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 $𝒞$. It is something like the “square root” of the Turaev-Viro construction on $𝒞$.

In the case that $C$ is a category of positive energy representation?s 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

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 TQFT from compact Lie groups – 3d Chern-Simons as a fully extended TQFT.

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

References

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