TOWARDS MIRROR SYMMETRY AS DUALITY FOR 2-DIMENSIONAL ABELIAN GAUGE-THEORIES

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 dime nsions. We formulate mirror symmetry for this class of Calabi-Yau spac es as a duality in the abelian gauge theory, giving the explicit mappi ng relating the two Lagrangians. The duality relates inequivalent theo ries which lead to isomorphic theories in the low-energy limit. This f ormulation suggests that mirror symmetry could be derived using abelia n duality. The application of duality in this context is complicated b y the presence of nontrivial dynamics and the absence of a global symm etry. We propose a way to overcome these obstacles, leading to a more symmetric Lagrangian. The argument, however, fails to produce a deriva tion of the conjecture.