gr.group theory - Schur Multipliers

The group $H^2(G,mathbb{C}^times)$ plays a rôle in orbifold conformal field theory and is usually known as the discrete torsion group. In fact, in this context one actually needs the explicit cocycle and for the case of a finite simple abelian group it is very easy to compute explicitly.

Let $varepsilon: G times G to mathbb{C}^times$ be the cocycle. Without loss of generality one can normalise it so that

$$varepsilon(0,g)=varepsilon(g,0) = 1$$
for all $g in G$. With this normalisation the cocycle conditions become, in addition, the following:
$$varepsilon(g,g)=1 quad varepsilon(g,g')= varepsilon(g',g)^{-1}$$
and
$$varepsilon(g_1+g_2,g) = varepsilon(g_1,g)varepsilon(g_2,g)$$
from where it follows that if $G$ has order $N$, then for all $g,g' in G$,
$$varepsilon(g,g')^N = 1$$

Let $G = mathbb{Z}/N_1 times cdots times mathbb{Z}/N_k$ be a finite simple abelian group and let $alpha_i$ be a generator of $mathbb{Z}/N_i$, so that we can write any element of $G$ as a sum $sum_i n_i alpha_i$ where $n_i = 0,1,ldots,N_i-1$.

Then one finds that all cocycles are given in terms of $B_{ij} = -B_{ji}$ taking the possible values $0,1,ldots,mathrm{gcd}(N_i,N_j)-1$, by the formula

$$varepsilon(sum_i n_ialpha_i,sum_j m_jalpha_j) = exp 2pisqrt{-1}sum_{i,j} frac{B_{ij} n_im_j}{mathrm{gcd}(N_i,N_j)}$$

It is the bilinear $B_{ij}/mathrm{gcd}(N_i,N_j)$ which is called the discrete torsion. It should be emphasised that torsion here is by analogy with the torsion of a connection in differential geometry and not with torsion as in group theory.

If you google "discrete torsion" and "orbifold" you might find suitable references, just like this paper of Vafa and Witten.

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