SDiff Gauge Theory and the M2 Condensate

We develop a general formalism for the construction, in $D$-dimensional Minkowski space, of gauge theories for which the gauge group is the infinite-dimensional group SDiff$_n$ of volume-preserving diffeomorphisms of some closed $n$-dimensional manifold. We then focus on the D=3 SDiff$_3$ superconformal gauge theory describing a condensate of M2-branes; in particular, we derive its ${\cal N}=8$ superfield equations from a pure-spinor superspace action, and we describe its relationship to the D=3 SDiff$_2$ super-Yang-Mills theory describing a condensate of D2-branes.