We show that any 2D open field theory F extends canonically to an open-closed field theory whose value at the circle is the Hochschild homology of the E_1-Frobenius algebra associated to F. In particular, the open-closed bordism category is obtained by formally adjoining iterated Hochschild homology to the open 2D-bordism category. As a corollary, we identify the space of universal formal operations on the Hochschild homology of E_1-Frobenius algebras to be moduli spaces of punctured surfaces. This is joint work with Barkan and Steinebrunner, and provides a space level refinement of work of Costello (over Q) and Wahl (over Z).
In dimension ∞, we establish analogous extensions of E_∞-Frobenius algebras, which are corepresented by the ∞-category Gr of graphs, to the ∞-category GrCob of graph cobordisms between spaces. In this case, we show that GrCob is obtained by formally adjoining to Gr the factorization homology of the universal E_∞-Frobenius algebra over any space X. Thus GrCob parametrizes universal operations among such factorization homologies. This is joint work with Andrea <a href="http://Bianchi.

https://indico.math.cnrs.fr/event/15404/ » target= »_blank » title= »Bianchi.

https://indico.math.cnrs.fr/event/15404/ »>Bianchi.

https://indico.math.cnrs.fr/event/15404/