Higgs fields, non-abelian Cauchy kernels and the Goldman symplectic structure

Abstract

We consider the moduli space of vector bundles of rank n and degree ng over a fixed Riemann surface of genus 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 ≥ 1. On the complement of this divisor we construct a non-abelian Cauchy kernel explicitly in terms of the Tyurin data. With the additional datum of a non-special divisor, we can construct a reference flat holomorphic connection which is also dependent holomorphically on the moduli of the bundle. This allows us to identify the bundle of Higgs fields, i.e. the cotangent bundle of the moduli space, with the affine bundle of holomorphic connections and provide a monodromy map into the GLn character variety. We show that the Goldman symplectic structure on the character variety pulls back along this map to the complex canonical symplectic structure on the cotangent bundle and hence also on the space of affine connections. The pull-back of the Liouville one-form to the affine bundle of connections is then shown to be a logarithmic form with poles along the non-abelian theta divisor and residue given by h 1 .

Nonlinear steepest descent approach to orthogonality on elliptic curves

“…However one quickly realizes [13] that det Y necessarily must have r g zeros (with r being the size of the matrix) if det J has index zero. This fact can be framed within the general theory of vector bundles on Riemann surfaces and this was extensively investigated in the recent [4].…”

“…For matrix problems the issue is compounded and in fact the relevant notion is that of a matrix Cauchy kernel that depends on r 2 g complex parameters (r the size of Y ) which go under the name of Tyurin data. We refer to [4] for a comprehensive description of these kernels.…”

“…Definition of the non-abelian Cauchy kernel. The non-abelian Cauchy kernel C 0 (p, q)dp [4] is a matrix-valued differential with respect to the variable p and meromorphic function with respect to the variable q satisfying the following properties 1. It has a simple pole for p = q and p = 0 and no other poles with respect to p;…”

“…These conditions determine the kernel uniquely as we now show. An explicit expression is given in [4] in arbitrary genus, but it will not feature prominently here.…”

“…This requires some level of sophistication both in the construction of the g-function (be it in the fixed measure or scaling measure case) and in the construction of the corresponding model problem. In this case the results of [4] on the construction of the non-abelian Cauchy kernel will become even more fundamental and moreover the analysis is nicely intertwined with the theory of vector bundles.…”