Abstracts for Workshop "Young Researchers in String Mathematics", November 27 - 30, 2017

I will talk about a joint work with Si Li on the computation of higher genus B-model for elliptic curves.

I will first formulate the Feynman amplitudes in the higher genus B-model (Kodaira-Spencer theory)
in terms of cohomological parings. Then I will discuss properties of the Feynman amplitudes, including
the origin of their quasi-modularity, the geometric Interpretation of their modular completions, etc.
Finally I explain the implication of the cohomological reformation in renormalization.

We discuss the recent Hodge--GUE correspondence conjecture on an explicit
relationship between special cubic Hodge integrals over the moduli space of stable
algebraic curves and enumeration of ribbon graphs with even valencies. We sketch
a proof of this conjecture based on the Virasoro constraints. We also discuss the
conjectural relationship between the cubic Hodge integrals satisfying the local
Calabi--Yau condition and the Bogoyavlensky--Toda hierarchy (aka fractional
KdV). The talk is based on a series of joint works with B. Dubrovin, S.-Q. Liu

Nekrasov, Rosly and Shatashvili observed that the generating function of a certain space
of ${\rm SL}(2)$ opers has a physical interpretation as the effective twisted superpotential
for a four-dimensional $\mathcal{N}=2$ quantum field theory. In this talk we describe the
ingredients needed to generalise this observation to higher rank. Important ingredients are
spectral networks generated by Strebel differentials and the abelianization method. As an
example we find the twisted superpotential for the $E_6$ Minahan-Nemeschansky theory.

We study two types of actions on King's moduli spaces of quiver representations over a
field $k$, and we decompose their fixed loci using group cohomology in order to give
modular interpretations of the components. The first type of action arises by considering
finite groups of quiver automorphisms. The second is the absolute Galois group of a
perfect field $k$ acting on the points of this quiver moduli space valued in an algebraic
closure of $k$; the fixed locus is the set of $k$-rational points, which we decompose