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Towards mirror symmetry as duality for two-dimensional abelian gauge theories. (English) Zbl 0957.81656
Summary: Superconformal sigma models with Calabi-Yau target spaces described as complete intersection subvarieties in toric varieties can be obtained as the low-energy limit of certain abelian gauge theories in two dimensions. We formulate mirror symmetry for this class of Calabi-Yau spaces as a duality in the abelian gauge theory, giving the explicit mapping relating the two Lagrangians. The duality relates inequivalent theories which lead to isomorphic theories in the low-energy limit. This formulation suggests that mirror symmetry could be derived using abelian duality. The application of duality in this context is complicated by the presence of nontrivial dynamics and the absence of a global symmetry. We propose a way to overcome these obstacles, leading to a more symmetric Lagrangian. The argument, however, fails to produce a derivation of the conjecture.

##### MSC:
 81T40 Two-dimensional field theories, conformal field theories, etc. in quantum mechanics 14J32 Calabi-Yau manifolds (algebro-geometric aspects) 32G81 Applications of deformations of analytic structures to the sciences 32J81 Applications of compact analytic spaces to the sciences
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