New Heat Kernel Method in Lifshitz Theories
NNew Heat Kernel Method in Lifshitz Theories
Kevin T. Grosvenor a , Charles Melby-Thompson b , Ziqi Yan ca Max-Planck-Institut f¨ur Physik komplexer Systeme andW¨urzburg-Dresden Cluster of Excellence ct.qmatN¨othnitzer Str. 38, 01187 Dresden, Germany b Institut f¨ur Theoretische Physik und Astrophysik, Julius-Maximilians-Universit¨at W¨urzburg,Am Hubland, 97074 W¨urzburg, Germany c Nordita, KTH Royal Institute of Technology and Stockholm UniversityRoslagstullsbacken 23, SE-106 91 Stockholm, Sweden
E-mail: [email protected] , [email protected] , [email protected] Abstract:
We develop a new heat kernel method that is suited for a systematic studyof the renormalization group flow in Hoˇrava gravity (and in Lifshitz field theories ingeneral). This method maintains covariance at all stages of the calculation, which isachieved by introducing a generalized Fourier transform covariant with respect to thenonrelativistic background spacetime. As a first test, we apply this method to computethe 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 togeneral scalar operators and evaluate their one-loop effective action. The covariant heatkernel method that we develop also directly applies to operators with spin structuresin arbitrary dimensions. a r X i v : . [ h e p - t h ] J a n ontents
1. Introduction 22. Covariant Heat Kernel Method on Riemannian Manifolds 4
3. Covariant Heat Kernel Method on Foliated Spacetimes 11
4. Lifshitz Scalar Field Theories in
Dimensions 20 z = 2 Lifshitz Point 28
5. Conclusions 32A. Procedural Example: Second Heat Kernel Coefficient 33B. Expression of (cid:101) E (4) for General Scalar Operators 38C. Sign Conventions in the Literature 42 – 1 – . Introduction Hoˇrava gravity has attracted much attention over the years since its original proposalin 2009 [1]. Allowing that Lorentz invariance be absent at high energies, higher spa-tial derivative terms are introduced in Hoˇrava gravity in such a way that the theoryexhibits an anisotropic scaling of space and time. This construction results in a classof unitary and power-counting renormalizable quantum theories of gravity, which arefundamentally nonrelativistic. It is then hoped that the theory is driven to a relativis-tic fixed point at low energies by classical renormalization group (RG) flow. Moreover,due to the existence of a preferred time direction in Hoˇrava gravity, it admits a rigidnotion of causality and its Hamiltonian and diffeomorphism constraints form a closedalgebra, at least for some versions of the theory. This is in contrast to the situationin General Relativity, where these constraints fail to form a Lie algebra, leading tointrinsic conceptual and technical difficulties in defining a relativistic quantum gravitynonperturbatively.Despite the desirable features of Hoˇrava gravity, it nevertheless still faces the chal-lenge of explaining the vast catalogue of experimental data that highly constrainsLorentz violation [4]. For example, it would require a great deal of fine-tuning tomatch the speeds of light for different species of low energy probes [5]. Different mech-anisms for the emergence of low-energy Lorentz invariance have been discussed in theliterature (see, e.g., [6] and references therein). One mechanism is to impose super-symmetry at high energies [7]. Another promising mechanism relies on the existenceof a strongly-coupled fixed point, in which case it is possible for the strong dynamicsto drive the theory sufficiently quickly towards the Lorentzian fixed point [8, 9].Regardless of whether or not Hoˇrava gravity is phenomenologically viable for de-scribing our universe in 3+1 dimensions, it still has a plethora of important applicationsin the context of the AdS/CFT correspondence for nonrelativistic systems [10–14], theCausal Dynamical Triangulation approach to quantum gravity [15, 16], the formulationof membranes at quantum criticality [2], the geometric theory of Ricci flow on Rieman-nian manifolds [17], and so on. For both phenomenological and formal interests, it isimportant to understand the quantum structure of Hoˇrava gravity.In recent years, there has been some intriguing progress made in mapping out theRG flow of the so-called “projectable” Hoˇrava gravity, in which the lapse function isindependent of space [18–22]. First, the perturbative renormalizability of projectableHoˇrava gravity has been proven in [18]. Then, it was shown in [19] that there is an This has been demonstrated for the projectable case [2], where an extra degree of freedom ispresent in addition to the ones in General Relativity. For the nonprojectable case, the situation ismore intricate (see, e.g., [3]). – 2 –symptotically free ultraviolet (UV) fixed point in 2 + 1 dimensions. In addition, theanomalous dimension of the cosmological constant in the same theory was evaluated in[20]. Finally, partial results of the RG flow in 3+1 dimensions were later obtained in [21].Nevertheless, knowledge of the RG flow in Hoˇrava gravity without the projectabilitycondition remains rather limited. To reveal this more general quantum structure, whichis essential for many phenomenological and formal applications of Hoˇrava gravity, it isurgent to look for more efficient methods.One powerful technique for evaluating the one-loop effective action for quantumfield theories (QFTs) on a curved background spacetime is the heat kernel method.Different heat kernel methods have been applied in the past to evaluate quantum cor-rections in QFTs around a Lifshitz fixed point [11, 23–25]. These heat kernel meth-ods rely heavily on the Zassenhaus formula, which is the inverse companion of theBaker–Campbell–Hausdorff formula, and in general involves a rather tedious proce-dure of evaluating nested commutators of covariant time and space derivatives. In somespecial cases, various simplifications are available. However, when the projectabilitycondition is lacking, the calculation of the full effective action can be inefficient andeven computationally impractical. We therefore devote this paper to the formulation of a new heat kernel method,which we find better suited to non-projectable cases. We aim to use this new methodto systematically study the RG flow in Hoˇrava gravity. In general, this method can beapplied to evaluate the one-loop effective action in Lifshitz QFTs on a nonrelativisticbackground geometry.The method we develop in this paper generalizes the algorithm for relativistic the-ories introduced by Gusynin in [26, 27] (also see [28–32]) to nonrelativistic models. InGusynin’s approach, general covariance is maintained at all stages of the calculationby introducing a generalized Fourier transform that is covariant with respect to thebackground geometry. This type of covariant Fourier transforms has been studied ex-tensively by mathematicians in the symbolic calculus for pseudodifferential operators Unlike General Relativity, which is topological in 2 + 1 dimensions, Hoˇrava gravity in 2 + 1dimensions has one propagating degree of freedom. For example, [24] utilizes a clever inverse Laplace transform method to express heat kernel co-efficients of anisotropic operators, whose spatial derivative term is just a power of the isotropic one,in terms of the heat kernel coefficients of the isotropic operator. However, as a consequence, in thismethod, one has to perform the Zassenhaus expansion twice: once before the inverse Laplace trans-form and again afterwards, essentially to undo the expansion in the first place. Except in the casewhen temporal and spatial covariant derivatives can be chosen to commute, for example if one isinterested only in spatial curvature terms in the effective action, this method is technically challengingto implement. Furthermore, it does not apply to anisotropic operators which are not simply powersof isotropic ones, such as the case we study in § – 3 –33, 34], which replaces the phase factor in the usual Fourier transform with a phasefunction that resembles the world function in the Schwinger-DeWitt technique (see § §
2, we give a detailed review of the heatkernel method of Gusynin [26, 27], which is designed for relativistic QFTs. In §
3, wegeneralize Gusynin’s method to Lifshitz field theories. In §
4, we apply the new heatkernel method developed in § z = 2 Lifshitz scalars on a curved background geometry in 2 + 1 dimensions,as a first step towards the evaluation of RG flows in nonprojectable Hoˇrava gravityin 2 + 1 dimensions. The full result of this calculation is recorded in Appendix B.We conclude our paper in §
5. In Appendix A, we show a procedural example thatdetails the evaluation of heat kernel coefficients. In Appendix C, we clarify some of thedifferent sign conventions used in the literature to clear up any potential confusion.
2. Covariant Heat Kernel Method on Riemannian Manifolds
We start with a review of Gusynin’s covariant heat kernel method for quantum fieldtheories on a Riemannian manifold, following closely [26, 27].
Consider a QFT on a d -dimensional Riemannian manifold M equipped with a metric g µν for a field configuration Φ A ( x ), where x µ , µ = 0 , · · · , d − M . We take the time to be imaginary. The field Φ A can have a generaltensor structure and A denotes collectively its indices. Splitting Φ A into its backgroundvalue Φ A and the fluctuation ξ A around this background value, we have the followingexpansion: Φ A = Φ A + √ (cid:126) ξ A + O ( (cid:126) ) . (2.1) More precisely, coordinates on a patch of M as part of an atlas on M . – 4 –e assume that the QFT is described by an action principle S [Φ], which expands as S [Φ] = S [Φ ] + √ (cid:126) (cid:90) d d x √ g S (1) A ( x ) ξ A ( x )+ (cid:126) (cid:90) d d x √ g ξ A ( x ) O AB ξ B ( x ) + O ( (cid:126) / ) , (2.2)where g = det g µν and √ g S (1) A ( x ) = δS [Φ] δ Φ A ( x ) , √ g O AB δ ( d ) ( x − x (cid:48) ) = δ S [Φ] δ Φ A ( x ) δ Φ B ( x (cid:48) ) . (2.3)Here, O AB is in general a pseudodifferential operator, which is a linear operator O AB = ∆ (cid:88) k =1 α µ ··· µ k AB ( x ) ∇ µ · · · ∇ µ k + α AB ( x ) , (2.4)where ∆ is a positive even integer and is the order of the differential operator and ∇ µ is the covariant derivative compatible with the metric g µν (i.e., ∇ µ g νρ = 0).To be specific, we assume that Φ A is bosonic. However, the derivation workssimilarly for fermionic fields. The associated (Euclidean) path integral with a source J A ( x ) is Z [ J ] = (cid:90) d Φ exp (cid:26) − (cid:126) − (cid:104) S [Φ] − (cid:90) d d x √ g J A ( x ) Φ A ( x ) (cid:105)(cid:27) . (2.5)Choosing the background value Φ A such that S (1) A = J A , we obtain that, in the semi-classical limit (cid:126) → Z [ J ] = exp (cid:26) − (cid:126) − (cid:104) S [Φ ] − (cid:90) d d x √ g J A ( x ) Φ A ( x ) (cid:105)(cid:27) (cid:0) det O AB (cid:1) − / . (2.6)We will drop the internal indices A and B in O AB in the following. We therefore readoff the quantum effective action,Γ[Φ] = S [Φ] + (cid:126) Γ + O (cid:0) (cid:126) (cid:1) . (2.7)We take the standard heat kernel representation [35] of the effective action Γ = 12 tr log (cid:0) O /µ (cid:1) Technically, O AB is a positive elliptic differential operator such that the log in the effective actionis well defined [26]. – 5 – − dds (cid:12)(cid:12)(cid:12) s =0 µ s Γ( s ) (cid:90) d d x √ g (cid:90) ∞ dτ τ s − K O ( x, x | τ ) . (2.8)Note that (2.8) is defined after analytically continuing the domain of s to include s = 0 .We have introduced the heat kernel for the operator O in (2.8), K O ( x, x | τ ) = (cid:104) x | e − τ O | x (cid:105) , (2.9)which satisfies the heat kernel equation (cid:0) ∂ τ + O (cid:1) K O = 0 .Note that the symbol K O without arguments and the name “heat kernel” willbe used interchangeably to refer to the operator e − τ O in general or to the specificcoordinate space representation of this operator, K O ( x, x | τ ) . The coefficients in the asymptotic expansion of the heat kernel K O in the high energyregime τ → + encodes important physical data such as the renormalization groupflow in the associated QFT. There exist different methods in the literature for ex-tracting these heat kernel coefficients. One most widely used method was developedby Schwinger and DeWitt, where an ansatz called the Schwinger-DeWitt expansion istaken for the heat kernel [37, 38]. The DeWitt expansion ensures the covariance ofthe calculation of the heat kernel coefficients. The original method of Schwinger andDeWitt is valid for second-order minimal operators, i.e. O = −∇ µ ∇ µ + V ( x ) . Thismethod was then generalized to higher-order nonminimal operators, for example in [39].However, this generalization was achieved at the cost of involving nonlocal operatorsin the intermediate steps of the calculations. On the other hand, the Seeley-Gilkey method for calculating the heat kernel coef-ficients uses techniques developed in the theory of pseudodifferential operators, which The asymptotic expansions of the heat kernel were first introduced by DeWitt, which resemblesHadamard’s expansion for the retarded Green’s function. It was also noted in [36] that the Hadamard-DeWitt expansion is closely related to an expansion given by Minakshisundaram in the study of ellipticoperators. Therefore, Gibbons proposed the name Hamidew (Hadamard-Minakshisundaram-DeWitt)for the heat kernel coefficients. The same type of heat kernel expansion was also developed by Seeleyand Gilkey in the study of pseudodifferential operators, which we will primarily follow in this paper.To avoid possible confusion regarding the terminology, we will stick to the neutral term “heat kernelcoefficient” in this paper. While the Schwinger-DeWitt method keeps covariance at all stages and invokes recursive methodsfor solving the heat kernel coefficients, sometimes it is also useful to evaluate the heat kernel in anon-covariant but more direct way, without relying on the recursive methods. This is usually doneby using the ordinary plane waves as in Fujikawa’s method of calculating the chiral anomaly. See, forexample, [40, 41]. – 6 –voids introducing nonlocal operators. A pseudodifferential operator O is defined byits symbol σ O ( x , k ) via (cid:104) x |O| x (cid:105) = (cid:90) d d k (2 π ) d (cid:112) g ( x ) e ik · ( x − x ) σ O ( x , k ) . (2.10)We defined k · x = k µ x µ . The heat kernel (2.9) is likewise defined by its symbol σ K O .We write the heat kernel in terms of its resolvent as e − τ O = i (cid:90) C dλ π e − τ λ (cid:0) O − λ (cid:1) − , (2.11)where C is a contour that bounds the spectrum of the operator O in the complexplane and is traversed in the counter-clockwise direction. Then, the symbol σ K O canbe exchanged with the symbol σ ( O− λ ) − of the resolvent ( O − λ ) − : K O ( x , x | τ ) = i (cid:90) C dλ π e − τλ G ( x , x | λ ) , (2.12)where G ( x , x | λ ) ≡ (cid:10) x (cid:12)(cid:12)(cid:0) O − λ (cid:1) − (cid:12)(cid:12) x (cid:11) = (cid:90) d d k (2 π ) d (cid:112) g ( x ) e ik · ( x − x ) σ ( x , k | λ ) . (2.13)Here, we have written σ ( O− λ ) − ( x, k ) as σ ( x, k | λ ) for ease of notation because this isthe only symbol that we will actually care about. Further note that (cid:2) O ( x , ∇ ) − λ (cid:3) G ( x, x | λ ) = 1 (cid:112) g ( x ) δ ( d ) ( x − x ) . (2.14)Using (2.13) and (2.14), we find e − ik · ( x − x ) [ O ( x , ∇ ) − λ ] e ik · ( x − x ) σ ( x , k | λ ) = 1 . (2.15)It is well known that the part of the heat kernel diagonal in x has the followingasymptotic expansion around τ → + [42, 43]: K O ( x , x | τ ) = ∞ (cid:88) m =0 E ( m ) ( x ) τ m − d ∆ . (2.16)The coefficients E ( n ) ( x ) are the heat kernel coefficients. Recall that ∆ denotes theorder of the operator O and is taken to be a positive even integer. To compute theheat kernel coefficients, we expand the symbol σ ( x , k | λ ) as σ = ∞ (cid:88) m =0 σ ( m ) , (2.17)– 7 –here σ ( m ) is a homogeneous function of λ and k µ , i.e., σ ( m ) ( x , b k | b ∆ λ ) = b − m − ∆ σ ( m ) ( x , k | λ ) . (2.18)Plugging (2.17) into (2.15) and then taking the rescalings λ → b ∆ λ and k µ → b k µ , thecoefficients σ ( m ) can be determined recursively by matching terms of different orders in b in the resulting equation.Finally, we take the coincidence limit x → x in (2.12), followed by the rescalings λ → τ − λ , k µ → τ − k µ . (2.19)Then, plugging in the expansions (2.16) and (2.17) into (2.12) and matching the powersin τ on both sides of the resulting equation, the heat kernel coefficients are computedto be E ( m ) ( x ) = (cid:90) d d k (2 π ) d (cid:112) g ( x ) (cid:90) C i dλ π e − λ σ ( m ) ( x , k | λ ) . (2.20)When m is odd, σ ( m ) is odd in k µ , and the corresponding E ( m ) vanishes. Therefore, E ( m ) is only nonzero when m is even, and K O ( x , x | τ ) = ∞ (cid:88) r =0 E (2 r ) ( x ) τ r − d ∆ . (2.21)Note that the phase factor k · ( x − x ) in (2.13) is not invariant with respect togeneral coordinate transformations. Therefore, while the heat kernel coefficients arecovariant with respect to general coordinate transformations in x , the definition of thesymbol σ in (2.13) is not. We will now describe Widom’s solution [33, 34], which leadsto a covariant definition of the symbol σ , and its application by Gusynin in [26] to thecomputation of heat kernel coefficients. As pointed out by Gusynin in [26], there is a covariant generalization of the Seeley-Gilkey method by applying the covariant Fourier transform as in [33, 34]. The idea isto generalize the phase factor k · ( x − x ) to a real-valued phase function (cid:96) : M × T ∗ ( M ) → R , (2.22)such that (cid:96) ( x ; x , k ) is linear in k ≡ k µ dx µ ∈ T ∗ x ( M ) for any fixed x , x ∈ M . Incontrast, there is no canonical meaning to linearity in the variable x ∈ M . Widomproposed in [33, 34] the following definition for linearity of (cid:96) in x µ in the vicinity of the– 8 –oint x : for each k ∈ T ∗ x ( M ) , the symmetrized n -th covariant derivative vanishes at x for n (cid:54) = 1 , namely, ∇ µ (cid:96) ( x ; x , k ) (cid:12)(cid:12) x = x = k µ , (2.23a) ∇ ( µ · · · ∇ µ n ) (cid:96) ( x ; x , k ) (cid:12)(cid:12) x = x = 0 , n = 0 , , , · · · . (2.23b)It is understood that ∇ µ acts on x . Here, ∇ ( µ · · · ∇ µ n ) has all the indices symmetrized.For example, T ( µν ) = (cid:0) T µν + T ν µ (cid:1) for any tensor T µν . Using (2.23), higher derivativesof the phase function (cid:96) ( x ; x , k ) with respect to x can be expressed in terms of fewerderivatives at x = x and thus all covariant derivatives of (cid:96) are also determined in thecoincidence limit x → x . Such an (cid:96) ∈ C ∞ ( M ) always exists in a neighborhood of x due to Borel’s theorem [34].The above definition of the phase function is designed such that a covariant gen-eralization of the Taylor series can be constructed. Since (cid:96) ( x ; x , k ) is linear in k forfixed x and x , we have (cid:96) ( x ; x , k ) = k µ (cid:96) µ ( x , x ) , (2.24)where (cid:96) µ ( x , x ) defines a tangent vector. It is then straightforward to show by using(2.23) that, for a given function f ( x ) ∈ C ∞ ( M ) [34], f ( x ) = ∞ (cid:88) n =0 n ! ∇ µ · · · ∇ µ n f ( x ) (cid:96) µ ( x , x ) · · · (cid:96) µ n ( x , x ) . (2.25)for x in a neighborhood of x . This generalizes the usual Taylor theorem, replacing ∂ µ with the covariant derivative ∇ µ .With the phase function (cid:96) in hand, we are able to replace the expressions in § O is now related to its symbol σ O ( k , x ; x ) via (cid:104) x |O| x (cid:105) = (cid:90) d d k (2 π ) d (cid:112) g ( x ) e i (cid:96) ( k, x ; x ) σ O ( x ; x , k ) , (2.26)which replaces (2.10) and defines a covariant Fourier transform. It then follows thatthe matrix element of the operator O resolvent introduced in (2.13) is replaced with G ( x , x | λ ) ≡ (cid:10) x (cid:12)(cid:12)(cid:0) O − λ (cid:1) − (cid:12)(cid:12) x (cid:11) = (cid:90) d d k (2 π ) d (cid:112) g ( x ) e i (cid:96) ( x ; x , k ) σ ( x ; x , k | λ ) . (2.27) More precisely, (2.25) holds for the analytic germ of the function f ( x ) . We take this for grantedthroughout this paper. – 9 –e defined σ ( x ; x , k | λ ) ≡ σ ( O− λ ) − ( x ; x , k ) . The heat kernel K O takes the sameform as in (2.12), i.e., K O ( x , x | τ ) = i (cid:90) C dλ π e − τλ G ( x , x | λ ) . (2.28)Identical to (2.14), we have (cid:2) O ( x , ∇ ) − λ (cid:3) G ( x, x | λ ) = 1 (cid:112) g ( x ) δ ( d ) ( x − x ) , (2.29)but with G represented as in (2.27). For (2.29) to be satisfied, we require the followinganalogue of (2.15): (cid:2) O ( x , ∇ + i ∇ (cid:96) ) − λ (cid:3) σ ( x ; x , k | λ ) = I ( x , x ) , (2.30)where the biscalar I ( x , x ) satisfies (cid:90) d d k (2 π ) d e i (cid:96) ( x ; x , k ) I ( x , x ) = δ ( d ) ( x − x ) . (2.31)There are a couple of things to keep in mind. Firstly, O may depend on the fieldsand their derivatives. The derivatives acting on fields are not shifted by ∇ → ∇ + i ∇ (cid:96) as in (2.30). Only the derivatives which act on G are shifted in this way; these areshifted for the simple reason that when a derivative acts on G , it acts on the symbol σ as well as the phase factor e i(cid:96) and pulls down a factor of i ∇ (cid:96) from e i(cid:96) .A second point to keep in mind is that O implicitly carries the bundle space indexstructure and should be written as O AB . Therefore, I also carries the index structure I AB ( x , x ) . The covariant derivative ∇ µ then also picks up the bundle space indicesand reads (cid:0) ∇ µ (cid:1) AB . For example, when Yang-Mills theory on a curved background isunder consideration, the index A contains both a spacetime index µ (the vector indexof the gauge field) and a gauge group index a . We have (cid:0) ∇ ρ (cid:1) µaνb = δ νµ (cid:2) δ ba ∇ ρ − i ( t c ) ab A cρ (cid:3) , (2.32)where t a are generators in the gauge group and A aµ the gauge field. From (2.31) and inanalogue of (2.23), as in [26], we require that I AB satisfy I AB ( x , x ) = AB , (2.33a) (cid:0) ∇ ( µ (cid:1) A B · · · (cid:0) ∇ µ n ) (cid:1) A n B n I B n C ( x , x ) (cid:12)(cid:12) x = x = 0 , n ≥ . (2.33b)Here, AB consists of Kronecker symbols with the bundle index A . The covariantderivatives only act on the first bundle index in I AB . The equations in (2.33) fully– 10 –etermine the coincidence limit x → x of any number of covariant derivatives actingon I .Finally, we take the expansion of σ as in (2.17), σ ( x ; x , k | λ ) = ∞ (cid:88) m =0 σ ( m ) ( x ; x , k | λ ) , (2.34)where σ ( m ) satisfies (2.18) and can be determined recursively. The heat kernel coeffi-cients are then determined as in (2.20).
3. Covariant Heat Kernel Method on Foliated Spacetimes
We will formulate a generalization of the covariant Fourier transform as well as thealgorithm for calculating heat kernel coefficients in cases when the spacetime carriesa foliation structure. Here, we will consider a foliation structure in which the leavesof the foliation have codimension one, but the discussion can be naturally generalizedto other cases. We start with introducing a few essential ingredients about foliatedspacetimes that will be useful for subsequent discussion. • Metric Decomposition
Consider the geometry of a ( D + 1)-dimensional spacetime manifold M . Let M be foliated by leaves Σ of codimension one, which means that M is equipped with anatlas of coordinate systems x µ = ( t , x i ) , i = 1 , · · · , D . The transition functions arerestricted to be foliation preserving, such that (cid:101) t = (cid:101) t ( t ) , (cid:101) x i = (cid:101) x i ( t , x ) . (3.1)This foliated spacetime is naturally described by a rank D degenerate symmetric tensor h µν together with a choice of a vector n µ such that h µν n ν = 0 . Given this data, thereexist unique objects h µν and n µ that satisfy the relations n µ n µ = 1 , h µν n ν = 0 , h µρ h ρν + n µ n ν = δ µν . (3.2)It is important to keep in mind that n µ is not retrieved from n µ by “lowering” withrespect to any metric. For one thing, there is no actual metric, and secondly, if we wereto “lower” the index on n µ using h µν , the result would be zero. Instead, h µν and n µ Here, we choose an “all positive” convention. However, note that another common convention isto choose n µ n µ = − h µρ h ρν − n µ n ν = δ µν . – 11 –re simply objects which are defined by the above relations. Define a two-tensor field g µν and its inverse g µν over M as follows: g µν = n µ n ν + h µν , g µν = n µ n ν + h µν . (3.3)Note that g µν does not always qualify as a metric field. Frequently, it is convenient to refer to these objects without reference to a coordi-nate system. For this purpose, we introduce the tensor notation h = h µν ∂ µ ⊗ ∂ ν , n = n µ ∂ µ , (3.4a) h = h µν dx µ ⊗ dx ν , n = n µ dx µ , (3.4b)where ∂ µ ∈ T x ( M ) and dx µ ∈ T ∗ x ( M ) . Moreover, ∂ µ ( dx ν ) = dx ν ( ∂ µ ) = δ νµ . Theoverline (underline) indicates that the object is a ( n, , n )-tensor) living inthe tensor product of n tangent (cotangent) spaces. In this notation, the “givens” are h and n with relation h ( . , n ) = 0 . The “derived quantities” are h and n , satisfyingthe relations n ( n ) = 1 , h ( . , n ) = 0 , h ( . , h ) + n ⊗ n = , , (3.5)where in h ( . , h ) only the first cotangent space over which the tensor product in h takesis acted on. Moreover, the (1,1)-tensor , is ∂ µ ⊗ dx µ when a coordinate systemis chosen. Similarly, the two-tensor g µν and its inverse g µν are associated with thequantities g and with its inverse g , respectively, satisfying g ( . , g ) = , . Then, wehave (3.3) cast in the coordinate independent form g = n ⊗ n + h , g = n ⊗ n + h . (3.6)Finally, n is assumed to be orthogonal to the hypersurface Σ , which means that thereexists an acceleration vector field a ∈ T ∗ ( M ) with a ( n ) = 0 such that dn + a ∧ n = 0 . (3.7)Now, let us pick a convenient set of coordinates adapted to the foliation structure.Choose a holonomic basis { e i } for T x (Σ) ⊂ T x ( M ) , satisfying[ e i , e j ] = 0 , n ( e i ) = 0 . (3.8) One famous example in which g µν , assembled in this way from n µ and h µν , does not form a bonafide metric field is realized in Newton-Cartan theory [44]. – 12 –ote that the index i is a label that runs from 1 to D and it does not denote thecoordinate components of each basis element. The dual basis { ω i } (generally non-holonomic) is defined by the relations ω i ( e j ) = δ ij , ω i ( n ) = 0 . (3.9)The spatial metric in this basis is h ij ≡ h ( e i , e j ) . (3.10)Then, { e µ } = { n, e i } and { ω µ } = { n, ω i } serve as bases for T ( M ) and T ∗ ( M ) , respec-tively. In coordinates ( t, x i ) adapted to this basis, we have ω i = dx i + N i dt n = N dt , h = h ij ω i ⊗ ω j , (3.11a) e i = ∂ i n = 1 N ( ∂ t − N i ∂ i ) , h = h ij e i ⊗ e j , (3.11b)where h ik h kj = δ ij and where N and N i are quantities defined by the above equations.In the case when g is a bona fide metric field, N and N i are, respectively, the lapsefunction and shift vector of the ADM metric decomposition and it is customary, evenin the Newton-Cartan literature, to continue referring to these quantities as the lapseand shift. • Covariant Derivatives
We introduce the Levi-Civita connection D with respect to the two-tensor g . Wealso require that the manifold is torsionless: for any vector fields X , Y ∈ T ( M ) , thetorsion tensor D X Y − D Y X − [ X , Y ] is set to zero. Define the Christoffel symbols via D e µ e ν = e λ Γ λµν . The Christoffel coefficients can be computed by noting the identity2 g (cid:0) D X Y , Z (cid:1) = X (cid:0) g ( Y , Z ) (cid:1) + Y (cid:0) g ( X, Z ) (cid:1) − Z (cid:0) g ( X, Y ) (cid:1) + g (cid:0) [ X, Y ] , Z (cid:1) + g (cid:0) [ Z, X ] , Y (cid:1) + g (cid:0) [ Z, Y ] , X (cid:1) , (3.12)where the vector fields X, Y , Z can run through { n , e i } . This yields, in components,Γ kij ≡ γ kij , Γ ijn = K ij , Γ inj = K ij + L ij , Γ inn = a i , Γ nij = − K ij , Γ njn = 0 Γ nnj = − a j , Γ nnn = 0 , (3.13)where γ kij denotes the Levi-Civita connection of h ij on T (Σ), the Latin indices areraised and lowered by h ij and h ij , and K ij = 12 N ( ˙ h ij − ∇ i N j − ∇ j N i ) , L ij = ∂ j N i N , a i = − ∂ i NN , (3.14)– 13 –re the extrinsic curvature of Σ in M , the shift variation, and the acceleration vector,respectively. Here, ∇ i is the covariant derivative with respect to h ij . The spatialRiemann curvature tensor is defined as usual: R ijk(cid:96) = ∂ k Γ i(cid:96)j − ∂ (cid:96) Γ ikj + Γ ikm Γ m(cid:96)j − Γ i(cid:96)m Γ mkj . (3.15)The Ricci tensor is defined as R ij = R kikj and the Ricci scalar is R = R ii .We spare a few words on the geometrical meaning of L ij . First note that thetorsionless condition 0 = ∇ n e i − ∇ e i n − [ n, e i ] implies [ n, e i ] = e j L ji − n a i . Therefore, e i L ij = Π([ n, e j ]) , (3.16)where Π is the projection operator from T ( M ) to T (Σ) that distributes over tensorproducts, with Π( e µ ) = δ iµ e i . This implies that L ij encodes the action of infinitesimaldiffeomorphisms by n on T (Σ).Now, we define covariant derivatives adapted to the foliation. We have alreadyintroduced the “covariant spatial derivative” ∇ i , which is related to D acting on M as ∇ i T ≡ Π( D e i T ) , (3.17)for an arbitrary tensor field T . On a C ∞ function, ∇ i acts as ∂ i . We also need anotion of “covariant time derivative,” which we denote by d n . For a C ∞ function f ,we simply take d n ( f ) ≡ n ( f ) = n µ ∂ µ f ; (3.18)for a given spatial tensor field T , it is convenient to define its covariant time derivativeto be the Lie derivative with respect to n , projected onto Σ , i.e., d n T ≡ Π (cid:0) L n T (cid:1) = Π (cid:0) [ n , T ] (cid:1) . (3.19) • Expressions in Components
For practical calculations, we will eventually need to write various derivatives actingon tensor fields in component form. To facilitate such calculations, we note that d n and ∇ i act on the (co)tangent space basis as ∇ i e j = γ kij e k , d n e i = L ji e j , (3.20a) ∇ i ω j = − γ jik ω k , d n ω i = − L ij ω j . (3.20b)Furthermore, the commutator [ d n , ∇ i ] acts on the same bases as[ d n , ω j ∇ j ] e i = (cid:0) − a j d n e i + M kij e k (cid:1) ⊗ ω j , (3.21a)– 14 – d n , ω j ∇ j ] ω k = (cid:0) − a j d n ω i − M kij ω i (cid:1) ⊗ ω j , (3.21b)where the (1,2)-tensor M is defined for any vector fields X and Y as M ( X , Y ) ≡ [ n , ∇ X ] Y − ∇ d n X Y + a ( X ) d n Y . (3.22)In components, we find M kij = ( ∇ i − a i ) K jk + ( ∇ j − a j ) K ik − ( ∇ k − a k ) K ij , (3.23)where we have lowered the first index on M using h ij for neatness.Using (3.20) and (3.21), it follows that, in components, ∇ k T i ··· i m j ··· j n = ∂ k T i ··· i m j ··· j n + m (cid:88) r =1 Γ i r k(cid:96) T i ··· i r − (cid:96)i r +1 ··· i m j ··· j n − n (cid:88) r =1 Γ (cid:96)kj r T i ··· i m j ··· j r − (cid:96)j r +1 ··· j n , (3.24a) d n T i ··· i m j ··· j n = n µ ∂ µ T i ··· i m j ··· j n + m (cid:88) r =1 L i r (cid:96) T i ··· i r − (cid:96)i r +1 ··· i m j ··· j n − n (cid:88) r =1 L (cid:96)j r T i ··· i m j ··· j r − (cid:96)j r +1 ··· j n , (3.24b)[ d n , ∇ k ] T i ··· i m j ··· j n = − a k d n T i ··· i m j ··· j n + m (cid:88) r =1 M i r k(cid:96) T i ··· i r − (cid:96)i r +1 ··· i m j ··· j n − n (cid:88) r =1 M (cid:96)kj r T i ··· i m j ··· j r − (cid:96)j r +1 ··· j n . (3.24c) We have learned from § k µ ( x µ − x µ ) and that isadapted to the foliation. The phase function (cid:96) defined in (2.22) does not qualify as anappropriate choice since it treats all coordinates equally, leading to differential symbolsthat do not naturally decompose with respect to the foliation. A phase function that isdesigned to already take into account the foliation structure will significantly simplifythe computation of heat kernel coefficients. For examples of (3.24b), we have d n h ij = 2 K ij and d n h ij = − K ij . – 15 –ecall that x µ = ( t , x i ) , i = 1 , · · · , D . In the following, we use x to denote x µ and x to denote x i . We start with defining a temporal phase function χ ( x ; x , ν ) ∈ R that generalizes the flat limit phase factor ν ( t − t ) , with ν the frequency. We defined ν = ν n ∈ T ∗ x ( M ) . (3.25)We require that χ ( x ; x , ν ) be linear in ν for fixed x , x ∈ M . It is natural to require χ to be constant on each spatial slice, which implies that all spatial derivatives vanishidentically, i.e., ∇ i · · · ∇ i k χ ( x ; x , ν ) = 0 , k ≥ . (3.26)It then follows that d (cid:96)n ∇ i · · · ∇ i k χ ( x ; x , ν ) = 0 , k ≥ , (cid:96) ≥ . (3.27)Together with (3.24c), these relations imply that D∇ i χ = 0 for any operator D builtout of d n ’s and ∇ j ’s, which constitutes a significant simplification.Moreover, we require that χ satisfy d n χ ( x ; x , ν ) (cid:12)(cid:12) x = x = ν , (3.28a) d kn χ ( x ; x , ν ) (cid:12)(cid:12) x = x = 0 , k = 0 , , , · · · , (3.28b)such that χ reduces to ν ( x − x ) in the flat limit. The conditions in (3.26) and (3.28)determine all d n and ∇ i derivatives of χ in the coincidence limit x → x . The existenceof such a function χ in a neighborhood of x is ensured by Borel’s theorem.Next, we define a spatial phase function ψ ( x ; x , q ) ∈ R that generalizes the flatlimit phase factor q i ( x i − x i ) , with q i the spatial momentum. We define q = q i ω i ∈ T ∗ x ( M ) . (3.29)We require that ψ ( x ; x , q ) be linear in q for fixed x , x ∈ M . On the spatial slice, wecan take the same conditions as in (2.23) for (cid:96) , with ∇ i ψ ( x ; x , q ) (cid:12)(cid:12) x = x = q i , (3.30a) ∇ ( i · · · ∇ i k ) ψ ( x ; x , q ) (cid:12)(cid:12) x = x = 0 , k = 0 , , , · · · . (3.30b)We further require that these conditions on ψ be trivially covariantly transported fromone leaf of the foliation to the next (i.e., by d n ), by demanding that d (cid:96)n ∇ ( i · · · ∇ i k ) ψ ( x ; x , q ) (cid:12)(cid:12) x = x = 0 , (cid:96) ≥ , k ≥ . (3.31)– 16 –gain, the above conditions determine all d n and ∇ i derivatives of ψ in the coincidencelimit x → x , and such a ψ exists in a neighborhood of x by Borel’s theorem.A covariant generalization of the Taylor series can be constructed. Since χ ( x ; x , ν )is linear in ν and ψ ( x ; x , q ) is linear in q i , we have χ ( x ; x , ν ) = ν Φ ( x , x ) , ψ ( x ; x , q ) = q i Φ i ( x , x ) , (3.32)where Φ µ ( x , x ) defines a tangent vector. We then find, for a given function f ( x ) ∈ C ∞ ( M ) and for x in a neighborhood of x , f ( x ) = ∞ (cid:88) k, (cid:96) =0 (cid:96) ! d kn ∇ i · · · ∇ i (cid:96) f ( x ) (cid:2) Φ ( x , x ) (cid:3) k Φ i ( x , x ) · · · Φ i (cid:96) ( x , x ) . (3.33)We define a covariant Fourier transform for a pseudodifferential operator O as (cid:104) x |O| x (cid:105) = (cid:90) dν πN d D q (2 π ) D (cid:112) h ( x ) e i Φ( x ; x , { ν, q } ) σ O ( x ; x , { ν, q } ) , (3.34)where σ O defines the symbol of O andΦ( x ; x , { ν , q } ) ≡ χ ( x ; x , ν ) + ψ ( x ; x , q ) , (3.35)which reduces to the phase factor ν ( t − t ) + q i ( x i − x i ) in the flat limit. The matrixelement of the operator O resolvent is G ( x , x | λ ) ≡ (cid:104) x | ( O − λ ) − | x (cid:105) = (cid:90) dν πN d D q (2 π ) D (cid:112) h ( x ) e i Φ( x ; x , { ν, q } ) σ ( x ; x , { ν, q } | λ ) , (3.36)where σ ( x ; x , { ν, q } | λ ) ≡ σ ( O− λ ) − ( x ; x , { ν, q } ) . As in (2.29), (3.36) implies that (cid:2) O ( x ; d n , ∇ ) − λ (cid:3) G ( x, x | λ ) = 1 N (cid:112) h ( x ) δ ( D +1) ( x − x ) . (3.37)Therefore, (cid:2) O ( x ; d n + id n Φ , ∇ + i ∇ ψ ) − λ (cid:3) σ ( x ; x , { ν, q } ) = I ( x , x ) , (3.38)where I ( x , x ) satisfies (cid:90) dν π d D q (2 π ) D e i Φ( x ; x , { ν, q } ) I ( x , x ) = δ ( D +1) ( x − x ) . (3.39)– 17 –sing the conditions on the phase functions χ and ψ introduced earlier in this subsec-tion, we find from (3.39) that I AB ( x , x ) = AB , (3.40)and (cid:0) d n (cid:1) A B · · · (cid:0) d n (cid:1) A (cid:96) B (cid:96) (cid:0) ∇ ( i (cid:1) C D · · · (cid:0) ∇ i k ) (cid:1) C k D k I D k E ( x , x ) (cid:12)(cid:12) x = x = 0 , (3.41)where (cid:96) + k ≥ I and the covariant derivatives.For the scalar case without internal gauge symmetries, we simply have I ( x , x ) = 1 .The above conditions on I determine all derivatives of I . Finally, we discuss how to compute the heat kernel K O ( x , x | τ ) = (cid:104) x | e − τ O | x (cid:105) = i (cid:90) C dλ π e − τλ G ( x , x | λ ) (3.42)for an operator O in the coincidence limit x = x , defined over a foliated spacetime.Here, C is a contour that bounds the spectrum of the operator O in the complex planeand is traversed in the counter-clockwise direction. In the following, we require that O have an anisotropic scaling exponent z at high energies, i.e. the UV fixed point isinvariant under the rescaling of the spacetime coordinates t → b − z t , x → b − x . (3.43)Typically, we consider O taking the form O = − d n − ( − z ζ ∇ z + · · · , (3.44)where we omitted the index structure of the spatial covariant derivatives and we notethat there can be different terms that involve the same number of ∇ ’s. Moreover,“ · · · ” denotes terms that contain fewer derivatives. We define the order of O to be thedimension of its highest order term measured in the dimension of ∇ . For example, theorder of O in (3.44) is ∆ = 2 z .We start with expanding the symbol σ (cid:0) x ; x , { ν , q } (cid:12)(cid:12) λ (cid:1) as σ (cid:0) x ; x , { ν , q } (cid:12)(cid:12) λ (cid:1) = ∞ (cid:88) m =0 σ ( m ) (cid:0) x ; x , { ν , q } (cid:12)(cid:12) λ (cid:1) , (3.45)– 18 –here σ ( m ) is a homogeneous function of λ , ν , and q i , satisfying σ ( m ) (cid:0) x ; x , { b z ν , b q } (cid:12)(cid:12) b ∆ λ (cid:1) = b − m − ∆ σ ( m ) (cid:0) x ; x , { ν , q } (cid:12)(cid:12) λ (cid:1) . (3.46)Taking the rescalings λ → b ∆ λ , ν → b z ν , and q i → b q i in (3.38), supplemented with χ → b z χ and ψ → b ψ , we find ∞ (cid:88) m =0 b − m − ∆ D b σ ( m ) ( x ; x , { ν , q } | λ ) = I ( x , x ) , (3.47)where D b ≡ O (cid:0) x ; d n + i b z d n χ + i b d n ψ , ∇ + i b ∇ ψ (cid:1) − b ∆ λ . (3.48)Expand D b with respect to b , such that D b = ∆ (cid:88) (cid:96) =0 b ∆ − (cid:96) D ( (cid:96) ) (cid:0) x ; d n , ∇ (cid:1) . (3.49)Plugging (3.49) back into (3.47), we find ∞ (cid:88) m =0 ∆ (cid:88) (cid:96) =0 b − m − (cid:96) D ( (cid:96) ) σ ( m ) = I . (3.50)Matching the order in b on both sides of (3.50) gives rise to a series of recursion relationsthat can be used to determine σ ( m ) in the coincidence limit. We will demonstrate howthis works in explicit detail when we apply this method to the case of a Lifshitz operatoron a foliated spacetime (see, e.g., (4.16)).Next, plugging (3.36) into (3.42), and taking the rescalings λ → τ − λ , ν → τ − z/ ∆ ν , q i → τ − / ∆ q i , (3.51)we find K O ( x , x | τ ) = ∞ (cid:88) m =0 E ( m ) ( x ) τ m − ( D + z )∆ , (3.52)where E ( m ) ( x ) are the heat kernel coefficients generalized to anisotropic operators,which are given by E ( m ) ( x ) = (cid:90) dν πN d D q (2 π ) D √ h (cid:90) C i dλ π e − λ σ ( m ) (cid:0) x ; x , { ν , q } (cid:12)(cid:12) λ (cid:1) . (3.53)– 19 –gain, if m is odd, then either the number of frequency factors or the number ofmomentum factors in the integrand of E ( m ) is odd. Thus, E ( m ) = 0 when m is odd andthe heat kernel can be written as K O ( x , x | τ ) = ∞ (cid:88) r =0 E (2 r ) ( x ) τ r − ( D + z )∆ . (3.54)Note that, when z = 1 , the asymptotic expansion (3.52) around τ → + reduces to(2.16). This concludes our formal discussion on the covariant heat kernel method forpseudodifferential operators that involves both d n and ∇ .
4. Lifshitz Scalar Field Theories in
Dimensions
In this section, we will use the simplest examples of scalars on an anisotropic grav-itational background in 2 + 1 dimensions to illustrate how the covariant heat kernelmethod proceeds in practice. However, the method developed in this paper is applicableto general dimensions as well as gauge vector fields and more general tensor fields.
We first consider a single real scalar field at a z = 2 Lifshitz fixed point in 2 + 1dimensions. We take the time to be imaginary. We focus on the following action: S = 12 (cid:90) dt d x N √ h (cid:104)(cid:0) d n φ (cid:1) + (cid:0) (cid:3) φ (cid:1) (cid:105) = 12 (cid:90) dt d x N √ h φ O φ , (4.1)where (cid:3) ≡ ∇ i ∇ i and O ( x ; d n , ∇ ) = − d n − K d n + 1 N ( x ) (cid:3) N ( x ) (cid:3) = − d n − K d n + (cid:3) − a i ∇ i (cid:3) + (cid:0) a i a i − ∇ i a i (cid:1) (cid:3) . (4.2)Here, x = ( t , x ) . As we remarked previously, the operator O can depend on the fieldsand their derivatives. Here, for example, O depends on time and spatial derivatives ofthe spatial metric, lapse, and shift, in the form of the extrinsic curvature K , accelerationvector a i , and the derivative of the acceleration vector, ∇ i a i . The derivatives in thoseexpressions do not continue on to act on the scalar field φ . Therefore, when the latter iscovariantly Fourier-transformed, only the derivatives that actually act on φ get shifted,not the derivatives acting on the background fields.– 20 –ake the engineering dimensions for space and time as[ t ] = − , [ x i ] = − . (4.3)The scaling dimensions of the derivatives, background, and scalar fields, and the oper-ator O then follow:[ d n ] = 2 , [ ∇ i ] = 1 , [ N ] = [ N i ] = [ h ij ] = [ φ ] = 0 , ∆ = [ O ] = 4 . (4.4)Note that the action (4.1) is classically invariant under the local anisotropic Weyltransformation, N → e − z Ω( x ) N , N i → e − x ) N i , h ij → e − x ) h ij , φ → φ . (4.5)The infinitesimal anisotropic Weyl transformation of the effective action Γ as introducedin (2.7) can be anomalous, with [10]Ω( x ) (cid:18) z N δδN + 2 N i δδN i + 2 h ij δδh ij (cid:19) Γ( x ) = A ( x ) , (4.6)where A denotes the Weyl anomaly. To determine the form of the anomaly, we intro-duce a ghost field c and the nilpotent BRST operator s = c (cid:18) z N δδN + 2 N i δδN i + 2 h ij δδh ij (cid:19) , (4.7)which acts on c trivially. Then, the Wess-Zumino consistency condition [45] requiresthat s (cid:90) dt d x N √ h A c = 0 . (4.8)If an anomaly term is BRST exact, i.e., this term can can be expressed as a BRSTvariation of a local operator, then it can be subtracted from the action, thus cancelingthe associated anomaly. As a result, the possible anomalies of interest are those in thecohomology of the BRST differential s . It turns out that all terms in this cohomologygroup are linear combinations of the following two terms [10, 11]: K ij K ij − K , (cid:0) R + ∇ i a i (cid:1) , (4.9)which are of dimension ∆ = 4 , measured in spatial momentum. On a two-dimensionalleaf, the Riemann and Ricci tensors are related to the Ricci scalar as R ijk(cid:96) = 12 R (cid:0) h ik h j(cid:96) − h i(cid:96) h jk (cid:1) , R ij = 12 R h ij . (4.10)– 21 –n [11, 46], it is shown that there is one time-derivative term that is BRST exact, K + d n K = 1 N √ h (cid:104) ∂ t (cid:0) √ h K (cid:1) − ∂ i (cid:0) √ h N i K (cid:1)(cid:105) . (4.11)This is a total derivative term. It has also been shown in [10, 11, 46] that there are fiveindependent BRST exact terms with only spatial derivatives, which are total derivativesthat take the form 1 N √ h ∂ i (cid:0) √ h F iI (cid:1) , I = 1 , · · · , , (4.12)where, in the basis chosen in [10], F i = ∇ i (cid:2) N ( R − ∇ i a i ) (cid:3) , (4.13a) F i = − N ( R − ∇ j a j ) a i , (4.13b) F i = −∇ i (cid:0) N ∇ j a j (cid:1) , (4.13c) F i = N ( a + ∇ j a j − a j ∇ j ) a i , (4.13d) F i = − N a a i . (4.13e)We are interested in the heat kernel coefficient associated with the anisotropic Weylanomalies (4.9) and the total derivative terms (4.11) and (4.12), and reproducing theresult in [10, 11] using the new method developed here. All such terms have scalingdimension four and are thus marginal. Therefore, the heat kernel coefficient of interestis E (4) . In fact, it is shown in [11] that A ( x ) = − E (4) ( x ) .In the following, we compute E (4) using the covariant heat kernel method. Westart by plugging (4.2) into (3.49), and setting z = 2 and ∆ = 4 , which gives D b = O ( x ; d n + i b z d n χ + i b d n ψ , ∇ + i b ∇ ψ ) − b ∆ λ = − ( d n + i b d n χ + i b d n ψ ) − K ( d n + i b d n χ + i b d n ψ )+ (cid:2)(cid:0) ∇ i + i b ψ i (cid:1)(cid:0) ∇ i + i b ψ i (cid:1)(cid:3) − (cid:2) a i (cid:0) ∇ i + i b ψ i (cid:1) − a i a i + ∇ i a i (cid:3) (cid:0) ∇ j + i b ψ j (cid:1)(cid:0) ∇ j + i b ψ j (cid:1) − b λ = (cid:88) (cid:96) =0 b − (cid:96) D ( (cid:96) ) ( x ; d n , ∇ ) , (4.14)where D (0) = ( d n χ ) + (cid:0) ψ i ψ i (cid:1) − λ , (4.15a) D (1) = 2 d n χ d n ψ − i (cid:104) ψ i ψ j ψ ij + ψ i ψ i (cid:0) ψ j ∇ j + (cid:3) ψ (cid:1)(cid:105) + 2 i a i ψ i ψ j ψ j , (4.15b)– 22 – (2) = − i d n χ d n − (cid:104) ψ i ψ j ∇ i ∇ j + ψ i ψ i (cid:3) + 2 (cid:0) ψ i ψ jj + 2 ψ j ψ ij (cid:1) ∇ i (cid:105) − (cid:0) ψ ii (cid:1) − i (cid:0) d n χ + Kd n χ + i d n ψ d n ψ (cid:1) − (cid:104) ψ ij ψ ij + ψ i (cid:0) ψ jji + ψ ijj (cid:1)(cid:105) + 2 a i (cid:104) ψ i ψ jj + ψ j (cid:0) ψ ij + 2 ψ i ∇ j + ψ j ∇ i (cid:1)(cid:105) + (cid:0) ∇ j a j − a (cid:1) ψ i ψ i , (4.15c) D (3) = − i (cid:0) d n ψ d n + d n ψ + K d n ψ (cid:1) + i ψ iijj + 2 i (cid:2) ψ i ( ∇ i (cid:3) + (cid:3) ∇ i ) + (cid:0) ψ jji + ψ ijj (cid:1) ∇ i + ψ ii (cid:3) + 2 ψ ij ∇ i ∇ j (cid:3) − i a i (cid:0) ψ ijj + ψ jj ∇ i + 2 ψ j ∇ i ∇ j + 2 ψ ij ∇ j + ψ i (cid:3) (cid:1) − i (cid:0) ∇ j a j − a (cid:1)(cid:0) ψ ii + 2 ψ i ∇ i (cid:1) , (4.15d) D (4) = − ( d n + K ) d n + (cid:3) − (cid:0) a i ∇ i − a i a i + ∇ i a i (cid:1) (cid:3) . (4.15e)We introduced the notation ψ i ··· i k ≡ ∇ i · · · ∇ i k ψ . Plugging (4.15) into (3.50), we find the following recursion relations: D (0) σ (0) = I , (4.16a) D (0) σ (1) + D (1) σ (0) = 0 , (4.16b) D (0) σ (2) + D (1) σ (1) + D (2) σ (0) = 0 , (4.16c) D (0) σ (3) + D (1) σ (2) + D (2) σ (1) + D (3) σ (0) = 0 , (4.16d) D (0) σ ( k ) + D (1) σ ( k − + D (2) σ ( k − + D (3) σ ( k − + D (4) σ ( k − = 0 , (4.16e)for k ≥
4. From this set of recursion relations, we solve for σ (0) , · · · , σ (4) and calculatethe associated heat kernel coefficients. Clearly, to calculate σ ( m ) , we must also calculatevarious derivatives acting on σ (0) up to σ ( m − . Generically, we will need up to m derivatives on σ (0) , m − σ (1) , and so on up to one derivative on σ ( m − (withtime derivatives counting as z spatial derivatives). To compute these, we simply takederivatives of the appropriate recursion relation. For example, to calculate σ (1) , we willneed ∇ i σ (0) , which we compute by taking ∇ i of (4.16a). We detail this procedure asfollows: Coincidence Limit of σ (0) . From (4.16a), using the coincidence limits of derivativesof χ , ψ and I given in § d n χ (cid:12)(cid:12) x = x = ν , ∇ i ψ (cid:12)(cid:12) x = x = q i , I (cid:12)(cid:12) x = x = 1 , (4.17) Covariant derivatives are sometimes denoted in the literature by a semicolon preceding the in-dices of the derivatives. In this case, the order of the indices reads left-to-right the order in whichthe covariant derivatives actually act , not the order in which they would be written left-to-right: ∇ i · · · ∇ i k ψ = ψ ; i k ··· i . This is, for example, the convention used in [43]. – 23 –e find σ (0) ( x ; x , { ν , q } | λ ) = 1 ν + | q | − λ ≡ G . (4.18)This is essentially the propagator of the theory.From (3.53), we find that the heat kernel coefficient E (0) is given by E (0) ( x ) = (cid:90) dν d q (2 π ) √ h (cid:90) C i dλ π e − λ σ (0) (cid:0) x ; x , { ν , q } (cid:12)(cid:12) λ (cid:1) = 116 π . (4.19) Coincidence Limit of σ (1) . Since ψ is a scalar, the two derivatives in ∇ i ∇ j ψ com-mute. Therefore, the defining relation ∇ ( i ∇ j ) ψ (cid:12)(cid:12) x = x in (3.30b) implies ∇ i ∇ j ψ (cid:12)(cid:12) x = x = 0 . (4.20)The defining relation (3.31), with the number of time derivatives set to (cid:96) = 1 and thenumber of spatial derivatives set to k = 0, reads d n ψ (cid:12)(cid:12) x = x = 0 . (4.21)Then, from (4.16b), we obtain σ (1) (cid:12)(cid:12) x = x = − G D (1) σ (0) (cid:12)(cid:12) x = x = 4 i G | q | q i ∇ i σ (0) (cid:12)(cid:12) x = x − i G a i q i | q | . (4.22)To derive the coincidence limit of ∇ i σ (0) in (4.22), we act ∇ i on both sides of (4.16a)and then take the coincidence limit. When ∇ i acts on the phase functions in D (0) ,it will produce factors involving ∇ i d n χ , whose coincidence limit can be derived from(3.27) by commuting ∇ i past d n using (3.24c). The result is ∇ i d n χ (cid:12)(cid:12) x = x = ν a i . (4.23)Moreover, ∇ i I (cid:12)(cid:12) x = x = 0 by (3.41), so the right hand side vanishes. In the end, we find ∇ i σ (0) (cid:12)(cid:12) x = x = − G ν a i . (4.24)Plugging (4.24) back into (4.22) gives σ (1) (cid:12)(cid:12) x = x = − i G a i q i | q | (cid:0) G ν (cid:1) . (4.25)As expected, σ (1) is odd in q i and will vanish upon integration over momentum. Thus,the associated heat kernel coefficient vanishes: E (1) ( x ) = 0 . (4.26)– 24 – oincidence Limit of σ (2) . From the conditions given in § d n ∇ i ∇ j χ = d n ∇ ψ (cid:12)(cid:12) x = x = ψ ( ijk ) (cid:12)(cid:12) x = x = 0 , (4.27)we find the following coincidence limits: ∇ i ∇ j d n χ (cid:12)(cid:12) x = x = ν (cid:0) a i a j + ∇ i a j (cid:1) , ∇ i d n ψ (cid:12)(cid:12) x = x = 0 , (4.28a) ψ ijk (cid:12)(cid:12) x = x = 16 R (cid:0) h ij q k + h ik q j − q i h jk (cid:1) , (4.28b)Using (4.16a) and (4.16b) and the above coincidence limits of χ and ψ , we derive d n σ (0) (cid:12)(cid:12) x = x = 4 G | q | q i q j K ij , ∇ i ∇ j σ (0) (cid:12)(cid:12) x = x = 23 G | q | (cid:0) q i q j − | q | h ij (cid:1) R − G ν (cid:104) ∇ i a j + 2 a i a j (cid:0) − G ν (cid:1)(cid:105) , ∇ i σ (1) (cid:12)(cid:12) x = x = − i G | q | q i R − i G | q | q j (cid:104) a i a j G ν (cid:0) − G ν (cid:1) + ∇ i a j (cid:0) G ν (cid:1)(cid:105) . Finally, plugging the coincidence limits that we have derived so far into (4.16c), we find σ (2) (cid:12)(cid:12) x = x = − G | q | (1 − G | q | ) R + i G ν (cid:0) h ij + 8 G q i q j | q | (cid:1) K ij − G ∇ i a i | q | (cid:0) G ν (cid:1) + 8 G ∇ i a j q i q j (cid:104) | q | − (1 − G | q | ) ν (cid:105) − G ( a · q ) (cid:104) | q | + 2 (cid:0) − G | q | (cid:1) ν − G (1 − G | q | ) ν (cid:105) + G a | q | (cid:104) − G ν (cid:0) − G ν (cid:1)(cid:105) . (4.29)We also give a step-by-step derivation of σ (2) in Appendix A. From (3.53), we obtainthe heat kernel coefficient E (2) , E (2) ( x ) = (cid:90) d q (2 π ) √ h (cid:90) C i dλ π e − λ σ (2) (cid:0) x ; x , { ν , q } (cid:12)(cid:12) λ (cid:1) = 148 π / (cid:0) R + ∇ i a i − a i a i (cid:1) , (4.30)where we used the integral I z, D, j, k, (cid:96) ≡ (cid:90) d D q (2 π ) D √ h q i · · · q i k (cid:90) dν π ν j (cid:90) C i dλ π e − λ ( ν + | q | z − λ ) (cid:96) – 25 – 2 − k (4 π ) D +1 z Γ (cid:0) j + (cid:1) Γ (cid:0) kz + D z (cid:1) Γ (cid:0) k + D (cid:1) Γ (cid:0) (cid:96) (cid:1) h i ··· i k . (4.31)This is a straightforward integral, which is essentially Gaussian once the integral over λ is performed using the residue theorem. It does require us to convert a product ofmomenta with free indices to a product of pairwise-contracted momenta. Provided thatthe rest of the integral is rotationally invariant in momentum space or, in other words,is a function only of | q | , then we can perform the following replacement: q i · · · q i k → Γ (cid:0) D (cid:1) k Γ (cid:0) k + D (cid:1) q k h i ··· i k , (4.32)where h i ··· i k is the symmetrized combination of h ij ’s. For example, h ijk(cid:96) ≡ h ij h k(cid:96) + h ik h j(cid:96) + h i(cid:96) h jk . (4.33)In the relativistic case, the appropriate integral is the same, but without the ν integralin (4.31) and setting z = 1. Notice that setting z = 1, the factor of Γ (cid:0) kz + D z (cid:1) in thenumerator of (4.31) cancels the factor of Γ (cid:0) k + D (cid:1) in the denominator and, besides theoverall power of 4 π , the dependence on D completely drops out of this integral. Thisis why the heat kernel coefficients in the relativistic case do not explicitly depend onthe dimension besides the overall power of 4 π . For other values of z , however, the heatkernel coefficients will explicitly depend on the spacetime dimension in addition to theoverall power of 4 π . Of course, we will not expose this extra dimension-dependence inthis work because our calculations are performed specifically in D = 2.Looking back at E (2) in (4.30), note that the combination N √ h ( ∇ i a i − a i a i ) isa total derivative and, in two dimensions, the term N √ h R is also a total derivative.Therefore, E (2) vanishes once integrated over a spacetime with no boundaries. Ourresults for E (0) and E (2) agree with the ones in Appendix D of [11]. Heat Kernel Coefficient E (4) . Similarly, a more involved process that we imple-mented using xAct [47] on Mathematica leads to the results E (3) = 0 and E (4) = − π (cid:18) K ij K ij − K (cid:19) − π (cid:0) d n K + K (cid:1) + 1 N √ h ∂ i f i , (4.34)where f i = 1960 π √ h (cid:110) N (cid:2) R + 2 (cid:3) ) − ∇ j a j − a j ∇ j (cid:3) a i (cid:111) = − √ h (cid:0) F i + 5 F i − F i − F i (cid:1) , (4.35)– 26 –nd F iI , I = 2 , · · · , ∇ i a j = −∇ i ∇ j ln N = ∇ j a i . (4.36)The combinations K ij K ij − K and d n K + K are BRST-invariant terms classifiedin (4.9) and (4.11). In particular, as indicated in (4.11), d n K + K is a total derivativeterm. Therefore, E (4) in (4.34) is a linear combination of the BRST-invariant terms in(4.9) ∼ (4.12), exactly reproducing the result in [10, 11]. This matching provides uswith a rather strong check of our method.The contribution to the diagonal heat kernel in (3.54) is K O ( x , x | τ ) = E (0) ( x ) τ − + E (2) ( x ) τ − / + E (4) ( x ) + O ( τ / ) . (4.37)While E (0) and E (2) contribute power law divergences to the one-loop effective action(2.8) that can be set to zero by adding in appropriate counterterms, E (4) contributes alog divergence. The one-loop effective action defined in (2.8) givesΓ = − dds (cid:12)(cid:12)(cid:12) s =0 µ s Γ( s ) (cid:90) dt d D x N √ h (cid:90) /M dτ τ s − E (4) + finite= − π (cid:18) log M µ − γ E (cid:19) (cid:90) dt d x N √ h (cid:18) K ij K ij − K (cid:19) + finite , (4.38)where γ E is the Euler-Mascheroni constant. We have introduced a cutoff for the τ -integral, with M effectively acting as a UV cutoff, while µ acts as an infrared regulator.The theory exhibits an anomaly under the local anisotropic scale transformation (inmomentum), A = 132 π (cid:18) K ij K ij − K (cid:19) . (4.39)The fact that the second term in (4.9) does not show up in the anomaly at this one-loop order can be explained by the detailed balance condition [10]. This corroboratesprevious results which were computed using the (non-covariant) plane wave methodfor evaluating the heat kernel [11] and also from the holographic renormalization cal-culation [10–12]. In contrast to the previous heat kernel calculation in [11], however,our method is fully covariant and does not assume any special ansatz for the spacetimemetric, which may prove useful for a systematic study of more complicated scenarios.Note that E (2) contributes a power-law divergence to the effective action. TheWess-Zumino consistency condition then requires E (2) to be zero up to total derivatives.There are four terms that can show up in E (2) : K , R , a i a i and ∇ i a i , which all havescaling dimension 2. It will be convenient to choose K , R , ∇ i a i − a i a i and a i a i as the– 27 –asis elements instead. The reason for this is that N K is a total time derivative andthe combination N √ h ( ∇ i a i − a i a i ) is a total space derivative and can be dropped.In two spatial dimensions, N √ h R also happens to be a total derivative and can bedropped as well. The only coefficient left is that of a i a i , which is forced to vanish bythe Wess-Zumino consistency condition (once part of it is combined with ∇ i a i in theform of ∇ i a i − a i a i ). Indeed, our result for E (2) in (4.30) bears this out.Table 1 in [24] reports some heat kernel coefficients for Lifshitz scalars with z = 2and z = 3 in D = 2 and D = 3 . In particular, for z = D = 2, they report that thecoefficient of a i a i is − . However, the operator studied in [24] is not the same as theone we have just studied here and which was studied previously in [10, 11]. The spatialderivative part of the classically Weyl-invariant operator is N − (cid:3) N (cid:3) . In contrast, theoperator studied in [24] is ∆ zx , where∆ x = − N √ h ∂ i N √ h h ij ∂ j = N − ∇ i N ∇ i . (4.40)For z = 2, the operator ∆ x is equal to N − ∇ i N ∇ i N − ∇ j N ∇ j , which is not the sameas N − (cid:3) N (cid:3) . However, in the next subsection, we will study the most general z = 2scalar operator in 2 + 1 dimensions and this will obviously include the one consideredin [24]. Thus, we will return to the question of their a i a i coefficient at the end of thenext subsection. z = 2 Lifshitz Point
We have tested our method in the last subsection for a well-known example. Now, wetake one step forward and consider the most general scalar operator in 2+1 dimensions,around a z = 2 Lifshitz fixed point. The associated action principle is S = 12 (cid:90) dt d x N √ h φ (cid:101) O φ , (4.41)where (cid:101) O = O + U d n + (cid:102) W ijk ∇ i ∇ j ∇ k + X ij ∇ i ∇ j + (cid:101) Y i ∇ i + Z . (4.42)Here, O is defined in (4.2), which we record here: O = − d n − K d n + 1 N (cid:3) N (cid:3) = − d n − K d n + (cid:3) − a i ∇ i (cid:3) + (cid:0) a i a i − ∇ i a i (cid:1) (cid:3) . (4.43) However, we will impose the condition that the operator does not mix time and space derivatives. – 28 –ote that (cid:102) W ijk = (cid:102) W ikj and X ij = X ji . Under the condition U = (cid:102) W ijk = X ij = (cid:101) Y i = Z = 0 , (4.44)the operator in (4.42) reduces to the one in (4.2). Note that the action in (4.41) typi-cally breaks the anisotropic Weyl invariance and thus we expect more general operatorsto appear in the one-loop effective action. One may also introduce a more general op-erator V ijk(cid:96) ∇ i ∇ j ∇ k ∇ (cid:96) that replaces (cid:3) in the operator (cid:101) O defined in (4.42). However,since the coefficeint V ijk(cid:96) is dimensionless and should be covariant under the foliationpreserving diffeomorphisms, the operator V ijk(cid:96) ∇ i ∇ j ∇ k ∇ (cid:96) can be reduced to the onesalready included in (cid:101) O . In more complicated theories, a richer structure may arise andrequire introducing a general V ijk(cid:96) .We further note that (cid:102) W ijk ∇ i ∇ j ∇ k φ = (cid:102) W ( ijk ) ∇ i ∇ j ∇ k φ − (cid:0)(cid:102) W ijj − (cid:102) W jji (cid:1) R ∇ i φ . (4.45)It is therefore convenient to define W ijk ≡ (cid:102) W ( ijk ) , Y i ≡ (cid:101) Y i − (cid:0)(cid:102) W ijj − (cid:102) W jji (cid:1) R . (4.46)Using (4.46), we rewrite (cid:101) O in (4.42) as (cid:101) O = O + U d n + W ijk ∇ i ∇ j ∇ k + X ij ∇ i ∇ j + Y i ∇ i + Z . (4.47)This change of basis allows us to have a much more succinct result for the heat kernelcoefficients.Following the same procedure described in § (cid:101) E (0) = E (0) and (cid:101) E (2) = E (2) + 1512 π / (cid:0) X ii − W ijj W ikk − W ijk W ijk (cid:1) + 1 N √ h ∂ i f i , (4.48)where E (2) is given in (4.30) and f i = − π / N W ijj . (4.49)A detailed derivation of (cid:101) E (2) can be found in Appendix A. We also find, using xAct [47],the result for (cid:101) E (4) in Figure 1. We will transcribe this result in Appendix B. Under thecondition (4.44), the result in Figure 1 reduces to (cid:101) E (4) = E (4) . The one-loop effectiveaction is related to (cid:101) E (4) similarly as in (4.38), withΓ = 12 (cid:18) log M µ − γ E (cid:19) (cid:90) dt d x N √ h (cid:101) E (4) + finite . (4.50)– 29 – igure 1 . Expression of (cid:101) E (4) copied from Mathematica. One particularly useful case is when W ijk = w (cid:0) a i h jk + a j h ki + a k h ij (cid:1) , (4.51)where w is a constant number. Let us write (cid:101) E (4) = E (4) + (cid:88) p =0 b p , (4.52)where E (4) is given in (4.34) and b p contains terms of p -th order in W , with p = 0 , · · · , b , · · · , b simplify significantly. The expressions for b p , p = 0 , · · · , b = 1256 π (cid:0) X ij X ij + X ii X jj (cid:1) − π (cid:0) U + 4 Z (cid:1) + 1 N √ h (cid:0) ∂ t f t + ∂ i f i (cid:1) ,b = − w π (cid:104) ∇ j X ij + 3 X jj (cid:0) ∇ i − a i (cid:1) + 2 X ij (cid:0) ∇ j − a j (cid:1) − Y i (cid:105) a i + wN √ h ∂ i f i ,b = w a i π (cid:104)(cid:0) R − X jj + 48 a − ∇ j a j (cid:1) a i − (cid:0) X ij a j + (cid:3) a i (cid:1)(cid:105) + w N √ h ∂ i f i , – 30 – = 3 w a π (cid:0) ∇ i a i − a (cid:1) + w N √ h ∂ i f i ,b = 15 w a π , (4.53)where f t = √ h π U , (4.54a) f i = √ h π (cid:110) − N i U + N (cid:2) Y i + (cid:0) a j X ij − ∇ j X ij + 3 ∇ i X jj (cid:1)(cid:3)(cid:111) , (4.54b) f i = √ h π N (cid:104)(cid:0) R + 6 a − ∇ j a j (cid:1) a i − ∇ i ∇ j a j + 4 a j ∇ i a j (cid:105) , (4.54c) f i = − √ h π N (cid:104)(cid:0) a j ∇ j − ∇ j a j + 9 a (cid:1) a i (cid:105) , (4.54d) f i = 3 √ h π N a a i . (4.54e)Let us now return to the discussion of the coefficient of a i a i reported in [24]. Aswe noted at the end of the previous subsection, the operator that they are studying isnot the classically Weyl-invariant one. In the language of this subsection, we can writethe operator considered in [24] as the Weyl-invariant one plus terms parametrized by U , W , X , Y and Z as U = W ijk = Z = 0 , (4.55a) X ij = ( ∇ i a i − a i a i ) h ij + a i a j − ∇ i a j , (4.55b) Y i = a j ∇ j a i − (cid:3) a i − R a i . (4.55c)It is easy to see that the trace of X ij is X ii = − a i a i and, therefore, the second heatkernel coefficient in (4.48) simplifies to (cid:101) E (2) = E (2) − π / a i a i . (4.56)The coefficients reported in Table 1 of [24] factors out a factor of (4 π ) − ( D +1) / . Thus,we would expect the coefficient of a i a i , which they denote by c , to be c = − π / (4 π ) / = − . (4.57)Instead, [24] gets − . Given the concerns voiced in [25] regarding the method used in[24], a resolution of this discrepancy would be most welcome.– 31 – . Conclusions In this paper we develop a new heat kernel method for calculating the one-loop effectiveaction in Lifshitz theories on a curved background geometry with anisotropic scaling.We tested this method by applying it to the computation of the anisotropic Weylanomaly for a (2 + 1)-dimensional scalar field theory around a z = 2 Lifshitz point,and corroborated the results previously found by other heat kernel methods and alsoby holographic renormalizaiton. In addition, since the form of the anisotropic Weylanomaly is highly constrained by the Wess-Zumino consistency condition, it serves asa strong check that our method applies to field theories on nonrelativistic backgroundgeometries without the projectability condition. We then took one step further andcomputed the effective action for the most general scalar operators around a z = 2fixed point in 2 + 1 dimensions.One interesting next step is to evaluate the effective action for the most general(3 + 1)-dimensional scalar operator around a z = 3 fixed point. In particular, thecomplete cohomologies for anisotropic Weyl anomalies in spacetime dimensions d ≤ z ≤ a i (i.e.,terms dependent on the spatial variation of the lapse function).Another interesting topic is the study of the Weyl anomalies of Hoˇrava gravity witha U (1) extension [48]. This type of theory was formulated covariantly using torsionalNewton-Cartan geometry in [49]. In [13, 14], boundary Weyl anomalies that take theform of Hoˇrava gravity for bulk torsional Newton-Cartan gravity are derived usingthe Fefferman-Graham expansion. It will be intriguing to reproduce and extend theseresults using our new method.We emphasize that our method directly applies to operators with spin structures,simply by turning on the bundle structure on the Lifshitz scalar field. This promisesmany applications to Yang-Mills theory and Hoˇrava gravity. For example, it will befascinating to generalize the RG flow results in [19] to non-projectable Hoˇrava gravityin 2 + 1 dimensions, which are essential ingredients for understanding the quantummembrane theory at quantum criticality proposed in [2]. Acknowledgments
We would like to thank Andrei Barvinsky, Diego Blas, Shira Chapman, Mario Herrero-Valea, and Sergey Sibiryakov for useful discussions. KTG and CMT would like tothank the hospitality of Perimeter Institute for Theoretical Physics, where part of this– 32 –ork was conducted. KTG and ZY would like to thank the hospitality of CERNfor stimulating discussions that partly motivated this work. ZY is grateful for thehospitality of Julius-Maximilians-Universit¨at W¨urzburg, where this work was initi-ated. KTG acknowledges financial support from the Deutsche Forschungsgemeinschaft(DFG, German Research Foundation) under Germany’s Excellence Strategy throughthe W¨urzburg-Dresden Cluster of Excellence on Complexity and Topology in QuantumMatter – ct.qmat (EXC 2147, project–id 390858490) as well as the Hallwachs-R¨ontgenPostdoc Program of ct.qmat. The work of CMT was supported through a researchfellowship from the Alexander von Humboldt foundation. This research is supportedin part by Perimeter Institute for Theoretical Physics. Research at Perimeter Instituteis supported in part by the Government of Canada through the Department of Inno-vation, Science and Economic Development Canada and by the Province of Ontariothrough the Ministry of Colleges and Universities.
A. Procedural Example: Second Heat Kernel Coefficient
Let us show how our method proceeds in complete detail for the calculation of E (2) .We first introduce some notation. We will use the standard semicolon notation forcovariant derivatives. However, keep in mind that because we are using the standardnotation, we will also adopt the standard way of ordering the indices: the indices areordered left-to-right in the order in which the derivatives act on the field, which isright-to-left. This is in contrast to the notation that we introduced for derivativesacting on ψ : for example, ψ ijk = ∇ i ∇ j ∇ k ψ , which can also be written as ψ ; kji . Wealso extend the notation we introduced for spatial derivatives to time derivatives: forexample, ψ ni ≡ d n ∇ i ψ . So as to reduce the clutter, let us introduce some notation.Firstly, note that the recursion relation for σ (2) involves only the operators D (0) , D (1) and D (2) , but not D (3) or D (4) , where D ( I ) , I = 0 , · · · , D (0) by A ≡ D (0) . (A.1)This will never contain any dangling derivatives that act on the scalar field. On theother hand, D (1) contains up to one derivative acting on the scalar field. So, we writeit as D (1) = B + B i ∇ i . (A.2)We will try to be as general as possible until the end, so we will not yet plug in specificexpressions for B and B i , but one can easily do so for the Weyl-invariant case or eventhe general case. – 33 –ext, D (2) contains up to two spatial derivatives or one time derivative acting onthe scalar field. So, we write it as D (2) = C + C i ∇ i + C ij ∇ i ∇ j + C n d n , (A.3)Again, we will not yet plug in specific expressions for the coefficient functions C , C i , C ij , and C n , but one could certainly do so for whichever operator is of interest.The σ (2) recursion relation now reads0 = A σ (2) + B i σ (1); i + B σ (1) + C ij σ (0); ji + C n σ (0); n + C i σ (0); i + C σ (0) . (A.4)From this equation, subtract the contraction of A − B i with the derivative of the σ (1) recursion relation:0 = A σ (1); i + A ; i σ (1) + B j σ (0); ji + B σ (0); i + B j ; i σ (0); j + B ; i σ (0) . (A.5)This will get rid of the σ (1); i term, leaving us with0 = A σ (2) + (cid:0) B − A − A ; i B i (cid:1) σ (1) + (cid:0) C ij − A − B i B j (cid:1) σ (0); ji + C n σ (0); n + (cid:2) C i − A − (cid:0) BB i + B j B i ; j (cid:1)(cid:3) σ (0); i + (cid:0) C − A − B i B ; i (cid:1) σ (0) . (A.6)We just keep doing this repeatedly, each time getting rid of the term with the largestnumber of derivatives acting on σ (0) , keeping mindful of the fact that σ ( m ) itself containsup to m derivatives acting on σ (0) by the recursion relations. For example, next, wewould subtract the contraction of A − ( C ij − A − B i B j ) with the derivatives ∇ i ∇ j acting on the σ (0) defining equation: I ; ji = A σ (0); ji + 2 A ;( j σ (0); i ) + A ; ji σ (0) . (A.7)We can simultaneously subtract A − C n multiplied by the time derivative of the σ (0) defining equation: I ; n = A σ (0); n + A ; n σ (0) . (A.8)This will remove the terms σ (0); ji and σ (0); n in (A.6) and leaves us with0 = A σ + A − ( C ij − A − B i B j ) I ; ji + A − C n I ; n + (cid:0) B − A − A ; i B i (cid:1) σ (1) + (cid:2) C i − A − (cid:0) BB i + B j B i ; j + 2 A ; j C ij (cid:1) + 2 A − A ; j B j B i (cid:3) σ (0); i + (cid:2) C − A − (cid:0) B i B ; i + A ; ji C ij + A ; n C n (cid:1) + A − A ; ji B i B j (cid:3) σ (0) . (A.9)Next, subtract A − (cid:0) B − A − A ; i B i (cid:1) multiplied by the σ (1) recursion relation0 = A σ (1) + B i σ (0); i + B σ (0) , (A.10)– 34 –hich leaves us with0 = A σ (2) + A − ( C ij − A − B i B j ) I ; ji + A − C n I ; n + (cid:2) C i − A − (cid:0) BB i + B j B i ; j + 2 A ; j C ij (cid:1) + 3 A − A ; j B j B i (cid:3) σ (0); i + (cid:2) C − A − (cid:0) B + B i B ; i + A ; ji C ij + A ; n C n (cid:1) + A − (cid:0) A ; i B i B + A ; ji B i B j (cid:1)(cid:3) σ (0) . (A.11)Now, we subtract the appropriate term involving ∇ i acting on the σ (0) defining equationto finally get an expression for σ (2) that depends only on σ (0) . Finally, we can replace σ (0) with A − I :0 = A σ + A − ( C ij − A − B i B j ) I ; ji + A − C n I ; n + A − (cid:2) C i − A − (cid:0) BB i + B j B i ; j + 2 A ; j C ij (cid:1) + 2 A − A ; j B j B i (cid:3) I ; i + A − (cid:2) C − A − (cid:0) B + B i B ; i + A ; i C i + A ; ji C ij + A ; n C n (cid:1) + A − (cid:0) A ; i B i B + A ; ji B i B j + A ; i B j B i ; j + 2 A ; i A ; j C ij (cid:1) − A − ( A ; i B i ) (cid:3) I . (A.12)Once we take the coincidence limit, all the terms with derivatives acting on I vanish, I just turns into unity, and A − turns into G , the propagator of the theory. Thus,the final expression for the coincidence limit of σ (2) is σ (2) (cid:12)(cid:12) x = x = − G (cid:2) C − G (cid:0) B + B i B ; i + A ; i C i + A ; ji C ij + A ; n C n (cid:1) + G (cid:0) A ; i B i B + A ; ji B i B j + A ; i B j B i ; j + 2 A ; i A ; j C ij (cid:1) − G ( A ; i B i ) (cid:3)(cid:12)(cid:12) x = x . (A.13)We emphasize that this is the general expression for the coincidence limit of σ (2) for any z = 2 pseudodifferential operator acting on a scalar field. For a specific operator,we can compute the coincidence limits of the operator coefficients remaining above.Explicit expressions for the operator coefficients for the general z = 2 operatorwritten in the specific form in (4.47), with the Weyl-invariant operator defined in (4.2),are A = ( χ n ) + ( ψ i ψ i ) − λ , (A.14a) B = 2 χ n ψ n − iψ i ψ i ψ jj − iψ i ψ j ψ ij + 2 ia i ψ i ψ j ψ j − iW ijk ψ i ψ j ψ k , (A.14b) B i = − iψ j ψ j ψ i , (A.14c) The coincidence limits of I ; n and I ; i vanish by definition. The term I ; ji is contracted with either C ij or B i B j , both of which are symmetric. Thus, we can symmetrize the derivatives and write I ;( ji ) the coincidence limit of which also vanishes by definition. – 35 – = − i ( χ nn + Kχ n ) + ( ψ n ) − ( ψ ii ) − ψ ij ψ ij − ψ i (cid:0) ψ jji + ψ ijj (cid:1) + ( ∇ · a − a ) ψ i ψ i + 2 a i (2 ψ ij ψ j + ψ i ψ jj )+ iU χ n − W ijk ψ i ψ jk − X ij ψ i ψ j , (A.14d) C i = − ψ ij ψ j + ψ i ψ jj ) + 2(2 a j ψ j ψ i + a i ψ j ψ j ) − W ijk ψ j ψ k , (A.14e) C ij = − ψ i ψ j + ψ k ψ k h ij ) , (A.14f) C n = − iχ n . (A.14g)Taking one derivative of A gives A ; i = 2 χ n χ in + 4 ψ k ψ k ψ j ψ ij . (A.15)Since ψ is a scalar, ∇ i ∇ j ψ = ∇ j ∇ i ψ , and thus the defining relation ∇ ( i ∇ j ) ψ (cid:12)(cid:12) x = x = 0implies ψ ij (cid:12)(cid:12) x = x = 0 . (A.16)Using the commutation relation [ d n , ∇ i ] χ = − a i χ n and the defining relation χ n (cid:12)(cid:12) x = x = ν and χ ni (cid:12)(cid:12) x = x = 0, we find χ in (cid:12)(cid:12) x = x = νa i . (A.17)Therefore, A ; i (cid:12)(cid:12) x = x = 2 ν a i . (A.18)Taking another derivative we can immediately ignore any terms with two spatial deriva-tives acting on ψ , which we denote by . . . : A ; ji = 2 χ n χ ijn + 2 χ in χ jn + 4 ψ (cid:96) ψ (cid:96) ψ k ψ ijk + . . . . (A.19)Using the commutation relations, χ ijn = χ inj − ∇ i [ d n , ∇ j ] χ = χ nij − [ d n , ∇ i ] χ j + ( a j χ n ) ; i = χ nij + a i χ nj + M kij χ k + a j χ in + a j ; i χ n . (A.20)The first three terms vanish in the coincidence limit and the rest give χ ijn (cid:12)(cid:12) x = x = ν ( a i a j + a j ; i ) . (A.21)Thus, A ; ji (cid:12)(cid:12) x = x = 2 ν (2 a i a j + a j ; i ) . (A.22)Next, let us take coincidence limit of B and B i : B (cid:12)(cid:12) x = x = iq i q j q k (2 a i h jk − W ijk ) , (A.23a)– 36 – i (cid:12)(cid:12) x = x = − i | q | q i . (A.23b)Next, we take one derivative of B . We will denote any terms containing one time ortwo space derivatives on ψ by . . . since we know that these vanish in the coincidencelimit: B ; i = 2 χ n ψ in − iψ k ψ k ψ ijj − iψ j ψ k ψ ijk + i (2 a j ; i h k(cid:96) − W jk(cid:96) ; i ) ψ j ψ k ψ (cid:96) + . . . . (A.24)Again, using the commutation relations, we have ψ in (cid:12)(cid:12) x = x = (cid:0) ψ ni + a i ψ n (cid:1)(cid:12)(cid:12) x = x = 0 . (A.25)Using the commutator of two covariant spatial derivatives, we can write12 ψ ( ijk ) = ψ ijk + ψ jki + ψ kij = 3 ψ ijk + 2 R (cid:96) ( k | i | j ) ψ (cid:96) . (A.26)In the coincidence limit, the left hand side vanishes by definition, and we find ψ ijk (cid:12)(cid:12) x = x = − q (cid:96) R (cid:96) ( k | i | j ) . (A.27)In 2 dimensions, this simplifies to ψ ijk (cid:12)(cid:12) x = x = 13 R ( h i ( j q k ) − h jk q i ) . (A.28)Plugging these back in B ; i gives B ; i (cid:12)(cid:12) x = x = 2 i R | q | q i + iq j q k q (cid:96) (cid:0) a j ; i h k(cid:96) − W jk(cid:96) ; i (cid:1) . (A.29)Since B i contains only single derivatives on ψ , the coincidence limit of B j ; i simplyvanishes: B j ; i (cid:12)(cid:12) x = x = 0 . (A.30)Next, we must take coincidence limits of the C operators: C (cid:12)(cid:12) x = x = iν ( U − K ) + (cid:18) ∇ · a − a − R (cid:19) | q | − k i k j X ij , (A.31a) C i (cid:12)(cid:12) x = x = 2 (cid:0) a j q j q i + a i | q | (cid:1) − W ijk q j q k , (A.31b) C ij (cid:12)(cid:12) x = x = − (cid:0) q i q j + | q | h ij (cid:1) , (A.31c) C n (cid:12)(cid:12) x = x = − iν . (A.31d)– 37 –ll that is left is to plug in these coincidence limits into the general expression for σ (2) in (A.13). For example, setting U = W ijk = X ij = 0 gives the expression for σ (2) inthe case of the Weyl-invariant operator, which we write out in (4.29). This is thenintegrated over frequency and momentum to get the heat kernel coefficien E (2) . Theresult in the Weyl-invariant case is in (4.30) and the the result in the general case is in(4.48).The process is exactly the same for the higher-order heat kernel coefficients. Thecomputation will clearly get much more tedious, but is simple in principle and amenableto automation. B. Expression of (cid:101) E (4) for General Scalar Operators In this appendix, we present the general results for (cid:101) E (4) in (4.50), transcribing fromFigure 1. We write (cid:101) E (4) = E (4) + (cid:88) p =0 b p , (B.1)where E (4) is given in (4.34) and b p , p = 0 , · · · , p -th order in W .We collect the results for b , · · · , b below: Terms Constant in W : Coefficient b . This result already appears in (4.53), whichwe write below again for completeness: b = 1256 π (cid:0) X ij X ij + X ii X jj (cid:1) − π (cid:0) U + 4 Z (cid:1) + 1 N √ h (cid:0) ∂ t f t + ∂ i f i (cid:1) , (B.2)where f t = 132 π √ h U , (B.3a) f i = 132 π (cid:110) −√ h N i U + N (cid:2) Y i + (cid:0) a j X ij − ∇ j X ij + 3 ∇ i X jj (cid:1)(cid:3)(cid:111) . (B.3b) Terms Linear in W : Coefficient b . We write b as b = (cid:88) q =0 b ,q , (B.4)where b , q includes all terms that are linear in W and that contain q derivatives actingon W . We find that b , = − π (cid:104)(cid:0) a j a k − ∇ j a k − X jk + 2 ∇ j ∇ k (cid:1) a i (cid:105) W ijk – 38 – 1512 π (cid:104)(cid:0) (cid:3) − R (cid:1) a i − (cid:0) a k ∇ i − ∇ i ∇ k (cid:1) a k + 3 a i X kk + 6 (cid:0) a k − ∇ k (cid:1) X ik + 6 Y i (cid:105) W ijj , (B.5a) b , = 1512 π (cid:104) (cid:0) ∇ i a j − a i a j (cid:1) ∇ i W jkk − X ii ∇ j W jkk + 2 (cid:0) a i a j − ∇ i a j − X ij (cid:1) ∇ k W ijk (cid:105) , (B.5b) b , = 1256 π a i (cid:0) ∇ k ∇ i W jjk − ∇ j ∇ k W ijk + 3 (cid:3) W ijj (cid:1) , (B.5c) b , = 1256 π (cid:0) ∇ i ∇ j ∇ k W ijk − (cid:3) ∇ i W ijj (cid:1) . (B.5d) Terms Quadratic in W : Coefficient b . We start with a classification of possibleterms that are quadratic in W , containing no derivatives acting on any of the W ’s.The result of b will be given later in (B.9). We will need the scalars (in which allsix indices in the two factors of W combined are contracted into three pairs) and thematrices (the ones with two free, un-conctracted, indices). It will also be convenient todefine the multiplicity of a particular term. The multiplicity of a term is determinedby all possible ways of contracting the six indices in the pair of W ’s to produce theterm. The full classification of all scalars and matrices with their multiplicity numbersare listed as follows: S (1) = W ikk W i(cid:96)(cid:96) multiplicity = 9 , (B.6a) S (2) = W ik(cid:96) W ik(cid:96) multiplicity = 6 , (B.6b) M (1) ij = W ikk W j(cid:96)(cid:96) multiplicity = 9 , (B.6c) M (2) ij = W ik(cid:96) W jk(cid:96) multiplicity = 18 , (B.6d) M (3) ij = W k(cid:96)(cid:96) W ijk multiplicity = 36 , (B.6e)We can then define the sum of all possible scalars (counting multiplicity): P = 3 (cid:0) S (1) + 2 S (2) (cid:1) . (B.7)Similarly, the sum of all possible matrices (counting multiplicity) is Q = 9 (cid:0) M (1) + 2 M (2) + 4 M (3) (cid:1) . (B.8)Using the notation introduced above, we write b = (cid:88) q =0 b ,q , (B.9)– 39 –here b , q includes all terms that are quadratic in W and that contain q derivativesacting on W ’s. We have b , = 2 − π (cid:104)(cid:0) R − a + 4 ∇ i a i − X ii (cid:1) P + 2 (cid:0) a i a j − ∇ i a j − X ij (cid:1) Q ij − (cid:0) R h ij + 20 a i a j + 4 ∇ i a j − X ij (cid:1) M (3) ij (cid:105) , (B.10a) b , = 2 − π a i (cid:104) W jk(cid:96) (cid:0) ∇ i W jk(cid:96) + 3 ∇ j W ik(cid:96) (cid:1) − W ijk (cid:0) ∇ (cid:96) W jk(cid:96) + 9 ∇ j W k(cid:96)(cid:96) (cid:1) − W ikk ∇ j W j(cid:96)(cid:96) + 3 W jkk (cid:0) ∇ i W j(cid:96)(cid:96) + ∇ j W i(cid:96)(cid:96) − ∇ (cid:96) W ij(cid:96) (cid:1)(cid:105) , (B.10b) b , = − − π (cid:104) W iij (cid:0) (cid:3) W jkk + 2 ∇ k ∇ j W k(cid:96)(cid:96) − ∇ k ∇ (cid:96) W jk(cid:96) (cid:1) + 2 W ijk (cid:0) (cid:3) W ijk + 6 ∇ (cid:96) ∇ k W ij(cid:96) − ∇ j ∇ k W i(cid:96)(cid:96) (cid:1)(cid:105) + 2 − π (cid:104) (cid:0) ∇ i W ik(cid:96) ∇ j W jk(cid:96) + ∇ i W ikk ∇ j W j(cid:96)(cid:96) (cid:1) − ∇ i W jkk (cid:0) ∇ i W j(cid:96)(cid:96) + 5 ∇ j W i(cid:96)(cid:96) − ∇ (cid:96) W (cid:96)ij (cid:1) − ∇ (cid:96) W ijk (cid:0) ∇ k W ij(cid:96) + ∇ (cid:96) W ijk (cid:1)(cid:105) . (B.10c) Terms Cubic in W : Coefficient b . We start with a classification of possible termsthat are cubic in W , containing no derivatives acting on any of the W ’s. The result of b will be given later in (B.13). We only need the vectors: V (1) i = S (1) W i(cid:96)(cid:96) multiplicity = 81 , (B.11a) V (2) i = S (2) W i(cid:96)(cid:96) multiplicity = 54 , (B.11b) V (3) i = M (1) k(cid:96) W ik(cid:96) multiplicity = 162 , (B.11c) V (4) i = M (2) k(cid:96) W ik(cid:96) multiplicity = 324 , (B.11d) V (5) i = M (3) k(cid:96) W ik(cid:96) multiplicity = 648 . (B.11e)Here, the multiplicity of a term is determined by all possible ways of contracting theeight indices in the three W ’s to produce the term. Let V be the weighted sum of allof the above terms (including multiplicity): V = 3 (cid:0) V (1) + 18 V (2) + 54 V (3) + 108 V (4) + 216 V (5) (cid:1) . (B.12)Using the notation introduced above, we write b = b , + b , , (B.13)– 40 –here b , q , q = 0 , W and that contain q derivativesacting on W ’s. We have b , = − − π a i (cid:0) V i − M (3) jk W ijk (cid:1) , (B.14a) b , = 32 π (cid:0) P h ij + 2 Q ij − M (3) ij (cid:1) ∇ k W ijk . (B.14b) Terms Quartic in W : Coefficient b . We start with a classification of possibleterms that are quartic in W , containing no derivatives acting on any of the W ’s: W (1) = S (1) S (1) multiplicity = 81 , (B.15a) W (2) = S (1) S (2) multiplicity = 108 , (B.15b) W (3) = S (2) S (2) multiplicity = 36 , (B.15c) W (4) = tr (cid:0) M (1) M (2) (cid:1) multiplicity = 648 , (B.15d) W (5) = tr (cid:0) M (1) M (3) (cid:1) multiplicity = 216 , (B.15e) W (6) = tr (cid:0) M (2) M (2) (cid:1) multiplicity = 648 , (B.15f) W (7) = tr (cid:0) M (2) M (3) (cid:1) multiplicity = 1296 , (B.15g) W (8) = W ijk W i(cid:96)m W j(cid:96)n W kmn multiplicity = 432 . (B.15h)Here, the multiplicity of a term is determined by all possible ways of contracting thetwelve indices in the four W ’s to produce the term. Let W be the weighted sum of allof the above terms (including multiplicity): W = 9 (cid:0) W (1) + 12 W (2) + 4 W (3) + 72 W (4) + 24 W (5) + 72 W (6) + 144 W (7) + 48 W (8) (cid:1) . (B.16)In terms of W in (B.16), we find b = 2 − π W . (B.17)Note that there is an interesting relation between b and b , b (cid:16) W ijk → w W ijk − (cid:0) a i h jk + a j h ki + a k h ij (cid:1)(cid:17) = w b ( W ) + w b , ( a , W ) + O ( w ) , (B.18)where b and b are given in (B.14a) and (B.17), respectively.– 41 – . Sign Conventions in the Literature As pointed out in [24], the heat kernel coefficient of the monomials built out of thespatial curvature in a Lifshitz theory in which the spatial derivative part of the operatoris essentially just the z -th power of the isotropic ( z = 1) case can be expressed as somenumber (a Mellin integral of some sort) times the z = 1 result. In the case of z = D = 2,for the R monomial, which shows up in E (6) , the appropriate Mellin integral evaluatesto −
2, which is the number reported in the R entry of the z = D = 2 column oftheir Table 1 (they denote the number of spatial dimensions by d ). The number to the left of that, in the a n column, is supposed to be the z = 1 coefficient of R in two spacetime dimensions and the z = 2 coefficient is simply the product of thesetwo numbers: − × = − . However, the result we get using our method is − = − × . We agree with the Mellin integral result of −
2, but we disagree withthe z = 1 coefficient of R : we get instead of .Due to this discrepancy, we were led to examine the relativistic heat kernel lit-erature. There are numerous different sign conventions being used in the literature,which can easily cause confusion. We will clarify some of these conventions, which weencountered in the course of verifying the heat kernel coefficients in the relativistic case.Firstly, we can confirm the correctness of the coefficients calculated by Gilkey[43] in arbitrary dimensions for the terms that contain only Riemann curvatures andtheir covariant derivatives. We reproduced these coefficients using Gusynin’s methodreviewed in Section 2. One just needs to be mindful of the fact that Gilkey’s definitionof the Riemann tensor is negative of the one we use here in (3.15). The coefficients E , E , and E are reported on page 610, just after Theorem 2.2, and E is given inTheorem 4.1 on page 613. Once evaluated in two dimensions with constant curvature,with the Ricci and Riemann tensors related to the Ricci scalar as in (4.10), and wherethe conventional sign for curvature is used (instead of the one used by Gilkey), onederives the result (4 π ) E = R . This factor of is precisely the same one as weget from our aforementioned z = 2 computation.In [35], Vassilevich calls these coefficients a to a and reports them in Eqns. (4.26-29). Here, one has to be careful keeping track of the sign conventions as well. Vassilevichdefines the Riemann tensor in his Eqn. (2.6), which is negative of the one we use here in(3.15). However, his definition of the Ricci tensor and Ricci scalar coincide with ours.Keeping these sign conventions in mind, the coefficients in Vassilevich are consistentwith those in Gilkey. The same results are found in App. B and C of [50] by independentcalculation by Gustavsson using the same sign conventions as in Vassilevich. We are grateful to Andreas Gustavsson for personal communications that led us to realize ourearlier misunderstanding of their sign conventions. – 42 –n [51], the authors independently computed E , though they denote the result by A in their Eqn. (2.16) (the overline indicates the coincidence limit because this workstudies the so-called “off-diagonal” heat kernel in which the coincidence limit x → x may or may not be taken). We focus on the pieces that depend only on the spatialcurvature and its derivatives (not on the endomorphism E or the field strength F µν ofthe vector bundle describing the non-metric field content of the field theory). Note thattheir definitions of the Riemann tensor, Ricci tensor, and Ricci scalar have the samesign as ours. The comparison with Gilkey is complicated slightly by the fact that theauthors chose to use a slightly different basis for the terms involving two derivativesand two factors of curvature: where Gilkey uses the term R µν ∇ ρ ∇ µ R ρν , the authorsof [51] instead use R µν ∇ µ ∇ ν R . However, we can use the commutator of two covariantderivatives acting on a tensor as well as the second Bianchi identity and the symmetriesof the Riemann tensor to write R µν ∇ ρ ∇ µ R ρν = 12 R µν ∇ µ ∇ ν R − R µν R µλ R ν λ + R µν R ρλ R µρνλ . (C.1)Thus, we can convert Gilkey’s result to the basis used in [51], remembering to switchthe sign of any terms that are linear or cubic in curvature. We must also keep in mindthat ∆ is defined in [51] with a minus sign: ∆ = − g µν ∇ µ ∇ ν . Doing so, we find justone discrepancy in the expression A in Eqn. (2.16) of [51]: the coefficient of R (∆ R )should be − , not − , which may just be a misprint.The coefficients of the terms cubic in curvature contain slight differences (in ad-dition to the overall sign difference) between [43] and [51] because of the change ofbasis (C.1), but they are consistent. In particular, in two dimensions with constantcurvature, one again gets the result (4 π ) E = R .More recently, Kluth and Litim [52] computed up to the R , R , and R coefficientsin E , E , and E , respectively, in 2 to 6 dimensions for a sphere (actually for anymaximally symmetric space). This is done using quite a different method, which lever-ages simplifications specific to maximal symmetry from the beginning. Sure enough,the coefficient of π R in two dimensions is .– 43 – eferences [1] P. Hoˇrava, Quantum gravity at a Lifshitz point , Phys. Rev. D (2009) 084008,[ arXiv:0901.3775 ].[2] P. Hoˇrava, Membranes at quantum criticality , JHEP (2009) 020,[ arXiv:0812.4287 ].[3] J. Bellorin and A. Restuccia, Closure of the algebra of constraints for a non-projectableHoˇrava model , Phys. Rev. D (2011) 044003, [ arXiv:1010.5531 ].[4] V. Kostelecky and N. Russell, Data Tables for Lorentz and CPT Violation , Rev. Mod.Phys. (2011) 11–31, [ arXiv:0801.0287 ].[5] P. Hoˇrava, General covariance in gravity at a Lifshitz point , Class. Quant. Grav. (2011) 114012, [ arXiv:1101.1081 ].[6] A. Coates, C. Melby-Thompson and S. Mukohyama, Revisiting Lorentz violation inHoˇrava gravity , Phys. Rev. D (2019) 064046, [ arXiv:1805.10299 ].[7] S. Groot Nibbelink and M. Pospelov,
Lorentz violation in supersymmetric fieldtheories , Phys. Rev. Lett. (2005) 081601, [ hep-ph/0404271 ].[8] M. M. Anber and J. F. Donoghue, The emergence of a universal limiting speed , Phys.Rev. D (2011) 105027, [ arXiv:1102.0789 ].[9] G. Bednik, O. Pujol`as and S. Sibiryakov, Emergent Lorentz invariance from StrongDynamics: Holographic examples , JHEP (2013) 064, [ arXiv:1305.0011 ].[10] T. Griffin, P. Hoˇrava and C. M. Melby-Thompson, Conformal Lifshitz gravity fromholography , JHEP (2012) 010, [ arXiv:1112.5660 ].[11] M. Baggio, J. de Boer and K. Holsheimer, Anomalous breaking of anisotropic scalingsymmetry in the quantum Lifshitz model , JHEP (2012) 099, [ arXiv:1112.6416 ].[12] T. Griffin, P. Hoˇrava and C. M. Melby-Thompson, Lifshitz gravity for Lifshitzholography , Phys. Rev. Lett. (2013) 081602, [ arXiv:1211.4872 ].[13] M. H. Christensen, J. Hartong, N. A. Obers and B. Rollier,
Torsional Newton-Cartangeometry and Lifshitz holography , Phys. Rev. D (2014) 061901, [ arXiv:1311.4794 ].[14] M. H. Christensen, J. Hartong, N. A. Obers and B. Rollier, Boundary stress-energytensor and Newton-Cartan geometry in Lifshitz holography , JHEP (2014) 057,[ arXiv:1311.6471 ].[15] J. Ambjørn, A. G¨orlich, J. Jurkiewicz and R. Loll, Quantum gravity via CausalDynamical Triangulations , arXiv:1302.2173 .[16] P. Hoˇrava, Spectral dimension of the Universe in quantum gravity at a Lifshitz point , Phys. Rev. Lett. (2009) 161301, [ arXiv:0902.3657 ]. – 44 –
17] A. Frenkel, P. Hoˇrava and S. Randall,
Perelman’s Ricci flow in topological quantumgravity , arXiv:2011.11914 .[18] A. O. Barvinsky, D. Blas, M. Herrero-Valea, S. M. Sibiryakov and C. F. Steinwachs, Renormalization of Hoˇrava gravity , Phys. Rev. D (2016) 064022,[ arXiv:1512.02250 ].[19] A. O. Barvinsky, D. Blas, M. Herrero-Valea, S. M. Sibiryakov and C. F. Steinwachs, Hoˇrava gravity is asymptotically free in 2 + 1 dimensions , Phys. Rev. Lett. (2017)211301, [ arXiv:1706.06809 ].[20] T. Griffin, K. T. Grosvenor, C. M. Melby-Thompson and Z. Yan,
Quantization ofHoˇrava gravity in 2+1 dimensions , JHEP (2017) 004, [ arXiv:1701.08173 ].[21] A. O. Barvinsky, M. Herrero-Valea and S. M. Sibiryakov, Towards the renormalizationgroup flow of Hoˇrava gravity in 3+1 dimensions , Phys. Rev. D (2019) 026012,[ arXiv:1905.03798 ].[22] D. Benedetti and F. Guarnieri,
One-loop renormalization in a toy model ofHoˇrava-Lifshitz gravity , JHEP (2014) 078, [ arXiv:1311.6253 ].[23] D. Nesterov and S. N. Solodukhin, Gravitational effective action and entanglemententropy in UV modified theories with and without Lorentz symmetry , Nucl. Phys. B (2011) 141–171, [ arXiv:1007.1246 ].[24] G. D’Odorico, J.-W. Goossens and F. Saueressig,
Covariant computation of effectiveactions in Hoˇrava-Lifshitz gravity , JHEP (2015) 126, [ arXiv:1508.00590 ].[25] A. O. Barvinsky, D. Blas, M. Herrero-Valea, D. V. Nesterov, G. P´erez-Nadal and C. F.Steinwachs, Heat kernel methods for Lifshitz theories , JHEP (2017) 063,[ arXiv:1703.04747 ].[26] V. Gusynin, New algorithm for computing the coefficients in the heat kernel expansion , Phys. Lett. B (1989) 233–239.[27] V. Gusynin,
Seeley-Gilkey coefficients for fourth-order operators on a Riemannianmanifold , Nuclear Physics B (1990) 296–316.[28] V. Gusynin,
Asymptotics of the heat kernel for nonminimal differential operators , Ukrainian Mathematical Journal (1991) 1432–1441.[29] V. Gusynin and E. Gorbar, Local heat kernel asymptotics for nonminimal differentialoperators , Phys. Lett. B (1991) 29–36.[30] V. Gusynin, E. Gorbar and V. Romankov,
Heat kernel expansion for nonminimaldifferential operators and manifolds with torsion , Nucl. Phys. B (1991) 449–471.[31] E. Gorbar,
Heat kernel expansion for operators of the type of the square root of theLAplace operator , J. Math. Phys. (1997) 1692–1699, [ hep-th/9602018 ]. – 45 –
32] V. Gusynin and V. Kornyak,
Computation of the DeWitt-Seeley-Gilkey coefficient E (4) for nonminimal operator in curved space , Nucl. Instrum. Meth. A (1997) 365–369.[33] H. Widom,
Families of pseudodifferential operators, topics in functional analysis (i.gohberg and m. kac, eds.) , 1978.[34] H. Widom,
Complete symbolic-calculus for pseudodifferential-operators , Bulletin desSciences Math´ematiques (1980) 19–63.[35] D. Vassilevich,
Heat kernel expansion: User’s manual , Phys. Rept. (2003) 279–360,[ hep-th/0306138 ].[36] G. Gibbons,
Quantum field theory in curved space-time , pp. 639–679. CambridgeUniversity Press, 1978.[37] J. Schwinger,
On gauge invariance and vacuum polarization , Physical Review (1951) 664.[38] B. de Witt, Dynamical theory of groups and fields , Relativity, groups and topology (1963) .[39] A. Barvinsky and G. Vilkovisky,
The generalized Schwinger-DeWitt technique in gaugetheories and quantum gravity , Physics Reports (1985) 1–74.[40] R. I. Nepomechie,
Calculating heat kernels , Phys. Rev. D (Jun, 1985) 3291–3292.[41] A. Ceresole, P. Pizzochero and P. van Nieuwenhuizen, Curved-space trace, chiral, andeinstein anomalies from path integrals, using flat-space plane waves , Phys. Rev. D (Mar, 1989) 1567–1578.[42] R. Seeley, Complex powers of an elliptic operators , in
Proc. Symp. Pure Math. , vol. 10,pp. 288–307, 1967.[43] P. B. Gilkey,
The spectral geometry of a Riemannian manifold , J. Diff. Geom. (1975) 601–618.[44] C. W. Misner, K. Thorne and J. Wheeler, Gravitation . W. H. Freeman, San Francisco,1973.[45] S. Weinberg,
The quantum theory of fields. Vol. 2: Modern applications . CambridgeUniversity Press, 8, 2013.[46] I. Arav, S. Chapman and Y. Oz,
Lifshitz scale anomalies , JHEP (2015) 078,[ arXiv:1410.5831 ].[47] J. M. Mart´ın-Garc´ıa et al ., xAct: Efficient tensor computer algebra for Mathematica , (2002-2020) .[48] P. Hoˇrava and C. M. Melby-Thompson, General covariance in quantum gravity at aLifshitz point , Phys. Rev. D (2010) 064027, [ arXiv:1007.2410 ]. – 46 –
49] J. Hartong and N. A. Obers,
Hoˇrava-Lifshitz gravity from dynamical Newton-Cartangeometry , JHEP (2015) 155, [ arXiv:1504.07461 ].[50] A. Gustavsson, Abelian M5-brane on S , JHEP (2019) 140, [ arXiv:1902.04201 ].[51] K. Groh, F. Saueressig and O. Zanusso, Off-diagonal heat-kernel expansion and itsapplication to fields with differential constraints , arXiv:1112.4856 .[52] Y. Kluth and D. F. Litim, Heat kernel coefficients on the sphere in any dimension , Eur. Phys. J. C (2020) 269, [ arXiv:1910.00543 ].].