Supersymmetric Non-singular Fractional D2-branes and NS-NS 2-branes

TL;DR

This work addresses constructing regular, supersymmetric fractional brane solutions in type IIA string theory by replacing the flat space transverse to D2-branes with complete $G_2$-holonomy manifolds. The authors use transgression couplings and harmonic 3-forms to generate wrapped brane fluxes, presenting two explicit regular constructions: (i) a D2-brane with wrapped D4-branes on a $G_2$ manifold whose principal orbits are $\mathbb{CP}^3$ or the flag $SU(3)/(U(1)\times U(1))$, and (ii) a deformed NS-NS 2-brane on the spin bundle of $S^3$, both preserving $1/16$ of the original supersymmetry. The warp factor solutions are regular at all distances and yield a discrete spectrum for minimally coupled scalars, signaling confinement in the dual ${\cal N}=1$, 3D gauge theories; in the UV the solutions approach the standard D2-brane behavior but with charges tied to the fractionally wrapped branes. The paper also broadens the study of fractional brane resolutions, derives criteria for regularity, and constructs new explicit complete Ricci-flat metrics, highlighting the rarity of fully regular supersymmetric fractional branes and suggesting avenues where stringy effects may be essential. Altogether, these results advance the holographic understanding of confining 3D gauge theories with reduced supersymmetry and enrich the landscape of explicit, complete Ricci-flat geometries relevant to AdS/CFT generalizations.

Abstract

We obtain regular deformed D2-brane solutions with fractional D2-branes arising as wrapped D4-branes. The space transverse to the D2-brane is a complete Ricci-flat 7-manifold of G_2 holonomy, which is asymptotically conical with principal orbits that are topologically CP^3 or the flag manifold SU(3)/(U(1) x U(1)). We obtain the solution by first constructing an L^2 normalisable harmonic 3-form. We also review a previously-obtained regular deformed D2-brane whose transverse space is a different 7-manifold of G_2 holonomy, with principal orbits that are topologically S^3 x S^3. This describes D2-branes with fractional NS-NS 2-branes coming from the wrapping of 5-branes, which is supported by a non-normalisable harmonic 3-form on the 7-manifold. We prove that both types of solutions are supersymmetric, preserving 1/16 of the maximal supersymmetry and hence that they are dual to {\cal N}=1 three-dimensional gauge theories. In each case, the spectrum for minimally-coupled scalars is discrete, indicating confinement in the infrared region of the dual gauge theories. We examine resolutions of other branes, and obtain necessary conditions for their regularity. The resolution of many of these seems to lie beyond supergravity. In the process of studying these questions, we construct new explicit examples of complete Ricci-flat metrics.