ ) depending only on the degree zero line bundle O ( D − e D ); in particular M descends to a map to the P GL n character variety which is independent of the divisor D . △ Equivalence of the (complex) symplectic structure on T ∗ V and theGoldman bracket on R ( C , GL n ) In this section we prove our main result in Thm. 7.5; it shows that the pull–back by the monodromy map M ofthe Goldman symplectic structure on R ( C , GL n ) coincides with the canonical symplectic structure on A D . Thislatter symplectic structure is the one induced by the identification of the A D with the cotangent bundle T ∗ V via the map I in (6.15).We first observe that the non-degeneracy of the pairing (5.3) implies that the Liouville one-form λ on T ∗ V can be expressed as λ ( E , Φ) = X t ∈ T res t tr (cid:0) Φ δP P − (cid:1) = X t ∈ T res t tr (cid:0) Φ δ ΨΨ − (cid:1) , (7.1)where in the second equality we use the first part of Thm. 6.1 to write Ψ = P H with H analytic and analyticallyinvertible in the neighbourhood of each point in T . Here we denote by δ the differential with respect to themoduli of the bundle, as opposed to d representing the differential on C . The expression (7.1) is independent ofthe framing. Therefore, δλ is the canonical (complex) symplectic structure on T ∗ V .The main tool for the proof of Thm. 7.5 is to rewrite the tautological pairing (5.3) or equivalently (7.1) asan integral over the canonical dissection of C . This rewriting yields the Malgrange–Fay one-form which we nowdefine.
The Malgrange-Fay form.
Consider a canonical dissection of the Riemann surface C along generators α , β , . . . , α g , β g of π ( C , ∞ ) satisfying α β α − β − · · · α g β g α − g β − g = Id . (7.2)Let us denote by Σ the boundary of the fundamental polygon of this dissection of C , and by Σ the disjoint unionof arcs obtained from Σ removing the points corresponding to ∞ (the vertices of the fundamental polygon).Given a connection A ∈ A D ( E , S ) , by Theorem 6.1 we shall regard a fundamental solution ∇ Ψ = 0, see (6.16),as a single valued meromorphic function on
C \
Σ with poles at D and such that z n p p Ψ( p ) analytic at p, for all p ∈ D , (7.3)where n p is the multiplicity of p in D and z p a local coordinate at p , z p ( p ) = 0, compare with (6.1). Note that since ∞ is the base-point of the dissection the normalization Ψ( ∞ ) = is intended in the sense that ∞ is approachedwithin the sector bounded by β g , α (see Fig. 2).We denote the boundary values of Ψ( p ) at Σ as Ψ ± ( p ), where + ( − , respectively) denotes the boundary valuefrom the left (right, respectively) of the oriented arcs α k , β k , see Fig. 2. Since points p on Σ are identified inpairs, we have the following jump relations Ψ + ( p ) = Ψ − ( p ) J ( p ) , p ∈ Σ , (7.4)where the matrix J is piecewise continuous on Σ . It is readily established that denoting J α k , J β k the values of J ( p ) when p ∈ α k , p ∈ β k (respectively) we have J α k = M γ ··· γ k β k γ − k − ··· γ − , J β k = M γ ··· γ k α − k γ − k − ··· γ − , (7.5)where γ k := α k β k α − k β − k ∈ π ( C , ∞ ), see Fig. 2. Definition 7.1
We define the
Malgrange–Fay one-form Ξ on A D by Ξ( E , ∇ ) := 12 π i Z Σ tr (cid:0) Ψ − − ∇ D Ψ − δJJ − (cid:1) . (7.6) where ∇ D = d − F D and Ψ is the fundamental matrix of flat sections satisfying ∇ Ψ = 0 , Ψ( ∞ ) = , as in (6.16).Here Σ is the boundary of the canonical dissection, as above. The expression (7.6) is independent of the framing used to compute it. Moreover, it does not depend on thechoice of D . Indeed, if we change D to e D , according to Remark 6.6 the matrix Ψ is multiplied by a scalar factorthat is independent of the moduli of ( E , S ). The monodromy representation is also multiplied by a scalar unitaryrepresentation of π ( C , ∞ ) (also independent of the moduli). These factors cancel in the integrand of (7.6).The definition of the Malgrange–Fay form (7.6) comes partially from prior works [2, 3] with necessary adjust-ments to account for the nontrivial topology of the curve C .16 k α k β k ··· β g α ··· α k p + Ψ + ( p ) p − Ψ − ( p ) ∞ Ψ( ∞ ) = p ∈ α k : p + = γ · · · γ k − β − k γ − k · · · γ − p − β k α k β k ··· β g α ··· α k p + Ψ + ( p ) p − Ψ − ( p ) ∞ Ψ( ∞ ) = p ∈ β k : p + = γ · · · γ k − α k γ − k · · · γ − p − Figure 2: The fundamental flat section Ψ satisfying (6.16) is analytic in the interior of the fundamentalpolygon. Points on the boundaries are identified in pairs p ± , related by the indicated deck transformations( γ k := α k β k α − k β − k ). The corresponding boundary values Ψ ± ( p ) are then related by the associated monodromytransformation. Proposition 7.2
The Malgrange-Fay form (7.6) is the pullback of the Liouville form λ on T ∗ V along the iden-tification (6.15) of Corollary 6.4; Ξ = I ∗ λ. (7.7) Proof.
From (7.4) it follows that ∆ Σ (cid:0) δ ΨΨ − (cid:1) = Ψ − δJJ − Ψ − − . (7.8)where we denote by ∆ Σ the jump across Σ,∆ Σ X ( p ) = X + ( p ) − X − ( p ) , p ∈ Σ . (7.9)Therefore we rewrite (7.1) as λ = X t ∈ T res t tr (cid:0) Φ δ ΨΨ − (cid:1) = 12 π i Z Σ tr (cid:0) Φ∆ Σ ( δ ΨΨ − ) (cid:1) = 12 π i Z Σ tr (cid:0) ΦΨ − δJJ − Ψ − − (cid:1) . (7.10)In the second equality we have used Cauchy residue theorem, and noted that the poles of Ψ at D do not contributebecause δ ΨΨ − is holomorphic at D thanks to (7.3), as the points of the divisor D do not enter the differential δ .The map I sends the connection ∇ to the Higgs field Φ = ∇ D − ∇ ; hence from the last expression of (7.10)we get I ∗ λ = 12 π i Z Σ tr (cid:0) Ψ − − ( ∇ D − ∇ )Ψ − δJJ − (cid:1) = 12 π i Z Σ tr (cid:0) Ψ − − ∇ D Ψ − δJJ − (cid:1) = Ξ . (7.11)Here we have used that Ψ are flat sections for the connection ∇ and thus ∇ Ψ = 0. (cid:4)
We next proceed with the computation of δ Ξ. To this end we recall the following notion of “two-formassociated to a graph” from [5]. Let Σ ⊂ C be an arbitrary (locally finite) graph consisting of smooth arcsmeeting transversally at the vertices. We denote by V (Σ) the set of vertices of Σ, by E (Σ) the set of orientededges of Σ (namely, each edge appears twice in E (Σ) with both possible orientations). An admissible jump matrixon Σ is a map J : E (Σ) → GL n ( C ) satisfying the following two requirements;1. For each oriented edge e ∈ E (Σ), we have J ( − e ) = J ( e ) − , where − e denotes the oppositely oriented edge.2. For each vertex v ∈ V (Σ), consider the oriented edges which are incident to v and oriented outwards; weenumerate them cyclically e , . . . , e n v counterclockwise, where n v is the valence of v . Then the requirementon J (independent of the cyclical order chosen) is that J ( e ) · · · J ( e n v ) = . (7.12)17 = α e = β − e = α − e = β e = α e = β − e = α − e = β Figure 3: Half edges of Σ incident at v = ∞ for g = 2. Definition 7.3 (Two form
Ω(Σ) ) Given a pair (Σ , J ) of an oriented graph and admissible jump matrix, weassociate the following two-form Ω(Σ) := X v ∈ V (Σ) n v X ℓ =1 tr (cid:0) K − ℓ δK ℓ ∧ J − ℓ δJ ℓ (cid:1) , K ℓ := J · · · J ℓ , (7.13) where J ℓ = J ( e ℓ ) are the jump matrices for the edges e , . . . , e n v incident at v , enumerated counterclockwisestarting from an arbitrary one and oriented away from v . The form is invariant under cyclic permutations of the labels of the incident edges at a vertex, thanks to (7.12).It is shown in [5] that Ω is invariant under natural transformations of the graph like edge contractions etc. Wewill not need them specifically here and thus do not recall the details.In our case we take Σ to be the canonical dissection. Then the jump matrix J in (7.4), given in (7.5), are anadmissible jump on Σ. Indeed the graph Σ has only one vertex at ∞ and so we label the half edges e , . . . , e g ofΣ, in counterclockwise sense, so that e corresponds to β − g , e to α − g , e to β − g , e to α g , and so on as in Fig.3; denoting J ℓ = J ( e ℓ ) we have J g − ℓ )+1 = J β ℓ , J g − ℓ )+2 = J − α ℓ , J g − ℓ )+3 = J − β ℓ , J g − ℓ )+4 = J α ℓ , ℓ = 1 , . . . , g. (7.14) Theorem 7.4
The exterior derivative of the Malgrange–Fay form Ξ of Def. 7.1 is given in terms of the mon-odromy matrices by − π i δ Ξ = Ω(Σ) = g X ℓ =1 tr (cid:0) K − ℓ δK ℓ ∧ J − ℓ δJ ℓ (cid:1) , K ℓ := J · · · J ℓ , (7.15) where Σ is the canonical dissection and Ω(Σ) is in (7.13). The matrices J are given by the holonomy of the flatconnection ∇ , see (7.5) and (7.14). This theorem is central to our main result on the Goldman symplectic structure, and its proof is deferred toAppendix A.It is appropriate to remark here the connection with the results in [19, 18], where the author considers the“universal symplectic structure”: this is indeed the same, with the proper identifications, as the exterior differentialof λ in (7.1). In the same papers the author shows that under the (extended) monodromy map, the universaltwo–form is mapped to an expression (Theorem 6.1 in loc.cit.) which is ultimately equivalent to Ω(Σ) for anappropriate Σ. Apparently the connection with the Goldman Poisson structure was not made in the paper,although a bridge with the results of [1] is theoretically possible to establish. Equivalence with the Goldman bracket.
The (complex) Goldman bracket [15] is a Poisson bracket { , } G on the character variety R ( C , GL n ); for two arbitrary γ, e γ ∈ π ( C , ∞ ) it is defined by (cid:26) tr M γ , tr M e γ (cid:27) G = X p ∈ γ ∩ e γ ν ( p )tr ( M γ p e γ ) (7.16)18here γ, e γ are representative of the corresponding homotopy classes which meet transversally, so that γ ∩ e γ is afinite set, ν ( p ) = ± γ and e γ at p , and γ p e γ is the homotopy class of the path obtainedtraversing γ starting from the intersection point p , then traversing the loop e γ .In the case under consideration, where C is without punctures and without boundary, the Poisson bracket(7.16) is known to be non-degenerate; the corresponding symplectic form ̟ G can be defined by the relation { f, g } G = ̟ G ( X f , X g ) , denoting X f = { , f } G the Hamiltonian vector fields, for any smooth functions f, g on R ( C , GL n ). Theorem 7.5
The pull-back to A D of the Goldman symplectic structure ̟ G along the monodromy map M : A D → R ( C , GL n ) coincides (up to a factor π i ) with the canonical symplectic structure on A D induced by theisomorphism I : A D → T ∗ V ; − π i I ∗ δλ = M ∗ ̟ G . (7.17) Proof.
The symplectic form, ̟ G , associated to the Goldman Poisson structure was provided in [1]. In [5], Thm.3.1 it is shown that the expression of Alekseev-Malkin, after appropriate transformations, equals Ω(Σ) where Σis the canonical dissection, and Ω(Σ) is the two-form of Definition 7.3; namely we have12 Ω(Σ) = M ∗ ̟ G . (7.18)The proof is completed by the chain of equalities − π i I ∗ δλ Prop. 7.2 = − π i δ Ξ Thm. 7.4 = 12 Ω(Σ) = M ∗ ̟ G . (cid:4) Remark 7.6 (Relationship with Fay’s [10])
In [10] Fay introduced, in a slightly different context, a similarone–form. The main difference is that Fay works with flat vector bundles tensored by a root of the canonicalbundle, and hence he works on vector bundles of degree n ( g − so that Riemann–Roch’s theorem states h = h and both are generically zero. The Cauchy kernel is ill-defined on the non-abelian theta divisor d (Θ), and hence we do not have a way to extendthe reference connection F D over d (Θ). In this last section we show that the Malgrange-Fay form Ξ on the bundleof connections A D has a first order pole with residue given by the index h ( E ).This is the content of the following theorem, which is an analogue of Theorems 3 and 4 in [10].More precisely, we consider a holomorphic family E ǫ , for | ǫ | <
1, such that E ∈ (Θ). On the family E ǫ weconsider framings S ǫ , and the associated polynomial normal forms P ǫ , Tyurin divisors T ǫ , and Brill-N¨other-Tyurinmatrices T ǫ .Recalling that (Θ) is defined by det T = 0, we call the family E ǫ transversal to (Θ) ifres ǫ =0 d ln det T ǫ = h ( E ) . (8.1)We correspondingly take any family of connections ∇ ǫ = d − A ǫ such that A ǫ satisfies the conditions (6.3),(6.4) identically in ǫ and is holomorphic for | ǫ | < Theorem 8.1
Denote by Ξ ǫ the pull-back of Ξ along a transversal holomorphic family ( E ǫ , ∇ ǫ ) ∈ A D , < | ǫ | < .Then Ξ ǫ has a simple pole at ǫ = 0 , with residue h ( E ) , namely Ξ ǫ = h ( E ) d ǫǫ + O (1) . (8.2)Before proving the theorem we need some remarks and a lemma. Equation (8.1) holds true if and only if˙ T := ∂ ǫ T ǫ | ǫ =0 restricts to a non-degenerate pairing between left and right null-spaces of T . Thanks to the19dentifications of the null-spaces of the Brill-N¨other-Tyurin matrix provided in Prop. 3.1, such restriction of ˙ T can be expressed (up to an inessential sign) by the pairing H ( E ( −∞ )) ⊗ H ( E ∗ ⊗ K ) → C : ( r , η )
7→ h r , η i = X t ∈ T res t r t ˙ P P − η , (8.3)where ˙ P = ∂ ǫ P ǫ | ǫ =0 . Lemma 8.2 [1]
For any E ∈ (Θ) and an arbitrary framing S , there exists a transversal deformation ( E ǫ , S ǫ ) ,i.e. a deformation such that the pairing (8.3) is non-degenerate. [2] Let ( E ǫ , S ǫ ) be a transversal deformation of ( E , S ) ∈ d (Θ) and let us denote C ∞ ( q, p ; ǫ ) the Cauchy kernelassociated to ( E ǫ , S ǫ ) . Let k = h ( E ) ≥ and r , . . . , r k and η , . . . , η k be arbitrary bases of H ( E ( −∞ )) and H ( E ∗ ⊗ K ) . Then C ∞ ( q, p ; ǫ ) has a simple pole at ǫ = 0 , whose residue is the kernel of rank k given by lim ǫ → ǫ C ∞ ( p, q ; ǫ ) = − k X a,b =1 Q ab η a ( p ) r tb ( q ) (8.4) where Q ab are the entries of the inverse Q − of the Gram matrix Q = ( Q ab ) ka,b =1 , Q ab = h r a , η b i . Proof. [1]
The proof follows along the lines of an argument by Fay [10] adapted to our setting. We start byobserving that if r ∈ H ( E ( −∞ )) and η ∈ H ( E ∗ ⊗ K ) then Φ = η r t is a Higgs field, see (5.1). The pairing (5.3)with Φ = η r t coincides with (8.3);(Φ , ˙ P ) = X t ∈ T res t tr (cid:16) Φ ˙ P P − (cid:17) = X t ∈ T res t r t ˙ P P − η = h r , η i . (8.5)For any [ r ] ∈ P H ( E ( −∞ )) let V [ r ] be the subspace of T E b V defined by h r , ·i = 0; equivalently, denoting k = h ( E ) ≥ η , . . . , η k of H ( E ⊗ K ), then the space V [ r ] is defined by the k linear equations( η i r t , ˙ P ) = 0 for i = 1 , . . . , k . Since ( , ) is a nondegenerate pairing by Proposition 5.3, these k linear equationsare independent and we have dim V [ r ] = n g − k . Therefore the set of deformations ˙ P for which h , i is degeneratecoincides with S [ r ] ∈ P H ( E ( −∞ )) V [ r ] , which is of dimension at most ( n g − k ) + ( k −
1) and thus cannot cover T E b V .Hence there exists ˙ P such that h , i is non-degenerate. [2] Looking at (4.7) it is clear that C ∞ ( q, p ; ǫ ) has at most a simple pole at ǫ = 0; indeed due to the transversalitycondition res ǫ =0 d ln det T ǫ = k it is easy to conclude that the numerator in (4.7) vanishes at ǫ = 0 of order at least k −
1, while the denominator vanishes of order exactly k . Hence we can write C ∞ ( q, p ; ǫ ) as a Laurent expansion: C ∞ ( q, p ; ǫ ) = X j ≥− C [ j ] ∞ ( q, p ) ǫ j . (8.6)Since the Cauchy kernel has a simple pole along the diagonal with residue (independent of ǫ ) it follows thatthe kernel C [0] ∞ ( q, p ) has a simple pole with residue along q = p while C [ j ] ∞ , j = 0 are analytic at q = p . As aconsequence of Corollary 4.3 we also have that: • the rows of C [ − ∞ ( q, p ), as meromorphic vector functions of p , are (transposition of) holomorphic sections of E ( −∞ ); • the columns of C [ − ∞ ( q, p ), as differentials in q , are holomorphic sections of E ∗ ⊗ K .Therefore we have C [ − ∞ ( q, p ) = − k X a,b =1 Q ab η a ( q ) r tb ( p ) (8.7)for some matrix ( Q ab ) (the minus sign is for convenience). To complete the proof we have to show that Q ab arethe entries of the inverse of the matrix Q ab = h r a , η b i . To this end we first note that C [ − ∞ ( q, p ) ˙ P ( p ) = lim ǫ → C ∞ ( q, p ; ǫ ) ( P ( p ; ǫ ) − P ( p ; 0)) . (8.8)20hoose now η ∈ H ( E ∗ ⊗ K ) and assume q ∈ C \ D : applying the Cauchy theorem we then obtain D C [ − ∞ ( q, · ) , η E = X t ∈ T res p = t C [ − ∞ ( q, p ) ˙ P ( p ) P − ( p ) η ( p ) = 12 π i I p ∈ ∂ D C [ − ∞ ( q, p ) ˙ P ( p ) P − ( p ) η ( p )= 12 π i lim ǫ → (cid:18)I p ∈ ∂ D C ∞ ( q, p ; ǫ ) P ǫ ( p ) P − ( p ) η ( p ) − C ∞ ( q, p ; ǫ ) η ( p ) (cid:19) , (8.9)where exchanging the limit and the integral is legitimate because (8.8) is uniform with respect to p ∈ ∂ D . Theexpression C ∞ ( q, p ; ǫ ) P ( p ; ǫ ) P − ( p ) η ( p ) is regular for p ∈ D (as P − ( p ) η ( p ) is regular for p → T ) and so doesnot contribute to the integral. The remaining part is − π i I p ∈ ∂ D C ∞ ( q, p ; ǫ ) η ( p ) = res p = q C ∞ ( q, p ; ǫ ) η ( p ) = − η ( q ) . (8.10)Therefore we have shown that for all q ∈ C \ D we have D C [ − ∞ ( q, · ) , η E = − η ( q ). We now apply this identity tothe basis { η c , c = 1 , . . . , k } of H ( E ∗ ⊗ K ); D C [ − ∞ ( q, · ) , η c E = − k X a,b =1 Q ab η a ( q ) h r b , η c i = − k X a,b =1 Q ab Q bc η a ( q ) = − η c ( q ) , (8.11)yielding P kb =1 Q ab Q bc = δ ac , and the proof is complete. (cid:4) We now are in a position to prove Theorem 8.1.
Proof of Theorem 8.1.
Using Proposition 7.2 and formula (7.1) can rewrite the pullback of the Malgrangeform as Ξ ǫ = X t ∈ T res t tr (cid:16) Φ ǫ ˙ P ǫ P − ǫ (cid:17) d ǫ, (8.12)where P ǫ are the polynomial normal forms associated to the framed bundles ( E ǫ , S ǫ ) as above, and the family ofconnections ∇ ǫ are trivialized as d − A ǫ with A ǫ = Φ ǫ + F D ,ǫ where F D ,ǫ denotes the Fay connection on E ǫ andhas a pole at ǫ = 0 according to Lemma 8.2, and in particular formula (8.4). The singular part is of the form F D ( z ; ǫ ) = − ǫ X a,b Q ab η a ( z ) r tb ( z ) + O(1) =: 1 ǫ Φ [ − ( z ) + O(1) , (8.13)where O(1) denotes terms regular at ǫ = 0. In particular the coefficient Φ [ − is a certain Higgs field on E . Since,by construction, A ǫ is chosen holomorphic in ǫ , near ǫ = 0 we haveΦ ǫ = − ǫ Φ [ − ( z ) + O(1) . (8.14)Inserting this expression into (8.12) we obtainΞ ǫ = − d ǫǫ X t ∈ T res t tr (cid:16) Φ [ − ˙ P P − (cid:17) + O(1) = d ǫǫ h ( E ) X a,b =1 Q ab = Q ba z }| { h r b , η a i +O(1) = h ( E ) d ǫǫ + O(1) . (8.15)This concludes the proof. (cid:4) In this paper we have extensively employed the holomorphic parametrization of vector bundles in terms of Tyurindata and the notion of non-abelian Cauchy kernel; to conclude, we comment on a few future directions andinteresting open questions that arise from these constructions.- The connection F D , as a function of the moduli, is meromorphic with poles along the non-Abelian Thetadivisor. In the setting of projective connections described in [4] it compares with the Wirtinger projectiveconnection. 21n the other hand in [4] the central object was the Bergman projective connection, which is holomorphicon the Teichmuller space but it is not single–valued on the moduli space M g . The question is then whetherthere is a similar reference connection which is holomorphic in the moduli of the bundles also on the Thetadivisor but not necessarily single valued.- We would like to define the notion of tau function in a similar spirit as [10]; since δ Ξ is an expression interms of the monodromy matrices alone, one can find a local antiderivative ϑ in terms of the same matrices,locally analytic at the theta divisor. Thus Ξ − ϑ is a locally defined closed one form which defines a “tau”function as δ ln τ = Ξ − ϑ . By Theorem 8.1 this function vanishes at the points E in the theta divisor (Θ)to order h ( E ). The concrete obstacle here is that one needs to construct an explicit local potential for theGoldman symplectic structure.- In perspective, one could consider next meromorphic connections; this is, implicitly, the setting of [19].However in loc. cit. the notion of tau function was not investigated, and thus neither its role as generatingfunction for change of Lagrangian polarization. The natural conjecture is now that one can introduce a taufunction by combining the logic indicated in the previous points. The resulting tau function should havethe properties of vanishing on the non-abelian Theta divisor.- A natural question arises in the context of Hecke modifications (see e.g. [8]): the Malgrange-Fay form willundergo a resulting modification but the main theorem that its exterior derivative is the pull-back of theGoldman structure should remain true. This implies that the difference of the Malgrange-Fay forms for theoriginal bundle and the Hecke-modified is the differential of a locally defined function. The question is thento explicitly characterize this function.- Fay’s identified (formula (10) in [10], proved in [11]) the one form with the holomorphic part of a suitablefactorization of the analytic Ray-Singer torsion (see (10) in [10] and also Corollary to Theorem 1 in [21]).In view of a generalization to meromorphic connections, the tau function described above will provide theanalog of a “determinant” for wild character varieties.- The Tyurin parametrization seems suited for studying the degeneration where the base curve develop oneor more nodes. The reason is that the main tool, the Cauchy kernel, is written in terms of holomorphicdifferentials and third–kind differentials, whose behaviour under degeneration is classically well studied.This may allow to extend results of [6] who studied in detail the rank two case.
A Proof of Theorem 7.4
There are two main components in the definition (7.6); the fundamental flat matrix Ψ and the reference connection F D . To compute the exterior derivative of Ξ we need variational formulas for both. Lemma A.1
The variational formula for Ψ takes the form δ Ψ( p )Ψ − ( p ) = 12 π i Z q ∈ Σ Ψ − ( q ) δJ ( q ) J − ( q )Ψ − − ( q ) C ∞ ( q, p ) . (A.1) Proof.
It follows from the jump condition Ψ + ( p ) = Ψ − ( p ) J ( p ) and the normalization Ψ( ∞ ) = that δ ΨΨ − satisfies the additive jump condition δ Ψ + Ψ − = δ Ψ − Ψ − − + Ψ − δJJ − Ψ − − , (A.2)as well as the normalization δ Ψ( ∞ ) = 0. It follows from the Sokhotskii-Plemelji formula that the expression inthe right side of (A.1) satisfies exactly the same conditions. Denote by R ( p ) their difference; since the divisor D is not part of the variation, it is holomorphic at D . Then it has poles only at the Tyurin divisor. At each point q ∈ T , the product R ( p ) P ( p ) is analytic because of Corollary 4.3 and Thm. 6.1 (part 1). Thus the rows of R ( p )are sections of E that vanish at ∞ . Since E is not on the non-abelian theta divisor, it follows that R ( p ) ≡ (cid:4) Lemma A.2 (Variational formulæ)
The variation of F D ( p ) is δ F D ( p ) = X t ∈ T res q = t C ∞ ( p, q ) δ Ψ( q )Ψ − ( q ) C ∞ ( q, p ) (A.3) and it is independent of the choice of D . roof. Observe that, as per Theorem 6.1 (part 1) the variational formula depends only on the polynomial normalform P and not the whole solution Ψ. The independence from D is clear because F D = F − d ln h D and thereference spinor h D is not subject to deformations, δh D = 0. Observe now that K ( q, p ) := Ψ − ( q ) C ∞ ( q, p )Ψ( p )has a simple pole at q = p , with residue identity and hence moduli-independent, and is holomorphic otherwise,with a zero at p = ∞ . Therefore δ K ( q, p ) is an analytic function of p ∈ C \ Σ with a zero at ∞ , K ( q, p = ∞ ) = 0,and on Σ it satisfies (for fixed q ∈ C \ Σ) δ K ( q, p + ) = δ K ( q, p − ) J ( p ) + K ( q, p − ) δJ ( p ) , p ∈ Σ . (A.4)It follows again from Sokhotskii-Plemelji formula, along the same lines as Lemma A.1 that δ K ( q, p ) = 12 π i Z s ∈ Σ K ( q, s − ) δJ ( s ) J − ( s ) K ( s − , p ) (A.5)because the expression on the right side satisfies the same jump condition and normalization at p = ∞ . Usingthe notation (7.9) and the Cauchy theorem the formula (A.5) can be written as a sum of residues at p, q, T asfollows δ K ( q, p ) = Ψ − ( q ) C ∞ ( q, p ) δ Ψ( p ) − Ψ − ( q ) δ Ψ( q )Ψ − ( q ) C ∞ ( q, p )Ψ( p )+ X t ∈ T res s = t Ψ − ( q ) C ∞ ( q, s ) δ Ψ( s )Ψ − ( s ) C ∞ ( s, p )Ψ( p )= K ( q, p )Ψ − ( p ) δ Ψ( p ) − Ψ − ( q ) δ Ψ( q ) K ( q, p )+ X t ∈ T res s = t Ψ − ( q ) C ∞ ( q, s ) δ Ψ( s )Ψ − ( s ) C ∞ ( s, p )Ψ( p ) . (A.6)Let z = ζ ( p ) and w = ζ ( q ) for a local coordinate ζ ; by a slight abuse of notation we shall indicate Ψ( z ) theevaluation of Ψ at p , and similarly for K and for the point q . Taking the regular part at w = z of the kernel K yields lim w → z (cid:18) K ( w, z )d w − w − z (cid:19) = Ψ − ( z ) F ( z )Ψ( z ) − Ψ − ( z ) dd z Ψ( z ) . (A.7)We now take a variation of the moduli and obtain (we denote for brevity ′ the derivative w.r.t z in the chosencoordinate and observe that δ K ( w, z ) is regular along w = z ): δ K ( z, z ) = (cid:20) Ψ − ( z ) F ( z )Ψ( z ) , Ψ − ( z ) δ Ψ( z ) (cid:21) − δ (Ψ − ( z )Ψ ′ ( z )) + Ψ − ( z ) δ F ( z )Ψ( z ) . (A.8)On the other hand, the limit q → p of (A.6) gives (all terms below are evaluated at p unless otherwise indicated) δ K ( p, p ) = (cid:20) Ψ − F Ψ , Ψ − δ Ψ (cid:21) − Ψ − ( δ Ψ) ′ + Ψ − δ ΨΨ − Ψ ′ + X t ∈ T res s = t Ψ − ( p ) C ∞ ( p, s ) δ Ψ( s )Ψ − ( s ) C ∞ ( s, p )Ψ( p )= (cid:20) Ψ − F Ψ , Ψ − δ Ψ (cid:21) − δ (Ψ − Ψ ′ ) + Ψ − ( p ) X t ∈ T res s = t C ∞ ( p, s ) δ Ψ( s )Ψ − ( s ) C ∞ ( s, p ) ! Ψ( p ) . (A.9)Then (A.3) follows by comparing (A.8) and (A.9). (cid:4) Proof of Thm. 7.4.
We split the Malgrange-Fay differential as follows2 π i Ξ = Z Σ tr (cid:18) Ψ − − dΨ − δJJ − (cid:19) − Z Σ tr (cid:18) F D Ψ − − δJJ − Ψ − (cid:19) =: Ξ − Ξ (A.10)We are going to compute separately δ Ξ and δ Ξ and we will show that δ Ξ = −
12 Ω(Σ) + 12 X t ∈ T res p = t tr (cid:18) d( δ ΨΨ − ) ∧ δ ΨΨ − (cid:19) , (A.11) δ Ξ = 12 X t ∈ T res p = t tr (cid:18) d (cid:0) δ ΨΨ − (cid:1) ∧ δ ΨΨ − (cid:19) , (A.12)from which the theorem will follow. 23 omputation of δ Ξ . For brevity we will use the notation ϑ := δJJ − . We have δ Ξ = Z Σ tr (cid:18) − Ψ − − δ Ψ − Ψ − − dΨ − ∧ ϑ + Ψ − − δ dΨ − ∧ ϑ + Ψ − − dΨ − δϑ (cid:19) . (A.13)Now use the identity δϑ = ϑ ∧ ϑ as well as the variational formulæ δ Ψ( p ) = Z q ∈ Σ Ψ − ( q ) ϑ ( q )Ψ − − ( q ) C ∞ ( q, p )Ψ( p ) ,δ dΨ( p ) = δ Ψ( p )Ψ( p ) − dΨ( p ) + Z q ∈ Σ Ψ − ( q ) ϑ ( q )Ψ − − ( q )d p C ∞ ( q, p )Ψ( p ) , (A.14)the first of which is the content of Lemma A.1 and the second is a consequence. Inserting (A.14) into (A.13) weobtain δ Ξ = χ + Z Σ tr (cid:18) Ψ − − dΨ − ϑ ∧ ϑ (cid:19) (A.15)where we denote χ := Z p ∈ Σ Z q ∈ Σ tr (cid:18) Ψ − ( q ) ϑ ( q )Ψ − ( q ) − d p C ∞ ( q, p − ) ∧ Ψ − ( p ) ϑ ( p )Ψ( p ) − − (cid:19) . (A.16)Here the subscript p − in the Cauchy kernel indicates that the integration is taken on the boundary value fromthe right of Σ, which is important to specify since the integral is a singular integral.To handle the term χ in (A.15) it is convenient to introduce the fundamental bi-differential B ( q, p ) on C ,which is the unique symmetric scalar bi-differential with a double pole at p = q with unit bi-residue and trivial α -periods [9, Ch. III]. The only property we are going to use is its symmetry B ( q, p ) = B ( p, q ). Then, we note thatd p C ∞ ( q, p ) = B ( q, p ) + H ( q, p ), where H ( q, p ) is a matrix bi-differential regular along q = p . Thus χ = χ + χ with χ := Z p ∈ Σ Z q ∈ Σ tr (cid:18) Ψ − ( q ) ϑ ( q )Ψ − ( q ) − B ( q, p − ) ∧ Ψ − ( p ) ϑ ( p )Ψ( p ) − − (cid:19) , χ := Z p ∈ Σ Z q ∈ Σ tr (cid:18) Ψ − ( q ) ϑ ( q )Ψ − ( q ) − H ( q, p ) ∧ Ψ − ( p ) ϑ ( p )Ψ( p ) − − (cid:19) . (A.17) Proposition A.3
The singular integral χ can be evaluated as χ = − Z Σ tr (cid:18) Ψ − − dΨ − ϑ ∧ ϑ (cid:19) −
12 Ω(Σ) . (A.18)We only sketch the proof, because the computation can be handled following identical steps (which we nowreview) as the proof of Thm. 2.1 in [3]. Indeed, we observe that the only properties used in the proof in loc. cit.are that the kernel of the integration has a singularity of type d z d w ( z − w ) − on the diagonal and it is symmetricin the two variables. Both properties are satisfied by the fundamental bidifferential B (Ch. III of [9]).We start by introducing a small disk D ǫ on C , centered at ∞ and of radius ǫ in some coordinate chart, andseparate the integration over Σ ∩ D ǫ and Σ \ D ǫ . Namely, we decompose χ = A ǫ + B ǫ + C ǫ , where A ǫ := Z p ∈ Σ \ D ǫ Z q ∈ Σ \ D ǫ tr (cid:18) Ψ − ( q ) ϑ ( q )Ψ − ( q ) − B ( q, p − ) ∧ Ψ − ( p ) ϑ ( p )Ψ( p ) − − (cid:19) , (A.19) B ǫ := Z p ∈ Σ ∩ D ǫ Z q ∈ Σ \ D ǫ tr (cid:18) Ψ − ( q ) ϑ ( q )Ψ − ( q ) − B ( q, p − ) ∧ Ψ − ( p ) ϑ ( p )Ψ( p ) − − (cid:19) + Z p ∈ Σ \ D ǫ Z q ∈ Σ ∩ D ǫ tr (cid:18) Ψ − ( q ) ϑ ( q )Ψ − ( q ) − B ( q, p − ) ∧ Ψ − ( p ) ϑ ( p )Ψ( p ) − − (cid:19) , (A.20) C ǫ := Z p ∈ Σ ∩ D ǫ Z q ∈ Σ ∩ D ǫ tr (cid:18) Ψ − ( q ) ϑ ( q )Ψ − ( q ) − B ( q, p − ) ∧ Ψ − ( p ) ϑ ( p )Ψ( p ) − − (cid:19) . (A.21)We compute separately these three contributions to χ ; each of them is treated in the same way as in loc. cit.Since Σ \ D ǫ is a disjoint union of smooth arcs without self-intersections, for the computation of A ǫ we can relyon the following result, which follows from the classical theory of singular integrals (see e.g. [12]).24 emma A.4 Let γ be an oriented smooth arc in C without self-intersection and let ϕ : γ × γ → C be a functionwhich is locally analytic in each variable and such that ϕ ( z, w ) = − ϕ ( w, z ) . Then Z z ∈ γ Z w ∈ γ B ( z − , w ) ϕ ( w, z ) = − Z γ d w ϕ ( w, z ) (cid:12)(cid:12)(cid:12)(cid:12) w = z . (A.22)This gives A ǫ = − Z Σ \ D ǫ tr (cid:18) d q (cid:0) Ψ − ( q ) ϑ ( q )Ψ − ( q ) − (cid:1) ∧ Ψ − ( p ) ϑ ( p )Ψ( p ) − − (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) q = p = − Z Σ \ Σ ǫ tr (cid:18) dΨ − ϑ ∧ ϑ Ψ − − − ϑ Ψ − − dΨ − ∧ ϑ + d ϑ ∧ ϑ (cid:19) = − Z Σ \ Σ ǫ tr (cid:18) Ψ − − dΨ − ϑ ∧ ϑ + d ϑ ∧ ϑ (cid:19) . (A.23)Thus A ǫ clearly admits the limit lim ǫ → + A ǫ = − Z Σ tr (cid:18) Ψ − − dΨ − ϑ ∧ ϑ + d ϑ ∧ ϑ (cid:19) . (A.24)Finally, the same reasoning in [3] applies to show that B ǫ = 0 because of skew-symmetry and that C ǫ yields thelocalized contribution lim ǫ → + C ǫ = −
12 Ω(Σ) . (A.25)This part of the computation is identical to the one in loc. cit. because it can be carried out entirely in localcoordinate and requires only the symmetry of B together with its singular behaviour along the diagonal. Finally,since J ( p ) in (7.9) is constant on each arc of Σ, we have d ϑ = 0 and (A.18) follows.To complete the computation of δ Ξ = −
12 Ω(Σ) + χ , (A.26)cf. (A.15), it remains to compute χ . Note that from (7.8) we obtain Ψ − ϑ Ψ − − = Ψ − δJJ − Ψ − − = ∆ Σ ( δ ΨΨ − ),where ∆ Σ is the jump operator across Σ, see (7.9); hence we can use Cauchy residue theorem to write χ = Z p ∈ Σ Z q ∈ Σ tr (cid:18) ∆ Σ ( δ Ψ( q )Ψ − ( q )) H ( q, p ) ∧ ∆ Σ ( δ Ψ( p )Ψ − ( p )) (cid:19) = X t ∈ T res q = t Z p ∈ Σ tr (cid:18) δ Ψ( q )Ψ − ( q ) H ( q, p ) ∧ ∆ Σ ( δ Ψ( p )Ψ − ( p )) (cid:19) (A.27)To proceed, we now show thatres q = t δ Ψ( q )Ψ − ( q ) H ( q, p ) = − res q = t δ Ψ( q )Ψ − ( q ) B ( q, p ) . (A.28)To this end we note that, by construction of the Cauchy kernel, Ψ − ( q ) C ∞ ( q, p ) is a differential with respect to q without poles at the Tyurin divisor T (this follows from Thm.–Def. 4.2 and Theorem 6.1). Therefore ∀ t ∈ T : res q = t δ Ψ( q )Ψ − ( q ) C ∞ ( q, p ) = 0 ⇒ res q = t δ Ψ( q )Ψ − ( q )d p C ∞ ( q, p ) = 0 . (A.29)Recalling that d p C ∞ ( q, p ) = B ( q, p ) + H ( q, p ) we obtain (A.28). Therefore in (A.27) we can replace the matrixkernel H with the (scalar) kernel − B , use Cauchy theorem again and obtain χ = − X t ∈ T res p = t X r ∈ T res q = r tr (cid:18) δ Ψ( q )Ψ − ( q ) ∧ δ Ψ( p )Ψ − ( p ) (cid:19) B ( q, z ) . (A.30)In the double sum over the points of the Tyurin divisor T only the diagonal part contributes because of theskew-symmetry of the expression. To compute the iterated residue at t ∈ T , we work in a local coordinate ζ centered at t and denote ϕ ( w, z ) := tr (cid:18) δ Ψ( w )Ψ − ( w ) ∧ δ Ψ( z )Ψ − ( z ) (cid:19) , (A.31)25ith w = ζ ( q ), z = ζ ( p ). This expression is jointly analytic in the punctured disks around z = 0 and w = 0, andskew symmetric, ϕ ( z, w ) = − ϕ ( w, z ). Using Lemma A.4 we obtainres z =0 res w =0 B ( z, w ) ϕ ( z, w )d z d w = −
12 res z =0 d w ϕ ( z, w ) (cid:12)(cid:12) w = z . (A.32)From this we finally obtain χ = − X t ∈ T res p = t res z = t tr (cid:18) δ ΨΨ − ( p ) B ( p, z ) ∧ δ ΨΨ − ( z ) (cid:19) = 12 X t ∈ T res p = t tr (cid:18) d p ( δ ΨΨ − ) ∧ δ ΨΨ − (cid:19) , (A.33)and equation (A.11) follows. Computation of δ Ξ . Using again that Ψ − − ϑ Ψ − = ∆ Σ ( δ ΨΨ − ) and Cauchy residue theorem, we obtainΞ = Z tr (cid:18) F D Ψ − ϑ Ψ − − (cid:19) = X t ∈ T res p = t tr (cid:18) F D δ ΨΨ − (cid:19) . (A.34)In the use of Cauchy theorem we have noted that F D ( p ) has a simple pole at p = ∞ but δ ΨΨ − has a simplezero such that there is no contribution to the residue theorem at p = ∞ . We remark, in passing, that this termΞ does not have a counterpart in the genus zero setting of [2, 3].We can use the variational formula for F D (see Lemma A.2) to complete the computation; δ Ξ = X t ∈ T res p = t (cid:20) tr (cid:18) δ F D ∧ δ ΨΨ − (cid:19) + tr (cid:18) F D δ ΨΨ − ∧ δ ΨΨ − (cid:19)(cid:21) ( A. ) = X t ∈ T res p = t X s ∈ T res r = s tr (cid:18) C ∞ ( p, r ) δ Ψ( r )Ψ − ( r ) C ∞ ( r, p ) ∧ δ Ψ( p )Ψ( p ) − (cid:19) + X t ∈ T res p = t tr (cid:18) F D δ ΨΨ − ∧ δ ΨΨ − (cid:19) . (A.35)Denote by ( D ) the double sum in (A.35). We observe (using the ciclycity of the trace) that the argument is oddunder the exchange p ↔ r with a simple pole at p = r . Then, using the same logic used in the previous part ofthe proof we easily obtain( D ) = − X t ∈ T res p = t res r = p tr (cid:18) C ∞ ( p, r ) δ Ψ( r )Ψ − ( r ) C ∞ ( r, p ) ∧ δ Ψ( p )Ψ − ( p ) (cid:19) = − X t ∈ T res p = t tr (cid:18) − d (cid:0) δ ΨΨ − (cid:1) ∧ δ ΨΨ − + F D δ ΨΨ − ∧ δ ΨΨ − − δ ΨΨ − F D ∧ δ ΨΨ − (cid:19) . (A.36)Note that in the above formula tr ( F D δ ΨΨ − ∧ δ ΨΨ − ) is actually a well defined (local) differential and not anaffine connection with respect to change of coordinates because tr ( δ ΨΨ − ∧ δ ΨΨ − ) = 0. Then( D ) = 12 X t ∈ T res t tr (cid:18) d (cid:0) δ ΨΨ − (cid:1) ∧ δ ΨΨ − (cid:19) − X t ∈ T res t tr (cid:18) F D δ ΨΨ − ∧ δ ΨΨ − (cid:19) (A.37)Finally, plugging (A.37) in (A.35) we obtain (A.12). Plugging (A.11) and (A.12) in the differential of the relation(A.10), 2 π i δ Ξ = δ Ξ − δ Ξ , gives the theorem. (cid:4) References [1] Alekseev, A. Yu. and Malkin, A. Z. “Symplectic structure of the moduli space of flat connection[s] on aRiemann surface”.
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