Vacuum Solutions of Classical Gravity on Cyclic Groups from Noncommutative Geometry

TL;DR

This paper constructs action functionals for classical gravity on cyclic groups using noncommutative geometry, revealing nontrivial vacuum solutions expressed with Chebyshev's polynomials across various coupling constants.

Contribution

It introduces new action functionals based on moduli of link variables that modify Connes' distance, leading to explicit vacuum solutions in a noncommutative geometric setting.

Findings

Identifies nontrivial vacuum solutions for gravity on cyclic groups.

Expresses solutions using Chebyshev's polynomials.

Demonstrates solutions exist over a broad range of coupling constants.

Abstract

Based on the observation that the moduli of a link variable on a cyclic group modify Connes' distance on this group, we construct several action functionals for this link variable within the framework of noncommutative geometry. After solving the equations of motion, we find that one type of action gives nontrivial vacuum solution for gravity on this cyclic group in a broad range of coupling constants and that such solutions can be expressed with Chebyshev's polynomials.