Holomorphic Yang-Mills fields on $B$-branes

TL;DR

This paper extends the concept of gauge fields and Yang-Mills functional to $B$-branes in complex geometry, establishing conditions for their existence and uniqueness, and characterizing the space of solutions algebraically.

Contribution

It introduces a novel framework for holomorphic gauge fields on $B$-branes, linking their existence to the Atiyah class and characterizing the Yang-Mills fields via algebraic sets.

Findings

Atiyah class obstructs gauge field existence on $B$-branes.

Unique holomorphic gauge field for $B$-branes over flag varieties.

Yang-Mills fields correspond to solutions of polynomial equations.

Abstract

Considering $B$-branes over a complex manifold $X$ as objects of the bounded derived category of coherent sheaves over $X$, we define holomorphic gauge fields on $B$-branes and introduce the Yang-Mills functional for these fields. These definitions extend well-known concepts in the context of vector bundles to the setting of $B$-branes. For a given $B$-brane, we show that its Atiyah class is the obstruction to the existence of gauge fields. When $X$ is the variety of complete flags in a $3$-dimensional complex vector space, we prove that any $B$-brane over $X$ admits at most one holomorphic gauge field. Furthermore, we establish that the set of Yang-Mills fields on a given $B$-brane, if nonempty, is in bijective correspondence with the points of an algebraic set defined by $m$ complex polynomials of degree less than four in $m$ indeterminates, where $m$ is the dimension of the space of morphisms from the brane to its tensor product with the sheaf of holomorphic one-forms.