Global anomalies, discrete symmetries and hydrodynamic effective actions

We derive effective actions for parity-violating fluids in both (3 + 1) and (2 + 1) dimensions, including those with anomalies. As a corollary we confirm the most general constitutive relations for such systems derived previously using other methods. We discuss in detail connections between parity-odd transport and underlying discrete symmetries. In (3+1) dimensions we elucidate connections between anomalous transport coefficients and global anomalies, and clarify a previous puzzle concerning transports and local gravitational anomalies.


Introduction
Through studies of free field theories [1][2][3][4], holographic duality [5][6][7], phenomenological arguments based on entropy current [8][9][10][11][12], and equilibrium partition functions [13][14][15], it has been recognized that systems with quantum anomalies exhibit novel transport behavior in the presence of rotation or in a magnetic field (for a recent review see [16]). Effects of anomalies on transport in superfluids, superconductors and topological insulators have also been discussed in [17][18][19][20]. (See also [21][22][23][24][25][26][27][28].) These anomalous transports could be relevant in a wide range of physical contexts: from the study of quark-gluon plasma at subnuclear JHEP01(2019)043 scales [29][30][31][32][33], to cosmology, where the dynamics of primordial magnetic fields plays an important role in the early stage of the universe [34,35], and astrophysical phenomena such as pulsar kicks [36,37]. In addition, there have been various experimental searches for the signatures of anomalies on transports in condensed matter systems, see [38][39][40]. Given their importance, it is of primary interest to incorporate anomalous transports in an effective field theory framework, which is the goal of this paper. Such a formulation has a number of advantages. Firstly, an effective field theory provides a framework where hydrodynamic fluctuations can be systematically incorporated, thus enabling one to search for new physical effects due to fluctuations in parity-violating systems. Secondly, the effective action approach provides a first-principle derivation of the constitutive relations which automatically incorporates all the phenomenological constraints. Indeed our derivation reproduces fully the constitutive relations of previous approaches. It also highlights some new insights which we will discuss momentarily.
Consider a parity-violating relativistic system in (3 + 1)-dimension with a global U(1) symmetry whose conserved current isĴ µ . Suppose the symmetry becomes anomalous in the presence of an external source A µ forĴ µ , where F is the field strength for A. Due to (1.1), the Euclidean partition function of the system in the presence of source A µ is not invariant under small gauge transformations of A. We will refer to (1.1) as a local U(1) anomaly, in contrast to a global anomaly in which case the partition function is invariant under small gauge transformations, but not under large gauge transformations when the system is put on a topologically nontrivial manifold.
To first order in the derivative expansion, the parity-odd part J µ o of charge current can be written in the Landau frame as [8,9,29,30] J µ o = ξ ω ω µ + ξ B B µ . (1. 2) The first term implies a contribution to the current that is induced by and parallel to, the vorticity ω µ ≡ µνλρ u ν ∂ λ u ρ (u µ is the local velocity field). This is called the chiral vortical effect (CVE). The second term is proportional to the magnetic field strength B µ ≡ 1 2 µναβ u ν F αβ , which is often referred to as the chiral magnetic effect (CME). The transport coefficients ξ ω and ξ B receive contributions from local anomaly (1.1) as follows [8,9,13] ξ ω = −3c µ 2 where a 1,2,3 are constants, and µ, T, n 0 , 0 , p 0 are local chemical potential, temperature, charge density, energy density and pressure respectively.

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It is curious that even in the absence of local anomaly (1.1), i.e. with c = 0, there can still be chiral vortical and magnetic effects, determined up to three constants. It has been pointed out that for a CTP invariant theory, only a 2 is allowed [13,18], whose physical origin has generated much recent interest. From holography and free theory examples, a 2 appears to be related to the coefficient λ of the local mixed gravitational anomalies ∇ µĴ µ = λ µνλρ R α βµν R β αλρ (1.6) as [41][42][43][44][45][46][47] a 2 = −32π 2 λ . (1.7) Relation (1.7) is puzzling from the perspective of anomaly matching in a low energy effective theory, as the right hand side of (1.6) contains four derivatives and thus should modify J µ only at the third derivative order while terms in (1.2) have only one derivative. Furthermore, matching with constitutive relations or partition functions as done in [8,9,13,14] will not lead to any multiplicative factor π as in (1.7). Arguments have been made in [48][49][50][51][52] which show that (1.7) should apply at least to field theory systems smoothly connected to free theories through continuous parameter(s). Alternatively, it has been hinted in [53] and subsequently explicitly worked out in various examples in [54,55] that the transport coefficient a 2 should be considered as being directly related to global mixed gravitational anomalies when putting the system on a topologically nontrivial manifold. It has also been known that relation like (1.7) is violated for systems with gravitinos [45,48,50,56].
In this paper we work out effective actions for parity-violating fluids in both (2 + 1) and (3 + 1) dimensions following the approach developed in [57][58][59] (see [60][61][62] for earlier attempts at an effective action for anomalous transports). We assume that at microscopic level the system has an underlying discrete symmetry Θ which includes time reversal. Here Θ can be the time reversal T itself, or any combinations of C, P with T , such as CPT . As a corollary we confirm (1.2)-(1.4) as the most general constitutive relation for a parity-violating system in (3 + 1)-dimensions, and in (2 + 1)-dimension we confirm the constitutive relations obtained earlier in [13,14,63]. In (2 + 1)-dimension the story is much richer, containing six independent functions of local temperature and chemical potential. The rest of the paper is devoted to detailed derivations of the effective actions. Here we highlight a couple of conceptual points related to (1.3)-(1.4). In particular, we offer an interpretation for (1.7) which reconciles various different perspectives. 1 We find: 1. In both (3 + 1) and (2 + 1) dimensions, possible parity-odd transport behavior sensitively depends on the underlying discrete symmetries. Hence hydrodynamic transports can be used to probe microscopic discrete symmetries. For example, given the form (1.2)-(1.4), when PT is conserved, then a 1,2,3 = 0 and c = 0, i.e. no chiral vortical or magnetic effects. If CPT is conserved, then a 1 = a 3 = 0. If only T is conserved, then all a 1,2,3 and c are allowed. Thus detection of possible existence of a 1 , a 3 can be used to test CPT violations.

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While CT P is preserved for all relativistic local field theories, searching for its possible violations through transports could be interesting. Some condensed matter systems exhibit emergent relativistic symmetries, and transport behavior can then be potentially used to probe whether there is emergent CPT as well.
2. All three constants a 1,2,3 in (1.3)-(1.4) are associated with global anomalies, respectively with pure gauge, mixed gauge, and pure gravitational anomalies. More explicitly, consider the partition function of the system on a spatial manifold S 1 × S 2 at a finite temperature, i.e. the full manifold is S 1 T × S 1 × S 2 , with S 1 T denoting the Euclidean time direction along which we put thermal boundary conditions. We also turn on the external metric and source A µ as where both f and g are independent of τ . Equation (1.9) aries from time diffeomorphism along the Euclidean time circle 3 while (1.10) is the stationary gauge transformation for A µ . It turns out, however, when a 1,2,3 are nonzero, the partition function is only invariant under transformations which are smoothly connected to the identity, but not invariant under large gauge transformations.
More explicitly, suppose b i has a magnetic flux along S 2 , then under a large gauge transformation of b i and v i along S 1 we find that the partition function transforms as where q is the minimal U(1) charge of the system. The term proportional to a 2 in (1.11) is fully consistent with the discussion of various examples in [54,55]. In (1.11) the term in the exponent proportional to a 1 is real; recall that the presence of a 1 breaks CPT . Similarly when only v i has a magnetic flux along S 2 , under a large gauge transformation of v i along S 1 we find that which is again real. The standard lore is that there can be no pure global gravitational anomaly in d = 4. But here CPT is broken and we are at a finite temperature.

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We thus see measuring parity-violating transports can also be used to probe global anomalies of a system. Note that a 2 appears in (1.11) in a phase, so the global anomaly (1.11) only captures the "fractional" part of a 2 , i.e. a 2 → a 2 + kq with k ∈ Z does not change the phase. In contrast, the factors associated with a 1 and a 3 in (1.11)-(1.12) are real. As a result the global anomalies associated with them are fully equivalent to the corresponding transport coefficients.
The relations between coefficients a 1,2,3 and global anomalies described above are universal relations which can be deduced solely at the level of low energy effective theory, without any knowledge of UV physics. Now let us come back to the relation (1.7) which from the light of the above discussion may be interpreted as the combination of the following: (a) the connection between a 2 -related transports in (1.3)-(1.4) to global gravitational anomaly (1.11) which is a universal low energy relation; (b) a relation between local mixed anomaly coefficient λ in (1.6) and the global mixed anomaly (1.11) which has been known to be valid for some class of systems. This relation goes beyond low energy physics.
This resolves the two puzzles mentioned below (1.7): equation (1.7) should not be viewed as a low energy relation. Indeed, from the perspective of low energy effective field theory, neither transport behavior in (1.3)-(1.4) nor the global anomaly in (1.11) has anything to do with (1.6). Nevertheless, when UV physics is taken into consideration, they are controlled by the same number in a large class of systems. In this light the discussion of [48][49][50][51] can be considered as establishing (b) for field theory systems smoothly connected to free theories through continuous parameter(s).
The plan of the paper is as follows. In section 2 we briefly review the formalism of [57][58][59] to set up the notations and the rules for derivations of later sections. In section 3 we obtain the effective action of a parity-violating fluid in (3 + 1)-dimension. In section 4 we discuss the connection between the effective action and thermal partition function, and connection with global anomalies. In section 5 we discuss the entropy current for (3 + 1)systems. In section 6 we repeat the analysis for (2+1)-dimensional parity-violating systems, obtaining the effective action, partition function and the entropy current. We have also included a number of appendices for technical details.

Review of hydrodynamical action in physical spacetime
In this section, we review the formulation of the hydrodynamical action introduced in [57][58][59] to set up the notations and formalism for deriving anomalous transports in later sections. 4

General setup
Consider the closed time path (CTP) generating functional W [g 1 , A 1 ; g 2 , A 2 ] for a system with a U(1) symmetry in some state specified by the density matrix ρ 0 where U(t 2 , t 1 ; g 1µν , A 1µ ) denotes the quantum evolution operator of the system from t 1 to t 2 in the presence of spacetime metric g 1µν and an external vector field A 1µ (sources for the U(1) current). The sources for two legs of the CTP contour are taken to be independent. We introduce the "on-shell" stress tensors and currents for each leg as 3) The expectation values T µν , J µ of the stress tensor and the U(1) current in the state ρ 0 in an external metric g µν and external background A µ are obtained by where g,A denotes setting g 1µν = g 2µν = g µν and A 1µ = A 2µ = A µ . In the absence of any gravitational and U(1) anomalies, W [g 1 , A 1 ; g 2 , A 2 ] should be invariant under independent gauge transformations of A 1 , A 2 and independent diffeomorphisms of g 1 A 1 and g 2 , A 2 , i.e.
where g ξ , A ξ denote diffeomorphisms of g, A generated by a vector field ξ µ . 5 Equations (2.5)-(2.6) in turn ensure that where ∇ 1 is the covariant derivative associated with g 1µν , and F 1µν is the field strength of A 1µ . Similarly for quantities with subscript 2. For slowly varying sources, we can express the generating functional (2.1) in terms of path integrals over slow degrees of freedom of the system where χ collectively denotes slow variables of the system which in general also come in two copies. The low energy effective action I EFT depends on ρ 0 and external sources which we have suppressed, and is assumed to be local.

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Fluid spacetime Physical spacetime Physical spacetime Figure 1. X µ 1,2 (σ A ) describe motion of a continuum of fluid elements labelled by σ A in two copies of physical spacetimes with coordinates X µ 1,2 respectively. σ A = (σ 0 , σ i ) are coordinates for a "fluid spacetime", where σ i are interpreted as labels of each fluid element while σ 0 is their "internal" time. The red straight line in the fluid spacetime with constant σ i is mapped by X µ 1,2 (σ 0 , σ i ) to physical spacetime trajectories (also in red) of the corresponding fluid element.
For ρ 0 describing a medium in local equilibrium, generically the only slow modes are those associated with conserved quantities (2.7), i.e. hydrodynamical modes, with I EFT the corresponding hydrodynamical action I hydro . We will limit ourselves to the generic situation. 6 The slow variables associated with the stress tensor can be chosen to be X µ 1,2 (σ A ) which describe motions of a continuum of fluid elements labelled by σ A in two copies of physical spacetimes with coordinates X µ 1,2 respectively. See figure 1. The slow variables associated with the U(1) currents are ϕ 1,2 (σ A ) which can be interpreted as U(1) phase rotations associated for each fluid elements. It is also convenient to introduce an additional scalar field β(σ A ) which gives the local inverse temperature in fluid spacetime. 7 X µ 1,2 and ϕ 1,2 are the Stuckelberg fields for diffeomorphisms and gauge transformations (2.5)-(2.6), and we require the hydrodynamical action I hydro to be a local action of pullbacks of g sµν and B sµ = A sµ + ∂ µ ϕ s , s = 1, 2 to the fluid spacetime i.e. (2.10) By construction h 1,2 and B 1,2 are invariant under independent diffeomorphisms and gauge transformations of the two legs of the CTP contour (s = 1, 2): which along with (2.10) immediately implies (2.5)-(2.6). Furthermore, the form of the action (2.10) implies that the equations of motion of X µ 1,2 and ϕ 1,2 are equivalent to the 6 The discussion can be readily generalized to systems such as near a critical point where one should also include the corresponding order parameter(s). See [59,76]. 7 Note that there is only one temperature field rather than two copies.

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conservations of the "off-shell" hydrodynamical stress tensors and currents defined as (2.14) As defined the path integrals (2.8) apply to a general quantum system. At sufficiently high temperatures it is often enough to consider the leading order in a small expansion. For this purpose we decompose 16) and the action I hydro can be expanded in as (2.17) In this limit the path integrals (2.8) survive and describe classical statistical averages. We will refer to variables with subscript a as a-variables and those without as r-variables. rvariables can be considered as describing physical quantities while a-variables correspond to noises. For example, X µ (σ A ) is interpreted as mapping fluid spacetime into the physical spacetime (now only one copy) with X µ a interpreted as the corresponding position noises. While the hydrodynamical action I hydro is naturally formulated in the fluid spacetime σ A , one can also formulate it in physical spacetime by inverting X µ (σ A ), i.e. use σ A (X) as dynamical variables and express all other variables accordingly as functions of X µ . In the physical spacetime formulation, the dynamical variables are then σ A (x), ϕ(x), β(x) and X µ a (x), ϕ a (x), while the background fields are g µν (x), A µ (x), g aµν (x), A aµ (x), where we have replaced X µ by x µ to emphasize they are now just coordinates for physical spacetime. The physical spacetime formulation has the advantage of being more physically intuitive and connects more directly with the traditional phenomenological approach.

Formulation of I hydro in physical spacetime
We now list various symmetries and consistency requirements which I hydro should satisfy when formulated in the physical spacetime to leading order in the -expansion [57][58][59]. They can be separated into the following categories: 1. Spacetime diffeomorphisms and gauge transformations. In the absence of any gravitational and charged current anomalies, the action I hydro should be invariant under physical spacetime version of (2.11)-(2.12). Invariance under these transformations implies that a-fields (including both background and dynamical variables) must appear through the combinations

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while A µ and ϕ must appear through The above variables are the physical spacetime version of (2.9).
2. Spatial and time diffeomorphisms in the fluid spacetime which define a fluid. We require the action I hydro be invariant under Furthermore we require the action be invariant under the diagonal shift where λ is a function of σ i only. Invariance under (2.23) defines a normal fluid. For a superfluid where the U(1) symmetry is spontaneously broken this symmetry should be dropped. The symmetries (2.21)-(2.23) involve only dynamical variables, yet they should be viewed as "global gauge symmetries," i.e. configurations related by such transformations are deemed physically equivalent.
Invariance under (2.21)-(2.23) implies that the only invariant which can be con- are invariant. To summarize, the only combinations of r-variables which can appear are It is often convenient to combine the first three variables further into where β µ is now unconstrained.
3. Classical remnants of constraints from quantum unitarity of (2.1), where Λ r,a collectively denote all r-and a-variables including both dynamical and background fields.

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4. Discrete spacetime symmetries. If the microscopic system is invariant under charge conjugation C, parity P or CP, such discrete symmetries should be imposed on I hydro and they can be imposed straightforwardly as usual.
5. We assume the microscopic Hamiltonian underlying the macroscopic many-body state ρ 0 is invariant under a discrete symmetry Θ containing time reversal. Θ can be time reversal T itself, or any combinations of C, P with T , such as CPT . Θ can also be a combination of T with some other internal discrete operations. Unlike C or P, Θ by itself can not be imposed directly on I hydro , since Θ does not take the generating functional W to itself, but to a time reversed generating functional W T . 8 The fact that the underlying Hamiltonian is invariant under Θ nevertheless leads to important constraints on I hydro as we will discuss in the next item.
6. We require I hydro to be invariant under a Z 2 dynamical KMS symmetrỹ where tilde denotes a Z 2 transformation which is a combination of Θ and the Kubo-Martin-Schwinger (KMS) transformation. 9 Equation (2.31) plays the dual role of imposing microscopic time-reversibility and local equilibrium. It should be understood as a mathematical characterization of a state ρ 0 in local equilibrium. The prototype of such a state is the thermal density matrix in slowly varying external sources, but (2.31) is more general, applicable also to pure states. It was found in [57][58][59] that (2.31) leads to Onsager relations, local first law, local second law, and local fluctuation-dissipation relations.
To leading order in , the tilde operation in (2.31) can be written schematically as where Φ r denotes certain combination of r-variables with total one derivative. More explicitly, in (2.33) we denoted Θ transformation of a tensor G(x) as where we have suppressed tensor indices for G, and η G should be understood as a collection of phases (±1) one for each component for G. Similarly for ηx. For example, for Θ = T and G = A µ The second set of equations in (2.33) for a-variables can be written explicitly as The explicit transformations for Θ = T , PT , CPT for various tensors are given in appendix A.
It is straightforward to write down the most general I hydro = d d x √ −g L consistent with the above prescriptions. We can expand the corresponding Lagrangian density L in terms of the number of a-variables and derivatives. The first few terms in the a-field expansion can be written schematically as where we have introduced notation andT µM , W µν,M N , · · · are covariant tensors constructed out of r-variables {β µ ,μ, F µν , ∆ µν } and covariant derivatives on G aµM . Given that G aµν = g aµν + · · · and C aµ = A aµ + · · · , we identifyT µν andĴ µ as the "off-shell" hydrodynamic stress tensor and U(1) current, and the equations of motion of X µ a , ϕ a give the standard hydrodynamic equations.
If we introduce n as the sum of the number of a-fields and the number of derivatives in a term, then since Φ r in (2.33) contains one derivative, the dynamical KMS transformation (2.31) preserves n, which implies that terms in the action which have the same value of n transform separately among themselves. We can thus write the action as where L n contains all terms with given n. They are separately invariant under (2.31). L 1 contains only zeroth derivative term inT µM while L 2 contains first derivative terms in T µM and zeroth derivative terms in W µν,M N . The explicit expressions for (2.41) to order L 2 for a parity-preserving fluid are given in [58]. 10 10 They are given to order L3 for conformal fluids.

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We now give a brief review of the derivation of the entropy current, whose details are given in [59]. Dynamical KMS invariance (2.31) implies that , and V µ k contains k factors of a-fields. The entropy current can then be defined as whereV µ 1 is V µ 1 with Λ a replaced by the corresponding Φ r as introduced in (2.33). It can be shown upon using equations of motion where R is a local non-negative expression.

Effective action for parity-violating systems in (3+1)-dimension
In this section we apply the formalism reviewed in the previous section to four-dimensional systems which break parity, including those with a local U(1) anomaly where constant c depends on specific systems. µνλρ is the fully antisymmetric tensor with 0123 = 1 √ −g . In (3.1) we have made manifest -dependence so as to be clear about the order in -expansion at which the corresponding anomalous transports appear in the hydrodynamical action. We assume that the system does not have any local mixed gravitational anomalies. We will see that the system can nevertheless possess global gravitational anomalies which are closely connected to certain novel transports.

Generating functional
From (3.1), under independent local transformations of A 1,2 , equation (2.5) should be replaced by 11 while (2.6) remains. Note that F ≡ 1 2 F µν dx µ ∧ dx ν = dA, and the second term on the right hand side is independent of metrics. Indeed, from (3.2) the consistent currents introduced in (2.2)-(2.3) now satisfy 12

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and from diffeomorphism invariance of W we also have we can write equations (3.3) and (3.4) as Note that the equation for T µν must be expressible in terms of covariant current J µ as T µν should be gauge invariant (the last term in (3.2) is independent of the metric). To leading order in -expansion, the anomalous piece in (3.2) becomes (see (2.15)-(2.16) and

Parity odd action
We now construct the hydrodynamic action for a parity-violating system with a local U(1) anomaly. We can write the action as I hydro = I even + I odd (3.9) where I even and I odd are parity even and odd parts respectively. I odd can be further decomposed as I odd = I o,inv + I anom (3.10) where I anom is responsible for generating the anomalous term in (3.2), and I o,inv is invariant under gauge transformations. Given that I even is invariant under gauge transformations we can also write I hydro = I inv + I anom , I inv = I o,inv + I even . (3.11) Note that I inv should depend on ϕ 1,2 only through B 1,2 introduced in (2.9), while I anom does not have to.
Since neither the diagonal shift (2.23) nor the dynamical KMS transformations (2.33) mix parity even and odd parts, I even and I odd can be treated independently. I even was discussed in detail in [57,58]. Here we focus on (3.12) and will construct L odd to order L 2 as defined in (2.43).

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Let us first look at I anom . To match with the anomalous term in (3.2), we take the anomalous action as (written in fluid spacetime) where X µ 1,2 are functions of σ A , F 1AB is the pull-back of F 1µν . Note that under gauge transformations (2.12) we precisely recover (3.2) from (3.13). To see this, for two terms in (3.13) one changes the integration variables to X 1 and X 2 respectively, which then become dummy variables.
Given (3.13) and that I inv depends only on B 1,2 , the equations of motion of ϕ s and X µ s lead to 14) where the off-shell stress tensors and consistent currents are defined in (2.13)-(2.14). Again we have suppressed s = 1, 2 and each equation should be understood to have two copies.
Defining the covariant off-shell currents aŝ whereĴ µ inv is defined as the off-shell currents corresponding to I inv , we then have The last equality of (3.16) makes it manifest thatĴ µ is invariant under gauge transformations.
Expanding in small and rewriting the resulting expressions in physical spacetime we find that (3.13) becomes

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and the terms on the right hand side may be further expanded in and derivatives. Let us first considerT µν o which as usual can be decomposed aŝ where q ν o and Σ µν o are transverse to u µ . Since terms proportional to G aµν will never generate a term of the form (3.21) under (2.23), T µν o should be diagonal shift invariant by itself. At zeroth derivative order there is no such term. At first derivative order the only non-vanishing quantity is q ν o which can be written as where As discussed above to first derivative order since there is no diagonal shift invariant scalar termĵ µ o should then be transverse and can be written aŝ where h 1 , h 2 are some functions of β(x) andμ(x). Now let us consider quadratic terms in a-fields (order O(a 2 )) to zeroth order in derivative, which should have the form where W µν,M N o is parity odd and is diagonal shift invariant. Such a term does not exist at zero derivative order so we conclude there are no new parity-odd terms at order O(a 2 ).
Collecting the above expressions, L odd can be written as where F aλρ is defined by (3.20), and q µ o ,ĵ µ o are given respectively by (3.24), (3.28). Using field redefinitions one can write L odd in the Laudau frame (see section VI of [58] for details) where 0 , p 0 , n 0 are respectively zeroth order energy, pressure and charge densities.

Dynamical KMS condition
We now impose the dynamical KMS condition (2.31) on the parity-odd action (3.10). We will consider respectively Θ = PT , T , CPT and will see that they lead to very different results.
Due to the presence of on the right hand side of (3.3), 1 I anom is of order O( ). In 1 I o,inv the first term in (3.26) is O( ) while g 1 , g 2 , h 1 , h 2 are undetermined at the moment.
We will later argue that they should also be O( ). Thus in our discussion below it is enough to consider the leading order terms in dynamical KMS transformations (2.33). 13

Θ = PT
We find in this case KMS invariance at O(a) then requires

From (3.19) we find that under dynamical KMS transformation, the anomalous action becomes (see appendix B for useful formulae)
We then find thatĨ For I odd to be invariant, we need the second term of (3.38) to be a total derivative. More explicitly, using (3.24)-(3.28), we find after some algebraic manipulations (see appendix B for useful formulae) In fact one can check that the structure of O( ) corrections in (2.33) are such that even if g1, g2, h1, h2 are of order 0 , at first derivative order for O(a) terms the leading terms in (2.33) are adequate.

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where u ≡ u µ dx µ . For the above expression to be a total derivative we find that h 1 , h 2 , g 1 , g 2 must arise from derivatives of two functions H 1 , H 2 and satisfy the following relations where a 1 is a constant. Note that one could add a constant to the right hand side of equation 2H Note that Q is defined only up to a closed three-form as such an addition will not change (3.42). The most general solutions to (3.40)-(3.41) can be written as where a 1 , a 2 , a 3 are constants. Thus to first derivative order I odd is fully determined up to three constants.

Θ = CPT
The analysis for Θ = CPT is very similar. Note that and equation (3.36) again applies. For I o,inv , we now havẽ

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The analysis for O(a 0 ) terms is the same as before and (3.40)-(3.43) apply. Imposing (3.50) on the solutions (3.44)-(3.46) we find that a 1 = a 3 = 0, and thus Thus for a macroscopic system whose underlying Hamiltonian is invariant under CPT to first derivative order I odd is fully determined up to a single constant.

Explicit expressions for q µ o andĵ µ o
We can now write down the explicit expressions for q µ o andĵ µ o to be used in (3.30) or (3.31). It is enough to do it for Θ = T . The expressions for Θ = CPT can be obtained by setting a 1 = a 3 = 0, while those for Θ = PT can be obtained by setting a 1 = a 2 = a 3 to zero.
From (3.45)-(3.46) we find that The frame independent quantity µ o (3.32) is then given by where we have introduced α ≡ n 0μ β( 0 + p 0 ) (3.56) and Equations (3.55)-(3.57) reproduce previous results in the literature obtained from entropy current [8,9] and equilibrium partition function [13], confirming that these methods indeed give the complete answer for the current problem. However, those methods did not pinpoint the exact discrete symmetry a system should have for (3.55)-(3.57). Ref. [13] did point out for CPT invariant theories one should set a 1 = a 3 = 0.

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We presented our results in terms of ω µ , B µ which were defined in (3.25) from respective "field strengths" of u µ and B µ . But note that u µ B µ = 0. We now present (3.53)-(3.55) in a slightly different basis which makes their expressions a bit more transparent. Introduce and Note that v µ b µ = 0, and Similarly Q of (3.43) can be written more transparently in the basis of (3.58) as where we have dropped an exact three-form as mentioned earlier Q is defined only up to a closed three-form.

Equilibrium partition function and global gravitational anomalies
In this section we first explain how to obtain the equilibrium partition function from the hydrodynamical effective action. We discuss two different ways of doing it. We then apply the procedures to I odd found in the last section to obtain the parity-odd part of the equilibrium partition function. We will see that in the absence local anomalies, i.e. c = 0, all the parity-odd transport terms are connected to global anomalies. When the underlying theory is only invariant under T , terms proportional to a 1 , a 2 , a 3 in (3.53)-(3.57) are respectively associated with global U(1), mixed gravitational, and gravitational anomalies. With CPT invaraince, only a global mixed gravitational anomaly is present. This connection also implies that a 1,2,3 should be proportional to .

Equilibrium partition function from effective action
We will now describe two methods of obtaining the equilibrium partition function from the effective action when ρ 0 in (2.1) is given by the thermal density matrix with an inverse temperature β 0 . By definition the generating functional W of (2.1) becomes identically zero when we set the external fields for the two legs to be the same. Nevertheless, as already indicated in [57][58][59], the equilibrium partition function can be extracted from the effective action with the help of the dynamical KMS condition. We will again work to leading order in small expansion. For notational simplicity we will now denote the sources collectively by φ i and their corresponding operators O i with index i labelling different operators/components. In [57] it was shown that a generating functional W satisfying the combined Θ and KMS transformation (2.32) can be "factorized" in the stationary limit. That is, when the sources φ 1i , φ 2i are time independent, to leading order in the a-field expansion we can write W as where · · · denotes terms of order O(a 2 ),W [φ i ( x)] is a functional defined on the spatial manifold of the spacetime, and satisfies where Θ here should be understood as the extension of (2.34) to time-independent field configurations. Equation (4.1) implies that Writing the equilibrium partition function Z as where F is the free energy, and doing analytic continuation ofW to Euclidean signature, 14 from (4.3) we can identify −W with β 0 F . The free energy F (and thusW ) should have a local expansion in terms of external sources, as the equilibrium partition function can be computed by putting the system on a Euclidean manifold with a periodic time circle, which generates a finite gap. As discussed in [57] we can obtainW from the contact terms in I hydro as follows. One first obtains the source action I s by setting the dynamical fields in I hydro to the following equilibrium values (4.6)

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All external fields are taken to be time independent. Then to leading order in the a-field expansion where · · · denotes terms of order O(a 2 ). That I s is factorizable at this order is warranted by the dynamical KMS condition. 15 There is also an alternative way to obtain the equilibrium free energy as follows. The dynamical KMS condition (2.31) implies that whereL is defined asL = L[ΘΛ a , ΘΛ r ] (see (2.33)). V µ can be further expanded in terms of a-fields as V µ = iV µ 0 + · · · (4.9) where V µ 0 contains r-fields only. From the discussion of the entropy current in [59], we can then identify 16 where V 0 0 eq denotes the expression obtained by setting dynamical fields in V 0 0 to equilibrium values (4.5).
The equivalence of the two methods can be considered as a consequence of equivalence of local KMS condition of [57] and the dynamical KMS condition (2.31) as shown in [58]. One can readily check that applied to the parity even part of the effective action I even the two methods indeed give the same answers and are equivalent to the results discussed in [13,14].

Parity-odd equilibrium partition function and global anomalies
We now obtain the parity-odd partition function from I odd following the procedures discussed in the previous subsection. It can be readily checked that the two approaches give the same answers. The second approach is significantly simpler technically, which we will describe here. Recall that from our analysis for Θ = PT there is no parity-odd contribution to the partition function at first derivative order. The results below are for Θ = T ; to obtain Θ = CPT one needs to take a 1,3 = 0 together with (3.50).
From (3.36), (3.43), and (4.10) we immediately obtain that where the integration is over the spatial manifold with A ≡ A i dx i , u ≡ u i dx i . Using the basis of (3.65), equation (4.11) can be written more transparently as In [57] the KMS condition on I hydro was imposed by requiring Is to satisfy the combination of Θ and KMS, dubbed the local KMS condition there. In [58] it was shown the dynamical KMS (2.31) and local KMS conditions are equivalent. 16 See equation (3.14) there. The second termV 0 1 vanishes in the stationary limit.
Let us now explore a bit further physical implications of (4.12). The background fields in (4.12) are those for a stationary Lorentzian manifold with and g 00 < 0. Note that (4.14) is preserved by time reparameterizations and time-independent U(1) transformations Below we will refer to (4.15) as time U (1) and (4.16) as flavor U(1). The thermal partition function is usually calculated by analytically continuing to Euclidean signature with t → −iτ (with τ on a circle with period β 0 ), and the background fields are taken so that they are real in Euclidean signature. We take the Euclidean metric and gauge field to be of form Here, g 00 > 0. Thus, under the analytic continutaion t → −iτ , we get the replacements after which (4.12) becomes Note that CPT invariant terms become pure imaginary while the terms proportional to a 1 and a 3 remain real. Now let us consider a system with no local anomalies, i.e. c = 0. Then in (4.20) we have three Chern-Simons terms, respectively, for flavor U(1), mixed time and flavor U(1), and time U(1). A defining feature of Chern-Simons terms is that they are not invariant under "large" gauge transformations i.e. those are not connected to the identity. Consider for example the flavor U(1) Chern-Simons term (4.21) Let us take the spatial manifold to have the topology of S 1 × S 2 , where S 1 has size L. We can choose b to have a monopole configuration on S 2 , i.e.

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where q is the minimal charge under U(1). A large gauge transformation of b x (x is the circle direction) is we then have [77,78] Z → e Under Kaluza-Klein reduction, v couples to matter as a U(1) gauge field with minimal "charge" 2π q a 2 Z . Note in (4.24) and (4.28) the partition function transforms by a real number rather than a phase. As mentioned earlier non-vanishing a 1 or a 3 breaks CPT . Thus we find in the absence of local anomaly, all the anomalous transports are associated with global gauge or gravitational anomalies for putting the system on a Euclidean four-manifold with a thermal time circle.
In the presence of a local anomaly, i.e. c = 0, then the transport coefficients in (3.61)-(3.63) are then mixed among local and global anomalies. The same thing happens to the partition function. But note that a 1 , a 3 are terms, being real, are not mixed with local anomalies.
Possible connections of the term proportional to a 2 with mixed global gravitational anomaly was first hinted in [53] and shown explicitly in [54,55] in some free theory models.

Entropy current
In this section we obtain the entropy current for a (3 + 1)-dimensional parity-violating fluid by applying (2.45). One thing to notice is that the anomalous action (3.19) does not have the same structure of the rest of the action. At O(a) the latter has the form (now also including the parity-even part, see (3.11))

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which is the form assumed in [59]. The fact that I anom has a different structure does not cause a problem, as I anom is KMS invariant by itself. We can then simply apply the procedure of (2.45) to I inv which will generate an entropy current with non-negative divergence. Now applying (2.45) we find that and with R even to be divergence of the entropy current of the parity-even part. 17 Equation (5.3) means that parity-odd part does not contribute to entropy dissipation. From (3.42)-(3.43), for the parity-odd part, V µ 0 is simply the dual of Q, giving the following odd-parity contribution to the entropy where we have dropped a term which is dual to an exact 3-form. Note that this expression is independent of c. The entropy current in the Landau frame is then given by The parts of the expression which involve the anomaly coefficient agree with the Landau frame entropy current given in [8] when a 1 = a 2 = a 3 = 0. Furthermore, there is also agreement with [9] when a 1 = 0. After dropping duals of exact three forms, the vector above can be written in the new basis introduced here as 6 Parity-violating action in 2 + 1-dimension Let us now consider the action for parity-violating terms in 2+1-dimension. The procedures are exactly parallel to those of the 3 + 1-dimensional story. So we will be brief, only giving JHEP01(2019)043 the main results. We will again work to the level of L 2 as defined in (2.43). The results below are fully consistent with the constitutive relations presented in [63] from entropy current analysis and those presented in [13,14] where q ν o , j µ o and Σ µν o are transverse to u µ . For this purpose let us list all the parity-odd scalars, vectors, and tensors which are diagonal shift invariant at first derivative order 18 scalars : We can then expand various quantities in (6.2)-(6.3) as where all coefficients g 1,2 , h 1,2 , f 1,2 , k 1,2,3,4 , l 1,2,3,4 , η o are functions of β,μ.

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Non-negativity of the imaginary part of the action, eq. (2.29), leads to various constraints among the coefficients of L (2) . The constraints on the parity even part (6.11)-(6.13) were analyzed in detail in [57]. Among other constraints we have When the parity-odd coefficient r is nonzero, the second inequality of the above becomes To summarize, to level L 2 the parity-odd action can be written as (6.18) Using field redefinitions one can write L odd as (see section VI of [58] for details) with frame independent quantities θ o , µ o defined by where 0 , p 0 , n 0 are respectively zeroth order energy, pressure and charge densities. Note that the coefficient r can be defined away using field redefinitions, so (6.17) does not lead to new constraints on transport coefficients. The outcome of the dynamical KMS condition (2.31) again depends very much on the choice Θ, which we will discuss separately.

Θ = T
In this case, we find all coefficients in (6.9)-(6.10) are zero, except for k 2 and l 1 which satisfy the relation The full parity-odd action to level L 2 can then be written as The above Lagrangian satisfiesL = L (6.23) which can be seen by noting the relation Due to (6.23), there is no parity-odd contribution to the thermal partition function to first derivative order. The entropy current is given by

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where p, T µν , J µ also include the parity-even part, and where R even is the parity-even expression. Note that the second term in the right hand side of (6.26) vanishes by ideal fluid equation of motion The dynamical KMS condition implies that the coefficients in (6.9)-(6.10) should satisfy The first equation of (6.30) implies that there exists a function Y such that while the second equation of (6.30) can be further written as which upon using (6.31) implies that there exists a function X such that Applying the above relations to (6.9)-(6.10) we then havê It can be checked that the above expressions satisfŷ which givesL odd − L odd = i∇ µ V µ 0 . (6.39)

B Some useful formulae
In this appendix we give some useful formulae used in deriving equations such as (3.36) and (3.39).

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which can be written in differential forms as where ξ is a vector field, F, G are two-forms, and V is a one-form. As an example, given u ≡ u µ dx µ , w = du, and β µ = βu µ , we then have It is also useful to recall that for a differential form λ and a vector field ξ d(ξ · λ) = L ξ λ − ξ · dλ . (B.4) It then follows that for some vector v µ which can be used to derive (3.36). To see (3.39), we note that: or in differential form where W is a one-form and G a two-form. Here are two examples: − µνλ u ν ∂u λ = µνλ u ν w λρ u ρ = 1 2 ∆ µ ν νλρ w λρ (B.12) where w = du.

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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.