Joint Mathematical Physics seminar at Chennai Mathematical Institute

Last time we defined the Verlinde algebra of TQFT as the Frobenius algebra of states on the torus. Using basic topological considerations we deduced an expression for Verlinde's algebra structure constants in the basis of primary fields in terms of the modular action by S-matrix.

This time I will sketch the reverse procedure, at least in an unitary theory of Chern-Simons type. Namely, we start with algebraic observation showing that, up to a rescaling, the S-matrix can be defined with the basis of canonical projectors in a semisimple commutative algebra. Then we will show how the S-matrix entries are encoded by the eigenvalues of the multiplication operators by the basis vectors.

The detailed discussion will be devoted to the Chern-Simons theory for SU(2) at level k. Using CFT-based identification of the Frobenius algebra with the Grothendieck ring of certain representations equipped with the fusion tensor product we will compute the S-matrix entries explicitly. As an application we get an expression of the dimension of conformal blocks in arbitrary genus, i.e. the global sections of powers of the determinant line bundle over the stack of SL(2,C)-bundles.

Last time we introduced the Verlinde algebra of TQFT: the partition functor value on the torus Z(S^1xS^1). The underlying vector space of this Frobenius algebra enjoys (projective) action of the torus mapping class group SL(2,Z). First we will prove the Verlinde formula: the normalized action by S in SL(2,Z) recovers the multiplicative structure constants N^k_{ij} in the basis of the Wilson loops in D^2xS^1 labeled by irreducible representations.

Modulo the link between (2+1)-TQFT states and CFT conformal blocks, Verlinde's conjecture tautologically says that N_{ijk} is equal to the dimension of 3-point conformal blocks Z(S^2,ijk). Moreover, the dimension of conformal blocks in higher genus admits an expression in terms of N_{ijk} via pantalons decomposition. The remaining part will be devoted to the computation of the S-matrix in the case of Chern-Simons/WZW theory for the group SU(2) at level k.

As was indicated last time, this physics result has a striking application to algebraic geometry. Namely, as we will see later, the Chern-Simons TQFT states for SU(N) can be described as the geometric quantization of the moduli stack of SL(n,C)-bundles on a complex curve. In particular this implies a formula for the dimension of generalized theta functions, i.e. global sections of powers of Quillen line bundle on the stack. Geometrically this is a non-abelian analog for the Riemann-Roch formula for line bundles on abelian varieties.

Last time we introduced the Verlinde algebra of TQFT - the partition functor value on the torus. Under semisimplicity assumption, we will show that the modular action recovers the multiplication in the Verlinde algebra. This provides a topological proof of Verlinde's conjecture.

Under Kontsevich's correspondence between TQFT and modular functors, the partition functor value is the space of conformal blocks of the modular functor. Using this, we will show that the Verlinde algebra recovers the dimension of conformal blocks on a surface via pantalons decomposition.

Combing with the representation theory of affine algebras, we will indicate further a striking application of this result to algebraic geometry: the dimension of generalized theta functions, i.e. the global sections of powers of Quillen line bundle on the moduli stack of SL(n,C)-bundles on a curve, admits an explicit expression by means of the Weyl character formula. This is a non-abelian generalization the Riemann-Roch formula for line bundles on abelian varieties.

Last time, we showed that 2+1-dimensional topological quantum field theories lead to nontrivial invariants of 3-manifolds (e.g. Jones polynomial). In turn, the very description of spaces of states in Chern-Simons theory involves many beautiful constructions from representations theory, differential and algebraic geometry. In this talk I will sketch a roadmap for the future directions.

Last time we covered abelian theory over S^3 using standard path integral technique. Namely, using the fact that U(1)-theory contains only quadratic kinetic term we computed the two-point function as Green function of the curl operator, and, as we saw, the corresponding Wilson correlators are given by Gauss integral formula for the linking number of pair of knots in S^3 modulo k.

This time we will turn to the non abelian case SU(N) to recover the value of Jones polynomial at the primitive (N+k)-root of unity using axiomatic TQFT approach modulo some finite amount of data predicted by the connection of Chern-Simons to the conformal blocks of the underlying CFT.

Last time we covered abelian theory over S^3 using standard path integral technique. Namely, using the fact that U(1)-theory contains only quadratic kinetic term we computed the two-point function as Green function of the curl operator, and, as we saw, the corresponding Wilson correlators are given by Gauss integral formula for the linking number of pair of knots in S^3 modulo k.

This time we will turn to the non abelian case SU(N) to recover the value of Jones polynomial at the primitive (N+k)-root of unity using axiomatic TQFT approach modulo some finite amount of data predicted by the connection of Chern-Simons to the conformal blocks of the underlying CFT.

Following Witten, we will show that non perturbative analysis of Chern-Simons in d=3 within Atiyah-Segal axiomatic approach to TQFT's, naturally leads to skein relations on Wilson correlators. Fix the level k. We will start with the gauge group U(1), where we recover Gauss integral formula for the linking number of a pair of knots in S^3 modulo k. Then we will turn to the non abelian case SU(N) to recover the value of Jones polynomial at the primitive (N+k)-root of unity.

We will review the definition of observables in quantum mechanics, in the Atiyah-Segal framework. Thereafter, we will review the Lagrangian field theory description of the Chern-Simons theory and discuss a variation of the standard Atiyah-Segal axioms needed to construct this theory.