Quantizing non-projectable Hořava gravity with Lagrangian path integral

TL;DR

The paper develops a Lagrangian path-integral formulation for non-projectable Hořava gravity, incorporating a field-dependent ultra-local measure for second-class constraints. Through a detailed one-loop analysis in 2+1 dimensions, it demonstrates the cancellation of irregular frequency divergences and computes the beta-functions for the essential couplings $\lambda$ and $G$, using both diagrammatic and heat-kernel methods as cross-checks. The results support perturbative control of the theory at this order and illuminate the conditions needed to establish full perturbative renormalizability, emphasizing the role of the measure's structure. The work provides a concrete, gauge-consistent framework and regularization strategy that can be extended to higher dimensions, laying groundwork toward a complete quantum theory of non-projectable Hořava gravity.

Abstract

We formulate the quantum version of non-projectable Hořava gravity as a Lagrangian theory with a path integral in the configuration space with an ultra-local in time, but non-local in space, field-dependent measure. Using auxiliary fields, we cast the measure into a local form satisfying several bosonic and fermionic symmetries. We perform an explicit one-loop computation in the theory in $(2+1)$ dimensions, using for the case study the divergent part of the action on a background with non-trivial shift vector; the background spatial metric is taken to be flat and the background lapse function is set to 1. No truncations are assumed at the level of perturbations, for which we develop a diagrammatic technique and a version of the heat-kernel method. We isolate dangerous linear-in-frequency divergences in the two-point function of the shift, which can lead to spatial non-localities, and explicitly verify their cancellation. This leaves a fully local expression for the divergent part of the quadratic effective action, from which we extract the beta functions for the Newton constant and the essential coupling $λ$ in the kinetic term of the metric. We formulate the questions that need to be addressed to prove perturbative renormalizability of the non-projectable Hořava gravity.

Figures (7)

• Figure 1: Graphic notation for the fields.
• Figure 2: Interaction vertices with one background shift vector $\bar{N}_i$.
• Figure 3: Interaction vertices with two background shift vectors $\bar{N}_i$.
• Figure 4: Bubble diagrams contributing into the two-point function of $\bar{N}_i$.
• Figure 5: Fish diagrams contributing into the two-point function of $\bar{N}_i$.