The Backreacted Kähler Geometry of Wrapped Branes

TL;DR

The paper addresses the problem of characterizing the backreacted Kähler geometry underlying AdS$_3$ from wrapped D3 branes and AdS$_2$ from wrapped M2 branes, beyond standard Sasaki–Einstein constructions. It adopts a Killing-spinor framework to derive a universal $U(1)$-fibration structure over a Kähler base, and introduces a general ansatz for the base in terms of $H^2$ fibered over a positive-curvature KE base, constrained by the master equation $\Box R - \frac{1}{2} R^2 + R_{ij}R^{ij}=0$. The authors explicitly rewrite known wrapped-brane solutions in canonical form, obtain D3/CY$_3$ and M2/CY$_4$ examples, and develop a higher-dimensional generalization that yields a family of regular AdS solutions with base geometries like $H^2\times KE^+_{2n}$ over a squashed $S^2$, including the $n=1$ and $n=2$ cases. This framework provides a geometric blueprint for constructing new AdS$_3$/AdS$_2$ vacua from wrapped branes, with potential implications for holographic duals and twisted field theories.

Abstract

For supersymmetric solutions of D3(M2) branes with AdS3(AdS2) factor, it is known that the internal space is expressible as U(1) fibration over Kähler space which satisfies a specific partial differential equation involving the Ricci tensor. In this paper we study the wrapped brane solutions of D3 and M2-branes which were originally constructed using gauged supergravity and uplifted to D=10 and D=11. We rewrite the solutions in canonical form, identify the backreacted Kähler geometry, and present a class of solutions which satisfy the Killing spinor equation.