Quantum mechanics and nonabelian theta functions for the gauge group ${\rm SU}(2)$
Volume 228 / 2015
Fundamenta Mathematicae 228 (2015), 97-137
MSC: Primary 81S10; Secondary 81R50, 57R56, 81T45, 57M25.
DOI: 10.4064/fm228-2-1
Abstract
We propose a direction of study of nonabelian theta functions by establishing an analogy between the Weyl quantization of a one-dimensional particle and the metaplectic representation on the one hand, and the quantization of the moduli space of flat connections on a surface and the representation of the mapping class group on the space of nonabelian theta functions on the other. We exemplify this with the cases of classical theta functions and of the nonabelian theta functions for the gauge group ${\rm \mathop {\rm SU}\nolimits }(2)$. The emphasis of the paper is on this analogy and on the possibility of generalizing this approach to other gauge groups, and not on the results, of which some have appeared elsewhere.
