Hidden sl(2,R) Symmetry in 2D CFTs and the Wave Function of 3D Quantum Gravity

TL;DR

This paper reveals a hidden sl(2,R) symmetry in all 2D conformal field theories by extending them with ghost fields and constructs candidate wave functions for 3D quantum gravity, offering new insights into their structure.

Contribution

It demonstrates the existence of a universal hidden sl(2,R) affine symmetry in 2D CFTs and develops a method to construct wave functions for 3D quantum gravity.

Findings

All 2D CFTs have a hidden sl(2,R) symmetry.

Constructed candidate wave functions for 3D quantum gravity.

Provided a BRST framework to relate extended theories to original CFTs.

Abstract

We show that all two-dimensional conformal field theories possess a hidden sl(2,R) affine symmetry. More precisely, we add appropriate ghost fields to an arbitrary CFT, and we use them to construct the currents of sl(2,R). We then define a BRST operator whose cohomology defines a physical subspace where the extended theory coincides with the original CFT. We use the sl(2,R) algebra to construct candidate wave functions for 3-d quantum gravity coupled to matter, and we discuss their viability.