Poisson geometry of flat connections for SU(2)-bundles on surfaces

Abstract

Abstract. In earlier work we have shown that the moduli space N of flat connections for the (trivial) SU(2)-bundle on a closed surface of genus ℓ ≥ 2 is a stratified symplectic space so that, in particular, the data give rise to a Poisson algebra (C ∞ N, {·, ·}) of continuous functions on N; furthermore, the strata are Kähler manifolds, and the stratification consists of an open connected and dense submanifold N Z of real dimension 6(ℓ − 1), a connected stratum N (T ) of real dimension 2ℓ, and 2 2ℓ isolated points. In this paper we show that, close to each point of N (T ) , the space N and Poisson algebra (C ∞ N, {·, ·}) look like a product of C ℓ endowed with the standard symplectic Poisson structure with the reduced space and Poisson algebra of the system of (ℓ − 1) particles in the plane with total angular momentum zero, while close to one of the isolated points, the Poisson algebra (C ∞ N, {·, ·}) looks like that of the reduced system of ℓ particles in R 3 with total angular momentum zero. Moreover, in the genus two case where the space N is known to be smooth we locally describe the Poisson algebra (C ∞ N, {·, ·}) and the various underlying symplectic structures on the strata and their mutual positions explicitly in terms of the Poisson structure.1991 Mathematics Subject Classification. 32G13, 32G15, 32S60, 58C27, 58D27, 58E15, 81T13.

Poisson geometry of certain moduli spaces for bundles on a surface

“…This spare and the related ones have been studied extensively in the literature [12][13][14][15]. This was re-proved in [10] within our framework. This has the following consequence, established in [10].…”

“…In a different guise, cf. This has the following consequence, established in [10]. This spare and the related ones have been studied extensively in the literature [12][13][14][15].…”

“…In this note, we present an extension of structures of this kind to the whole space, its singularities included, phrased in terms of the symplectic or, more generally, Poisson geometry of certain related classical constrained systems [5][6][7][8][9][10], which are formally similar to what are called "collective Hamiltonian systems" in the mathematical physics literature. It is known that the spaces N(n, k) have structure of a normal projective variety [15,16].…”

“…Roughly speaking, the local models amount to certain deformation spaces for families of bundles with connection carrying local symplectic or, more generally, Poisson information about the moduli space. In [8][9][10] it is shown that the model locally comprises all the geometric structures of interest. This remark can be made precise in the language of momentum mappings and Poisson structures.…”

Poisson structures on certain moduli spaces for bundles on a surface

“…It is the fifth of a series of papers about a program revealing the structure of these moduli spaces by means of the symplectic or more generally Poisson geometry of certain related classical constrained systems. Its predecessors are [7] - [10], and it will be followed by [11] and [12].…”