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The Cartan model computes the equivariant cohomology of a smooth manifold X with
differentiable action of a compact Lie group G, from the invariant functions on
the Lie algebra with values in differential forms and a deformation of the de Rham
differential. Before extracting invariants, the Cartan differential does not square
to zero. Unrecognised was the fact that the full complex is a curved algebra,
computing the quotient by G of the algebra of differential forms on X. This
generates, for example, a gauged version of string topology. Another instance of
the construction, applied to deformation quantisation of symplectic manifolds,
gives the BRST construction of the symplectic quotient. Finally, the theory for a
X point with an additional quadratic curving computes the representation category
of the compact group G.