Free fermion

introduction

• $c=1/2$ (for $\psi$ real)
• $c=1$ (for \psi complex)

action

\(S= \int\!d^2x\, \psi^\dagger \gamma^0 \gamma^\mu \partial_\mu \psi= \int\!d^2z\, \psi^\dagger_R \bar\partial \psi_R + \psi_L^\dagger \bar\partial \psi_L\,\)

 

OPE of fermionic fields

• $\psi(z)\psi(w) \sim \frac{1}{(z-w)}$
• $\partial \psi(z) \psi(w) \sim -\frac{1}{(z-w)^2}$
• $\partial \psi(z) \partial \psi(w) \sim -\frac{2}{(z-w)^3}$

 

energy-momentum tensor

• $T(z)=-\frac{1}{2}:\psi(z)\partial \psi(z):=-\frac{1}{2}\left(\lim_{w\to z}\psi(z)\partial \psi(z)+\frac{1}{(z-w)^2}\right)$
• $T(z)\psi(w) \sim \frac{\psi(w)}{2(z-w)^2}+\frac{\partial \psi(w)}{(z-w)}$
• $T(z)\partial \psi(w) \sim \frac{\psi(w)}{2(z-w)^3}+\frac{3\partial \psi(w)}{2(z-w)^2}+\frac{\partial^2 \psi(w)}{(z-w)}$
• $T(z)T(w) \sim \frac{1}{4(z-w)^4}+\frac{2T(w)}{(z-w)^2}+\frac{\partial T(w)}{(z-w)}$

related items

• free massless boson
• ghost fields

computational resource

• https://drive.google.com/file/d/0B8XXo8Tve1cxc0c0WC1Xb0l1MDg/view

 

expositions

• exercise 5