It was never clear to me how to actually obtain the modes of an energy-momentum tensor in string theory.

Let's consider the energy-momentum tensor $$ T_{\psi}(z) = \frac{1}{2}: (\partial_z \psi^{\mu}(z)) \psi_\mu (z): = \sum_{m \in z} L_m z^{-m-2}.$$

How do I derive that the modes are given by $$ L_{m \neq 0} = \frac{1}{2} \sum_{r \in z} \psi_{m-r}^\mu \psi_r ^\nu \eta_{\mu \nu}.$$

First I was thinking about just computing the OPE $$ \frac{1}{2}: (\partial_z \psi^{\mu}(z)) \psi_\mu (z):$$ but this feels weird since both fields are a function of $z$.

  • $\begingroup$ Plug in the mode expansion for $\psi$. $\endgroup$
    – and008
    Mar 7 at 21:02

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