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

The complete eleven-dimensional supergravity solutions with 16 supersymmetries on manifolds of the form $AdS_3 \times S^3 \times S^3 \times \Sigma$, 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 $\Sigma$ 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 $\Sigma$, and an integral transform of two further harmonic functions on $\Sigma$. 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.