Branes as Stable Holomorphic Line Bundles On the Non-Commutative Torus

TL;DR

This paper explores the correspondence between B-branes and holomorphic line bundles on non-commutative tori, translating stability conditions into non-commutative instanton equations and analyzing localized instantons with non-constant field strength.

Contribution

It provides a detailed translation of brane stability conditions into non-commutative instanton equations, including the treatment of non-constant field strengths using the Seiberg--Witten map.

Findings

Stable topological branes correspond to non-commutative instantons.

Localization of instantons is crucial for the identities derived.

Non-linearities are managed via the rank-one Seiberg--Witten map.

Abstract

It was recently suggested by A. Kapustin that turning on a $B$-field, and allowing some discrepancy between the left and and right-moving complex structures, must induce an identification of B-branes with holomorphic line bundles on a non-commutative complex torus. We translate the stability condition for the branes into this language and identify the stable topological branes with previously proposed non-commutative instanton equations. This involves certain topological identities whose derivation has become familiar in non-commutative field theory. It is crucial for these identities that the instantons are localized. We therefore explore the case of non-constant field strength, whose non-linearities are dealt with thanks to the rank-one Seiberg--Witten map.