The non-abelian open superstring effective action through order $α'{}^3$

TL;DR

The paper addresses constructing the non-abelian open superstring effective action up to $O(\alpha'^{3})$ by deforming ten-dimensional $U(n)$ Yang–Mills theory around stable holomorphic bundles and enforcing DUY stability, including derivative terms that are essential in the non-abelian case. A computer-aided scheme systematically classifies invariants, field redefinitions, and DUY deformations to determine allowed corrections order by order, yielding no $O(\alpha')$ deformation, a unique $O(\alpha'^{2})$ deformation, and a one-parameter family at $O(\alpha'^{3})$ with derivative terms fixed by open-string amplitudes. The derivative sector at $O(\alpha'^{3})$ agrees with results from open-string disk amplitudes (e.g. $\lambda = -\tfrac{2\zeta(3)}{\pi^3}$), while the non-derivative $F^8$ structure shows basis-dependent discrepancies with some literature, illustrating the subtle interplay between dual descriptions. Altogether, the work provides a concrete candidate non-abelian Born–Infeld action through $O(\alpha'^{3})$ and demonstrates a viable path toward higher-order determinations.

Abstract

Using the method developed in {\tt hep-th/0103015}, we determine the non-abelian Born-Infeld action through ${\cal O}(α'{}^3)$. We start from solutions to a Yang-Mills theory which define a stable holomorphic vector bundle. Subsequently we investigate its deformation away from this limit. Through $ {\cal O}(α'{}^2)$, a unique, modulo field redefinitions, solution emerges. At $ {\cal O}(α'{}^3)$ we find a one-parameter family of allowed deformations. The presence of derivative terms turns out to be essential. Finally, we present a detailed comparison of our results to existing, partial results.