Theta functions and mirror symmetry

TL;DR

This survey explores the generalization of theta functions through mirror symmetry, highlighting their construction on degenerations of varieties and applications to homological mirror symmetry and mirror variety construction.

Contribution

It introduces a new perspective on theta functions beyond abelian varieties, connecting them with mirror symmetry and providing methods for their construction on degenerations.

Findings

Theta functions can be constructed on degenerations of varieties.

Applications to homological mirror symmetry are demonstrated.

New methods for constructing mirror varieties are outlined.

Abstract

This is a survey covering aspects of varied work of the authors with Mohammed Abouzaid, Paul Hacking, and Sean Keel. While theta functions are traditionally canonical sections of ample line bundles on abelian varieties, we motivate, using mirror symmetry, the idea that theta functions exist in much greater generality. This suggestion originates with the work of the late Andrei Tyurin. We outline how to construct theta functions on the degenerations of varieties constructed in previous work of the authors, and then explain applications of this construction to homological mirror symmetry and constructions of broad classes of mirror varieties.