Gauge Equivalence in Two--Dimensional Gravity

TL;DR

This paper reformulates two-dimensional quantum gravity as a first-class system using the BF procedure, derives gauge actions from first principles, and explores the role of BF degrees of freedom and symmetries in different gauges.

Contribution

It introduces a general gauge formulation of 2D gravity via the BF method and clarifies the dynamical roles of degrees of freedom in various gauges.

Findings

Derivation of conformal gauge action from first principles

Identification of the Liouville mode as a BF degree of freedom

Revelation of $SL(2,R)$ Kac-Moody symmetry in light-cone gauge

Abstract

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 most general class of gauges. The conformal gauge action suggested by David, Distler and Kawai is derived from a first principle. We find a local, light-cone gauge action whose Becchi-Rouet-Stora-Tyutin invariance implies Polyakov's curvature equation $\partial_{-}R=\partial_{-}^{3}g_{++}=0$, revealing the origin of the $SL(2,R)$ Kac-Moody symmetry. The BF degree of freedom turns out be dynamically active as the Liouville mode in the conformal gauge, while in the light-cone gauge the conformal degree of freedom plays that r{\^o}le. The inclusion of the cosmological constant term in both gauges and the harmonic gauge-fixing are also considered.