Functional equations for quantum theta functions

Abstract

Quantum theta functions were introduced by the author in [Ma1]. They are certain elements in the function rings of quantum tori. By definition, they satisfy a version of the classical functional equations involving shifts by the multiplicative periods. This paper shows that for a certain subclass of period lattices (compatible with the quantization form), quantum thetas satisfy an analog of another classical functional equation related to an action of the metaplectic group upon the (half of) the period matrix. In the quantum case, this is replaced by the action of the special orthogonal group on the quantization form, which provides Morita equivalent tori. The argument uses Rieffel's approach to the construction of (strong) Morita equivalence bimodules and the associativity of Rieffel's scalar products.