Quantum field theory is the unification of quantum theory and special theory of relativity. It provides the framework for the standard model, a theory of all the observed forces (except gravity) of nature. String theory is the leading candidate for a unified theory of all the four forces. It yields a consistent theory of quantum gravity, at least in the perturbative, weak field domain. For several decades, quantum field theory and string theory have been the major sources of inspirations formathematics. There have been deep interactions between them with mathematics through algebra and geometry.

The aim of the proposed program is to bring together world experts working at the interfaces of QFT, string theory and mathematical physics. The program naturally divides into two parts: the first part (roughly the first 4 weeks) will focus on the fundamentals of string theory and recent development of string theory, and the second part (the last 4 weeks) will focus on the algebraic and geometric aspects in mathematical physics.

Topics of interest will include pure spinor superstring, flux compactification of string theory, twistor string theory and QCD amplitude computation, the representation theory of infinite Lie algebra and the generalized complex geometry, etc.

Lecture Series :

(I) July 12—July 16 Introduction to quantum groups, by Professor Hechun Zhang, Tsinghua University , China

(II) July 19---July 23: Representations of some Lie algebras related to the Virasoro algebra, by Professor Kaiming Zhao, Wilfrid Laurier University, Canada, and Chinese Academy of Sciences, China

(III) July 26—July 30 Introduction to Classical Lie superalgebras and their representations, by Professor Yucai Su, University of Science and Technology of China

(IV) August 2---August 6: Central extensions of Lie algebras, by Professor Erhard Neher, University of Ottawa, Canada

(I)Introduction to quantum groups

By Professor Hechun Zhang, Tsinghua University, China

Abstract: Quantum groups introduced independently by Drinfeld and Jimbo, are certain families of Hopf algberas that are deformations of universal enveloping algebras of Kac-Moody algberas. In the first half of this short introductary we will introduce some basics of quantum groups to graduate students and young researchers. The second half will focus on the theory of canonical basis and crystal basis introduced by Kashiwara and Lusztig.

Tentative plan:

Lecture 1. Lie algebars and Kac-Moody algebras.

Lecture 2. Quantum groups: Basics in quantum groups, braid group action and R-matrix. Category ${\mathcal O}_{int}$.

Lecture 3. Crystal basis and canonical basis: Kashiwara operators, crystal lattice and balanced triple. $B(\infty)$.

Lecture 4. Quantized function algebra (quantum coordinate ring) Peter-Weyl theorem. Quantum matrices, elementary construction of dual canonical basis.

Lecture 5. Canonical basis of tensor product of based modules. Quasi R-matrix and stability. Canonical basis of modified quantum enveloping algebra $\dot{U}$.

(II) Representations of some Lie algebras related to the Virasoro algebra

By Professor Kaiming Zhao, Wilfrid Laurier University, Canada, and Chinese Academy of Sciences, China

Abstract: Lie algebra theory has become more and more widely used in many branches of mathematics and physics, including, associative algebra theory, algebraic combinatorics, theory of 2-dimensional statistical models, string theory, conformal field theory, soliton theory, and other mathematical physics. Besides being useful in many subjects, Lie algebra theory is inherently attractive, combining a great depth and a satisfying degree of completeness in its basic theory. It also presents many interesting and important problems.

The Virasoro algebra Vir is the universal central extension of the derivation Lie algebra of the commutative associative Laurent algebra. In this series of talks, I will start with representations of Vir, describing properties, giving known results with unsolved problems. Then show you some methods to use them to other interesting related algebras including generalized Virasoro algebras, and Witt algebras.

Tentative plan:

Lecture 1: Representations of the Virasoro algebra.

Lecture 2: Representations of generalized Virasoro algebras.

Lecture 3: Shen's Representations of Witt algebras.

Lecture 4: Irreducible representations of Witt algebras with weight multiplicity 1.

Lecture 5: Harish-Chandra representations of Witt algebras.

(III) Abstract: Finite dimensional classical simple Lie superalgebras over the complex

field are widely used in mathematical physics. In this lecture, we will mainly give an

introduction to the classical Lie superalgebras of types A-D (with focus on general

linear Lie superalgebras), their root systems, Weyl groups, atypicality of dominant integral weights, finite dimensional representations, generalized Verma modules or

Kac-modules, irreducible modules, formal characters, cohomology groups, Kazhdan-Lusztig polynomials, etc.

Tentative plan:

Section 1: classical Lie superalgebras of types A-D

Section 2: atypicality of weights and atypicality matrix

Section 3: Generalized Verma modules and irreducible modules

Section 4: Kazhdan-Lusztig polynomials and inverse Kazhdan-Lusztig polynomials

Section 5: characters, cohomology groups

(IV) Title: Central extensions of Lie algebras

By Professor Erhard Neher, University of Ottawa, Canada

Abstract: A central extension of a Lie algebra L is a Lie algebra homomorphism from a Lie algebra K onto L whose kernel is contained in the centre of K. Central extensions are important for any class of Lie algebras. For example, they are basic for the construction of affine Lie algebras. Moreover, they often turn up quite naturally when one studies representations of L. It is therefore not surprising that there are many papers devoted to describing central extensions of particular classes of Lie algebras.

This lecture series will give an introduction to central extensions of Lie algebras, starting from the very beginning. The emphasis will be on general methods, but many examples will also be presented.

We will start by introducing various types of central extensions, in particular universal central extensions, and present constructions of the universal central extension, for example via certain derivations and 2-cocycles. We will describe how central extensions behave with respect to automorphisms, derivations, semi direct products, direct limits, gradings, and comment briefly on the case of Lie superalgebras.

We will then consider central extensions of root-graded Lie algebras and show how they are related to homology groups of their coordinate algebras, e.g. cyclic homology. In particular we will describe central extensions of Lie tori. We will also look at central extensions of fixed point subalgebras, which includes as examples multiloop algebras and, more generally, Lie algebras obtained by Galois descent.

The prerequisite of the lecture series is a basic introductory course on Lie algebras. No prior knowledge of central extensions will be assumed.