Coherent state transforms and vector bundles on elliptic curves

Abstract

We extend the coherent state transform (CST) of Hall to the context of the moduli spaces of semistable holomorphic vector bundles with fixed determinant over elliptic curves. We show that by applying the CST to appropriate distributions, we obtain the space of level k, rank n and genus one non-abelian theta functions with the unitarity of the CST transform being preserved. Furthermore, the shift k → k + n appears in a natural way in this finite-dimensional framework.

Quantization of some moduli spaces of parabolic vector bundles on $${{\mathbb C}{\mathbb P}^1}$$

“…For this, we need to apply the framework of [10] for non-abelian theta functions over the moduli space of rank 2 vector bundles, with trivial determinant, over X (see Sect. 4).…”

“…In turn, a comparison between real and Käh-ler quantizations for abelian varieties was obtained using a coherent state transform [4,9,10].…”

Coherent State Transforms and Abelian Varieties

“…The case with a general polarization is treated analogously [12]. We will address the CST transform for the general case of abelian varieties in [10], in the context of genus 1 non-abelian theta functions. We can always find a basis l 1 ; .…”

“…. ; n g Þ with 04n a 5k, can be chosen freely while the others are fixed by the quasi-periodicity conditions (10). In fact, the general level k theta function, y 2 H O;k , is of the form yðzÞ ¼ X ðkz; kOÞ (see [17]).…”

“…The CST on Lie groups appears in this context through the use of the Schottky uniformization of the moduli space [5,9] and it provides a purely finite-dimensional framework for the explicit study of non-abelian theta functions. The application of these techniques to genus one non-abelian theta functions will appear in [10]. Applications to higher genus will appear in [11].…”