Siqi He (Chinese Academy of Sciences)

Title: Z/2 Harmonic 1-Forms and Related Problems in Geometry

Abstract: This mini-course will focus on Z/2 harmonic 1-forms and their connections to gauge theory, topology, and compactification problems. The lectures will be divided into four parts:

Gauge Theory with SL(2,C) structure group: Gauge-theoretic equations with SL(2,C) structure group, including flat connection equations and the Kapustin–Witten equations. We will discuss their basic properties and related geometric and topological problems.

Compactness of Flat SL(2,C) Connections: Taubes’ compactness theorem and the role of Z/2 harmonic 1-forms in describing the ideal boundary. Basic properties of Z/2 harmonic 1-forms will also be introduced.

Deformation of Z/2 Harmonic 1-Forms: Donaldson’s work on the deformation of Z/2 harmonic 1-forms, along with possible geometric applications of these deformations.

Relations to Low-Dimensional Topology: Connections between Z/2 harmonic 1-forms and classical objects in low-dimensional topology, including Thurston’s compactification of Teichmüller space, measured foliations, and the Morgan–Shalen compactification.

The lectures are intended for PhD students and early-career researchers with a background in differential geometry or gauge theory.

References:

S. K. Donaldson, Deformations of multivalued harmonic functions, arXiv:1912.08274.

C. H. Taubes, PSL(2,C) connections on 3-manifolds with L² bounds on curvature, arXiv:1205.0514.

C. H. Taubes, Compactness theorems for SL(2,C) generalizations of the 4-dimensional anti-self-dual equations, arXiv:1307.6447.

H. Siqi, R. Wentworth, B. Zhang, Z/2 harmonic 1-forms, R-trees, and the Morgan–Shalen compactification, arXiv:2409.04956.

Johanna Knapp (University of Melbourne)

Title: The Physical Mathematics of Gauged Linear Sigma Models

Abstract: Gauged linear sigma models (GLSMs), first introduced by Witten in 1993, are supersymmetric gauge theories in two dimensions. They provide a powerful tool to study properties of extra dimensions in string theory and the mathematical structures behind them. The aim of these lectures is to show how a physics analysis of GLSMs (vacuum configurations, low-energy effective theories, D-branes, path integrals etc.) leads to advanced mathematics (GIT quotients, categorical equivalences, enumerative invariants etc.). The main focus will be on GLSMs that are related to Calabi-Yau compactifications of string theory.

A rough outline of the lectures is as follows (we may not cover all of it):

1.    GLSMs: physics definition and phases

•    Field content and symmetries
•    Phases of GLSMs
•    Higgs vs Coulomb branches
•    Examples

2.    (B-type) D-branes in GLSMs

•    B-branes in GLSMs
•    D-brane transport and categorical equivalences (physics perspective)

3.    GLSM partition functions and what they compute

•    (Hemi-)sphere partition function
•    B-brane central charges and enumerative invariants

Literature: There aren't really any good recent reviews and premliminary reading is not necessary, but many of the GLSM papers of the past few years contain useful summaries. Some good older references are:

•    Witten "Phases of N=2 theories in two-dimensions" hep-th/9301042: Witten's original paper on GLSMs
•    Hori et el "Mirror symmetry" (available at http://www.claymath.org/library/monographs/cmim01.pdf): The big mirror symmetry book, Chapter 15 is on GLSMs; other chapters may be useful too
•    Herbst,Hori,Page "Phases Of N=2 Theories In 1+1 Dimensions With Boundary" arXiv:0803.2045[hep-th]: 265 page-long exhaustive discussion on B-branes in abelian GLSMs, Chapters 4,5,7,10 are the most relevant
•    Jockers et al "Two-Sphere Partition Functions and Gromov-Witten Invariants" arXiv:1208.6244[hep-th]: sphere partition function and enumerative invariants

Yixuan Li (Australian National University)

Title: Mirror Symmetry of Type A Affine Grassmannian Slices

Abstract: This mini-course is about a mirror symmetry result central to Mina Aganagic’s ICM 2022 talk [1] on two categorifications of Jones polynomials. Recall that the Jones polynomial of a knot can be calculated via the fundamental representation V of the quantum group U_q(sl_2) roughly in the following way: First present the knot as the closure of a braid with n strands by Alexander’s theorem. Then associated to the n strands, we have the weight spaces of the tensor product of n copies of V. Associated to the braid, we have a product of R-matrices acting on each weight space. Jones polynomial is related to the trace of this product of R-matrices on the weight spaces of this tensor product. Thus to category this picture, we need to upgrade the weight spaces to categories and the R matrices to certain braid group actions on these categories. Taking trace would be interpreted as taking a homomorphism between certain objects in the category.

Via the geometric Satake equivalence[2][3], weight spaces of tensor products of fundamental representations of gl(m) are related to the geometry of certain slices in the affine grassmannian of Gl(m). These slices are conical symplectic singularities. There are two ways to smoothen this singularity: One can consider the semi-universal symplectic deformation or the symplectic resolution. Hence the weight spaces will be categorified into certain Fukaya categories of deformed affine grassmannian slices or the category of coherent sheaves on the symplectic resolution of these slices. The braid group action will in fact be provided by the monodromy action of the semi-universal symplectic deformation.

Mirror symmetry is a relation between Fukaya category of a symplectic manifold X and the derived category of coherent sheaves on a complex manifold X^. In fact these two categorifications will be related to each other by a conjectural homological mirror symmetry. In [4] we proved a partial result saying that the coherent side embeds into the symplectic side. In fact, as another evidence, we can show manually that both the quantum connection on X^ and a Gauss-Manin connection on X can be identified with certain Knizhnik-Zamolodchikov connections, following the work [5] of Danilenko.

These talks are prepared for PhD students and early career researchers with a background in representation theory or geometry/topology. Time permitting, I will mention the connection of these slices with certain monopole moduli spaces as predicted by [6]. What we need in order to prove these predictions is some control over Kapustin-Witten equations.

References:

[1] Aganagic, Homological Knot Invariants from Mirror Symmetry, arxiv.org/pdf/2207.14104

[2] Mirkovic-Vilonen, Geometric Langlands duality and representations of algebraic groups over commutative rings, Annals of Mathematics, 166 (2007), 95–143

[3] Zhu, An introduction to affine Grassmannians and the geometric Satake equivalence, arxiv.org/abs/1603.05593

[4] Aganagic-Danilenko-Li-Shende-Zhou, Quiver Hecke algebras from Floer homology in Coulomb branches, https://arxiv.org/abs/2406.04258

[5] Danilenko, Quantum Differential Equation for Slices of the Affine Grassmannian, https://arxiv.org/abs/2210.17061

[6] Kapustin-Witten, Electric-Magnetic Duality And The Geometric Langlands Program, https://arxiv.org/abs/hep-th/0604151