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

Question

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.

Answer 1

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 $\mathbb{C}$).

• F.M. Lev, Finiteness of physics and its possible consequences.
• F.M. Lev, Modular representations as a possible basis of finite physics.
• F.M. Lev, Why is quantum physics based on complex numbers?

Answer 2

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.