Let $T_\tau$ be the 2-dimensional torus, with the complex structure induced by the lattice generated by $1$ and $\tau$. Then for a line bundle $L_k$ over $T$ with level $k$, there is an orthonormal basis whose sections lift to theta functions on the complex plane.

My question is, what if we try to compare the Hilbert spaces obtained for different $\tau$? One method would be to look at the Hitchin connection. However, if we instead tried to use the Blattner-Kostant-Souriau (BKS) pairing between polarizations, would this just amount to doing a Fourier transform of sorts? In particular will this just give us something proportional to the Hermite-Jacobi action?

`Tata Lectures on Theta III', is not just that`

sections` are theta functions. This question drops some big words, but no mathematics. Not yet, anyway. Anybody who wants to understand one iota $(\iota)$ of theta functions has to start bottom-up everytime, my opinion. Plus what this pairing between polarizations? Why does the OP not generously recall this? Either they don't know, or they're too lazy. – J. Martel Aug 29 '12 at 15:38