##
Geometry and Physics Seminar

## 报告摘要 Abstract

**2019-8-20**

**Speaker: **Brian Williams (Northeastern University)

**Title: **Twisted superconformal theories in four dimensions

**Abstract:** We present an infinite
dimensional enhancement of the minimal, holomorphic, twists of the 4d N=1,2,4
superconformal algebras. When N=1, this is just the Lie algebra of holomorphic
vector fields on two dimensional complex space, and when N=2,4 we find graded
versions built from holomorphic vector fields. As an application, we study
superconformal deformations of twists of familiar four-dimensional
supersymmetric objects, including symmetry multiplets. As a particular example,
we present a mathematical formulation of recent work of Beem, et. al. on chiral
algebras obtained from superconformal 4d N=2 theories, using the language of
factorization algebras.

**-----------------------------History-----------------------------**

**2019-7-10**

**Speaker: **Zhengfang Wang (Max Planck Institute for Mathematics)

**Title: **Landau-Ginzburg model, Fukaya category, and homological mirror
symmetry

**Abstract: **Recall that a B-infinity algebra is an
A-infinity algebra such that the bar construction (a dg coalgebra) can be
extended to a dg bialgebra. In this talk, we will discuss a natural
construction of B-infinity algebras from Kontsevich-Soibelman's minimal operad,
which is a dg model of the little 2-discs operad. For instance, the B-infinity algebra
structure on the Hochschild cochain complex can be induced from this construction.

Recall that the Tate-Hochschild cohomology of an algebra is defined as the Yoneda algebra of the identity bimodule in the singularity category (in the sense of Buchweitz and Orlov) of bimodules. Analogous to Hochschild cohomology, it is proved that the Tate-Hochschild cohomology at the complex level has a natural action of Kontsevich-Soibelman's minimal operad.

Recently, Keller proves that the Tate-Hochschild cohomology of an algebra A is isomorphic as graded algebras to the Hochschild cohomology of the dg singularity category of A. He also conjectures that this isomorphism lifts to a B-infinity quasi-isomorphism at the complex level. Using an explicit homotopy between two projective resolutions construed by W. He-S. Li-Y. Li, we will give a proof of Keller's conjecture for radical square zero algebras. This is a joint work with X.Chen and H. Li.

** **

**2019-5-28**

**Speaker: **蒋文峰（中山大学）

**Title: **Landau-Ginzburg model, Fukaya category, and homological mirror
symmetry

**Abstract: **In this talk, we will give an introduction to the global picture
of the homological mirror symmetry, especially the mirror symmetry of
Landau-Ginzburg models. We will also give an introduction to the recent
progress in Landau-Ginzburg A model, mainly the work of the speaker, together
with Huijun Fan and Dingyu Yang.

** **

**2019-5-13**

**Speaker: **John Alexander Cruz Morales (Universidad Nacional de Colombia)

**Title: **Quantum cohomology for isotropic Grassmannians

**Abstract: **We will discuss the big quantum cohomology ring of isotropic
Grassmannians IG(2,2n). After introducing the basic notions we will show that
these rings are regular. In particular, by “generic smoothness”, we will give a
conceptual proof of generic semisimplicity of the big quantum cohomology for these
Grassmannians. We will also relate certain decomposition of the ring with a
exceptional collection of the derived category of IG(2,2n). This is based on
joint work with A. Mellit, A. Kuznetsov, N. Perrin and M. Smirnov.

** **

**2019-5-7**

**Speaker: **Shan Hu (Hubei University)

**Title: **S-duality transformation of N＝4 SYM theory at
the operator level

**Abstract: **I will discuss the S-duality transformation of physical operators
and states in N=4 SYM theory with the gauge group U(N). The transformation is
implemented via a unitary operator S, which is a canonical transformation in
loop space exchanging the Wilson and t' Hooft operators. S-duality invariance
of the theory is equivalent to the requirement that such S makes the
superconformal charges and their S-duals differ by a U(1)_{Y} phase.

** **

**2019-4-23**

**Speaker: **Satoshi Nawata (Fudan University)

**Title: **New TQFTs from DAHA

**Abstract: **The first part of my talk will be devoted to geometric
representation theory of DAHA by using Hitchin moduli space. Second, I will
explain how the finite-dimensional modules of DAHA are connected to the Hilbert
space of modular tensor categorties.

** **

**2019-4-16**

**Speaker: **Xiang Tang (WUSTL)

**Title: **Analytic Grothendieck Riemann Roch Theorem

**Abstract: **In this talk, we will introduce an interesting index problem
naturally associated to the Arveson-Douglas conjecture in functional analysis.
This index problem is a generalization
of the classical Toeplitz index theorem, and connects to many different
branches of Mathematics. In particular, it can be viewed as an analytic version
of the Grothendieck Riemann Roch theorem. This is joint work with R. Douglas，M. Jabbari, and G. Yu.

**2019-4-9**

**Speaker: **Eugene Rabinovich (UC Berkeley)

**Title: **The chiral anomaly in the Batalin-Vilkovisky formalism

**Abstract: **We use the methods of Costello and Gwilliam to study the chiral
anomaly of the massless free fermion. Costello and Gwilliam have developed a
general and mathematically rigorous formalism for studying the perturbative
quantization of field theories and their symmetries. After reviewing the
relevant methods, we introduce the massless free fermion field theory
associated to a Dirac operator. This theory has a symmetry known as the chiral
symmetry. We explain how the quantization of this symmetry is obstructed by the
index of the corresponding Dirac operator.

** **

**2019-4-2**

**Speaker: **Zijun Zhou

**Place: **Jing Zhai 304

**Title: **3d mirror symmetry and elliptic stable envelopes for T^*Gr

**Abstract: **3d mirror symmetry, also known as symplectic duality, is a duality
originated from physics between the so-called Higgs branch and Coulomb branch
of 3d supersymmetric gauge theories. In this talk, I will discuss the
exposition of this duality in the case of the cotangent bundle of Grassmannians,
for certain geometric invariants called elliptic stable envelopes, introduced
by Aganagic--Okounkov. The restriction matrix to fixed points of elliptic
stable envelopes for T^*Gr and its mirror turn out to be related to each other
under transposition and an exchange of equivariant and K\"ahler
parameters. In terms of explicit formulas, the duality gives rise to infinitely
many nontrivial identities of theta functions. This work is joint with R.
Rim\'anyi, A. Smirnov and A. Varchenko.

** **

**2019-3-26**

**Speaker: **JunWu Tu (ShanghaiTech University)

**Title: **Bogomolov-Tian-Todorov Theorem of cyclic A-infinity algebras

**Abstract: **In this talk, we will discuss the categorical analogue of the
Bogomolov-Tian-Todorov Theorem of cyclic A-infinity algebras. Furthermore,
given a splitting of the Hodge filtration, we discuss the associated flat
structure on the formal moduli space. As an application, we give a categorical
construction of the classical Saito's theory of primitive forms.

**2019-3-12**

**Speaker: **Jian Zhou (Tsinghua University)

**Title: **Emergent geometry of topological 2D gravity

**Abstract:** Mirror symmetry and Witten-Kontsevich Theorem are two paradigms in the
mathematical study of string theory.

In an attempt to unify them we have come to realize that even though both have been developed following the reductionist point of view, but the latter can be also reformulated from an emergent point of view. This leads to a funny notion of "mirror symmetry of a point" and it suggests a possibility to develop mirror symmetry also from an emergent point of view.