Reshetikhin-Turaev construction

The Reshetikhin-Turaev construction is the FQFT construction of a 3d TQFT from the data of a modular tensor category $\mathcal{C}$. It is something like the “square root” of the Turaev-Viro model on $\mathcal{C}$.

In the case that $C$ is a category of positive energy representations of a loop group $\Omega G$ of a Lie group $G$, then this algebraically defined QFT is thought to be the result of quantization of Chern-Simons theory over the group $G$.

The Reshetikhin-Turaev model is a boundary field theory of the 4d TQFT Crane-Yetter model (Barrett&Garci-Islas&Martins 04, theorem 2) Related discussion is in Freed “4-3-2 8-7-6”.

The RT-construction for group $G$ is expected to be the FQFT of $G$-Chern-Simons theory, though a fully explicit proof of this via quantization is currently not in the literature.

See at *quantization of Chern-Simons theory* for more on this.

The Fuchs-Runkel-Schweigert-construction builds from the RT-construction explicitly the rational 2-dimensional 2d CFT boundary theory (see at *holographic principle*).

Original articles include

- Reshetikhin; Turaev,
*Invariants of 3-manifolds via link polynomials and quantum groups*. Invent. Math. 103 (1991), no. 3, 547–597. (pdf)

A standard textbook account is

- B. Bakalov & Alexandre Kirillov,
*Lectures on tensor categories and modular functors*AMS, University Lecture Series, (2000) (web).

(See the dedicated page *Help me! I'm trying to understand Bakalov and Kirillov* for help with understanding the computations in this book.)

See also

- Kevin Walker,
*On Witten’s 3-Manifold Invariants*, (old version draft of new version)

Discussion that relates the geometric quantization of $G$-Chern-Simons theory to the Reshetikhin-Turaev construction of a 3d-TQFT from the modular tensor category induced by $G$ is in

- Jørgen Andersen,
*A geometric formula for the Witten-Reshetikhin-Turaev Quantum Invariants and some applications*(arXiv:1206.2785)

and references cited there.

- Alain Bruguières, Alexis Virelizier,
*Hopf diagrams and quantum invariants*, math.QA/0505119;*Categorical centers and Reshetikhin-Turaev invariants*, arxiv/0812.2426

The relation to the Crane-Yetter model was discussed in

- John Barrett, J. Garcia-Islas, João Faria Martins,
*Observables in the Turaev-Viro and Crane-Yetter models*, J. Math. Phys. 48:093508, 2007 (arXiv:math/0411281)

Revised on December 27, 2015 17:36:31
by Manuel Bärenz?
(91.17.130.248)