The non-abelian D-brane effective action through order $α'{}^4$

TL;DR

This work determines the bosonic non-abelian D-brane effective action through order ${\alpha'}^4$ by leveraging BPS-like holomorphic bundle solutions and a generalized symmetrized trace organization that confines non-abelian structure to covariant derivatives. The authors construct the action order by order in ${\alpha'}$, fix coefficients by demanding that deformed holomorphic stability and equations of motion are satisfied, and present an explicit decomposition ${\cal L}=\frac{1}{g^2}( {\cal L}_0+{\cal L}_2+{\cal L}_3+{\cal L}_4 )$ with detailed forms for ${\cal L}_0$, ${\cal L}_2$, ${\cal L}_3$, and ${\cal L}_4={\cal L}_{4,0}+{\cal L}_{4,2}+{\cal L}_{4,4}$. The result confirms that derivative corrections are essential at this order, fixes previously ambiguous abelian limits, and suggests all-order structures, including constrained communications between derivative orders separated by multiples of four. The methodology and explicit coefficients provide a stringent basis for comparing with string-theoretic results in backgrounds and for exploring higher-order and all-order extensions.

Abstract

Requiring the existence of certain BPS solutions to the equations of motion, we determine the bosonic part of the non-abelian D-brane effective action through order $α'{}^4$. We also propose an economic organizational principle for the effective action.