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. 

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The Classification of Two-Dimensional Extended Topological Field Theories

Ph.D. Dissertation (UC Berkeley) 2009

current version | arXiv

 

This work proves the cobordism hypothesis in dimension two. Specifically, this work classifies 2-dimensional extended topological field theories in terms of generators and relations. In this context, a topological field theory is a functor from a bordism category to some target category and an extended field theory is a higher categorical version of this. Why use higher categories? They allow you to mathematically encode the locality of the theory, something which is predicted from the physical point of view and which is very helpful mathematically.

The methods combined algebraic results on symmetric monoidal bicategories with a generalization of Cerf theory. This provides and alternate approach to the cobordism hypothesis to the one given by Hopkins and Lurie, and to date (Feb 2012) remains the only complete account.