From Multiplicative to Additive Geometry: Deformation Theory and 2D TQFT

In this paper, we present a theory of Poisson deformation of Hamiltonian quasi-Poisson manifoldsto Hamiltonian Poisson manifolds that include degenerate cases. More significantly, this theory extendsto singular cases arising from symplectic implosion: we introduce a generalized Hamiltonian deformationtheory and we show that the imploded cross section of the double D(G)imp deforms to the implosion of thecotangent bundle T∗Gimp with applications to the master moduli space of G-flat connections.In parallel, we construct a topological quantum field theory N : Cob2 → QHam, where QHam is thecategory of quasi-Hamiltonian manifolds. To each cobordism Σ, we associate a quasi-Hamiltonian spaceN(Σ) built from the fusion product of copies of the double D(G). We show that these spaces are invariantunder the quiver homotopy and that the composition of cobordisms corresponds to a quasi-Hamiltonianreduction. This provides a multiplicative version of the 2D Hamiltonian TQFT of [30].