Local normal form theorems for smooth equivariant maps between infinite-dimensional manifolds are established. These normal form results are new even in finite dimensions. The proof is inspired by the Lyapunov–Schmidt reduction for dynamical systems and by the Kuranishi method for moduli spaces. It uses a slice theorem for Fréchet manifolds as the main technical tool. As a consequence, the abstract moduli space obtained by factorizing a level set of the equivariant map with respect to the group action carries the structure of a Kuranishi space, i.e., such moduli spaces are locally modeled on the quotient by a compact group of the zero set of a smooth map. The general results are applied to the moduli space of anti-self-dual instantons, the Seiberg–Witten moduli space and the moduli space of pseudoholomorphic curves.

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Let M be a manifold with a closed, integral (k+1)-form ω⁠, and let G be a Fréchet–Lie group acting on (M,ω)⁠. As a generalization of the Kostant–Souriau extension for symplectic manifolds, we consider a canonical class of central extensions of g by R⁠, indexed by Hk−1(M,R)∗⁠. We show that the image of Hk−1(M,Z) in Hk−1(M,R)∗ corresponds to a lattice of Lie algebra extensions that integrate to smooth central extensions of G by the circle group T⁠. The idea is to represent a class in Hk−1(M,Z) by a weighted submanifold (S,β)⁠, where β is a closed, integral form on S⁠. We use transgression of differential characters from S and M to the mapping space C∞(S,M) and apply the Kostant–Souriau construction on C∞(S,M)⁠.@en