A free fermion formulation of the (small) quantum cohomology ring

Christian Korff (Glasgow University)

Wednesday 23rd September, 2009 16:00-17:00 Mathematics Building, Room 204


An alternative title for this talk could be: "How to count rational curves intersecting Schubert varieties by creating and destroying particles". This talk should be accessible, even to those who do not know what a "free fermion" (think Clifford algebra) or "quantum" (think q-deformed) cohomology is. Both terms will be explained. Using a simple physics model, particles hopping on a circluar lattice, I will define in a purely combinatorial manner a ring with integer structure constants. The latter turn out to be the celebrated (3-point, genus 0) Gromov-Witten invariants, which count the number of rational curves intersecting Schubert varieties of the complex Grassmannian. Several identities for Gromov-Witten invariants, some of which are new, can be easily derived in this formalism. Time permitting I shall present a concise reduction formula for Gromov-Witten invariants which relates them to the structure constants of another ring, the so-called fusion or Verlinde ring. The structure constants of the latter equal the dimensions of moduli spaces of generalized theta functions. Part of this seminar will be based on joint work with C. Stroppel, Uni Bonn.

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