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

TL;DR

This work identifies π-stability as the physically appropriate stability notion for D-branes in the geometric, large-volume limit of Calabi–Yau compactifications and derives a string-induced deformation of the Hermitian Yang–Mills equations whose solvability exactly characterizes π-stable bundles. By connecting π-stability with a non-linear YM-instanton equation, the authors generalize the Donaldson–Uhlenbeck–Yau correspondence beyond μ- and γ-stability and show that solutions exist if and only if the underlying bundle is π-stable in the large-volume limit. This framework formalizes a string-theoretic stability criterion and suggests that π-stability could yield new, canonical stable objects with potential applications to moduli space constructions in algebraic geometry, especially in settings where a canonical stability notion is absent. The results rely on a GIT-based existence proof and a π-stability–adapted Jordan–Hölder filtration, highlighting how central charges and Td(X) encode stability in the geometric, large-volume regime.

Abstract

We show that B-type $Π$-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 $π$-stable, a geometric large volume version of $Π$-stability. This shows that $π$-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.