Higher Derivative Corrections to Charged Fluids in 2n Dimensions

We study anomalous charged fluid in $2n$-dimensions ($n\geq 2$) up to sub-leading derivative order. Only the effect of gauge anomaly is important at this order. Using the Euclidean partition function formalism, we find the constraints on different sub-leading order transport coefficients appearing in parity-even and odd sectors of the fluid. We introduce a new mechanism to count different fluid data at arbitrary derivative order. We show that only the knowledge of independent scalar-data is sufficient to find the constraints. In appendix we further extend this analysis to obtain fluid data at sub-sub-leading order (where both gauge and gravitational anomaly contribute) for parity-odd fluid.


Introduction and Summary
In past few years there has been much interest and progress in further understanding of relativistic, charged, dissipative fluid in presence of some global anomalies. Presence of quantum anomalies play a crucial role in transport properties of fluid. The first evidence of quantum anomaly in fluid transport was holographically observed in [1,2]. The authors found a new parity-odd term (and hence a new transport coefficient) in the charge flavour current. The origin of this new term can be traced back to gauge Chern-Simons term in the dual supergravity theory. Soon after these results were published, it was shown that the new parity-odd term in the charge current is essential because of the triangle flavour anomalies and the second law of thermodynamics [3]. In general the second law of thermodynamics (or equivalently the positivity of divergence of entropy current) imposes constraints on different transport coefficients. The same constraint can also be obtained from the equilibrium partition function of fluid [4,5]. Equilibrium partition function provides an alternate and a microscopically more transparent way to derive the constraints on these transport coefficients. A generalization of this approach for charged U (1) anomalous fluid in arbitrary even dimensions up to leading order has been considered in [6].
In [7] Bhattacharyya et.al. studied parity odd transport for a four dimensional non-conformal charged fluid at second order in derivative expansion. In four spacetime dimensions the effect of anomaly appears at one derivative order and the parity-odd transport coefficients at this order are determined in terms of anomaly coefficient. In this paper the authors studied the transport properties at second order and found that out of 27 transport coefficients 7 are fixed in terms of anomaly and lower order transport coefficients. The goal of our current paper is to generalize this work to arbitrary even dimensions. In 2n spacetime dimensions the leading effect of anomaly appears at (n − 1) derivative order. Hence the subleading corrections appear at nth derivative order. The aim of this paper is to study the constraints on transport coefficients appearing at subleading order. We innovate a systematic mechanism to compute different fluid data at arbitrary derivative order (parity odd or even). We list all possible scalars, vectors and tensors at any arbitrary derivative order in this paper. It seems to be rather difficult to find the independent sets. However, we argue that it is possible to get the correct constraint relations between transport coefficients even without knowing the independent sets of fluid data.
Our analysis is not valid in two spacetime dimensions. In two dimensions the parity odd terms appear at zero derivative order itself, and hence parity-odd and parity even sectors are not independent at any arbitrary order. Independence of these two sectors is important in our computation.
In the parity-even sector, the leading correction appears at first order in derivative expansion, e.g. shear viscosity and bulk viscosity terms in energy momentum tensor etc. In this paper we have extended our calculation to include the sub-leading order correction (i.e. second order corrections) to parity-even sector in constitutive relations in arbitrary even dimensions in presence of U (1) gauge anomaly. This completes the description of fluid dynamics up to sub-leading order in derivative expansion (both in parity-odd and even sectors) in arbitrary even dimensions with abelian gauge anomaly.
The organization of our paper is as following. In § 2 we explain our notation and perturbation scheme which we use in this paper. In § 3 we construct the partition function for both gauge invariant and non-invariant sectors and compute the constitutive relations from the partition function. We also describe the construction of the anomalous entropy current. § 4 is the most important section of this paper. Here we first describe how to construct fluid data at arbitrary derivative order. Next, we list all the leading and sub-leading order scalars, vectors and tensors which may appear in constitutive relations up to sub-leading order in derivative expansion both in parity-even and odd sectors. Although, we have not been able to find the 'independent' parity-odd vectors and tensors at sub-leading order, this does not inhibit us from finding the constraints on the transport coefficients. We elaborate this issue in § 4.4. Finally, in § 5 we list the constraint on the transport coefficients up to sub-leading order. In appendices we explain the Kaluza-Klien decomposition (appendix (A)) and sub-sub-leading order counting (appendix (B)).

Scheme and the Perturbative Expansion
We consider a 2n-dimensional spacetime manifold M (2n) with metric ds 2 = G µν dx µ dx ν and gauge field 1-form A = A µ dx µ . We want to study fluid dynamics in this background. A fluid is a statistical system in local thermodynamic equilibrium, which is generally characterized in terms of (covariant) energy-momentum tensorT µν , (covariant) charge currentJ µ and their constitutive equations∇ T ν only starts getting values at subsubleading derivative order. Let us explain our notation here.
• All the fluid quantities (like currents, transport coefficients, independent terms etc.) appearing in parity-odd sector, are denoted by 'tilde' (e.g.Ã). On the other hand we use no special notation for parity-even sector (e.g. A). Wherever applicable,Ā = A +Ã denotes the total quantity (parity-odd and parity-even).
•∇ and ∇ denote the covariant derivative and on M (2n) and the equilibrium manifold M (d−1) respectively. We use ∧ and as wedge product and Hodge Dual on all manifolds, as no confusion is possible.
Due to dissipative nature of fluid, it is not possible to write an exact generating functional W (or action) for fluids from which one can derive the energy-momentum tensorT µν and charge currentJ µ . Therefore we write their most generic forms, allowed by symmetries, in terms of fundamental fluid variables and their derivatives in a particular thermodynamic ensemble. In our analysis we consider the fluid variables to be temperature ϑ, chemical potential 1 ν and fluid four-velocity u µ with u µ u µ = −1.
We prefer to work in Landau Frame, where all the dissipation terms are transverse to the direction of the fluid flow. Hence, we can decomposeT µν andJ µ as whereΠ µν andῩ µ are the most generic symmetric tensor and vector made out of fluid variables. In the Landau frame The easiest way to implement this is to project all vectors or tensors appearing inῩ µ orΠ µν , transverse to u µ using the projection operator Since fluid is a low energy fluctuation about the local thermodynamic equilibrium,Π µν and Υ µ can be expanded in derivatives of fundamental fluid variables (ϑ, ν, u µ ): whereΠ µν (N ) andῩ µ (N ) involves N number of derivatives on fluid variables. The terms on RHS can have the most generic form as, where S (N )t , V µ (N )t and T µν (N )t are a collection of all possible gauge invariant scalars, vectors and symmetric traceless tensors (collectively known as data) respectively, made out of fluid variables and source fields at N derivative order. t corresponds to sum over independent terms at any particular derivative order. The data which is required for our computation has been enlisted in § 4.
In eqn. (2.6), the expression forΠ µν (N ) andῩ µ (N ) are fixed up to some undetermined coefficients appearing at each derivative order. Therefore, a fluid is characterized by an infinite set of such unknown functions (τ (N )t , σ (N )t , ν (N )t ), known as transport coefficients. Fluid up to a particular derivative order is characterized by a finite number of such transport coefficients. In general, these transport coefficients are not all independent. The second law of thermodynamics (or equivalently, positivity of local entropy current) imposes restrictions on different transport coefficients 2 [9]. Such relations among various transport coefficients are known as constraints.
[4] uses a different mechanism to find 'some' of these constraints. The idea is to write an equilibrium partition function for the fluid and derive the energy-momentum tensor and charge current from that partition function. Because of dissipation it is not possible to write a generating functional (W ) for the fluid. However, one can still write a generating functional in equilibrium configuration, which we denote by W eqb . Using W eqb one can find all the constraint relations involving transport coefficients which comes with data that survives at equilibrium.
More precisely, if the theory has a timelike Killing vector ω µ , we can write an Euclidean generating functional using the background fields and Killing equation on the decomposed manifold S 1 × M (d−1) . Here S 1 is the euclidean time circle along ω µ with time periodβ, and M (d−1) is the spacetime transverse to ω µ . [4] has conveniently chosen ω µ = ∂ 0 . Therefore, one can decompose the background in Kaluza-Klein form, For more details please refer appendix (A). Using this choice along with the Landau Gauge conditions and velocity normalization, the most-generic energy-momentum tensor and charge current in eqn. (2.3) on M (d) can be decomposed into scalars, vectors and tensors on S 1 × Indices on M (d−1) are raised and lowered using g ij . Details of Kaluza-Klein decomposition of fluid variables and background fields have been given in appendix (A).
Since the fluid we are considering is in local thermodynamic equilibrium, we can write the fluid variables as a spatial derivative expansion about their equilibrium values The terms with subscript 'o' are the equilibrium values, while∆ (N ) designates the N th derivative corrections 3 . The zeroth component of fluid velocity u 0 = v also gets derivative corrections which are determined by the corrections to v i using the four-velocity normalization. Similarly all the transport coefficients can also be expanded using the Taylor Series expansion Therefore the energy-momentum tensor and charge current receive two fold derivative corrections. First of all we write these expressions as a derivative expansion in terms of fluid variables in eqn. (2.5). Secondly, each term in that expansion can be further expanded around the equilibrium values of fluid variables according to eqn. (2.9). Thus we finally get (2) . . . , Expansion of time components can be determined from these using Landau gauge condition eqn. (2.4).
We choose the equilibrium convention for ϑ and ν by identifying their equilibrium values to be the red-shifted temperature and Wilson loop in the lower dimensional theory In the next section we construct the equilibrium partition function and obtain energy-momentum tensor and charge current in terms of background data following [4]. After that, we compare these stress tensor and current with the fluid stress tensor and current order by order in derivative expansion to find the constraints among the transport coefficients at any particular derivative order. A typical constraint will connect transport coefficients at equilibrium {α o (ϑ o , ν o )} and their derivatives with respect to ϑ o and ν o (up to a particular derivative order) We can extrapolate this constraint to non-equilibrium configurations: C {α(ϑ, ν)} , {∂α(ϑ, ν)} = 0, (2.14) while doing this, we are making an error of at least one derivative order higher, which will be compensated at next derivative order computation. This is how we find the generic constraints among fluid transport coefficients. Please note that while the equality constraints determined by this procedure are generic, the inequality constraints are not determined by this method.

Equilibrium Partition Function
The equilibrium partition function 4 W eqb of the theory can generally be disintegrated into two parts: The first one is the 'conserved' partition function which is gauge and diffeomorphism invariant, and generates conserved part of currents denoted byT µν (C) ,J µ (C) . The other piece is not gaugeinvariant and is referred to be 'anomalous' partition function. It generates anomalous piece of 'consistent currents' which will not be gauge-invariant in general. By defining a consistent subtraction scheme (Bardeen-Zumino currents), we can make these anomalous currents gauge invariant (see [11] for details) which we denote byT µν (A) ,J µ (A) . Their value at equilibrium is fixed by anomaly, and upto subleading order is given by: Here {X 1 , X 2 } are {−ϑdu, dA + ϑνdu} projected transverse to u µ , and their equilibrium values upon KK reduction reduce to: {f 1 , f 2 } = θ da, dA . Hence we have 5 : Let us now concentrate on W eqb (C) . It's variation on background (2.7) will determine the conserved currents: Kaluza-Klein decomposition of eqn. (3.7) gives, Here we have switched the basis to ϑ o = e −σ /β and ν o =βA for later convenience. a i is the Kaluza-Klein gauge field. Note that while W eqb (C) is gauge invariant, its integrand does not need to be. We can include a typical Chern-Simons term to it, which is defined such that its integral is gauge invariant 6 : Here C m 's are constants. This is indeed a valid Chern-Simons form as at equilibrium l i o(m) is just made of Chern classes of f 1 and f 2 : For the gauge-invariant integrand, we assume that curvature scales of the background M (d−1) is much much larger than the mean free path of the fluid, therefore the whole manifold can be thought of as union of various flat patches. The system can be thought of in thermal equilibrium in each local patch. On each patch we can define the euclidean partition function locally, hence giving us where P ( x) is local thermodynamic pressure and β( x) is local thermodynamic temperature. Given pressure, we can use the thermodynamic relations in local patch dP = + P ϑ dϑ + ϑqdν, + P = ϑs + νq, (3.12) to define energy density , entropy density s and charge density q of the fluid. All are functions of ϑ and ν. We can expand W eqb (C) around its equilibrium value as Derivative correction to the ideal fluid partition function is denoted by∆W eqb (C) , which will contain all the possible gauge invariant scalars made out of background metric and gauge field components at a particular derivative order. We have computed these scalars (till the derivative level of our interest) in § 4.
Collating together the conserved currents in eqn. (3.8) and the anomalous pieces in eqns. (3.4) and (3.5), and varying the Chern-Simons terms in W eqb (C) i.e. eqn. (3.9), we can finally write:T (3.14) Comparing these to the most generic fluid expressions in eqn. (2.8) we can compute the constraints. Thus, we see that it is only the gauge invariant part W eqb (C) of the partition function that we need to evaluate at any desired order.

Anomalous Entropy Current
In last section we reviewed a procedure to get equality type constraints among fluid transport coefficients. It is generally known that these very constraints can also be get by demanding existence of an entropy current whose divergence is positive semi-definite. The most generic Entropy Current can be written as: whereJ µ S(A) is the part which captures the explicit dependence on anomaly coefficients. However, the other pieceJ µ S(C) can get implicit dependence on the anomaly coefficients through the fluid equations of motion. We need to demand this current to be positive semi-definite, (3.16) whenever EOM are satisfied. For equilibrium fluid configuration, both the pieces can be demanded to be positive semi-definite separately. Such decoupling is not always possible, as the fluid equations of motion depend on anomaly coefficients, which can induce some implicit anomaly dependence inJ µ S(C) . However, for equilibrium fluid configurations, the equations of motion are trivially satisfied and thus entire information of anomaly can be incorporated in J µ S(A) . Hence, if any part of∇ µJ µ S(C) couple to∇ µJ µ S(A) , the respective transport coefficients will be determined in terms of anomaly coefficients, and hence will be present inJ µ S(A) at the first place. Therefore all the information about constraints among fluid transport coefficients is encoded in the existence of J µ S(C) . In [12,13] the author gives an explicit construction of entropy current from Eqb. Partition Function. Now concentrating on the second term: at equilibrium,∇ µJ µ S(A) ≥ 0, since it does not have any independent coefficients, just constants, one cannot apply any constraints for it to be satisfied. Therefore J µ S(A) must be exact. But any current is always ambiguous upto some exact terms, and hence we can choose J µ S(A) = 0 equally well. We can hence write in a generic hydrodynamic frame 7 Note that in the expression forT µν (A) we have used the fact that anomalies are parity-odd. Now depending on the choice of hydrodynamic frame, these conditions can be used to determine anomalous dissipative parts of the various currents. For example, if we define u µ such that it does not contribute to anomaly, i.e. u µ (A) = 0, we will get: (3.20) This is the neatest frame for anomalies. Similar results for U (1) anomaly were derived in [14], however these expressions are also applicable to gravitational anomalies 8 . Here we present explicit expressions for the anomalous parts of currents, in presence of both U (1) and gravitational anomaly. Following the generic expressions given in [8], these can be computed directly from the anomaly polynomial. The anomaly polynomial in 2n dimensions up to (n + 1) derivative order is given as [15], where, C (2n) is gauge anomaly coefficient which we have already introduced in the last section and c m is gravitational anomaly coefficient. The two form R is defined in terms of the Riemann tensor as, Taking appropriate derivative of the above, one can find explicit expressions for anomalous parts of the currents. The leading part of the currents proportional to the gauge anomaly coefficient C (2n) have already been given in eqns. (3.2) and (3.3). Here we present the subleading order contributions to currents coming due to the gravitational anomaly, (3.23) 7 We have used the thermodynamic functions , q, s here, which will be explicitly proved in § 5. 8 restricted to equilibrium configurations where, The heat current has the form, Finally, the stress tensor looks like, Instead if we are working in Landau Frame, whereq µ (A) =q µ (C) = 0, we will get condition: We need to invert G µν +Π µν (C) , which can be done perturbatively in derivatives. To leading order: and hencẽ As showed by [14], in presence of just U (1) anomaly, it gives the exact result of Son-Sorowka [3]. To write a similar expression for gravitational anomaly in Landau Frame, one will need to find anomalous velocity to subsubleading order, which might be non-trivial.

Counting of Independent Terms
This section is dedicated to develop a systematic procedure to compute independent fluid data (vectors, tensors transverse to velocity and scalars). First we will review the counting in parity-even sector in generic dimensions. Then we will extend this idea to parity-odd sector in generic dimensions at arbitrary derivative order through a procedure we call 'derivative counting'.
After describing the generic procedure, we explicitly construct leading and sub-leading order parity-odd and even terms which are important for our current work. Many of these terms vanish in equilibrium. In tables tables (1), (2) and (5) we list all the leading and subleading terms both parity-odd and even and check if they survive at equilibrium. Further in appendix (B) we extend this counting procedure to parity-odd subsubleading derivative order fluid. For this reason we will keep our illustrations in the construction explicit to subsubleading order.

Parity-even Counting
In this subsection we present the parity-even counting in generic dimensions. One can always count independent data in the local rest frame (LRF) of the fluid, which turns out to be easier. We can later covariantize the terms to a generic reference frame by following simple (and generic) rules 9 . In LRF, the fundamental quantities are • Temperature -ϑ, Chemical Potential -ν.
• Derivatives of fluid velocity 10 All other quantities are merely derivatives of these fundamental quantities. Since LRF is locally flat, we are using the coordinate derivatives ∂ o and ∂ i . We introduce a notation for parity-even terms which will be useful later in parity-odd counting. Terms with d derivatives and i indices will be denoted collectively as ( i d , i, d). When working at equilibrium, it is also 9 The rules can be summarized as: replace 1) '0' index with contraction with u µ , 2) 'i' indices with a projection along P µν , 3) ∂ with∇, 4) ki 2 j 2 ...injn with µ 1 ν 1 µ 2 ν 2 ...µnνn uµ 1 , and finally 5) put all extra factors of projectors and velocities on left-most, so no derivatives act on them. 10 u µ uµ = −1 would imply u µ ∂uµ = 0 and hence in local rest frame ∂u0 = 0.
convenient to define 11 : The purpose of above notation is revealed in Kaluza Klein formalism: at equilibrium only spatial components of X µν Λ survive which land exactly to f ij Λ defined by: In the same spirit we define Only spatial components of K µνρσ survive at equilibrium, and they exactly match R ijkl . The usage of index 'Λ' is purely to facilitate counting and computations. Similarly we define ϑ Λ = {ϑ, ν}.
Bianchi identity: In counting, we will extensively use the Bianchi identity to get rid of many terms, so it would be worth to spend some time on it. The Bianchi Identities for Field Tensor, Vorticity and Riemann Tensor take the form: However our redefined variables X Λ and K do not satisfy Bianchi Identities. But nevertheless we can always use these identities to relatê to other terms, and hence we can safely get rid of these in the following computation. In rest frame especially (or at equilibrium in any generic frame), one can check that X Λ and K also satisfy Bianchi Identities.
Killing equation: 11 Our conventions are: If the theory has a unit Killing direction ω µ we have the following Killing equation for a general tensor which in local rest frame becomes Therefore if we are considering a theory at equilibrium, we do not have to consider the ∂ 0 derivatives. Secondly, the Killing equation for metric G µν is given bŷ Taking ω µ √ −ω 2 = u µ and using Killing Equation for scalars this translates to:

First Derivative Order
Below, we compute all possible terms at first derivative order in LRF.
However all these first derivative terms are not independent on-shell. Using first order equations of motion one can eliminate some of them. The equations of motion are given by eqn. (2.1) (at equilibrium) Using these equations we have killed the boxed terms in the counting.

Second Derivative Order
Below we list all possible pure second derivative terms. By pure we mean they are not product of two first derivative terms. Product of two lower derivative terms are called composite terms.
Here also all the terms are not independent because of equations of motion. The second order equations of motion are given by, Again we have killed boxed terms in the counting using equations of motion. We have provided a list of all terms till second order (also composites) in covariant form and their equilibrium values in tables (1) and (2). We can iterate this procedure to further derivative orders as required by the cause. Note that, for a pure term at N th derivative order, the maximum number of indices possible are N + 2; we will need it later.

Parity-odd Counting
In this section we shall compute the parity-odd leading and sub-leading derivative fluid data. Calculation in parity-odd sector is a lot more cumbersome, even in LRF. We introduce here a scheme called 'derivative counting' to compute these terms step by step. Any parity-odd term in (2n)-dimension must have a (2n − 1)-dim Levi-Civita involved in LRF (4.11) We are interested in constructing all possible scalars, vectors and symmetric tensors using it. A bit of thinking will reveal that one needs at least (2n − 2)-rank parity-even tensors to be combined with ii 2 j 2 ...injn for this purpose. One can subsequently form a list of parity-odd data types: 1. V : Vectors with free index on (2n − 2 rank parity-even tensor contracted with ).
4. V f : Vectors with free index not on (2n rank parity-even tensor contracted with ).
5. V C : Vectors formed of contraction of two non-indices with free index on (2n rank parity-even tensor contracted with ).
6. T f : Tensors with no free index on (2n + 1 rank parity-even tensor contracted with ).
7. S C : Scalars formed of contraction of T f (2n+1 rank parity-even tensor contracted with ).
8. T C : Tensors formed of contraction of two non-indices with one free index on (2n + 1 rank parity-even tensor contracted with ).
9. V C f : Vectors formed of contraction of two non-indices with one free index not on (2n + 2 rank parity-even tensor contracted with ).
10. V CC : Vectors formed of contraction of four non-indices with free index on (2n + 2 rank parity-even tensor contracted with ) . . . and so on.
Here we note that given D derivatives, one cannot construct a parity-even term, pure or composite, with more than 2D indices, because (2, 2, 1) and (2,4,2) have the highest index to derivative ratio, which is 2. Therefore, if we are interested in a fluid at (n − 2 + s) derivative order (s = 1 corresponds to parity-odd leading order and so on), we can get at most 2(n − 2 + s) indices. The list of parity-odd data types we gave above is complete till subsubleading derivative order (s = 3).

Independent Data Types
We should emphasise that not all parity-odd data-types listed above are independent. The dependence comes from that fact that when we are working in 2n−1 dimensions, any antisymmetrization over 2n or more indices will vanish. Given that we are dealing with parity-even tensors of arbitrary rank which are to be contracted with , there are a whole lot of these antisymmetrizations possible. Hence, to find the independent data-types becomes highly non-trivial.
Let's look at a special case of this dependence. We construct a 2n-antisymmetrization, therefore, The consequence of this is that the data types [ ] f (i.e. ones with a free index not on ) can be expressed in terms of [ ] C (i.e. the ones with a free index on and an extra contraction).
Note that this result is only based on a specific form of 2n-antisymmetrization (eqn. (4.12)).
One can in principle go on with any random antisymmetrizations over 2n or more indices and find relations among the data, which as it turns out, is not a trivial task to do. We will come back to this issue in § 4.4. For now we continue with the counting.

Derivative Counting
We have classified parity-odd terms in data-types based on the number of parity-even indices required. We want to construct all allowed parity odd terms with D derivatives. We observe that it is not required to include all parity-even data type of the form (r, i, d) in this construction. We will show this below.
For a parity-odd fluid at D = (n − 2 + s) derivative order, we need to construct all the D derivative parity-even terms with number of indices ranging from 2D (the maximum possible) to 2(D + 1 − s) (= 2n − 2, the minimum required), i.e.
These D-derivative parity-even terms can be constructed out of pure derivative terms. We need not consider pure terms with self contractions in parity-even data types as they have been included in our counting procedure.
We now want to argue that not all parity-even data-types are required for this construction. For a data-type ( i N , i, N ) to be included at least once, the following combination with (2D − 2N + i) indices must be included: Since the minimum rank of this term must be 2n − 2 = 2(D + 1 − s) and maximum possible rank is N + 2, therefore we get, (4.14) For this equation to have a solution N ≤ 2s. So we need at max 2s derivative order parityeven terms, to construct parity-odd terms till (n − 2 + s) derivative order. For example at leading order, s = 1, only pure terms with at max 2 derivatives are required. The parity even terms required till s = 3 are enlisted in tables (3) and (4). Further, if we were only interested in finding terms that survive at equilibrium, we can use the Killing condition and drop all terms with ∂ 0 derivatives.
Some of the combinations constructed by this procedure using table (3)  and so on... The counting can be extended arbitrarily to the derivative order we need. In next section we will construct terms till subleading order, and later in appendix (B) we will extend it to subsubleading order. We will suppress the usage of data-type (2, 4, 2) for brevity; combinations involving it can always be reached by exchanging (2, 4, 2) with two (2, 2, 1)'s.

The Basis of Independent Data
As we discussed in § 4.2, the data we have enlisted in the preceding sections is a 'complete set' but not independent. There might exist numerous relations among them through anti- symmetrizations of 2n or more indices. If we look back at § 2, the need of all independent data arose to write down the most generic form of the constitutive relations. We write the energy-momentum tensor and charged current as a combination of all independent tensors and vectors respectively up to some undetermined coefficients which are called transport coefficients. We then determine the same quantities from equilibrium partition function and compare with the fluid results. It turns out that the transport coefficients which destroys the positivity of entropy current divergence are set to zero by this procedure. We call these transport coefficients unphysical. Put differently, the partition function generates only the physical transport coefficients in the constitutive relations (eqn. (3.14)) at equilibrium. Now if we relax the condition 'independence' while writing fluid constitutive relations, i.e., add more terms to these relations which could have been determined in terms of others; they can be regarded as redundant transport coefficients in our system. Since the charge current and the energy-momentum tensor we derive from the partition function remain unchanged, we get relations between the transport coefficients (including the redundant coefficients) and the coefficients appearing in partition function. However, we still have our answers -the independent transport coefficients and distinct constitutive relations.
Let us explain with an example. Suppose at some particular derivative order, we have total I number of vectors V µ i . We can write charge current at this order as, J µ = I i=1 a i V µ i , where a i 's are transport coefficients. On the other hand, suppose our partition function has X number of independent coefficients C j 's, and it generates a charge current are some functions of C j 's. By comparison we will get a i = c i (C j ). These are I relations with X free parameters, and thus imposes I −X constraints on a i . Now let's add to our set K more vectors V µ α , α = I + 1, . . . , I + K which could in principle be determined as: Then we would have guessed our ansatz to be J µ = K+I i=1 b i V µ i , and by varying partition function we will get J µ = K+I i=1 d i (C j )V µ i . d i (C j ) are some functions of C j 's determined by relation c i = d i − I+K α=I+1 d α C αi , as our partition function is still the same. By comparison we will get b i = d i (C j ). These are K + I relations with X free parameters, and thus imposes K + I − X constraints on b i . We hence get exactly K extra constraints, to kill the K extra degrees of freedom we added in the system. But once we have imposed these constraints, we will only be left with X independent transport coefficients.
However, note that we still need independent set of scalars that enters the equilibrium partition function, for our arguments to make sense. We check it here before we proceed. At leading order there are no scalars. At subleading order the scalars do not have enough indices for 2n or more antisymmetrizations, as a result all the scalars we get are independent. At higher order however, it may not be so easy to find out all the independent set of scalars.
Lets look at an example of such residual 2n-antisymmetrization conditions. In eqn. (4.13) if we chose B to be of the form Sg ij , we will get: where denotes the traceless symmetric part of a matrix. Hence one of these matrices of type T (after making traceless) is not independent for a given A. A similar argument is valid on other tensors like T C using S C g ij . But as we are treating all symmetric traceless tensors (of type [ ] ) to be independent, this should reflect in our final constraints, and as we will see, it will. It turns out that till subleading order, eqn. (4.17) is the only remaining residual constraint, and thus we can construct an independent basis; but this issue might turn more subtle at higher derivative orders. To illustrate the procedure we will not start with the independent basis even for subleading order, and show that we get consistent results at the end.

Fluid Constitutive Relations
Having all the data we require, we are ready to find the constitutive relations for fluid. We start with the results which are already known in literature, i.e. fluid up to leading derivative order. We revisit the results in our notation. Later we consider charged fluid at subleading order in § 5.3. We also set up the notation and architecture for subsubleading order parityodd fluid in this formalism in appendix (B). However we do not compute the constitutive relations explicitly, as we will discuss, the calculation becomes a lot non-trivial.

Ideal Fluid
At zero derivative order only energy-momentum gets a transverse contribution: where A is some arbitrary function of ϑ and ν. Now comparing eqn. (2.8) with eqn. (3.14) we can write at ideal order, The identifications will then give Note that we have identified A, Q, E exactly, and not just at equilibrium, as we explained in § 2. Therefore, the energy-momentum tensor and charge current for ideal fluid can be written as,

Leading Order Fluid
One can divide the constitutive relations in hydrodynamics in two different sectors -parity even and parity odd. Ideal fluid belongs to the first sector (in d > 2). The first non trivial derivative corrections in parity-even sector appears at the first derivative order e.g. shear viscosity term in energy-momentum tensor. Whereas in the parity-odd sector, the leading terms appear at (n − 1) derivative order for a fluid in 2n dimensions. All these terms and the corresponding transport coefficients (at leading order) have already been found in [6]. We shall discuss their result in our notation.

Parity-odd
Since there is no parity odd scalar and transverse symmetric traceless tensor at (n − 1) derivative order (see table (5)), only charge current gets parity-odd corrections: The combinatorial factor is introduced for convenience. It also ensures we do not surpass the limits of m. The fluid variables receives following corrections, Further, there is no parity-odd gauge invariant scalar at equilibrium on M 2n−1 , implying that ∆ (n−1) W eqb (C) = 0. Now comparing eqn. (2.8) with eqn. (3.14) we will find the constraints at parity-odd leading derivative order: And the corrections to fluid variables, where, Here we present these relations for completion as well as to set up our notations and conventions. We would also like to make some interesting observations about these functions. One can verify that Here pressure P (ϑ, ν) is function of temperature ϑ and redefined chemical potential ν. For any function Q(ϑ, ν) we define Q (m,n) = ∂ m+n ∂ m ϑ∂ n ν Q. These will come handy in subleading order calculation.

Parity-even
The most generic current corrections at parity-even leading derivative order are (see table (1)): (5.12) while at equilibrium the only surviving contributions are: There are no gauge-invariant parity-even scalars at equilibrium that appear at this order. Therefore, ∆ (n−1) W eqb (C) = 0. Now comparing eqn. (2.8) with eqn. (3.14) we will find at parity-even leading derivative order that all corrections vanish We hence get the constraints: So finally the form of currents is We also get to know that no fluid quantities (ϑ, ν, v i ) get order one parity-even correction.

Subleading Order Fluid
In this section, we shall describe the constraints on charged fluid in arbitrary even dimensions at subleading derivative order (i.e. n order), in presence of U (1) anomaly. Where as, the subleading correction to parity-even sector comes at second order in derivative expansion. Some aspects of four dimensional fluids at sub-leading order have already been performed in [7,16].

Parity-odd
Sub-leading order parity-odd fluid dynamics in four spacetime dimensions has already been discussed in [7]. Here, we generalize the results in arbitrary even dimensions and find the constraints on the transport coefficients. We see that, much like in [7], the higher dimensional transport coefficients depend on first order transport coefficients η, ζ.
From counting we can see that the n order parity-odd corrections (at eqb.) are given by (see table (5) Now, Therefore from eqn. (5.19) and eqn. (5.20) we can write, where, Similarly comparing theT i equations in (2.8) and (3.14) we get, which can be written as,∆ where, One can check that E ΛΓ is symmetric matrix and ∂ Λ = ∂ ∂ϑ Λ . We would like to emphasize that these are purely notations, to make the calculations tractable and easy to digest. There is a summation on repeated Λ, Γ indices. Now comparingT ij andJ i in eqn. (2.8) with eqn. (3.14) at parity-odd subleading derivative order, we have corrections to constitutive relations where, Λ swaps the value of Λ : 1 ↔ 2. The generating functional∆ (n) W eqb (C) contain all scalarsS oΛm . But one can check thatS o1m can be connected toS o2m by a total derivative. So we take the partition function∆ We compute the variation of generating functional with respect to different fields and find that Using the form of lower order currents corrections from eqn. (5.16) we can write, One can now use the results, obtained in eqn. (5.30) and eqn. (5.32) in eqn. (5.27) and comparing these expressions with eqn. (5.18) to get the constraints, Hence everything is determined in terms of a known function α m and a new coefficientχ m . Note that if we had used the 2n-assymetrization condition eqn. (4.17) to get rid of one traceless symmetric tensor to start with; a consistent choice would have been to removeT µν 3,Λ1m entirely andT µν 3,Λ2m for m = 1 (see table (5)). The coefficients of these terms are set to zero already by our constraints, which means the other leftover constraints are independent.
Finally we get the corrections to fluid variables using eqn. (5.25) as

Parity-even
Next, we present the results for sub-leading order (two-derivative) parity even sector for the fluid. From counting we can verify that at the second order, parity-even corrections (at eqb.) are given by (see table (2)): # refers to sum over all relevant indices. Now comparing eqn. (2.8) with eqn. (3.14) at parityeven subleading derivative order, and performing a similar manipulation as last section, we have corrections to constitutive relations: while the fluid variables get the corrections: Notice that the boxed terms only contribute for four dimensional fluids (n = 2). Out of the scalars enlisted in table (2), S o1Λ can be related to others by a total derivative. Hence ∆ (2) W eqb (C) is given by: Now we can find the variations of ∆ (2) Using the form of lower order corrections from eqn. (5.5) for n = 2 we can write, We can now put the variations of generating functional along with lower order corrections worked out above in eqn. (5.38). Using eqns. (5.10) and (5.11) and eliminating partition function coefficients S's we will find following 7 constraints, Coincidently none of the constraints depend on n = 2 special contributions. On the other hand fluid variables corrections are given by eqn. (5.39): This completes our calculation of subsubleading derivative order fluid.

Conclusions
In this paper we computed the energy momentum tensor and charge current for a fluid system in 2n dimensions with U (1) anomaly up to subleading order in derivative expansion (for both parity odd and parity even sectors) from the equilibrium partition function of the fluid. We described a novel counting prescription to construct the fluid data. However, an important issue we encountered here is that it is non-trivial to find independent vectors and tensors at arbitrary derivative order. But we were still able to find the independent transport coefficients and distinct constitutive relations. We showed that the knowledge of independent scalars at the required derivative order is sufficient for this purpose. This is a powerful observation and it enables us to carry on the computation at (n + 1) derivative order, where, we could find the independent scalars. We observe that the parity odd transport coefficients which appear at n derivative order in constitutive relations are constrained and some of them depend on the first order transport coefficients like η, ζ etc. It would be interesting to find the similar dependence in a holographic set up [17]. We plan to explore the holographic computation in future.
It is also interesting to find the fluid constitutive relations in presence of both U (1) and gravitational anomaly in arbitrary 2n dimensions. But, since the gravitational anomaly appears at two higher derivative level compared to the U (1) anomaly, it requires to carry on our analysis to one higher derivative (sub-sub-leading) order, i.e. to (n + 1) derivative order. Fortunately, as mentioned earlier, even at this order, we could determine the independent scalars and hence, in principle, the computation is possible. We have carried a large part of it in appendix (B). initial collaboration. AJ would also like to thank IISER Bhopal, IISER Pune and Durham University for their support, which made this project possible. NB would like to acknowledge DST for Ramanujan fellowship. Finally, we are indebted to the people of India for their support.

A Kaluza-Klein Decomposition
If a (d+1)-dim spacetime M (d+1) has a preferred time-like direction ω µ , it can be decomposed into S 1 × M (d) , where S 1 is the euclidean time circle. A k-rank tensor decompose in 2 k parts in this scheme: 1. S does not decompose.
and so on. Where P µν o = G µν − ω µ ω ν G αβ ω α ω β is the projection operator. If we are studying theory at equilibrium, we already have a preferred direction along the Killing vector of the theory ω µ = ∂ 0 . In this case we know that a (d + 1)-vector V µ will yield a scalar: and a (d)-vector: Hence we see that a U (1) gauge field A µ will be decomposed in A 0 ( x), A i ( x) . Similarly a tensor T µν decomposes in T 00 , T i 0 , T i 0 , T ij . It is the similar way the metric G µν on M (d+1) decomposes, hence we define: where we define g ij as metric on M (d) . Now using the diffeomorphic invariance one can work out the full form of G µν where time redefinition invariance requires that a i is an independent gauge field, named as Kaluza-Klein gauge field. Using the euclidean time periodβ we can define the local equilibrium temperature of the theory as: ϑ o = 1/β o = e −σ /β. Our higher dimensional metric is hence disintegrated in a scalar (Temperature), a gauge field and a lower dimensional metric.
We can now use the metric G µν to raise/lower the components of vectors: which are not Kaluza-Klein gauge invariant. From here we read out the (d)-covectors: Determinant of metric in two spaces can be related as: We have the Levi-Civita symbol in lower spatial dimensions: It is useful to see how higher dimensional contractions behave in lower dimensions:

A.1 Derivatives of Metric
Once the metric is known we can reduce the derivatives of metric, i.e. the Christoffel Symbol and the Riemann Tensor. The Christoffel Symbol is defined by: Pretending it to be a tensor at the moment, if we define its indices to be raised and lowered with the metric G µν . We can reduce it for Kaluza-Klein form of the metric: where Γ k ij is Christoffel Symbol on M (d) , which is raised and lowered by g ij . Also we define KK field tensor: Γ ij 0 is not KK gauge invariant, even though it has time index down and spatial index up, which is the manifestation ofΓ not being a tensor.
Lets define the higher dimensional covariant derivative as∇ and lower dimensional as ∇, whereas the usual derivative is given by ∂. We can check that: Finally the Riemann Curvature Tensor is defined using an arbitrary vector X µ as: using which we can define: Now a straight away computation will yield: Here R ijkl is defined to be lower dimensional Riemann tensor, and R ij = R ikj k , R = R i i .

A.2 Derivatives of Gauge Field
Now let us have a look at derivatives of gauge field A µ . Being a vector it decomposes as: The gauge transformation A µ → A µ + ∂ µ Λ translates to: Hence A i is a gauge field on M (d) , while A is a scalar. Usingβ (euclidean temperature) we define the local equilibrium potential ν o =βA. Higher dimensional field tensor however decomposes as: where, Now we define the four vector electric field:

B Subsubleading Order Fluid
In this appendix we extend the counting discussed in § 4 to subsubleading order fluid. We form a complete set of data at this order and classify the respective scalars, vectors and symmetric tensors. Later using the independent scalars at this order we construct an equilibrium partition function and compute its variation. We were however unable to process the constraints explicitly, as the calculations are analytically intractable.
2D Family: 2D family was already discussed in § 4.3.1, but this time since four indices are free from , two (2,4,2) can appear with two antisymmetric indices of R ijkl contracted. We 2D-1 Family: 2D − 1 family was already discussed in § 4.3.2, but this time three indices are free from . So only one among (2, 4, 2) and ( 5 3 , 5, 3) can appear, and not more that once. We will find 5 combinations of type T C : .
All the subsubleading parity-odd data surviving at equilibrium has been summarized in tables (6) to (8). Name Term Equilibrium will determine S 3Λm and S 4ΛΓm in terms of S 1Λm and S 2ΛΓm respectively: Each of S 1Λm and S 2ΛΓm , on the other hand is a unique scalar per choice of the parity-even tensor used to construct it by contracting with . Since antisymmetrization conditions cannot alter the tensor structure, these scalars are independent.

B.2 Attempt for Fluid Constraints
In the equilibrium partition function∆ (n+1) W eqb (C) we can include the scalars:S o1Λm ,S o2ΛΓm . But it can be checked that antisymmetric part ofS o2[ΛΓ]m can be related through a total derivative toS o1Λm . So we have: Sum over relevant indices is understood. Varying the partition function we will get:    On the other hand from counting we can see that the third order parity-odd corrections (at eqb.) are given by (see tables (6) to (8) # corresponds to all the relevant indices. Similar to subleading order, here also we will have special contributions for n = 2, as 3 leading order (n − 1) parity odd corrections can combine to give a 3n − 3 order parity-odd corrections, which will be equal to n + 1 only at n = 2 (Remember we are not considering n = 1 case). Now comparing eqn. (2.8) with eqn. (3.14) at parity-odd subsubleading derivative order, we have corrections to constitutive relations: while the fluid variables get corrections:

(B.9)
From here onwards in principle the way would be to solve eqn. (B.8) and find constraints for transport coefficients appearing in eqn. (B.7). To solve we would need to plug in the fluid variable corrections to all lower orders, along with corrections to lower order constitutive relations due to fluid variable corrections. The terms which were zero at equilibrium at lower orders will also start to contribute by gaining the fluid variable corrections. Leaving aside terms specifically for n = 2, still we would have to deal with a large mess in eqn. (B.8) which is analytically not quite tractable. So we leave these expressions at this point for reference.
Readers are advised that expressions eqns. (B.8) and (B.9) does not contain contributions from gravitational and mixed anomaly, and conserved Chern Simons form. Recall that while