Poisson geometry of certain moduli spaces and beyond in the light of the work of A. Tyurin

Moduli spaces of holomorphic vector bundles on a Riemann surface were extensively studied by A. Tyurin [Tyu08]. Such a moduli space carries, beyond its projective variety structure, a certain real Poisson structure that is symplectic when there are no (real) singularities, the real singularity structure actually being finer than the standard complex analytical one. A special case is the degree zero rank two moduli space with trivial determinant over a Riemann surface of genus two. I had discussed that particular case with A. Tyurin.

I will explain how, in the general case, the real structure arises by finite-dimensional ordinary symplectic reduction from an extended moduli space. Thereafter I will explain a general lattice gauge theory construction of which the extended moduli space construction is a special case. This includes a finite-dimensional construction of the Chern-Simons function on a 3-manifold and answers a question raised by Atiyah. Prompted by Atiyah’s question, A. Tyurin had studied theta functions in a lattice gauge theory framework [Tyu02]. I also plan to explain how, for the moduli space situation, the complex analytic and Poisson structures fit together to form a stratified Kähler space. In the presence of classical phase space singularities, the standard methods are insufficient to attack the problem of quantization. In certain situations, these difficulties can be overcome by means of stratified Kähler spaces [Hue11]. Related quantization issues were among A. Tyurin’s central research themes, see, e. g., [Tyu03].

References:

[Hue11]

Johannes Huebschmann. Singular Poisson-Kähler geometry of stratified Kähler spaces and quantization. In Geometry and quantization, volume 19 of Trav. Math., pages 27-63, arxiv:1103.1584 [math.DG]. Univ. Luxemb., Luxembourg, 2011.

[Tyu02]

A. N. Tyurin. Lattice gauge theories and the Florentino conjecture. Izv. Ross. Akad. Nauk Ser. Mat., 66(2):205-224, 2002.

[Tyu03]

Andrei Tyurin. Quantization, classical and quantum field theory and theta functions, volume 21 of CRM Monograph Series. American Mathematical Society, Providence, RI, 2003. With a foreword by Alexei Kokotov.

[Tyu08]

Andrey Tyurin. Vector bund les. Universitaetsverlag Goettingen, Goettingen, 2008. Collected works. Volume I, Edited by Fedor Bogomolov, Alexey Gorodentsev, Victor Pidstrigach, Miles Reid and Nikolay Tyurin.

Anhang	Größe
Extended version of the talk (pdf).	9.86 MB