Superfluid Kubo Formulas from Partition Function

Linear response theory relates hydrodynamic transport coefficients to equilibrium retarded correlation functions of the stress-energy tensor and global symmetry currents in terms of Kubo formulas. Some of these transport coefficients are non-dissipative and affect the fluid dynamics at equilibrium. We present an algebraic framework for deriving Kubo formulas for such thermal transport coefficients by using the equilibrium partition function. We use the framework to derive Kubo formulas for all such transport coefficients of superfluids, as well as to rederive Kubo formulas for various normal fluid systems.


Introduction and outlook
Hydrodynamics is the long wavelength effective description of a dynamical system at local thermal equilibrium. The fluid dynamics is governed by the conservation laws of the stressenergy tensor and charge currents, whose dependence on the thermal parameters, such as the fluid velocity, temperature and chemical potentials, is given by constitutive relations. These, supplemented by an equation of state define the hydrodynamics completely. It has been recently demonstrated in [1,2], that the non-dissipative properties of hydrodynamic systems are captured by the equilibrium partition function on curved stationary backgrounds. The most general gauge and diffeomorphism invariant equilibrium partition function on such backgrounds consists of thermal functions, i.e. functions of the temperature and the chemical potentials. The hydrodynamic transport coefficients can be expressed in terms of the thermal functions. This yields relations among the transport coefficients, since there are in general more transport coefficients than thermal functions. These relations between hydrodynamic transport coefficients coincide with the equality type constraints on the transport coefficients that are obtained by imposing the local second law of thermodynamics.
Linear response theory relates hydrodynamic transport coefficients to retarded correlation functions of the stress-energy tensor and charge currents of the microscopic theory by Kubo formulas. Thus, the Kubo formulas provide means to calculate the properties of field theories in their hydrodynamic regime. A way to derive these Kubo formulas is to consider the hydrodynamic stress-energy tensor and charge currents on an external gauge and gravity background and differentiate with respect to the metric and gauge fields perturbations (for recent relevant works see e.g. [3,4]). This typically requires to solve the hydrodynamic equations for the various fields (velocity, temperature, chemical potentials etc.) in terms of the background metric and gauge fields, and substitute the solution into the constitutive relations for the stress-energy tensor and the charge currents.
As an alternative to this differential method, we propose in this work a new algebraic framework for deriving Kubo formulas for the thermal functions and the transport coefficients, by using the equilibrium partition function on stationary gauge and gravity backgrounds. The partition function encodes the stress-energy tensor and the charge currents and their dependence on the metric and the gauge fields, which can be used in the linear response theory in order to derive the Kubo formulas.
A study of hydrodynamic transport coefficients in parity non-preserving superfluids 1 using the local version of the second law of thermodynamics was performed in [5] to first dissipative order, and generalized in the parity odd sector for an arbitrary number of unbroken charges in [6].
In [7] the partition function analysis was carried out for relativistic superfluids with one important difference in the formalism. Instead of using the equilibrium partition function, JHEP04(2014)186 the authors of [7] used the local effective action for the massless Goldstone field. The reason for using the effective action rather than the partition function itself in the analysis of superfluid transport coefficients is that the equilibrium partition function is not a local functional of the external fields, while the effective field theory for the Goldstone mode is local. 2 We use the same framework and the results of [1,7] to derive Kubo formulas for thermal transport coefficients of superfluids, as well as to rederive Kubo formulas for various normal fluid systems. Since Kubo formulas are eventually evaluated on a flat background with no external gauge fields, at the final stage of our analysis the solution for the gradient of the Goldstone phase will no longer be non-local, but rather a constant independent thermal parameter, such as the temperature T and chemical potential µ. We will denote it by ξ µ , and its transverse part by ζ i . To first order in derivatives the effective action in the presence of a background gauge potential A i and metric with g 0i = −a i , g 00 = −e 2σ , takes the form: whereζ is the Goldstone field. All the integrals are carried over the three dimensional volume element. All the vectors are oriented in the spatial directions and are contracted using the transverse part of the metric. C is the anomaly coefficient, P is the thermodynamic pressure function and f = −2(∂P/∂ζ 2 ). Using our method we derive the Kubo formulas for the three non-dissipative parity even thermal functions c i , i = 1, 2, 3 and the two parity odd thermal functions g 1 , g 2 . 3 These functions can be used to express all the superfluid non-dissipative transport coefficients (see subsection 3.4 for details). They enter in the constitutive relations of the current as (ν = µ/T ): where q, s, ǫ are the charge, entropy and energy densities respectively, u µ is the fluid velocity, P µν is the transverse projector, ω µ is the vorticity and B µ is the magnetic field. The terms proportional to the c i coefficients in the first line of (1.2) look similar to conductivities, JHEP04(2014)186 but, as opposed to conductivities in a normal fluid, they appear at thermal equilibrium in the constitutive relations of the current derived from the effective action. However, they are canceled by the first order corrections to the superfluid velocity once the equation of motion of the Goldstone field has been solved. The Kubo formulas that we derive in the parity even sector make use of correlation functions of the Goldstone and another (composite) operator. 4 The reason for this slightly different approach in the parity even sector is due to influence of the non-local terms mentioned earlier. We get the following Kubo formulas in the parity even sector: where every correlation function has to be calculated at ζ 0 ⊥ k (the zero momentum limit should be taken at the last step). In the parity odd sector we have: G(k n , −k n ) is the correlator of stress-tensors and currents according to its superscripts with external momentum k in the n-th direction only (the exact definition is given in equation (2.15)), the tilde stands for a correlator obtained from a variation of the covariant current and wherever the subscripts /⊥ appear, the spatial momenta is taken to be perpendicular/parallel to the direction of the superfluid phase gradient ζ i . The formula for g 1 found above seems to reinforce the suggestion of [6] that g 1 (α ab in [6]) may be related to a JJT type anomaly. We however cannot prove directly that it vanishes. This will be explained in detail in the discussion. Finally, we note that the same thermal functions can be often extracted from different components of the stress-tensor or currents. The correlators obtained should be consistent with one another, therefore we get identities between different retarded correlators of the stress-energy tensor and charge currents. We present examples throughout the text. This paper is organized as follows. In section 2, we present our method, and implement it for a charged anomalous fluid in 3+1 dimensions at first order in the derivative expansion. In section 3 we derive Kubo formulas for superfluid transport coefficients. In the discussion we comment on the interpretation of the Kubo formulas for the parity odd transport coefficients in superfluids. In addition to the material presented in the main text, in the last two appendices we consider all the other cases of [1] and derive the relevant Kubo formulas.

JHEP04(2014)186 2 Anomalous charged fluid in 3+1 dimensions
In this section we study 3 + 1 dimensional charged fluid dynamics up to first order in the derivative expansion. We take into account the effect of quantum anomalies. We will derive Kubo formulas for the hydrodynamic transport coefficients of such a fluid using the most general equilibrium partition function. We start with some preliminaries (see [1] for a detailed discussion).

Preliminaries
We will be working with the most general stationary metric and gauge-connection background: in the notations of [1]. The most general (CPT invariant) equilibrium partition function for such a system is: where,

4)
A is the gauge field, µ 0 and T 0 are the equilibrium chemical potential and temperature used to evaluate the partition function. P (T, µ) is the thermal pressure function, and Since we are working on a stationary background, the partition function can be written as a three dimensional local integral. The local values of the temperature and chemical potential are T ( x) ≡ T 0 e −σ , µ( x) ≡ A 0 e −σ , respectively. C ,C 2 are constants. C is the anomaly coefficient of the triangle diagram of three currents. It has been argued that C 2 is related to mixed gauge-gravitational anomaly [8].
Using the equilibrium partition function one derives the equilibrium stress-energy tensor and charge current: (2.6) -5 -

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When regarding the partition function as a functional of: these can be recast as: .
Note that the formulas are preferably presented with upper spatial and lower temporal indexes. This is due to the fact that tensors with such an index structure are invariant under Kaluza-Klein gauge Plugging the most general partition function for a 3+1 dimensional charged fluid (equation (2.3)) into the relations (2.8), the authors of [1] found the following results for the stress-energy tensor and charge current: where a ≡ e −σ T 0 , b ≡ e −σ A 0 , and P a , P b are the partial derivatives of P with respect to a and b respectively. Some T 0 factors were missing in equation (3.9) of [1] and are added here. The covariant form of the current: is given by: (2.14) Using the metric and gauge field dependence of the stress-energy tensor and the charge current, which is fully revealed in eqs. (2.9)-(2.14), it is straightforward to find Kubo formulas for the thermal constants C and C 2 . In this case C and C 2 must be constants rather than functions of the temperature and chemical potential in order for the partition function to have the required anomaly and invariance properties. One needs now to vary the stress-energy tensor and charge current with respect to the appropriate component of the metric/gauge-field to get the retarded correlation functions that constitute the Kubo formulas for the thermal constants.

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Since we work with a stationary background, the Kubo relations we shall find will only allow us to determine the thermal non-dissipative transport coefficients, i.e. those coefficients that affect the fluid dynamics at equilibrium. These will be determined by the correlation functions evaluated at zero frequency. Up to powers of i, zero frequency retarded correlators equal zero frequency Euclidean correlators. Equivalently, the Kubo relations can be worked out directly in Euclidean space as in [4], relating the thermal constants and, as a consequence, the non-dissipative transport coefficient, to Euclidean correlation functions.
It should be noted that if we wish to keep the independent variables as in (2.7), i.e. e σ , A 0 , a i , A i , g ij , when varying w.r.t. the gauge field and metric perturbation, we must vary according to equation (2.8) type formulas. Special attention must be paid when raising/lowering stress-tensor/charge-current indexes, since these operations normally involve extra metric factors and as a consequence do not in general commute with a variation w.r.t. the metric. Equivalently, one can translate back e σ , A 0 , a i , A i , g ij into the original gauge field and metric variations δA µ , δg µν . The variation needed to obtain Kubo-formula is then immediate. We will be using both methods alternately depending on which is simpler for the case studied. For the second order fluid studied in appendix D for example, corrections from raising/lowering indexes using the set of variables e σ , A 0 , a i , A i , g ij become involved, so the second method is preferable. For the cases studied in this section and the next however, this set of variables will suffice.
We will be using the following definition for the Green function: This is very similar to the Euclidean n-point function defined in [4], with a small difference, we differentiate w.r.t. the "Lorentzian" metric, which is a factor of i different for each t index compared to the definition in [4]. Otherwise, the definitions are the same (for a Lorentzian definition see [3]). To evaluate this type of Green functions using Feynman diagrams (cf. [9]), one passes to Euclidean space. We therefore find it advantageous to work with this Euclidean definition all along (up to the above mentioned factors of i).
Our definition for the Green function (2.15) involves multiple metric/gauge-field derivatives acting on the partition function of our system. This partition function can be thought of as the Euclidean action of the system with the metric given in (2.1) and with time coordinate compactified to a circle of length 1/T 0 . Since the system is stationary, we are allowed to replace time integration with 1/T 0 factor and time functional derivative with a T 0 factor. We can thus content ourselves with 3-integration and 3-differentiation in equation (2.15). One extra T 0 factor is present since we have one extra differentiation. The first differentiation stage was already performed in eqs. (2.9)-(2.14)), which we will use.

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Two comments are in order. First, when repeatedly differentiating the energy functional, each derivative can either pull an extra factor of T µν /J µ or it can act on a factor of T µν /J µ , already pulled down by the previous g µν /A µ derivatives. This is the origin of the contact terms (c.t.) on the last line of equation (2.15). Second, what we get by differentiation in the intermediate steps is not really the stress tensor T µν but rather the stress-energy tensor density √ −g 4 T µν . One can check that when evaluated in flat space, none of the Kubo formulas presented in this paper change due to the additional contact terms implied by the differentiation of the extra √ −g 4 factor.

Extracting the Kubo relations
Let us start by varying T 0j with respect to the i-th component of the gauge field A i . Using the set of variables from equation (2.7), this would mean varying T 0j = (T j 0 − g 0k T kj )/g 00 with respect to the Kaluza-Klein gauge invariant "gauge field" A i . Since T kj does not depend on A i , (and neither does g 00 ), upon setting the metric and gauge field perturbation to zero we obtain (in momentum space): G is the Euclidean Green function of stress tensors and currents (2.15) evaluated in flat space. The zero frequency limit removes any dissipative contribution which might not be accounted for by our equilibrium partition function analysis. Since we have set the metric and gauge field perturbation to zero, T = T 0 is the equilibrium temperature. Similarly µ = µ 0 = A 0 is the equilibrium chemical potential.
We have thus obtained a Kubo relation for C 2 : where k n is the external momentum and C is the chiral anomaly coefficient. The identification of C with the anomaly coefficient can be inferred from the expected transformation properties of the equilibrium partition function under gauge transformation. Alternatively, one can vary the divergence of the current (2.12) twice, with respect to both A 0 and A k , restoring the anomaly non conservation equation. Note, that (2.12) is the consistent form of the current. Similar Kubo relations follow from varying the current J j given in equation (2.12) (or its covariant counterpartJ j given in equation (2.14)) with respect to i-th component of the gauge field A i : where we have again used the set of independent variables (2.7) when varying. G refers to a correlator that is obtained from the variation of the covariant current, G i,j = δJ j /δA i (rather than the consistent current as in (2.15)). This is usually the type of Green functions obtained in hydrodynamic analysis of Kubo-relations (see e.g. [10]).

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Some more Kubo formulas can be obtained by varying the stress tensor T 0j w.r.t. the metric component g 0i (for i = j). Upon setting the metric and gauge field perturbation to zero we get: where we have used T 0j = (T j 0 − g 0k T kj )/g 00 , and the variation w.r.t. g 0i was performed using the set of independent variables of equation (2.7), according to equation (2.8) type differentiation rules: (2.20)

Hydrodynamic transport coefficients
The most general allowed form for the hydrodynamic stress-energy tensor and charge current can be found on symmetry grounds to be: with ǫ the energy density, P the pressure, q the charge density, s (which we will use later) the entropy density, u µ the normalized (u µ u µ = −1) fluid velocity, P µν = g µν + u µ u ν the transverse projector, σ µν = P µα P νβ ∇αu β +∇ β uα 2 − ∇αu α 3 P µν the shear tensor, ω µ = 1 2 ǫ µνρσ u ν ∂ ρ u σ the vorticity vector, E µ = F µν u ν the electric field and B µ = 1 2 ǫ µνρσ u ν F ρσ the magnetic field. All the hydrodynamic expressions in this paper will be presented in the Landau frame, i.e. the frame in which the stress-energy tensor and current corrections are transverse to the fluid velocity.
One can then write the most general equilibrium solution for the fluid fields (T, µ, u µ ) as a function of the external fields. The zeroth order solution consists of the local red shifted values:T (2.23) The first order solution consists of any addition to the above (δT, δµ, δu µ ) which is allowed by symmetry and is of first order in derivatives of the external sources. Plugging these into the stress-energy tensor and charge current (2.21)-(2.22) and evaluating them on the equilibrium configuration (2.1)-(2.2) one obtains a general expression for the stress-energy tensor as a function of the external background fields.
Comparing this form with the stress-energy tensor and current obtained by varying the equilibrium partition function (2.9)-(2.14), one can express the non-dissipative hydro-

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dynamic transport coefficients ξ ω , ξ B in terms of the partition function constants C 2 , C [1]: (2.25) Expressing these as Kubo formulas for the chiral transport coefficients using (2.16)-(2.19): we reproduce the Kubo formulas of [10]. 5 When equating the hydrodynamic stress-energy tensor on the most general equilibrium fluid solution to the one derived from the equilibrium partition function one in fact solves for the fluid profile in this very special equilibrium case. The k n → 0 limit is taken in order to get rid of terms of higher order in derivatives. For an example of how to evaluate these formulas see [11].

Superfluid dynamics in 3+1 dimensions
In this section we derive new Kubo formulas for the non-dissipative transport coefficients associated with the flow of a 3+1 dimensional relativistic superfluid up to first order in the derivative expansion. The authors of [5] found that for time reversal invariant superfluids all the transport coefficients can be expressed in terms of fourteen independent functions in the parity even sector, and six independent functions in the parity odd sector. All the parity even functions and one of the parity odd functions are dissipative in the sense that they result in entropy production. We are therefore left with five parity odd entropically non-dissipative independent functions. Of these, only two (σ 8 and σ 10 in the notations of [5]) multiply terms that do not vanish at equilibrium. These two functions (and their derivatives) can be used to express all the (thirteen) superfluid transport coefficients that affect the superfluid dynamics in equilibrium. In the absence of time reversal invariance three more thermal functions are needed to express all the non-dissipative superfluid transport coefficients. We require our superfluid to be neither parity preserving nor time reversal invariant. However, we require that our fluid is CPT invariant. Our analysis is divided into two parts. In the first part we derive Kubo formulas for the parity even transport coefficients. In the second we analyze the parity odd transport coefficients. The Kubo formulas for each sector (even/odd) receive no mixed contribution from the other sector, as will be shown JHEP04(2014)186 throughout the analysis. Therefore the study could have been carried out separately for each sector. In the discussion we draw general conclusions from the Kubo analysis about the nature of the parity odd superfluid transport coefficients. We also present new identities that are revealed when performing the analysis.

Preliminaries I -superfluid hydrodynamics
A superfluid is the fluid phase of a system with a spontaneously broken global symmetry. For 's' wave superfluids the symmetry breaking manifests itself in the appearance of a vacuum expectation value of a charged scalar operator. The phase of the charge condensate induces a new massless Goldstone mode into the theory. Being massless the Goldstone mode participates in the hydrodynamics. The motion of a superfluid consists of two distinct flows. The first is the flow of the normal part of the fluid which is encoded in the fluid velocity u µ . The second is the flow associated with the condensate (superfluid) part. This part has a velocity in the direction of the gradient of the Goldstone phase. When considering a background gauge field A µ as well, it is the covariant derivative of the Goldstone phase φ that points in the direction of the superfluid velocity and thus enters the hydrodynamic description of the system: The superfluid velocity is then given by u µ It is sometimes convenient to replace these eight fields by nine hydrodynamic fields subject to a single constraint. The additional field in that description is the local chemical potential µ(x) related to the other fields by the "Josephson equation": where µ diss (x) is a function of derivatives of the fluid variables. At zeroth order in the derivative expansion this relation simply equates the component of the 'generalized' gauge field ξ µ in the direction of the normal-fluid velocity with the chemical potential µ. It is sometimes convenient to use the definition: for the component of the ξ µ orthogonal to u µ . The equations of superfluid dynamics are: where the stress tensor and current are given by: (3.5)

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The superfluid constitutive relations are expressions for π µν , j µ diss and µ diss in terms of derivatives of the superfluid dynamical fields (u µ , ξ µ , T, µ) and background fields (metric/gauge field). All the thermal coefficients in equation (3.5) are functions of the three scalars: T, µ, ξ. They are not independent but rather given in terms of a single thermodynamical pressure function P (T, µ, ξ) through the thermodynamic relations: The equations of superfluid dynamics change their detailed form under field redefinitions. The temperature, chemical potential and (normal) fluid velocity field, are only well defined at the zeroth order in the derivative expansion. At higher orders in derivatives they are ambiguous. This means that a redefinition u µ → u µ + δu µ , T → T + δT, µ → µ + δµ accompanied by an appropriate adaptation of the constitutive relations can provide an equivalent description of superfluid dynamics. This is not true for the Goldstone phase gradient ξ µ (x) which is microscopically well defined. To completely fix the equations of superfluid dynamics we therefore need to specify a 'frame' (that is, a non ambiguous definition of u µ , T and µ). This is achieved by specifying certain conditions on the derivative corrections to the constitutive relations (i.e. on π µν , j µ diss and µ diss ). For example the 'Transverse Frame' is defined to be the frame in which: As mentioned above, the superfluid constitutive relations are expressions for the derivative corrections to the stress tensor π µν , charge current j µ diss and chemical potential µ diss in terms of derivatives of the fluid dynamical fields (and background data). It is sometimes convenient to specify the constitutive relations in terms of field redefinition invariant combinations of π µν and j µ diss and µ diss instead of specifying the full π µν and j µ diss and µ diss in a specific frame. In such a case the full constitutive relations are completely determined after adding five frame fixing conditions such as (3.7).
It is possible to obtain various constraints on the most general form allowed for the constitutive relations by requiring the existence of an entropy current of positive divergence. This was done in [12] for parity preserving time reversal invariant superfluids, in [5] for parity non-preserving (but still time reversal invariant) superfluids, in [6] for the parity odd sector of superfluids with multiple unbroken charges, and finally in subsection 3.1 of [7] for a single charge without assuming parity/time reversal invariance. We will present some of these results in the following sections where needed.

Preliminaries II -superfluid effective action
The partition function analysis for superfluids was carried out in [7] with one major difference compared to the partition function analysis of [1] that we used in the previous section. Instead of considering the partition function for superfluids as a function of the external sources (A µ , g µν ), the local effective action for the Goldstone phase gradient was used. To

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get from the local effective action to the full partition function one has to integrate out the Goldstone boson. In the classical limit this amounts to solving the equation of motion for the Goldstone mode and plugging back the solution into the effective action. We will be working in this limit.
The theory admits a degenerate set of vacua which break spontaneously the symmetry, the Goldstone mode is an excitation along these vacua. Integrating out this massless mode therefore results in a highly non-local expression for the partition function as a function of the external fields. It is therefore easier to use the effective action for the Goldstone phase gradient directly to derive the stress tensor and current instead of using the full partition function, integrating out the Goldstone phase at the last step of the calculation. This has the advantage that the Goldstone phase can be treated as independent of the external sources at the step in which the stress tensor and charge current are obtained by differentiation. One therefore doesn't have to deal with solving the equation of motion for the Goldstone phase and recovering its exact dependence on the background fields. This is no longer true when computing higher correlation functions, since they are not just determined by the variation of the action evaluated at the solution, but can also receive contributions from the variation of the solution itself.
In our Kubo formula derivation we will use the results of [7] for the stress tensor and current obtained as explained above. We will also solve the Goldstone equation of motion (minimize the effective action) to find the expectation value of the Goldstone field in the classical limit. We will then vary these quantities with respect to the external background sources to obtain Kubo formulas for the transport coefficients. We will have to pay careful attention to the variation of the Goldstone solution ζ eq w.r.t. the external sources, because of the corrections induced by the variation of the Goldstone solution.
When comparing the hydrodynamic stress-energy tensor and charge current with the ones obtained from the effective action, the authors of [7] regarded the equilibrium solution for the Goldstone phase gradient as independent of the other background fields. This is due to the non-locality of the classical solution, which lead to the conclusion that cancelations between the Goldstone phase gradient and other local functionals of the background fields are impossible, except for those implied by the equation of motion of the Goldstone phase gradient.
Since Kubo formulas are eventually evaluated on a flat background with constant gauge fields, at the final stage of our analysis, after setting the sources to zero, the solution for the gradient of the Goldstone phase becomes a constant independent thermal equilibrium parameter. We will denote the component of the equilibrium Goldstone phase gradient in the direction perpendicular to the normal fluid velocity in the absence of sources ζ i 0 . The addition of the equilibrium Goldstone phase gradient strongly resembles the addition of a finite chemical potential to the normal fluid. In the absence of sources we therefore set A µ = (µ 0 , ζ i 0 ). As we mentioned in the first part of those preliminaries, an equilibrium solution for superfluid dynamics in flat space is fixed by eight thermal parameters. The general form we were using for the metric (2.1) made use of the coordinate freedom to fix the alignment of the time-like killing vector with the t coordinate. This alignment fixes the equilibrium JHEP04(2014)186 velocity in the absence of sources to be u µ = (1, 0, 0, 0). This still leaves us with five free parameters (T 0 , µ 0 and ζ i 0 ) at thermal equilibrium in flat space. In a stationary setup these are the values that the temperature, chemical potential (zero component of the gauge field) and spatial components of the Goldstone phase gradient will obtain after setting the sources to zero. They are all constants. When evaluating the Kubo formulas, that will be presented in the next subsection, in terms of Euclidean (flat space) thermal QFT Feynman diagrams we expect a change in the fermion propagators of the form iω → iω n + µ 0 , q i → q i + ζ i 0 , where ω n are the Matsubara frequencies and q is the spatial momenta of the fermion line. This should be accompanied by an appropriate change of the stress-tensor/charge-current vertex operators (to account for the superfluid contribution).
Every part of our analysis will be carried out in two steps. First, only parity even contributions to the effective action and superfluid constitutive relations will be considered. Kubo formulas for the parity even thermal functions c 1 , c 2 , c 3 of [7] will be presented along with the associated transport coefficients. Time reversal invariance is not assumed. In the second step we will consider the parity odd sector. Kubo formulas for the parity odd thermal functions g 1 , g 2 and the associated transport coefficients will be presented. In the discussion we present conclusions drawn from the Kubo analysis about the nature of the parity odd superfluid transport coefficients.
We start by presenting parts of the effective action analysis of [7] that we will need to derive the Kubo formulas for both the parity even and parity odd transport coefficients.

Parity even effective action
The most general parity even equilibrium effective action one can build from the Goldstone phase gradient and external sources up to first order in derivatives (keeping Kaluza Klein, gauge and 3d diff-invariance intact) is given by: where the background metric and gauge field were defined in (2.1)-(2.2) and (2.4),T and µ where defined in (2.23),ν

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ξ µ is the superfluid phase gradient of (3.1) and, is the Kaluza Klein gauge invariant combination of the superfluid phase gradient. By convention ζ i 's index is raised and lowered with the three dimensional metric g ij . On the zeroth order solution (2.23) the above ζ i indeed turns out to be the orthogonal component

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of ξ µ as implied by (3.3). We therefore use the same symbol for these two quantities. All the functions c i are given in terms of the independent variables: f is defined through (3.6): where the differentiation with respect to ζ 2 is carried out at constantT andν, after the appropriate change of variables. After comparing the hydrodynamic stress-energy tensor and current to the ones derived from the effective action at zeroth order, it can be demonstrated that P from (3.8) and (3.6) are the same thermal pressure function. The leading derivative order equation of motion for the Goldstone phase can be obtained by varying the action S 0 with respect to φ and is given by [7]: where the derivative is covariant with respect to the 3 dimensional spatial metric g ij (the next order corrections to the equation of motion originating from S 1 are given in appendix A). We will denote the solution to this equation φ eq , and the associated ζ will be denoted ζ eq . It will be in general a functional of the external sources. In the classical limit φ eq is the expectation value of the Goldstone phase. The solution to the equation of motion (including the appendix A corrections) at linear order in the sources is given in appendix B. In momentum space, we have for the special case of ζ 0 ⊥ k and δg ij = 0: where we used the following definitions δA 0 ≡ A 0 − µ 0 , δA i ≡ A i − (ζ 0 ) i and δ 2 stands for any contribution which is of second order in the variation of the sources. For ζ 0 k we have: Using these to express the transverse superfluid velocity in momentum space gives: where in the absence of background source variation A i = ζ i 0 . The stress-tensor and charge-current are obtained by varying the effective action with respect to the various sources according to eqs. (2.8). For our analysis we will only need JHEP04(2014)186 J i and T i 0 . We list their explicit expressions as given in [7]: 6 where all the functions f, c i are evaluated on ζ = ζ eq . We will find Kubo formulas for the c i 's in the next subsection, right after reviewing the parity odd effective action results of [7].

Parity odd effective action
The most general parity odd (CPT invariant) first order effective action is given by: where g 1 = g 1 (T ,ν, ψ) ; g 2 = g 2 (T ,ν, ψ) ; (3.21) C is the anomaly coefficient andν The conversion between the ci's and the fi's of [7] is given by:

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The corrections to J i and T i 0 from the parity odd sector are given by [7]: where and all the thermal functions and their derivatives are evaluated at ζ = ζ eq . A comma followed by a subscript indicates derivative w.r.t. the appropriate thermal parameter. The parity odd one derivative contribution to the covariant current is given by: (3.25) All the non-dissipative parity odd superfluid transport coefficients can be expressed in terms of the thermal functions g 1 and g 2 . We will find Kubo formulas for those thermal functions in the next subsection.

Extracting the Kubo relations
In this subsection we will use our procedure to extract Kubo formulas for the parity even thermal functions c 1 , c 2 and c 3 . We will also present Kubo formulas for the parity odd thermal functions g 1 and g 2 .

Kubo formulas for the parity even thermal functions
Due to the addition of non-local terms to the Goldstone solution it turns out that in the parity even sector one should adopt a slightly different approach to derive Kubo formulas. It is possible to express the Kubo formulas in terms of correlation functions of the Goldstone and another (composite) operator by varying (3.14) according to (2.8).

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We get the following Kubo formulas: where every correlation function has to be calculated at ζ 0 ⊥ k (the zero momentum limit should be taken at the last step). It is also useful to get a Kubo formula for the zeroth order thermal function f from a variation of the zeroth order current. Let us start by varying J j with respect to A i . After setting the external sources to their flat space constant values (T 0 , µ 0 , ζ i 0 ), we get in momentum space for ζ 0 ⊥ k: where we have only used the lowest order solution for the gradient of the Goldstone phase.
If we now take i = j = x, k in theẑ direction and ζ 0 in theŷ direction (when evaluating correlators in terms of Feynman diagrams, this is our choice to make), we end up with the following Kubo formula for f : It is essential that the zero momentum limit is taken after evaluating the formula with ζ 0 ⊥ k.
One may wonder about the consistency of the derivative expansion when considering non-local terms (of negative momentum powers). Fortunately, if we take the momenta in the direction of one of the axes only, and since ζ i eq starts at zeroth order in momenta (3.16), we can still count powers of derivatives in a consistent way. 7 It is understood that our Green functions are evaluated in flat space with compactified time coordinate. We can therefore lose the 0 subscripts on T, µ, ζ and present the Kubo formulas as in the introduction (eq. (1.3)).
It is possible to use a similar calculation to obtain Kubo formulas for all the zeroth order thermal functions (energy density, pressure, charge density, entropy density, charge susceptibility, etc.) in all the cases studied in this paper.
Note, that the parity even Kubo formulas derived in this section received no contributions from the parity odd sector. Similarly, the parity odd Kubo formulas that will be JHEP04(2014)186 derived in the next subsection will receive no parity even contributions. We could have therefore treated the two sectors separately.
We will give some details on how the hydrodynamic transport coefficients relate to c 1 , c 2 and c 3 in the next subsection, right after extracting Kubo formulas for the thermal function g 1 , g 2 of the parity odd sector.

Kubo formulas for the parity odd thermal functions
In this subsection we obtain Kubo formulas for the thermal functions g 1 and g 2 from the parity odd effective action of equation (3.20). For this purpose we will vary the covariant current (3.18), (3.25), and stress tensor (3.19), (3.23) with respect to the gauge field A i and metric perturbation a i . In appendix B equations (B.5), (B.6), (B.7), (B.8) we have solved for ζ eq up to first order in variation of the background fields including the non-local contributions (of negative derivative powers). It will be useful in the following to have expressions for the variation of the superfluid velocity w.r.t. to A i and a i . In the special case of ζ 0 ⊥ k we have after setting the source fields to zero: 8 and for ζ 0 k: In both cases we have δζ i eq /δa j = 0 at first order in momenta. In the absence of sources we set ζ eq = ζ 0 .
Let us start by varying the covariant current with respect to A j . We get in momentum space, after setting the external sources to zero: where all the functions and their derivatives are evaluated in terms of the flat space parameters (T 0 , ν 0 , ζ 2 0 ). This evaluates to: for ζ 0 ⊥ k, and to:

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for ζ 0 k. It should be noted that the derivatives of the thermal variables T 0 , µ 0 and ζ i 0 vanish at thermal equilibrium in flat space, although the functional derivatives may be non-zero. Note that these expressions are symmetric under i ↔ j and k ↔ −k as they should. Relating to the current-current Green function using (2.8) and contracting with the Levi-Civita symbol we get for ζ 0 ⊥ k: and for ζ 0 k: where the ⊥ / subscripts are there to remind us that the Green functions are to be evaluated with superfluid velocity thermal parameter ζ 0 perpendicular/parallel to the external momentum k. Setting ω = 0 allows us to disregard any dissipative contribution that may arise.
We can now find Kubo formulas using both the perpendicular and the parallel Green functions. Pursuing both ways will lead us to a new type of identities. First let us pick k ζ. Dividing by k n and taking the zero momentum limit we get: The expression we get for g 1 is therefore: . (3.37) Had we chosen k ⊥ ζ 0 we would have gotten: There is a slight abuse of notation in the last formula (and similar formulas above) in the sense that it is not clear what exactly we mean by the k n division in the last equation. What we mean is that the momentum in the Green function should be taken in the n direction (which is our choice to make), we then divide by the same k n and take the zero momentum limit. An explicit calculation could use for example G x,y with k in theẑ direction (n = z), and with perpendicular ζ 0 in thex orŷ directions. No summation over n is implied, but we could have used a very similar formula with summation over n. The ψ differentiation was taken at constant T and ν, so integrating back, and losing all the 0 subscripts everywhere, we get

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where F (T, ν) could be any arbitrary function of T and ν and the correlator is evaluated in flat space with temperature T , chemical potential µ and transverse superfluid velocity ζ i . This is not a full determination of g 1 , but we nevertheless find it interesting because of the identity that follows from it. Comparing this to the last formula we got for g 1 we reach the conclusion that The fact that this combination of Green functions does not depend on the superfluid transverse velocity component ζ 2 is curious and deserves further study.
Let us now proceed to obtain the Kubo formula for the thermal function g 2 . We will keep using the parallel limit ζ 0 k which leads to simpler Kubo formulas. Looking at the G 0i,j = − (δJ j /δa i ) − A 0 (δJ j /δA i ) correlator (evaluated in flat space) we get: Using the expression we already found for g 1 we get: .

(3.42)
For the clarity of structural arguments that we intend to make later, let us consider what would change in our analysis when including the CPT violating term [7] δS in the parity odd superfluid effective action (3.20). When C 1 is a dimensionless constant this term respects all the required symmetries except CPT. Such a term would not change the charge current, and would have therefore no effect on the Kubo formulas derived above. It would, however, change the stress-energy tensor T i 0 by the additional term: The ǫ ijn G 0i,0j correlator will allow us to derive a Kubo formula for C 1 . Since T 0j = (T j 0 − g 0k T jk )/g 00 and since T jk is symmetric and g 0k vanishes in the absence of sources, no contribution to the correlator comes from the second term. We are therefore left with: Evaluating the expression with k ζ 0 we get: Isolating C 1 : where all correlators are evaluated with (k n , −k n ) external momenta.

Generalization to multiple superfluid charges
A generalization of our analysis to superfluids with multiple unbroken charges (but only one broken charge) seems straightforward at least in the parity odd sector. The same case was treated in [6]. For simplicity we will only be considering multiple Abelian charges (a tensor product of multiple U(1)-s). A non-Abelian generalization is very likely possible. Our goal in this subsection is to reveal the charge-index structure of the formulas we have presented in the previous subsection. This by no means constitutes a full treatment of superfluids with multiple broken charges. First, we have to replace the first order parity odd effective action with a multiplecharge extension of the form: where a, b, c are charge indexes. The index associated with the broken charge is a = 0. Only one superfluid transverse velocity exist which is associated with the broken charge ζ a=0 i . All normal charges are related to appropriate gauge covectors ζ a =0 i = A a i . g ab 1 should vanish for a = 0, and g a 2 should become a constant in that case. The requirement of CPT invariance of the partition function forces C 1 = 0. This would result in the following Kubo formulas for g (ab) 1 , g a 2 and C 1 : , (3.50) where G ai,bj and G 0i,aj are defined in a similar way to the one described in (2.15), adding the appropriate charge indexes on the A µ derivatives. C abc is the completely symmetric anomaly coefficient of three currents. g (ab) 1 is the symmetric part of g ab 1 . 9 We have replaced the k n division of equations (3.37), (3.42), (3.47) by a ∂ kn differentiation in the above formulas. We find this form more likely to be generalized to the case of superfluid with multiple broken charges since the differentiation makes sure that we get rid of any zeroth order contribution that may arise. The above Kubo formulas reveal the full charge-index structure of the formulas derived in the last subsection.
In the case of more than one broken charge a bunch of new scalars are available at zeroth order for constructing the effective action due to mixed products of different-charge superfluid transverse velocities of the form ζ a · ζ b . Therefore a generalized new analysis JHEP04(2014)186 is needed, even at zeroth order, to constitute a full treatment of superfluids with multiple broken charges. It is important to emphasize that we have not listed all the possible contribution to the effective action of a superfluid with multiple broken charges in equation (3.48), even in the parity odd sector ( d 3 x √ g 3 κ abc 1 ǫ ijk ζ a i ζ b j ζ c k , d 3 x √ g 3 κ ab 2 ǫ ijk ζ a i ζ b j ∂ kT were ignored, just to name a few). In addition, for the case of multiple broken charges the full relation between g ab 1 , g a 2 and C 1 and the (non-dissipative) hydrodynamic superfluid transport coefficients hasn't been studied yet. For a non-Abelian analysis one has to furthermore extend the derivatives to covariant derivatives and check the influence of this change. We leave this for a future study.

Hydrodynamic transport coefficients
In this subsection we present the relations between the thermal functions c 1 , c 2 , c 3 , g 1 , g 2 and the non-dissipative part of the superfluid constitutive relations. The Kubo formulas for the thermal functions were already found in the previous subsection. Having this in hand, and stating the constitutive relations, we can identify Kubo formulas for any of the superfluid non-dissipative transport coefficients.

Parity even transport coefficients
We start with the parity even sector. The parity even superfluid constitutive relations are the first order parity even corrections to stress tensor π µν , charge current j µ diss and "Josephson equation" µ diss . The expressions are given in terms of the hydrodynamic fields T, µ, ζ µ , u µ and derivatives thereof.
The superfluid constitutive relation we shall present are given in terms of some special combinations of π µν , j µ diss , µ diss that are invariant under frame redefinitions (this is sometimes more convenient as was explained at the end of subsection 3.1). To completely specify the constitutive relations one has to specify five additional frame fixing conditions. Transforming between two fluid frames is a simple task (see section 2.4 of [12] for a detailed discussion).
The constitutive relations are expressed in terms of the thermal functions c 1 , c 2 , c 3 . Since the Kubo formulas for these thermal functions were already found (3.26), we now have in hand Kubo formulas for all the parity even non-dissipative superfluid transport coefficients. The results for the constitutive relations are taken from [7].
The frame redefinition invariant combinations that are used to present the constitutive relations are: where a = {1, 2, 3} and: A minor typo in S a (minus sign in the first term) was corrected here (compared to [7]). Using these, the constitutive relations are (we only present the non-dissipative part which is fixed by the equilibrium partition function): where diss stands for additional dissipative terms. A minor typo of [7] was corrected here by an additional f T factor in the last term of the constitutive relations for S a .
Using this in the transverse frame one obtains the following expression for the current: where the derivative with explicit subscripts s and q/s is taken as constant s and q/s (in [6] it was suggested that the set of variables (s, q/s, ζ 2 ) are better suited to describe some properties of superfluid hydrodynamics than (T, ν, ζ 2 )). A summation over a is implied.

Parity odd transport coefficients
We now move to the parity odd part of the first order superfluid constitutive relations. We present them in the transverse frame of (3.7). We find it easier to identify the physical significance of each transport term this way. The constitutive relations are given in terms of the thermal functions g 1 , g 2 , C 1 by the following formulas: These were derived in [6]. 10 After correcting for this term, the results match precisely those of [5,7]. The partial derivatives with respect to s, q/s and ζ 2 are taken with (s, q/s, ζ 2 ) as the independent thermal parameters. For the full charge-index structure one may refer to [6]. To get these formulas we had to use the following matching rules: g 1 = α = σ 8 , g 2 = −β = −σ 10 + 2νσ 8 + 1 2 Cν 2 + 2hν, 2C 1 = γ = s 9 to match between the different conventions of [5][6][7] respectively.
The chiral magnetic and chiral vortical conductivities (i.e. the coefficients of magnetic field and vorticity in the charge current) take the form: and can therefore be expressed (based on our analysis in the previous subsection) using 10 We have noticed a typo in [6], the ζ µ term of the charge current is missing.

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the following Kubo formulas: . (3.62) These formulas strongly resemble the Kubo formula we got for the case of the normal fluid (2.26), (2.27). The only difference is that the correlators should be evaluated on a thermal background with finite value of the superfluid transverse velocity ζ. As we mentioned earlier, when evaluating a diagram, a finite value of ζ is expected to influence the propagators as well as the vertices. We will make this statement more precise in the next subsection. The momenta should be taken parallel to ζ.

Discussion
There are various open issues that deserve further study, and we list some of them below. It would be interesting to evaluate in field theory models the Kubo formulas that we derived for superfluid transport. Of particular interest are the Kubo formulas for the chiral magnetic and chiral vortical effects. A simple relativistic model where some of the zeroth order thermodynamic coefficients were obtained using linear response theory was studied in [13], it would be interesting to extend this work to higher order coefficients. Evaluating superfluid Kubo formulas using Feynman diagrams requires the consideration of the new thermal parameter ζ i 0 . The addition of a superfluid velocity strongly resembles the addition of a finite chemical potential to the problem. Both always appear in the hydrodynamic description accompanied by the appropriate gauge field component (see (3.1)). This suggests that the new thermal parameter ζ 0 should be introduced to the thermal QFT description the same way that the thermal chemical potential µ 0 is. That is, through an adjusted definition of the grand canonical partition function obtained from the original partition function by the substitution rule A µ → A µ + (µ 0 , ζ i 0 ) in the functional integral of the original Lagrangian of the theory.
The partition function (and all derived correlation functions) could therefore be calculated using the path integral formalism with time coordinate compactified on a circle of radius 1/T 0 , and where derivatives (momentum vectors in momentum space) are subject to the following substitution rule: k µ → (iω n + µ 0 , k + ζ 0 ), where ω n = πT 0 (2n + [1]) are the bosonic [fermionic] Matsubara frequencies. Propagators will exhibit a suitable change.
This change is in addition to the usual changes that have to be made when evaluating Feynman diagrams in theories that have a spontaneous symmetry breaking. These include developing the theory in terms of new fields around the vacuum expectation value of the charged scalar operator and using those fields as the new elementary fields of the theory.
One may also wish to evaluate the second order non-conformal normal fluid transport coefficients obtained in appendix D in the strong coupling limit using AdS/CFT. It would be interesting to see the effect of these new non-conformal coefficients on observables such JHEP04(2014)186 as the elliptic flow and multiplicities in numerical hydrodynamic simulations of Heavy-Ion collisions such as [14].
It would be interesting to generalize our results and derive Kubo formulas for the first order non-dissipative transport coefficients of anomalous fluids in arbitrary dimensions using the equilibrium partition function [15]. A similar extension of our analysis will enable the derivation of Kubo formulas for Rindler hydrodynamics at second order using the partition function of [16]. Another required generalization of our work is the derivation of Kubo formulas for superfluids with more than one broken charge.
In [6] it was suggested that the hierarchy of charge indexes of the thermal functions/constants C abc , g ab 1 , g a 2 and C 1 and the associated factors of µ and T in expressions of the form (3.36), (3.41), (3.46), suggests that our thermal functions/constants may be related to anomaly coefficients of triangular diagrams with the appropriate number of charge current vertices.
The fact that C abc is the anomaly coefficient of the triangular diagram with three currents already came about from entropic constraints ( [17,18]). The relation between g a 2 and the coefficient of mixed chiral gravitational JT T anomaly was subject to intense debate recently (see e.g. [8,9,[19][20][21][22]). The relation between C 1 and the coefficient of the T T T anomaly is motivated by the fact they both vanish (in the case of C 1 , due to CPT invariance). The authors of [6] conjectured that in light of the progression of the charge-index structure and the associated factors of µ and T in the hydrodynamic constitutive relations, g ab 1 (their α ab ) should be related to the coefficient of the JJT anomaly. This led them to conjecture that g ab 1 should in fact vanish. This has been proven for the case of a normal fluid (see [23]) as it must from CP T . 11 We have tried to repeat the proof of [23] for the case of a superfluid. 12 Here, due to the possibility of including non local terms (with various powers of momenta in the denominator), we find that it is no longer possible to prove that g ab 1 = 0. One should take into account that the presence of a Goldstone mode allow for long range correlations. We find that the most general form of the current three-point function is where Σ 0,abc 2 encodes dg ab 1 /dµ c . In general we could use an analysis similar to the one in the previous subsections to relate JJT and g 1 motivated by the fact that temperature differentiation is related to 11 In [23] g ab 1 was named f AB 1 . 12 In [23] the author constrains the structure of the J i J j J 0 three point function using arguments of symmetry and the standard anomalous (non)-conservation equation. The author then relates it to a variation of the J i J j two point function (Kubo formula for the magnetic conductivity) with respect to the chemical potential. Invoking CPT invariance one can then rule out the presence of ∼ µT term in the magnetic conductivity of a normal fluid.

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T 00 insertion: 13 We therefore cannot find a general reason why g 1 should vanish in theories with nonfinite correlation length. We can however generally relate it to the JJT diagram as suggested by [6]. Finally, it would also be interesting to derive the Kubo formulas for the dissipative hydrodynamic coefficients. This requires to study time dependent dynamics as was done in [3]. One can repeat our analysis of the parity odd sector omitting the non-local terms, and get precisely the same Kubo formulas as we got in subsection 3.3.2. Drawing the conclusions from this, it is possible that the Goldstone phase gradient could be treated as an independent parameter without having to solve for it in terms of the external sources in the parity odd sector even in the dissipative case. All this is true up to an arbitrary addition that vanishes using the Goldstone equation of motion. This might facilitate the Kubo derivation for the dissipative superfluid transport coefficients. One such transport coefficient of special interest is the chiral electric conductivity of [6].

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form. The equation of motion up to this order in derivatives reads: where we have used H a = (T ,ν, ζ 2 ) for a = 1 . . . 3. The derivative is covariant w.r.t. to the three dimensional metric.
In the next appendix we will try and solve this equation. It should be noted that in general for non-local terms the derivative expansion fails. But since all our Kubo formulas will be evaluated with momenta directed along one of the axes only, in our case we can still rely on the consistency of an expansion in powers of momenta (momenta in numerator and denominator must either cancel or vanish).

B Solving the Goldstone E.O.M for non-local terms
In this appendix we want to solve the Goldstone equation of motion for the expectation value of the Goldstone phase φ defined through: We will do this in two steps. First we will solve the E.O.M at lowest order in derivatives: Then we will add the next order derivative corrections (A.2) to the E.O.M and correct our solution accordingly. We can solve the E.O.M order by order in the variation of the sources. For our Kubo formulas we only need to solve up to first order in the metric and gauge field perturbation. This is due to the fact that all our Kubo formulas are given in terms of two point function. We will not be interested in correlators including spatial components of the stress tensor. We may therefore immediately set g ij = δ ij . Let us expand (B.2) to linear order in the other external sources: 14

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This is the expectation value for the field φ. Differentiating w.r.t. the various sources and setting the sources to zero we will be able express c 1 ,c 2 and c 3 in terms of correlation functions of the Goldstone phase gradient and another (composite) operator. In the special case ζ 0 ⊥ k we have a simpler expression: whereas for ζ 0 k we have:

C First order charged fluid dynamics in 2+1 dimensions
In this appendix we use our method to rederive Kubo formulas for a 2+1 dimensional parity violating charged fluid up to first order in the derivative expansion.

C.1 Preliminaries
The most general partition function for such a fluid is given in terms of two thermodynamical functions α and β as follows: and the dependence of α and β on T 0 is hidden in their σ, A 0 dependence as follows [1]: Using equations (2.8) one is able to extract expressions for the stress tensor and charge current (up to first order in the derivative expansion) consistent with this partition function [1]: T 00 = −e 2σ (P − aP a − bP b ) − T 0 e σ ∂α ∂σ ǫ ij ∂ i A j + T 0 ∂β ∂σ ǫ ij ∂ i a j , (C.5) J i = T 0 e −σ ∂α ∂σ ǫ ij ∂ j σ + ∂α ∂A 0 ǫ ij ∂ j A 0 . (C.8)

D.1 Preliminaries
The most general equilibrium partition function for the fluid described above is given by: , where the (zeroth order) local value of the temperature is T ≡ T 0 e −σ (formerly denoted a), R is the Ricci scalar of the 3 dimensional metric g ij , f ij = ∂ i a j − ∂ j a i and we shall often use P i (σ) ≡P i (T 0 e −σ ).
Using the uncharged analog of eq. (2.8) the authors of [1] were able to find the stresstensor components: where ' denotes derivatives with respect to σ, T subscript denotes derivatives with respect to the zeroth order temperature T = T 0 e −σ , ∇ is the covariant 3-derivative and R stands for the three dimensional Ricci Tensor/Scalar of g ij .

D.2 Extracting the Kubo relations
To extract the Kubo relations, one has to vary equations (D.2)-(D.4) with respect to the various sources. Some of the Kubo relations we present in this section include three point functions. Because of this reason, using the set of independent variables of equations (2.7)-(2.8) will involve multiple instances of raising/lowering indexes, as well as careful surveillance of the point at which the differentiation is carried out. This encouraged us to use δg µν = g µν − η µν ≡ h µν as the independent set of variables instead, differentiating according to (2.6) directly. Differentiating according to (2.7)-(2.8) accompanied by a careful bookkeeping of indexes and momenta gives precisely the same results.
Here and in what follows we replace sub/superscripts (0, 1, 2, 3) with (t, x, y, z). Similar relations allow us to express a i and g ij as a function of the various components of h µν . Plugging these expressions into (D.2)-(D.4) gets us to our starting point of our Kubo formula analysis. We have revealed the full dependence of the stress tensor on the metric perturbation without having to solve the equation of motion for the fluid velocity and temperature first. Our analysis follows closely the one in [4], significantly shortened by using the results of [1].

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Open Access. This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.