MZ-TH/12-56

A functional renormalization group equation

[1.5ex] for foliated spacetimes

[2cm]

Stefan Rechenberger and Frank Saueressig

[1.5ex] PRISMA Cluster of Excellence & Institute for Physics (THEP)

University of Mainz, Staudingerweg 7, D-55099 Mainz, Germany

Abstract

We derive an exact functional renormalization group equation for the projectable version of Hořava-Lifshitz gravity. The flow equation encodes the gravitational degrees of freedom in terms of the lapse function, shift vector and spatial metric and is manifestly invariant under background foliation-preserving diffeomorphisms. Its relation to similar flow equations for gravity in the metric formalism is discussed in detail, and we argue that the space of action functionals, invariant under the full diffeomorphism group, forms a subspace of the latter invariant under renormalization group transformations. As a first application we study the RG flow of the Newton constant and the cosmological constant in the ADM formalism. In particular we show that the non-Gaussian fixed point found in the metric formulation is qualitatively unaffected by the change of variables and persists also for Lorentzian signature metrics.

## 1 Introduction

The construction of a consistent and predictive quantum theory for gravity constitutes one of the major challenges for theoretical high energy physics to date. Within the realm of quantum field theory there are currently two proposals that receive a lot of attention. Quantum Einstein Gravity (QEG) [1, 2, 3, 4] is based on Weinberg’s Asymptotic Safety scenario [5, 6] and suggests that gravity could be a non-perturbatively renormalizable quantum field theory. In this case the UV-completion of the theory is provided by a non-Gaussian fixed point (NGFP) of the renormalization group (RG) flow. For RG trajectories that are captured by the NGFP in the UV, this construction ensures that physical quantities are free from unphysical UV divergences. Provided that the UV fixed point comes with a finite number of relevant directions, this construction is predictive. Starting from the seminal work [7] there is now a solid body of evidence supporting the existence and predictivity of this NGFP [8, 9, 10, 11, 12, 13, 14, 15, 16, 17]. The more recent proposal advocated by Hořava [18] suggests that the UV completion of gravity is an action which is anisotropic in the space and time coordinates. Higher powers of the spatial momenta entering into the gravitational propagators could render this construction perturbatively renormalizable [19, 20] but lead to difficulties when trying to restore Lorentz invariance in the low energy regime, see [21, 22, 23] for selected reviews and further references.

A crucial ingredient in investigating both scenarios is the RG flow of the theory. For practical computations, this flow can conveniently be encoded by the Wetterich equation [24] which captures the scale dependence of the effective average action and can schematically be written as

(1.1) |

Here, denotes the second functional derivative of the effective average action with respect to the fluctuation fields at fixed background and Tr contains a sum over all fields of the theory and an integration over loop-momenta. Furthermore, is a matrix-valued IR cutoff, which provides a -dependent mass term for fluctuations with momenta . The interplay between the full regularized propagator and ensures that the Tr receives contributions from a small -interval around only, rendering the trace contribution finite.

The flow equation (1.1) is defined on the so-called theory space. This space contains all action functionals that can be build from a given field content and are compatible with the symmetries of the theory under consideration. Considering metric QEG, for example, the theory space contains all action functionals build from the spacetime metric , which are invariant under general coordinate transformations Diff().

An interesting open question at this stage is if the metric indeed provides the correct formulation for the gravitational degrees of freedom. Coupling gravity to fermionic matter degrees of freedom suggests that a first-order formalism based on the vielbein may be more fundamental. A first investigation of this scenario was initiated in [25] utilizing the Wetterich equation tailored to the fundamental fields of Einstein-Cartan gravity, i.e., the vielbein and the spin connection. Along a different line, the existence of time points at a preferred direction which in terms of Euclidean geometry may reflect itself in a foliation structure of spacetime. Such a structure is naturally captured by the ADM decomposition of the metric [26]. While all these formulations may be on-shell equivalent at the classical level, it is not clear if they describe the same quantum theory. In particular it is a priori unclear if the NGFP underlying Asymptotic Safety in the metric formulation also appears when using different fields for encoding gravity and, if so, weather these descriptions fall into the same universality class, in the sense that the universal critical exponents associated with the NGFPs actually coincide.

In this paper, we address these issues by adapting the Wetterich equation (1.1) to the ADM decomposition, thereby implementing a foliation structure on the underlying quantum spacetime. In this case the metric is decomposed into a lapse function , a shift-vector and a metric on spatial slices

(1.2) |

The lapse function and the shift vector essentially describe how the spatial slices are welded together, thus imprinting the (Euclidean) spacetime with a preferred direction. This decomposition then naturally entails a spacetime structure that is topologically , where are the -dimensional leaves of the foliation. The direction singled out by the allows to Wick-rotate between Euclidean and Lorentzian signature, and will thus be referred to as (Euclidean) time direction. Notably, the ADM construction is very close to the geometric setting underlying the Monte-Carlo simulations of Causal Dynamical Triangulations (CDT) [27, 28, 29, 30, 31].

theory | theory space | gravitational fields | symmetry |
---|---|---|---|

metric QEG | Diff() | ||

foliated QEG | Diff() | ||

projectable Hořava-Lifshitz | Diff() | ||

non-projectable Hořava-Lifshitz | Diff() |

Depending on the precise field content and symmetry groups there are several theory spaces that can naturally be associated with the decomposition (1.2), see Table 1. Firstly, one can insist that the symmetry group acting on the ADM fields is the full Diff() symmetry. In this case eq. (1.2) constitutes a non-linear field-redefinition of the gravitational degrees of freedom. The resulting theory space of foliated QEG, , is equivalent to the one of metric QEG, . Its interaction monomials are constructed from the ADM fields and preserve Diff(). The foliation structure of naturally defines a subgroup of Diff(), the foliation-preserving diffeomorphisms Diff(), eq. (3.8). Adopting this subgroup as the symmetry group of the theory space gives rise to the theory spaces underlying Hořava-Lifshitz (HL) gravity [18]. In its projectable version (pHL), the lapse function depends on time only. In terms of the field content, this can be understood as a partial gauge fixing eliminating the space dependence of the lapse function living on . The weaker symmetry requirements on allow to write additional interaction monomials for , which are invariant under Diff() but break Diff() invariance. Thus is embedded into the theory space of projectable HL theory . Finally, the non-projectable version of HL theory also includes a spatial dependence in the lapse function. Thus contains additional interaction monomials not present in . Based on these different symmetry requirements the theory spaces thus satisfy

(1.3) |

As it will turn out, the natural symmetry of the Wetterich equation formulated in terms of the ADM fields are foliation-preserving diffeomorphisms. The key observation underlying this assessment is that Diff acts non-linearly on the ADM fields. Since it is a key requirement in the derivation of a Wetterich-type flow equation that its regulator is quadratic in the fluctuation fields, it is impossible to retain invariance with respect to a non-linearly realized symmetry. Thus our functional renormalization group equation (FRGE) is invariant under background foliation-preserving diffeomorphisms only, since Diff is the maximal subgroup that is realized by linear transformations. This restriction of the symmetry group has a direct consequence for the theory space on which our FRGE is formulated: besides containing all interaction monomials that are invariant under the Diff(), the theory space also contains interactions which are invariant under foliation preserving diffeomorphisms and thus break the full diffeomorphism invariance explicitly. Thus our flow equation naturally encodes the RG flow on .^{1}^{1}1As it will turn out, the off-shell setting of the FRGE implies that foliated QEG and projective Hořava-Lifshitz gravity are actually described by the same flow equation.

The relation (1.3) indicates that can be embedded into . An important result following from [7] is that the subspace is actually closed under the RG flow. If the RG flow starts from an action functional preserving full diffeomorphism invariance, , integrating out quantum fluctuations will not generate interactions that violate Diff() dynamically. This leaves the phenomenologically interesting scenario that serves as an IR-attractor within , leading to a dynamical restoration of Lorentz symmetry at low energies.

The foliation structure will actually lead to a FRGE which closely resembles the ones encountered for quantum field theories at finite temperature. In this interpretation, the system is seen as -dimensional gravity coupled to a scalar and vector at finite temperature proportional to the radius of the . In this setting quantum fluctuations are split into two classes: spatial fluctuations of the -dimensional system at (zero temperature) and thermal fluctuations associated with the circle . The latter are captured by the Matsubara modes of the system. The FRGE constructed in this paper integrates out both quantum and thermal fluctuations, essentially relying on the “imaginary time” formalism of finite temperature quantum field theory. This analogy allows to parallel the construction of Wilsonian flow equations for thermal field theories [32, 33, 34], adapting it to the gravitational system at hand.

The rest of this work is organized as follows. We start by reviewing the classical ADM construction [26] in Sect. 2. The Wetterich equation capturing the RG flow on is constructed in Sect. 3. This flow equation respects foliation-preserving diffeomorphisms as a background symmetry, which severely restricts the interaction monomials that can be generated dynamically by the RG flow. As a first application we consider a truncation of the effective average action which does not contain anisotropic couplings and derive the signature-dependent beta functions of the ADM-decomposed Einstein-Hilbert truncation in Sect. 4.^{2}^{2}2First results on the RG flow on including Diff()-breaking interactions will be reported elsewhere [36]. Their properties and fixed point structure, which have partially been reported previously [35], are studied in
in Sect. 5. We close with a summary and an outlook on possible applications in Sect. 6. The technical details entering the construction of the beta functions have been collected in two appendices.

## 2 Decomposing spacetime into space and time

In order to make the present work self-contained and fix our notation, we start by reviewing the classical ADM decomposition of the metric field [26]. This construction plays an essential role when studying gravitational RG flows on spacetimes carrying a foliation structure or exhibiting anisotropic-scaling effects between space and time.

The ADM construction starts from a -dimensional Riemannian manifold with metric . The spacetime signature can either be Euclidean, , or Lorentzian, and we use the signature parameter to distinguish the two cases. On this manifold we introduce a time function relating a real number, which we call time , to every spacetime point . This function equips with a vector field which, for Lorentzian signature, is taken to be timelike. The -dimensional spacetime manifold can then be seen as a stack of spatial slices with spatial dimension . The hypersurfaces come with a normal vector , which we take as future-directed () and normalized to unity (). It can be related to the time function by where the Lapse function acts as a normalization factor.

On each spatial slice we introduce coordinates , . On neighboring slices these coordinate systems will be related by the integral curves along . Explicitly we choose to be constant along such a curve. The Jacobian relating the coordinate systems and is given by

(2.1) |

Note that holds, since is normal to the hypersurfaces. The vector can be decomposed into its components tangential and perpendicular to the spatial slices as

(2.2) |

Here, the -dimensional shift vector is purely spatial.

For the coordinate one-forms the change of coordinates (2.1) implies

(2.3) |

Consequently, the infinitesimal squared line element is given by

(2.4) |

where is the induced metric on the spatial slices and the component fields depend on the spacetime coordinates

(2.5) |

From this expression we read off the relation between the spacetime metric and the ADM fields

(2.6) |

Here the scalar products are with respect to the spatial metric . For completeness, we note that the decomposition of the determinant, appearing in the spacetime volume, is given as .

## 3 The RG equation for foliated spacetimes

In the sequel, we will adopt the viewpoint that the gravitational degrees of freedom are carried by the Lapse function , the shift vector and the spatial metric and derive the Wetterich equation for the component fields. We start with a discussion of symmetries and possible gauge fixings in subsection 3.1 before deriving the actual flow equation in subsection 3.2. This construction will lead to a FRGE which intrinsically implements the foliation structure of spacetime and captures the RG flow of the projectable case of Hořava-Lifshitz gravity [18].^{3}^{3}3As discussed in the introduction, the theory space underlying the non-projectable version of Hořava-Lifshitz gravity admits an additional class of interaction invariants constructed from the new building block . Since in this case there are additional subtleties concerning the closure of the constraint algebra [37, 38] it will not be discussed further at this point.

### 3.1 Symmetries and gauge fixing

Since symmetries play a crucial role when constructing a FRGE, we start with a systematic discussion of the diffeomorphism symmetry in the ADM framework.

#### 3.1.1 Classical gravity in the ADM formalism

Under a general coordinate transformation Diff() the spacetime metric transforms according to with

(3.1) |

Here denotes the Lie derivative of with respect to the -dimensional vector . The decomposition (2.2) then allows to write in terms of its time and spatial parts and

(3.2) |

Combining this split and the ADM decomposition (2.6) allows us to determine the transformation behavior of the component fields under Diff()

(3.3) | ||||

For completeness, we note

(3.4) |

One observes that, while Diff() acts linearly on the metric , the non-linearity of the ADM decomposition (2.6) leads to a non-linear transformation law for the component fields.

The gauge freedom can be exploited to adopt the proper-time gauge [39]

(3.5) |

This gauge choice still leaves the freedom to choose coordinate transformations which satisfy and . Note that these equations encode the freedom to choose a coordinate system on the initial slice of the foliation. At the level of the path integral they are typically fixed by the corresponding boundary conditions, see [40, 41] for a more detailed discussion. Since we are not interested in surface effects, we will neglect this point in the subsequent discussion.

#### 3.1.2 Hořava-Lifshitz gravity: the projectable case

Hořava-Lifshitz gravity [18] encodes the gravitational degrees of freedom in terms of the ADM fields. Considering the projectable version of the theory, the key difference is that here only the metric on the spatial slices and the shift vector are spacetime dependent fields, while the lapse function depends on time only and is constant along

(3.6) |

Moreover, the symmetry group is restricted to foliation-preserving diffeomorphisms Diff(). In this case the vector appearing in the Lie derivative (3.1) is restricted to the form

(3.7) |

The variations of the component fields under foliation-preserving diffeomorphisms are those of (3.1.1) restricted by the fact that and are space independent

(3.8) |

while

(3.9) |

In contrast to the full diffeomorphisms (3.1.1) the foliation-preserving diffeomorphisms act linearly on the component fields. This will lead to considerable simplifications when applying the background-field method later on.

Again, we can use the gauge freedom to adopt the proper-time gauge (3.5). This fixes the Diff() up to the residual transformations and , i.e., the choice of coordinate system on the initial slice. Thus, upon gauge-fixing, the field content of projectable Hořava-Lifshitz gravity and foliated QEG are identical. In this sense the transition from the field content (2.5) and gauge-symmetries (3.2) of the classical ADM formalism to (3.6) and (3.7) can be understood as a partial gauge-fixing of the former. Owed to their identical off-shell field content, the Wetterich equation for the ADM formulation of classical gravity and projectable Hořava-Lifshitz gravity will look identical. We stress, however, that the two theories are different in the sense that the latter allows a larger class of admissible interaction functionals, since the requirement of invariance under foliation-preserving diffeomorphisms is less restrictive then demanding invariance under Diff(). At this stage we proceed by constructing the Wetterich equation on , which retains Diff() as a background symmetry.

#### 3.1.3 Gauge fixing in the background field formalism

The consistent quantization of the theory requires gauge fixing the symmetries (3.8) in order to restrict the integration in the path integral to physically inequivalent configurations. For our purpose it is most convenient to implement this gauge fixing via the background field method [42], see [1] for a detailed review. This construction ensures that the resulting effective action contains interaction monomials that are invariant under (3.8) only.

When implementing the background field method the quantum fields (3.6) are split into fixed but arbitrary background fields and fluctuations around this background

(3.10) |

Note that the fluctuation fields are not assumed to be small in any sense. No expansion in powers of is implied in this split.

The central element of the background field method is that the symmetry transformations can be distributed between the background and fluctuation fields in different ways. Quantum gauge transformations leave the background invariant and attribute (3.8) completely to the fluctuation field,

(3.11) |

Here the variations are given by (3.8) with substituted by . It is this set of symmetries that have to be gauge-fixed.

Background gauge transformations on the other hand act on both the background and fluctuation fields

(3.12) |

This transformation plays the role of an auxiliary symmetry, which is retained by all the terms entering into the path integral. Its purpose is to ensure that the effective action contains interaction monomials that are invariant under Diff() only. It thereby suffices that the background transformations become identical to the ones of the quantum field once the fluctuations are set to zero. This leaves some freedom in the choice of . Explicitly, we adopt

(3.13) |

for the background fields while the fluctuations are taken to transform as

(3.14) |

Here denotes the Lie derivative on the spatial slices which contains only spatial derivatives. Note that here the -terms have all been incorporated in the transformation of the background field. While this implies that the background fields in (3.13) do not transform as the corresponding tensors, this still constitutes a choice of the background split since it reduces to the transformations (3.8) in the limit of vanishing fluctuation fields. Combining these transformation laws shows that is given by (3.8). Here it is essential that Diff() acts linearly on .

In the next step, we construct a gauge fixing term that implements the proper-time gauge in the background field formalism. By definition, this term has to break the quantum gauge transformation (3.11) while being invariant under the background gauge transformation (3.12). A straightforward computation then establishes that

(3.15) |

indeed satisfied these requirements. In the Landau-limit, , becomes a delta-distribution which eliminates the fluctuations of the lapse and shift vector, implementing the background proper-time gauge

(3.16) |

The ghost action, exponentiating the Faddeev-Popov determinant arising from (3.15) can then be found in the standard way

(3.17) |

The vector ghosts are functions of the spacetime, while the scalar ghosts depend on time only. The background lapse function has been distributed in such a way that the ghost action is invariant under background gauge transformations with the ghosts transforming as scalars and vectors, respectively,

(3.18) |

### 3.2 The Functional RG Equation

After discussing the symmetries of our gravitational theory, we are now in the position to derive the Wetterich equation encoding the RG flow on . The construction follows the standard derivation [24, 7], see, e.g., [43, 44] for pedagogical reviews in the context of gravity.

#### 3.2.1 Defining the effective average action

Our starting point is a generic action build from the multiplet (3.6) and invariant under foliation-preserving diffeomorphisms (3.8). We then consider the scale-dependent generating functional for the connected Green functions

(3.19) |

Here and are given by (3.15) and (3.17), respectively and the measure consists of the integration over the gravitational fluctuations and the ghost contributions. The source term

(3.20) |

is invariant with respect to background-Diff() if the sources transform covariantly with respect to time-reparametrizations and via the Lie derivative with respect to spatial diffeomorphisms . Here indices are raised and lowered with the spatial background metric. For convenience, we absorb the extra powers of appearing in the lapse and shift terms into the sources, setting and . The sources can then be collectively written as

(3.21) |

The essential piece in eq. (3.19) is the IR cutoff for the gravitational multiplet and the ghosts

(3.22) |

The cutoff operators and serve the purpose of discriminating between the high-momentum and low-momentum fluctuations. Following [7], we use the eigenvalues of the Laplacian constructed from the background multiplet to discriminate these modes. Eigenmodes of with eigenvalues are integrated out without suppression whereas modes with small eigenvalues are suppressed by a momentum dependent mass term. Generally, the have the structure

(3.23) |

Here is a background-field dependent matrix that ensures the invariance of (3.22) with respect to the background-gauge transformations. The dimensionless shape function interpolates between and . Convenient choices are, e.g., the exponential cutoff or the optimized cutoff (A.12).

In the next step, we construct the -dependent classical fields

(3.24) |

as the expectation value of the fluctuation fields. These are easily found as variations of with respect to the sources

(3.25) |

The extra factors included in (3.21) ensure that the classical lapse and shift transform as the corresponding quantum fields

(3.26) |

As usual, we assume that one can invert the relations (3.25) and solve for the sources as functionals of the classical fields and, parametrically, of the background. The Legendre transform of reads

(3.27) |

The effective average action is then obtained from by subtracting the cutoff action with the classical fields inserted

(3.28) |

#### 3.2.2 The Wetterich equation

The derivation of the Wetterich equation encoding the -dependence of starts with the connected two-point function

(3.29) |

which is matrix-valued in field space. Correspondingly, the Hessian of (3.27) is given by

(3.30) |

where the index takes the value zero for commuting and one for anti-commuting fields , respectively. The Legendre transform (3.27) implies that (3.29) and (3.30) are each others inverse in a functional sense

(3.31) |

Taking the derivative of (3.19) with respect to the RG “time” then gives

(3.32) |

Here denotes the two-point correlator of the fluctuation fields (including the ghost fields) and includes an integration over loop momenta and a sum over internal indices. The two-point correlator is related to the connected two-point function and the classical fields, (3.25), via

(3.33) |

Expressing the flow (3.32) in terms of the effective average action (3.28) then yields the desired functional renormalization group equation for Hořava-Lifshitz gravity

(3.34) |

Here the Hessian denotes the second derivative of with respect to the fields . Both and are matrix valued in field space and the supertrace includes an integral over loop momenta and a sum over field space. The FRGE (3.34) constitutes the central result of this section.

We close this subsection by discussing the limits of the RG flow implied by (3.34). Since the ’s vanish for , the limit of brings us back to the standard effective-action functional which still depends on two sets of fields and . The ordinary effective action with one argument is obtained from this functional by setting the expectation value of the fluctuation fields [42]

(3.35) |

Besides the FRGE (3.34) the effective average action also satisfies an exact integro-differential equation, which can be used to find the limit of the average action [7]:

(3.36) |

Intuitively, this limit can be understood from the observation that for all quantum fluctuation in the path integral are suppressed by an infinite mass term. Thus, in this limit no fluctuations are integrated out and agrees with the microscopic action supplemented by the gauge fixing and ghost actions, also see [45] for more details.

#### 3.2.3 Symmetries preserved by the RG flow

The main advantage in the use of the background field method is that the functional and, as a result, also is invariant under background foliation-preserving diffeomorphisms, when all its arguments transform according to their transformation rules

(3.37) |

Note that here, contrary to the quantum-gauge transformation (3.11), also the background fields transform according to their corresponding symmetries. The invariance (3.37) is a consequence of

(3.38) |

which in turn follows from (3.32) if one uses the invariance of , , and under the background transformations (3.12) and (3.18). At this point, we assume that the functional measure in (3.32) is invariant under Diff(.

The background-gauge invariance of , expressed in eq. (3.37), is of enormous practical importance. It implies that a RG flow, starting from a background foliation-preserving diffeomorphism invariant at a scale , will not generate interactions that violate this background symmetry dynamically. Nevertheless, even if the initial action is simple, the RG flow will generate all sorts of local and non-local terms in which are consistent with the symmetries.

We close this section with the following important remark. Owed to the requirement that the cutoff must be quadratic in the fluctuation fields, it generically seems impossible to construct a Wetterich-type FRGE which employs a linear background split and preserves a non-linear symmetry as a background symmetry. In principle it should be possible, however, to construct a FRGE for the gravitational multiplet (2.5) where the full Diff() is realized as a background symmetry. Since the corresponding symmetry transformations (3.1.1) are non-linear, we expect that this construction will either require a generalization of the background gauge fixing procedure to non-linear symmetries [46] or a non-linear background split [1]. From the flow equation for gravity in the metric formalism [7], we expect that this will lead to a functional renormalization group equation on a subspace which is closed under RG transformations. While constructing the corresponding flow equation is certainly interesting we will not embark on this construction at this point.

## 4 Signature effects in the gravitational RG flow

In principle the Wetterich equation (3.34) constitutes an exact RG equation on which is, however, notoriously difficult to solve. A common technique to find approximate solutions of the equation which do not rely on an expansion in a small parameter consists in truncating the effective average action by restricting to a finite set of running coupling constants. In this section we will derive the beta functions in the simplest gravitational setting, the foliated Einstein-Hilbert truncation. This setup allows the direct comparison of the approximate RG flows on and . Owed to the foliated background, our derivation uncovers many structures that are well-known in the context of quantum field theory at finite temperature [32, 33, 34].

### 4.1 The ADM-decomposed Einstein-Hilbert truncation

The ansatz for corresponding to the ADM-decomposed Einstein-Hilbert truncation is of the general form^{4}^{4}4In the terminology of [47] this corresponds to a single-metric computation. Results for metric flows that take into account the effect of higher order terms in the fluctuation fields have recently been reported in [48, 49].

(4.1) |

where we have approximated the gauge-fixing and ghost part of the effective average action by their classical expressions (3.15) and (3.17). For the gravitational part we adopt the Einstein-Hilbert action in spacetime dimensions. Expressing

(4.2) |

in terms of the gravitational multiplet (3.6) the corresponding action reads

(4.3) |

Here the are the classical counterparts of the full quantum fields

(4.4) |

Moreover

(4.5) |

is the extrinsic curvature, , and and denote the covariant derivative and the intrinsic curvature of the -dimensional spatial slices constructed from , respectively. The functional form (4.3) can easily be established by substituting the ADM decomposition (2.6) into the standard Einstein-Hilbert action written in terms of the covariant metric . As in the standard Einstein-Hilbert case (4.3) contains two scale-dependent coupling constants, the Newton constant and the cosmological constant .

The action (4.3) is, by construction, invariant under the full diffeomorphism group Diff(). This symmetry fixes the relative coefficients between the extrinsic and intrinsic curvature terms to the form (4.3). When restricting the symmetry group to the foliation-preserving diffeomorphisms (3.8), the term, the term and the term are invariant on their own. Thus the anisotropic case of Hořava-Lifshitz gravity allows to introduce further coupling constants in front of the terms. The study of their RG flow will be subject to a future publication [36].

### 4.2 Constructing the functional traces

Substituting the ansatz (4.1) into the FRGE (3.34), we observe from its left hand side that the scale dependence of and can be read off from the coefficients of the spacetime volume and the curvature on the spatial slices . Thus it is sufficient to project the traces appearing on the right hand side onto these two curvature invariants. We stress that when computing the beta functions of the theory the geometric quantities merely act as a bookkeeping devices. No physical meaning should be attached to them.

The first step in evaluating the functional traces arising from (4.1) constitutes in computing the Hessian with respect to the fluctuation fields. This calculation can be simplified in a number of ways. Firstly, we adopt Landau gauge by sending , . In this limit the gauge-fixing term (3.15) is converted to a -function which freezes the fluctuation fields , . Thus and decouple and do not contribute to the traces on the right hand side of the flow equation. Thus,

(4.6) |

where the Hessian in the ghost trace is given by

(4.7) |

The second simplification originates from the observation that we can set the fluctuation fields once the second variations are computed. Thus we can use the background covariance of the construction to carry out the computation in a specific background geometry. The choice of background has to be general enough to distinguish the two interaction monomials carrying the information of the RG flow. A natural choice uses

(4.8) |

i.e., the direct product of a “time”-circle with periodicity and a time-independent maximally symmetric sphere . In terms of the background multiplet, this choice implies

(4.9) |

Owed to the maximal symmetry of the background curvatures on the spatial slices satisfy

(4.10) |

In order to lighten our notation, we will drop the prefix of the intrinsic curvature from now on.