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Anomaly Induced Transport in Arbitrary Dimensions

Motivated by the consistency of a global anomaly with the second law of thermodynamics, we propose a form for the anomaly induced charge/energy transport in arbitrary even dimensions. In a given dimension, this form exhausts all second law constraints on anomaly induced transport at any given order in hydrodynamic derivative expansion. This is achieved by solving the second law constraints off-shell without resorting to hydrodynamic equations at lower orders. We also study various possible finite temperature corrections to such anomaly induced transport coefficients.


Introduction
One of the interesting features to come out of studying hydrodynamics of quantum field theories via holography [1,2] is the discovery that there are macroscopic transport coefficients in the hydrodynamics which probe the global anomalies in the microscopic theory [3,4,5,6]. Since the anomalies are exact objects in field theory, this leads to the hope that there are certain transport coefficients of interacting field theories which can be exactly computed by relating them to underlying anomalies.
Son and Surowka in [7] constrained various such first order transport coefficients using the consistency of the (3+1)d-anomaly with the second law of thermodynamics. Motivated by the form of their answer, we propose a generalization of the anomaly induced charge and energy transport in arbitrary dimensions and show that the form is consistent with the second law 1 .
We will argue that the simple form of anomaly induced transport presented in this paper exhausts all constraints on anomaly induced transport (at arbitrary order in hydrodynamic derivative expansion) that can ever be derived from second law considerations 1 Recently, there have been indications that there might be additional transport [8] induced at finite temperature by the mixed gravitational anomalies [9,10] in (3+1)dimensions at least in free theories. All these results generalize the results of various old calculations in the literature [11,12,13,14,15,16]. However, a general argument linking these transport to mixed anomalies in an arbitrary interacting system is presently lacking (at least the author is unaware of any such argument in the literature).
In this article, we will neglect such additional contributions and focus only on purely global anomalies. However it is possible that the succinct forms of the global anomaly induced transport presented in this paper will lead to a suitable ansatz taking into account gravitational anomalies. alone (in a flat spacetime). This is achieved by solving the second law constraints off-shell without resorting to hydrodynamic equations at lower orders in derivative expansion 2 .
The particular form of anomaly induced transport that we propose has various structural features and we resort to a specific notation which exhibits these features clearly. We begin by establishing the relevant notation/formalism in section §2 . In particular, we introduce a useful notation which streamlines the entire discussion.
We present the advertised generalization in the section §3 and prove its consistency with the second law. In the next section §4 we restrict to (3+1)dimensions and show that under particular frame redefinition, the results in [7] are recovered. We conclude in §5 with the discussion of various issues and future directions.
In the appendix §A, we discuss possible finite temperature corrections to the anomalyinduced transport . These corrections represent other possible non-dissipative transport which are very similar in their structure to anomaly induced transport phenomena and are interesting in their own right.
In the next appendix §B, we present various expressions relevant for dimensions upto d = 10 for ready reference. The last appendix §C is a summary of various notations employed in this paper.
Note: As this paper was readied for submission, a preprint addressing similar questions [17] appeared on arXiv.

Basic formalism
We are interested in hydrodynamics of field theories in arbitrary even spacetime dimensions. Let us denote the spacetime dimension by 2n. We will assume that the field theory has a global symmetry which has a global anomaly -by this we mean, the covariant divergence of the covariant global current 3 is given by a series of terms involving the non-dynamical gauge fields associated with this global symmetry. This in turn means that we can use the standard 'covariant anomaly' in our discussion -with the anomaly symmetrically shared between the currents 4 .
We want to study hydrodynamics at finite temperature and cartan chemical potentials for the global symmetry. We want to do this in the presence of the non-dynamical gauge fields also turned on in the same cartan subgroup 5 . When a global symmetry has an anomaly, the constitutive relations in the ideal hydrodynamics are modified -we will find it convenient to present the modifications in the following form for the energy, charge and the entropy currents 6 where P µν ≡ g µν + u µ u ν , pressure of the fluid is p and {ǫ, n i , s} are the energy,charge and the entropy densities respectively. u µ is the velocity of the fluid under consideration which obeys u µ u µ = −1. We have denoted by {q µ anom , J iµ anom , J µ S,anom } the anomalous heat/charge/entropy currents and by {T µν diss , J iµ diss , J µ S,diss } the dissipative currents. These are assumed to satisfy 7 where D is the covariant derivative including gauge fields and A i is the global anomaly.
So, whenever a subgroup is gauged, all currents get corrected due to a Bardeen current coming from these additional terms which is a function of various gauge fields. These Bardeen counterterms depend on the details of the subgroup that is gauged and the effect of these terms on the final hydrodynamics depends further on the details of the quantum gauge dynamics at finite temperature/chemical potential.
The results derived here might be of some relevance to such mixed anomalies between global symmetries and gauge redundancies only if this gauge dynamics is perturbative, i.e., if the theory is 'weakly gauged' in which case the effect of Bardeen terms can be accounted for perturbatively. 5 Note that these non-dynamical gauge fields violate the conservation of the global charges. Thus, by introducing chemical potentials for these charges, we are making an implicit assumption that this nonconservation is in some sense 'small' enough that it makes sense to talk about local thermal/chemical equilibrium within each fluid element. 6 Note that we do not include any tensor corrections to the energy momentum tensor due to the anomalous transport. As will be seen later, the second law considerations do not constrain such terms and hence such terms do not fall into the ambit of the anomalous transport terms considered in this paper. It is of course possible that more microscopic considerations lead to additional transport phenomena which a second law argument is blind to. Further, in specific theories, anomalies might, in consonance with other features of the theory, lead to specific phenomena which cannot be captured by any such general analysis. All subsequent analysis is subject to these caveats. The author wishes to thank Jyotirmoy Bhattacharya for emphasizing this point. 7 Note that we do not insist on particular frame conventions that need to be used for the dissipative parts of the constitutive relation. So, most of what we say would work in any frame provided the frame convention does not explicitly forbid the forms in which anomalous pieces appear in our equation. For example, one of the commonly employed frames is the Landau frame where terms like q (µ anomu ν) are explicitly forbidden. To figure out the anomaly-induced transport in such frames, one needs to first shift to a frame which allows such terms, add the transport in the new frame and then shift back. The author wishes to thank Jyotirmoy Bhattacharya and Sayantani Bhattacharyya for discussions related to this point. Now the rate of entropy production is given by where E iµ ≡ F iµν u ν is the rest frame electric field and a µ ≡ (u.D)u µ is the acceleration field.
The second law of thermodynamics is the assertion that the RHS of the above equation be positive. If we assume that the anomalous transport terms do not contribute to the entropy production then we get the standard second law constraint on the dissipative part when there is no anomaly, viz., assuming we get the second law constraint as Note that if we can find even one solution that satisfies equation(2.4), we can use that to reduce the second-law to the usual non-anomalous case .
To be useful this way the solution should be an 'off-shell' solution i.e., we should be able to show that it is a solution without any further use of the conservation equations which involve dissipative contributions from various orders 8 . The main result of this paper is to demonstrate by constructing a solution that this can always be done in arbitrary dimensions 9 .
It is convenient to write the equation(2.4) in terms of forms. To this end, introduce the hodge dual 2n − 1 forms {J S,anom ,q anom ,J i anom }. We remind the reader that in 2n dimensions, given any 2n − 1 formV hodge-dual to V µ and a 1-form A µ , we have So the equation(2.4) can be recast as whereĀ i ≡ A i Vol 2n is the 2n-form Hodge dual to the 0-form A i . Now, we turn to a more detailed analysis of the anomaly. Let F i be the field-strength 2-form and we have already defined the electric 1-form via E iµ ≡ F iµν u ν . We can do an electric-magnetic decomposition or in the language of forms where B i is the magnetic 2-form completely transverse to u µ , i.e., B iµν u ν = 0. In fact if our spacetime dimension is 2n, in the rest frame of the fluid B i can be thought of as 2-form in 2n − 1 spatial dimensions. This in particular means that where i k s are the flavor indices. We will also use the standard decomposition of the velocity gradients in terms of the shear strain rate σ µν , the vorticity ω µν , the acceleration a µ and the expansion rate θ of the fluid. This in particular means the exterior derivative of the velocity 1-form has the decomposition where ω is the vorticity 2-form. Further, using DF i = D 2 u = 0, we get We can write the anomaly in 2n dimensions in the form where ..in is the anomaly coefficient which is completely symmetric in all its indices. All our subsequent expressions will be linear in the anomaly coefficient. We will exploit this by introducing a notational trick which simplifies the presentation. We will begin by doing a replacement where we have replaced the anomaly coefficient with a bunch of fictitious parameters. This turns the above equation tō we can always restore the original expression by doing the reverse replacement We will further find it convenient to do a formal sum over all the even dimensions, writing 10 Hence, in the following, to obtain the formulae for 2n dimensions we have to Taylor-expand in t i s, take the terms with n + 1 number of t i s and do the replacement ..in is the anomaly-coefficient of the theory.

Anomalous transport in General dimensions
We now want to present a solution for the equation(2.7) which we repeat here We remind the reader that this is physically the statement that the anomalous transport leads to no entropy production. One simple solution of the above equation valid in arbitrary dimensions is 11q where we have used the notation introduced in the previous section to present the solution in arbitrary dimensions in a nice succinct form. This is the central result of this paper and we will now demonstrate that this is indeed a solution. In our notation, this is straightforward . Using the results in equation(2.13), we have Note that in the last line every term is a 2n form in 2n dimensions made only out of the 'spatial' forms ω and B i and any such form vanishes. Hence, we conclude that this is indeed the required 'offshell' solution.
Further it is interesting to note that our solution satisfies a First law-type relation and a generalized Onsager-type Reciprocity relation This means we can think of both the heat and the charge current as generated from a formal object V anomq Note that V anom is a formal spatial 2n-form in 2n dimensions whose role is similar to the anomaly polynomial which is a formal 2n + 2 form in 2n dimensions.
Further, we can think of these currents as derived from a single expression for the anomaly induced Gibbs free energy currentḠ anom

Comparison with Son-Surowka result
In d = 4 case (when n = 2) we get the following result for the anomalous transport we take the Hodge-duals to get To compare with [7], we first define Son and Surowka presented their results in the Landau frame (where q µ = 0) to first order in derivative expansion. The most general frame-change at this order leads to

(4.4)
Hence to set q µ = 0 we choose δu µ = − q µ ε+p which gives which is exactly the expression obtained in [7]. Certain comments are in order -as the manipulations which lead to the above expressions make it clear, the expressions above get corrected at the next order in derivative expansion and in all subsequent orders. To find these corrections in the Landau frame, one needs to know the higher derivative pieces in the constitutive relation and th resultant corrections to the equations of motion. -hence, this is an 'on-shell' solution unlike the 'off-shell' solution we started with 12 .The beauty of the solution presented in our frame is that the solution is independent of all such detailed transport coefficients at lower orders in derivative expansion.
This exercise can be repeated in arbitrary dimensions in the Landau frame and the conclusions are similar. Whereas the leading order pieces in the anomalous transport can be determined via leading order equations of motion, the subleading orders require the knowledge of more pieces in the constitutive relation. This difficulty is again an artifact of working in the Landau frame : our solution gives a way of relaxing the second law constraints back to the non-anomalous case in most other frames. Given the complete constitutive relation, one can always frame-transform our solution to the Landau frame to get the anomalous transport to any order one wants.

Discussion
In this paper, we have proposed a form for the anomaly-induced transport in arbitrary even dimensions motivated by consistency with the second law. This transport takes a very suggestive form and it is tempting to speculate that this is related to some sort of an index-like object, especially since the anomalies themselves are computed by various indices.
We will present a particular suggestion for what this index might be. In presence of anomaly, the rotational equilibrium states get deformed because of the anomaly-induced transport. By taking a thermal density matrix of free chiral fermions on S 2n−1 × R with chemical potentials for angular momentum/charges turned on, we should be able to see the transport proposed in this paper 13 . This suggests that we are looking for a related index of some bundle on S 2n−1 × S 1 with various appropriate twists. It will be nice if we could make this proposal precise. By fluid-gravity correspondence, similar rotational equilibrium states in large N strongly coupled theories are dual to blackhole solutions of Einstein-Maxwell-Chern-Simons actions in global AdS 2n+1 . It is an interesting question whether these gravity solutions can be constructed exactly 14 .
As we have mentioned at various points in the paper, we have completely ignored the issue of pure and mixed gravitational anomalies in our discussion. Hopefully, the simple solution given in this paper and the methods used to construct it can be generalized to gravitational anomalies. The structure of various finite temperature corrections (discussed in Appendix §A ) and the 3 + 1d free-theory computation of [10] suggest relations between gravitational anomalies and various finite temperature corrections , but there is no definitive argument (apart from the statement that they have similar symmetry structure) to link these two. It would be instructive to see whether for free theories in arbitrary dimensions we still have a relation between finite temperature corrections to anomaly-induced transport on one hand and the coefficients of anomalies involving gravity on the other hand 15 . It is also worthwhile to independently understand these finite temperature corrections in various theories, as some of them violate CP and hence might of some phenomenological relevance.
The discovery of anomalies and their various effects have led us to a deeper understanding of QFTs with Lorentz-invariant ground states. While various effects of anomalies in finite temperature and finite density have been explored by now, we still lack a complete framework to encompass various anomaly-induced finite temperature/density transport. We hope that this work is a small step towards that broader goal. hospitality during the workshop on string theory and its applications organised at TIFR Mumbai. This work was supported by the Harvard Society of Fellows through a junior fellowship. Finally, I would like to thank various colleagues at the society for interesting discussions.

A. Finite temperature corrections to anomaly-induced transport
In the main text of this article we presented a solution to the equation (2.7) In this appendix, we will examine the uniqueness of this solution. Since we are solving an inhomogeneous linear equation, the difference between various solutions of this equation satisfies the homogeneous equation We have used the suffix rev to denote that physically these denote transport processes which do not result in entropy production and hence are reversible. Evidently, it is too ambitious to try to classify all possible reversible transport processes which can happen in hydrodynamics, so we will content ourselves with describing some classes of solutions to the above equations which are close in form to the transport due to global anomalies.
In examining the solution-space of the equation (A.1), it is insightful to first prove the following theorem Theorem. Consider a V anom of the form where f and g are some Taylor-expandable functions 16 at the origin (i.e., are analytic at the origin). Construct from this the following currents Then they satisfy Notice that the main solution of this paper appears as a special case of this general theorem (with f[B i ] = e t.B and g = 0). At least in this subspace of solutions this is the unique solution at zero temperature.
It can be checked by explicit computation that these class of solutions satisfy first-law type relations and a generalized Onsager-type Reciprocity relation Now, it is clear that if we take f = 0, we get a large class of homogeneous solutions that we want. In 2n dimensions, g takes the general form where g i 1 i 2 ...i k k is some invariant tensor of the global symmetry 17 . Note that this construction which we just outlined is a generalization of the finite temperature corrections in (3+1)d proposed by the authors of [8]. In (3+1)d, for example, n = 2, we can take and these exactly correspond to the finite temperature corrections proposed in [8], provided we shift to the Landau frame using the procedure described in section §4. The (3+1)d free fermion calculation 18 in [10] gives g i 1 ∝ tr T i , i.e., g i 1 is non-zero in free fermion theories if and only if there is a mixed gravitational anomaly. It is unknown what happens in interacting theories. As far as the author is aware g 0 has never been calculated in free theory or otherwise. Given that the associated transport violates CP, it might be of some phenomenological relevance as a possible measure of CP violation (say in the early universe).
A similar exercise can be repeated in arbitrary dimensions to get the finite temperature corrections from the equations (A.5) and (A.9) .

B. Explicit expressions in various cases
In this appendix, we present the explicit solutions for B = 0 and ω = 0 followed by the explicit solutions for all even dimensions till d=10. 17 Curiously, such invariant tensors inevitably occur for odd n − k (only odd n − k occur if g is taken to be an even function of T ω ) whenever there is a mixed global-gravitational anomaly of the form hence it is interesting to speculate that theories with mixed anomalies generically have this kind of transport -in fact, something like this happens for free fermions in (3+1)dimensions [10] but the question of whether such a relation continues to hold for strongly coupled theories with same coefficient is open. More generally, there is the question of finding the transport consistent with the second law in the presence of mixed anomalies which is yet to be answered. 18 Also see old calculations by Vilenkin in the context of neutrinos [11,12,13,14,15,16].
All these follow from our solution -keeping only parts relevant to d = 2n dimensions

B=0 case : Chiral Vortical Effect
We now present the answer in arbitrary dimensions when the gauge fields are turned of This gives the chiral vortical effect in arbitrary dimensions due to the anomaly. Note that it is still present when the anomaly itself has been turned off.
ω=0 case : Chiral Magnetic effect We now present the answer in arbitrary dimensions when the vorticity is turned of This gives the chiral magnetic effect in arbitrary dimensions due to the anomaly. We now present the explicit solutions for all even dimensions till d=10 for ready reference.

Table of Notation Symbol Definition
Symbol Definition u µ , u Fluid velocity, 1-form g µν Spacetime metric P µν g µν + u µ u ν ε Fluid energy density p Fluid pressure n i Fluid charge density s Fluid entropy density µ i Chemical potentials T Fluid temperature T µν Energy-momentum J iµ Charge currents tensor of the fluid with anomalies Entropy current 2n − 1 form F iµν , F i non-dynamical gauge E µ i , E i Rest frame electric field strength, 2-form field F iµν u ν , 1-form B iµν , B i Rest frame magnetic A i Anomaly in the fields F i − u ∧ E i ith current DJ i ≡ A i a µ , a Acceleration field D µ Flavor/Lorentz (u.D)u µ , 1-form Covariant derivative σ µν Shear strain rate ω µν , ω Fluid vorticity, 2-form V anom A formal spatial 2n-formḠ anom Anomaly-induced encoding anomalous transport the Gibbs current (Hodge dual)