June 14 -- 16, 2010

Research Institute for Mathematical Sciences, Room 111,

Kyoto Univ.

June 14 (Mon)

crystal limit and the WZNW fusion ring

a free fermion formulation of quantum cohomology

I will present a brief introduction to the quantum cohomology ring of the Grassmannian. It first appeared in works by Gepner, Vafa, Intriligator and Witten and a particular specialisation of it can be identified with the fusion ring of the gauged $\widehat{gl}(n)$-WZNW model. The talk will focus on how one derives known (geometric) results about the quantum cohomology ring in a simple combinatorial setting using well-known techniques from quantum integrable systems. For instance, performing a Jordan-Wigner transformation one derives the Vafa-Intriligator formula for Gromov-Witten invariants. The free fermion formalism also allows one to derive new results such as recursion relations for Gromov-Witten invariants and a fermion product formula.

crystal limit and the WZNW fusion ring

The second talk will focus on a closely related ring, the fusion ring of the $\hat{sl}(n)$-WZNW model. It will be discussed how this ring arises from the crystal limit of the $U_q\hat{sl(2)}$-vertex model with "infinite" spin and $n$ lattice sites. Its transfer matrix can be interpreted as the generating function of complete symmetric polynomials in a noncommutative alphabet, the generators of the affine plactic algebra. (The latter is an extension of the finite plactic algebra first introduced by Lascoux and Sch\"utzenberger.) Exploiting the Jacobi-Trudy formula one introduces noncommutative Schur polynomials and defines the fusion product in a purely combinatorial manner. In close analogy to the discussion of the quantum cohomology ring one derives the Verlinde formula for the fusion coefficients (the structure constants of the fusion ring) via the algebraic Bethe ansatz. I shall conclude by stating the precise relationship between the quantum cohomology and fusion ring in terms of a simple projection formula which directly relates Gromov-Witten invariants and fusion coefficients. The former count rational curves of finite degree, while the latter are dimensions of spaces of generalized theta functions over the Riemann sphere with three punctures.

Organizing committee

Michio Jimbo (Rikkyo), Atsuo Kuniba (Tokyo), Tetsuji Miwa (Kyoto), Tomoki Nakanishi (Nagoya),

Masato Okado (Osaka), Yoshihiro Takeyama (Tsukuba)