Quantum surface holonomies for loop quantum gravity and their application to black hole horizons

TL;DR

This paper introduces new flux operators in loop quantum gravity that help solve key equations related to black hole horizons, offering a quantum deformation of surface holonomies with potential applications in quantum geometry.

Contribution

The paper defines novel flux operators in loop quantum gravity that enable solving curvature equations and interpreting surface holonomies as quantum deformations, advancing the understanding of quantum horizons.

Findings

New flux operators are constructed and analyzed.

They provide solutions to curvature equations in LQG.

Potential to define quantum states satisfying horizon boundary conditions.

Abstract

In this work we define a new type of flux operators on the Hilbert space of loop quantum gravity. We use them to solve an equation of the form $F(A)=c\,\Sigma$ in loop quantum gravity. This equation, which relates the curvature of a connection $A$ with its canonical conjugate $\Sigma=*E$, plays an important role for spherically symmetric isolated horizons, and, more generally, for maximally symmetric geometries and for the Kodama state. If the equation holds, the new flux operators can be interpreted as a quantization of surface holonomies from higher gauge theory. Also, they represent a kind of quantum deformation of SU(2). We investigate their properties and discuss how they can be used to define states that satisfy the isolated horizon boundary condition in the quantum theory.