From multiplicative to additive geometry: Deformation theory and 2D TQFT

In this paper, we present a theory of Poisson deformation of Hamiltonian quasi-Poisson manifolds to Hamiltonian Poisson manifolds that include degenerate cases. More significantly, this theory extends to singular cases arising from symplectic implosion: we introduce a generalized Hamiltonian deformation theory and we show that the imploded cross section of the double $D(G{)}_{\text{imp}}$ deforms to the implosion of the cotangent bundle $T^{\ast }{G}_{\text{imp}}$ with applications to the master moduli space of $G$-flat connections.\ In parallel, we construct a topological quantum field theory $\mathbb{N}:{\text{Cob}}_2\to \mathbf{Q}\mathbf{H}\mathbf{a}\mathbf{m}$, where $\mathbf{Q}\mathbf{H}\mathbf{a}\mathbf{m}$ is the category of quasi-Hamiltonian manifolds. To each cobordism $\mathrm{\Sigma }$, we associate a quasi-Hamiltonian space $\mathbb{N}(\mathrm{\Sigma })$ built from the fusion product of copies of the double $D(G)\mathrm{.}$ We show that these spaces are invariant under the \emph{quiver homotopy} and that the composition of cobordisms corresponds to a quasi-Hamiltonian reduction. This provides a multiplicative version of the 2D Hamiltonian TQFT of Maiza-Mayrand.