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The Verlinde algebra and the cohomology of the Grassmannian. (English) Zbl 0863.53054

Yau, S.-T. (ed.), Geometry, topology and physics for Raoul Bott. Lectures of a conference in honor of Raoul Bott’s 70th birthday, Harvard University, Cambridge, MA, USA 1993. Cambridge, MA: International Press. Conf. Proc. Lect. Notes Geom. Topol. 4, 357-422 (1995).
The main goal of these notes is to elucidate a formula of Doron Gepner, which relates the quantum cohomology ring of the Grassmannian \(G(k,N)\) of complex \(k\)-planes in \(N\)-space and the Verlinde algebra, which computes the Hilbert polynomial of the moduli space of vector bundles on a curve. Gepner’s formula says that the quantum cohomology ring of \(G(k,N)\) coincides with the Verlinde algebra of the group \(U(k)\) essentially at level \(N-k\). The author seeks an explanation of this fact more conceptual than the original one given by Doron Gepner who discovered his formula by computing the left- and right-hand side and observing that they were equal. The author shows that a more conceptual explanation can be found by means of a quantum field theory. He represents the quantum cohomology ring of the Grassmannian in a quantum field theory and reduces that quantum field theory at low energies to another quantum field theory which is known to compute the Verlinde algebra. The field theories the author uses are the two-dimensional supersymmetric sigma model with target space \(G(k,N)\) and the gauged WZW model of \(U(N)/U(k)\). The author explains the relation between the Verlinde algebra and the gauged WZW model of \(U(N)/U(k)\).
For the entire collection see [Zbl 0826.00038].
Reviewer: V.Abramov (Tartu)

MSC:

53Z05 Applications of differential geometry to physics
81T60 Supersymmetric field theories in quantum mechanics
17B37 Quantum groups (quantized enveloping algebras) and related deformations
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