# RST model

Short description: Conformal anomaly free CGHS 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

$\displaystyle{ S_{\text{CGHS}} = \frac{1}{2\pi} \int d^2x\, \sqrt{-g}\left\{ e^{-2\phi} \left[ R + 4\left( \nabla\phi \right)^2 + 4\lambda^2 \right] - \sum^N_{i=1} \frac{1}{2}\left( \nabla f_i \right)^2 \right\} }$, the following term
$\displaystyle{ S_{\text{RST}} = - \frac{\kappa}{8\pi} \int d^2x\, \sqrt{-g} \left[ R\frac{1}{\nabla^2}R - 2\phi R \right] }$

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.

## References

1. Russo, Jorge; Susskind, Leonard; Thorlacius, Lárus. Physical Review. D 46 (8): 3444–3449. doi:10.1103/PhysRevD.46.3444. PMID 10015289. Bibcode1992PhRvD..46.3444R.