In Chern-Simons theory, one has modular fusion categories that are labelled by a Lie algebra and a "level", e.g. $SU(2)_2$ ("$SU(2)$ level $2$").

Physically this modular fusion category describes the TQFT/anyon statistics of the Chern-Simons gauge theory.

Now I think of a modular fusion category as an $F$ and an $R$ tensor, where the $F$ tensor has $6$ indices labelled by simple objects and $4$ fusion space indices, and $R$ has $3$ indices labelled by simple objects and $2$ fusion space indices. Similarly, one can think of a Lie algebra in terms of the "structure coefficients" of the Lie bracket in some chosen basis, yielding a tensor with three indices. The level I'm not sure how to formulate in such a constructive language (this is somehow part of the question).

My question: Is there a constructive way to calculate the $F$ and $R$ tensor from the Lie algebra structure coefficients? Constructive in the sense that e.g. there is a computer program that takes tensors as inputs and tensors as outputs. And what would that construction look like?

  • 2
    $\begingroup$ The F-matrix goes by the name “quantum 6-j symbols.” The R-matrix you can compute from, well, the “R-matrix.” Neither is particularly easy to do without spending a lot of time with combinatorial representation theory. For su(2) this is all nicely worked out with pictures in Kauffman-Lins. $\endgroup$ – Noah Snyder Sep 13 '18 at 22:13

There are two constructions of the modular fusion category. The conformal field theory approach is to take representations of the affine Kac-Moody algebra of given level and define a tensor product. This is not the naive one where you would add the levels. Then make it modular.

The other is to take the category of type I representations of the quantised enveloping algeba and specialise q to a root of unity (depending on the level). Then take the semisimplification of this category; i.e. quotient by the tensor ideal of negligible morphisms.

I believe Lusztig proved that these give the same result.

  • $\begingroup$ "Tensor structures arising from affine lie algebras" has the equivalence $\endgroup$ – Phil Tosteson Nov 2 '18 at 23:07

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.