Canonical structure of minimal varying $\Lambda$ theories

TL;DR

This paper analyzes the canonical structure of minimal varying  theories, revealing the number of degrees of freedom varies with background and parameters, and explores special cases like self-dual actions and de Sitter backgrounds.

Contribution

It provides a detailed canonical analysis of minimal varying  theories, identifying degrees of freedom and their dependence on parameters and background geometry.

Findings

Five degrees of freedom on generic backgrounds and parameters.

Three degrees of freedom when parameters satisfy certain conditions.

Two complex degrees of freedom in the self-dual case.

Abstract

Minimal varying $\Lambda$ theories are defined by an action built from the Einstein-Cartan-Holst first order action for gravity with the cosmological constant $\Lambda$ as an independent scalar field, and supplemented by the Euler and Pontryagin densities multiplied by $1/\Lambda$. We identify the canonical structure of these theories which turn out to represent an example of irregular systems. We find five degrees of freedom on generic backgrounds and for generic values of parameters, whereas if the parameters satisfy a certain condition (which includes the most commonly considered Euler case) only three degrees of freedom remain. On de Sitter-like backgrounds the canonical structure changes, and due to an emergent conformal symmetry one degree of freedom drops from the spectrum. We also analyze the self-dual case with an holomorphic action depending only on the self-dual part of the connection. In this case we find two (complex) degrees of freedom, and further discuss the Kodama state, the restriction to de Sitter background and the effect of reality conditions.