The Moore-Tachikawa conjecture via shifted symplectic geometry

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.