Exact $\mathrm{M}$-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 $\mathrm{M2}$ branes ending on, or intersecting with, $\mathrm{M5}$ and $\mathrm{M5'}$ 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 $\Sigma$, and are invariant under the exceptional Lie superalgebra $D(2,1;\gamma) \oplus D(2,1;\gamma)$, where $\gamma $ is a real continuous parameter and $|\gamma|$ is governed by the ratio of the number of $\mathrm{M5}$ and $\mathrm{M5'}$ branes. The construction proceeds by mapping the reduced BPS equations onto an integrable field theory on $\Sigma$ which is of the Liouville sine-Gordon type. Families of regular solutions are distinguished by the sign of $\gamma$, and include a two-parameter Janus solution for $\gamma >0$, and self-dual strings on $\mathrm{M5}$ as well as asymptotically ${\rm AdS}_4/{\mathbb Z}_2$ solutions for $\gamma <0$.