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

Question

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? — ACuriousMind, Oct 15 '14 at 10:36

score 0 · Accepted Answer · answered Nov 06 '14 at 08:16

0

I've got the answer by myself.

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 $T_B$.

answered Nov 06 '14 at 08:16

Han Yan

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

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