Lax-Kirchhoff moduli spaces and Hamiltonian 2D TQFT
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| author | Maiza, Mohamed Moussadek Mayrand, Maxence |
| author_facet | Maiza, Mohamed Moussadek Mayrand, Maxence |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_23567 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Lax-Kirchhoff moduli spaces and Hamiltonian 2D TQFT Maiza, Mohamed Moussadek Mayrand, Maxence Differential Geometry Symplectic Geometry 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. |
| title | Lax-Kirchhoff moduli spaces and Hamiltonian 2D TQFT |
| topic | Differential Geometry Symplectic Geometry |
| url | https://arxiv.org/abs/2510.23567 |