# 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 construction 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

### 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 include

• Reshetikhin; 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.)

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