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Extra resources for Communications In Mathematical Physics - Volume 269
In this paper we suggest an interpretation of mirror symmetry for toric varieties. We show that there is a certain conformal field theory (the “I–model”) that is intermediate between the type A twisted sigma model and the type B twisted Landau-Ginzburg model. On the one hand, this model is equivalent to the sigma model of a toric variety in the infinite volume limit, considered as a conformal field theory, and on the other hand its BPS sector is closely related to the BPS sector of the corresponding Landau-Ginzburg model.
The linear sigma model corresponding to M0 is a free superconformal field theory, and we wish to describe the non-linear model with the target M in terms of this theory. Let us observe that a generic holomorphic map : → M will take values in C at a finite set of points x1 , . . , xn , and generically we will have (x j ) ∈ Ck j and (x j ) ∈ Cl , l = k j . To account for such maps we need to insert some vertex operators k j corresponding to the compactification divisors Ck j at the points x j , j = 1, .
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.