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.
Categories: math.SG math.AG math.RT