Theta functions on the Kodaira-Thurston manifold

The Kodaira-Thurston manifold $M$ is a compact, $4$-dimensional nilmanifold which is symplectic and complex but not Kähler. We describe a construction of $\vartheta$-functions associated to $M$, which parallels the classical theory of $\vartheta$-functions associated to the torus (from the point of view of representation theory and geometry), and which yields pseudoperiodic complex-valued functions on $\mathbb{R}^4.$

There exists a three-step nilpotent Lie group $\widetilde{G}$ which acts transitively on the Kodaira-Thurston manifold and on the universal cover of $M$ in a Hamiltonian fashion. The $\vartheta$-functions discussed in this paper are intimately related to the representation theory of $\widetilde{G}$ in much the same way that the classical $\vartheta$-functions are related to the Heisenberg group. One aspect of our results is a connection between the representation theory of $\widetilde{G}$ and the existence of Lagrangian and special Lagrangian foliations and torus fibrations in $M$; in particular, we show that $\widetilde{G}$-invariant special Lagrangian foliations can be detected by a simple algebraic condition on certain subalgebras of the Lie algebra of $\widetilde{G}.$

Crucial to our generalization of $\vartheta$-functions is the spectrum of the Laplacian $\Delta$ acting on sections of certain line bundles over $M$. One corollary of our work is a verification of a theorem of Guillemin-Uribe describing the structure (in the semiclassical limit) of the low-lying spectrum of $\Delta$.