# What is the dictionary between Witten's (Chern-Simons) invariants and Reshetikhin-Turaev (quantum group) invariants?

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Using Chern-Simons theory, Witten defined invariants of a framed link in a 3-manifold that are intrinsically 3-dimensional. From a mathematical perspective there is some ambiguity surrounding how well-defined these invariants are (they involve a path integral over the space of flat connections), but are thought to generalise the Jones polynomial. Around the same time Reshetikhin and Turaev defined invariants of tangles/links and 3-manifolds using quantum groups (I assume anyone who attempts to answer this question knows how these are defined). In one of their papers they state: "We believe that our invariants may be viewed as a mathemematical realisation of Witten's program". Does anyone know to what extent the dictionary taking me from one viewpoint to the other is now established?

edited May 31, 2016

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A possible answer extracted from https://ncatlab.org/nlab/show/quantization+of+3d+Chern-Simons+theory

The 3d TQFT candidate for quantum CS theory in the form of the Reshetikhin-Turaev construction and the corresponding modular tensor category data is discussed in

Reshetikhin; Turaev, Invariants of 3-manifolds via link polynomials and quantum groups. Invent. Math. 103 (1991), no. 3, 547–597.

B. Bakalov & Alexandre Kirillov, Lectures on tensor categories and modular functors AMS, University Lecture Series, (2000)

Stephen Sawin, Quantum groups at roots of unity and modularity J. Knot Theory Ramifications 15 (2006), no. 10, 1245–1277 (arXiv:0308281)

Victor Ostrik, MO comment August 2014

answered Jun 1, 2016 by (1,105 points)

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