.

Liouville gravity[1][2] is a model of gravity in one spatial and one time dimension with a dilaton. It has the specific coupling of the form

\( S = \frac{1}{4\pi}\int d^2x\sqrt{-g} \left[ \left( b+ b^{-1} \right)\Phi R + \left( \nabla\Phi \right)^2 + 4\pi \mu e^{2b\Phi} \right], \)

where Φ is the dilaton field. It should not be confused[3][4] with the CGHS model or Jackiw–Teitelboim gravity. After taking quantum conformal anomalies into account, it turns out to be a conformal field theory. This model is used in the study of non-critical string theory.
References

D'Hoker, Eric; Jackiw, Roman (15 Dec 1982). "Classical and quantal Liouville field theory". Physical Review. D 26 (12): 3517–3542. Bibcode:1982PhRvD..26.3517D. doi:10.1103/PhysRevD.26.3517.
Nakayama, Yu (2004). "Liouville Field Theory – A decade after the revolution". International Journal of Modern Physics. A 19 (17–18): 2771–2930. arXiv:hep-th/0402009. Bibcode:2004IJMPA..19.2771N. doi:10.1142/S0217751X04019500. Archived from the original on 10 Dec 2004.
Grumiller, Daniel; Kummer, Wolfgang; Vassilevich, Dmitri (October 2002). "Dilaton Gravity in Two Dimensions". Physics Reports 369 (4): 327–430. arXiv:hep-th/0204253. Bibcode:2002PhR...369..327G. doi:10.1016/S0370-1573(02)00267-3. Archived from the original on 4 Jan 2008.

Grumiller, Daniel; Meyer, Rene (2006). "Ramifications of Lineland". Turkish Journal of Physics 30 (5): 349–378. arXiv:hep-th/0604049. Bibcode:2006TJPh...30..349G. Archived from the original on 1 June 2006.