PIRSA:18010072

Non-abelian Hodge theory in dimension one, Fukaya categories and periodic monopoles.

Soibelman, Y. (2018). Non-abelian Hodge theory in dimension one, Fukaya categories and periodic monopoles.. Perimeter Institute. https://pirsa.org/18010072

Soibelman, Yan. Non-abelian Hodge theory in dimension one, Fukaya categories and periodic monopoles.. Perimeter Institute, Jan. 08, 2018, https://pirsa.org/18010072 

          @misc{ pirsa_PIRSA:18010072,
            doi = {10.48660/18010072},
            url = {https://pirsa.org/18010072},
            author = {Soibelman, Yan},
            keywords = {Mathematical physics},
            language = {en},
            title = {Non-abelian Hodge theory in dimension one, Fukaya categories and periodic monopoles.},
            publisher = {Perimeter Institute},
            year = {2018},
            month = {jan},
            note = {PIRSA:18010072 see, \url{https://pirsa.org}}
          }

Yan Soibelman

Kansas State University

Talk number
PIRSA:18010072
Collection
Talk Type
Subject
Abstract

By the non-abelian Hodge theory of Carlos Simpson, harmonic bundles
interpolate between  bundles with connections on a curve and
 Higgs bundes (precise formulations requires some additional data like parabolic structure and stability structure).

I will explain the framework for a generalization of the non-abelian Hodge theory
which unifies Simpson's story ("rational case") with those  for q-difference
equations ("trigonometric case") and elliptic difference equations
("elliptic case").

This unification leads to a class of examples of the new notion of ``twistor families of categories".
In the rational,trigonometric and elliptic cases twistor families of categories  involve partially wrapped Fukaya categories of certain complex symplectic surfaces, categories of
holonomic modules over quantizations of these surfaces and categories of coherent sheaves on the surfaces
with certain restrictions on the support.

In the trigonometric and elliptic cases doubly and triply periodic monopoles give an alternative description of harmonic objects, hence playing the same role
as harmonic bundles play in the case of Simpson theory.