Worldsheet Green function

A worldsheet Green function inverts the quadratic kinetic operator of a scalar field on a string worldsheet, after removing any constant worldsheet zero mode. For one Euclidean target coordinate, fix $z=x+iy$, area measure $dA=dxdy$, andThe Polyakov action isThe local planar Green function isIndeed, $\partial \bar{\partial }\log |z-w{|}^2=\pi {\delta }_A(z-w)$ for the stated area measure. The arbitrary scale $\ell$ changes only an additive constant.

A common complex-coordinate convention uses $d^{2}z=2dA$ and ${\delta }_{d^{2}z}={\delta }_A/2$. With that convention the same kernel is inverted using $-(\pi {\alpha }^{\prime }{)}^{-1}\partial \bar{\partial }$ and the correspondingly normalized delta function. Keeping the area and delta conventions together prevents a factor-of-two mismatch. ^[ref]

On a closed compact worldsheet, the constant worldsheet zero mode obstructs an unrestricted inverse. Its removal replaces the delta source by the delta source minus the normalized zero mode projection. Boundary conditions and global geometry determine the remaining harmonic part.