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

Mohamed Moussadek Maiza

TL;DR

The paper develops a cohesive deformation-theoretic framework that unifiedly connects multiplicative (quasi-Poisson, quasi-Hamiltonian) and additive (Poisson, Hamiltonian) geometries, extending to stratified and singular spaces such as imploded cross-sections and master moduli spaces. It proves Poisson deformations exist for key objects (e.g., $G \to \mathfrak g^{*}$, $D(G) \to T^{*}G$) and introduces generalized deformations for stratified spaces, ensuring compatibility with fusion, reduction, and partial fusion. A new family of quasi-Hamiltonian moduli spaces $\mathcal N_G(\Gamma)$ is constructed from quivers, shown to be invariant under quiver homotopy and to satisfy a gluing formula under quiver composition. These ingredients culminate in a 2D topological quantum field theory $\\mathcal N: \mathbf{Cob}_2 \to \mathbf{QHam}$ valued in quasi-Hamiltonian spaces, where cobordisms correspond to fused/reduced quasi-Hamiltonian data built from copies of $D(G)$. The framework provides a multiplicative-to-additive passage for important geometric objects and yields a robust TQFT structure linking quiver combinatorics to cobordism data.

Abstract

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.

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

TL;DR

The paper develops a cohesive deformation-theoretic framework that unifiedly connects multiplicative (quasi-Poisson, quasi-Hamiltonian) and additive (Poisson, Hamiltonian) geometries, extending to stratified and singular spaces such as imploded cross-sections and master moduli spaces. It proves Poisson deformations exist for key objects (e.g., , ) and introduces generalized deformations for stratified spaces, ensuring compatibility with fusion, reduction, and partial fusion. A new family of quasi-Hamiltonian moduli spaces is constructed from quivers, shown to be invariant under quiver homotopy and to satisfy a gluing formula under quiver composition. These ingredients culminate in a 2D topological quantum field theory valued in quasi-Hamiltonian spaces, where cobordisms correspond to fused/reduced quasi-Hamiltonian data built from copies of . The framework provides a multiplicative-to-additive passage for important geometric objects and yields a robust TQFT structure linking quiver combinatorics to cobordism data.

Abstract

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 deforms to the implosion of the cotangent bundle with applications to the master moduli space of -flat connections.\\ In parallel, we construct a topological quantum field theory , where is the category of quasi-Hamiltonian manifolds. To each cobordism , we associate a quasi-Hamiltonian space built from the fusion product of copies of the double 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.
Paper Structure (26 sections, 25 theorems, 81 equations, 2 figures)

This paper contains 26 sections, 25 theorems, 81 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Main results
  3. Notations and conventions
  4. Deformation of Hamiltonian quasi-Poisson $G$-spaces
  5. Deformation of quasi-Poisson manifolds
  6. Compatibility with fusion
  7. Fundamental examples
  8. The group $G$
  9. Non-degenerate Hamiltonian quasi-Poisson manifolds
  10. Deformation theory of imploded quasi-Hamiltonian manifolds
  11. Context
  12. Partial fusion product
  13. Deformation of partial fusion products
  14. Deformation of reductions
  15. Examples of generalized deformations
  16. ...and 11 more sections

Key Result

Theorem 1.1

Let $G$ be a compact Lie group equipped with an $\mathrm{Ad}$-invariant symmetric bilinear form $\langle\cdot,\cdot\rangle_{\mathfrak{g}}$. Then $G$ admits a smooth Poisson deformation to its dual $\mathfrak{g}^*$.

Figures (2)

  • Figure 1: The gluing operation $\star$ on oriented quivers: $\partial\Gamma_1^+ = \partial\Gamma_2^-$ becomes an interior vertex in $\Gamma_1 \star \Gamma_2$.
  • Figure 2: The homotopy move on quivers: contracting an edge $e_0$ between two interior vertices $v_1, v_2 \in \Gamma_{\mathrm{int}}$ yields a homotopy-equivalent quiver.

Theorems & Definitions (54)

  • Theorem 1.1
  • Definition 1.2
  • Definition 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Definition 2.1
  • Remark 2.2
  • Proposition 2.3
  • Lemma 2.4
  • ...and 44 more