The quantum non-linear $\sigma$-model RG flow and integrability in wormhole geometries

TL;DR

This paper investigates the renormalization group flow of wormhole geometries in non-linear sigma models, revealing non-integrability and singularity formation through numerical evolution and analyzing associated physical quantities.

Contribution

It provides a numerical analysis of the RG flow of spherically symmetric wormholes, demonstrating non-convergence to fixed points and exploring the evolution of singularities and physical energies.

Findings

All wormhole metrics pinch off at finite flow time.

The flow does not reach a fixed point, indicating non-integrability.

Evolution of Hamilton's entropy and Brown-York energy was characterized.

Abstract

The target space of the non-linear $\sigma$-model is a Riemannian manifold. Although it can be any Riemannian metric, there are certain physically interesting geometries which are worth to study. Here, we numerically evolve the time-symmetric foliations of a family of spherically symmetric asymptotically flat wormholes under the $1$-loop renormalization group flow of the non-linear $\sigma$-model, the Ricci flow, and under the $2$-loop approximation, RG-2 flow. We rely over some theorems adapted from the compact case for studying the evolution of different wormhole types, specially those with high curvature zones. Some metrics expand and others contract at the beginning of the flow, however, all metrics pinch-off at a certain time. This is related with the fact that the flow does not converge to a fixed point when its starting geometry is the spatial sections of a Morris-Thorne wormhole, and therefore the corresponding non-linear $\sigma$-model is non-integrable/renormalizable. We present a numerical study of the evolution of wormhole singularities in three dimensions extending the theoretical estimations. Finally, we compute the evolution of the Hamilton's entropy and the Brown-York energy.