I am trying to reproduce result results from an article about genus 2 partition function in conformal field theory.
At some point, the author state that, for a general conformal field theory, the currents satisfy the usual commutation relations, with a normalization of the current :
$[J_m^a,J_n^b] = if_{abc}J_{m+n}^c + m\delta_{ab}\delta_{m,-n}$
Just after, he states that one find, in terms of those structure constant,
$\sum_{a}Tr_{\mathcal{H}_{1}}(J_{-1}^aJ_1^a)=N = dim(\mathcal{H_1})$
And
$\sum_{a}Tr_{\mathcal{H}_{1}}(J_{0}^aJ_0^a)=-\sum_{abc}f_{abc}f_{acb}$
I tried many different things in order to recover those results but I don't find a convincing way. The most promising way was to use the OPE of two CFT currents inside a double integral as :
$J_n^aJ_m^a = \oint\frac{dz}{2\pi i}z^nj^a(z)\oint\frac{dw}{2\pi i}w^mj^a(w)$
with the OPE given by
$j^a(z)j^b(w)\sim\frac{1}{(z-w)^2}+\sum_{c}\frac{if^{abc}}{z-w}j^c(w)$
I arrived to $J_n^aJ_m^a = m\delta_{m+n,0}$ but I am really not sure about the result. Nevertheless, I really don't understand how a sum over two structure constant can appear .
