RST model

The Russo–Susskind–Thorlacius model^[1] or RST model in short is a modification of the CGHS model to take care of conformal anomalies and render it analytically soluble. In the CGHS model, if we include Faddeev–Popov ghosts to gauge-fix diffeomorphisms in the conformal gauge, they contribute an anomaly of -24. Each matter field contributes an anomaly of 1. So, unless N=24, we will have gravitational anomalies. To the CGHS action

${\displaystyle S_{\text{CGHS}}={\frac {1}{2\pi }}\int d^{2}x\,{\sqrt {-g}}\left\{e^{-2\phi }\left[R+4\left(\nabla \phi \right)^{2}+4\lambda ^{2}\right]-\sum _{i=1}^{N}{\frac {1}{2}}\left(\nabla f_{i}\right)^{2}\right\}}$, the following term

${\displaystyle S_{\text{RST}}=-{\frac {\kappa }{8\pi }}\int d^{2}x\,{\sqrt {-g}}\left[R{\frac {1}{\nabla ^{2}}}R-2\phi R\right]}$

is added, where κ is either ${\displaystyle (N-24)/12}$ or ${\displaystyle N/12}$ depending upon whether ghosts are considered. The nonlocal term leads to nonlocality. In the conformal gauge,

${\displaystyle S_{\text{RST}}=-{\frac {\kappa }{\pi }}\int dx^{+}\,dx^{-}\left[\partial _{+}\rho \partial _{-}\rho +\phi \partial _{+}\partial _{-}\rho \right]}$.

It might appear as if the theory is local in the conformal gauge, but this overlooks the fact that the Raychaudhuri equations are still nonlocal.