Topological and conformal field theory as Frobenius algebras
Ingo Runkel, Jens Fjelstad, Jurgen Fuchs, Christoph Schweigert
备注：
23 pages, several figures, proceedings to the Streetfest (Canberra 07/2005); v2: section 2.4 expanded, version accepted for publication in Contemp. Math
Two-dimensional conformal field theory (CFT) can be defined through its correlation functions. These must satisfy certain consistency conditions which arise from the cutting of world sheets along circles or intervals. The construction of a (rational) CFT can be divided into two steps, of which one is complex-analytic and one purely algebraic. We realise the algebraic part of the construction with the help of three-dimensional topological field theory and show that any symmetric special Frobenius algebra in the appropriate braided monoidal category gives rise to a solution. A special class of examples is provided by two-dimensional topological field theories, for which the relevant monoidal category is the category of vector spaces.
In this article, we establish a connection between two models for $r$-spin structures on surfaces: the marked PLCW decompositions of Novak and Runkel-Szegedy, and the structured graphs of Dyckerhoff-Kapranov. We use these models to describe $r$-spin structures on open-closed bordisms, leading to a generators-and-relations characterization of the 2-dimensional open-closed $r$-spin bordism category. This results in a classification of 2-dimensional open closed field theories in terms of algebraic structures we term "knowledgeable $Λ_r$-Frobenius algebras". We additionally extend the state sum co…
We consider Frobenius objects in the category Span, where the objects are sets and the morphisms are isomorphism classes of spans of sets. We show that such structures are in correspondence with data that can be characterized in terms of simplicial sets. An interesting class of examples comes from groupoids. Our primary motivation is that Span can be viewed as a set-theoretic model for the symplectic category, and thus Frobenius objects in Span provide set-theoretic models for classical topological field theories. The paper includes an explanation of this relationship.
