An explicit description of the symplectic structure of moduli spaces of flat connections (original) (raw)

A symplectic structure for connections on surfaces with boundary

Communications in Mathematical Physics, 1996

For compact surfaces with one boundary component, and semisimple gauge groups, we construct a closed gauge invariant 2-form on the space of flat connections whose boundary holonomy lies in a fixed conjugacy class. This form descends to the moduli space under the action of the full gauge group, and provides an explicit description of a symplectic structure for this moduli space.

On moduli spaces of flat connections with non-simply connected structure group

We consider the moduli space of flat G-bundles over the two-dimensional torus, where G is a real, compact, simple Lie group which is not simply connected. We show that the connected components that describe topologically non-trivial bundles are isomorphic as symplectic spaces to moduli spaces of topologically trivial bundles with a different structure group. Some physical applications of this isomorphism which allows to trade topological non-triviality for a change of the gauge group are sketched.

The Moduli Space of Yang–Mills Connections Over a Compact Surface

Reviews in Mathematical Physics, 1997

Yang–Mills connections over closed oriented surfaces of genus ≥1, for compact connected gauge groups, are constructed explicitly. The resulting formulas for Yang–Mills connections are used to carry out a Marsden–Weinstein type procedure. An explicit formula is obtained for the resulting 2-form on the moduli space. It is shown that this 2-form provides a symplectic structure on appropriate subsets of the moduli space.

Universal moduli spaces of surfaces with flat connections and cobordism theory

2008

Given a semisimple, compact, connected Lie group G with complexification G^c, we show there is a stable range in the homotopy type of the universal moduli space of flat connections on a principal G-bundle on a closed Riemann surface, and equivalently, the universal moduli space of semistable holomorphic G^c-bundles. The stable range depends on the genus of the surface. We then identify the homology of this moduli space in the stable range in terms of the homology of an explicit infinite loop space. Rationally this says that the stable cohomology of this moduli space is generated by the Mumford-Morita-Miller kappa-classes, and the ring of characteristic classes of principal G-bundles, H^*(BG). We then identify the homotopy type of the category of one-manifolds and surface cobordisms, each equipped with a flat G-bundle. We also explain how these results may be generalized to arbitrary compact connected Lie groups. Our methods combine the classical techniques of Atiyah and Bott, with the new techniques coming out of Madsen and Weiss's proof of Mumford's conjecture on the stable cohomology of the moduli space of Riemann surfaces.

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

Inventiones mathematicae, 2017

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.

Moduli space of symplectic connections of Ricci type on T2nT^{2n}T2n; a formal approach

2002

We consider analytic curves nablat\nabla^tnablat of symplectic connections of Ricci type on the torus T2nT^{2n}T2n with nabla0\nabla^0nabla0 the standard connection. We show, by a recursion argument, that if nablat\nabla^tnablat is a formal curve of such connections then there exists a formal curve of symplectomorphisms psit\psi_tpsit such that psitcdotnablat\psi_t\cdot\nabla^tpsitcdotnablat is a formal curve of flat invariant symplectic connections and so nablat\nabla^tnablat is flat for all ttt. Applying this result to the Taylor series of the analytic curve, it means that analytic curves of symplectic connections of Ricci type starting at nabla0\nabla^0nabla0 are also flat. The group GGG of symplectomorphisms of the torus (T2n,omega)(T^{2n},\omega)(T2n,omega) acts on the space E\EE of symplectic connections which are of Ricci type. As a preliminary to studying the moduli space E/G\E/GE/G we study the moduli of formal curves of connections under the action of formal curves of symplectomorphisms.

Group cohomology construction of the cohomology of moduli spaces of flat connections on 2-manifolds

Duke Mathematical Journal, 1995

We use group cohomology and the de Rham complex on simplicial manifolds to give explicit differential forms representing generators of the cohomology rings of moduli spaces of representations of fundamental groups of 2-manifolds. These generators are constructed using the de Rham representatives for the cohomology of classifying spaces BK where K is a compact Lie group; such representatives (universal characteristic classes) were found by Bott and Shulman. Thus our representatives for the generators of the cohomology of moduli spaces are given explicitly in terms of the Maurer-Cartan form. This work solves a problem posed by Weinstein, who gave a corresponding construction (following Karshon and Goldman) of the symplectic forms on these moduli spaces. We also give a corresponding construction of equivariant differential forms on the extended moduli space X, which is a finite dimensional symplectic space equipped with a Hamiltonian action of K for which the symplectic reduced space is the moduli space of representations of the 2-manifold fundamental group in K.

Remarks on Symplectic Connections

Letters in Mathematical Physics, 2006

This note contains a short survey on some recent work on symplectic connections: properties and models for symplectic connections whose curvature is determined by the Ricci tensor, and a procedure to build examples of Ricci-flat connections. For a more extensive survey, see [5]. This note also includes a moment map for the action of the group of symplectomorphisms on the space of symplectic connections, an algebraic construction of a large class of Ricci flat symmetric symplectic spaces, and an example of global reduction in a non symmetric case.

On the symplectic form of the moduli space of projective structures

Journal of Symplectic Geometry, 2008

Let S be a C ∞ compact connected oriented surface whose genus is at least two. Let P(S) be the moduli space of isotopic classes of projective structures associated to S. The natural holomorphic symplectic form on P(S) will be denoted by Ω P. The natural holomorphic symplectic form on the holomorphic cotangent bundle T * T (S) of the Teichmüller space T (S) associated to S will be denoted by Ω T. Let e : T (S) −→ P(S) be the holomorphic section of the canonical holomorphic projection P(S) −→ T (S), given by the Earle uniformization. Let T e : T * T (S) −→ P(S) be the biholomorphism constructed using the section e. We prove that T * e Ω P = π • Ω T. This remains true if e is replaced by a large class of sections that include the one given by the Schottky uniformization.