Theta-functions on the Kodaira-Thurston manifold

TL;DR

This paper constructs theta-functions on the Kodaira-Thurston manifold, linking representation theory, symplectic geometry, and complex analysis, revealing new connections with Lagrangian foliations and torus fibrations.

Contribution

It introduces a novel construction of theta-functions on a non-Kähler symplectic manifold using representation theory of a nilpotent Lie group, extending classical theory.

Findings

Theta-functions are pseudoperiodic complex functions on R^4.

The construction relates to the representation theory of a three-step nilpotent Lie group.

Connections are established between representation theory and Lagrangian foliations.

Abstract

The Kodaira--Thurston M manifold is a compact, 4-dimensional nilmanifold which is symplectic and complex but not Kaehler. We describe a construction of theta-functions associated to M which parallels the classical theory of theta-functions associated to the torus (from the point of view of representation theory and geometry), and yields pseudoperiodic complex-valued functions on R^4. There exists a three-step nilpotent Lie group G which acts transitively on the Kodaira--Thurston manifold M in a Hamiltonian fashion. The theta-functions discussed in this paper are intimately related to the representation theory of G in much the same way the classical theta-functions are related to the Heisenberg group. One aspect of our results which has not appeared in the classical theory is a connection between the representation theory of G and the existence of Lagrangian and special Lagrangian foliations and torus fibrations in M.