Abstract
This note is devoted to report some results proven in [5, 8] and some work in progress [6] concerning the relation between groupoids and Frobenius algebras specialized in the case of Poisson sigma models with boundary. We prove a correspondence between groupoids in Set and relative Frobenius algebras in Rel, as well as an adjunction between a special type of semigroupoids and relative H*-algebras. The connection between groupoids and Frobenius algebras is made explicit by introducing what we called weak monoids and relational symplectic groupoids, in the context of Poisson sigma models with boundary and in particular, describing such structures in the extended symplectic category and the category of Hilbert spaces. This is part of a joint work with Alberto Cattaneo and Chris Heunen.
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Notes
- 1.
\(\mathbf{Symp }^{ext}\) is not properly speaking a category, since the composition of canonical relations is not in general a smooth manifold; some transversality conditions are required. For our purposes, the smoothness of the composition of canonical relations will be guaranteed from the defining axioms of the relational symplectic groupoid.
- 2.
A dagger Frobenius algebra on the category Hilb of finite dimensional Hilbert spaces corresponds to the usual notion of Frobenius algebra.
- 3.
The symbol \(\nrightarrow \) denotes that we are considering relations instead of maps as morphisms.
- 4.
More precisely, in \(\mathbf {Symp} ^{ext}\), by a morphism between two symplectic manifolds \((M,\omega _M)\) and \((N, \omega _N)\) we mean a pair \((X, p)\) where \(X\) is a smooth manifold, \(p\) is a smooth map from \(X\) to \(M\times N\), such that \(dp\) is surjective and \(T_x(\mathfrak {Im}(p))\) is a Lagrangian subspace of \(T(p(x))((M,\omega _M)\times (N, -\omega _N)), \forall x\in X\).
- 5.
Here \(*\) denotes path concatenation.
- 6.
In this case that we are considering complex valued functions we set \(\varepsilon = i\hbar / 2\).
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Contreras, I. (2015). Groupoids, Frobenius Algebras and Poisson Sigma Models. In: Calaque, D., Strobl, T. (eds) Mathematical Aspects of Quantum Field Theories. Mathematical Physics Studies. Springer, Cham. https://doi.org/10.1007/978-3-319-09949-1_12
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