generalized tangle hypothesis

**functorial quantum field theory**
## Contents
* cobordism category
* cobordism
* extended cobordism
* (∞,n)-category of cobordisms
* Riemannian bordism category
* cobordism hypothesis
* generalized tangle hypothesis
* classification of TQFTs
* FQFT
* extended TQFT
* CFT
* vertex operator algebra
* TQFT
* Reshetikhin–Turaev model / Chern-Simons theory
* HQFT
* TCFT
* A-model, B-model, Gromov-Witten theory
* homological mirror symmetry
* FQFT and cohomology
* (1,1)-dimensional Euclidean field theories and K-theory
* (2,1)-dimensional Euclidean field theory
* geometric models for tmf
* holographic principle of higher category theory
* holographic principle
* AdS/CFT correspondence
* quantization via the A-model

The *generalized tangle hypothesis* is a refinement of the cobordism hypothesis.

The original *tangle hypothesis* was formulated in

- John Baez and James Dolan,
*Higher-dimensional Algebra and Topological Quantum Field Theory*1995 (arXiv)

as follows:

The $n$-category of framed $n$-tangles in $n+k$ dimensions is $(n+k)$-equivalent to the free weak $k$-tuply monoidal $n$-category with duals on one object.

In the limit $k \to \infty$, this gives:

The $n$-category $n Cob$ of cobordisms is the free stable $n$-category with duals on one object (the point).

In extended toplogical quantum field theory, which is really the representation theory of these cobordism $n$-categories, we expect:

An $n$-dimensional unitary extended TQFT is a weak $n$-functor, preserving all levels of duality, from the $n$-category $n Cob$ of cobordisms to $n Hilb$, the $n$-category of $n$-Hilbert spaces?.

Putting the extended TQFT hypothesis and the cobordism hypothesis together, we obtain:

An $n$-dimensional unitary extended TQFT is completely described by the $n$-Hilbert space it assigns to a point.

Further discussion can be found here:

- Bruce Bartlett,
*On unitary 2-representations of finite groups and topological quantum field theory*. PhD thesis, Sheffield (2008) (arXiv)

More recently Mike Hopkins and Jacob Lurie have claimed (see Hopkins-Lurie on Baez-Dolan) to have formalized and proven this hypothesis in the context of (infinity,n)-categories modeled on complete Segal spaces. See:

- Jacob Lurie,
*On the classification of topological field theories*(pdf)

where an (infinity,n)-category of cobordisms is defined and shown to lead to a formalization and proof of the *cobordism hypothesis*. Lurie explains his work here:

- Jacob Lurie,
*TQFT and the cobordism hypothesis*, videos of 4 lectures at the Geometry Research Group, Mathematics Department, University of Texas Austin.

Lecture notes for Lurie’s talks should eventually appear at the Geometry Research Group website.

The $k$-tuply monoidal $n$-category of $G$-structured $n$-tangles in the $(n + k)$-cube is the fundamental $(n + k)$-category with duals of $(M G,Z)$.

- $M G$ is the Thom space of group $G$.
- $G$ can be any group equipped with a homomorphism to $O(k)$. (comment)

- Jacob Lurie,
*TQFT and the Cobordism Hypothesis*(video, notes)

Revised on November 29, 2009 20:56:00
by Toby Bartels
(173.60.119.197)