A pathway to non-perturbative Quantum Affine Gravity

TL;DR

This work advances a non-perturbative approach to quantum gravity by formulating gravity as a purely affine theory with the connection as the fundamental field and the metric emerging only under suitable conditions. It develops a concrete lattice implementation of the Eddington-type action, defines torsionful and torsionless ensembles, and provides a public Monte Carlo code to study these ensembles in arbitrary dimensions. In two dimensions, where a metric does not emerge, the authors still obtain a well-behaved statistical model and observe monotonic, phase-transition-free behavior of diffeomorphism-invariant observables as a function of the bare coupling. The results establish a solid computational baseline for exploring higher dimensions, emergent metric constraints, and potential continuum or Lorentzian extensions within purely affine gravity frameworks.

Abstract

We explore a new route toward a non-perturbative quantization of gravity based on a purely affine formulation, where the affine connection is the fundamental field and the metric, when it exists, emerges as a derived quantity. Starting from the Palatini formulation of General Relativity, we recall how an equivalent Eddington-type purely affine action arises at the classical level under mild assumptions. A key feature for the non-perturbative program is that, in the pure gravity case, for a positive cosmological constant, the action is bounded below, allowing one to define a well-posed statistical ensemble of connections. We discretize this theory on a fixed hypercubic lattice and construct the corresponding partition function, including torsionful and torsionless ensembles. We provide an open C++ Monte Carlo implementation that can simulate these ensembles in arbitrary dimension, and we present proof-of-principle results in two dimensions.

Figures (4)

• Figure 1: Expectation value of the observable $\overline{V^{\text{lat}}_{R}}$ over $2000$ sampled path integral configurations from the Monte Carlo simulations, as a function of the bare parameter $\alpha$, for three different lattice sizes. The results are overlapping monotonically decreasing curves, with no signs of a phase transition. Statistical error bars of below $0.01\%$ of the reported values, and are too small to be seen.
• Figure 2: Expectation value of the observable $\overline{V^{\text{lat}}_{Q}}$ over $2000$ sampled path integral configurations from the Monte Carlo simulations, as a function of the bare parameter $\alpha$, for three different lattice sizes. The results are overlapping monotonically decreasing curves, with no signs of a phase transition. Statistical error bars of below $0.01\%$ of the reported values, and are too small to be seen.
• Figure 3: Expectation value of $f_{d_{R}<0}$ over $2000$ sampled path integral configurations from the Monte Carlo simulations, as a function of the bare parameter $\alpha$, for three different lattice sizes. The results are overlapping monotonically decreasing curves, with no signs of a phase transition, but always above $0.5$, meaning that the configurations have a majority of points with a symmetric part of the Ricci tensor with $(-,\,+)$ signature, instead of $(-,\,-)$ or $(+,\,+)$.
• Figure 4: Wilson loop on a square plaquette with edge length $\varepsilon$.