A framework for geometric field theories and their classification in dimension one
This paper introduces a framework in terms of stacks to define bordisms with geometric structures in quantum field theory. A ``geometry'' on \(d\)-manifolds is a stack \(\mathcal{G}\rightarrow \operatorname{Man}^d\) on the site \(\operatorname{Man}^d\) of manifolds of dimension \(d\). The geometric bordism category \(\mathcal{G}\operatorname{Bord}\) has as objects \(X\in \operatorname{Man}^d\) together with a \(\mathcal{G}\)-structure on \(X\), that is, an object of the groupoid \(\mathcal{G}(X)\) (the subcategory lying over \(\operatorname{id}_X\)). The geometric bordism category becomes ``smooth'' considering simplicial stacks or a complete Segal object in the category of stacks. Therefore, \(\mathcal{G}Bord_n\) consists of \(d\)-dimensional manifolds \(X\) together with a collection of marked hypersurfaces \(Y_k\subset X\), for \(k=0,\ldots,n\), where the composition is encoded by forgetting the marked hypersurfaces. The site \(\operatorname{Man}^d\) is promoted to a category of submersions \(X\rightarrow S\) and morphisms of submersions in order to have a suitable Grothendieck topology. A \(d\)-dimensional field theory with geometry \(\mathcal{G}\) is a symmetric monoidal smooth functor \(Z:\mathcal{G}\operatorname{Bord}\rightarrow \operatorname{Vect}\), where \(\operatorname{Vect}\) is the smooth category of finite dimensional vector bundles on \(\operatorname{Man}\). In dimension one, the papers shows that the groupoid of \(1\)-dimensional oriented topological field theories over \(M\) is equivalent to the groupoid of finite-dimensional vector bundles with connections over \(M\). A similar result is proved for the unoriented case.
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