nLab Reshetikhin-Turaev construction

Idea

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 $G$.

Properties

As a boundary of the Crane-Yetter model

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”.

Relation to Chern-Simons theory

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.

Relation to conformal field theory

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

References

Original articles:

• Nikolai Reshetikhin, Vladimir Turaev, Invariants of 3-manifolds via link polynomials and quantum groups. Invent. Math. 103 (1991), no. 3, 547–597. (doi:10.1007/BF01239527, pdf)

• Vladimir Turaev, Modular categories and 3-manifold invariants, International Journal of Modern Physics B Vol. 06, No. 11n12, pp. 1807-1824 (1992) (doi:10.1142/S0217979292000876)

  (introducing modular tensor categories)

Textbook accounts:

• Vladimir Turaev, Quantum invariants of knots and 3-manifolds, de Gruyter Studies in Mathematics 18 Walter de Gruyter & Co., Berlin, 1994 (doi:10.1515/9783110435221)

• Bojko Bakalov, Alexandre Kirillov, Lectures on tensor categories and modular functors AMS, University Lecture Series, (2000) (ISBN:978-1-4704-2168-7, webpage).

Review:

• Wikipedia, Reshetikhin-Turaev invariant

See also:

• Jørgen Ellegaard Andersen, William Petersen, Construction of Modular Functors from Modular Tensor Categories (arXiv:1602.04907)

• Jørgen Ellegaard Andersen, Construction of the Witten-Reshetikhin-Turaev TQFT from conformal field theory (arXiv:1110.5027)

• 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

Relation to the Crane-Yetter model:

• 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)