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Victoria Hoskins: Lectures

January 9, 2023
TITLE: Motivic mirror symmetry for Higgs bundles

ABSTRACT: Moduli spaces of Higgs bundles for Langlands dual groups are conjecturally related by a form of mirror symmetry. For SLn and PGLn, Hausel and Thaddeus conjectured a topological mirror symmetry given by an equality of (twisted orbifold) Hodge numbers, which was proven by Groechenig-Wyss-Ziegler and also Maulik-Shen. We lift this to an isomorphism of Voevodsky motives, and thus in particular an equality of (twisted orbifold) rational Chow groups. Our method is based on Maulik and Shen’s approach to the Hausel-Thaddeus conjecture, as well as showing certain motives are abelian, in order to use conservativity of the Betti realisation on abelian motives. This is joint work with Simon Pepin Lehalleur.

Oscar Garcia-Prada: Lectures

January 10, 2023
TITLE: Vinberg pairs and Higgs bundles

ABSTRACT: A finite order automorphism of a complex semisimple Lie group determines a cyclic grading of its Lie algebra. Vinberg’s theory is concerned with the geometric invariant theory associated to this grading. Important examples include the case of involutions and representations of cyclic quivers. After reviewing some basic facts about Vinberg’s theory, in this talk I will discuss about its relation to the geometry of moduli spaces of Higgs bundles over a compact Riemann surface.

Slides of Lecture

Katrin Wendland: Lectures

January 11, 2023
TITLE: An application of folding ADE to BCFG

ABSTRACT: We consider families of Calabi-Yau threefolds which are obtained from the deformation spaces of ADE type surface singularities. For these non-compact Calabi-Yau threefolds, Diaconescu, Donagi and Pantev discovered in 2007 that the associated Calabi-Yau integrable systems agree with the ADE type Hitchin integrable systems. In joint work with Beck and Donagi we show that these integrable systems allow `folding´ by automorphisms of the underlying ADE root systems, and we investigate the corresponding orbifoldings of Calabi-Yau threefolds.

Timo Weigand: Lectures

January 12, 2023
TITLE: Tower Counting for the Weak Gravity Conjecture

ABSTRACT: This talk presents recent advances in our understanding of the Tower Weak Gravity Conjecture (WGC) in string compactifications with minimal supersymmetry. The underlying mathematics involves aspects of the Kahler and enumerative geometry of Calabi-Yau manifolds, in particular modular properties of partition functions of certain D4-D2-D0 bound states. The Tower Weak Gravity Conjecture predicts that any consistent gauge theory coupled to quantum gravity should exhibit an infinite tower of so-called super-extremal particles, i.e. of states whose charge-to-mass ratio exceeds that of an extremal black hole. While BPS states are automatically super-extremal, the Tower WGC is less obvious in those directions in the charge lattice that do not support towers of BPS states.
For time constraints we focus in this talk on M-theory compactifications on Calabi-Yau threefolds, but similar results hold for F-theory compactifications on Calabi-Yau three- or fourfolds. To deduce the presence of super-extremal towers, we first classify all weak coupling limits in M-theory compactifications on Calabi-Yau threefolds, extending an earlier classification of the possible infinite distance limits in the classical Kahler moduli space. We then show that every direction in the charge lattice dual to a gauge group with a weak coupling limit admits a tower of BPS or of superextremal non-BPS states at least asymptotically. To this end we translate the problem into a counting problem for certain D4-D2-D0 bound states and make use of the modular properties of their partition function and results from Noether-Lefschetz theory. From a physics perspective, the asymptotic Tower WGC can be viewed as a consequence of the Emergent String Conjecture.
Slides of Lecture

Matt Turner: Lectures

January 9, 2023 (jointly with Johannes Nordström)
TITLE: Examples of asymptotically conical G2-instanstons

ABSTRACT: We present examples of G2-instantons with dilation-invariant asymptotics on the “C7” asymptotically conical G2-metric on the anticanonical bundle of CP1 x CP1. The examples have cohomogeneity one which reduces the problem to solving an ordinary differential equation. We find solutions to these equations using a dynamical systems approach. This is joint work with Karsten Matthies.

Slides of Lecture (second part)

Boris Pioline: Lectures

January 11, 2023
TITLE: Modularity of BPS indices on Calabi-Yau threefolds

ABSTRACT: Unlike in cases with maximal or half-maximal supersymmetry, the spectrum of BPS states in type II string theory compactified on a Calabi-Yau threefold with generic SU(3) holonomy remains partially understood. Mathematically, the BPS indices coincide with the generalized Donaldson-Thomas invariants associated to the derived category of coherent sheaves, but they are rarely known explicitly. String dualities indicate that suitable generating series of rank 0 Donaldson-Thomas invariants counting D4-D2-D0 bound states should transform as vector-valued mock modular forms, in a precise sense. I will spell out and test these predictions in the case of one-modulus compact Calabi-Yau threefolds such as the quintic hypersurface in P4, where rank 0 DT invariants can (at least in principle) be computed from Gopakumar-Vafa invariants, using recent mathematical results by S. Feyzbakhsh and R. Thomas. Work in progress with S. Alexandrov, S. Feyzbakhsh, A. Klemm and T. Schimannek.

Slides of Lecture

Miguel Moreira: Lectures

January 11, 2023
TITLE: Virasoro constraints: vertex algebras and wall-crossing

ABSTRACT: This talk will be the second one concerning the Virasoro constraints in moduli spaces of sheaves (see Woonam’s abstract), based on joint work with A. Bojko and W. Lim. In this talk, I will focus on the connection between Virasoro constraints and the vertex algebra that D. Joyce recently introduced to study wall-crossing. It turns out that this vertex algebra can be endowed with a conformal element that induces the Virasoro operators that had appeared previously in the literature. In this language, our conjectures/results say that moduli of sheaves define physical/primary states in this vertex operator algebra. From this point of view and Joyce’s theory, we can prove that the Virasoro constraints are compatible with wall-crossing. This is the main new technical tool that allows to prove the constraints for torsion-free sheaves on curves and surfaces by reducing everything to rank 1.

Woonam Lim: Lectures

January 11, 2023
TITLE: Virasoro constraints; history and moduli of sheaves

ABSTRACT: Virasoro constraints were first conjectured for the moduli of stable curves (the Witten conjecture) and stable maps. These conjectures provide a set of universal relations among descendent invariants described by a representation of half of the Virasoro algebra. Recently, the analogous constraints were conjectured in several sheaf theoretic contexts. In joint work with A. Bojko and M. Moreira, we provide a unifying viewpoint to Virasoro constraints for general moduli of sheaves and prove the conjecture for torsion-free sheaves on curves and surfaces.

Thibault Langlais: Lectures

January 12, 2023
TITLE: An introduction to some aspects of the swampland distance conjectures

ABSTRACT: The swampland program aims at distinguishing the quantum field theories which can be consistently coupled to quantum gravity at high energies from those which cannot. It leads to the formulation of many interesting problems at the intersection of geometry and physics. The first part of this talk will be an introduction to the distance conjectures, concerning the moduli spaces of vacua of the theories which admit a consistent quantum gravity completion. In the second part I will present ongoing work on twisted connected sum G2-manifolds related to the distance conjectures.
Slides of Lecture

Thomas Grimm: Lectures

January 12, 2023
TITLE: Quantum gravity conjectures and asymptotic Hodge theory

ABSTRACT: In this talk I will explain how asymptotic Hodge theory can be used to provide general evidence for some of the quantum gravity conjectures. In particular, I will describe how the orbit theorems of Hodge theory can be used in addressing the so-called Distance Conjecture. For Calabi-Yau manifolds, I will sketch the implied classification of asymptotic regions of the moduli space and comment on the special properties of infinite distance boundaries. I will highlight that the quantum gravity conjectures can actually lead to new mathematical theorems by presenting a finiteness theorem generalizing a famous result of Cattani, Deligne, and Kaplan on Hodge classes. This new result is based on work with B. Bakker, C. Schnell, and J. Tsimerman.

Slides of Lecture