1:00pm SCGP: Gauged linear sigma model and gauged Witten equation Where: 313When: Tue, Jan 21 1:00pm — 2:30pm Abstract: (Review of symplectic vortex equation) In this talk, I will explain the basic theory of the symplectic vortex equation as the first part of the preparation for the symplectic geometry construction of the gauged linear sigma model (GLSM). If V is a symplectic manifold with a Hamiltonian group action, then analogous to the holomorphic curve equation, one has a first order elliptic system over a Riemann surface. Depending on different choices of domain metrics, one can (in principle) either define Gromov--Witten type invariants on the equivariant cohomology or the cohomology of the quotient space.
2:30pm SCGP: Talk by Alba Grassi: String Dualities and Quantum Geometry Where: 102When: Tue, Jan 21 2:30pm — 3:30pm “String Dualities and Quantum Geometry”
Wednesday, January 22
1:00pm SCGP: Gauged linear sigma model and gauged Witten equation Where: 313When: Wed, Jan 22 1:00pm — 2:30pm Abstract: (Review of FJRW theory) In this talk, I will recall the detailed construction of FJRW theory (following Fan-Jarvis-Ruan) as the second part of the preparation for the symplectic geometry construction of the gauged linear sigma model (GLSM). The FJRW theory, also called the orbifold Landau--Ginzburg A-model theory, is a cohomological field theory associated to a pair $(W, G)$, where $W$ is a nondegenerate quasihomogeneous polynomial and $G$ is a symmetry group. The correlation functions are defined by counts of solutions to the Witten equation over higher spin curves in a way analogous to the construction of Gromov-Witten invariants. I will explain how to set up the Witten equation, how to properly perturb it, and if time permits, how to construct the virtual fundamental cycle and prove axioms of FJRW invariants.
Thursday, January 23
11:00am SCGP: Neural Networks Program Seminar: Andrej Risteski Where: 313When: Thu, Jan 23 11:00am — 12:00pm
1:00pm SCGP: Gauged linear sigma model and gauged Witten equation Where: 313When: Thu, Jan 23 1:00pm — 2:30pm Abstract: (A mathematical theory of gauged linear sigma model in geometric phase) In this talk, I will overview the symplectic geometric construction of Witten’s gauged linear sigma model (in a geometric phase). The construction is based on the analysis of the gauged Witten equation, which is a combination of the Witten equation as in FJRW theory and the vortex equation in gauged Gromov-Witten theory. The upshot is that the counts of solutions to the gauged Witten equation defines a cohomological field theory on the cohomology of the classical vacuum (e.g. the quintic threefold). If time permits, I will briefly explain how to prove the relation between GLSM correlation functions and Gromov--Witten invariants. This is a joint work with Gang Tian.
1:00pm Comprehensive Exams: January Comps - Part I - January Comps - Part I Where: Math Tower P-131When: Thu, Jan 23 1:00pm — 5:00pm Title: January Comps - Part I
11:00am SCGP: Neural Networks Program Seminar: Andrej Risteski Where: 313When: Fri, Jan 24 11:00am — 12:00pm
1:00pm SCGP: Gauged linear sigma model and gauged Witten equation Where: 313When: Fri, Jan 24 1:00pm — 2:30pm Abstract: (Adiabatic limit of gauged Witten equation and mirror map) In this talk, I will review the adiabatic limit analysis of both the vortex equation and the gauged Witten equation. If one rescales the metric on the domain curves by a large constant, then as proved by Gaio-Salamon, vortices should approximate holomorphic curves in the symplectic quotient, modulo bubbling of “point-like instantons.” This picture leads to an isomorphism between the gauged Gromov-Witten theory and the usual Gromov-Witten theory of the quotient up to a change of variable (the quantum Kirwan map). I will show you how to extend this picture to the gauged Witten equation and how to obtain the precise relation between the GLSM CohFT and the Gromov-Witten CohFT of the classical vacuum. This is a joint work in progress with Gang Tian.
1:00pm Comprehensive Exams: January Comps - Part II - January Comps - Part II Where: Math Tower P-131When: Fri, Jan 24 1:00pm — 5:00pm Title: January Comps - Part II