4d Lorentzian Holst action with topological terms

TL;DR

This paper investigates how adding topological invariants to the Holst action in a first-order tetrad–connection formulation of general relativity reshapes the canonical structure and the space of SU(2) connection formulations. Through a 3+1 (time-gauge) Hamiltonian analysis, it shows that Pontrjagin, Euler, and Nieh-Yan terms are boundary contributions that act as generating functions for canonical transformations, shifting momenta without altering the classical equations of motion. The work identifies two branches of formulation: a real-$\gamma$ sector with ${^{+}\omega}=\Gamma+\gamma \hat{K}$ and $\alpha_3=\alpha_4=0$, and a complex-$\gamma$ sector with $\gamma=\pm i$, each with explicit Dirac brackets and reduced constraints, clarifying when a nontrivial $\theta$-like term can arise. It further shows that realizing a genuine $\theta$ term in the real theory requires a Lorentz-symmetry-breaking boundary term, and discusses the quantum implications and the existence of an infinite family of classically equivalent but potentially quantum-inequivalent connection formulations.

Abstract

We study the Hamiltonian formulation of the general first order action of general relativity compatible with local Lorentz invariance and background independence. The most general simplectic structure (compatible with diffeomorphism invariance and local Lorentz transformations) is obtained by adding to the Holst action the Pontriagin, Euler and Nieh-Yan invariants with independent coupling constants. We perform a detailed canonical analysis of this general formulation (in the time gauge) exploring the structure of the phase space in terms of connection variables. We explain the relationship of these topological terms, and the effect of large SU(2) gauge transformations in quantum theories of gravity defined in terms of the Ashtekar-Barbero connection.