Reshetikhin-Turaev construction



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

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


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 Freed4-3-2 8-7-6”.

Relation to Chern-Simons theory

The RT-construction for group GG is expected to be the FQFT of GG-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).


Original articles include

  • N. Reshetikhin, V. 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

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

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

Last revised on June 1, 2016 at 14:42:29. See the history of this page for a list of all contributions to it.