Symplectic geometry of the moduli space of projective structures in homological coordinates

Abstract

We introduce a natural symplectic structure on the moduli space of quadratic differentials with simple zeros and describe its Darboux coordinate systems in terms of so-called homological coordinates. We then show that this structure coincides with the canonical Poisson structure on the cotangent bundle of the moduli space of Riemann surfaces, and therefore the homological coordinates provide a new system of Darboux coordinates. We define a natural family of commuting "homological flows" on the moduli space of quadratic differentials and find the corresponding action-angle variables.The space of projective structures over the moduli space can be identified with the cotangent bundle upon selection of a reference projective connection that varies holomorphically and thus can be naturally endowed with a symplectic structure. Different choices of projective connections of this kind (Bergman, Schottky, Wirtinger) give rise to equivalent symplectic structures on the space of projective connections but different symplectic polarizations: the corresponding generating functions are found. We also study the monodromy representation of the Schwarzian equation associated with a projective connection, and we show that the natural symplectic structure on the the space of projective connections induces the Goldman Poisson structure on the character variety. Combined with results of Kawai, this result shows the symplectic equivalence between the embeddings of the cotangent bundle into the space of projective structures given by the Bers and Bergman projective connections.

Quantization of the Quantum Hitchin System and the Real Geometric Langlands Correspondence

“…The definition of the generating function W given below will use the results of [34,35,36] comparing two natural holomorphic symplectic structures on P(S). The first is defined using the symplectic structure (5.47) on the character variety via the holonomy map.…”

The monodromy of meromorphic projective structures

“…Gunning [26] proved that when g = g(S) > 1, this point lies in the open substack X * (S) consisting of representations with non-commutative image. This substack is a (possibly non-Hausdorff) complex manifold, and there is a holomorphic map F : P(S) → X * (S) (7) sending a marked projective structure to its monodromy representation.…”

“…Meromorphic projective structures. In this paper we study a monodromy map analogous to (7) but for meromorphic projective structures. The notion of a meromorphic projective structure has meaning because of the above-mentioned fact that the set of projective structures on a fixed Riemann surface S is an affine space for the space of quadratic differentials.…”

Tau functions, Hodge classes, and discriminant loci on moduli spaces of Hitchin’s spectral covers

“…Since the space of covers (1.5) forms an open subspace in the cotangent bundle T * M g of the moduli space M g of curves of genus g, it possesses a canonical symplectic structure. This symplectic structure, including a natural system of period, or homological, Darboux coordinates was studied in detail in the recent paper [3]. An immediate generalization of (1.5) is given by the family of Z n -invariant covers v n + Q n = 0 ,…”