# Relations between quantum groups at roots of unity, modular representation theory, and physics

I understand that quantum groups at roots of unity are related to physics because they are used in the construction of Reshetikhin-Turaev invariants, conjectured by Witten. Are there other relations of quantum groups at roots of unity to physics? Also, modular representation theory of Lie algebras is related to quantum groups at roots of unity via Andersen-Jantzen-Soergel. Modular representation theory is a very active area of research (cf. work of Lusztig, Bezrukavnikov, Williamson, and others), and I am wondering if there are relations between results/questions in this area and physics.

• Witten conjectured that every unitary modular tensor category would come from a Chern-Simons-Witten TQFT. Quantum groups at roots of unity provide an algebraic setting for studying anyons and topological phases of matter. Such phenomena are (expected to be) relevant to various physical systems; most notably the fractional quantum Hall effect. A further application of these ideas is in the construction of a 'topological quantum computer' or memory. There's a lot of literature on this topic and several related questions on MO. – S Valera Jun 1 at 14:58
• @SValera Thank you very much! 1) Would you be able to give some links (to preprints, papers, books, questions on MO) about these applications (to anyons, topological phases of matter, fractional quantum Hall effect, topological quantum computers) of quantum groups at roots of unity that you mention? 2) Would you be able to tell if/how the existing research/questions on modular representation theory of Lie algebras have a bearing on these physical applications via the equivalence of certain categories of representations of Lie algebras in pos. char. and quantum groups at roots of unity? – Yellow Pig Jun 1 at 16:07

Modular representations (representations in spaces over a field of nonzero characteristics) have been used in physics by Felix Lev to construct a quantum theory that is based on a finite number field (rather than on $$\mathbb{C}$$).

• Thank you very much! I am sorry, my education is in mathematics rather than physics, but may I ask if there is much work in physics which uses these ideas of F. Lev? – Yellow Pig Jun 1 at 11:17
• Also, sorry, I saw that some articles on this topic by F. Lev are posted on viXra, which seems to suggest that they may not reflect a very mainstream point of view in physics. Again, my understanding of physics is quite limited, so it would be great to receive a clarification from an expert. – Yellow Pig Jun 1 at 11:38
• The articles which Carlo mentions are all published in journals to be fair, I think the majority of prolific academics have some preprints which never see publication for whatever reason (usually just because of not being able to find the right journal for them). – Hollis Williams Jun 1 at 13:53
• While true, the first two of those papers have about 30 citations, and only two of them are by authors other than Lev. I don't think there's much evidence that other authors have adopted these ideas. Chasing through the citations of the final paper seems more promising. – Mike Miller Jun 1 at 17:30

The area seems to be very broad, let me just give some remarks, which are somewhat close to me.

Many interesting conformal field theories are "rational", in some simple cases it means that some parameter like a central charge is rational/integer $$k$$. By some reasons people consider expressions like $$(P)\exp( 2\pi i/k J(x) )$$, where $$J(x)$$ are some generators of symmetries of field theories—currents. And they appeared to be related to quantum groups. So if $$k$$ is an integer, then $$\exp(2\pi i /k)$$ is a root of unity. So the point is that roots of unity are related to "rationality" of some field theories.

The oversimplified example is just to consider canonical commutation relations: $$[X,Y] = 2\pi i/k$$, which after exponentiation gives a quantum-group-style relation: $$\exp(X)\exp(Y) = q\exp(Y)\exp(X)$$, for $$q=\exp(2\pi i/k)$$—so $$q$$ will be a root of unity for $$k$$ integral. (Although this is oversimplified, from some very high-level point of view the whole story is about it.)

Probably the most famous example is the Kazhdan–Lusztig equivalence between the category of certain

• integrable representations of the Kac–Moody algebra at a negative level

• and the category of (algebraic) representations of the "big" (a.k.a. Lusztig's) quantum group.

So here you see that we need "negative level", i.e. central charge $$k$$ is a negative integer, and what you get is a quantum group at $$q = \exp(2\pi i/k)$$—a root of unity. (I might forget to shift $$k$$ by dual Coxeter number.)

I guess mathematical proof does not use explicitly calculations like I mentioned above—just take "currents" $$J(x)$$—and show that $$P\exp( J(x)/k)$$ generate quantum group, but that's what some physicists were doing.

To put that idea in the right framework - let us think of the famous Drinfeld–Kohno theorem which says that the monodromy of the representation of the Knizhnik–Zamolodchikov equation is given by the corresponding quantum group. Again you can see that integer values of $$k$$ would correspond to root of unity for $$q$$, by the trivial reason that monodromy locally is given by exponent. In some sense that statement is closely related to Kazhdan–Lusztig theorem—the KZ-equation is given by "currents" $$J(x)$$ in tensor product of evaluation modules, one considers its Pexp (i.e. monodromy) and gets the quantum group.

Another example: in Integrable Structure of Conformal Field Theory III. The Yang-Baxter Relation, V. V. Bazhanov, S. L. Lukyanov, A. B. Zamolodchikov explicitly construct quantum-group-like relations from the conformal field theory operators. For certain special values of parameters one can get quantum groups at roots of unity. If I remember correctly they exploit that in some papers.

• Would it be all right with you if I TeXed this? – LSpice Jun 3 at 21:54
• @LSpice you are welcome – Alexander Chervov Jun 3 at 21:55