nLab Reshetikhin-Turaev construction

Contents

Context

Functorial quantum field theory

functorial quantum field theory

Contents

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 Freed4-3-2 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:

Textbook accounts:

Review:

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

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.