Gauge fields on $B$-branes over $\mathbb{CP}^n$

TL;DR

This paper proves that for $B$-branes over complex projective space, there is at most one holomorphic gauge field, and it exists only under specific decomposability conditions of the brane components.

Contribution

It establishes a uniqueness result for holomorphic gauge fields on $B$-branes over ${ m f P}^n$, linking existence to the structure of the brane components.

Findings

At most one holomorphic gauge field per $B$-brane.

Existence of a gauge field iff components are sums of ${ m f O}_{{ m f P}^n}$.

Characterization of gauge fields in terms of direct sum decompositions.

Abstract

Considering the $B$-branes over the complex projective space ${\mathbb P}^n$ as the objects of the bounded derived category $D^b({\mathbb P}^n)$, we prove that the cardinal of the set of holomorphic gauge fields on a given $B$-brane ${\mathscr G}^{\bullet}$ is $\leq 1$. Moreover, the cardinal is $1$ iff each ${\mathscr G}^p$ is isomorphic to a direct sum of copies of ${\mathscr O}_{{\mathbb P}^n}$.