Conformal Field Theory and Hyperbolic Geometry

TL;DR

This paper explores the deep connection between conformal field theory and hyperbolic geometry, revealing new insights into boundary operators, three-point functions, and a series of central charges with physical interpretations.

Contribution

It introduces a reformulation of conformal covariance equations, provides a factored form for three-point functions, and uncovers a series of central charges related to hyperbolic geometry.

Findings

Discovered a doubly-infinite series of central charges with a limit at c=-2.

Established a correspondence between anomalous dimensions and hyperbolic angles.

Reformulated conformal covariance with a new physical interpretation of scale factors.

Abstract

We examine the correspondence between the conformal field theory of boundary operators and two-dimensional hyperbolic geometry. By consideration of domain boundaries in two-dimensional critical systems, and the invariance of the hyperbolic length, we motivate a reformulation of the basic equation of conformal covariance. The scale factors gain a new, physical interpretation. We exhibit a fully factored form for the three-point function. A doubly-infinite discrete series of central charges with limit c=-2 is discovered. A correspondence between the anomalous dimension and the angle of certain hyperbolic figures emerges. Note: email after 12/19: kleban@maine.maine.edu