Multiply charged magnetic black branes

We discuss analytic solutions describing magnetically charged black branes in $d$ dimensional AdS space. Focusing on $d=5$, we study the response of the brane to an external short lived electric field. We argue that when the theory possesses an 't Hooft anomaly then at sufficiently low temperature a long lived oscillatory current will be observed long after the electric field has been turned off. We demonstrate this ``anomalous resonance'' effect via a numerical study.

Thermally equilibrated states in the presence of an external magnetic (and/or electric) field on R d−1,1 of theories with a holographic dual are equivalent to asymptotically anti de Sitter (AdS) magnetically (and/or electrically) charged black brane solutions to Einstein-Maxwell theory with a negative cosmological constant.The value of the magnetic (and/or electric) field on the asymptotic boundary of AdS space represents the external magnetic (and/or electric) field in the dual theory.
Often, it is convenient to have an analytic description of the black brane metric.This allows for a better understanding of its dynamics and its response to perturbations.While analytic expressions for magnetically (and electrically) charged black branes exist in 3 + 1 dimensional spacetimes, their higher dimensional analog does not seem to be available.The reason for the absence of analytic solutions in higher dimensions is likely associated with the lack of symmetry: In 3 + 1 dimensions a constant magnetic field will pierce the (planar) event horizon so that rotational invariance (in the spatial directions associated with the asymptotic boundary) is maintained.In higher dimensions, a non vanishing magnetic field will necessarily break rotation invariance.
With the above symmetry considerations in mind, an analytic expression for magnetically charged black brane configurations can be obtained by allowing for several U (1) gauge fields, such that each of their (equal in magnitude) magnetic fields pierce the horizon on a different plane so that rotation invariance is retained.Explicitly, the equations of motion following from have charged black brane solutions of the form where for d ̸ = 4 with M ∈ R, Q 2 = i,j (q ij ) 2 and F ij = dA ij .The Greek indices µ, ν = 0, . . ., d run over the spacetime coordinates and i, j = 1, . . .d − 1 count gauge fields.The solution for d = 4 (AdS 5 ) has been obtained in [22] and will be discussed in detail in section 2. In the remainder of this work we will set L = 1.
Clearly, the number of gauge fields required to support (1.2) grows as d 2 and therefore seems unrelated to physical applications.Fortuitously, for d = 3 we require only one gauge field and then (1.2) reproduces the well known magnetically charged AdS 4 black brane solution [41].In four dimensions, 3 gauge fields are required which is somewhat unrelated to the electromagnetic field whose behavior one often tries to capture using holographic duality.However, N = 2 five dimensional gauged supergravity does contain three Abelian gauge fields.In the context of N = 4 super Yang Mills theory these can be thought of as sources for the maximal Abelian subgroup of the SO(6) R-symmetry.Indeed, in the remainder of this work we will focus on (d = 4) five dimensional magnetically charged black brane solutions of the above type.
One interesting characteristic of magnetically charged black branes is their quasi normal modes (QNM's).Similar to their non rotationally invariant counterpart [25,29,[42][43][44][45], the magnetically charged black branes we study possses quasi normal modes which approach the real axis as the magnetic field becomes very large compared to the temperature (in the presence of a Chern-Simons term).Given the analytic control we have over the black hole solution we are able to compute these quasi normal modes at perturbatively small temperatures (or large magnetic fields).We do this in section 3.
As discussed in [29], since the quasi normal modes are long lived, perturbing the black brane at an appropriate frequency will excite this QNM and its oscillations may then be observed long after the perturbation has been turned off.We refer to this effect as an anomalous resonance and demonstrate its manifestation explicitly in section 4.

Magnetically charged black branes in AdS 5
Our starting point is the action which is related to the STU ansatz for the D = 5, N = 2 gauged supergravity action [46] once λ = 1/4.The equations of motion resulting from (2.1) are given by and even permutations of the last equation for F 2 µν and F 3 µν .The action (2.1) possesses an SO(3) f flavor symmetry rotating the three U (1) fields which is broken by the Chern-Simons term.The ansatz preserves a diagonal SO(3) B subgroup of the SO(3) f symmetry and spatial rotations once a i = 0.That is, (2.3) (with a i = 0) is invariant under a simultaneous spatial rotation in the x, y, z directions, R µ ν (θ, ϕ), and a rotation of the three U (1) fields, R i j (θ, ϕ), viz.
Note that this ansatz is also invariant under parity in the x i coordinates.
After fixing the residual r → r + c symmetry by setting Σ = r + O(r −1 ), we find that, for λ = 0, is the most general solution to (2.2).The horizon is located at A(r 0 ) = 0 where we may choose q i = µ i r 2 0 in order that the gauge fields are not singular at the bifurcation point.We will refer to this solution as the charged, magnetic black brane.When λ ̸ = 0 we must set µ i = q i = 0 in order for (2.4) to solve the equations of motion.In this case the solution reproduces the one found in [22] in the context of gauged supergravity.We refer to the latter as an uncharged magnetic black brane.Note that in the absence of charges, r 0 = r h .
The Hawking temperature for the charged solution is given by and the Bekenstein entropy per unit volume is Using the AdS/CFT dictionary [47], the stress tensor and charge currents associated with these solutions are given by with r c a renormalization scheme dependent parameter (see appendix A).Here, the roman indices m, n = 0, . . ., 3 are spacetime coordinates of the boundary theory.In the dual theory µ i is the chemical potential associated with the i'th current and B is an external magnetic field.Indeed, once the magnetic field doesn't vanish we find, as expected [47].
The thermodynamic constitutive relations for the stress tensor and current in the presence of a magnetic field have been computed in [48][49][50][51].It is straightforward to generalize the former expressions to configurations with the SO(3) B spatio-magnetic symmetry under discussion.In the notation of [51] we find that in equilibrium where ∆ mn = η mn + u m u n is the projection orthogonal to the velocity field and B m i = 1 2 ϵ mnrs u n F i rs are the magnetic fields as seen in a local rest frame.The Gibbs-Duhem relation is given by (2.10) For the case at hand we have α i = α for all i from symmetry and B m i = δ m i B so that the last term proportional to α on the right of (2.9) vanishes.Comparing (2.9) and (2.10) to (2.7) and using (2.6) and u m ∂ m = ∂ t we find (2.11) 3 Quasi normal modes of uncharged black branes The uncharged magnetic black brane solutions given by (2.4) with q i = 0 possess a resdiual SO(3) B symmetry (the diagonal of SO(3) f and SO(3) rotational symmetry).When studying perturbations of these solutions it is convenient to decompose the perturbations into representations of this SO(3) d symmetry.We denote the metric perturbations by δg µν = r 2 e −iωt γ µν and the gauge field perturbations by δA i µ = e −iωt a i µ .We will work in a gauge where γ µr = 0 and a i r = 0.In what follows we will use roman indices i, j = 1, 2, 3 to denote both spatial indices and SO(3) f indices.Thus, for example, a i j denotes the j'th spatial component of the i'th flavor type of the gauge field fluctuation.Metric perturbations can be decomposed into symmetric traceless modes of the form γ T ij = g ij − 1  3 g kj δ kl δ ij , vector perturbations of the form γ V i = γ 0i r h , and scalar perturbations, γ S 1 = γ 00 and γ S 2 = γ jk δ jk .We decompose gauge field perturbations in a similar way: symmetric traceless modes a T ij = a ij + a ji − 2 3 a kl δ kl δ ij , vector and pseudovector modes a V i = a i 0 and ãV i = ϵ i jk a jk (respectively), and a scalar mode a S = δ ij a ij .
Since the Chern-Simons term (which breaks the residual SO(3) B symmetry) does not affect the Einstein-Maxwell equations we find that the symmetric traceless modes of the metric, γ T (ρ), decouple and satisfy where we have defined On the other hand, the tensor modes satisfy The expression on the right hand side of this equation is due to the presence of the Chern-Simons term which breaks the SO(3) B symmetry; it vanishes for λ = 0.
For the vector sector we find The non tensorial nature of this equation is, as before, a result of the Chern-Simons term.
The remaining vector equations are Finally, for the scalar sector we have and In order to find the quasi normal modes associated with the magnetic black hole we must solve these equations with boundary conditions such that the modes are ingoing at the horizon and are not sourced on the boundary.
The equations for the metric and gauge field perturbations (3.1) together with the ingoing boundary conditions reduce to Sturm-Liouville-type equations and can be satisfied only for particular values of the frequency ω.For generic values of β and λ one must resort to numerics in order to compute the quasi normal frequencies.However, for particular values of β and λ some analytic solutions are available.

The limit of zero magnetic field
Consider the β → 0 (zero magnetic field) limit of equations (3.1).In this case the quasi normal modes reduce to those of the Schwarzschild AdS black brane which have been studied in [52].One finds that the modes for the gauge field decouple and can be solved for analytically.In particular, one finds that a 0 (ρ), (with a denoting either a T , ãV or a S at β = 0) satisfies where and Ω 0 = lim β→0 Ω.The solution to (3.2) with ingoing boundary conditions is given by , where n ≥ 1 is an integer and C 0 a constant.There is an additional family of solutions with Ω = 2n(−1 − i) and an associated a(ρ) given by the conjugate of the one in (3.4).
One can now work perturbatively in β to obtain corrections to the quasinormal frequency Ω.In particular, let us write where S modes and T modes with i = j. .
and γ 1 is the leading component of an expansion of γ V in β, viz., ).The expression for the source term of the pseudo-vector components is supplemented by Since a 0 satisfies the boundary conditions for a, we can look for a solution where (Note that we could also set a 2 (1) to take on any finite non zero value by adding to a 2 the solution to the homogenous equation (3.4).) To solve (3.6) we compute the relevant Green's function.The solutions to the homogenous equation (ph ′ ) ′ + qh = 0 with p and q as in (3.3) and These solutions satisfy near the horizon and near the asymptotic boundary.Thus, the most general solution to (3.6) is given by where C is a constant which can be read off the Wronskian of h 1 and h 2 and A and B are integration constants.
For the tensor modes we find that and The first integral on the right hand side of (3.14) is well defined and converges in the ρ → ∞ limit while the second integral on the right hand side of (3.14) vanishes as we near the boundary.Thus, lim suggesting that we must set B = 0. Near the horizon the first integral on the right hand side of (3.14) vanishes but the second integral is divergent.To properly evaluate a 2 (ρ) in the ρ → 1 limit we write That is, the sum on the right of (3.18) leads to a divergent integral when evaluated close to x = 1.We can now regulate the last expression on the right hand side of (3.14) by writing To avoid outgoing modes at the horizon we must set the term in parenthesis on the right had side of (3.20) to zero.This gives a constraint of the form where, unfortunately, we have not managed to obtain an analytic expression for N n , L n and W n .Numerical estimates for these parameters can be found in table .
The tensor quasi normal modes are the most interesting for our current purposes since they are sensitive to the Chern-Simons term we have introduced.From the point of view of the dual boundary theory they get modified due to an 't Hooft anomaly.The vector and scalar quasi normal modes may also be computed by similar means.

Large Chern-Simons coupling
Another interesting limit one can take is λ → ∞ while keeping Bλ = M 2 fixed (see [29]).This corresponds to taking the λ → ∞ limit of (2.1) but keeping A µ λ fixed.In this case, the Maxwell and Chern-Simons terms become subleading relative to the Einstein-Hilbert term, leading to a probe limit where the gauge field explores a background AdS black brane geometry.If we further take the zero temperature limit, ) we find that the symmetric tensor field fluctuations satisfy and we have defined with ã1 denoting a T ij λ with i ̸ = j in the λ → ∞ limit, and Ω0 = lim λ→∞ Ω. Solving (3.22) with the boundary conditions ã1 (∞) = 0 and lim λ→∞ ã1 where L ν m (x) is a generalized Laguerre polynomial.Working perturbatively in λ we expand where ã represents a T ij with i ̸ = j, + . . .with T the temperature.As we will see shortly, the powers of λ in the perturbative expansion are dictated by the structure of the perturbative equations of motion.
The resulting equations of motion for a 2 are pã ′ The ingoing boundary conditions now translate to where the last two equalities follow from the asymptotic behavior of ã1 at large ρ.As in the previous limit, we compute the subleading corrections to the quasi normal modes by constructing the Greens function for the solution.The two linear solutions to (3.22) are given by h1 = e where Ei(x) is the exponential integral function Ei z dz and is a polynomial of its argument with coefficients c k satisfying the recursion relation The general solution to (3.28) is C.f., (3.14).
A short computation reveals that so the second integral converges near the horizon.In order for the boundary conditions to be satisfied we must set It is straightforward though somewhat tedious to solve (3.39).We find where k are harmonic numbers and γ represents Euler's constant, γ = 0.5772 . ... The expressions for N 1 and N 2 can be obtained from the appropriate limits of N n , viz., The subleading corrections to the quasinormal mode frequencies can be read off from (3.40).Note that they are real to the order we are working in.

Full solution
Starting from the analytic solutions for small values of β (3.4), we use a shooting algorithm to solve (3.1c) for increasingly larger values of β.Typical behavior of the quasi normal is exhibited in table 1.When comparing the numerical value of the quasi normal modes to the theoretical, large λ and small temperature (small √ 8 > β > 0) prediction (equations (3.25) and (3.40)), we find a surprisingly good match down to very low temperatures, and for values of λ larger than, roughly, 1/2.See figure 2.
At small values of the magnetic field, we find that the approximation given in (3.4) and (3.21) provides a good match to the data as long as the Chern-Simons coupling λ is small.Once λ is of order 1 the small magnetic field approximation seems to break down rapidly.See figure 3.

Response to an electric field
Our main goal in this work is to use the gauge gravity duality to understand how the boundary theory responds to an external electric field.Indeed, recall that a U (1) gauge field in the bulk, A i , is dual to a conserved current, J i in the boundary.These conserved currents are associated with global symmetries of the (boundary) theory, e.g., the U (1) 3   of the SO(6) R symmetry in N = 4 super Yang-Mills is dual to the three U (1) currents in the STU ansatz of [46]. 1eaning on the results of [47] (which rely, in turn on [53]), having an external electric field on the boundary implies that the boundary value of the associated gauge field A µ should support a non trivial electric field.The current J µ dual to A µ in the presence of said electric field can then be read off the asymptotic behaviour of the A µ as it approaches the boundary.Thus, to compute J µ in the presence of an electric field we need to perturb the black hole solution of (2.4) such that the boundary value of the gauge field A i µ supports an electric field.Before proceeding with the computation outlined above we would like to point out that the late time behavior of the current can be gleaned from the quasi normal mode analysis we have carried out in the previous section.Consider a charged black hole of the form (2.4) which is perturbed at some time t = 0 by an external, time dependent, boundary electric field.Long after the black hole is perturbed we expect it to asymptote to its equilibrium solution (2.4) with deviations characterized by its quasi normal modes.Hence, if the temperature is small enough, and λ large enough we expect to observe the long lived quasi normal modes of the black hole.From the perspective of the boundary theory we expect that after exciting the equilibrated system by a localized (in time) electric field, an oscillatory current will be observed even long after the lifetime of the excitation.Following [29] we refer to this effect as anomalous resonance.
Demonstrating that the anomalous resonance effect takes place as suggested in the previous paragraph requires knowledge of the dynamics of perturbed black holes.Explicit solutions to the Einstein equations describing perturbed black holes are scarce and often rely on the existence of an underlying symmetry.Therefore, in order to observe the anomalous resonance effect we resort to numerics.The remainder of this section is divided in two.In 4.1 we provide some technical details regarding the numerical solution, and in 4.2 we present our results.

Setting up the numerical problem
We will consider a setup where the black hole (2.4) is perturbed by non vanishing boundary electric fields ⃗ E 1 and ⃗ E 2 whose non vanishing components are given by E = E 1 2 = E 2 1 where, we remind the reader, the upper index is a flavor index and the bottom index is a spacetime index.This type of boundary electric field breaks the SO(3) B × Z 2 symmetry of the black hole ansatz (2.3) (with a i = 0 and Z 2 denoting parity) to a discrete Z 2 subgroup involving rotations by π/2 in the (x, y) plane and parity in the spatial coordinates.
To implement the Z 2 symmetry described above, we use an ansatz for the line element and for the gauge fields.
we are interested in won't be modified from the current ones.
Using the boundary conditions ensures that the boundary metric is the Minkowski metric.Likewise, ensures a homogenous magnetic field B, and an electric field E which breaks the SO(3) × Z 2 symmetry as described above.With these boundary conditions, inserting the ansatz (4.1) into the equations of motion (2.2) yields a near boundary expansion of the form where we have defined the dimensionless quantities β = Br −2 h , ρ = r/r h and t h = tr h (recall that we are working in units where r L 2 scales as energy) and we have used the residual gauge freedom of our ansatz, r → r + λ(t), to set ∂ r Σ = 1 + O(r −1 ).
The undetermined integration parameters in (4.4), a (0) , a (2) , c, g (4) and A (2) determine the boundary electric field E = E 1 2 = E 1 2 , the associated expectation value of the (covariant) flavor currents, J m i cov (t), and the expectation value of the energy momentum tensor, T mn (t) with the other components vanishing.(See appendix A; in writing (4.5) and (4.6) we have also chosen a scheme where the logairthms in (A.17) vanish.)Note that (as is always the case) energy momentum conservation and current conservation are compatible with the equations of motion, The values of a (0) , c, a (2) , g (4) and A (2) control the thermal expectation values of the currents and stress tensor.The coefficient a (0) is determined from the boundary condition that E = −∂ t a (0) .Likewise, requiring that the covariant current vanish before turning on the electric field implies c = 0.The remaining functions can not be determined by a near boundary expansion and one must resort to numerics to determine their explicit values.
To numerically solve the resulting set of equations we use the methods of [54] to rewrite the Einstein equations as a set of nested linear equations.Indeed, if we replace time derivatives with outgoing null derivatives, then, given values for Σ(t 0 , r), g(t 0 , r) a(t 0 , r) and A (2) (t 0 ) at some time t 0 , the equations for d + Σ(t 0 , r), d + g(t 0 , r), d + a(t 0 , r), A(t 0 , r) and ∂ t A (2) (t 0 ) become a set of nested linear equations which we can easily solve sequentially.With the solution for the above variables available, we can step forward in time using (4.8).Note that ã can be completely removed from the equations of motion using In slightly more detail, we define the barred variables, Our initial data involves values for Σ, g, a and A (2) at some time t 0 .Given these inital values we can solve the first order linear ordinary differential equation for d + Σ with the boundary conditions Next we solve for d + g and d + a.These form a set of two coupled first order ordinary differential equations.The boundary conditions we use are Next up is a second order differential equation for A, as boundary conditions we use and A (0) = 0. Finally, we evolve A (2) using (4.7).
In practice at each time we used 21 Gauss-Lobatto grid points in the radial direction which covered 0.91 ≤ ρ < ∞ and expanded all functions in Chebyshev cardinal functions, supported on those grid points.We have checked that ρ = 0.91 is always on or inside the event horizon.We used a fourth order Runge-Kutta algorithm to integrate forward in time.

Results
To demonstrate the existence of an anomalous resonance effect we excite the system at some initial time t by turning on an electric field (We have used (4.14) instead of a Gaussian to disentangle the anomalous resonance effect from another effect associated with the anomaly referred to as an "anomalous trailing" effect.See [29].)Going to Fourier space, this corresponds to an electric field peaked around a frequency of ±1/τ .Thus, we expect to observe an anomalous resonance effect whenever Re(ω n ) ∼ 1 τ with ω n the n'th quasi normal mode, and −Im(ω n ) ≪ 1.In practice, we found that the 1st quasi normal mode is excited as long as Re (ω 1 ) ≲ 4 τ .Our initial run involves an electric field of strength E 0 = 0.5T 2 and width τ T = 2/5, a magnetic field of B = 2T 2 , and an anomaly with λ = 3/2 where we have introduced T = 2πT 0 with T 0 the initial temperature prior to the excitation.For these parameters we find that the final temperature of the system, long after the electric field has been turned on, asymptotes to T f ∼ 1.01T 0 .In this case the lowest quasi normal mode of the late time solution is given by ω 1 = 7.87 − 0.003i, so that Re(ω 1 ) ∼ 3.1/τ .This is sufficient to generate an anomalous resonance as exhibited in figure 4.
Since the resulting late time temperature change is small, (T f − T 0 )/T 0 ∼ 0.01, the backreaction of the geometry to the electric field is almost negligible outside the transient region where the electric field is non zero.Away from the transient region the spatial components of the metric fluctuate at a small amplitude, and at a frequency which is double that of the QNM of the gauge field, probably due to the non-linear nature of the Einstein equations.See figure 5. We have checked that these modes decay at a very low rate, as expected.
If the value of the Chern-Simons coupling λ is too small the quasi normal modes will not drift towards the real axis and we don't expect to see long lived oscillations of the current .We plot components of the stress tensor (purple and pink) relative to the stress tensor for an unpertrubed magnetic black brane at λ = 3/2, B = 2T 2 , and an electric field described by (4.14) with E 0 = 1/2T 2 and τ T = 2/5 (dashed red).It seems that T 33 exhibits small oscillations at a frequency which is double that of the first quasi normal mode of the gauge field.
and stress tensor at late times.Nevertheless, we do expect to see a decaying excitation whose lifetime is proportional to the imaginary component of the quasi normal frequency.To this end, we consider the response of the current and stress tensor an electric field of strength E 0 = 0.5T 2 and width τ T = 11/40 (with T as before), a magnetic field of B 0 = 2T 2 and an anomaly with λ = 1/2.For these parameters we find that the final temperature of the system, long after the electric field has been turned on, asymptotes to T f ∼ 1.65T 0 .In this case the lowest quasi normal mode of the late time solution is given by ω 1 = 3.90 − 1.14i, so that Re(ω 1 ) ∼ 1.1/τ .The behavior of the current in this setup is depicted in figure 6 and that of the stress tensor in figure 7. Here, while the current decays quickly to zero, the spatial components of the stress tensor remains excited at intermediate times 1 ≲ t ≲ 3 after the electric field has been turned off.As before, this transient behavior involves strong gravitational effects where non linear gravity is important in capturing the correct dynamics.

Discussion
In this work we've studied quenches in magnetically charged anomalous thermal states.The momentum independent quasi normal modes associated with the dual black hole description of these thermal states exhibit long lived excitations which may be triggered by an appropriately tuned external electric field.Long lived excitations of this type have been previously observed in [25,29,55] and it seems likely that they are generic, at least for theories with a holographic dual.In this context, there are several directions one may pursue.
If the anomalous resonance effect is robust, it stands to reason that such an effect will manifest itself in other, non holographic, systems where the anomaly is sufficiently strong.In this case, the effect has the potential of being observed experimentally in systems whose effective field theory description involves chiral fermions, e.g., Weyl semi-metals.
On a somewhat different note, one may also attempt to observe the anomalous resonance effect in a fully controllable holographic dual pair (where the field theory description and the gravitational dual are well understood).As discussed earlier, the action 2.1 is related to the STU ansatz for the D = 5, N = 2 gauged supergravity action [46].One may check that the solution (2.4) with q i = 0 is a solution to the gauged supergravity action of [46] (first discussed in [22]) but quasi normal modes of these magnetically charged black brane solutions will neccessarily involve fluctuations of the scalar fields of the STU ansatz.
Our expectation is that the quasi normal modes associated with the anomalous resonance effect computed here will be unmodified when including the aforementioned additional scalar fields.The reason being that the quasi normal modes of interest are vector modes which should decouple from the scalar quasi normal modes.
In parallel to the anomalous resonance effect the authors of [22] discussed spatially modulated instabilities of the magnetically charged black brane solution (see also, e.g., [56,57] for a general discussion).In [22] it was argued that the magnetically charged black brane solution is unstable to a spatially modulated phase with the critical temperature, T c , for this instability being rather low.In the presence of such a phase transition one might worry that the anomalous resonance effect is unobservable since the spatially modulated phase dominates the dynamics at low temperatures where the imaginary component of the quasi normal modes becomes sufficiently small.Of course, this is a matter of scales: whether the imaginary part of the quasi normal mode at T > T c is negligible.A full numerical analysis of the critical temperature and the associated quasi normal modes of the homogenous phase should be able to resolve this issue.
Apart from the anomalous resonance effect, an additional effect associated with the anomaly, referred to as an anomalous trailing effect was discussed in [29] and previously in [43,[58][59][60].As it turns out, if the late time value of the (anomalous) background gauge field differs from its value at early times then the expectation value of the late time current will be sensitive to this (possibly gauge dependent) shift.The analysis of the anomalous trailing effect associated with the action (2.1) is almost identical to the one provided in [29] and will not be repeated here.We have checked, in several examples, that a sufficiently small Gaussian electric field perurbation indeed generates an anomalous trailing effect.A full study of anomalous trailing for strong background electric fields is left for future work.
m and g (2) mn are determined by the equations of motion and we have used F (0) = dA (0) , and In what follows we will raise and lower indices of the field strength F (0) mn using the boundary metric, g (0)mn , e.g., F (0)m n = g (0)m ′ m F (0) m ′ n .The coefficients α i are scheme dependent undetermined coefficients associated with finite counterterms which can be added in the holographic renormalization program associated with the trace anomaly and the field strength squared [47].There are other counterterms which can be added which will modify the expectation value of the stress tensor in a non flat background metric g (0) mn [61].We have refrained from writing these since we will be setting the boundary metric to the Minkowski metric so they won't be relevant to our discussion.
We point out that when the background metric is flat the aforementioned counterterms are proportional to each other and vanish whenever α 1 + 8α 2 = 0. To see this we note that choosing g (0) mn = η mn considerably simplifies the expressions for h where ∇ is a covariant derivative associated with the boundary metric g mn .Inserting these expressions into (A.3)brings the stress tensor and current into the form 8πG N ⟨T mn ⟩ = 2g (4)  mn − 2 48 F (0) 2 g (0) mn + where indices have been raised with the (inverse) boundary metric g (0)mn .The first term on the right hand side of (A.7) is the standard Joule heating term associated with an external electromagnetic field F (0)µν and a current J m .The second term on the right of (A.7) can be thought of as a Joule heating term associated with a conserved current proportional to ∇ m F (0)mn .The last term on the right of (A.7) indicates that the consistent current J µ is not gauge invariant [62].
If we want to obtain a canonical Joule heating term on the right hand side of the energy momentum conservation equation (A.7) to be associated with a single covariant current, we (A.9) We point out that while the unique covariant current J m cov is gauge invariant and satisfies the canonical Ward identity associated with a Joule heating term, the current J m is what will be evaluated by, say, a diagrammatic evaluation of the U (1) current associated with the symmetry of the problem.
In our setup there are three U (1) fields and a mixed Chern-Simons term.The resulting one point function can be deduced from (A.6) via symmetry.We find 8πG N ⟨T mn ⟩ = 2g (4)  mn − 1 24 (A.10) The metric and gauge field we are interested in are given by the Eddington-Finkelstein coordinate system (4.1) and not the Fefferman-Graham coordinate system of (A.10).In order to relate the asymptotic coefficients, A

Figure 1 .
Figure1.Complex values of the first quasi normal mode in units of the magnetic field, B, as a function of inverse temperature, T .As the temperature decreases the quasi normal modes with λ ≥ 1 approach the real axis.Here λ specifies the strength of the Chern-Simons term.

Figure 2 .Figure 3 .
Figure2.A comparison of the absolute value of the quasi normal modes obtained numerically, |ω|, denoted by circles, to the theoretical prediction obtained by taking the large λ limit, ω λ , denoted by a solid line, c.f.,(3.25)and(3.40).Here λ specifies the strength of the Chern-Simons coupling and n the number of the quasi normal mode.Note that ω λ is real while ω has a complex component.

Figure 4 . 4 ΔT 33 E 4  2 Figure 5
Figure 4.The expectation value of the current as a function of time (blue) in the presence of an external electric field (dashed red) for λ = 3/2, B = 2T 2 , and an electric field described by (4.14) with E 0 = 1/2T 2 and τ T = 2/5.The Fourier transform of the external electric field is supported at 1/τ ∼ Re(ω 1 )/3 with ω 1 the quasi normal mode (QNM) associated with the gauge field.The excitation of the quasi normal mode manifests in the boundary theory as an oscillatory current.

Figure 6 . 2 Figure 7 .
Figure 6.The expectation value of the current as a function of time (blue) in the presence of an external electric field (dashed red) for λ = 1/2, B = 2T 2 , and an electric field described by (4.14) with E 0 = 1/2T 2 and τ T = 11/40.The Fourier transform of the external electric field is supported at 1/τ ∼ Re(ω 1 ) with ω 1 the quasi normal mode (QNM) associated with the gauge field.The excitation of the quasi normal mode manifests in the boundary theory as a decaying oscillatory current.
mn appearing in (A.10) to the expansion in (4.4) we need to carry out a coordinate transformation from (4.1) to (A.2).Recall that (4.4) reads r −1 * a = a (0

Table 1 .
1. Solving (3.21)for Ω 2 gives us the subleading correction (in B) to the tensor quasi normal modes.Numerical evaluation of the expressions in (3.21) which determine subleading corrections to the quasinormal frequencies of the tensor modes in(3.4)