Holographic Gravitational Anomaly and Chiral Vortical Effect

Holographic Gravitational Anomaly and Chiral Vortical Effect

Karl Landsteiner, Eugenio Megías, Luis Melgar, Francisco Pena-Benitez
Instituto de Física Teórica UAM/CSIC, C/ Nicolás Cabrera 13-15,
karl.landsteiner@csic.eseugenio.megias@csic.esluis.melgar@estudiante.uam.esfran.penna@uam.es
Abstract:

We analyze a holographic model with a pure gauge and a mixed gauge-gravitational Chern-Simons term in the action. These are the holographic implementations of the usual chiral and the mixed gauge-gravitational anomalies in four dimensional field theories with chiral fermions. We discuss the holographic renormalization and show that the gauge-gravitational Chern-Simons term does not induce new divergences. In order to cancel contributions from the extrinsic curvature at a boundary at finite distance a new type of counterterm has to be added however. This counterterm can also serve to make the Dirichlet problem well defined in case the gauge field strength vanishes on the boundary. A charged asymptotically AdS black hole is a solution to the theory and as an application we compute the chiral magnetic and chiral vortical conductivities via Kubo formulas. We find that the characteristic term proportional to is present also at strong coupling and that its numerical value is not renormalized compared to the weak coupling result.

Gauge-gravity correspondence, Anomalies, Chern-Simons terms, Transport theory
preprint: IFT-UAM/CSIC-11-45

1 Introduction

Anomalies belong to the most interesting and most subtle properties of relativistic quantum field theories. They are responsible for the breakdown of a classical symmetry due to quantum effects. The Adler-Bardeen non-renormalization theorem guarantees that this breakdown is saturated at the one-loop level. Therefore the presence of anomalies can be determined through simple algebraic criteria on the representations under which the chiral fermions of a particular theory transform. In vacuum the anomaly appears as the non-conservation of a classically conserved current in a triangle diagram with two additional currents. In four dimension two types of anomalies can be distinguished according to whether only spin one currents appear in the triangle [1, 2] or if also the energy-momentum tensor participates [3, 4]. We will call the first type of anomalies simply chiral anomalies and the second type gravitational anomalies. To be precise, in four dimension we should actually talk of mixed gauge-gravitational anomalies since triangle diagrams with only energy-momentum insertions are perfectly conserved (see e.g. [5]). In a basis of only right-handed fermions transforming under a symmetry generated by the presence of chiral anomalies is detected by the non-vanishing of whereas the presence of a gravitational anomaly is detected by the non-vanishing of .

It recently has been emphasized how at finite temperature and density anomalies give rise to new non-dissipative transport phenomena in the hydrodynamics of charged relativistic fluids [6, 7, 8, 9]. More precisely magnetic fields and vortices in the fluid induce currents via the so-called chiral magnetic and chiral vortical conductivities. Although there have been many early precursors that found manifestations of this phenomena in the physics of neutrinos [10, 11, 12, 13], the early universe [14], condensed matter systems [15], the recent surge of interest is clearly related to the physics of the quark gluon plasma. It has been suggested that the observed charge separation in heavy ion collisions is related to a particular manifestation of these anomalous transport phenomena, the chiral magnetic effect [16]. The latter describes how a usual (i.e. electro-magnetic) B-field induces via the axial anomaly an electric current parallel to the magnetic field. The first application of holography to the anomalous hydrodynamics is [17] where the anomalous transport effects due to R-charge magnetic fields have been examined. Later studies showed that there is also a related vortical effect [7, 8], i.e. a vortex in the fluid induces a current parallel to the axial vorticity vector , and related effects of the presence of angular momentum had been discussed before in a purely field theoretical setup in [12] and [18]. Studies of the chiral magnetic effect using holography have appeared in [19, 20, 21, 22, 23, 24, 25, 26, 27] and using lattice field theory in [28, 29, 30]. A related effect is the so called chiral separation effect that induces an axial current in a magnetic field [31]. Chiral magnetic waves have been shown to arise through the interplay of chiral magnetic effect and chiral separation effect in [32]. Experimental signatures of anomalous transport beyond the charge separation effect have been proposed in [33, 34, 35, 36]. The experimental status of the observed charge separation in heavy ion collision is discussed in [37, 38].

A first principals approach to transport theory is via Kubo formulas. In general the reaction of a system to external perturbations can be studied via linear response theory. The basic objects of linear response theory are the retarded Green functions. Hydrodynamic transport coefficients can be extracted from the long-wavelength and low-frequency limits of the retarded Green functions. A typical example is the Kubo formula for the shear viscosity

 (1)

Electric or thermal conductivities can be calculated in a similar fashion.

Also the transport coefficients related to the presence of anomalies can be computed via Kubo formulas. For the chiral magnetic effect this has been done in [39] and has been further studied in [40]. The Kubo formula for the chiral vortical conductivity was derived in [41]. These formulas can easily be generalized to the case of a general non-abelian symmetry group, generated by matrices . For the chiral magnetic conductivities and chiral vortical conductivities they are

 σBAB = limpn→0i2pc∑a,bϵabc⟨JaAJbB⟩(ω=0,→p), (2) σVA = limpc→0i2pc∑a,bϵabc⟨JaAT0b⟩(ω=0,→p), (3)

with .

The properties of these Kubo formulas for the anomalous conductivities are rather different from the ones for the dissipative conductivities. Whereas the Kubo formulas for the usual transport coefficients are given by the derivatives with respect to the frequency at zero momentum, the ones for the anomaly related transport coefficients are given by derivatives in momentum space at zero frequency. For this reason the anomalous conductivities are contained in the hermitian part of the correlators. In contrast the dissipative transport coefficients are contained in the spectral densities, i.e. the anti-hermitian parts of the correlators. In the presence of external sources that couple to operators the rate of dissipation is describe in terms of the spectral density , i.e. the anti-hermitian part of the retarded correlator , by [42]

 dWdt=12ωfI(−ω)ρIJ(ω)fJ(ω). (4)

This shows that the anomaly related currents due to the conductivities (2) and (3) do no work on the system and are therefore examples of dissipationless transport. Let us also note that dissipative transport breaks time reversal invariance whereas anomaly induced dissipationless transport preserves 111This point has recently also been emphasized in [43]..

In [44] these general Kubo formulas were evaluated for a theory of free chiral fermions. The results showed a somewhat surprising appearance of the anomaly coefficient for the gravitational anomaly. More precisely the chiral vortical conductivity for the symmetry generated by was found to have two contributions, one depending only on the chemical potentials and proportional to the axial anomaly coefficient and a second one with a characteristic temperature dependence proportional to the gravitational anomaly coefficient . At weak coupling the anomalous magnetic and vortical conductivities were found to be

 σBAB = dABC4π2μC, (5) σVA = dABC8π2μBμC+bA24T2. (6)

This characteristic behavior had appeared already previously in neutrino physics [13, 12, 11, 10]. It furthermore shows up via undetermined integration constants in effective field theory inspired approaches to hydrodynamics. A purely hydrodynamic approach to anomalous transport has been initiated in [9] (see also [45]). There it was shown that the anomalous transport coefficient can be fixed by construction of an appropriate entropy current whose divergence is positive definite. Later in [46] it was shown that there are additional integration constants proportional to and . CPT invariance forbids however the terms [47] so that at least in systems that can be described by local quantum field theories these terms are absent. Recently these studies have been extended to superfluids [48, 47, 49], to second order hydrodynamics [43] and to higher dimension [50] where similar undetermined integration constants were found.

The usage of Kubo formulas has here a clear advantage, it fixes all integration constants automatically. In this way it was possible in [44] to show that the coefficient in front of the term in the chiral vortical conductivity is essentially given by the gravitational anomaly coefficient . The disadvantage of Kubo formulas is of course that we have to calculate the potentially complicated correlations functions of a quantum field theory. They are easy to evaluate only in certain limits, such as the weak coupling limit considered in [44]. In principle the results obtained in this limit can suffer renormalization due to the model dependent interactions.

The gauge-gravity correspondence [51, 52, 53, 54] makes also the strong coupling limit easily accessible. Strongly coupled non-abelian gauge theories can be described via their gravity duals, more precisely in the large and infinite ’t-Hooft coupling limit, , allows a weakly coupled gravitational description. The drawback is that only some special and supersymmetric gauge theories, such as the maximally supersymmetric Yang-Mills theory have well understood gravity duals.

We would like to understand the effects anomalies have on the transport properties of relativistic fluids. Anomalies are very robust features of quantum field theories and do not depend on the details of the interactions. Therefore a rather general model that implements the correct anomaly structure in the gauge-gravity setup is sufficient for our purpose even without specifying in detail to which gauge theory it corresponds to. Our approach will therefore be a “bottom up” approach in which we simply add appropriate Chern-Simons terms that reproduce the relevant anomalies to the Einstein-Maxwell theory in five dimensions with negative cosmological constant. 222Very successful holographic bottom up approaches to QCD have been studied recently, either to describe non-perturbative phenomenology in the vacuum, see e.g. [55, 56], or the strongly coupled plasma [57, 58, 59].

In this paper we will introduce a model that allows for a holographic implementation of the mixed gauge-gravitational anomaly via a mixed gauge-gravitational Chern-Simons term of the form

 SCS=∫d5x√−gϵMNPQRAMRABNPRBAQR. (7)

Gravity in four dimensions augmented by a similar term with a scalar field instead of a vector field has attracted much interest recently [60] (see also the review [61]). A four dimensional holographic model with such a term has been shown to give rise to Hall viscosity in [62]. The quasinormal modes of this four dimensional model have been studied in [63].

In section 2 we will define the Lagrangian of our model, derive its equations of motion and study how the gravitational anomaly arises. We will find it necessary to add a particular boundary counterterm that cancels dependences on the extrinsic curvature of the gauge variation of the action. In section 3 we will study the holographic renormalization assuming the existence of an asymptotically AdS solution and show that the gravitational Chern-Simons terms does not introduce new divergencies.

In section 4 we will compute the equations of motions for metric and gauge perturbations in the shear sector. At zero frequency and to lowest order in the momentum these equations can be solved analytically. This allows us to compute the anomaly related conductivities in our model. We find that the contribution due to the gravitational anomaly is not renormalized compared to the weak coupling result. As is well-known, in a charged fluid a current necessarily induces also an energy flux. Kubo formulas for the energy flux induced by magnetic fields and vortices related to energy-momentum correlators. We also evaluate these and find that no terms proportional to appear. This is of course consistent with CPT conservation. We also show that the shear-viscosity to entropy ratio is unchanged.

We conclude with a discussion of our results and an outlook towards possible future directions in section 5. Several technical details of the calculations such as the Gauss-Codazzi form of the equations of motion, the details of the holographic renormalization, and the equations of motion for the shear sector and their solutions are collected in the appendices.

2 Holographic Model

In this section we will define our model. We start by fixing our conventions. We choose the five dimensional metric to be of signature . The epsilon tensor has to be distinguished from the epsilon symbol. The symbol is defined by whereas the tensor is defined by . Five dimensional indices are denoted with upper case latin letters. We define an outward pointing normal vector to the holographic boundary of an asymptotically AdS space with unit norm so that the induced metric takes the form

 hAB=gAB−nAnB. (8)

In general a foliation with timelike surfaces defined through can be written as

The Christoffel symbols, Riemann tensor and extrinsic curvature are given by

 ΓMNP = 12gMK(∂NgKP+∂PgKM−∂KgNP), (10) RMNPQ = ∂PΓMNQ−∂QΓMNP+ΓMPKΓKNQ−ΓMQKΓKNP, (11) KAV = hCA∇CnV=12\poundsnhAB, (12)

where denotes the Lie derivative in direction of .

Finally we can define our model. The action is given by

 S = 116πG∫d5x√−g[R+2Λ−14FMNFMN (13) +ϵMNPQRAM(κ3FNPFQR+λRABNPRBAQR)]+SGH+SCSK, SGH = 18πG∫∂d4x√−hK, (14) SCSK = −12πG∫∂d4x√−hλnMϵMNPQRANKPLDQKLR, (15)

where is the usual Gibbons-Hawking boundary term and is the covariant derivative on the four dimensional boundary. The second boundary term is needed if we want the model to reproduce the gravitational anomaly at general hypersurface. To study the behavior of our model under the relevant gauge and diffeomorphism gauge symmetries we note that the action is diffeomorphism invariant. The Chern Simons terms are well formed volume forms and as such are diffeomorphism invariant. They do depend however explicitly on the gauge connection . Under gauge transformations they are therefore invariant only up to a boundary term. We have

 δS = 116πG∫∂d4x√−hξϵMNPQR(κ3nMFNPFQR+λnMRABNPRBAQR)− (16) −λ4πG∫∂d4x√−hnMϵMNPQRDNξKPLDQKLR.

This is easiest evaluated in Gaussian normal coordinates (see next section) where the metric takes the form . All the terms depending on the extrinsic curvature cancel thanks to the contributions from ! The gauge variation of the action depends only on the intrinsic four dimensional curvature of the boundary and is given by

 δS=116πG∫∂d4x√−hϵmnkl(κ3^Fmn^Fkl+λ^Rijmn^Rjikl). (17)

This has to be interpreted as the anomalous variation of the effective quantum action of the dual field theory. The anomaly is therefore in the form of the consistent anomaly. Since we are dealing only with a single symmetry the (gauge) anomaly is automatically expressed in terms of the field strength. We could also express the anomaly in terms of an anomalous current conservation equation. One has to be however careful about the definition of the current since it is always possible to add a Chern-Simons current and redefine . This redefined current can not be expressed as the variation of a local functional of the fields with respect to the gauge field. In particular the so-called covariant form of the anomaly differs precisely in such a redefinition of the current333Note that the effective field thoery hydrodynamic approaches following [9] typically use the covariant form of the anomaly [46].. A good general reference for anomalies is Bertlmann’s book [64] where the consistent form of the anomaly for chiral fermions transforming under a symmetry group generated by is quoted as

 DmJmA=ηH124π2ϵijklTr[TA∂i(Aj∂kAl+12AjAkAl)], (18)

with for for right-handed and left-handed fermions respectively. We use this to fix to the anomaly coefficient for a single chiral fermion transforming under a symmetry. To do so we simply set in (18) which fixes the anomaly coefficient and therefore

 −κ48πG=196π2. (19)

Similarly we can fix by matching to the gravitational anomaly of a single left-handed fermion

 DmJm=1768π2ϵijkl^Rmnij^Rnmkl, (20)

and find

 −λ16πG=1768π2. (21)

As a side remark we note that the gravitational anomaly could in principle also be shifted into the diffeomorphism sector. This can be done by adding an additional (Bardeen like) boundary counterterm to the action

 Sct=∫d4x√−hAmIm, (22)

with fulfilling . Since this term depends explicitly on the four dimensional Christoffel connection it breaks diffeomorphism invariance.

The bulk equations of motion are

 GMN−ΛgMN = 12FMLFNL−18F2gMN+2λϵLPQR(M∇B(FPLRBN)QR), (23) ∇NFNM = −ϵMNPQR(κFNPFQR+λRABNPRBAQR), (24)

and they are gauge and diffeomorphism covariant. We note that keeping all boundary terms in the variations that lead to the bulk equations of motion we end up with boundary terms that contain derivatives of the metric variation normal to the boundary. We will discuss this issue in more detail in the next section where we write down the Gauss-Codazzi decomposition of the action.

3 Holographic Renormalization

In order to go through the steps of the holographic renormalization program within the Hamiltonian approach [65, 66], first of all we establish some notations. Without loss of generality we choose a gauge with vanishing shift vector , lapse and . So we can use four dimensional (boundary) indices and denote them by small latin letters. We therefore also write and . In this gauge the bulk metric can be written as

 ds2=dr2+γijdxidxj. (25)

The non vanishing Christoffel symbols are

 −Γrij = Kij=12˙γij, (26) Γijr = Kij, (27)

and are four dimensional Christoffel symbols computed with . Dot denotes differentiation respect . All other components of the extrinsic curvature vanish, i.e. . Another useful table of formulas is

 ˙^Γlki = DkKli+DiKlk−DlKki, (28) Rrirj = −˙Kij+KilKlj, (29) Rkrjr = −˙Kkj−KklKlj, (30) Rrijk = DkKij−DjKik, (31) Rlkri = DkKli−DlKik, (32) Rijkl = ^Rijkl−KikKjl+KilKjk. (33)

Note that indices are now raised and lowered with , e.g. , and intrinsic four dimensional curvature quantities are denoted with a hat, so is the intrinsic four dimensional Riemann tensor on the surface. Finally the Ricci scalar is

 R=^R−2˙K−K2−KijKij. (34)

Now we can calculate the off shell action. It is useful to divide it up in three terms. The first one is the usual gravitational bulk and gauge terms with the usual Gibbons-Hawking term. After some computations we get

 S0 = 116πG∫d5x√−γ[^R+2Λ+K2−KijKij−12EiEi−14^Fij^Fij], (35) S1CS = −κ12πG∫d5x√−γϵijklAiEj^Fkl, (36) S2CS = −8λ16πG∫d5x√−γϵijlk[Ai^RnmklDnKmj+EiKjmDkKml+12^FikKjm˙Kml]. (37)

We have used implicitly here the gauge and denoted . The purely four dimensional field strength is denoted with a hat.

Of particular concern is the last term in which contains explicitly the normal derivative of the extrinsic curvature . For this reason the field equations will be generically of third order in -derivatives and that means that we can not define a well-posed Dirichlet problem by fixing the and alone but generically we would need to fix also . Having applications to holography in mind we can however impose the boundary condition that the metric has an asymptotically AdS expansion of the form

 γij=e2r(g(0)ij+e−2rg(2)ij+e−4r(g(4)ij+2r~g(4)ij)+⋯). (38)

Using the on-shell expansion of obtained in the appendix B we can show that the last term in the action does not contribute in the limit . Therefore the boundary action depends only on the boundary metric but not on the derivative . This is important because otherwise the dual theory would have additional operators that are sourced by the derivative. Similar issues have arisen before in the holographic theory of purely gravitational anomalies of two dimensional field theories [62, 67, 68]. Alternatively one could restrict the field space to configurations with vanishing gauge field strength on the boundary. Then the last term in is absent. We note that the simple form of the higher derivative terms arises only if we include in the action. An analogous term in four dimensional Chern-Simons gravity has been considered before in [69].

The renormalization procedure follows from an expansion of the four dimensional quantities in eigenfunctions of the dilatation operator

 δD=2∫d4xγijδδγij. (39)

We explain in much details the renormalization in appendix B. The result one gets for the counterterm coming from the regularization of the boundary action is

 Sct = −(d−1)8πG∫∂d4x√−γ[1+1(d−2)P (40) −14(d−1)(PijPji−P2−14^F(0)ij^F(0)ij)loge−2r],

where

 P=^R2(d−1),Pij=1(d−2)[^Rij−Pδij]. (41)

As a remarkable fact there is no contribution in the counterterm coming from the gauge-gravitational Chern-Simons term. This has also been derived in [70] in a similar model that does however not contain .

4 Kubo formulas, anomalies and chiral vortical conductivity

We are now going to evaluate the Kubo formulas for anomalous transport in our holographic model. First we note that in a charged fluid a charge current is always accompanied by an energy current through . Therefore charge transport is always accompanied by energy transport. Kubo formulas for the energy transport coefficients can easily be obtained as well. In [41] it was shown that the chiral vortical conductivity for charge and energy transport can be obtained from the retarded Green functions

 σV = limkc→0i2kc∑a,bϵabc⟨JaT0b⟩|ω=0, (42) σϵV = limkc→0i2kc∑a,bϵabc⟨T0aT0b⟩|ω=0, (43)

where is the (anomalous) current and is the energy momentum tensor, the chiral vortical conductivity and the vortical conductivity of energy current. The chiral magnetic conductivity and the magnetic conductivity for energy current are given by

 σB = limkc→0i2kc∑a,bϵabc⟨JaJb⟩|ω=0,A0=0, (44) σϵB = limkc→0i2kc∑a,bϵabc⟨T0aJb⟩|ω=0. (45)

As explained in [23, 41] we also have to set the background value of the temporal component of the gauge field to zero. Hydrodynamic constitutive relations depend however on a particular definition of the fluid velocity. In the case of the anomalous conductivities this frame dependence has been addressed in [41] where it was shown how the Landau frame conductivities used by Son & Surowka [9] can be obtained from a combination of the charge and energy transport coefficient. This combination emerges because of the change of coordinates from the laboratory rest frame to a local comoving frame on a element of fluid in which there is no energy flux. Applying this change of frame we arrive to the transport coefficients in Landau frame

 ξB = (46) ξV = (47)

The frame dependence has also recently been discussed in [50]. The relevant parts of the hydrodynamic constitutive relations are

 δTmn = σϵB(umBn+unBm)+σϵV(umωn+unωm), (48) δJm = σBBm+σVωm, (49)

whereas in Landau frame demanding we have no contribution to the energy momentum tensor but instead

 δJm=ξBBm+ξVωm, (50)

where .

The AdS/CFT dictionary tells us how to compute the retarded propagators [71, 72]. Since we are interested in the linear response limit, we split the metric and gauge field into a background part and a linear perturbation,

 gMN = g(0)MN+ϵhMN, (51) AM = A(0)M+ϵaM. (52)

Inserting these fluctuations-background fields in the action and expanding up to second order in we can read the second order action which is needed to get the desired propagators [73]. If we construct a vector with the components of and and Fourier transforming it

 ΦI(r,xμ)=∫ddk(2π)dΦIk(r)e−iωt+i→k→x, (53)

it is possible to write the complete second order action on-shell as a boundary term

 δS(2)ren=∫ddk(2π)d{ΦI−kAIJΦ′Jk+ΦI−kBIJΦJk}∣∣r→∞, (54)

where derivatives are taken with respect to the radial coordinate.

Now we can compute the holographic response functions from (54) by applying the prescription of [71, 72, 74, 73]. For a coupled system the holographic computation of the correlators consists in finding a maximal set of linearly independent solutions that satisfy infalling boundary conditions on the horizon and that source a single operator at the AdS boundary. To do so we can construct a matrix of solutions such that each of its columns corresponds to one of the independent solutions and normalize it to the unit matrix at the boundary. Therefore, given a set of boundary values for the perturbations, , the bulk solutions are

 ΦIk(r)=FIJ(k,r)φJk. (55)

Finally using this decomposition we obtain the matrix of retarded Green functions

 GIJ(k)=−2limr→∞(AIM(FMJ(k,r))′+BIJ). (56)

The system of equations (23)-(24) admit the following exact background AdS Reissner-Nordström black-brane solution

 ds2 = ¯r2L2(−f(¯r)dt2+d→x2)+L2¯r2f(¯r)d¯r2, A(0) = ϕ(¯r)dt=(β−μ¯r2H¯r2)dt, (57)

where the horizon of the black hole is located at and the blackening factor of the metric is

 f(¯r)=1−ML2¯r4+Q2L2¯r6. (58)

The parameters and of the RN black hole are related to the chemical potential and the horizon by

 M=¯r4HL2+Q2¯r2H,Q=μ¯r2H√3. (59)

The Hawking temperature is given in terms of these black hole parameters as

 T=¯r2H4πL2f(¯rH)′=(2¯r2HM−3Q2)2π¯r5H. (60)

The pressure of the gauge theory is and its energy density is due to the underlying conformal symmetry .

Without loss of generality we consider perturbations of momentum in the -direction at zero frequency. To study the effect of anomalies we just turned on the shear sector (transverse momentum fluctuations) and , where .444Since we are in the zero frequency case the fields completely decouple of the system and take a constant value, see appendix C. For convenience we redefine new parameters and radial coordinate

 ¯λ=4μλL3¯r2H;¯κ=4μκL3¯r2H;a=μ2L23¯r2H;u=¯r2Hr2. (61)

Now the horizon sits at and the AdS boundary at . Finally we can write the system of differential equations for the shear sector, that consists on four second order equations. Since we are interested in computing correlators at hydrodynamics regime, we will solve the system up to first order in . The reduced system can be written as

 0 = hα′′t(u)−hα′t(u)u−3auB′α(u)+i¯λkϵαβ[(24au3−6(1−f(u)))Bβ(u)u (62) +(9au3−6(1−f(u)))B′β(u)+2u(uhβ′t(u))′], 0 = B′′α(u)+f′(u)f(u)B′α(u)−hα′t(u)f(u) (63) +ikϵαβ( 3uf(u)¯λ(2a(f(u)−1)+3u3)hβ′t(u)+¯κBβ(u)f(u)),

with the gauge field redefined as . The complete system of equations depending on frequency and momentum is showed in appendix C. This system consists of six dynamical equations and two constraints.

In order to get solutions at first order in momentum we expand the fields in the dimensionless momentum such as

 hαt(u) = h(0),α(u)t+ph(1)α(u)t, (64) Bα(u) = B(0)α(u)+pB(1)α(u). (65)

The relevant physical boundary conditions on fields are: , ; where the ‘tilde’ parameters are the sources of the boundary operators. The second condition compatible with the ingoing one at the horizon is regularity for the gauge field and vanishing for the metric fluctuation [41].

After solving the system perturbatively (see appendix D for solutions), we can go back to the formula (56) and compute the corresponding holographic Green functions. If we consider the vector of fields to be

 Φ⊤k(u)=(Bx(u),hxt(u),Bz(u),hzt(u)), (66)

the and matrices for that setup take the following form

 A=¯r4H16πGL5Diag(−3af,1u,−3af,1u), (67)
 BAdS+∂=¯r4H16πGL5⎛⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜⎝0−3a4κikμ2ϕL53r4H00−3u200−4κikμ2ϕL53r4H00−3a000−3u2⎞⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟⎠, (68)
 BCT=¯r4H16πGL5⎛⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜⎝\vspace0.15cm0000\vspace0.15cm03u2√f00\vspace0.15cm0000\vspace0.15cm0003u2√f⎞⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟⎠, (69)

where . Notice that there is no contribution to the matrices coming from the Chern-Simons gravity part, the corresponding contributions vanish at the boundary. These matrices and the perturbative solutions are the ingredients to compute the matrix of propagators. Undoing the vector field redefinition introduced in (62) and (63) the non-vanishing retarded correlation functions at zero frequency are then

 Gx,tx = Gz,tz=√3Q4πGL3, (70) Gx,z = −Gz,x=i√3kQκ2πG¯r2H+ikβκ6πG, (71) Gx,tz = Gtx,z=−Gz,tx=−Gtz,x=3ikQ2κ4πG¯r4H+2ikλπT2G, (72) Gtx,tx = Gtz,tz=M16πGL3, (73) Gtx,tz = −Gtz,tx=+i√3kQ3κ2πG¯r6H+4πi√3kQT2λG¯r2H. (74)

Using the Kubo formulas (46) and (47) and setting the deformation parameter to zero we recover the conductivities

 σB = −√3Qκ2πG¯r2H=μ4π2, (75) σV = σϵB=−3Q2κ4πG¯r4H−2λπT2G=μ28π2+T224, (76) σϵV = −√3Q3κ2πG¯r6H−4π√3QT2λG¯r2H=μ312π2+μT212. (77)

The first expression is in perfect agreement with the literature and the second one shows the extra term predicted in [44]. In fact the numerical coefficients coincide precisely with the ones obtained in weak coupling. This we take as a strong hint that the anomalous conductivities are indeed completely determined by the anomalies and are not renormalized beyond one loop. We also point out that the term that appears as undetermined integration constant in the hydrodynamic considerations in [49] should make its appearance in . We do not find any such term which is consistent with the argument that this term is absent due to CPT invariance.

It is also interesting to write down the vortical and magnetic conductivity as they appear in the Landau frame,

 ξB = −√3Q(ML2+3¯r4H)κ8πGML2¯r2H−√3QλπT2GM (78) = 14π2⎛⎜⎝μ−12n(μ2+T212)ϵ+P⎞⎟⎠, ξV = −3Q2κ4πGML2−2πλT2(¯r6H−2L2Q2)GML2¯r2H (79) = μ28π2(1−23nμϵ+P)+T224(1−2nμϵ+P).

Finally let us also note that the shear viscosity is not modified by the presence of the gravitational anomaly. We know that , so we should solve the system at for the fluctuations but the anomalous coefficients always appear with a momentum as we can see in (124), therefore if we switch off the momentum, the system looks precisely as the theory without anomalies. In [75] it has been shown that the black hole entropy doesn’t depend on the extra mixed Chern-Simons term, therefore the shear viscosity entropy ratio remain the same in this model555For a four dimensional holographic model with gravitational Chern-Simons term and a scalar field this has also been shown in [63]..

5 Discussion and conclusion

We have defined a holographic bottom up model that implements the mixed gauge-gravitational anomaly in four dimensional field theory via a mixed gauge-gravitational Chern-Simons term. We have discussed its holographic renormalization and have shown that the Chern-Simons term does not introduce new divergencies. As a first application of this theory we have computed the anomalous magnetic and vortical conductivities from a charged black hole background and have found the terms characteristic for the contribution of the gravitational anomaly.

The most important result is certainly that the numerical values of the conductivities coincide precisely with the ones obtained at weak coupling in [44]. This is a strong hint towards a non-renormalization theorem for the anomalous conductivities including the contributions from the gravitational anomaly.

We have studied a holographic system with only one anomalous symmetry. It should however present no problem to generalize our calculation to the case with additional non-abelian symmetries and various types of mixed anomalies, e.g. mimicking the usual interplay of axial and vector symmetries where gauge Bardeen counterterms are necessary to implement the correct anomaly structures in the currents [21, 23].

So far the contributions of the gravitational anomaly have shown up in the calculations of Kubo formulas. It is however also possible to calculate directly the constitutive relations of the hydrodynamics of anomalous currents via the fluid/gravity approach of [76]. This very interesting question can be addressed within the model presented in this paper [77].

Appendix A Codazzi form of Equations of Motion

We project the equations of motion (23) and (24) into the boundary surface and the orthogonal direction and rewrite them in terms of quantities at the regulated boundary. Doing so we get a set of two dynamical equations

 0 = ˙Ei+KEi+Dj^Fji−4ϵijkl(κEj^Fkl+4λ˙KsjDlKsk+2λ^RstklDsKtj (80) +4λKksKtlDtKsj+4λKstKtjDlKsk),
 0 = ˙Kij+KKij−^Rij+12EiEj+12^Fim^Fjm−δij(d