arXiv subjects

Mohamed Moussadek Maiza

Publications and source records attributed to Mohamed Moussadek Maiza.

5 recordsLinked to original sources

arXiv2026

Nahm-Kirchhoff Moduli Spaces, Hyperkähler Quotient and 2D Field theories

To an oriented quiver with boundary $Γ$ and a compact connected Lie group $G$ we associate a Nahm-Kirchhoff moduli space $\mathcal{M}_{\mathbb{H}}(Γ)$: solutions of Nahm's equations on the edges with Kirchhoff matching at interior vertices, modulo boundary-trivial gauge transformations. For connected $Γ$, $\mathcal{M}_{\mathbb{H}}(Γ)$ is a smooth manifold of dimension $4(|E|-|Γ_{\mathrm{int}}|)\dim G$ with a hyperkähler structure arising as an infinite-dimensional hyperkähler quotient. Via a Kempf-Ness argument on the Cartan-Hadamard manifold $G_{\mathbb{C}}/G$, we prove a quiver analogue of Donaldson's theorem identifying it with the complex-symplectic quotient $\mathcal{M}_{\mathbb{C}}(Γ)$ of the complex Nahm equations by the complexified gauge group. This quotient is independent of edge lengths and isomorphic to a finite-dimensional complex-symplectic quotient of $(T^*G_{\mathbb{C}})^E$, while the hyperkähler metric depends on them: uniform rescaling acts by a homothety, and the transported metrics genuinely vary. By cutting and gluing, $Γ\mapsto\mathcal{M}_{\mathbb{C}}(Γ)$ is a 2D TQFT valued in complex Hamiltonian manifolds, à la Moore-Tachikawa, whereas the hyperkähler refinement is not functorial on $\mathbf{Cob}_2$: spines of a cylinder with different lengths give non-isomorphic spaces. Retaining lengths, these spaces form a metric field theory, a functor of non-unital monoidal categories from metric cobordisms to hyperkähler manifolds composed by hyperkähler reduction; over each cone of $M_{g,n}^{\mathrm{trop}}$ they give a family with fixed complex-symplectic fibre and varying metric. Finally, harmonic gauge representatives converge as an internal edge collapses, and on a long edge with regular semisimple Higgs field the Kempf-Ness minimizer is exponentially confined near its centralizer; the limiting metrics are left open.

math.DG↗

arXiv2026

Dirac geometry, deformation theory and shifted symplectic geometry

We develop a Dirac deformation theory that interpolates between twisted Dirac geometry and Poisson geometry, and prove that this deformation is compatible with the principal structural operations of Dirac geometry: reduction along strong Dirac maps, integration to quasi-symplectic groupoids, and Morita equivalence of Lie algebroids. The fundamental example is the deformation of the Cartan--Dirac structure $L_G$ on a compact Lie group~$G$ to the Kirillov--Kostant--Souriau Poisson structure on $\mathfrak{g}^*$, and its lift to a deformation of the quasi-symplectic groupoid $D(G)\rightrightarrows G$ to the symplectic groupoid $T^*G\rightrightarrows\mathfrak{g}^*$. As applications, we obtain a uniform deformation theory recovering, as special cases, the deformation of quasi-Hamiltonian to Hamiltonian reduction along conjugacy classes, the Steinberg and Sevostyanov slices to their additive (Kostant, Slodowy) counterparts, the multiplicative parabolic and unipotent reductions, the quasi-Hamiltonian implosion to symplectic implosion, and the multiplicative Moore--Tachikawa varieties to their additive analogues. In the language of quasi-symplectic groupoid presentations of $1$-shifted symplectic stacks, our main reduction theorem yields a smooth deformation of the corresponding reduced spaces.

math.SG↗

arXiv2026

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 $D(G)_\imp$ 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.

math.SG↗

arXiv2025

Lax-Kirchhoff moduli spaces and Hamiltonian 2D TQFT

We introduce the Lax-Kirchhoff moduli space associated with a finite quiver $Γ$ and a compact connected Lie group $G$. On each oriented edge we consider the Lax equation $\dot{A}_1 + [A_0, A_1] = 0$ and impose a Kirchhoff-type matching condition for the fields $A_1$ at interior vertices. Modulo gauge transformations trivial on the boundary, this yields a moduli space $\mathcal{M}(Γ)$. We prove that $\mathcal{M}(Γ)$ is a finite-dimensional smooth symplectic manifold carrying a Hamiltonian action of $G^{\partialΓ}$ whose moment map records the boundary values of $A_1$. Analytically, we construct slices for the infinite-dimensional gauge action and realize $\mathcal{M}(Γ)$ by Marsden-Weinstein reduction. For the quiver consisting of a single edge, we recover the classical identification $\mathcal{M} \cong T^*G$. In general, we identify $\mathcal{M}(Γ)$ with a symplectic reduction of $T^*G^E$ by $G^{Γ_{\mathrm{int}}}$, where $E$ is the set of edges and $Γ_{\mathrm{int}}$ is the set of interior vertices. We further show that $\mathcal{M}(Γ)$ is invariant under quiver homotopies, implying that it depends only on the surface with boundary obtained by thickening $Γ$. We then assemble these spaces into a two-dimensional topological quantum field theory valued in a category of Hamiltonian spaces.

math.DG↗

arXiv2025

Deformations of quasi-Hamiltonian spaces

We introduce a notion of deformations of quasi-Hamiltonian $G$-spaces to Hamiltonian $G$-spaces and provide several examples. In particular, we show that the double $G \times G$ of a Lie group, viewed as a quasi-Hamiltonian $G \times G$-space, deforms smoothly to the cotangent bundle $T^*G$. Likewise, any conjugacy class of $G$ sufficiently close to the identity deforms to a coadjoint orbit. We further show that the moduli space of flat $G$-connections on a compact oriented surface of genus $g$ with $r+1$ boundary components deforms to $T^*G^{r+g}$.

math.SG↗