Kronecker foliation, D1-branes and Morita equivalence of Noncommutative two-tori

TL;DR

This paper explores the geometric and algebraic structures of D1-branes on two-tori, revealing their relation to noncommutative geometry, Morita equivalence, and mirror symmetry through Kronecker foliation and crossed product algebras.

Contribution

It provides a geometric realization of Morita equivalence of noncommutative two-tori and connects D-brane physics with homological mirror symmetry.

Findings

Identifies the algebra of open strings on D1-branes with crossed product algebras of noncommutative tori.

Shows Morita equivalence can be understood geometrically via Kronecker foliation.

Relates Heisenberg modules and tensor products to homological mirror symmetry.

Abstract

It is known that the physics of open strings on a D2-brane on a two-torus is realized from the viewpoint of deformation quantization in the Seiberg-Witten limit. We study its T-dual theory, i.e. D1-brane physics on two-tori. Such theory is described by Kronecker foliation. The algebra of open strings on the D1-brane is then identified with the crossed product representation of a noncommutative two-torus. The Morita equivalence of noncommutative two-tori is also realized geometrically along this line. As an application, Heisenberg modules and the tensor product between them are discussed from these geometric viewpoints. We show they are related to the homological mirror symmetry of two-tori.