Question

I was reading the abstract of a recent preprint (Division Algebras and Supersymmetry III by Juhn Huerta), and I wondered if something much simpler than what he was talking about had been worked on: have Lie $2$-groups been applied to the resolution of differential equations, in the same manner that Lie groups originated from the study of differential equations?

In other words, do Lie 2-groups arise as symmetries for (certain kinds of) differential equations, and can these in turn be used for the integration/resolution of those same differential equations? If they do not, then in what setting can a 2-group be understood as a symmetry (if any), and to what 'use' can this information be put?

My motivation here is to expand the toolset I can use to solve problems in classical analysis (like differential equations), and not to explore the other areas where Lie groups have developed in to (like Lie algebras and their classification, etc). For the purposes of this question, these issues are out-of-scope. In-scope are applications (to classical analysis) of generalizations going all the way to $\infty$-Lie groupoids.

Answer 1

A well-studied special case of higher symmetries of differential equations is that of differential equations that arise as Euler-Lagrange equations of local action functionals. The symmetries and symmetries-of-symmetries and symmetries-of-symmetries-of-symmetries of such a system of equations form an $\infty$-groupoid whose infinitesimal version is encoded by the corresponding BRST complex -- which is the Chevalley-Eilenberg algebra of the corresponding L-∞ algebroid. In simple cases (or else locally) this is the global quotient by a smooth ∞-group: the "ghosts" in the BRST complex are the cotangents to the local symmetries, the "ghosts-of-ghosts" are the cotangents to the local symmetries-of-symmetries, and so on.

For instance

• for the action functional of the Yang-Mills field the symmetries form an ordinary Lie group;

• for the action functional of the Kalb-Ramond field the symmetries form the circle 2-group $\mathbf{B}U(1) = (U(1) \to 1)$, (or rather the 2-group of functions with values in the circle 2-group);

• for the action functional of the supergravity C-field the symmetries are governed by the circle 3-group $\mathbf{B}^2 U(1) = (U(1) \to 1 \to 1)$;

• the higher abelian Chern-Simons theory in dimension $4k+3$ has the circle (2k+1)-group $\mathbf{B}^{2k} U(1)$ as its gauge group;

• the symmetries of full string field theory form a general $\infty$-group (not an $n$-group for any finite $n$) the structure of which nobody really understands, I think.

• every ∞-Chern-Simons theory (or equivalently its Euler-Lagrange equations) has a higher group of symmetries. In general, this is not just a higher gauge group, but even a higher gauge groupoid .

  • the gauge groupoid of the Poisson sigma-model is controled by the Lie integration of a [Poisson Lie algebroid](http://ncatlab.org/nlab/show/Poisson+Lie algebroid), which is a [symplectic+groupoid](http://ncatlab.org/nlab/show/symplectic groupoid);

  • the gauge 2-groupoid of the Courant sigma-model is controled by the Lie integration of a Courant Lie 2-algebroid, which is a symplectic Lie 2-algebroid;

  • the gauge $n$-groupoid of a grade $n$ AKSZ sigma-model is similarly controled by a symplectic Lie n-groupoid.

  • the 7-dimensional "fivebrane Chern-Simons theory" has string 2-group-symmetries