Exact Half-BPS Flux Solutions in M-theory I, Local Solutions

TL;DR

This work delivers exact local half-BPS flux solutions in M-theory with geometry $AdS_3\times S^3\times S^3\times \Sigma$, warped over a Riemann surface and with boundary behavior $AdS_4\times S^7$ or $AdS_7\times S^4$. By mapping the reduced BPS system to a sine-Gordon/Liouville-type integrable equation and subsequently to a linear equation, the authors construct the full local solution from a harmonic function on $\Sigma$ and an integral transform of two further harmonic functions, ensuring all supergravity equations are satisfied. They identify a holomorphic form $\kappa$ on $\Sigma$, organize the solution around three real constants $c_1,c_2,c_3$ with $c_1+c_2+c_3=0$, and demonstrate that cases with two coincident $c_i$ yield the known maximally supersymmetric backgrounds, while other parameter choices describe fully back-reacted M2/M5-brane configurations. The analysis also shows a discrete symmetry relating the different cases and provides explicit local expressions for the metric factors, fluxes, and Killing spinors, together with a robust linearization framework for the remaining non-linear BPS equation. This work lays the groundwork for global regularity studies and richer solution classes in the M-theory AdS/CFT context.

Abstract

The complete eleven-dimensional supergravity solutions with 16 supersymmetries on manifolds of the form $AdS_3 \times S^3 \times S^3 \times Σ$, with isometry $SO(2,2) \times SO(4) \times SO(4)$, and with either $AdS_4 \times S^7$ or $AdS_7 \times S^4$ boundary behavior, are obtained in exact form. The two-dimensional parameter space $Σ$ is a Riemann surface with boundary, over which the product space $AdS_3 \times S^3 \times S^3$ is warped. By mapping the reduced BPS equations to an integrable system of the sine-Gordon/Liouville type, and then mapping this integrable system onto a linear equation, the general local solutions are constructed explicitly in terms of one harmonic function on $Σ$, and an integral transform of two further harmonic functions on $Σ$. The solutions to the BPS equations are shown to automatically solve the Bianchi identities and field equations for the 4-form field, as well as Einstein's equations. The solutions we obtain have non-vanishing 4-form field strength on each of the three factors of $AdS_3 \times S^3 \times S^3$, and include fully back-reacted M2-branes in $AdS_7 \times S^4$ and M5-branes in $AdS_4 \times S^7$. No interpolating solutions exist with mixed $AdS_4 \times S^7$ and $AdS_7 \times S^4$ boundary behavior. Global regularity of these local solutions, as well as the existence of further solutions with neither $AdS_4 \times S^7$ nor $AdS_7 \times S^4$ boundary behavior will be studied elsewhere.

Figures (2)

• Figure 1: The space of parameters $c_1,c_2,c_3$. Assignments differing only by an overall sign have been identified in this representation.
• Figure 2: Allowed ranges of $G$ in the complex plane for case I, and cases II and III.