June 28 - July 2, 2010

Institute for Advanced Study, Tsinghua University

In recent years, there has been considerable interest in quantum systems which show novel behaviors emerging from the collective motion and interaction of the constituent local degrees of freedom. The novel phenomena include fractional quantum number, fractional statistics, topologically protected gapless bulk/edge excitations, etc. It appears that the long-range quantum entanglement in those system plays a critical role in generating those novel phenomena. The goal of this summer school is the introduce the physical background and the mathematical frame work for those novel behaviors. The intended audience is the graduate students in mathematics and physics.

Prof Yi-Zhi Huangplan to give four lectures on

1. Commutative associative algebras, Lie algebras and theirrepresentation theories. Vertex operator algebras as naturalanalogues of these algebras. Meromorphic operator product expansion. Examples of vertex operator algebras fromHeisenberg, Virasoro and affine Lie algebras.

2. Modules for vertex operator algebras. Chiral vertexoperators or intertwining operators. Examples fromHeisenberg, Virasoro and affine Lie algebras. Operatorproduct expansions for intertwining operators. Differentialequations of regular singular points satisfied by productsof intertwining operators. Fusing and braiding matrices.Intertwining operator algebras and nonabelian anyons.

3. Fusion rules. Tensor products of modules for vertexoperator algebras. Associativity and braiding isomorphisms.Intertwining operator algebras and braided tensorcategories. Modular invariance, the Verlinde formula forfusion rules, and rigidity. Modular tensor categorystructures on categories of modules for suitable vertexoperator algebras.

Prof. Yong-Shi Wu plan to give two lectures:

The first is on fractional statistics (Abelian and non-Abelian anyons, Relationship with braid groups and Chern-Simons gauge theory, Realization to the quantum Hall systems etc.) The second is on Ground State Degeneracy of Topological Phases (Topological argument for FQHE, Theoretical proofs (Chern-Simon theory, Braid group), Methods for determining GSD (Gauge theory, Topological algebra, the Levin-Wen Model) etc).

Prof. Zhenghan Wangplan to give 5 lectures on

Lecture I: TQFTs and Realization (Mainly Levin--Wen model and 6j symbols)

Lecture II: Algebraic theory of anyons (Tensor catgeory theory)

Lecture III: Interacting anyons (Anyon chains and some ladders theories)

Lecture IV: Anyon condensation

Lecuture V: Remnant in High dimensions (Zhenghan's recent paper for 3Dnon-abelian statistics.)

Prof. Xiao-Gang Wenplan to give four lectures, mainly to discuss simple physical systems that realize the topological orders:

1. What kind of data should we use to describe a topological ordered many-body states. (will mainly discuss the pattern of zeros and tensorproduct state, where pattern of zero and tensors will be our data.)

2. From those data, how do we calculate the topologicalproperties. (will mainly raise the questions and introducethose topological properties. The lectures by other three professors will present the actual discussions of the framework that allow us to do some calculations.)

3. Given a generic ground state or Hamiltonian, how do wecalculate those data that characterize the topologicalstate. How to directly calculate topological properties of those system.

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