I am working out some commutation relations within a CFT. We know that the modes of the currents within a CFT commute as

$$ [J_m^a,J_n^b] = if_{abc}J_{m+n}^c + m\delta_{ab}\delta_{m,-n}, $$ Now, we know that a commutator is antisymmetric. Therefore, the relation

$$[J_m^a,J_n^b] = -[J_n^b,J_m^a]$$ must be true. It implies

$$\begin{align}[J_n^b,J_m^a] &= if_{bac}J^{c}_{m+n} + n\delta^{ab}\delta_{m,-n}\\ & = -if_{abc}J_{m+n}^c + n\delta^{ab}\delta_{m,-n}\end{align}$$, which imply $f_{abc}=-f_{bac}=\dots$.

Now, I have also a basis of states at weight 2 that I denote with W. I worked out the commutation relations of their mode using the formula (2.54) of the book "Introduction to Conformal Field Theory" by Blumenhagen and Plauschinn, that is

$$ [\phi_{(i)m},\phi_{(j)n}] = \sum\limits_kC^{k}_{ij}p_{ijk}(m,n)\phi_{(k)m+n} + d_{ij}\delta_{m,-n}\left(\matrix{m+h_i-1\\2h_1-1}\right)$$.

The definition of $p_{ijk}(m+n)$ is just a polynomial in m and n.

They go as follows

$$ \begin{eqnarray} [W_m^i,W_n^j] &=& \frac{c}{12}\delta^{ij}m(m^2-1)\delta_{m,-n}+(m+n)h_{ijk}W_{m+n}^k \\ &+&\frac{1}{6}\left(\frac{c}{2}\right)^2(m^2-mn+n^2-1)P_{ijk}J_{m+n}^{k}+ig_{\alpha}^{ij}V^{a}_{m+n}, \end{eqnarray} $$

Here, V is an orthonormal basis of states at weight $3$ and I introduced two set of structure constants $h_{ijk}$ and $P_{ijk}$. Now, since $[W_m^i,W_n^j]= -[W_n^j,W_m^i]$, we obtain clearly

$$h_{ijk}=h_{jik},\qquad P_{ijk}=-P_{jik},\qquad g^{ij}=-g^{ji}$$.

Does that make sense ??

Now, I also worked out the commutation relation between a current and one state at weight 2. I arrive to

$$ [J^i_m,W^j_n] = \frac{c}{2}P_{ijk}W_{m+n}^k + mK_{ijk}J^k_{m+n}, $$

Following the method from Blumenhagen, we obtain

$$\begin{align} [W^j_n,J^i_m] &= \frac{c}{2}P_{jik}W_{m+n}^k - mK_{jik}J^k_{m+n},\\ & = -\frac{c}{2}P_{ijk}W_{m+n}^j - mK{jik}J^k_{m+n}, \end{align} $$ using $P_{ijk}=P_{ijk}$, but it would imply that $K_{ijk}=K{jik}$. Does that make sense ?

My question is the following: I have trouble knowing which of those structure constants are anti-symmetric. I know for sure that the f's are, so we have $f_{abc}=-f_{bac}=\dots$.

Now, what about $P_{ijk}$ and $K_{ijk}$, are they also antisymmetric ?Are any structure constants of the theory antisymmetric or some may be symmetric under the exchange of two indices?

Thank you very much!


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