Generalized Higher Gauge Theory and M5-brane dynamics. Proceedings: Higher structures in String and M-theory, Tohoku Forum for Creativity, March 7-11 2016

We give a review of truncated L$_\infty$ algebras, as used in the study of higher gauge theory. These structures are believed to hold the correct properties to adequately describe gauge theory of extended objects. We discuss how to construct topological higher-gauge-invariant theories and how their solutions relate to multisymplectic geometries. We also show how Courant algebroids fit into this formalism, so as to be able to study higher gauge theory on generalized geometric bundles, i.e. on $T\Sigma \oplus T^*\Sigma$, for some space-time $\Sigma$. We will see that via this formalism we can match and explain a recently proposed M5-brane model, arrived at in a more heuristic way, whose field content seemed difficult to interpret but finds a natural motivation in this framework.