Beijng Geometry and Physics Colloquium VIII

Time: 2016-10-18 Published By:

Speaker(s): Young-Hoon Kiem(Seoul National University); Jinwon Choi(Sookmyung Women's University)

Time: April 23, 2016

Venue: Room 77201，Jingchunyuan 78,BICMR

Beijing Geometry and Physics Colloquium at April 23, 2016

The gauged theory has been studied widely in mathematics and physics. Its algebra-geometric counterpart is stability condition. It appears in the recent mathematical formulation of gauged linear sigma model. Our next colloquium is devoted to the two types of stability conditions in gauged linear sigma model.

All the lecture will be held at 77201 of 78 yard of BICMR

Schedule:

9:00am-10:00am

Young-Hoon Kiem

Curve counting via sheaf theory

10:00am-10:30am

Tea Time

10:30am-11:30am

Jinwon Choi

Moduli space of $\delta$-stable pairs and its application

2:00pm-3:00pm

Young-Hoon Kiem

LG/CY correspondence via quasimaps

3:00pm-3:30pm

Tea Time

3:30pm-4:30pm

Jinwon Choi

Genus zero wall-crossing for quasi-maps

1. Curve counting via sheaf theory (Y.-H. Kiem)

Gromov-Witten invariant enumerates holomorphic maps from algebraic curves C to a smooth projective variety Y. In general it is very difficult to compute and sheaf theory is often useful. There are two ways to enter sheaf theory through the domain or the target.

A map f from C to Y gives us the direct image sheaf F on Y together with a section s of F. Intersection theory on moduli spaces of such pairs gives us the Pandharipande-Thomas invariant or the Donaldson-Thomas invariant. These sheaf theoretic perspectives led to a remarkable recent proof of the full Katz-Klemm-Vafa conjecture by Maulik-Pandharipande-Thomas, as well as a mathematical theory of the Gopakumar-Vafa invariant by categorifying the DT invariant by Kiem-Li.

A map f from C to a projective space is equivalent to a quasimap which is a line bundle L on C together with sections of L. By adding p-fields and a suitable stability condition, we obtain a moduli space which admits a virtual fundamental class via Witten's equation, K-theory or cosection localization. Integration on the virtual class gives us the famous Fan-Jarvis-Ruan-Witten invariant.

One benefit of taking sheaf theory point of view is that there are lots of stability conditions and corresponding moduli spaces which give us lots of invariants. Wall crossing means comparison of these invariants as we vary the stability condition. I will talk about the stack of quasimaps with p-fields and show that there are stability conditions for both GW and FJRW invariants. Moreover we will see that there are enough stability conditions that can possibly connect the GW and FJRW invariants.

2. Moduli space of $\delta$-stable pairs and its application (J. Choi)

The moduli space of $\delta$-stable pairs is the main tool used in the $\delta$-wall crossing. A pair refers to a pair of a pure sheaf and its nonzero (multi-)section. For each positive number $\delta$, there is a stability condition on pairs which gives a projective moduli scheme. I will talk about the GIT construction of the moduli space.

The moduli space of stable pairs has been used in many applications, to name a few, in the proof of Verlinde formula by Thaddeus, in Mochizuki's formula for the Donaldson invariants in terms of Seiberg-Witten invariants and in the calculation of PT-BPS correspondence. I will discuss some of these applications.

3. LG/CY correspondence via quasimaps (Y.-H. Kiem)

I will talk about two sequences of stability conditions on the stack of quasimaps with p-field. The first sequence, called the epsilon sequence, is geometric and arises from allowing base points of quasimaps. By allowing at most 1/epsilon base points and adding an ampleness condition, we obtain the epsilon stability whose wall crossing was studied by Ciocan-Fontanine-Kim on the GW side and by Fan-Jarvis-Ruan and Ross-Ruan on the FJRW side. The second sequence, called the delta sequence, is more sheaf theoretic and arises from Le Potier's stability

condtions on pairs.

I will discuss these stabilities and wall crossing in detail. When delta is zero, the invariants we get on the GW and FJRW sides are given by the same residue formula, differing only in ranges for the degree of the line bundle. Computing the wall crossings in the epsilon and delta lines may lead us to the Landau-Ginzburg/Calabi-Yau correspondence.

4. Genus zero wall-crossing for quasi-maps (J. Choi)

In this talk, I will discuss more explicit and concrete examples of $\epsilon$ and $\delta$ wall crossing for genus zero on the CY side. When $\delta$ is zero, the moduli space is a projective bundle over the moduli space of stable curves. The cosection localized virtual cycle in this case is given by the Euler class of the obstruction bundle. So after wall crossing, the calculation reduces to the intersection theory on the moduli space of stable curves.

I will describe the wall crossing with the example when $d=1$. When $d=1$, each $\epsilon$ and $\delta$ wall crossing is a blowup. When the target space is $\mathbb{P}^1$, this gives a new construction of the Fulton-MacPherson configuration space which is isomorphic to the the moduli space of stable maps. Moreover, the invariants can be computed by applying the blowup formula.