nLab
Reshetikhin-Turaev construction

Contents

Contents

Idea

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.

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 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).

References

Original articles:

Textbook accounts:

Review:

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

Relation to the Crane-Yetter model:

Last revised on May 23, 2021 at 01:17:32. See the history of this page for a list of all contributions to it.