Extrinsic Curvature Induced 2-d Gravity

TL;DR

This paper derives an effective 2D gravity action from fermions coupled to extrinsic geometry of surfaces, revealing Virasoro symmetry and calculating central charges, thus connecting geometry, gauge theory, and conformal symmetry.

Contribution

It introduces a novel approach to 2D gravity by coupling fermions to extrinsic curvature and deriving an explicit Virasoro-invariant action.

Findings

Derived a WZNW action involving extrinsic curvature.

Obtained an explicit Virasoro-invariant gravity action.

Calculated the central charges of the theory.

Abstract

Abtract: 2-dimensional fermions are coupled to extrinsic geometry of a conformally immersed surface in ${\bf R}^3$ through gauge coupling. By integrating out the fermions, we obtain a WZNW action involving extrinsic curvature of the surface. Restricting the resulting effective action to surfaces of $h\sqrt g=1$, an explicit form of the action invariant under Virasaro symmetry is obtained. This action is a sum of the geometric action for the Virasaro group and the light-cone action of 2-d gravity plus an interaction term. The central charges of the theory in both the left and right sectors are calculated.