Event Details

This two-week program aims to explore the intersection of gauge theory and string geometry, two of the most profound and rapidly developing areas of modern mathematics and theoretical physics. The first week will feature a series of five mini-courses taught by internationally acclaimed leaders in the field, while the second week will be an international conference bringing together experts and early-career researchers to foster discussions and new collaborations.

Gauge theory and string geometry have been at the forefront of mathematical and physical research for decades. Gauge theory, originating from physics, provides a framework for understanding fundamental forces and has inspired a range of mathematical advances. Notably, Donaldson’s work using gauge theory has led to significant progress in 4-manifold topology, and the development of Seiberg-Witten theory further refined our understanding of smooth 4-manifolds.

In parallel, string theory emerged as a candidate for unifying all known fundamental forces, leading to a deep interaction with mathematics. It inspired the concept of mirror symmetry, which reveals a duality between seemingly distinct Calabi- Yau manifolds and has motivated many developments in symplectic geometry and enumerative geometry.

The program will provide ample opportunities for participants to interact with experts, discuss open problems, and potentially initiate collaborations. We aim to foster an inclusive and interactive environment that encourages the exchange of ideas between researchers at different stages of their careers.

More information can be found on our webpage.

Mini Courses (24-29 Nov 2025):

Daemi Aliakbar (Washington University in St. Louis)
Title and abstract TBC.

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.

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

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.

Emanuel Scheidegger (BICMR, Peking University)
Title and abstract TBC.

Conference 01-05 Dec 2025
Invited speakers:

Organising Committee

Scientific Committee

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