Functional Equations for Quantum Theta Functions

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. §0. Introduction and summary 0.1. Theta functions and theta vectors. This paper is a contribution to the theory of quantum theta functions introduced in [Ma1] and further studied in [Ma2], [Ma3]. It addresses two interrelated questions: (a) What is the connection between quantum theta functions and theta vectors? This question was repeatedly raised by A. S. Schwarz, see e. g. [Sch]. (b) Does there exist a quantum analog of the classical functional equation for thetas (related to the action of the metaplectic group, see e. g. [Mu], §8)? Briefly, the (partial) answers we give here look as follows. (i) Schwarz’s theta vectors are certain elements of projective modules over C– or C– rings of unitary quantum tori. When such a module is induced from the basic Heisenberg representation by a lattice embedding into a vector Heisenberg group, the respective theta vectors fT are parametrized by the points T of Siegel upper half space, and in different models of the basic representation take the form of a “quadratic exponent” e tTx, a classical theta, or Fock’s vacuum state: see Theorem 2.2 in [Mu]. To the contrary, quantum thetas are certain elements of the C function ring itself. (For this reason, partial multiplication of quantum thetas studied in [Ma3] does not seem to be directly related to the tensor product of projective (bi)modules). The basic relationship between the two classes of objects is this. For a lattice embedded in a vector Heisenberg group, Rieffel’s scalar products of theta vectors 1