# Pseudo-topological Quasi-local Energy of Torsion Gravity

## TL;DR

This paper constructs a pseudo-topological quasi-local energy in torsion gravity using Wald's covariant method, revealing its properties, evaluating it in specific solutions, and discussing implications for holography and gravitational wave tests.

## Contribution

It introduces a formal expression for quasi-local energy in torsion gravity that is independent of coupling constants and explores its pseudo-topological nature and physical implications.

## Key findings

- The quasi-local energy is formally independent of torsion-fermion coupling.
- It matches holographic relative entropy in specific solutions.
- The energy can be non-positive, indicating swampland examples.

## Abstract

Torsion gravity is a natural extension to Einstein gravity in the presence of the fermion matter sources. In this paper we adopt Wald's covariant method of Noether charge to construct the quasi-local energy of the Einstein-Cartan-fermion system, and find that its explicit expression is formally independent of the coupling constant between torsion and axial current. This seemingly topological nature is unexpected and is reminiscent of similar nature of quantum Hall effect and topological insulator. However, the coupling dependence does enter when evaluating it on-shell, and thus the topological nature is pseudo. Based on the expression of the quasi-local energy, we evaluate it for a particular solution on the entanglement wedge and find the agreement with the holographic relative entropy obtained before. This shows that the equivalence of these two quantities in the Einstein-Cartan-fermion system. Moreover, the quasi-local energy in this case is not always positive definite so that it provides an example of swampland in torsion gravity. Based on the covariant Noether charge, we also derive the nonzero fermion effect on Komar angular momentum. The implication of our results to the tests of torsion gravity in the future gravitational wave astronomy is also discussed.