Gauged Maps, Vortices and Their Moduli

Minicourses

1) Jean-Michel Bismut (Université Paris-Saclay, France):

   Quillen metrics and applications

  Given a compact manifold X and a holomorphic vector bundle E on X, one can define the Dolbeault cohomology groups H = H^·(X,E) and consider the determinant line λ = det H. Given Hermitian metrics on TX and E, there is canonical Hermitian metric on λ, the Quillen metric, that is constructed using the corresponding holomorphic Ray-Singer analytic torsion, a spectral invariant of the associated Hodge Laplacian, which extends in arbitrary dimensions the idea of the determinant of the Laplacian.
  The Quillen metrics have a number of remarkable properties. They are compatible with corresponding constructions in algebraic geometry. Under a suitable Kähler condition, the curvature of the Quillen metric can be computed locally, in terms of Chern-Weil data. The Polyakov anomaly formulas are a special case of this curvature theorem. More generally, when one compares the Quillen metrics on canonically isomorphic determinant lines, the corresponding ratio can be evaluated by an explicit local formula, involving Bott-Chern classes, instead of the more familiar Chern-Simons classes.
  In the course, I will explain the basic known results on Quillen metrics, sketch some of the proofs, and explain some applications.

[slides]

videos: [1] [2] [3]

2) Dan Halpern-Leistner (Cornell University, USA):

   Introduction to Θ-stratifications of moduli stacks

  The moduli of vector bundles on a curve has a beautiful structure called the Harder-Narasimhan stratification, which has many applications to studying the geometry of this classical moduli problem. The theory of Θ-stratifications provides an intrinsic description of this stratification that generalizes to other moduli problems.

  I will give an overview of the main results in the theory of Θ-stratifications. Then I will explain how to apply these general results to the moduli stack of gauged maps from a curve to a linear representation. Ultimately, this leads to an explicit formula for the K-theoretic gauged Gromov-Witten invariants, in arbitrary genus, of a linear representation (or a complete intersection therein).

videos: [1] [2] [3]

3) Justin Hilburn (Perimeter Institute, Canada):

   The 3d A-model: generalized Seiberg-Witten equations, vortices and monopoles

  Three-dimensional mirror symmetry predicts an equivalence between pairs of 3d topological quantum field theories associated to hyper-Kähler manifolds equipped with hyper-Hamiltonian actions of compact Lie groups. The first TQFT, which is known as the 3d B-model or Rozansky-Witten theory, is of an algebro-geometric flavor and has been studied extensively by Kapustin, Rozansky and Saulina. The 3d A-model, which is is governed by the 3d Seiberg-Witten equations, is more mysterious.

  In this lecture series, I will explain what is known about this TQFT, including the rigorous construction of the ring of local operators by Braverman-Finkelberg-Nakajima, the category of line operators, and the 2-category of boundary conditions.

 

[notes/slides]

videos: [1] [2] [3]