  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.
