Holomorphic Yang-Mills fields on B-branes

Considering the B-branes over a complex manifold Y as objects of the bounded derived category Dᵇ(Y), we define holomorphic gauge fields on B-branes and the Yang-Mills functional for these fields.These definitions are a generalization to B-branes of concepts that are well known in the context of vector bundles. Given F•∈ Dᵇ(Y), we show that the Atiyah class a( F•)∈ Ext¹( F•,\,Ω¹( F•)) is the obstruction to the existence of gauge fields on F•. When Y be either the projective space Pⁿ or the variety of complete flags in C³, we prove that the cardinal of the set of holomorphic gauge fields on any B-brane over Y is ≤ 1. We prove that the set of Yang-Mills fields on the B-brane F•, if it is nonempty, is in bijective correspondence with the points of an algebraic subset of Cᵐ defined by m polynomial equations of degree ≤ 3, where m= dim\, Ext⁰( F•,\,Ω¹( F•)).