# Central charges c and topological ground state degeneracy GSD

A 2+1D topological field theory (topologically ordered states), implies that the topological ground state degeneracy (GSD) on $T^2$ torus (2D manifold without boundary). For example a level k U(1) Chern-Simons theory implies a GSD$=k$.

If we put the topological field theory on a 2D manifold with 1D boundary, we expect 1+1D gapless edge states; and there are central charges $c$, which roughly measures the degree of freedom of the gapless edge states.

My question is: are there some explicit formula relates: topological ground state degeneracy GSD and central charges c?

Say,

$$\text{GSD}=\text{GSD}(c,\dots)$$

and

$$c=c(\text{GSD},\dots)$$

here $\dots$ are other possible data. RHS are the desired functions of my questions. It will be better to take some examples of non-Abelian topological field theory (topologically ordered states) to test its formula's validity.

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Aren't there two different notions of central charge involved here - one is the chiral central charge of the boundary and the other is the topological central charge of the bulk TQFT? –  Matthew Titsworth Dec 26 '13 at 6:11