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) x S-3 x S-3 x Sigma, with isometry SO( 2, 2) x SO( 4) x SO( 4), and with either AdS(4) x S-7 or AdS(7) x 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) x S-3 x 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) x S-3 x S-3, and include fully back-reacted M2-branes in AdS(7) x S-4 and M5-branes in AdS(4) x S-7. No interpolating solutions exist with mixed AdS(4) x S-7 and AdS(7) x S-4 boundary behavior. Global regularity of these local solutions, as well as the existence of further solutions with neither AdS(4) x S-7 nor AdS(7) x S-4 boundary behavior will be studied elsewhere.