Delve into the future of research at MiS with our preprint repository. Our scientists are making groundbreaking discoveries and sharing their latest findings before they are published. Explore repository to stay up-to-date on the newest developments and breakthroughs.
Theta functions on the Kodaira-Thurston manifold
William Kirwin and Alejandro Uribe
The Kodaira--Thurston manifold $M$ 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, and on the universal cover of $M$ 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 that the classical theta-functions are related to the Heisenberg group. One aspect of our results is a connection between the representation theory of $G$ and the existence of Lagrangian and special Lagrangian foliations and torus fibrations in $M$; in particular, we show that $G$-invariant special Lagrangian foliations can be detected by a simple algebraic condition on certain subalgebras of the Lie algebra of $G$.
Crucial to our generalization of theta-functions is the spectrum of the Laplacian 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 this Laplacian.