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

Alternatively have a look at the program.

## Categorification of Chern characters

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Speaker:
Sarah Scherotzke
Affiliation:
Universität Münster
Date:
Mon, 2017-11-27 09:00 - 10:00
Location:
MPIM Lecture Hall

The Chern character is a central construction with incarnations in algebraic topology, representation
theory and algebraic geometry. It is an important tool to probe $K$-theory, which is notoriously hard
to compute. In my talk, I will explain, what the categorification of the Chern character is and how we
can use it to show that certain classical constructions in algebraic geometry are of non-commutative
origin. The category of motives plays the role of $K$-theory in the categorified picture. The categorification

## Large $N$ limits from a BV perspective

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Speaker:
Owen Gwilliam
Affiliation:
MPI
Date:
Mon, 2017-11-27 10:30 - 11:30
Location:
MPIM Lecture Hall

Starting with 't Hooft, physicists have used a ribbon graph expansion to understand certain integrals over spaces
of $N \times N$ matrices in the large $N$ limit. This expansion can be deduced from the Feynman diagram
expansion, which relies on the nice structure of moments of a Gaussian measure. We provide a homological
perspective on this situation: the Batalin-Vilkovisky formalism (which we will outline) provides a homological
approach to computing moments, and the Loday-Quillen-Tsygan theorem (which we will explain) gives a

## From classical Weierstraß elliptic functions to quantum invariants

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Speaker:
Jie Zhou
Affiliation:
Universität Köln
Date:
Mon, 2017-11-27 11:30 - 12:30
Location:
MPIM Lecture Hall

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.

## tba

Posted in
Speaker:
Miranda Cheng
Affiliation:
University of Amsterdam
Date:
Mon, 2017-11-27 16:30 - 17:30
Location:
MPIM Lecture Hall

## On the Hodge-GUE correspondence

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Speaker:
Di Yang
Affiliation:
MPI
Date:
Mon, 2017-11-27 17:30 - 18:30
Location:
MPIM Lecture Hall

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.~Dubro\-vin, S.-Q.~Liu

## Exact results for class $S_k$

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Speaker:
Elli Pomoni
Affiliation:
DESY
Date:
Tue, 2017-11-28 09:00 - 10:00
Location:
MPIM Lecture Hall

We will introduce a large class of $\mathcal{N}=1$ superconformal theories, called
$S_k$, which is obtained from Gaiotto's $\mathcal{N}=2$ class $S$ via orbifolding. We
will study the Coulomb branch of the theories in the class by
constructing and analyzing their spectral curves. Using our experience
from the $\mathcal{N}=2$ \textsc{agt} correspondence we will search for a 2d/4d relations
(\textsc{agt}${}_{k}$) for the $\mathcal{N}=1$ theories of class $S_k$. From the curves we will
identify the 2d \textsc{cft} symmetry algebra and its representations, namely

## Geometric recipe for superpotentials

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Speaker:
Lotte Hollands
Affiliation:
Heriot-Watt University, Edinburgh
Date:
Tue, 2017-11-28 10:30 - 11:30
Location:
MPIM Lecture Hall

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.

## Tropical Hurwitz and GW numbers

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Speaker:
Johannes Rau
Affiliation:
Universität Tübingen
Date:
Tue, 2017-11-28 11:30 - 12:30
Location:
MPIM Lecture Hall

Tropical geometry has been proved successful to study various types of enumerative
numbers, including Gromov-Witten invariants for toric surfaces and Hurwitz numbers
with at most two special points. In my talk I will try to give an overview on
some showcase results, recent developments (counting "real'' curves) and relations
to other approaches.

## Group actions on quiver varieties and application to branes

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Speaker:
Victoria Hoskins
Affiliation:
Humboldt Universität Berlin
Date:
Tue, 2017-11-28 16:30 - 17:30
Location:
MPIM Lecture Hall

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

## Line defects in $\mathcal{N}=2$ QFT and framed quivers

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Speaker:
Michele Cirafici
Affiliation:
IST Lisboa
Date:
Tue, 2017-11-28 17:30 - 18:30
Location:
MPIM Lecture Hall

I will discuss a certain class of line defects in four dimensional supersymmetric theories
with $\mathcal{N}=2$. Many properties of these operators can be rephrased in terms of
quiver representation theory. In particular one can study BPS invariants of a new kind, the
so-called framed BPS states, which correspond to bound states of ordinary BPS states with
the defect. Such invariants determine the IR vev of line operators. I will discuss how these
invariants arise from framed quivers. Time permitting I will also discuss a formalism to study

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