Brane Resolution Through Transgression

TL;DR

This work introduces a general transgression-based mechanism to resolve brane singularities by replacing flat transverse spaces with Ricci-flat manifolds $M_n$ that admit covariantly constant spinors and turning on fluxes sourced by harmonic forms $L_{(p)}$ so that $dF_{(n)} = F_{(p)}\wedge F_{(q)}$. When the harmonic form is $L^2$-normalisable, the deformed brane solutions can be completely non-singular and preserve some supersymmetry, providing gravity duals for gauge theories with less than maximal SUSY. The authors apply the method to D3-branes on Ricci-flat Kähler 6-manifolds, heterotic 5-branes on 4-manifolds, dyonic strings, M2-branes on Spin(7) holonomy 8-manifolds, D2-branes on G2 holonomy 7-manifolds, and further Type II examples on 8-manifolds and D4-branes on 5-manifolds, each time exploiting normalisable harmonic forms to source transgression flux and control singularities. The resulting non-singular solutions yield novel holographic duals with potential insights into confinement, chiral symmetry breaking, and domain-wall physics in lower-dimensional theories.

Abstract

Modifications to the singularity structure of D3-branes that result from turning on a flux for the R-R and NS-NS 3-forms (fractional D3-branes) provide important gravity duals of four-dimensional N=1 super-Yang-Mills theories. We construct generalisations of these modified p-brane solutions in a variety of other cases, including heterotic 5-branes, dyonic strings, M2-branes, D2-branes, D4-branes and type IIA and type IIB strings, by replacing the flat transverse space with a Ricci-flat manifold M_n that admits covariantly constant spinors, and turning on a flux built from a harmonic form in M_n, thus deforming the original solution and introducing fractional branes. The construction makes essential use of the Chern-Simons or ``transgression'' terms in the Bianchi-identity or equation of motion of the field strength that supports the original undeformed solution. If the harmonic form is L^2 normalisable, this can result in a deformation of the brane solution that is free of singularities, thus providing viable gravity duals of field theories in diverse dimensions that have less than maximal supersymmetry. We obtain examples of non-singular heterotic 5-branes, dyonic strings, M2-branes, type IIA strings, and D2-branes.