PHYSICAL REVIEW D 98, 106023 (2018)

Universality of the Chern-Simons diffusion rate

† ‡ Francesco Bigazzi,1,* Aldo L. Cotrone,1,2, and Flavio Porri1,2, 1INFN, Sezione di Firenze, Via G. Sansone 1, I-50019 Sesto Fiorentino (Firenze), Italy 2Dipartimento di Fisica e Astronomia, Universit`a di Firenze, Via G. Sansone 1, I-50019 Sesto Fiorentino (Firenze), Italy

(Received 20 August 2018; published 27 November 2018)

We prove the universality of the Chern-Simons diffusion rate—a crucial observable for the chiral magnetic effect—in a large class of planar strongly correlated gauge theories with dual string description. When the effects of anomalies are suppressed, the diffusion rate is simply given in terms of temperature, entropy density and gauge coupling, with a universal numerical coefficient. We show that this result holds, in fact, for all the top-down holographic models where the calculation has been performed in the past, even in the presence of magnetic fields and anisotropy. We also extend the check to further well-known models for which the same computation was lacking. Finally we point out some subtleties related to the definition of the Chern-Simons diffusion rate in the presence of anomalies. In this case, the usual definition of the rate—a late time limit of the imaginary part of the retarded correlator of the topological charge density—would give an exactly vanishing result, due to its relation with a nonconserved charge correlator. We confirm this observation by explicit holographic computations on generic isotropic black hole backgrounds. Nevertheless, a nontrivial Chern- Simons relaxation time can in principle be extracted from a quasinormal mode calculation.

DOI: 10.1103/PhysRevD.98.106023

I. INTRODUCTION techniques, as first performed in [3]. Thus, we consider the planar, strong coupling regime of quantum theories The Chern-Simons diffusion rate Γ is a fundamental CS which can model (but always present some differences observable in the study of chiral effects in the Quark-Gluon Plasma (QGP). Its value, setting the rate of (local) change of from) real world QCD. Holography has proven to give the Chern-Simons number, determines, through the axial fruitful indications about the physics of transport coeffi- anomaly, the (local) change rate of chirality imbalance in cients in QCD. In particular, when some universality can be the plasma. Chirality imbalance is then responsible, in the found among different holographic models, the result for presence of a magnetic field, for the chiral magnetic effect [1], the observable under scrutiny can be used as a benchmark which can be in principle measured in current experiments. for the experiments, since clearly it does not depend on Being inherently nonperturbative, the Chern-Simons many of the details of the theory. The shear viscosity is the diffusion rate cannot be computed in QCD from first most notable example of such a situation [4,5]. principles. In the literature there are interesting effective We are going to prove that at leading order in the planar field theory calculations, setting for example the behavior strong coupling limit, the Chern-Simons diffusion rate is with N (the number of colors of the theory) to be OðN0Þ universal in a relevant class of top-down holographic N 4 [2]. But it is fair to say that we lack a solid estimate models which includes, among the others, ¼ SYM (analogous to what can be obtained from the lattice for [3], even in the presence of anisotropies [6,7] and magnetic Γ fields [8], the Witten-Sakai-Sugimoto model [9–11] and the other observables) for the value of CS in the realistic strong coupling regime of the QGP. Maldacena-Nuñez model [12,13], for which the computa- Γ In this paper we address the problem of the calculation of tion of CS was lacking. In all the cases the field theory the Chern-Simons diffusion rate by means of holographic comes either from D3-branes or wrapped Dp-branes with p>3. More specifically, at leading order in the large N limit *[email protected] † where axial anomalies can be neglected, the Chern-Simons [email protected] ‡ [email protected] diffusion rate is given, with a universal coefficient, in terms of the temperature T, entropy density s and the gauge Published by the American Physical Society under the terms of α 2 4π coupling s ¼ gYM=ð Þ of the field theory the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to 2 Γ α ðTÞ the author(s) and the published article’s title, journal citation, CS ¼ s : ð1:1Þ and DOI. Funded by SCOAP3. sT 23π3

2470-0010=2018=98(10)=106023(14) 106023-1 Published by the American Physical Society BIGAZZI, COTRONE, and PORRI PHYS. REV. D 98, 106023 (2018) Z Δ 2 At present there are no counterexamples to this formula in hð NCSÞ i 4 Γ ¼ ¼ d xhQðxÞQð0Þi; ð1:2Þ top-down holographic models. So, it is not excluded that its CS Vt validity goes beyond the class of models considered in this where the variation in the Chern-Simons number is paper. Thus, the holographic behavior of this observable Z Z can be used with a certain degree of confidence as a 1 Δ 4 4 ˜ benchmark for the QGP. For example, a very rough NCS ¼ d xQðxÞ¼ d x 16π2 TrFF: ð1:3Þ estimate of the critical temperature values αsðTcÞ ∼ 3 Here F is the Yang-Mills field strength, Q the topological 1=2;sðTcÞ ∼ 10Tc (lattice results without magnetic field Γ 4 ∼ 0 01 charge density operator [14,15]) gives the value CS=Tc . . This would have quite a small effect in the QGP. 1 Q FF;˜ : From a technical point of view, the derivation of the ¼ 16π2 Tr ð1 4Þ result (1.1) relies on the fact that the quadratic part of a b the five dimensional bulk action for the field dual to the and we normalize the SUðNÞ generators so that Tr½t t ¼ topological charge density operator has a universal form in δab=2 for the fundamental representation. this class of models. It is the action of a massless scalar. Its The two-point function in formula (1.2) is the sym- kinetic term depends on a function which, in the models metrized Wightman one—everything is defined in real Γ at hand, is the dual of the [Eq. (2.13)]. time. In a state at thermal equilibrium at temperature T, CS The universality of this result can be useful beyond the can be given by a Kubo formula determination of the Chern-Simons diffusion rate. 2 The most interesting correction to the holographic result Γ − T ω ⃗ 0 CS ¼ lim ImGRð ; k ¼ Þ: ð1:5Þ for the Chern-Simons diffusion rate in the planar expansion ω→0 ω comes from possible anomalies proportional to the topo- Thus its computation boils down to the determination of the logical charge density operator. In the bulk, these are small frequency, zero momentum retarded correlator described by a Stueckelberg action for the scalar mentioned ω ⃗ 0 above and a vector field dual to the anomalous current [16]. GRð ; k ¼ Þ of the operator Q. The retarded correlator We are going to show that, in this case, the usual holo- is precisely what can be directly computed in holography Γ from the standard prescriptions [3], once the bulk field dual graphic prescription for the calculation of CS gives an identically vanishing result.1 This fact is very general for to the operator Q is identified. holographic field theories and does not rely on the details of The paper is organized as follows. In Sec. II we prove the the dual gravity backgrounds. universal formulas (1.1), (2.13) for the Chern-Simons dif- As we are going to discuss, this feature is not related to the fusion rate and the gravity action of the field dual to the holographic limit, but it simply relies on the fact that the usual corresponding operator in a relevant class of top-down Γ holographic models in absence of anomalies. Then, in definition of CS, as the late time behavior of the (imaginary part of the) retarded correlator of the topological charge Sec. III we discuss the effects of anomalies confirming, density, gives zero if the latter is proportional, as it happens through a holographic computation on generic isotropic black due to the anomaly relation, to the retarded correlator of a hole backgrounds, that the standard prescription for the Γ nonconserved (axial) charge. The relaxation time of the latter calculation of CS gives an identically vanishing result. In can, instead, be consistently defined and expressed in terms turn, we discuss how to extract the Chern-Simons relaxation of the Chern-Simons diffusion rate computed with a cutoff in time from the quasinormal modes of the dual black hole time, i.e., effectively turning off the anomaly. The rate backgrounds. The Appendices include the analysis of the Γ computed with this prescription is not automatically vanish- direct calculation of CS in specific examples, including, for ing. Some considerations along this line can be already found the first time, the Maldacena-Nuñez model. Moreover they in [2,18] where the same issue has been pointed out for QCD. contain a sketch of the derivation of a linear response formula – Chiral effects in the presence of anomalies can be inves- for the axial relaxation time as well as specific anti de Sitter tigated, from a holographic point of view, through the (AdS) examples of the generic holographic results discussed analysis of quasinormal modes of the dual black hole in Sec. III. backgrounds. These quasinormal modes, in fact, allow extraction of the above mentioned relaxation time. II. HOLOGRAPHIC CHERN-SIMONS Before concluding this section, let us define the relevant DIFFUSION RATE quantities for our purposes. The Chern-Simons diffusion rate is the probability of fluctuation of the Chern-Simons We are going to show that in the plasma phase of a large Δ class of planar strongly correlated gauge theories with dual number NCS per unit time t and unit volume V string description, the Chern-Simons diffusion rate reads 1 Γ All the results concerning CS present in the literature boil Γ 4 down to the leading planar contribution, i.e., the effects of CS gYMðTÞ ¼ 7 5 ; ð2:1Þ anomaly are not fully included in the calculation [17,18]. sT 2 π

106023-2 UNIVERSALITY OF THE CHERN-SIMONS DIFFUSION RATE PHYS. REV. D 98, 106023 (2018) Z   1 θ where the Yang-Mills coupling gYM and the entropy density 4 μν YM ˜ μν S − d x FμνF FμνF ; : s are evaluated at the temperature scale T of the plasma. YM ¼ 2 2 Tr þ 16π2 Tr ð2 3Þ gYM This result holds as long as any possible anomaly is ignored—effects of anomalies are discussed in the follow- with ing section. In the derivation below, we are going to prove Z another general result for the same class of models: the 1 τ pffiffiffiffiffiffiffiffiffi p 2πα0 2 p−3 −ϕ 2 ¼ ð Þ d xe det g; quadratic part of the five dimensional Lagrangian for the 2 Ω gYM p−3 field dual to the topological charge density operator Q is Z θ 2π 2τ 2πα0 2 universal, being the one of a massless scalar times a YM ¼ð Þ pð Þ Cp−3: ð2:4Þ Ω function of the background metric which is precisely the p−3 (squared) coupling in the dual field theory. In these formulas g is the (pull-back of the) metric, F the field strength of the gauge field on the brane A. Derivation of the result world-volume and Cn are RR n-form potentials; τp ¼ The main difficulty in deriving a general result for the −p 0−ðpþ1Þ=2 −1 ð2πÞ α gs is the brane tension. The discussion is Chern-Simons (CS) diffusion rate in holography is that it limited to models with vanishing Neveu-Schwarz-Neveu- requires a precise identification of the gravity fields dual to Schwarz B field and RR C −5 potential along the cycle the topological charge density operator Q and the Yang- p Ω −3, on the black hole background. This condition is not Mills coupling g . Unlike what happens for the shear p YM very restrictive and it is always obeyed in all the examples viscosity, whose related gravity field is a component of the where the CS diffusion rate has been holographically metric which has a universal Lagrangian, the form of the computed so far. Notice that in the case p ¼ 3 (unwrapped five dimensional action for the gravity field dual to Q is in D3-branes) the above relations give the usual identifica- principle model dependent. 2 tions g ¼ 4πg and θ ¼ 2πC0, where C0 is the type Nevertheless, there exists a class of models where the YM s IIB axion. identification of the field theory quantities needed for the We are going to consider Einstein frame actions in five computation is quite solid, allowing for the derivation of dimensions, so we rewrite the coupling as the result (2.1). The models we refer to are built up by Z wrapping Dp-branes over (p − 3)-cycles, giving rise at low τ pffiffiffiffiffiffiffiffiffiffiffiffi 1 p 2 −3 p−7ϕ 2πα0 p 4 energies to four dimensional gauge theories. The case 2 ¼ ð Þ d xe det gE; ð2:5Þ g 2 Ω p ¼ 3, i.e., N ¼ 4 SYM, is included in the discussion as YM p−3 well. In these cases, we are going to show that the five where g is the Einstein frame metric. Of course on shell dimensional action has indeed a universal form. E the coupling does not depend on the frame. In the probe All the calculations of the Chern-Simons diffusion rate in brane spirit, the identifications above are going to be holographic top-down models in the literature, and other considered as the definitions of the couplings in the full ones we are going to discuss, are performed in represent- theory where the branes have been replaced by geometry. atives of this class of models. As such, at present for- Inspection of formulas (2.3) and (2.4) dictates that the mula (2.1) has no counter-example. It is tempting to 1 ˜ coupling of Q ¼ 2 TrFF with the dual gravity field C is conjecture that it is valid for every theory with a gravity R 16π precisely QC if dual in the strong coupling regime. As stated above, the difficulty in checking this statement in other models is due Z 2 0 2 to the uncertain identification of the coupling and the field C ¼ τpð2πÞ ð2πα Þ Cp−3 Ω dual to Q. It would be obviously interesting to provide a p−3 2 0 2 ˜ more general proof of the result (2.1) or to find counter- ≡ τpð2πÞ ð2πα Þ VolðΩp−3ÞC; ð2:6Þ examples within different classes of holographic models. Let us now prove (2.1). The main observation is that, in where VolðΩp−3Þ is the volume of the cycle and we have holographic models coming from Dp-branes wrapped on introduced the reduced field C˜ which is the scalar typically − 3 Ω (p )-cycles p−3, the reduction of the brane DBI þ WZ present in the five dimensional reduced gravity action. action Thus, the gravity field dual to Q is the Cp−3 RR field − 3 Z pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi integrated on the (p )-cycle. pþ1 −ϕ 0 Let us derive its five dimensional action. Consider the S ¼ −τpTr d xe −detðg þ 2πα FÞ Z Einstein frame ten dimensional action for the correspond- X ing field strength þ τ Tr C ∧ e2πα0F; ð2:2Þ p n Z   n 1 10 pffiffiffiffiffiffiffiffiffiffi 7−pϕ 1 2 d x −g10e 2 − F ; : 2κ2 2 p−2 ð2 7Þ includes the Yang-Mills action 10

106023-3 BIGAZZI, COTRONE, and PORRI PHYS. REV. D 98, 106023 (2018) Z     together with the reduction ansatz 1 ffiffiffiffiffiffiffiffi 1 g2 2 5 p− − ∂ ∂M YM 2 d x g5H MC C ;H¼ 2 : 2κ5 2 8π ds2 ¼ efds2 þ ds2 ; ð2:8Þ 10 5 int ð2:14Þ where f is a function which encodes the dependence of the Thus, in the wrapped brane models, the quadratic internal volume on the external coordinates. The reduced part of the five dimensional action for the field dual to Einstein action has canonical form if Q is in a universal form. This is the main result of this Z ffiffiffiffiffiffi section. 5 p −3f=2 d y gint ¼ Vinte ; ð2:9Þ Now, for an action of the form (2.14)—on a diagonal black hole metric with components depending only on the i 1 …5 radial coordinate—with H a function of the background where y , i ¼ ; are the internal coordinates and Vint is the constant part of the compactification volume. fields and C the field dual to the operator Q, the holo- Now, one of the indices of the (p − 2)-form must be in graphic calculation of the CS diffusion rate (following the the five-dimensional directions, while the other ones are original prescription in [3]) gives [19] Ω along the p−3 cycle. Thus, assuming that the metric 1 Γ functions have a trivial dependence on the internal direc- CS ¼ 2π HhsT; ð2:15Þ tions, we have

2 ˜ M ˜ −1 −f where in Hh the fields are evaluated at the horizon and s is F −2 ¼ ∂MC∂ C½detðgE;Ω0 Þ e : ð2:10Þ p p−3 the Bekenstein-Hawking entropy density. Since in our case 4 gYMðTÞ (2.13) we have H ¼ 2 2 , where the coupling is at the 0 h ð8π Þ With detðgE;Ω −3 Þ we denote the determinant of the Einstein p temperature scale T, (2.15) gives immediately the frame metric along the Ω −3 cycle modulo its volume form, R p qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi result (2.1). i.e., g Ω ¼ VolðΩ −3Þ detðg Ω0 Þ. “M” is a Ωp−3 E; p−3 p E; p−3 Let us conclude this section by noting that the form of five dimensional index. the action (2.12) reproduces all the cases for which the In formula (2.10) we have assumed that there is no calculation of the CS diffusion rate has been performed. ˜ N 4 mixing of the field ∂MC with any vector potential: this For ¼ SYM it is immediate: there is no reduction and 2 ϕ the dilaton is trivial, so e ¼ 1 ¼ detðgΩ0 Þ and we have amounts to assuming that anomalies are subleading effects, p−3 as it will become clear in the next sections. the usual minimally coupled scalar studied in the original Since paper [3]. As it is shown in Appendix A, the same is true in Z Z 1 ffiffiffiffiffiffiffiffiffiffi 1 ffiffiffiffiffiffiffiffi the presence of a magnetic field [8], whose only effect is to 10 p− 5 p− f change the explicit form of the entropy density s but not the 2κ2 d x g10 ¼ 2κ2 d x g5e ; ð2:11Þ 10 5 form of the relation (2.1). The same relation is also

1 Vint precisely satisfied by the Chern-Simons diffusion rate where 2κ2 ¼ 2κ2 , the reduction of (2.7) gives 5 10 of the anisotropic N ¼ 4 plasma of [6] as computed in [7]. Z    1 ffiffiffiffiffiffiffiffi 1 1 This is a relevant nontrivial example since the background 5 p ˜ M ˜ d x −g5 − ∂ C∂ C −7 ; solution has a running dilaton. Details are provided in 2κ2 2 M p ϕ 5 e 2 detðg Ω0 Þ E; p−3 Appendix B. ’ ð2:12Þ We review the case of the wrapped D4-brane Witten s Yang-Mills (WYM) model [9] in Appendix C and N 1 or equivalently derive (for the first time) the result for the ¼ Super Z    Yang-Mills model by Maldacena and Nuñez (MN) 1 ffiffiffiffiffiffiffiffi 1 Ω 2 [12,13], coming from wrapped D5-branes, in 5 p ˜ M ˜ Volð p−3Þ d x −g5 − ∂ C∂ C R ffiffiffiffiffiffiffiffiffiffiffiffi 2 M p−7ϕp Appendix D. In all these cases the coupling, and so the 2κ5 2 4 Ω −3 e det gE 2ϕ Z  p ratio e = detðgΩ0 Þ is a constant in the deconfined phase. 1 ffiffiffiffiffiffiffiffi 1 1 p−3 5 p− − ∂ ∂M Finally, in the flavored version of the N ¼ 4 SYM plasma ¼ 2κ2 d x g5 2 MC C 2π 4 5 ð Þ [20,21],detðgΩ0 Þ¼1 and the dilaton factor in (2.13) is   p−3 1 2 precisely what is given by the consistent reduction of the × R −7 ffiffiffiffiffiffiffiffiffiffiffiffi : ð2:13Þ 2ϕ 4 2 p ϕp h 0 4 model [22],soH ∼ e ∼ g ðTÞ. τpð2πα Þ Ω e det gE h YM p−3 It is worth noticing that the result (2.1) holds at leading λ ≫ 1 λ 2 From (2.5) one can recognize that the term in round order in the holographic limits N; , where ¼ gYMN parenthesis is nothing else than the fourth power of the 2 coupling (times a number), so that we get the action In our conventions gs is included in τp.

106023-4 UNIVERSALITY OF THE CHERN-SIMONS DIFFUSION RATE PHYS. REV. D 98, 106023 (2018) is the ’t Hooft coupling. When the first higher derivative transforming the term in square brackets back to position (α0) corrections3 are added on the gravity side—which space we obtain 3 2 amounts to include 1=λ = corrections in the dual quantum Z — N 1 þ∞ field theory the Chern-Simons diffusion rate of the ¼ DD iωt∂ 0 4 limGR ðkÞ¼i dte thQAðtÞQAð ÞiR: ð3:4Þ plasma is given by [7] ω k⃗→0 −∞   g4 ðTÞ π2 45 ζð3Þ ω → 0 Γ ¼ YM N2T4 1 − þ ; ð2:16Þ Taking now the limit we get CS 27π5 2 8 λ3=2 1 lim limGDDðkÞ¼ihQ ðt → þ∞ÞQ ð0Þi : ð3:5Þ while the entropy density is (see [24]) ω→0 R A A R ω k⃗→0   2 π 2 3 15 ζð3Þ s ¼ N T 1 þ þ : ð2:17Þ Since the charge QA is not conserved the above retarded 2 8 3=2 λ correlator vanishes as t → ∞. Therefore, in the presence of a chiral anomaly, (1.5) does not provide a good definition α0 The result is that corrections reduce the Chern-Simons of the Chern-Simons diffusion rate. diffusion rate with respect to the value in (2.1). This could In other words, the modes associated to the noncon- led to conjecture that Eq. (2.1) sets an upper bound for served charge Q are gapped and do not survive the Γ =sT A CS for generic gauge theories. hydrodynamic limit. Indeed, the equilibrium, time- A last comment is in order. We have restricted the independent value of a nonconserved charge necessarily discussion to models with no Neveu-Schwarz-Neveu- vanishes and the decay in time of the charge is obscured by Schwarz B field turned on. A nontrivial B field is usually the limit. associated with multiple gauge groups, with more that one In order to study the explicit decay of the total chiral topological charge operator. The prototype models are the charge one should look at the retarded correlator GttðωÞ¼ Klebanov-Witten one [25] and its nonconformal extension, R t t ⃗ 0 ω the Klebanov-Strassler one [26], which have two gauge hJAJAiR at zero momentum k ¼ and finite frequency . groups. It is immediate to verify that the field dual to the sum Generically, the retarded correlator can be written as a of the two topological charge operators has the same sum of poles plus an analytic, scheme-dependent part. universal Lagrangian as above. In particular, for a nonconserved charge one expects the singular part of the correlator to be of the form4 III. CHERN-SIMONS DIFFUSION iR WITH U(1)A ANOMALIES tt ω ∼ GRð Þ i ; ð3:6Þ ω þ τ In a theory like QCD with Nf massless quarks, the topological charge density operator Q enters the anomaly with R and τ real and constant. The above correlator models equation for the axial current the gapped decay mode of the axial charge and, indeed, ∂ μ − taking the Fourier transform of both sides one obtains μJA ¼ qQ; ð3:1Þ −t ∼ hQ ðtÞQ ð0Þi ∼ RθðtÞe τ; ð3:7Þ where q Nf is the anomaly coefficient.R This equation A A R implies that the axial charge Q ¼ dx3Jt is not con- A A τ served and that its mean square change (e.g., on a thermal where is immediately identified with the axial relaxa- ensemble) is related to that of the Chern-Simons number as tion time. Notice that the imaginary part of the Green’s function Δ 2 2 Δ 2 (3.6) reads hð QAÞ i¼q hð NCSÞ i: ð3:2Þ R In turn, using the anomaly relation, the definition of the tt ω ∼ ω ImGRð Þ 2 −2 ; ð3:8Þ Chern-Simons diffusion rate (1.5) could be rewritten as ω þ τ   Γ 1 1 so that, if q ≠ 0, CS ¼ − limIm GDDðω; k⃗¼ 0Þ ; ð3:3Þ 2T q2 ω→0 ω R 1 ω 1 R ImGQQ ¼ ImGtt ∼ ω2 : ð3:9Þ DD ’ ω R 2 R 2 ω2 τ−2 where GR ðkÞ is the retarded Green s function of the q q þ divergence of the nonconserved axial current. Fourier 4There might be additional poles and branch cuts in the 3See [23] for holographic computations of the Chern-Simons correlators. We focus on the pole closest to the origin in the complex diffusion rate in a Gauss-Bonnet setup. ω plane, which is the one that dictates the late time behavior.

106023-5 BIGAZZI, COTRONE, and PORRI PHYS. REV. D 98, 106023 (2018) Z   ω → 0 1 ffiffiffiffiffiffi Hence, the limit of the left hand side, which one − 5 p− a 2 2 S ¼ 2 d x g jFAj þ HðdC þ qAÞ ; would take to get the Chern-Simons diffusion rate in 4κ5 2 absence of anomalies, would thus give zero. The above observations suggest that a correct way to FA ¼ dA: ð3:14Þ define the Chern-Simons diffusion rate in the presence of ≪ τ anomalies would be by means of a cut-off t in the We omit possible Chern-Simons terms since these are more time integration entering the related correlator in (1.2) than quadratic in the vector field A and cannot contribute to [2,18], whenever the microscopic time scales involved in the two-point function of the dual operator. The parameter the CS number fluctuation are much smaller than τ. q is constant while the kinetic coefficient a and the mass H A nice hydrodynamical model (realized holographically are in general functions of background fields. The action in the Witten-Sakai-Sugimoto theory) for axial charge (3.14) is found in all the dimensional reductions to five diffusion and relaxation can be found in [18]. Taking into dimensions of top-down holographic models with axial account the thermal fluctuations for the average squared anomaly, q being precisely the anomaly coefficient (see axial charge, which at equilibrium are related to the axial e.g., [16,18,22] and Appendix D). susceptibility χA, one can write, for t ≪ τ Here we will treat A and C as fluctuating fields over a fixed background. In the simplest case where the metric is 2χ 2 −2t AT 2 Δ ∼χ 1− τ ≈ ≡ Γ the only background field, the coefficients will all be hð QAÞ i AT½ e V Vt q CSVt; ð3:10Þ τ constant and can be set to any desired value rescaling where V is the spatial volume and in the last step we have the fields A and C. However, we will keep them unfixed to used (3.2). Crucially, this expression can make sense if the find a more general expression for the equations of motion. The action above is invariant under the transformation Chern-Simons diffusion rate is that of the q ¼ 0 theory Γ ≡ Γ 0 CS CSðq ¼ Þ: ð3:11Þ δA ¼ dλ; δC ¼ −qλ; ð3:15Þ The above formulas imply, in turn, that the frequency gap, which, enforced at the boundary on the couplings i.e., the inverse relaxation time, is given by (see also [2] and R μ Appendix E for an alternative derivation) AμJA þ CQ, precisely implies the operator relation

1 2Γ μ q CS ∂ ≃ 0 ¼ : ð3:12Þ μJA þ qQ ð3:16Þ τ 2χAT

Consistently, when q → 0, τ → ∞, hence the definition in the dual field theory, where JA is the current dual to the (3.11) can be read as the one which arises in the limit in vector A and Q the operator dual to the scalar C. which the cut-off in the time integration is sent to infinity. The combination B ¼ dC þ qA is invariant under the This framework is akin to the Witten-Veneziano formula above local transformation. In addition we have dB ¼ qdA, so when q ≠ 0 the action can be rewritten in terms of B only 2 χ 2 Nf g mWV ¼ 2 ; ð3:13Þ Z   fπ 1 ffiffiffiffiffiffi − 5 p− a 2 2 2 S ¼ 2 2 d x g jFBj þ q HB : ð3:17Þ 4κ5q 2 which, in large N QCD with Nf massless flavors at T ¼ 0, gives the squared η0 mass (driven by the axial anomaly) in terms of the topological susceptibility χg (the Euclidean In this formulation the holographic map reads Γ 0 counterpart of CS) of the unflavored (Nf ¼ ) theory (the topological susceptibility being zero for q ≠ 0 with mass- μ 1 μ Bμ ↔ J ¼ J ; ð3:18Þ less flavors). B q A As we are going to show in the following subsections, holography consistently implements the above observa- so that we have tions. Moreover, it also allows, in principle, to work beyond the hydrodynamical limit and to extract the complete ∂ μ ≃ 0 quasinormal mode spectrum of the correlators. μJB þ Q ð3:19Þ

A. The holographic approach inside correlation functions. In particular, the retarded correlator of the operator Q satisfies The holographic dual of a 4d quantum field theory with a Uð1Þ anomalous (axial) current includes a universal sector 0 ∂ μ ∂ ν 0 described by a Stueckelberg action [16] hQðxÞQð ÞiR ¼h μJBðxÞ νJBð ÞiR: ð3:20Þ

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B. Equations of motion which for q → 0 becomes the projector onto the subspace μ −k Let us consider a 5d background metric of the form transverse to k . From (3.26) and its counterpart for Bμ ,we find that 2 μ ν 2 ds gμνdx dx g du ; : ¼ þ uu ð3 21Þ ffiffiffiffiffiffi d p uu μν −k 0k k 0−k ½a −gg K ðBμ Bν − BμBν Þ ¼ 0; ð3:28Þ where the metric components are taken to be functions of du the radial variable u ∈ u ;u0 . The metric is assumed to ½ h so the quantity in parenthesis is conserved along radial have an horizon at u ¼ uh and to be asymptotically (i.e., at → motion. u u0) described by radial slices of Minkowski 4d The on-shell action reads spacetime. Notice the crucial assumption g μ ¼ 0.We u μ ν Z 4 denote as M the five-dimensional index, , the four 1 d k ffiffiffiffiffiffi u0 − −k p− uu μν 0k dimensional ones, i, j the three spatial directions Son-shell ¼ 4κ2 2 2π 4 Bμ ða gg K ÞBν : 5q ð Þ u and k ¼ðω; k⃗Þ. h Since eventually we will compute Green’s function in ð3:29Þ momentum space, it is convenient to use the Fourier transformed fields from the beginning. Defining Z C. Retarded Green’s functions d4k ≡ ˜ k −ik·x Thanks to the anomaly relation (3.20), the longitudinal BMðx; uÞ 2π 4 BMðuÞe ; ð3:22Þ ð Þ Green’s function of the current operator JB is equal to the retarded Green’s function of the topological operator Q and substituting inside the action (3.17) we find (dropping tildes) LL ≡ − μν − μ − ν GR ðkÞ kμkνGR ðkÞ¼ kμkνhJBð kÞJBðkÞiR Z − 1 d4k ¼hQð kÞQðkÞiR: ð3:30Þ S ¼ − 4κ2q2 ð2πÞ4 Z 5 The Chern-Simons diffusion rate is defined in terms of the ffiffiffiffiffiffi ’ p uu μν 0−k 0k uu ˆμ 0−k k retarded Green s function of the topological operator Q as × du −gfag g Bμ Bν þ iag k Bμ B þ u follows − uu ˆμ −k 0k −k ˆ2 2 μν − ˆμ ˆν k iag k Bu Bμ þ Bμ ½ðak þ q HÞg ak k Bν − Im½hQð kÞQðkÞiR ˆ2 2 − Γ ¼ −2T lim lim uu k k CS ω→0 þðak þ q HÞg Bu Bug; ð3:23Þ k⃗→0 ω Im½GLLðkÞ where 0 denotes the derivative along the radial direction and ¼ −2T lim lim R : ð3:31Þ ω→0 ⃗→0 ω hatted quantities are contracted using the metric gμν. The k equations of motion resulting from varying with respect to ’ 0−k The holographic computation of the Green s functions is Bμ are formally done through the following steps: a) setting k νk k k ffiffiffiffiffiffi ffiffiffiffiffiffi BμðuÞ¼bμ ðuÞbν, where bν ¼ kν can be chosen to imple- p− uu μν 0k 0 p− uu ˆμ k 0 ða gg g Bν Þ þ iða gg k BuÞ → ∂ ffiffiffiffiffiffi ment the boundary condition Bμ μC in momentum p ˆ2 2 μν ˆμ ˆν k νk − −g½ðak þ q HÞg − ak k Bν ¼ 0; ð3:24Þ space [both bμ ðuÞ and C are taken to be normalized to one at the boundary]; b) taking the functional (second) deriva- −k −k whereas varying with respect to Bu yields tive of the on-shell action (3.29) with respect to bμ (and k ⃗ tω ω bν). Setting k ¼ 0, with b ≡ b (and zero for the other ˆ2 2 k ˆμ 0k t t ðak þ q HÞBu ¼ iak Bμ : ð3:25Þ components) we get

k 2 The above equation can be used to remove Bu from (3.24), iω pffiffiffiffiffiffi Im½GLLðω; 0⃗Þ ¼ − lim ½a −gguu thus obtaining R 2 2 → 4κ5q u uh ffiffiffiffiffiffi ffiffiffiffiffiffi tt −ω 0ω ω 0−ω p uu μν 0k 0 p ˆ2 2 μν k × K ðb b − b b Þ; ð3:32Þ ða −gg K Bν Þ − −gðak þ q HÞK Bν ¼ 0; ð3:26Þ t t t t with where the right hand side, being independent on the radial variable [see (3.28)], can be equally computed at the μ ν horizon (as we do above) or at the boundary (as according kˆ kˆ μν ≡ μν − to the standard holographic prescription [3]). This allows K g a ˆ2 2 ; ð3:27Þ ðak þ q HÞ deduction of the generic behavior of the Green’s function

106023-7 BIGAZZI, COTRONE, and PORRI PHYS. REV. D 98, 106023 (2018) just by considering universal features of the near-horizon 0 ð0Þ 1 In the q ¼ case, jbh j¼ reproduces the relation (2.15). solution. Crucially, however, in the generic case equation (3.36) holds. In order to compute (3.32), we need to extract the generic Since we are interested in the ω → 0 limit, we can just look near-horizon behavior of a massive vector field in the at the first term in that equation. From this we immediately background (3.21). We will just make use of the following 0 1 see that if we want bð Þ to be finite, then bð Þ → ∞ in the assumptions: a) the metric is diagonal and depends on the h h radial coordinate u only; b) the uu component has a simple limit. Otherwise, if we want to avoid this singular behavior ð1Þ pole at u ¼ uh ≡ 1 (we suitably rescale our coordinates so keeping bh finite, we need to assume that that the latter equivalence holds); c) the tt component has a 1 ð0Þ ∼ ω → 0 simple zero at u ¼ ;d)theii components are proportional to bh ; ð3:38Þ the identity and finite at u ¼ 1; e) the mass of the vector is finite at u ¼ 1; f) the kinetic term of the vector is finite so that, consistently with the nonconservation of the axial at u ¼ 1. charge, the Chern-Simons diffusion rate as defined in (3.31) Let us consider the equation of motion for the t- vanishes. component of the vector field in momentum space setting We have checked this general result in the simplest setup ki ¼ 0 where A and C are treated as fluctuations over a fixed Schwarzschild-AdS background (see Appendix F for  ffiffiffiffiffiffi  p− tt uu details). Solving the related equations of motion numeri- gaHg g ω pffiffiffiffiffiffi ω ∂ ∂ − tt ð0Þ u 2 2 tt ubt ¼ gHg bt ; ð3:33Þ 0 ≠ 0 Hq þ aω g cally, the analysis confirms that bh ¼ whenever q as shown in Fig. 1. Note that for q → 0 but nonvanishing, ð0Þ ω 2 0 and consider the following expansions jbh ð Þj approaches the behavior of the q ¼ case for ω > 0, but eventually it always drops to zero at ω ¼ 0. X∞ X∞ aðuÞ¼ aðnÞð1−uÞn;HðuÞ¼ HðnÞð1−uÞn; h h D. Quasinormal modes n¼0 n¼0 X∞ X∞ Holography implies that the location of the poles of a xx ðnÞ n tt −1 ðnÞ n g ¼ cx ð1−uÞ ;g¼ −ð1−uÞ ct ð1−uÞ ; two-point retarded correlator are given by the frequencies n¼0 n¼0 of the quasinormal modes of the corresponding bulk field X∞ [3,27], so their study allows extraction of the relaxation uu 1− ðnÞ 1− n g ¼ð uÞ cu ð uÞ ; time τ. The quasinormal mode spectrum can be computed n¼0 in the following way. X∞ Let Φ be the bulk field dual to an operator O in the field bωðuÞ¼ð1−uÞα bðnÞð1−uÞn; ð3:34Þ t h theory, and let Φ fluctuate over a background with an event n¼0 horizon. The fluctuations satisfy second order linear 0 0 0 equations of motion whose solution near the horizon will with cð Þ > 0;cð Þ > 0;cð Þ > 0. Solving the equation of t x u be the sum of incoming and outgoing waves. As above, motion (3.33) and choosing the incoming wave solution at the horizon sets sffiffiffiffiffiffiffi 0 cð Þ α −iω t : 3:35 ¼ ð0Þ ð Þ cu

Moreover one finds

1 0 0 bð Þ q2 Hð Þ q2Hð Þ h −i qffiffiffiffiffiffiffiffiffiffiffiffiffiffih h O ω : 3:36 ð0Þ ¼ þ ð0Þ ð0Þ þ ð Þ ð Þ ω ð0Þ ð0Þ ð0Þ bh ah cu ah ct cu

Now the point is that the Chern-Simons diffusion rate as defined in (3.31) turns out to be given by ð0Þ 2 ω FIG. 1. The coefficient jbh j as a function of for an AdS5 ð0Þ 1 T H 0 black hole background with unit radius and a ¼ H ¼ for Γ h ð Þ 2 CS ¼ 2 jb j : ð3:37Þ different values of q ¼ 0.04, 0.44, 3 (top to bottom at small κ ð0Þ 3=2 h 5 ðcx Þ ω) and for q ¼ 0 (dashed line).

106023-8 UNIVERSALITY OF THE CHERN-SIMONS DIFFUSION RATE PHYS. REV. D 98, 106023 (2018) Z   since we are after the retarded correlator, we pick the 1 ffiffiffiffiffiffi 12 − 5 p− MN − S ¼ d x g R þ F FMN 2 ingoing solution. The solution close to the boundary, placed 16πG5 L Z at u ¼ 0, will have the form 1 þ pffiffiffi A ∧ F ∧ F; ðA1Þ ⃗ ⃗ Δ ⃗ Δ 6 3πG5 Φðu; ω; kÞ¼Aðω; kÞu − þ Bðω; kÞu þ ; ð3:39Þ 1 where A and B are completely determined up to a supplemented by standard boundary terms. In units L ¼ , the solution follows from the ansatz (see e.g., [8] and common nonvanishing multiplicative factor fðω; k⃗Þ ≠ 0. references therein) Quasinormal modes are solutions satisfying ingoing boun- dary conditions at the horizon that vanish at infinity. Their 2 2VðrÞ 2WðrÞ ω ⃗ 2 2 dr e 2 2 e 2 frequencies, nðkÞ, are thus defined implicitly by the ds ¼ −UðrÞdt þ þ ðdx1 þ dx2Þþ dx3; equation UðrÞ v w B ⃗ ⃗ F ¼ dx1 ∧ dx2; ðA2Þ AðωnðkÞ; kÞ¼0: ð3:40Þ v

In order to see the connection with the poles of the retarded where U, V, W are functions of the radial variable r, the correlator, recall that the correlator is [3] coordinates are rescaled in such a way that the black hole horizon is at r ¼ 1 where Uð1Þ¼0;U0ð1Þ¼1;Vð1Þ¼ B Wð1Þ¼0 and v, w are functions of the magnetic field B. hOOi ∼ þ …; ð3:41Þ R A The Chern-Simons diffusion rate for the dual magnetized plasma has been computed in [8] finding where … stand for scheme dependent contact terms. The zeros of A are exactly the poles of the correlator. According g4 N2 T to (3.8) we need to find the first zero of Γ ¼ YM pffiffiffiffi ; ðA3Þ CS 27π5 2π v w   BðωÞ −1 Im ð3:42Þ 1 4π AðωÞ where T is the temperature (T ¼ = in the rescaled coordinates defined above). It is easy to realize that this along the imaginary ω axis. Applying this logic to the AdS formula, which holds for any value of the magnetic field, black hole background, one finds [28] that the axial precisely matches our Eq. (2.1), since the black hole relaxation time τ scales like q−2 for small q. Since q entropy density is given by enters the equations of motion only as q2 even in the most 2 general case (3.33), we expect the relation τ ∼ q−2 to be A N 1 s ¼ h ¼ pffiffiffiffi ; ðA4Þ universal for small nonzero q. This relation would thus 4V3G5 2π v w reproduce the phenomenological formula (3.12).

where Ah is the area of the horizon, V3 is the (infinite) 3d- 2 ACKNOWLEDGMENTS space volume and the holographic relation G5 ¼ π=2N has been used. We are grateful to Riccardo Argurio, Matteo Baggioli, Luca Griguolo, Carlos Hoyos, Matthias Kaminski, Dmitri Kharzeev, Karl Landsteiner, Javier Mas, Daniele Musso, APPENDIX B: CHERN-SIMONS DIFFUSION Domenico Seminara and Javier Tarrio for relevant com- RATE IN THE ANISOTROPIC N = 4 PLASMA ments and discussions. A. L. C. is partly supported by the Florence University Grant “Fisica dei plasmi relativistici: The anisotropic N ¼ 4 SYM plasma considered in [6] teoria e applicazioni moderne”. corresponds to thermal N ¼ 4 SYM deformed by a linear space-dependent topological theta angle

APPENDIX A: CHERN-SIMONS DIFFUSION θðx3Þ¼2πax3; ðB1Þ RATE IN THE N = 4 PLASMA WITH EXTERNAL MAGNETIC FIELD where x3 is one of the space directions and a is a The effective 5d holographic description of the N ¼ 4 dimensionful anisotropic parameter which can be read as SYM plasma in the presence of a constant magnetic field is the density of homogeneously smeared D7-branes along provided by a magnetically charged black hole solution the x3 direction. The dual black hole solution arises from arising from the Einstein-Maxwell-Chern-Simons action the effective 5d action (Einstein frame)

106023-9 BIGAZZI, COTRONE, and PORRI PHYS. REV. D 98, 106023 (2018) Z   2 2 1 1 1 16π3 1 ffiffiffiffiffiffi ∂ϕ ∂C0 5 p ð Þ 2ϕ ð Þ 2πR4 S4 ; S ¼ d x −g R þ 12 − − e ; 2κ2 ¼ 2κ2 Volð Þ¼2κ2 3 ðC5Þ 16πG5 2 2 5 10 10 MKK ðB2Þ where R4 is the radius of the circle of the cigar and VolðS4Þ is the volume of the unit radius four sphere. where ϕ is the dilaton and C0 is the axion dual to the The reduction of the term of the ten dimensional action topological charge density operator of the quantum field containing C1, whose integral on the circle (with coordinate theory. The background has a metric of the form  x4) is dual to the operator Q (up to a constant), −ϕðuÞ=2 2 2 e −F B 2 du Z ds ¼ 2 ðuÞ ðuÞdt þ F 1 10 pffiffiffiffiffiffiffiffiffiffi 3ϕ 2 u ðuÞ − 2  2 d x g10e F2; ðC6Þ 2κ10 2 2 2 þ dx1 þ dx2 þ HðuÞdx3 ; ðB3Þ gives supported by a running dilaton ϕðuÞ and the axion Z C0 ¼ ax3. The horizon is at r ¼ rh. The Chern-Simons 1 ffiffiffiffiffiffiffiffi 5 p 3ϕ−2f−8w 2 d x −g5e2 F ; diffusion rate, as computed in [7], has, for any value of 2κ2 1 ðC7Þ a=T, precisely the same form given in (2.1), provided we 5 recall that in this case where F1 ¼ dCx4 . The functions f, w are the metric 2 ϕ 4π ðrhÞ gYMðTÞ¼ e : ðB4Þ functions in the reduction ansatz

10 2 − 3 f 2 2f 8w 2 −2w 2 APPENDIX C: CHERN-SIMONS DIFFUSION ds10 ¼ e ds5 þ e ½e dx4 þ e dS4: ðC8Þ RATE IN WYM Amazingly enough, on the background5 In this Appendix we review the calculation of the Chern- Simons diffusion rate in Witten’s Yang-Mills model [9] and 3ϕ−2 −8 e2 f w 1: C9 show that it obeys the universal formula (2.1). The ¼ ð Þ diffusion rate has been calculated in [11] (in string frame) and reads Thus, the five dimensional action is the one of a minimally coupled scalar.6 Considering that the field dual to Q is not 3 2πR4 1 λ4 2 6 Γ 4 2π C1 but g l C1 after reduction on the circle of the D4-brane CS ¼ 6 2 ð R4Þ T ; ðC1Þ s s 2π 3 π action, as shown in [11], one obtains that (C7), (C9) give where the overall factor 4 (instead of the 1 in [11]) the result (C4) for the CS diffusion rate in the WYM theory. originates from the difference between our conventions A remark is in order. While the WYM has a running (2.4) and (2.6) and the corresponding ones adopted in [11]. coupling in the confined phase, the gravity combination of In this formula T is the temperature, fields above dual to the coupling in the deconfined case is again a constant. This is due to the fact that the theory in the 1 2 deconfined case is basically a six dimensional conformal λ4 ¼ 2πg Nl M ¼ g N; ðC2Þ s s KK 2 YM field theory. In fact, one can calculate the four dimensional is proportional to the “UV ’t Hooft coupling” and M ¼ coupling from the action of a probe D4-brane wrapped KK around the circle of the cigar, as in [30], obtaining in the 1=R4 is the mass scale of the theory. Since the entropy density is deconfined phase

8 4 2 5 Z 2 π λ4N T 1 1 −ϕpffiffiffiffiffiffi β4 s ; dx4e g44 ; ¼ 36 2 ðC3Þ 2 ¼ 8π2 ¼ 8π2 ðC10Þ MKK gYM;dec ls lsgs one can write where β4 is the length of the circle. Thus, the ’t Hooft 4 g coupling is constant and equals precisely 2λ4, Γ YM sT; C4 CS ¼ 27π5 ð Þ 2 2λ which is exactly in the universal form (2.1). gYM;decN ¼ 4: ðC11Þ It is instructive to see how the result above is generated in Einstein frame. The five dimensional reduction of the 5 We use the convention in which gs is in the brane tension and WYM background has been performed in [29]. The in κ10, so there are no factors of gs in the dilaton solution. 6 2ϕ gravitational constant reads Referring to the general action (2.12), we have e ¼ g44.

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Φ Φ APPENDIX D: CHERN-SIMONS g1 ¼ dc − qA: ðD6Þ DIFFUSION RATE IN MN In the action it enters in a quadratic term and some mixing In this appendix we calculate the Chern-Simons diffusion terms with other fields, with specific coefficients depending rate in the Maldacena-Nuñez (MN) model of N ¼ 1 Super on the metric. On the background the only nonzero Yang-Mills [12,13] and show that it obeys the universal coefficient is the one of the quadratic term (which is 1), formula (2.1). The nonextremal solution corresponding to so the action reduces to7 N 1 Z the deconfined, chirally symmetric phase of the dual ¼ 1 1 1 SYM theory is a linear dilaton background [31], i.e., the dual S − gΦ 2⋆1: c ¼ 2κ2 62 2 ð 1 Þ ðD7Þ of the little representing the UV completion of 5 the model. The internal metric is the product of a two sphere One can also check that A appears always in terms which and a three sphere. In Einstein frame the solution reads are at least of third order in fluctuating fields, apart from its ds2 ¼ eϕ=2½−νðrÞdt2 þ dx dxi þ νðrÞ−1dr2 þ dS2 þ dS2; kinetic term E i 2 3 Z 1 1 4 3 4ϕ 2 ðD1Þ − 54 = 3 ⋆1 SA ¼ 2 ð Þ e ðdAÞ : ðD8Þ 2κ5 2 F3 ¼ P½Ω3 þ w3 ∧ Ω2; ðD2Þ ϕ ∼ r: ðD3Þ Now, by comparing formula (2.49) in [32] with the notation in [35] one gets that the field dual to Q is8 In this formula P is proportional to the number of colors, Ω2 3   ; 8π2 cΦ are the two and three-sphere volume forms and w3 ¼ dψ þ ↔ ≡ Q CðrÞ 2 6 : ðD9Þ cos θ2dϕ2 is the third left-invariant one-form of SUð2Þ (the gYM other two will be denoted as w1;2). The two and three spheres Thus, if the effects of anomaly are ignored, i.e., we drop the have radii equal to 2. terms in A, the action (D7) reads The identification of the gravity field dual to Q has Z   – 1 ffiffiffiffiffi 2 2 1 been done directly in [13], see also [32 34].Itisa p gYM 2 S − g5 dC ; D10 component of the RR two-form potential C2, whose UV c ¼ 2 2 ð Þ ð Þ 2κ5 8π 2 asymptotics give, upon integration on a two-sphere, the θ — YM term, which is in the form (2.13), implying that the CS diffusion rate is in the universal form (2.1). Q ↔ qðrÞðw1 ∧ w2 − Ω2Þ ≡ qðrÞΦ; with As an aside, note that the function H of formula (2.14) is 9 qðrÞ → ψ − ψ 0 for r → ∞: ðD4Þ a constant in the MN model. The latter has no AdS UV asymptotics but in the deconfinement phase the coupling is Note that ψ is a coordinate of the three-sphere, while ψ 0 is a anyway constant. In fact, if we calculate it as in formula θ constant proportional to YM. (4.2) of [32], but on the nonextremal solution, we get The five dimensional consistent reduction of 1 1 type IIB containing the relevant fields for our purposes : 2 ¼ 4π2 ðD11Þ can be found in [35]. The relevant component of C2 is gYM;dec gs termed cΦ. The full action is quite involved. But we are interested only in the quadratic fluctuations around the APPENDIX E: NOTES ON THE AXIAL solution (D1). It can be checked that the only terms RELAXATION TIME contributing are in the form of the usual Stueckelberg coupling of cΦ with the Reeb vector A. Let us sketch the In this Appendix a quick derivation of formula (3.12) is derivation. presented. Let us turn off the anomaly for a moment. Comparing the metric (3.3) in [35] with (D1) we get that Diffusion of the axial current implies Fick’slaw the only metric fields which are turned on are ⃗ J⃗ ¼ −D∇Jt ; ðE1Þ ϕ ϕ 1 A A 3 6 v ¼ 4 þ log ;u¼ 4 þ 2 log ; where D is the diffusion constant. In momentum space

1 4ϕ −1 gμν ¼ e3 diagð−ν; 1; 1; 1; ν Þ: ðD5Þ Jx ¼ −ikDJt ; ðE2Þ 9 × 24=3 A A

Apart from the dilaton, the other quantity in [35] which is 7 1 We correct a factor of 2 in [35]. different from zero is the flux q ¼ 18P. 8We work in units α0 ¼ 1 here. Φ 9 2ϕ The only five dimensional field where c enters, which Referring to the notation in (2.12), e ¼ detðgΩ0 Þ and we get 1 2 is linear in the fluctuating fields, is (D7) once we take into account the 6 normalization in [35].

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x 0 ≤ ≤ where k ¼ kx. Now, turning on a source Ax for JA we have, where z zh, the horizon is at z ¼ zh and the black 0 in linear response, hole temperature is given by 4πT ¼jb ðzhÞj ¼ 4=zh. Another useful description is obtained using the dimen- x − xx ∈ 0 1 hJAi¼ GR Ax; ðE3Þ sionless radial coordinate u ½ ;   where Gxx is the retarded correlator of Jx . At zero spatial z2 l2 −bðuÞdt2 þjdxj2 du2 R A u ; ds2 ; momentum, considering the electric field ¼ 2 ¼ 2 þ 4 zh u zh ubðuÞ bðuÞ¼1 − u2: ðF2Þ Ex ¼ −iωAx; ðE4Þ the previous relation reproduces Ohm’s law with Let us now solve the equation of motion (3.26) on the above AdS-BH background. Gxxðω; 0⃗Þ¼iωσ; ðE5Þ R 1. Near-boundary expansion where σ is the conductivity. Close to the boundary at u ¼ 0 the equation of motion Using (E2) we thus get (3.26) becomes (assuming H and a to go to 1 at the boundary) x xx t hJAi GR hJ i¼ ¼ Ax: ðE6Þ 2 00k 2 k A −ikD ikD 4u Bμ − ðqlÞ Bμ ≃ 0; ðF3Þ

Now, the anomaly, through the gauge invariant combina- therefore we have the following expansion close to the tion ∂μC þ qAμ, implies that a source C for the topological boundary: charge density QðxÞ, is induced by the electric field as k Ξ kν k BμðuÞ ≡ u fμ ðuÞbν; ikC ¼ qAx; ðE7Þ kν ν 1−2Ξ k ν fμ ðuÞ¼δμ þ OðuÞþu ðb2δμ þ OðuÞÞ; ðF4Þ so that pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2 where the exponent Ξ ¼ 2 ð1 − 1 þðqlÞ Þ is strictly GQQ Γ ω negative. hQi¼−GQQC ¼ − R qA ¼ CS qA ; ðE8Þ R ik x 2kT x 2. Near-horizon expansion where we have used Close to the horizon at u ¼ 1 (again assuming H and a to Γ go to constant values Hh, ah)wehave QQ − CS ω O 2 GR ¼ i þ ðk Þ: ðE9Þ 2T μν 0k 0 2 k K ½16ð1 − uÞðð1 − uÞBν Þ þ w Bν¼0; ðF5Þ All in all, from where w ¼ zhω and the matrix K is evaluated at the hQi iωqΓ D qΓ D 1 horizon: ¼ CS ¼ CS þ OðkÞ ≡ þ OðkÞ; ðE10Þ hJt i 2TGxx 2Tσ qτ A R z2 Kij → h δij; σ χ−1 l2 and using D= ¼ A we get the desired relation z2 ki it → h 1 q2Γ K l2 ω ; ¼ CS : ðE11Þ τ 2 χ 2 2 2 T A ql H a z k⃗ tt → ð Þ h þ h hj j K 2 2 : ðF6Þ ahl ω APPENDIX F: STUECKELBERG ACTION ON The matrix is nonsingular at the horizon, and the near- THE AdS-BH BACKGROUND horizon solution has the form A C In the simplest possible setup the fields and defined ν ν − w k k 1 − i4 1 O 1 − in (3.14) are treated as fluctuations over a fixed fμ ðuÞ¼fhμ ð uÞ ½ þ ð uÞ ν w Schwarzschild-AdS background k 1 − þi4 1 O 1 − þ ghμ ð uÞ ½ þ ð uÞ: ðF7Þ   l2 2 4 2 − 2 2 dz 1−z We pick the in-falling solution which corresponds to ds ¼ 2 bðzÞdt þjdxj þ ;bðzÞ¼ 4 ; ðF1Þ z bðzÞ z k ν 0 h setting ghμ ¼ .

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3. Retarded Green’s functions and Chern-Simons diffusion rate

In order to define the retarded Green’s function for our current operator, JB, we need to evaluate the action on a generic solution of the equations of motion. Since we are after the imaginary part of the correlator, we do not need to renormalize explicitly the action, the relevant part being finite. From (3.29) and (F4) we have Z 4 1 1 d k ffiffiffiffiffiffi − μ − −k p− uu 2Ξ−1 k ρσ Ξ kν 0kν k Son-shell ¼ 2 2 4 bμ ½ gag u fρ K ð fσ þ ufσ Þ bν: ðF8Þ 4κ5q ð2πÞ 0

The Green’s function then reads 1 ffiffiffiffiffiffi μ ν μν p uu 2Ξ−1 −kμ ρσ kν 0kν hJ ð−kÞJ ðkÞi ¼ G ðkÞ¼ lim½ −gag u fρ K ðΞfσ þ ufσ Þ; ðF9Þ B B R R 2 2 →0 2κ5q u and its imaginary part ffiffiffiffiffiffi μν i p uu 2Ξ ρσ −kμ 0kν kμ 0−kν Im½G ðkÞ ¼ − lim½ −gag u K ðfρ fσ − fρ fσ Þ; ðF10Þ R 2 2 →0 4κ5q u is essentially (3.28), and therefore independent from u. Here we are interested in the longitudinal component (3.30) of the above correlator. The imaginary part of this quantity coincides with the conserved current in (3.28) when the latter is k computed on solutions satisfying the boundary condition bμ ¼ kμ. In position space the boundary condition would read Bμðu → 0;xÞ¼∂μCðxÞ. Since the right hand side of (F10) is constant we can evaluate the longitudinal correlator at the horizon u ¼ 1. We obtain a ωl LL h F μν Im½GR ðkÞ ¼ 2 2 3 kμkν × ; 2κ5q z  h  2 2 ⃗2 i ðqlÞ H þ a z jkj − μ − μ k − μ μ F μν ≡ h h h f k fkν z2f k fkν z2 f k fkν f−kνfk ; 2 ht ht þ h hi hi þ h ω ð ht hi þ hi ht Þ ahω l π3l3 ah μν Hh 4 kt 2 Γ ¼ lim kμkν × F ¼ T jf j ; ðF11Þ CS ⃗ πκ2 2 4 κ2 ht jkj;ω→0 5q zh 5 where we assumed all the f’s are analytic in ki (before taking the zero-frequency limit). All we need to compute the CS kt ⃗ ω 0 diffusion rate is the norm of the coefficient fht at jkj¼ ¼ . However, it turns out that for the temporal component of a kt ω massive bulk vector the coefficient fht always goes to zero with (at zero momentum), and therefore formula (3.31) gives zero. In fact, the solution to the equations of motion admits the following expansion close to the horizon10 (this is model independent):

− w 2 k 1 − i4 k 1 − k O 1 − f ¼ð uÞ ½fh þð uÞfhð1Þ þ ð uÞ : ðF12Þ

On the AdS-Schwarzschild background one then finds

ð−16ΞðΞ − 1Þþ4iΞðΞ þ 1Þw þð4Ξ − 1Þw2 − iw3ÞH fk ¼ h fk: ðF13Þ hð1Þ 4wðw þ 2iÞ h For small w (and q ≠ 0) the above formula gives

q2H fk ∼ i h fk; ðF14Þ hð1Þ 2w h

k ∼ 11 Γ 0 which explodes unless fh w, so that from (F11) it follows that CS ¼ .

10We suppress the super and subscript t. 11 k k The expressions for fhð2Þ and fhð3Þ all suffer from the same issue.

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