An Introductory Workshop to 3D Mirror Symmetry and AGT Conjecture

An Introductory Workshop to 3D Mirror Symmetry and AGT Conjecture 

Date:  September 30, Monday- October 04, Friday 

Location：Lecture Hall of Institute for Advanced Study in Mathematics, 1st floor of east 7th teaching building, Zhejiang University (Zijingang Campus) 

Invited Speakers: 

Organizing Committee: 

Yongbin Ruan (University of Michigan, USA)

Yefeng Shen (University of Oregon, USA) 

Schedule: 

Abstract: 

Luis F. Alday

Title: 4D/2D correspondences

Abstract:

4d/2d correspondences have been a very active field over the last decade. In this set of lectures we will introduce the basic ideas and ingredients. In the first lecture we will introduce the relevant two-dimensional theories, namely q-deformed 2d YM and Liouville theory. In the second lecture we will introduce the Gaiotto construction of four-dimensional N=2 super-symmetric gauge theories. In the third lecture we will show how partition functions in each side are related. These lectures are intended to be very pedagogical, and the simplest cases will be analysed. If time permits, generalisations will be briefly discussed. 
 

Tudor Dimofte

Title: Boundaries, defects, and dualities in SUSY gauge theory

Abstract:

I will discuss the physics and mathematics of BPS boundaries and defects of various kinds, in the "dimensional tower" of theories that includes 1) 1d N=2 quantum mechanics; 2) 2d N=(2,2) gauge theory; 3) 3d N=4 gauge theory; and 4) 4d N=4 SYM. Part of this (1d and 2d) will be a quick survey of necessary concepts. The star of the lecture series will be 3d N=4, and the interplay of boundaries & defects therein with 3d mirror symmetry. This is an area of active current development, both in mathematics and in physics. 
 

Kentaro Hori

Title: 3d Mirror symmetry

Abstract:

Lecture 1: Intorduction to quantum field theory, supersymmetry, and 3d N=4 mirror symmetry

Lecture 2: Branes in superstring theory and 3d N=4 mirror symmetry

Lecture 3: From 3d mirror symmetry to 2d mirror symmetry 
 

Hans Jockers

Title: Enumerative geometry from low-dimensional supersymmetric gauge theories

Abstract:

We will discuss certain aspects of supersymmetric gauge theories in low dimensions focussing on their role in

the context of enumerative geometry. We will connect such gauge theories to the concepts of quantum cohomology and

quantum K-theory in mathematics. 
 

Jie Zhou

Title: Introduction to mock modular forms

Abstract:

Part 1. Basics on modular forms and Jacobi forms: analytic and algebraic aspects:

In this part I will introduce the concepts of elliptic, Siegel modular forms and Jacobi forms from both the analytic and algebraic point of view.

I will also explain some results on the description and classfication of vector bundles on elliptic curves. These will be useful later in

understanding the notion of mock modular forms.

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Part 2. Mock modular forms: examples and definition:

In this part I will present various sources and examples of classical mock modular forms. I will also explain how functional relations satisfied by a class of these mock modular forms are related to automorphy factors of higher rank vector bundles on elliptic curves. I will finally discuss the sheaf-theoretic definition of mock modular forms and how the notion of "propagator" (and thus many quantities arising from physics) naturally enters the picture.

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Part 3. Application: counting planar polygons and homological mirror symmetry:

Mock modular forms has recently found a few interesting applications such as in homological mirror symmetry and quantum K-theory.

In some cases there is an underlying elliptic curve supporting the mock modular forms, while in others their occurence is still mysterious.

In this part, I will present an example belonging to the former case. I will explain how the A_\infty structure constants of the Fukaya category of the elliptic curve

can be reduced to the enumeration of planar polygons, and hence to indefinite theta functions which is one of the main sources of mock modular forms.

In this case, homological mirror symmetry relates the structure constants to sections of higher rank vector bundles on the mirror elliptic curve

and hence offers an explanation of mock modularity.