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

### APA

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

### MLA

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

### BibTex

@misc{ 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

## 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.