TCFT Elliptic Objects – arXiv:1202.4118

This paper by Yasha Savelyev caught my eye. It appears to blend together ideas from the Segal-Stolz-Teichner approach to elliptic cohomology and Costello’s notion of topological conformal field theories. 

The Stolz-Teichner approach to elliptic cohomology (which builds on Graeme Segal’s original proposal) in short is that (supersymmetric) extended 2D quantum field theories (of some correct flavor such as conformal or Euclidean) give a model tmf (topological modular forms). Specifically by passing to concordance classes we get a homotopy functor which is supposed to be tmf, at least conjecturally.

This paper builds what you might call a toy model of the Stolz-Teichner extended 2D quantum field theories by working with a target consisting of doctored up dg-bicategories. It adds in an open-closed structure and is very reminiscent of Costello’s topological conformal field theories. It is also constructed over a space X. Instead of passing to concordance classes, Savelyev passes to equivalence classes of field theory, then quotients by the field theories over a point, and finally group completes.

I haven’t fully digested this work yet. The main result seems to be that this functor is a homotopy functor which is representable. Of course one would want to go a bit further with this. For example do these spaces assemble into a cohomology theory? It is not clear.  It is also not clear to me that this can be directly connected to the Stolz-Teichner program, but this work does bring together a number of the important ingredients.

ArXiv:1202.4118

abstract

This paper develops a topological conformal field theory analogue of the Segal-Stolz-Teichner project for geometric construction of elliptic cohomology. In our context, we replace the theory Von-Neuman algebra bimodules and Connes fusion by differential graded category bimodules, and their natural categorical tensor product. At the same time, we extend the notion of bicategory in a geometrically natural fashion, and using this we show that some functors on the homotopy category of topological spaces coming from topological conformal field theory are representable.

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