Worldsheet Green function

A worldsheet Green function inverts the scalar kinetic operator after its zero modes have been removed. In a locally flat complex coordinate it satisfies $-(\pi {\alpha }^{\prime }{)}^{-1}\partial \bar{\partial }G(z,w)={\delta }^{(2)}(z-w)$ and is $G(z,w)=-({\alpha }^{\prime }/2)\log |z-w{|}^2$ up to harmonic and constant terms.