Tudor Dimofte: 3d gauge theories: vortices and vertex algebras
3d N=4 gauge theories admit two topological twists, often called A and B, that are expected to lead mathematically to fully extended 3d topological quantum field theories (TQFTs). I will review some aspects of these putative TQFTs, some of their known and expected connections to representation theory, and (especially) connections to moduli spaces of vortices and their cohomology. I will then present some recent work on accessing these 3d TQFTs via boundary vertex algebras -- much as was done for Chern-Simons TQFT using the WZW model in the '80s and '90s. In particular, I will discuss using boundary vertex algebras to define braided tensor categories of line operators and to prove their equivalence under 3d mirror symmetry. (These developments in joint work with Andrew Ballin, Thomas Creutzig and Wenjun Niu.)
David Favero: Cohomological field theories from GLSMs
Gauged linear sigma models (GLSMs) serve as a means of interpolating between Kähler geometry and singularity theory. In enumerative geometry, they should specialize to both Gromov–Witten and Fan–Jarvis–Ruan–Witten theory. In joint work with Bumsig Kim (see arXiv:2006.12182), we constructed such enumerative invariants for GLSMs. Furthermore, we proved that these invariants form a cohomological field theory (CohFT). In this lecture sequence, I will describe GLSMs and CohFTs, review the history of their development in enumerative geometry, and discuss the construction of these general invariants. Briefly, the invariants are obtained by forming the analogue of a virtual fundamental class which lives in the twisted Hodge complex over a certain “moduli space of maps to the GLSM”. This virtual fundamental class roughly comes as the Atiyah class of a “virtual matrix factorization” associated to the GLSM data.
Jeongseok Oh: Quasimaps, their wall-crossings and mirror symmetry*
Inspired by the role of loop spaces in the proof of the mirror theorem by his advisor (Alexander Givental), Bumsig Kim and his collaborators developed the theory of spaces of quasimaps. Interestingly, the change of their generating functions of invariants (defined by integration over these spaces) according to their stability conditions, known as a wall-crossing formula, has exactly the same form as mirror symmetry for certain potential functions predicted by physics. So it gives a geometric interpretation of mirror symmetry. A huge computational advantage is that the moduli spaces of quasimaps have less boundary components than those of stable maps, so that they become easier to work with. Together with the wall-crossing formula, we can compute genus g Gromov-Witten invariants for quintics, up to a few low degree invariants which remain unknown.
In this tribute talk to Bumsig Kim, I shall review his achievements in mirror symmetry through quasimaps.
Constantin Teleman: Gromov-Witten theory and gauge theory
We will review the close relation between 3-dimensional gauge theory (N=4 SUSY 3d gauge theory, in physics language) and the 2-dimensional A-model, focusing on the role of the Toda integrable system. The latter plays the role, in 3d, that the classifying space BG of a compact Lie group G plays in lower dimension. Compact symplectic manifolds with Hamiltonian G-action give boundary conditions for the 3d theory. We will review the Floer-theoretic construction of the latter, as well as a key geometric application to the quantum cohomology of GIT quotients of Fano manifolds. Much of this material is joint work with Dan Pomerleano. In the last portion of the lectures, we will quickly review the GLSM construction of Coulomb branches for 3d gauge theory and their relation to Gromov-Witten theory, along with more speculative comments on the case of quaternionic matter.
*Tribute to Bumsig Kim