# FRG Special Lecture on April 21 2017, 3-4pm

## University of Michigan, EH 3866

Andrey Smirnov, UC-Berkeley
Title: Quasimaps and geometric representation theory I

# FRG Workshop on April 22 2017

## University of Michigan, EH 4096

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 14:00-15:30 Jinwon Choi Stable quasimaps and wall crossing phenomena 15:30-16:00 Coffee break 16:00-17:00 Andrey Smirnov Quasimaps and geometric representation theory III

Supported by an FRG grant.

## Abstracts

### Young-Hoon Kiem (Seoul National University)

#### Stability conditions on quasimaps and FJRW theory

The 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 phenomena

In 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-Berkeley

#### Quasimaps and geometric representation theory

The goal of the following three lectures is to give an overview and a short introduction into quantum geometry of Nakajima varieties.

Lecture 1:
I define quasimaps to Nakajima varieties and discuss their relations with representation theory, DT-invariants of threefolds and other parts of mathematical physics. We will compute some quasimap invariants for cotangent bundles over grassmannians explicitly.

Lecture 2:
I will discuss quantum difference equations which governs the quasimap counts of a Nakajima variety and do explicit computations for the case of contangent bundles over grassmannians and Hilbert scheme of points on a plane.

Lecture 3:
In this lecture, starting from quasimap counting, I will define the quantum K-theory of a Nakajima variety and quantum tautological bundles. I will explain the relation of quantum K-theory with spin chains and Bethe ansatz.