Modular Tensor Categories from Chern-Simons Plus Simple Current Condensation


Unpublished work with Alexander Verhaeghe.


The question we wanted to ask is whether all modular tensor categories (MTCs) could be constructed from products of the classic Chern-Simons theories group G at level k (G=A_n,B_n,C_n,D_n for any n, or E_6,E_7,E_8,F_2,G_2 or U(1)_k with even k) plus condensation of simple currents only. A complete list of MTCs is is given here up to rank 12.

Using the program Kac we were able to generate all MTCs up to rank 7, and only a few are missing at rank 8. Higher rank we did not pursue too much for lack of time, and we list only partial results.

We highly suspect that it is not possible to generate *all* of the MTCs this way, but there is no proof that we cannot do so, and we were surprised how many we could generate.

Here is a list of how these MTCs can be constructed using the notation of the program Kac. Roughly the structure of the command is ('[h or g] G n k) g meaning with positive level, h is negative level. G_n is the group. k is the absolute value of the level. With the exception if U(1) is the group, "u" is listed and n is not listed and only the level k is given. If multiple commands are issued these Chern-Simons theories are multiplied together. When "Condensation" is listed as true, then a simple current (or more than one simple current is condensed). The particular currents to condense are listed by number given by the program Kac.

In the following table, each target MTC is listed by its T-matrix and central charge. For such small rank system, the T matrix and central charge (mod 8) give a unique descritpion of the MTC.

For example, consider the entry in the table

Matrix 12: T = (0, 1/7, 5/7)
Central Charge = 48/7
Matched with: Command: ('g c 5 1', 'h u 18')
Condensation: True, labels: ['0 12', '1 3']

This is the 12th MTC of rank 3 in the list. It has three simple objects, identity plus two fields with twists exp(2 pi i h) with h=1/7 and 5/7. (mod 1) and has central charge 48/7 (mod 8). It can be generated on Kac by issuing the commands

g c 5 1
h u 18

This generates the theory (C5 level 1) times (u1 level 18). Then we issue the condensation
current 0 12
current 1 3

Which then gives the desired MTC. Note this is not a unique construction. There are many other constructions that might give the same final result.

The complete table of our results is here .