题目：Vafa--Witten Invariants, Fixed Loci, and Ruled Surfaces

报告人：赵鹭天 研究员，东京大学IPMU研究所

时间：2026年5月20日，14:00-15:00

地点：西大楼308

摘要：Vafa--Witten invariants are enumerative invariants of algebraic surfaces that arise from the study of (N=4) supersymmetric gauge theory. Mathematically, they are defined by applying virtual localization to moduli spaces of Higgs pairs on a surface. These invariants sit at the intersection of algebraic geometry, gauge theory, moduli spaces, and modular forms.

In this talk, I will give an introduction to Vafa--Witten theory using ruled surfaces as a concrete example. After recalling the basic moduli problem of sheaves and Higgs fields, I will explain the distinction between the instanton branch, where the Higgs field vanishes, and the monopole branch, where the Higgs field is nonzero. Ruled surfaces provide a useful setting because their geometry contains a natural torus action, allowing the fixed loci in localization to be described in terms of simpler curve-theoretic data.

The main goal is to show how complicated surface invariants can sometimes be reduced to explicit geometry on the base curve: Hilbert schemes of points, fixed sheaves, nested Hilbert schemes, and chains of divisors. I will also indicate how these geometric descriptions lead to generating series with modular or quasimodular behavior.

This is an ongoing joint work with Amin Gholampour.

注：本报告为《几何、代数与数学物理讨论班》第6讲

邀请人：黄鹏飞 老师