Two-dimensional quantum gravity is identified as a second-class system which we convert into a first-class system via the Batalin-Fradkin (BF) procedure. Using the extended phase space method, we then formulate the theory in the most general class of gauges. The conformal gauge action suggested by David, Distler, and Kawai is derived from first principles. We find a local, light-cone gauge action whose Becchi-Rouet-Stora-Tyutin invariance implies Polyakov's curvature equation ${\partial }_{-}R={\partial }_{-gg++}^3=0\mathrm{}$, revealing the origin of the $\mathrm{SL}(2, R)$ Kac-Moody symmetry. The BF degree of freedom turns out to be dynamically active as the Liouville mode in the conformal gauge, while in the light-cone gauge the conformal degree of freedom plays that role. The inclusion of the cosmological constant term in both gauges is also considered.

• Received 5 April 1993

DOI:https://doi.org/10.1103/PhysRevD.48.1736