Mirror symmetry at genus one of elliptic curves

Abstract

This master's thesis is concerned with mirror symmetry at genus one for elliptic curves. Mirror symmetry stems from string theory in physics and conjectures a relation between the symplectic (the A-model) and complex structures (the B-model) of a Calabi-Yau manifold and its mirror manifold. At genus one, the B-model calculation can be defined using analytic torsion introduced by Bershadsky, Cecotti, Ooguri and Vafa. For elliptic curves, this is calculated using the Kronecker limit formula, which is also derived in detail. The A-model is concerned with the generating series of genus one Gromov-Witten invariants which is also calculated for elliptic curves. Then the mirror symmetry correspon- dence is shown using the derivatives of the A- and B-model calculations.