Central charge counting free Boson, and free fermions

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 No time to write trivial mathematical details. Might update this with details in the future.

 Recipe:

Consider first the Free boson first, apply a similar technique to Majorana, then get Dirac directly

    - Recall:

           - Two point function  equations:

 $G^{2} = \langle \phi(x) \phi(y) \rangle$

$ -g \partial_x^2 G^{(2)} (x,y) = \delta( x-y)$

$G^{2} = \langle \phi(x) \phi(y) \rangle =  \frac{1}{4 pi g} ln^2( x-y)$


The energy-momentum tensor can be computed:

$T = g(\partial_\mu \phi \partial_nu \phi - \frac{1}{2} \eta_{\mu \nu} \partial_\rho \phi \partial^\rho \phi )$


OPE is calculated as

$T(z)T(w) = (2 \pi  g)^2:\partial \phi(z) \partial \phi(z)::\partial \phi(w) \partial \phi(w):$


Read the central charge from the general form!


$T(z)T(w) = \frac{c/2}{ (z-w)^4} + \frac{2}{(z-w)^2} T(w) + \frac{\partial  T(w)}{z-w} +  . . $


Read the conformal dimension!

$T(z) \Phi(w, \bar(w)) =  (\frac{h}{(z-w)^2} + \frac{\partial}{z -w}) \phi(w,\bar(w))$



For free fermion


- Write Majorana action

- Get Majorana field

- Write OPE of Majorana field


c = 1 for free boson

c = 1/2 for free fermion






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