Symmetry of Quantum Torus with Crossed Product Algebra

TL;DR

This paper explores the symmetry properties of quantum tori using crossed product algebra, analyzing how the symplectic group acts on quantum theta functions and identifying subgroups compatible with orbifolding.

Contribution

It introduces a quantum version of orbifolding for quantum tori via crossed product algebra and examines the symplectic group's role in this framework, extending classical symmetry concepts.

Findings

Symplectic group Sp(2n,Z) satisfies the algebraic consistency condition.

Only a subgroup of Sp(2n,Z) is compatible with quantum orbifolding.

A variant of classical theta functions invariant under group action is constructed.

Abstract

In this paper, we study the symmetry of quantum torus with the concept of crossed product algebra. As a classical counterpart, we consider the orbifold of classical torus with complex structure and investigate the transformation property of classical theta function. An invariant function under the group action is constructed as a variant of the classical theta function. Then our main issue, the crossed product algebra representation of quantum torus with complex structure under the symplectic group is analyzed as a quantum version of orbifolding. We perform this analysis with Manin's so-called model II quantum theta function approach. The symplectic group Sp(2n,Z) satisfies the consistency condition of crossed product algebra representation. However, only a subgroup of Sp(2n,Z) satisfies the consistency condition for orbifolding of quantum torus.