Hyperbolicity of Partition Function and Quantum Gravity

TL;DR

This paper explores the geometric structure of the partition function related to quantum gravity, revealing that hyperbolic geometry naturally emerges in its classical limit and connecting it to string theory via ideal tetrahedra.

Contribution

It introduces a novel geometric interpretation of the partition function using hyperbolic structures and links quantum gravity concepts with string theory through ideal tetrahedra.

Findings

Hyperbolic geometry arises in the classical limit of the partition function.

The partition function can be associated with 3D hyperbolic or Euclidean AdS_3 structures.

Oriented ideal tetrahedra are connected to the string partition function.

Abstract

We study a geometry of the partition function which is defined in terms of a solution of the five-term relation. It is shown that the 3-dimensional hyperbolic structure or Euclidean AdS_3 naturally arises in the classical limit of this invariant. We discuss that the oriented ideal tetrahedron can be assigned to the partition function of string.