# OPE of fermionic field bosonization in string theory, in Polchinski 10.3.12

In Polchinski's String Theory Vol. 2, equations 10.3.12 are

$$e^{iH(z)}e^{-iH(-z)}~=~\frac{1}{2z}+i\partial H(0)+2zT_B^H(0)+O(z^2)\tag{10.3.12a}$$

$$\psi(z)\bar\psi(-z)~=~\frac{1}{2z}+\psi\bar\psi(0)+2zT_B^\psi(0)+O(z^2)\tag{10.3.12b}$$

How are these two OPEs calculated, especially the second and third terms?

• Please consider explaining the notation in the equations and adding minimal context to make it accessible to people without access to Polchinski. Also, do you know how such OPE are calculated in general contexts, e.g. generic CFTs? Oct 15, 2014 at 10:36
• Sep 5, 2020 at 11:28

Simply do Taylor expansion of the left hand side. Expand both the exponential, and the field around $H(0)$ or $\psi(0)$, then the right hand follows naturally after plugging in definitions of $T_B$.