Exact M-Theory Solutions, Integrable Systems, and Superalgebras

In this paper, an overview is presented of the recent construction of fully back-reacted half-BPS solutions in 11-dimensional supergravity which correspond to near-horizon geometries of M2 branes ending on, or intersecting with, M5 and M5$'$ branes along a self-dual string. These solutions have space-time manifold ${\rm AdS}_3 \times S^3 \times S^3$ warped over a Riemann surface $Σ$, and are invariant under the exceptional Lie superalgebra $D(2,1;γ) \oplus D(2,1;γ)$, where $γ$ is a real continuous parameter and $|γ|$ is governed by the ratio of the number of M5 and M5$'$ branes. The construction proceeds by mapping the reduced BPS equations onto an integrable field theory on $Σ$ which is of the Liouville sine-Gordon type. Families of regular solutions are distinguished by the sign of $γ$, and include a two-parameter Janus solution for $γ>0$, and self-dual strings on M5 as well as asymptotically ${\rm AdS}_4/{\mathbb Z}_2$ solutions for $γ<0$.

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