For supersymmetric solutions of $D3\mathrm{\text{−}}(M2\mathrm{\text{−}})$ branes with ${\mathrm{AdS}}_3({\mathrm{AdS}}_2)$ 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.

• Received 10 June 2012

DOI:https://doi.org/10.1103/PhysRevD.86.067901