Andrey Smirnov, UC-Berkeley
Title: Quasimaps and geometric representation theory I
Andrey Smirnov, UC-Berkeley
Jinwon Choi, Sookmyung Women's University
Young-Hoon Kiem, SNU
| 9:30-10:30 | Andrey Smirnov | Quasimaps and geometric representation theory II |
| 10:30-11:30 | Coffee and bagels | |
| 11:30-13:00 | Young-Hoon Kiem | Stability conditions on quasimaps and FJRW theory |
| 14:30-16:00 | Jinwon Choi | Stable quasimaps and wall crossing phenomena |
| 16:00-16:30 | Coffee break | |
| 16:30-17:30 | Andrey Smirnov | Quasimaps and geometric representation theory III |
Supported by an FRG grant.
AbstractsYoung-Hoon Kiem (Seoul National University)Stability conditions on quasimaps and FJRW theoryThe Fan-Jarvis-Ruan-Witten theory constructs a virtual cycle on the moduli space of spin curves which generates a cohomological field theory, given a quasi-homogeneous polynomial $w$. When $w=\sum_{i=1}^5x_i^5$ is the Fermat quintic, the Landau-Ginzburg/Calabi-Yau correspondence conjectures an equivalence of the Gromov-Witten invariant of the quintic CY 3-fold with the FJRW invariant of $(\mathbb{C}^5/\mathbb{Z}_5,w)$. The FJRW moduli space for the Fermat quintic is an open substack of the Artin stack $\mathfrak{X}$ of quadruples $(C,L,x,p)$ of an orbifold curve $C$, a line bundle $L$, sections $x\in H^0(L)^{\oplus 5}$ and $p\in H^0(L^{-5}\omega_C^{\mathrm{log}})$. In fact, this stack is big enough to contain both the FJRW moduli space and the moduli space of stable maps to $\mathbb{P}^4$ together with $p$-fields, which gives the GW invariant of the Fermat quintic CY 3-fold. Many more stability conditions are expected to be found in this stack $\mathfrak{X}$ which should give us enough invariants to interpolate the GW and FJRW invariants by wall crossing. There is a line of stability conditions, called $\epsilon$-stability, which allow base points of the $x$-field on the CY side and of the $p$-field on the LG side.In this talk, based on a joint work with Jinwon Choi, I will describe another line of stability conditions which narrows the gap between the GW and FJRW stabilities. These stability conditions arise from the theory of stable pairs, which was much studied in 1990s and led to celebrated results like (1) a proof of the Verlinde formula by Thaddeus, (2) a remarkable progress in the Brill-Noether theory of stable vector bundles and (3) a formula comparing the Donaldson invariant with the Seiberg-Witten invariant for algebraic surfaces by T. Mochizuki. Jinwon Choi (Sookmyung Women's University)Stable quasimaps and wall crossing phenomenaIn the previous talk of Young-Hoon Kiem, the moduli spaces of $\delta$-stable quasimaps are introduced as an interpolation of GW and FJRW theories. In this talk, we discuss in more detail how the moduli space changes as we cross the walls. We also present another line of stability conditions which connect the $\epsilon=0^+$ and $\delta=\infty$-stability conditions. This talk is based on joint work (in progress) with Young-Hoon Kiem.Andrey Smirnov, UC-BerkeleyQuasimaps and geometric representation theoryThe goal of the following three lectures is to give an overview and a short introduction into quantum geometry of Nakajima varieties.Lecture 1: Lecture 2: Lecture 3: |