Exact Half-BPS Flux Solutions in M-theory with $D(2, 1; c'; 0)^2$ Symmetry: Local Solutions

TL;DR

This paper constructs a broad class of exact local solutions in M-theory with specific supersymmetry, characterized by a complex PDE and parameters, generalizing known AdS solutions.

Contribution

The authors derive explicit local solutions invariant under $D(2, 1; c'; 0)^2$ symmetry for all $c'$, extending previous solutions and providing a framework for deformations of AdS backgrounds.

Findings

Solutions depend on a harmonic function and a complex function on a Riemann surface.

Explicit metric and field strength expressions are obtained, satisfying supergravity equations.

One-parameter deformations of AdS$_7$×S$^4$ and AdS$_4$×S$^7$ are constructed.

Abstract

We construct local solutions to 11-dimensional supergravity (or M-theory), which are invariant under the superalgebra $D(2, 1; c'; 0)\oplus D(2, 1; c'; 0)$ for all values of the parameter $c'$. The BPS constraints are reduced to a single linear PDE on a complex function $G$. The PDE was solved in 0806.0605 modulo application of boundary and regularity conditions. The physical fields of the solutions are determined by $c'$, a harmonic function $h$, and the complex function $G$. $ h$ and $G$ are both functions on a 2-dimensional compact Riemannian manifold. The harmonic function $ h$ is freely chosen. We obtain the expressions for the metric and the field strength in terms of $G$, $h$, and $c'$ and show that these are indeed valid solutions of the Einstein, Maxwell, and Bianchi equations. Finally we give a construction of one parameter deformations of $AdS_7 \times S^4$ and $AdS_4 \times S^7$ as a function of $c'$.