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Mirror Symmetry in Two Steps: A–I–B

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Abstract

We suggest an interpretation of mirror symmetry for toric varieties via an equivalence of two conformal field theories. The first theory is the twisted sigma model of a toric variety in the infinite volume limit (the A–model). The second theory is an intermediate model, which we call the I–model. The equivalence between the A–model and the I–model is achieved by realizing the former as a deformation of a linear sigma model with a complex torus as the target and then applying to it a version of the T–duality. On the other hand, the I–model is closely related to the twisted Landau-Ginzburg model (the B–model) that is mirror dual to the A–model. Thus, the mirror symmetry is realized in two steps, via the I–model. In particular, we obtain a natural interpretation of the superpotential of the Landau-Ginzburg model as the sum of terms corresponding to the components of a divisor in the toric variety. We also relate the cohomology of the supercharges of the I–model to the chiral de Rham complex and the quantum cohomology of the underlying toric variety.

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Authors and Affiliations

  1. Department of Mathematics, University of California, Berkeley, CA, 94720, USA

    Edward Frenkel

  2. Institute of Theoretical and Experimental Physics, B. Cheremushkinskaya 25, Moscow, 117259, Russia

    Andrei Losev

Authors
  1. Edward Frenkel
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  2. Andrei Losev
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Correspondence to Edward Frenkel.

Additional information

Communicated by L. Takhtajan.

Partially supported by the DARPA grant HR0011-04-1-0031 and the NSF grant DMS-0303529.

Partially supported by the Federal Program 40.052.1.1.1112, by the Grants INTAS 03-51-6346, NSh-1999/2003.2 and RFFI-04-01- 00637.

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Frenkel, E., Losev, A. Mirror Symmetry in Two Steps: A–I–B. Commun. Math. Phys. 269, 39–86 (2007). https://doi.org/10.1007/s00220-006-0114-1

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