  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.
