The Cocycle of the Quantum HJ Equation and the Stress Tensor of CFT

TL;DR

This paper explores the mathematical structure of the Quantum Hamilton-Jacobi Equation in conformal field theory, revealing how the cocycle condition determines the Schwarzian derivative and links energy quantization to duality transformations.

Contribution

It demonstrates that the cocycle condition uniquely defines the Schwarzian derivative and connects energy quantization to the quantum Hamilton-Jacobi equation within CFT.

Findings

Cocycle condition uniquely determines the Schwarzian derivative.

Infinitesimal stress tensor variation exponentiates to the Schwarzian derivative.

Energy quantization follows from the quantum Hamilton-Jacobi equation under duality.

Abstract

We consider two theorems formulated in the derivation of the Quantum Hamilton-Jacobi Equation from the EP. The first one concerns the proof that the cocycle condition uniquely defines the Schwarzian derivative. This is equivalent to show that the infinitesimal variation of the stress tensor "exponentiates" to the Schwarzian derivative. The cocycle condition naturally defines the higher dimensional version of the Schwarzian derivative suggesting a role in the transformation properties of the stress tensor in higher dimensional CFT. The other theorem shows that energy quantization is a direct consequence of the existence of the quantum Hamilton-Jacobi equation under duality transformations as implied by the EP.