QM Research Seminar: Canonical local to global extensions of topological field theories

Speaker: Adela Zhang (​University of Copenhagen)

Abstract:
With Barkan and Steinebrunner, we show that any open field theory extends canonically to an open-closed field theory whose value at the circle is the Hochschild homology object of the E_1-Frobenius algebras (i.e. E_1-Calabi-Yau objects) A associated to F. In particular, we show that the open-closed bordism category is obtained by formally adjoining iterated Hochchild homology to the open bordism category. As a corollary, we determine that the moduli spaces of surfaces is the space of universal formal operations on the Hochschild homology object of E_1-Frobenius algebras. This provides a space level refinement of previous work of Costello (over Q) and Wahl (over Z).
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Time permitting, I will also explain work with Andrea Bianchi on canonical local to global extensions of graph TFTs associated with E_\infty-Frobenius objects, i.e. in dimension infinity. In this case, we show that the graph cobordism category is obtained by formally adjoining to the \infty-category of graphs the factorization homology of the universal E_\infty-Frobenius algebra over any sapce X.