The Moore-Tachikawa conjecture via shifted symplectic geometry

Peter Crooks, Maxence Mayrand
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Abstract

We use shifted symplectic geometry to construct the Moore-Tachikawa topological quantum field theories (TQFTs) in a category of Hamiltonian schemes. Our new and overarching insight is an algebraic explanation for the existence of these TQFTs, i.e. that their structure comes naturally from three ingredients: Morita equivalence, as well as multiplication and identity bisections in abelian symplectic groupoids. Using this insight, we generalize the Moore-Tachikawa TQFTs in two directions. The first generalization concerns a 1-shifted version of the Weinstein symplectic category $\mathbf{WS}_1$. Each abelianizable quasi-symplectic groupoid $\mathcal{G}$ is shown to determine a canonical 2-dimensional TQFT $\eta_{\mathcal{G}}:\mathbf{Cob}_2\longrightarrow\mathbf{WS}_1$. We recover the open Moore-Tachikawa TQFT and its multiplicative counterpart as special cases. Our second generalization is an affinization process for TQFTs. We first enlarge Moore and Tachikawa's category $\mathbf{MT}$ of holomorphic symplectic varieties with Hamiltonian actions to $\mathbf{AMT}$, a category of affine Poisson schemes with Hamiltonian actions of affine symplectic groupoids. We then show that if $\mathcal{G} \rightrightarrows X$ is an affine symplectic groupoid that is abelianizable when restricted to an open subset $U \subseteq X$ statisfying Hartogs' theorem, then $\mathcal{G}$ determines a TQFT $\eta_{\mathcal{G}} : \mathbf{Cob}_2 \longrightarrow \mathbf{AMT}$. In more detail, we first devise an affinization process sending 1-shifted Lagrangian correspondences in $\mathbf{WS}_1$ to Hamiltonian Poisson schemes in $\mathbf{AMT}$. The TQFT is obtained by composing this affinization process with the TQFT $\eta_{\mathcal{G}|_U} : \mathbf{Cob}_2 \longrightarrow \mathbf{WS}_1$ of the previous paragraph. Our results are also shown to yield new TQFTs outside of the Moore-Tachikawa setting.
通过移位交映几何的摩尔-立川猜想
我们利用移调交映几何在哈密顿方案范畴中构建了摩尔-塔奇卡瓦拓扑量子场论(TQFTs)。我们的首要新见解是用代数解释这些 TQFTs 的存在,即它们的结构自然地来自三个元素:莫里塔等价性,以及无边交映群中的乘法和同分异构。利用这一洞察力,我们从两个方向推广了摩尔-立川 TQFT。第一个泛化涉及韦恩斯坦交映范畴 $\mathbf{WS}_1$ 的 1 移位版本。每个可abelianizable准交映群$\mathcal{G}$都被证明决定了一个规范的二维 TQFT$\eta_{mathcal{G}}:\mathbf{Cob}_2\longrightarrow\mathbf{WS}_1$。我们将开放的摩尔-立川 TQFT 及其乘法对应物作为特例加以复原。我们的第二个广义化是 TQFT 的亲和过程。我们首先把摩尔和立川的具有哈密顿作用的全形交映变量范畴 $\mathbf{MT}$ 放大到$\mathbf{AMT}$,这是一个具有哈密顿作用的仿射交映群的仿射泊松方案范畴。维特恩证明,如果 $\mathcal{G}\那么 $\mathcal{G}$ 就决定了一个 TQFT$eta_{\mathcal{G}} : \mathbf{Cob}_2 \longrightarrow \mathbf{AMT}$。更详细地说,我们首先设计了一个affinization过程,将$\mathbf{WS}_1$中的1-移位拉格朗日对应方案发送到$\mathbf{AMT}$中的哈密顿泊松方案。通过将这个affinization过程与前一段的TQFT $\eta_{\mathcal{G}|_U} : \mathbf{Cob}_2 \longrightarrow\mathbf{WS}_1$ 组合起来,就得到了TQFT。我们的结果还显示了在摩尔-立川设定之外的新 TQFT。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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