New heat kernel method in Lifshitz theories

Abstract

We develop a new heat kernel method that is suited for a systematic study of the renormalization group flow in Hořava gravity (and in Lifshitz field theories in general). This method maintains covariance at all stages of the calculation, which is achieved by introducing a generalized Fourier transform covariant with respect to the nonrelativistic background spacetime. As a first test, we apply this method to compute the anisotropic Weyl anomaly for a (2 + 1)-dimensional scalar field theory around a z = 2 Lifshitz point and corroborate the previously found result. We then proceed to general scalar operators and evaluate their one-loop effective action. The covariant heat kernel method that we develop also directly applies to operators with spin structures in arbitrary dimensions.

Heat kernel expansion for higher order minimal and nonminimal operators

“…One of the motivations for this project is the problem of UV renormalization in Horava gravity theory, which is computationally extremely complicated because of Lorentz invariance violation. Extension of the heat kernel method to Lorentz violating models is also possible [32][33][34] and includes a recent application of the Fourier method [35] along the lines of the Gusynin method for the operator resolvent [22,23]. In four dimensions this problem involves non-minimal operators of the sixth order [15] and requires the whole set of special methods involving the above mentioned universal functional traces [3], 3-dimensional reduction on a static background, square root extraction for the sixth-order operator having hundreds of terms, etc.…”

Strings in bimetric spacetimes

“…We start with a quick review of the heat kernel method and derive the general form of the effective action, following [26][27][28][29][30] but with adaptions to the bimetric sigma models. First, we define the associated path integral for the covariantly expanded action (3.30), with respect to a reference metric G µν [X 0 ] ,…”

“…However, the analysis of the dilaton beta-functionals requires a more thorough understanding of the worldsheet geometry and higher-loop calculation [32], for which other techniques are needed. For example, evaluating the Weyl anomalies on a curved worldsheet with a foliation structure requires the method developed in [30], which we leave for future studies.…”

“…The main focus of this paper has been the beta-functionals of the spacetime bimetric fields. However, to map out the complete RG structure of the two-dimensional Lifshitztype sigma model, one also needs to determine the beta-functionals for the dilaton fields, for which detailed RG properties of the worldsheet Hořava gravity is required (some useful techniques have been developed in the literature, see, e.g., [30,[54][55][56][57][58]). This piece of calculation will be essential for understanding the notion of critical dimensions in Lifshitztype sigma models.…”

“…On a curved worldsheet, there exists a covariant generalization of the phase function e i ω(τ −τ 0 )+i κ(x−x 0 ) adapted to the foliation, defined using the symbolic calculus of pseudodifferential operators[28,30,31]. We will not need this covariant generalization of the phase function in this paper, but it will play an important role when it comes to the beta-functionals of the dilaton fields.…”