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Title:
The Verlinde formula from mirror symmetry
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Abstract:
The quantization problem starts with an input data of a symplectic manifold, i.e. an even-dimensional manifold equipped with a non-degenerate closed 2-form, from which one aims to construct the Hilbert space, the non-commutative algebra of operators, etc. Another area of mathematics where the same input data is used as a starting point is "mirror symmetry" where, given a symplectic manifold, one then constructs a number of sophisticated invariants, including Gromov-Witten invariants, Fukaya category, quantum cohomology, etc. Could it be that the two areas are related and the answer to the quantization problem is already contained among these sophisticated gadgets used in mirror symmetry? The answer turns out to be "yes". Moreover, by embedding the quantization problem in the framework of mirror symmetry one finds a natural solution to various puzzles, e.g. a simple geometric explanation why minimal orbits of SO(p,q) in Cartan type B are "quantizable" only when p<4 or q<4. Applying this approach to the quantization of moduli space of flat G-connections, one finds that some of the coefficients in the Verlinde formula have a simple interpretation via classical geometry of the moduli space associated with the Langlands dual group.
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