The n-Category Café

Lurie on Extended TQFT

Posted by Urs Schreiber

Over at the Secret Blogging Seminar Noah Snyder is reporting on talks Jacob Lurie gave on extended TQFT, which is the theory of representations of $\infty$-categories of cobordisms (in contrast to ordinary TQFT, which is just representations of mere 1-categories of cobordisms):

Noah Snyder
Jacob Lurie on 2-d TQFT

From Noah’s notes, the talks mostly centered on the observation by John Baez and James Dolan

John C. Baez, James Dolan
Higher-dimensional Algebra and Topological Quantum Field Theory
arXiv:q-alg/9503002v2

that this should be essentially about representing the free stable $\infty$-groupoid and that hence such a representation is fixed by choosing just one object, the image of the point, with suitable dualities on it. See also our recent discussion about that here.

The extended TQFT would then be an assignment of this object to the point, and of the $n$-fold higher “trace” on this object to closed $n$-dimensional manifolds.

In his talk Lurie apparently talked about some new classification results on this. With a little luck, more details will percolate through to us eventually.

Incidentally, while typing this I am on my way back from Edinburgh to Sheffield to meet Bruce Bartlett again. Bruce is writing his PhD thesis, advised by Simon Willerton, on a beautiful description of (higher) Dijkgraaf-Witten theory – a finite group version of Chern-Simons theory – in this extended sense. He finds plenty of fascinating relations between these “higher traces” and finite group representation theory, providing a useful blueprint for and nice insights into what fully extended Chern-Simons theory has to eventually look like.