Quantization via mirror symmetry

Creators: Gukov, Sergei

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Abstract

When combined with mirror symmetry, the A-model approach to quantization leads to a fairly simple and tractable problem. The most interesting part of the problem then becomes finding the mirror of the coisotropic brane. We illustrate how it can be addressed in a number of interesting examples related to representation theory and gauge theory, in which mirror geometry is naturally associated with the Langlands dual group. Hyperholomorphic sheaves and (B, B, B) branes play an important role in the B-model approach to quantization.

Additional Information

© 2011 The Mathematical Society of Japan and Springer. Received: 5 November 2010; Revised: 1 November 2011; Accepted: 8 November 2011. Published online: 25 December 2011. This article is based on the 8th Takagi Lectures that the author delivered at Research Institute for Mathematical Sciences, Kyoto University on November 23, 2010. I would like to thank the organizing committee of the Takagi Lectures for inviting me, and to acknowledge helpful discussions with E. Frenkel, T. Hausel, and E. Witten. I also would like to thank E. Witten for collaboration on the A-model approach to quantization reviewed in Section 2. Special thanks are in order to referees for careful reading of the manuscript and suggesting many useful improvements. This work is supported in part by DOE Grant DE-FG03-92-ER40701 and in part by NSF Grant PHY-0757647. Opinions and conclusions expressed here are those of the author and do not necessarily reflect the views of funding agencies.

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