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Fabio Apruzzi: Lectures

March 13, 2023
TITLE: Generalized symmetries from string theory

ABSTRACT: String theory provides a systematic way of constructing quantum field theories (QFTs) via geometric engineering. In particular, this can involve non-compact Calabi-Yau spaces in various dimensions, as well as other special holonomy manifolds. I will describe the dictionary between the generalized symmetry data of the QFTs and some specific cohomology of the geometric engineering spaces by focusing on explicit examples.

Lakshya Bhardwaj: Lectures

March 13, 2023
TITLE: Overview of Generalized Symmetries

ABSTRACT: I will provide an overview of various types (higher-form, higher-group and non-invertibles) of generalized symmetries and how they arise in gauge theories. This will set stage for later talks that will describe how generalized symmetries are encoded in geometric engineering.

Federico Bonetti: Lectures

March 14, 2023
TITLE: SymTFTs, Differential Cohomology, and Geometric Engineering

ABSTRACT: The symmetry data of a quantum field theory (QFT) in d spacetime dimensions is conveniently captured by an auxiliary topological field theory in d+1 spacetime dimensions, referred to as the Symmetry Topological Field Theory (SymTFT). After a brief introduction to the SymTFT, I will focus on the following question: how can we compute the SymTFT for a QFT engineered geometrically in string theory/M-theory? The formalism of differential cohomology provides systematic tools to address this problem, as I will illustrate in some examples.

Saman Habibi Esfahani: Lectures

May 16, 2024
TITLE: On the Donaldson-Scaduto conjecture

ABSTRACT: Motivated by G2-manifolds with coassociative fibrations in the adiabatic limit, Donaldson and Scaduto conjectured the existence of associative submanifolds homeomorphic to a three-holed 3-sphere with three asymptotically cylindrical ends in the G2-manifold X×ℝ3, or equivalently similar special Lagrangians in the Calabi-Yau 3-fold X×C, where X is an A2-type ALE hyperkähler 4-manifold. We prove this conjecture by solving a real Monge-Ampère equation with a singular right-hand side, which produces a potentially singular special Lagrangian. Then, we prove the smoothness and asymptotic properties for the special Lagrangian using inputs from geometric measure theory. The method produces many other asymptotically cylindrical U(1)-invariant special Lagrangians in X×C, where X arises from the Gibbons-Hawking construction. This talk is based on a joint work with Yang Li.

Slides of Lecture

March 16, 2023
TITLE: Towards a Monopole Fueter Floer Homology 

ABSTRACT: Monopoles appear as the dimensional reduction of instantons to 3-manifolds. An interesting feature of the monopole equation is that it can be generalized to certain higher-dimensional spaces. The most interesting examples appear on Calabi-Yau 3-folds and G2-manifolds. Monopoles, conjecturally, can be used to define invariants of 3-manifolds, Calabi-Yau 3-folds, and G2-manifolds. These monopole invariants, conjecturally, are related to certain counts of calibrated submanifolds, similar to the Taubes’ theorem, which relates the Seiberg-Witten and Gromov invariants of symplectic 4-manifolds.  
Motivated by this conjecture, we propose a Floer theory for 3-manifolds, generated by Fueter sections on hyperkähler bundles with fibers modeled on the moduli spaces of monopoles on R3.  A major difficulty in defining these homology groups is related to the non-compactness problems. We prove partial results in this direction, examining the different sources of non-compactness, and proving some of them, in fact, do not occur.  

Constantin Teleman: Lectures

March 13, 2023
TITLE: Introduction to topological symmetries and higher groups

ABSTRACT: I will review the setting of an algebra of symmetries acting on a QFT. Special emphasis will be placed on symmetries arising from finite homotopy types (aka higher finite groups) and the way homotopical calculations quantize to a ‘higher categorical group ring’ of a space.

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)