Representations, Moduli and Duality

Week 2

Most of week 2, 28.07 - 01.08,  is dedicated to discussions and collaborations. Below is the schedule for the talks and the reception. 

 

	Monday	Tuesday	Wednesday	Thursday	Friday
10:00-11:00	Kaubrys	Braverman	Botta		Tomasini
18:00 -	Reception

 

 

Titles and Abstracts:

 

Kaubrys: Langlands duality for critical cohomology of local systems on the 3-torus

In this talk I will explain a proof of Langlands duality of critical cohomology
(cohomological DT invariants) of SLn/PGLn local systems on the 3-torus for prime n.
This duality is expected to hold for all compact oriented 3 manifolds.
From the physics point of view it arises as S duality of topological twists of 4D N=4 Yang-Mills theories
and can be viewed as a Geometric Langlands type statement for 3-manifolds.
The main tools of the proof are the use of an exponential map relating CoDT invariants of
local systems to "additive" CoDT invariants, a decomposition result called cohomological integrality
and a computation of BPS cohomology for the 3-torus. I will also explain work in progress concerning
applications of the exponential map construction to nonabelian Hodge theory,
which leads to a multiplicative version of nonabelian Hodge theory for stacks for GLn.

 

Braverman:  Koszul duality in the framework of relative Langlands duality.

 

Botta: Enumerative 3d mirror symmetry and bow varieties.

 

Abstract: As advocated by Aganagic and Okounkov, mirror symmetry in three dimensions admits an enumerative interpretation in terms of quasimap counts to mirror dual symplectic varieties. Specifically, the generating series of the counts, known as vertex functions, are expected to match up to some distinguished class in elliptic cohomology, known as the elliptic stable envelope. The latter also satisfies a remarkable mirror symmetry statement. After reviewing this general picture, I will focus on the case of bow varieties, which can be thought of as the type A version of mirror symmetry, and discuss the main ingredients of its proof (joint with subsets of R. Rimanyi and H. Dinkins). 

 

Tomasini: Elliptic Hochschild Homology

In this talk we will survey the topic of elliptic Hochschild homology,
an algebraic approximation of elliptic cohomology.
We will see how it is constructed and how it recovers elliptic cohomology in the complexified equivariant setting.
Based on joint work with Sarah Scherotzke and Nicolo Sibilla.