Quantum mechanics and nonabelian theta functions for the gauge group ${\rm SU}(2)$

Răzvan Gelca¹ ; Alejandro Uribe²

¹Department of Mathematics and Statistics Texas Tech University Lubbock, TX 79409, U.S.A.
²Department of Mathematics University of Michigan Ann Arbor, MI 48109, U.S.A.

Fundamenta Mathematicae, Tome 228 (2015) no. 2, pp. 97-137

Résumé

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.

Détail

DOI : 10.4064/fm228-2-1

Keywords: propose direction study nonabelian theta functions establishing analogy between weyl quantization one dimensional particle metaplectic representation quantization moduli space flat connections surface representation mapping class group space nonabelian theta functions other exemplify cases classical theta functions nonabelian theta functions gauge group mathop nolimits emphasis paper analogy possibility generalizing approach other gauge groups results which have appeared elsewhere