Non-linear Yang-Mills instantons from strings are $\pi$-stable D-branes

TL;DR

This paper demonstrates that $ abla$-stability, derived from string theory, correctly characterizes solutions of non-linear Yang-Mills equations related to D-branes, extending the mathematical concept of stability to a string-theoretic context.

Contribution

It establishes that $ abla$-stability (or $ abla$-Pi stability) is the appropriate physical stability condition for non-linear Yang-Mills instantons in string theory, connecting string stability with geometric equations.

Findings

Solutions exist iff they are $ abla$-stable.

$ abla$-stability aligns with solutions of almost Hermitian Yang--Mills equations.

Proposes $ abla$-stability as a canonical stability concept in algebraic geometry.

Abstract

We show that B-type $\Pi$-stable D-branes do not in general reduce to the (Gieseker-) stable holomorphic vector bundles used in mathematics to construct moduli spaces. We show that solutions of the almost Hermitian Yang--Mills equations for the non-linear deformations of Yang--Mills instantons that appear in the low-energy geometric limit of strings exist iff they are $\pi$-stable, a geometric large volume version of $\Pi$-stability. This shows that $\pi$-stability is the correct physical stability concept. We speculate that this string-canonical choice of stable objects, which is encoded in and derived from the central charge of the string-\emph{algebra}, should find applications to algebraic geometry where there is no canonical choice of stable \emph{geometrical} objects.