From multiplicative to additive geometry: Deformation theory and 2D TQFT
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866908775463518208 |
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| author | Maiza, Mohamed Moussadek |
| author_facet | Maiza, Mohamed Moussadek |
| contents | 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 $D(G)_\imp$ deforms to the implosion of the cotangent bundle $T^*G_\imp$ with applications to the master moduli space of $G$-flat connections.\\ In parallel, we construct a topological quantum field theory $\N: \text{Cob}_{2}\to \mathbf{QHam}$, where $\mathbf{QHam}$ is the category of quasi-Hamiltonian manifolds. To each cobordism $Σ$, we associate a quasi-Hamiltonian space $\N(Σ)$ built from the fusion product of copies of the double $D(G).$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_13455 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | From multiplicative to additive geometry: Deformation theory and 2D TQFT Maiza, Mohamed Moussadek Symplectic Geometry Differential Geometry 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 $D(G)_\imp$ deforms to the implosion of the cotangent bundle $T^*G_\imp$ with applications to the master moduli space of $G$-flat connections.\\ In parallel, we construct a topological quantum field theory $\N: \text{Cob}_{2}\to \mathbf{QHam}$, where $\mathbf{QHam}$ is the category of quasi-Hamiltonian manifolds. To each cobordism $Σ$, we associate a quasi-Hamiltonian space $\N(Σ)$ built from the fusion product of copies of the double $D(G).$ 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. |
| title | From multiplicative to additive geometry: Deformation theory and 2D TQFT |
| topic | Symplectic Geometry Differential Geometry |
| url | https://arxiv.org/abs/2601.13455 |