# How do I calculate the modular fusion category from a given Lie algebra and level in Chern-Simons theory?

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?

• 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. – Noah Snyder Sep 13 '18 at 22:13