Integrability from categorification and the 2-Kac-Moody Algebra

The theory of Poisson-Lie groups and Lie bialgebras plays a major role in the study of one dimensional integrable systems; many families of integrable systems can be recovered from a Lax pair which is constructed from a Lie bialgebra associated to a Poisson-Lie group. A higher homotopy notion of Poisson-Lie groups and Lie bialgebras has been studied using Lie algebra crossed-modules and $L_2$-algebras, which gave rise to the notion of (strict) Lie 2-bialgebras and Poisson-Lie 2-groups . In this paper, we use these structures to generalize the construction of a Lax pairs and introduce an appropriate notion of {higher homotopy integrability}. Within this framework, we introduce a higher homotopy version of the Kac-Moody algebra, with which the 2-Lax equation can be rewritten as a zero 2-curvature condition in 2+1d. An explicit characterization of our higher Kac-Moody algebra will be given, and we also demonstrate how it governs the 2-Lax pairs and the symmetries of a 3d topological-holomorphic field theory studied recently. This 3d theory thus serves as an example of a physical system that exhibits the sort of 2-graded integrability that we have defined here.