I was reading the abstract of a recent preprint ([Division Algebras and Supersymmetry III](http://arxiv.org/abs/1109.3574) by Juhn Huerta), and I wondered if something much simpler than what he was talking about had been worked on: have [Lie $2$-groups](http://ncatlab.org/nlab/show/Lie+2-group) been applied to the resolution of differential equations, in the same manner that Lie groups [originated](http://rads.stackoverflow.com/amzn/click/0387950001) 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](http://ncatlab.org/nlab/show/smooth+infinity-groupoid).



Lie $2$-groups and differential equations
A well-studied special case of higher symmetries of differential equations is that of differential equations that arise as [Euler-Lagrange equations](http://ncatlab.org/nlab/show/Euler-Lagrange+equation) 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](http://ncatlab.org/nlab/show/BRST+complex) -- which is the [Chevalley-Eilenberg algebra](http://ncatlab.org/nlab/show/Chevalley-Eilenberg+algebra) of the corresponding [L-∞ algebroid](http://ncatlab.org/nlab/show/Lie+infinity-algebroid). In simple cases (or else locally) this is the global quotient by a [smooth ∞-group](http://ncatlab.org/nlab/show/smooth+infinity-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](http://ncatlab.org/nlab/show/Yang-Mills%20field) the symmetries form an ordinary Lie group;
* for the action functional of the [Kalb-Ramond field](http://ncatlab.org/nlab/show/Kalb-Ramond%20field) the symmetries form the [circle 2-group](http://ncatlab.org/nlab/show/circle%20n-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](http://ncatlab.org/nlab/show/supergravity%20C-field) the symmetries are governed by the [circle 3-group](http://ncatlab.org/nlab/show/circle%20n-group) $\mathbf{B}^2 U(1) = (U(1) \to 1 \to 1)$;
* the [higher abelian Chern-Simons theory](http://ncatlab.org/nlab/show/higher+dimensional+Chern-Simons+theory) in dimension $4k+3$ has the [circle (2k+1)-group](http://ncatlab.org/nlab/show/circle%20n-group) $\mathbf{B}^{2k} U(1)$ as its gauge group;
* the symmetries of full [string field theory](http://ncatlab.org/nlab/show/string%20field%20theory) 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](http://ncatlab.org/schreiber/show/infinity-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](http://ncatlab.org/nlab/show/Poisson%20sigma-model) is controled by the [Lie integration](http://ncatlab.org/nlab/show/Lie%20integration) 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](http://ncatlab.org/nlab/show/Courant%20sigma-model) is controled by the Lie integration of a [Courant Lie 2-algebroid](http://ncatlab.org/nlab/show/Courant+Lie+2-algebroid), which is a [symplectic Lie 2-algebroid](http://ncatlab.org/nlab/show/symplectic+infinity-groupoid);
	+ the gauge $n$-groupoid of a grade $n$ [AKSZ sigma-model](http://ncatlab.org/nlab/show/AKSZ+sigma-model) is similarly controled by a [symplectic Lie n-groupoid](http://ncatlab.org/nlab/show/symplectic+infinity-groupoid).
	+ the 7-dimensional "[fivebrane Chern-Simons theory](http://ncatlab.org/nlab/show/differential+fivebrane+structure)" has [string 2-group](http://ncatlab.org/nlab/show/string%202-group)-symmetries


