A Conformal Field Theory of Extrinsic Geometry of 2-d Surfaces

TL;DR

This paper develops a conformal field theory describing the extrinsic geometry of 2D surfaces in R^3, revealing Virasoro symmetry and linking it to intrinsic 2D gravity, with implications for quantization.

Contribution

It constructs an explicit extrinsic curvature action with Virasoro symmetry, connecting extrinsic and intrinsic 2D gravity theories and enabling quantization in this geometric context.

Findings

Derived an extrinsic curvature analog of the WZNW action.

Identified Virasoro symmetry and SL(2,C) currents in the theory.

Established the quantum theory's universality class with intrinsic 2D gravity.

Abstract

In the description of the extrinsic geometry of the string world sheet regarded as a conformal immersion of a 2-d surface in $R^3$, it was previously shown that, restricting to surfaces with $h\surd{g}\ =\ 1$, where $h$ is the mean scalar curvature and $g$ is the determinant of the induced metric on the surface, leads to Virasaro symmetry. An explicit form of the effective action on such surfaces is constructed in this article which is the extrinsic curvature analog of the WZNW action. This action turns out to be the gauge invariant combination of the actions encountered in 2-d intrinsic gravity theory in light-cone gauge and the geometric action appearing in the quantization of the Virasaro group. This action, besides exhibiting Virasaro symmetry in $z$-sector, has $SL(2,C)$ conserved currents in the $\bar{z}$-sector. This allows us to quantize this theory in the $\bar{z}$-sector along the lines of the WZNW model. The quantum theory on $h\surd{g}\ =\ 1$ surfaces in $ R^3$ is shown to be in the same universality class as the intrinsic 2-d gravity theory.