Higher-form symmetries and 't Hooft anomalies in non-equilibrium systems
aa r X i v : . [ h e p - t h ] J a n Higher-form symmetries and ’t Hooftanomalies in non-equilibrium systems
Michael J. Landry
Department of Physics, Center for Theoretical Physics,Columbia University, 538W 120th Street, New York, NY, 10027, USA
Abstract
We investigate the role of higher-form symmetries and two kinds of ’t Hooft anomalies in non-equilibrium systems. To aid our investigation, we extend the coset construction to account for p -form symmetries at zero and finite temperature. One kind of anomaly arises when a p -formsymmetry is spontaneously broken: in a d + 1 -dimensional spacetime there often exists an emergent d − p − -form symmetry with mixed ’t Hooft anomaly. That is, the p -form and d − p − -formsymmetries cannot be gauged simultaneously. At the level of the coset construction, this mixedanomaly prevents the Goldstones for the p - and d − p − -form symmetries from appearing in the sameMaurer-Cartan form. As a result, whenever such a mixed anomaly exists, we find the emergenceof dual theories—one involving the p -form Goldstone and the other involving the d − p − -formGoldstone—that are related to each other by a kind of Legendre transform. Such an anomalycan exist at zero and finite temperature. The other kind of ’t Hooft anomaly can only arise innon-equilibrium systems; we therefore term it the non-equilibrium ’t Hoof anomaly . In this case,an exact symmetry of the non-equilibrium effective action fails to have a non-trivial, conservedNoether current. This anomalous behavior arises when a global symmetry cannot be gauged inthe non-equilibrium effective action and can arise in both open and closed systems. We constructactions for a number of systems including chemically reacting fluids, Yang-Mills theory, Chern-Simons theory, magnetohydrodynamic systems, and dual superfluid and solid theories. Finally, wefind that the interplay of these two kinds of anomalies has a surprising result: in non-equilibriumsystems, whether or not a symmetry appears spontaneously broken can depend on the time-scaleover which the system is observed. ONTENTS
I. Introduction 3II. Higher-form symmetries: a review 5A. Superfluids at zero temperature 6B. Electromagnetism 8III. Non-equilibrium EFT: a review 8A. The basics 8B. Non-equilibrium coset construction 10IV. Non-equilibrium ’t Hooft anomalies 12A. Reactive fluids 13V. Higher-form symmetries and the coset construction 16A. The zero-temperature limit 18B. Gauge theories 191. Yang-Mills 192. Chern Simons 203. Magnetohydrodynamics 21VI. Higher-form symmetries and ’t Hooft anomalies 23A. Gauge fields and the Maurer-Cartan form 23B. Dual theories and mixed ’t Hooft anomalies 25C. Non-equilibrium higher-form ’t Hooft anomalies 26D. Quadratic examples 271. Hot Maxwell theory 282. Magnetohydrodynamics I 293. Magnetohydrodynamics II 304. Gapped diffusion 315. Legendre transform for unbroken symmetries 32VII. Dual superfluids 34A. Zero temperature 34B. Finite temperature 36VIII. Dual solids 37A. Zero temperature 37B. Finite temperature 38IX. Discussion 40References 422 . INTRODUCTION
The physics of many-body systems out of finite-temperature equilibrium is a broad areaof study with applications to almost all aspects of physics. Such systems typically resist anyattempt at an exact description; they are simply too complicated and chaotic. If, however,we are interested only in the extreme infrared (IR) then we can use various approximationschemes that make the descriptions of non-equilibrium many-body systems tractable. Inparticular, it is often possible to construct an effective field theory (EFT) consisting of asmall number of IR degrees of freedom. The primary principle guiding the construction ofsuch EFTs is symmetry; in particular, once the field content is specified, an effective actioncan be constructed by writing down a linear combination of all symmetry-invariant terms ata given order in a derivative expansion [1–5]. In fact, Landau’s classification defines statesof matter according to their spontaneous symmetry breaking (SSB) pattern.Ordinary actions necessarily describe conservative systems, meaning they cannot ade-quately describe finite-temperature systems, which are inherently dissipative. However, re-cent work has extended the action principle by employing the in-in formalism, which enablesthe construction of a non-equilibrium effective action defined on the Schwinger-Kedysh (SK)contour [6–32]. Such non-equilibrium effective actions account for dissipation and statisticalfluctuations arising from thermal and quantum effects. The price we pay for dissipation isa doubling of the field content and an action with complex-valued coefficients.While ordinary symmetries provide a powerful guiding principle for constructing EFTs,they are not the whole story. This is particularly true for non-equilibrium systems. In thispaper we focus on two main extensions of ordinary symmetry:• There are many phases of matter that are characterized by topological features, knownas topological phases of matter . It was long believed that these topological phases couldnot be classified according to symmetry principles alone; however, recent work has chal-lenged this idea [33]. In particular, it is suggested that a kind of generalized symmetryprinciple—known as higher-form symmetries —can be employed to describe topologicalfeatures. In this way, topological properties of many-body systems can be classifiedaccording to Landau’s approach. The Noether charges corresponding to these higher-form symmetries describe the conservation of higher-dimensional extended objects.For example a p -form charge counts the number of charged p -dimensional objects.Such higher-form symmetries arise in many areas including gauge theories, magneto-hydrodynamics, dual theories of superfluids and solids, and many others [33–44]. Infact, a common feature of systems with spontaneously broken U (1) symmetries—likesuperfluids and solids [32]—is the existence of an emergent higher-form symmetry withmixed ’t Hooft anomaly [45]. Further, when a higher-form symmetry is spontaneouslybroken, a generalized Goldstone theorem guarantees the existence of a gapless mode.• The role of symmetries is somewhat different for non-equilibrium systems than forsystems at zero-temperature that are described by ordinary actions. In particular, forordinary actions, Noether’s theorem furnishes a one-to-one correspondence betweenphysical symmetries and conserved quantities. In non-equilibrium effective actions,3owever, no such one-to-one correspondence is guaranteed. There can be exact sym-metries of the action that have no corresponding conserved charge. We will see thatthese situations arise when there is a particular kind of obstruction to introducinggauge fields for a given symmetry. We therefore term this kind of non-conservation a non-equilibrium ’t Hooft anomaly. It is the aim of this paper to systematically investigate how higher-form symmetriesand non-equilibrium ’t Hooft anomalies can be used to construct non-equilibrium effectiveactions. To aid in the systematization of our approach, we employ the coset construction,which is a powerful tool that facilitates the formulation of effective actions for Goldstoneand Goldstone-like excitations [32, 46–51]. The basic idea behind the coset construction isthat the Maurer-Cartan one-form, parameterized by Goldstone fields, furnishes symmetry-invariant building-blocks, which are then used to formulate an effective action. We find thatfor spontaneously broken p -form symmetries, the corresponding equivalent of the Maurer-Cartan form is no longer a one-form but is instead a p +1 -form. When an ordinary or higher-form symmetry is spontaneously broken, a (generalized) Goldstone theorem guarantees theexistence of a gapless mode [34]. For non-equilibrium systems, even unbroken symmetriescan have corresponding gapless modes that resemble Goldstone excitations. At the level ofthe coset construction, these modes corresponding to unbroken symmetries can be accountedfor by introducing certain gauge-redundancies [32]. We extend these gauge-redundancies toaccount for gapless modes corresponding to unbroken p -form symmetries. Further, we findthat any action constructed exclusively using terms furnished by the Maurer-Cartan formwill have a one-to-one correspondence between physical symmetries and conserved quantities.In order to destroy this one-to-one correspondence, we find that there is a Maurer-Cartan-like object that can be used to furnish terms that lead to non-equilibrium anomalies of theeffective action.To build the readers’s intuition for higher-form symmetries and ’t Hooft anomalies, weinvestigate the non-equilibrium properties of finite-temperature electromagnetism in vari-ous settings at the level of quadratic effective actions. We use these various examples toillustrate key features of non-equilibrium systems: (1) whether or not a symmetry appearsspontaneously broken depends on the time-scale over which the system is observed, (2) thecurrents associated with p - and d − p − -form symmetries with mixed anomaly have a defi-nite mathematical relationship, and (3) there is close connection between mixed anomalies,non-equilibrium anomalies, SSB, and propagative versus diffusive dispersion relations.Throughout this work, we use the ‘mostly plus’ convention for the Minkowski spacetimemetric η µν = diag ( − , + , . . . , +) . We term such phenomena non-equilibrium ’t Hooft anomalies because they exists when a symmetry cannotbe gauged; however, there are important differences with standard ’t Hooft anomalies. Usually, a ’t Hooftanomaly does not out-right kill the conservation of a given Noether current. Instead, any attempt togauge a symmetry leads to non-conservation. Non-equilibrium ’t Hooft anomalies, by contrast, kill theconservation of the current even when no gauge fields have been introduced. I. HIGHER-FORM SYMMETRIES: A REVIEW
Suppose a relativistic quantum field theory in dimensions has conserved current J µ , ∂ µ J µ = 0 . (1)The usual way to define a conserved charge is by integrating the charge density over thevolume of space, that is Q = Z d xJ , ddt Q = 0 , (2)but there is another perspective that involves differential forms. In particular, consider theHodge star dual of the current ⋆ J µνλ = ǫ µνλρ J ρ , (3)where ǫ µνλρ is the Levi-Civita tensor. Notice that ⋆J is a three-form, which can naturallybe integrated over a three-dimensional hyper-surface in spacetime. Let Σ be such a hyper-surface. Then we can define the charge as a function of the hyper-surface by Q [Σ] = Z Σ ⋆J. (4)If we take Σ to be a spatial volume at fixed time, then we recover the original expressionfor the charge (2). Thus the differential form definition of charge contains within it thestandard formulation. Notice that the conservation of J implies that d ⋆ J = 0 , that is, ⋆J is closed. Therefore, if we continuously deform the hyper-surface Σ → Σ ′ in such a waythat it never intersects any charged operator insertions, then Q [Σ] = Q [Σ ′ ] . In this sense,the charge’s dependence on Σ is purely topological. Moreover, this topological dependencecan be interpreted as a conservation law in the sense that we can translate Σ throughtime without changing the total charge as long as we do not encounter any sources (i.e.charged operator insertions). Finally, we should comment on the dimensionality of theobjects that are conserved. The charge Q [Σ] counts the total number of point-particles onthe slice Σ . From another perspective, however, point-particles travel through spacetimealong their respective worldlines, so they ought to be thought of as one-dimensional objects(or D -branes). Thus the charge Q [Σ] counts the number of worldlines that intersect thehyper-surface Σ .At this point, we can generalize the notion of current and charge. In the interest ofcomplete generality, we will now work in a d + 1 dimensional spacetime. Suppose a currentwith p + 1 indices J µ ...µ p , ∂ µ J µ ...µ p = 0 (5)that is antisymmetric under interchange of any adjacent pair of indices. Then the Hodgestar dual is ⋆ J µ ...µ d − p = ǫ µ ...µ d − p ν ...ν p J ν ...ν p . (6)Since ⋆ J is a d − p -form, it is naturally integrated over an d − p -dimensional hyper-surface Σ d − p . We therefore define the charge by Q [Σ d − p ] = Z Σ d − p ⋆ J . (7)5otice that d ⋆ J = 0 , meaning that the dependence of Q on the choice of hyper-surface Σ d − p is topological, which in turn implies that Q is conserved. Finally, since we are inte-grating ⋆ J over a surface of dimension d − p , the dimension of the conserved objects willnot in general be point-particles traveling along worldlines (unless p = 0 ). Instead, Q [Σ d − p ] counts the number of charged p + 1 -dimension objects—or D p -branes—that intersect Σ d − p .We say that a system with conserved current J has an p -form symmetry. Thus, in thislanguage, ordinary symmetries are zero-form symmetries.We have made reference to charged operator insertions, but so far have not identified whatthey are. The objects charged under a p -form symmetry must themselves be p -dimensionalobjects that cannot be composed of lower-dimensional operators. Suppose we have a p -formfield ϕ p that enjoys some sort of gauge symmetry. For the sake of concreteness, supposethat we have the gauge symmetry ϕ p → ϕ p + dγ for p − -form γ . Then, given a closed p -dimensional manifold C p , we may construct the p -dimensional generalization of a Wilsonloop by W [ C p ] = e i R Cp ϕ p . (8)The charge Q [Σ d − p ] can detect the presence of the p -dimensional Wilson surfaces if and onlyif Σ d − p and C p are linked. In this way, Q [Σ d − p ] counts the number of closed p -dimensionalWilson surfaces that link with Σ d − p . A. Superfluids at zero temperature
A relativistic superfluid is a system that, in addition to Poincaré symmetry, enjoys aspontaneously broken U (1) internal symmetry N , which corresponds to particle numberconservation. Further, superfluids exist at finite charge density. We assume that in the deepinfrared, the only gapless mode is the Goldstone ψ associated with the broken charge N .The leading order-action is therefore S = Z d xP ( X ) , X = p − ∂ µ ψ∂ µ ψ, (9)and the current corresponding to N is J µU (1) = − P ′ ( X ) X ∂ µ ψ. (10)Notice that in equilibrium, since charge density is non-zero, J µU (1) ∝ δ µ , meaning that ∂ µ ψ ∝ δ µ . Thus, ψ has a time-dependent background h ψ i = µ t . It turns out that µ can beinterpreted as the equilibrium chemical potential.By virtue of possessing a U (1) Goldstone, superfluids also enjoy a two-form symmetrywith corresponding current K µνλ = ǫ µνλρ ∂ ρ ψ. (11)Notice that ∂ µ K µνλ = 0 identically as partial derivatives commute. The fact that K isconserved as a mathematical identity may give some readers pause: is there any physical6ontent to this conservation equation? It turns out the answer is emphatically, “yes!” To seehow this can be, we will recast the theory of superfluids in terms of a dual theory involvinga two-form gauge field A µν .Begin by replacing ∂ µ ψ → V µ , for some arbitrary one-form V . Then we can define anauxiliary action S AUX = Z d x (cid:18) P ( V ) − ǫ µνλρ V µ ∂ ν A λρ (cid:19) , (12)where V is the magnitude of V µ . Notice that the equations of motion for A µν are ∂ [ µ V ν ] = 0 , (13)which, if spacetime is topologically trivial, implies V µ = ∂ µ ψ for some scalar ψ . We thereforerecover the original superfluid action (9). Alternatively, we can integrate out V µ , in whichcase we find the equations of motion − P ′ ( V ) V V µ = F µ , (14)where we have defined the field strength by F µ = 12 ǫ µνλρ ∂ ν A λρ . (15)We can then algebraically solve (14) to find V in terms of F . Plugging this result backinto (12), we obtain a dual action that only depends on Y = p − F µ F µ . Explicitly, we have S DUAL = Z d xL ( Y ) , (16)for some function L , where L ( Y ) ≡ P ( V ( Y )) . Notice that this action is invariant under thegauge symmetry A µν → A µν + ∂ [ µ λ ν ] ( x ) . (17)The equations of motion of the dual action are ∂ µ ∂S DUAL ∂ ( dA ) µνλ = 0 . (18)But this equation is just the conservation law for a higher-form current. We therefore identify K µνλ = ∂S DUAL ∂ ( dA ) µνλ . (19)We now see that the conservation of K is no longer a mere mathematical identity, but is amanifestly physical equation describing the dynamics of superfluids. But what has happenedto J µU (1) ? Comparing equations (10) and (14), we have the identification J µU (1) = F µ . (20)But notice that ∂ µ F µ = 0 identically owing to the fact that partial derivatives commute.Thus, in the dual picture the particle-number conservation equation is a mere mathematicalidentity. In general the equation to be solved may be very complicated, but all we require is that a solution exist. . Electromagnetism We now consider the theory of electromagnetism as a system involving higher-form sym-metries. Letting A be the photon field and F = dA be the field-strength tensor, the actionfor free Maxwell theory in 3+1 dimensions is S = − Z d xF . (21)The equations of motion are ∂ µ F µν = 0 , meaning that we have a one-form symmetry withcorresponding conserved current J µν el ≡ F µν . Then given a closed two-dimensional surface Σ , we have the conserved charge Q [Σ ] = Z Σ ⋆F. (22)Taking Σ to exist at constant time, we therefore have Q [Σ ] = Z Σ ˆ n · ~E, (23)where ˆ n is normal to the surface Σ and E i = F i . We thus recognize this one-form chargeas counting the number of electric field lines passing through the Gaussian surface Σ .What are the charged objects? As we mentioned earlier, they are Wilson loops. Forclosed curve C , we have W [ C ] = e i R C A . (24)Physically, C corresponds to the worldline of a massive charged particle which is a sourcefor the electric field lines. In this way, if C links with Σ , then the amount of electric fluxthrough Σ changes according to Gauss’s law. We can see, therefore, that Q [Σ ] counts thenumber of Wilson loops that link with Σ .Electromagnetism has another one-form symmetry with corresponding current J µν mag = ⋆F µν , which is conserved as an identity. This charge counts the number of magnetic fieldlines passing through a given Gaussian surface. When dealing with the full, sourced Maxwelltheory, J µν el is no longer exactly conserved, whereas, in the absence of magnetic monopoles, J µν mag is always exactly conserved. As a result, when we consider magnetohydrodynamics, J µν mag will play a key role. III. NON-EQUILIBRIUM EFT: A REVIEW
In this section, we review the fundamentals of non-equilibrium EFT. We will merelyhighlight the key points of this formalism; for more in-depth discussions consult [7, 32].
A. The basics
The goal is to construct effective actions that capture the long-distance and late-timebehavior of systems out of thermal equilibrium. As a result, the equilibrium density matrix8bout which we will perturb is the standard thermal matrix give by ρ = e − β P tr ( e − β P ) , (25)where P is the time-translation generator and β is the inverse equilibrium temperature.Because the equilibrium state is mixed (i.e. not a pure quantum state), we must use thein-in formalism and define our effective action on the Schwinger-Keldysh (SK) contour. Forevery quantum field ϕ , we have two copies corresponding to the two legs of the SK contour: ϕ lives on the forward contour and ϕ lives on the backward contour. We write ϕ s for s = 1 , . It is often convenient to work in the retarded-advanced basis ϕ r = 12 ( ϕ + ϕ ) , ϕ a = ϕ − ϕ . (26)It turns out that ϕ r acts as a classical field (or the expectation value of a quantum field) and ϕ a contains information about thermal and quantum fluctuations. Denoting the effectiveaction by I EFT [ ϕ , ϕ ] , we claim without proof that I EFT is subject to the conditions thatfollow.• The coefficients of I EFT are complex. There are three important constraints that comefrom unitarity, namely I ∗ EFT [ ϕ , ϕ ] = − I EFT [ ϕ , ϕ ] Im I EFT [ ϕ , ϕ ] ≥ , for any ϕ , ϕ I EFT [ ϕ = ϕ ] = 0 . (27)• Any symmetry of the UV theory is a symmetry of I EFT , except for time-reversingsymmetries. If the equilibrium density matrix ρ takes the form of a thermal matrix, ρ ∝ e − β ¯ P , and the UV theory possesses some kind of anti-unitary time-reversingsymmetry Θ , then our EFT will enjoy the symmetries ϕ ( x ) → Θ ϕ ( t − iθ, ~x ) ϕ ( x ) → Θ ϕ ( t + i ( β − θ ) , ~x ) , (28)for any θ ∈ [0 , β ] , known as the dynamical KMS symmetries. In the classical limit,which is formally given by ~ → , the dynamical KMS symmetries become ϕ r ( x ) → Θ ϕ r ( x ) ,ϕ a ( x ) → Θ ϕ a + i Θ[ β∂ t ϕ r ] . (29)Because the change in ϕ a involves a temporal derivative of ϕ r , we consider ϕ a and ∂ ϕ r to contribute at the same order in the EFT derivative expansion.• In the distant future, the fields on the 1 and 2 legs of the SK contour must coincide,that is ϕ (+ ∞ ) = ϕ (+ ∞ ) . In the retarded-advanced basis, these SK boundaryconditions are equivalent to requiring that ϕ a (+ ∞ ) = 0 . As a result, there is just onecopy of every global symmetry and every time-independent gauge-transformation. Bycontrast, there are two copies of gauge symmetries with arbitrary time dependence.9 . Non-equilibrium coset construction Suppose we have a d +1 -dimensional relativistic system with global (zero-form) symmetrygroup G , which includes both internal and spacetime symmetries and is spontaneously brokento the subgroup H . We denote the generators by ¯ P µ = unbroken translations, T A = other unbroken generators, τ α = broken generators . (30)The unbroken subgroup H is then generated by ¯ P µ and T A . We assume the existence of d + 1 unbroken translation generators ¯ P µ , but do not require them to be the spacetime translationgenerators of the Poincaré group.We construct the effective action on the ‘fluid worldvolume’ coordinates σ M for M =0 , . . . , d and parameterize the most general elements (one for each leg of the SK contour) of G by g s ( σ ) = e iX µs ( σ ) ¯ P µ e iπ αs ( σ ) τ α e ib As ( σ ) T A , s = 1 , . (31)Under a global symmetry transformation γ ∈ G , transformations of the Goldstone fields canbe computed according to g s [ X s , π s , b s ] → g s [ X ′ s , π ′ s , b ′ s ] ≡ γ · g s [ X s , π s , b s ] . (32)In order to distinguish the broken and unbroken symmetries, we require that the EFTbe invariant under certain time-independent gauge transformations. Throughout the restof the paper, we let indices
M, N, P, Q = 0 , . . . , d be coordinate indices of σ and we let I, J, K, L = 1 , . . . , d be spatial coordinate indices of σ . First, since we are assuming the exis-tence of d +1 translation generators, impose the time-independent diffeomorphism symmetryon the coordinates σ → σ + f ( σ I ) , σ I → Σ I ( σ J ) , (33)for arbitrary f and Σ I . If there were fewer than d + 1 unbroken translations, then this set ofcoordinate symmetries could be reduced; see [31]. Next, because T A are unbroken, we havethe right-acting, time-independent gauge symmetries g s ( σ ) → g s ( σ ) · e iλ A ( σ I ) T A . (34)Finally, we compute the Maurer-Cartan form, which is a Lie algebra-valued one-formgiven by g − s dg s . Because it is Lie algebra-valued, it is always expressible as a linear combi-nation of symmetry generators. In particular, we have g − s dg s = idσ M (cid:2) E µsM (cid:0) ¯ P µ + i ∇ µ π αs τ α (cid:1) + B AsM T A (cid:3) . (35)It can be checked that under the gauge symmetries (33) and (34), E µsM transform as vielbeins, ∇ sµ π α transform linearly and hence we call them ‘covariant derivatives,’ and B AsM transform We say that these transformations are gauge symmetries because they do not change the state of thesystem, but are instead redundancies of description.
10s gauge fields. The effective action is then constructed by forming manifestly symmetry-invariant terms out of the vierbeins, covariant derivatives, and connections.Finally, suppose that Θ is some anti-unitary, time-reversing symmetry of the ultraviolettheory and that the equilibrium density matrix describes a state in thermal equilibrium,namely ρ = e − β ¯ P tr e − β ¯ P , (36)where β is the equilibrium inverse temperature. Then, the effective action must be invariantunder the so-called dynamical KMS symmetries, which are non-local Z symmetries whoseactions are g ( σ ) → Θ e θ ¯ P g ( σ − iθ, ~σ ) ,g ( σ ) → Θ e − ( β − θ ) ¯ P g ( σ + i ( β − θ ) , ~σ ) , (37)for arbitrary θ ∈ [0 , β ] [7]. In the classical limit, these symmetries reduce to a single local Z transformation.To take the classical limit, it is convenient to work in the retarded-advanced basis. Inparticular for a given set of fields ϕ s ( σ ) , s = 1 , defined on the SK contour, we define theretarded-advanced fileds by ϕ r = 12 ( ϕ + ϕ ) , ϕ a = ϕ − ϕ . (38)It will be useful to define the a -derivative denoted by δ a whose action is given by δ a ϕ r = ϕ a ,for any retarded-advanced pair of fields ϕ r and ϕ a . We also require that δ a = 0 and that itsatisfy the Leibnitz rule, namely δ a ( ϕ r ϕ ′ r ) = ϕ a ϕ ′ r + ϕ r ϕ ′ a . (39)Then, the way to construct classical invariant building-blocks is as follows. Parameterizethe most general group element with retarded Goldstones g r ( σ ) = e iX µr ( σ ) ¯ P µ e iπ αr ( σ ) τ α e ib Ar ( σ ) T A , (40)and construct the Maurer-Cartan form g − r dg r , which is given by (35) with replacement s → r . Then, we can construct retarded building-blocks from this Maurer-Cartan form inthe usual way. To construct advanced building-blocks, we need only act with δ a on terms ofthe retarded Maurer-Cartan form. The classical dynamical KMS symmetry is then g r ( σ ) → Θ g r ( σ ) , g a ( σ ) → Θ g a ( σ ) − i Θ β ∂g r ( σ ) ∂σ + Θ β ¯ P g r ( σ ) , (41)where g a ≡ δ a g r . The factors e θ ¯ P and e − ( β − θ ) ¯ P arise because the field X s ( σ ) transform as a time coordinates underdynamical KMS symmetries. V. NON-EQUILIBRIUM ’T HOOFT ANOMALIES
In non-equilibrium systems, it is possible to have a symmetry with no correspondingnon-trivial Noether current. As a simple example, consider a non-relativistic point particlein a uniform fluid undergoing Brownian motion. The effective action for the point particleis given by I p.p. = Z dt (cid:2) − νx a ˙ x r + ˜ M ˙ x r ˙ x a + i σx a (cid:3) , (42)where the retarded field x r ( t ) gives the classical position of the particle, x a ( t ) is the cor-responding advanced field, and the dynamical KMS symmetry requires that ν = σβ ≥ ,which is just a statement of the fluctuation-dissipation theorem. Evidently, the physics ofthe point particle is independent of its position in space and time as the action is invariantunder t → t + c , x r → x r + c , (43)for constants c and c . In other words, it enjoys spacetime translation symmetry. Giventhat it exists in a medium, however, the particle is subject to friction and hence its energyand momentum are not conserved. In particular, the equations of motion are ˜ M ¨ x r = − ν ˙ x r . (44)Thus, there are no conserved currents associated with the spacetime translation symmetry.This finding may puzzle some readers: we have an action with a global symmetry, meaningthat the assumptions of Noether’s theorem are satisfied. So how can there be no conservedcurrent? The answer is that at the level of the mathematics, there is a conserved currentassociated with every symmetry, but it need not contain any physical information. To seehow this is so, let us compute the Noether current associated with spatial translations. Wehave j = − νx a + ˜ M ˙ x a . (45)But when the equations of motion are satisfied, all advanced terms vanish, so we have x a = 0 and as a result j = 0 . Thus the conservation equation ∂ t j = 0 , while true, providesno physical information about the system.At the level of non-equilibrium EFTs, the existence of a conserved current is closely re-lated to gaugeability. In particular, if a given global symmetry can be gauged by introducingtwo copies—one for each leg of the SK contour—of a gauge field, then there exists a corre-sponding conserved current. For further explanation, see §VI A or consult [29–32]. We referto such anomalous charges as non-equilibrium ’t Hooft anomalies because they arise when asymmetry of a non-equilibrium system cannot be gauged.In the point-particle example, the fact that the energy and momentum exhibit a non-equilibrium ’t Hoof anomaly owes to the fact that the particle is an open system and canfreely exchange energy and momentum with an environment. Such anomalous behavior,however, does not always arise because of interactions with an environment. For example, Notice that j has no indices because we are working with a -dimensional QFT. U (1) shift symmetry withno corresponding Noether current [30]. We will see other examples of non-equilibrium ’tHooft anomalies that arise in closed systems in subsequent sections of this work.It can be shown that if we build the the effective action exclusively from terms furnished bythe coset construction, then our Goldstone fields are gaugeable; see §VI A. This observationraises the questions: what kinds of terms are invariant under the global symmetry groupbut are not gaugeable? To answer this question, consider g a ( σ ) ≡ g − ( σ ) · g ( σ ) . (46)Evidently g a transforms by conjugation under the gauge symmetry defined by (34) and isinvariant under all global (i.e. physical) G -symmetry transformations. Notice that g a ∈ G ,while the Maurer-Cartan forms are Lie-algebra valued. To turn g a into a Lie algebra-valuedobject, we need only take the log, that is G a ≡ log g a . (47)We can then expand G a as a linear combination of symmetry generators G a = i Φ µ ¯ P ¯ P µ + i Φ ατ τ α + i Φ AT T A . (48)If a given symmetry exhibits a non-equilibrium ’t Hooft anomaly, we must include thecorresponding Φ as a building-block in the effective action. For example if we want τ α tobe a symmetry with no corresponding conserved current, then we need only use Φ ατ as acovariant building-block in the EFT.Taking the classical limit, we have G a = g − r δ a g r . Thus, G a becomes like a Maurer-Cartanform with respect to the “derivative” defined by δ a . A. Reactive fluids
We will now demonstrate how to use the coset construction to formulate an EFT forfluids that exhibit reactive flows. There are, in addition to Poincaré symmetry, n differentunbroken U (1) charges that we denote by N , . . . , N n . Each U (1) charge counts the numberof particles of a given species. To allow for chemical reactions, that is, the transformationsof certain types of particles into other types of particles, we must kill the conservation ofsome subset of these charges. But first, we will construct the action for which there are nochemical reactions.It turns out that we will need to use the full SK contour to describe these dynamics, sowe must construct an action with doubled field content. We will work in the classical limit,so we can use the δ a trick to simplify matters. The most general retarded group element is g r ( σ ) = e iX µr ( σ ) P µ e iψ Ar ( σ ) N A e iη ir ( σ ) K i e iθ ir ( σ ) J i , (49)where A = 1 , . . . , N . Because translations are unbroken, we impose the fluid diffeomorphismsymmetry (33) and because J i and N A are unbroken we have the gauge symmetries g r ( σ ) → g r ( σ ) · e iλ i ( σ I ) J i , g r ( σ ) → g r ( σ ) · e ic A ( σ I ) N A . (50)13he resulting retarded Maurer-Cartan form is then g − r dg r = iE µr ( P µ + ∇ µ η ir K i ) + iω ir J i + i B Ar N A , (51)where E µrM = ∂ M X νr [Λ r R r ] νµ , ∇ µ η ir = ( E − r ) Mµ [Λ − r ∂ M Λ r ] j R jir ,ω irM = 12 ǫ ijk [(Λ r R r ) − ∂ M (Λ r R r )] jk , B arM = ∂ M ψ r , (52)such that Λ µr ν = ( e iη ir K i ) µν and R ijr = ( e iθ ir J i ) ij .To remove Lorentz Goldstones, we impose the IH constraints E ir = 0 = ⇒ η ir η r tanh η r , (53)and ǫ ijk ( E − r ) Mj E kaM = 0 , ǫ ijk ( E − r ) Mj ∂ E krM = 0 , (54)where E µaM = δ a E µrM . This second IH constraint can be solved to remove the rotationGoldstones, but it is not necessary at this level in the derivative expansion; see [32].After imposing these IH constraints, we are left with the invariant building-blocks T = 1 √− G r , µ A = 1 √− G r ∂ψ Ar ∂σ , (55)and the invariant integration measure is d σ p − G r . (56)As before, we have defined the fluid metrics by G rMN = E µrM η µν E νrN .To construct advanced building-blocks, we need only act with δ a on retarded building-blocks. At leading order, the dynamical KMS symmetries allow us to write I EFT = δ a S, (57)where S = Z d σ p − G r P ( T, µ A ) . (58)At this point, it is convenient to transform to the physical spacetime, so letting x µ ≡ X µr ,we have S = Z d xP ( T, µ A ) . (59) The fluid coordinates are now dynamical fields σ M ( x ) .
14o be explicit about the form of the non-equilibrium effective action, we may write I = Z d x (cid:2) T µν ∂ µ X aν + J Aµ ∂ µ ψ A (cid:3) , (60)where the stress-energy tensor is T µν = ǫ ( T, µ A ) u µ u ν + p ( T, µ A )∆ µν , (61)such that ∆ µν = η µν + u µ u ν and the U (1) currents associated with N A are J Aµ = n A ( T, µ B ) u µ . (62)The (classical) dynamical KMS symmetry imposes the relations ǫ + p = T ∂p∂T + µ A ∂p∂µ A ,n A = ∂p∂µ A , (63)which are equivalent to the local first law of thermodynamics. The equations of motion arejust the conservation of energy, momentum and charge, namely ∂ µ T µν = 0 , ∂ µ J Aµ = 0 . (64)Since J Aµ are all independently conserved, we see that our action describes fluids with n species of particles that are not permitted to undergo any chemical reactions.To include chemical reactions, we must kill certain linear combinations of U (1) conser-vation equations, while preserving other linear combinations. Such non-conservation canbe accomplished by introducing non-equilibrium ’t Hooft anomalies for certain charges.Without loss of generality, let N ˆ A for ˆ A = 1 , . . . , k < n be representatives of the coset ofnon-conserved U (1) charges. Then we have invariant building-blocks furnished by g − r δ a g r ⊃ iψ ˆ A N ˆ A . (65)We therefore find that new terms can be added to (60) and obtain I = Z d x (cid:20) T µν ∂ µ X aν + J Aµ ∂ µ ψ A − Γ ˆ A ψ ˆ Aa + i M ˆ A ˆ B ψ ˆ Aa ψ ˆ Ba (cid:21) , (66)where the dynamical KMS symmetry imposes Γ ˆ A = 12 M ˆ A ˆ B β µ ∂ µ ψ ˆ Br . (67)The equations of motion for X µa are still the conservation of the stress-energy tensor. Butthe equations for ψ Aa are now ∂ µ J Aµ = − Γ A , (68)where Γ A ≡ δ ˆ AA Γ ˆ A . But this is just the equation for k non-conserved U (1) currents and n − k conserved U (1) currents. Thus, chemical reactions may take place. These results agreewith those of [29]. 15 p A p V p +1 FIG. 1. This figure depicts a p + 1 -dimensional surface V p +1 (gray) and two p -dimensionalsurfaces A p (red) and B p (blue). The boundary of V p +1 is equal to the union of A p and B p , that is ∂V p +1 = A p ∪ ( − B p ) . V. HIGHER-FORM SYMMETRIES AND THE COSET CONSTRUCTION
When a zero-form symmetry is spontaneously broken, there exists a continuum of vacuumstates. The value of the Goldstone field at a given point A can be interpreted as definingthe local vacuum near that point. If we move to a different point, B , then there will be anew local vacuum that is related to the local vacuum at A by a symmetry transformation.The Maurer-Cartan form tells us how the local vacuum changes from point to point. Inparticular, suppose Ω is the Maurer-Cartan one-form and V is a path connecting A and B . In other words, ∂V = A ∪ ( − B ) . We can represent the local vacuum at arbitrarypoint x by an element of the symmetry group g ( x ) ∈ G . Then, we have P e R V Ω = g − ( B ) g ( A ) , (69)where P indicates path-ordering.Higher-form symmetries admit a straight-forward generalization of the Maurer-Cartanform. In particular, given a spontaneously broken p -form symmetry, let A p and B p be p -dimensional surfaces and V p +1 be a p + 1 -dimensional surfaces such that ∂V p +1 = A p ∪ ( − B p ) .Then, we define the Maurer-Cartan p + 1 -form Ω p +1 by e R Vp +1 Ω p +1 = g − p ( B p ) g p ( A p ) , (70)where g − p ( A p ) = e i Q p R Ap ϕ p and g − p ( B p ) = e − i Q p R Bp ϕ p , such that path-ordering is implicit for p = 1 , are p -dimensional generalizations of Wilson lines. Here, ϕ p is the p -form Goldstonecorresponding to the p -form charge Q p . Notice that for p > we have no notion of path-ordering for the l.h.s., which is related to the fact that all p > -form symmetries are abelian;however, if p = 1 then the r.h.s. involves Wilson lines, which must be path-ordered. In eithercase, we have for p > , e R Vp +1 Ω p +1 = g p ( ∂V p +1 ) , (71) The minus sign on B indicates a reversal of orientation. p = 1 .Supposing we have a mixture of various higher-form symmetries of differing p . Let A = { A p } dp =0 , B = { B p } dp =0 , and V = { V p +1 } dp =0 such that ∂V p = A p ∪ ( − B p ) for all p . Then, wedefine the Maurer-Cartan to be the mixed-form Ω = P p Ω p such that e R V Ω = g − ( B ) g ( A ) , (72)where we have left the path-ordering for one-dimensional integrals implicit and we define Z V Ω ≡ X p Z V p Ω p , g ( A ) ≡ Y p g p ( A p ) , g ( B ) ≡ Y p g p ( B p ) . (73)We are primarily interested in non-equilibrium systems, so we must double field contentand define our action on the fluid coordinates σ . As usual we let s = 1 , indicate on whichleg of the SK contours the fields live. Thus letting the zero-form symmetry generators begiven by (30) and letting Q p be the p > -form charges, we have g s (Σ) = Y p> e i Q p R Σ p ϕ ps ! e iX µs ( σ ) ¯ P µ e iπ αs ( σ ) τ α e ib As ( σ ) T A , (74)where we used Σ = { Σ p } dp =0 and we identified σ = Σ . If a given Q p is not spontaneouslybroken then we impose the time-independent gauge symmetry g s (Σ) → g s (Σ) · e i Q p R Σ p κ p , (75)where κ p = κ p ( σ I ) for I = 1 , . . . , d is a time-independent p -form such that κ p M ...M p = 0 .With the group elements (74) in mind, we may now compute the Maurer-Cartan mixed-forms Ω s = iE µs (cid:0) ¯ P µ + i ∇ sµ π α τ α (cid:1) + B As T A + F ps Q p , (76)where, E µs = E µsM dσ M , B As = B AsM dσ M and F ps = p +1)! F psM ...M p dσ M ∧ · · · ∧ dσ M p . The p > -form symmetry associated with charge Q p acts by g s (Σ) → e i Q p R Σ p α ps ( σ ) · g s (Σ) , (77)for any closed p -forms α ps . Notice that since α ps can have arbitrary time-dependence, theSK future-time boundary conditions permit the s = 1 , transformations to be different;however, all topological properties of α p and α p must be the same. Thus, α p − α p must beexact. In this way, there is just one copy of each global p -form symmetry. Finally, once theMaurer-Cartan form is computed, we can go through the usual procedure of constructingmanifestly-invariant terms built from the coefficients of the Maurer-Cartan form. We choose Q p to be defined on a surface that “links” with Σ p . . The zero-temperature limit There are two meanings of the zero-temperature limit. The first is the most straight-forward. We merely formulate an effective action using the in-out formalism; as a resultwe only have one set of fields (as opposed to the SK doubled field content). The secondmeaning, by contrast, involves a thermal system at very low temperature such that alltemperature fluctuations can be neglected. Essentially, we are interested in a regime inwhich thermal activity is suppressed, but not truly absent. As a result, we still need the SKcontour and doubled field content. To distinguish these two possibilities, we use T = 0 to in-dicate the true zero-temperature case and T → to indicate the very small temperature case. T=0:
The coset construction for systems at zero temperature is well-studied, so the onlynovel comments we can make have to do with higher-form symmetries. With the zero-formgenerators given by (30) and the p > -form generators Q p , the most general coset elementis g (Σ) = Y p> e i Q p R Σ p ϕ ps ! e ix µ ¯ P µ e iπ α ( x ) τ α , (78)where we identify x = Σ and we have supposed that Q p are spontaneously broken. Then,the Maurer-Cartan mixed-form is defined in the usual way and we can extract from itsymmetry-covariant building-blocks. Notice that in the zero-temperature limit, there is nopossibility of having unbroken Goldstones or symmetries without corresponding Noethercurrents. T → Now we take the small-temperature limit. To do this, we retain all of the samemachinery as the finite-temperature case, but with two small alterations. We begin by mod-ifying the fluid symmetry (33). To determine how to modify this diffeomorphism symmetry,we must first understand where it comes from. Suppose that we have full diffeomorphismsymmetry σ M → Σ M ( σ ) . For unbroken translations, this is the most natural symmetry tosuppose because it allows us to gauge-fix by σ M = x µ δ Mµ , which means our EFT is definedon the physical spacetime [51]. At finite temperature, we introduce the inverse temperaturefour-vector field β M ( σ ) , which transforms as a contravariant vector under diffeomorphismsof the fluid worldvolume. Under a coordinate transformation, we have β M → β N ∂ Σ M ∂σ N . (79)Thus, if we gauge-fix β M = β δ M , we find that the residual gauge symmetries are (33), asexpected. But now we wish to take the zero-temperature limit, which is equivalent to taking β → ∞ . As a result, the magnitude of β M is no longer meaningful, but the direction isstill important. Therefore we impose the gauge-fixing condition β M ∝ δ M , where we allowtransformations that alter the magnitude of β M without changing its direction. We are thusleft with the residual gauge symmetries σ → σ + f ( σ , σ I ) , σ I → Σ I ( σ J ) . (80)18e take this expanded set of symmetries to be the low-temperature fluid worldvolumesymmetries.Now we must modify the dynamical KMS symmetries. In the β → limit, the transfor-mations (37) are no longer well-defined. Instead, we impose continuous symmetries of theform (37) or in the classical limit (41) but now allow β to be a continuous real parame-ter. Notice that if we take β = θ = 0 , we find that our EFT is invariant under Θ alone.Therefore our action is invariant under the following symmetries that do not involve timereversal g ( σ ) → g ( σ − iθ, ~σ ) ,g ( σ ) → g ( σ + i ( β − θ ) , ~σ ) , (81)for arbitrary θ and β such that | θ | ≤ β . In the classical limit, we have g r → g r , g a → g a + iβ ∂g r ∂σ , (82)for arbitrary β . Notice that this classical KMS symmetry is a U (1) symmetry that actsonly on the advanced fields. As a result, there must be a retarded Noether current that isconserved. This Noether current has the interpretation of entropy and is conserved in the T → limit, as expected. In the T = 0 limit, it is no longer conserved as the dynamicalKMS symmetries are no longer continuous [7]. B. Gauge theories
In this subsection, we will construct the actions for systems involving gauge bosons. Webegin by woking at zero-temperature and construct the familiar pure Yang-Mills SU ( N ) gauge theory. Then we construct the Chern-Simons action, a purely topological theory,from symmetry considerations alone. Finally we turn our attention to finite temperatureand reproduce the effective action for magnetohydrodynamics that was first presented in [42].
1. Yang-Mills
As a simple example, we construct the action for pure Yang-Mills in dimensionsusing this new, higher-form coset construction. We consider the zero-temperature, T = 0 ,case, meaning there is only one copy of the fields. To construct an action with a higher-formsymmetry, we must define the Wilson loops. Let t a be the generators of a compact Lie group G such that [ t a , t b ] = if abc t c , (83)where f abc are the structure constants. We define the Wilson loop by g ( C ) = P e iq R C A , (84) The fact that the Noether current is retarded means that it is physical. C is a closed path, q is the charge of the Wilson loop, and A ≡ A aµ dx µ t a . For Yang-Mills theory, we take G = SU ( N ) ; then we will have a global Z n one-form symmetry [39].Letting S be a two-dimensional surface with ∂S = C , define the Maurer-Cartan two-form Ω implicitly by e i R S Ω = P e iq R ∂S A . (85)Taking S infinitesimal, we can expand both sides of the above equation to find Z S Ω = iq Z ∂S A − q Z dλ l Z dλ dz µ dλ dz ν dλ A aµ A bν (cid:0) t a t b θ ( λ − λ ) + t b t a θ ( λ − λ ) (cid:1) . (86)Using Stoke’s theorem, the r.h.s. can be simplified to yield Z S Ω = iq Z S ( dA + iqA ∧ A ) , (87)indicating that Ω = iq ( dA + iqA ∧ A ) . Recognize F ≡ dA + iqA ∧ A as the field strengthtensor for Yang-Mills theory. At leading order in the derivative expansion, there is only oneterm that is invariant under the global higher-form symmetry, namely S = − Z d xF aµν F aµν , (88)where the factor of − / is a convention that fixes field normalization. We have thusconstructed the familiar Yang-Mills action.
2. Chern Simons
One of the original motivations for considering higher-form symmetries was to understandtopological phases of matter from symmetry principles alone. Here we will construct theleading-order action for Chern-Simons theory in dimensions. We will consider a zero-temperature system so we need only one copy of the fields.In addition to Poincaré symmetry, which is unbroken, we postulate the existence of aspontaneously broken one-form symmetry with Goldstone A . Let (83) be the generators ofthe compact Lie group G such that A = A aµ dx µ t a . The Maurer-Cartan two-form, Ω , is thendefined by e R Σ2 Ω = e R ∂ Σ2 A . (89)Following the Yang-Mills construction, we arrive at the covariant building-block F = dA + A ∧ A , which we interpret as the field strength.Chern-Simons theory involves a term that is symmetry-invariant up to total derivativeterms. Unfortunately, the coset construction only furnishes terms that are exactly symmetry-invariant. To circumvent this difficulty, we can construct the necessary term by working in We drop the factor of i and the charge from the exponent on the r.h.s. so that our conventions matchstandard results. + 1 dimensions and only consider terms that are total derivatives. The leading-order such -dimensional action is then S = k π Z M tr ( F ∧ F ) = k π Z M tr (cid:20) d (cid:18) dA + 23 A ∧ A ∧ A (cid:19)(cid:21) , (90)where M is a -dimensional manifold and k is a phenomenological constant. Then,using Stoke’s theorem, we obtain the -dimensional action S = k π Z M tr (cid:18) A ∧ dA + 23 A ∧ A ∧ A (cid:19) , (91)for -dimensional manifold M . We thus have derived a purely topological field theoryfrom symmetry principles alone!
3. Magnetohydrodynamics
Magnetohydrodynamics (MHD) is the study of electromagnetism and fluids. In particu-lar, it is the study of fluids that can support the flow of electric current but cannot supportelectric charge density. Any regions of non-vanishing charge density locally equilibrate tozero exponentially fast. We will restrict our considerations to MHD in dimensions.Let A µ be the photon field, F µν be the field strength given by F = dA , and j µ be theelectromagnetic four-current. Then the electromagnetic equations of motion are ∂ ν F µν = j µ . (92)One might be tempted to say that the symmetries of electromagnetism are the gauged U (1) symmetry, but gauge symmetries are mere redundancies of description and are thereforeunphysical. Because this zero-form U (1) symmetry is not a genuine symmetry of nature, itneed not survive in our EFT. The true symmetry of electromagnetism is the one-form U (1) symmetry with corresponding current J = ⋆F , or in index notation, J µν = 12 ǫ µνλρ F λρ . (93)Notice that ∂ µ J µν = 0 is a mathematical identity meaning this current is automaticallyconserved off-shell. Thus, at the level of the coset construction, we have a one-form symmetrygenerated by some Q . The SSB pattern is simply that boosts are spontaneously broken andall other symmetries including Q are unbroken.We are only interested in constructing the leading-order action, which, it turns out,factorizes as the difference of two ordinary actions. As a result, we will use just one copy ofthe fields. The most general group element is g (Σ) = e iX µ ( σ ) P µ e iη i ( σ ) K i e iθ i ( σ ) J i e i Q R Σ2 ϕ . (94)Because translations are unbroken, we have the fluid symmetries (33) and because rotationsand the one-form symmetry are unbroken, we have the gauge symmetries g (Σ) → (Σ) · e iλ i ( σ I ) J i , g (Σ) → g (Σ) · e i Q R Σ2 κ , (95)21here λ i ( σ I ) is arbitrary and κ = κ M ( σ I ) dσ M is a time-independent one-form such that κ = 0 . And finally, we have the one-form symmetry ϕ → ϕ + α ( σ ) , for any closed one-form α .From this group element, we find that the Maurer-Cartan form is Ω = iE µ ( P µ + ∇ µ η i K i ) + i Ω i J i + i F Q , (96)where E µM = ∂ M X ν [Λ R ] νµ , ∇ µ η i = ( E − ) Mµ [Λ − ∂ M Λ] j R ji , Ω iM = 12 ǫ ijk [(Λ R ) − ∂ M (Λ R )] jk , F MN = ∂ [ M ϕ N ] , (97)such that Λ µν = ( e iη i K i ) µν and R ij = ( e iθ i J i ) ij .We can now impose IH constraints to remove the Lorentz Goldstones. First, we removethe boost Goldstones by setting E i = 0 , (98)which can be solved to give η i η tanh η = − ∂ X i ∂ X t , (99)where η ≡ p ~η . Second, to remove the rotation Goldstones, we may fix ǫ ijk ( E − ) Mi ∂ E jM = 0 . (100)This IH constraint can be used to remove θ i from the invariant building-blocks, but thesolution is not necessary for the construction of the leading-order action. For the sake ofbrevity, we therefore will not solve it; interested readers can consult [32] for more details.With these IH constraints imposed, the leading-order building-blocks are as follows. De-fine the fluid worldvolume metric by G MN = E µM η µν E νN = ∂X µ ∂σ M η µν ∂X ν ∂σ N . (101)Then, we have that T = 1 E t = 1 √− G (102)is an invariant building-block, which can be interpreted as the local temperature. Addition-ally, we have the invariant building-block µ = p F M G MN F N , (103)where G MN is the inverse of the fluid worldvolume metric. We may interpret µ as thelocal chemical potential. There are no other invariant building-blocks at this order in the22erivative expansion. Performing a coordinate transformation from σ M to the physicalspacetime x µ ≡ X µ , we have the leading-order action S = Z d xP ( T, µ ) , (104)which exactly matches the results of [42].Finally, we take the T → limit, which can be accomplished by expanding the fluidworldvolume symmetry and allowing full time diffeomorphisms, namely σ → σ + f ( σ , σ I ) .The only effect this enlarged gauge symmetry has is the removal of T as an invariant building-block. We therefore have S T → = Z d xP ( µ ) , (105)which again matches the results of [42]. VI. HIGHER-FORM SYMMETRIES AND ’T HOOFT ANOMALIESA. Gauge fields and the Maurer-Cartan form
We previously stated that the way to ensure all symmetries have corresponding conservedNoether currents is to construct the action using two distinct Maurer-Cartan one-forms—onefor each leg of the SK contour—that transform under a single copy of the global symmetrygroup G . In this section, we will use a Stückelberg trick inspired by [7, 32] to demonstratethat this approach is correct.We begin by introducing sources for the Noether currents corresponding to each symmetrygenerator of G , which amounts to introducing external gauge fields. Let the zero-formsymmetry generators be ¯ P m = unbroken translations ,T A = other unbroken generators ,τ α = broken generators , (106)and let Q p = p > -form charges. (107)We now use m, n = 0 , . . . , d to denote Lorentz indices and µ, ν = 0 , . . . , d to denote physicalspacetime coordinate indices. We wish to introduce external sources corresponding to eachof these symmetries so that we can construct a generating functional for conserved quantities.In particular these external sources are gauge fields corresponding to each symmetry. Theyare as follows:• Let ε ms ( x ) = ε msµ ( x ) dx µ be the vierbeins, which can be thought of as the gauge fieldscorresponding to unbroken translations ¯ P m . The metric tensors are then given by g sµν ( x ) = ε msµ ( x ) η mn ε nsν ( x ) . It is now necessary to distinguish between Lorentz indices m, n and physical spacetime coordinate indices µ, ν because the Stückelberg trick requires that we gauge all symmetries including Lorentz.
23 Let A s ( x ) ≡ A Asµ ( x ) dx µ T A be the gauge fields (or spin connections if the Lorentzgroup is involved) corresponding to the unbroken zero-form symmetries other thantranslations.• Let c s ( x ) ≡ c αsµ ( x ) dx µ τ α be the gauge fields (or spin connections) corresponding to thebroken zero-form symmetries.• Let H ps = p +1)! H psµ ...µ p dx µ ∧ · · · ∧ dx µ p Q p be the gauge fields corresponding to the p > -form symmetries.On each leg of the SK contour, we can combine these fields into a single object, θ s ( x ) = iε ms ( x ) ¯ P m + ic αs ( x ) τ α + i A As ( x ) T A + iH ps ( x ) Q p , (108)where the factors of i are included as a matter of convention. Now, letting U ( t, t ′ ; θ s ) for s = 1 , be the time-evolution operator from t ′ to t in the presence of external sources θ s ,the generating functional for the conserved currents is e W [ θ ,θ ] = tr (cid:2) U (+ ∞ , −∞ ; θ ) ρU † (+ ∞ , −∞ ; θ ) (cid:3) . (109)Since θ s couple to conserved currents, W [ θ , θ ] must be invariant under two independentcopies of the gauge symmetries [7]; that is, for gauge parameters ζ ( x ) and ζ ( x ) we have W [ θ µ , θ µ ] = W [ θ ζ µ , θ ζ µ ] . (110)We can therefore ‘integrate in’ both the broken and unbroken Goldstone fields using theStückelberg trick. In particular, define Θ s ( σ ) ≡ θ ζ s s ( σ ) , ζ s ≡ { X µs , π αs , b As , ϕ ps } , (111)where σ M for M = 0 , . . . , d are the fluid worldvolume coordinates. Now we can implicitlydefine the non-equilibrium effective action by e W [ θ ,θ ] ≡ Z D [ X µs π αs ǫ As ϕ ps ] e iI EFT [Θ , Θ ] . (112)Notice that if we remove the external source fields by fixing ε msµ ( x ) = δ mµ and A Asµ = c αsµ = H ps = 0 , we find that Θ s are nothing other than the Maurer-Cartan mixed-forms (76) definedon each leg of the SK contour. Since ε msµ ( x ) = δ mµ we can identify the Lorentz indices m, n with the physical spacetime coordinate indices µ, ν , allowing us to use ¯ P µ to refer to theunbroken translation generators. Moreover because these fields live on the SK contour, theirvalues must match up in the infinite future, meaning that even though there were two copiesof the gauge fields and gauge symmetries, there is only one copy of the global symmetrygroup G .Since W [ θ , θ ] is the generating functional for conserved quantities, we see that anyeffective action constructed exclusively with building-blocks furnished by the Maurer-Cartanform can be gauged and will have conserved Noether currents associated with each of itssymmetry generators. 24 . Dual theories and mixed ’t Hooft anomalies We now investigate dualities among higher-form currents. Recall the example of the -dimensional superfluid from §II A in which there were two conserved charges. One was the U (1) symmetry associated with particle number conservation and the other was associatedwith the higher-form current K µνλ = ǫ µνλρ ∂ ρ ψ . Notice that K µνλ is conserved automaticallyas an identity. However, suppose we gauge the U (1) symmetry by introducing gauge field A µ . This can be accomplished by replacing ∂ µ ψ → A µ + ∂ µ ψ . Then, the higher-form currentbecomes K µνλ = ǫ µνλρ ( A ρ + ∂ ρ ψ ) , which is no longer conserved; in particular, we have ∂ µ K µνλ = ⋆F νλ , (113)where F = d A . Because the introduction of a gauge field for the U (1) symmetry interfereswith the conservation of the higher-form charge, we say that there exists a mixed ’t Hooftanomaly . In particular, it is impossible to gauge the U (1) symmetry and the two-formsymmetry simultaneously. But recall how the arguments of the previous subsection reliedon the assumption that all symmetries could be gauged simultaneously. Thus, if there is amixed anomaly, we may only include Goldstones for one of the anomalous symmetries. Forthe example of the superfluid, we may choose to parameterize the symmetry group elementwith either the U (1) symmetry generator N and its corresponding Goldstone or the two-form generator Q and its corresponding Goldstone, but not both. It turns out that thepresence of a mixed ’t Hooft anomaly is equivalent to SSB. Moreover, this mixed anomalycan be used to prove a kind of Goldstone theorem [38].Statements regarding mixed anomalies can be generalized to more complicated scenarios.Working in d + 1 spacetime dimensions, suppose that Q and R are p -form and d − p − -form U (1) symmetry generators, respectively and let them have corresponding conserved currents J µ ...µ p and K µ ...µ d − p . Suppose further that Q and R enjoy a mixed anomaly such that ifwe introduce an external gauge field H µ ...µ p for Q , we have ∂ µ K µ ...µ d − p = 1( p + 1)! ǫ µ ...µ d − p ν ...ν p ( dH ) ν ...ν p . (114)Then, we may either include the Goldstone ϕ for Q or the Goldstone χ for R in the Maurer-Cartan form, but not both. As we will see, the resulting effective actions will be relatedby a Legendre transformation in much the same way that the ordinary and dual superfluidactions are related to one another.Supposing that we construct our action with ϕ as the Goldstone field associated with Q .Then, we have that the d − p − -form current, which is identically conserved, is given by K = ⋆dϕ, (115)and the p -form current J , which is conserved on-shell, is obtained by a Noether procedure.Conversely, if we construct an action with χ as the Goldstone associated with R , we findthat J = ⋆dχ (116)25s identically conserved and K , which is conserved on-shell, is obtained via a Noetherprocedure. Finally, if we are constructing a non-equilibrium EFT, then we must of coursedouble the fields: ϕ s and χ s for s = 1 , .To see explicitly how this duality works, suppose we have the non-equilibrium action I EFT [ dϕ , dϕ ] , which depends on the Goldstones ϕ s associated with Q and contains noGoldstones associated with R . This action may depend on other fields, but they will not beimportant for present considerations. Then, we may replace dϕ s → V s for generic p +1 -forms V s and define an auxiliary action by I AUX = I EFT [ V , V ] − Z V ∧ dχ + Z V ∧ dχ . (117)Notice that the equations of motion for χ s give d V s = 0 , which can be solved—assuming atopologically trivial spacetime—to give V s = dϕ s for some p -forms ϕ s . We thus recover theoriginal action I EFT . Conversely, we may compute the equations of motion for χ s , δI EFT δ V = dχ , δI EFT δ V = − dχ . (118)These equations of motion can then be solved for V s in terms of dχ s and any other fieldsthat may have appeared in I EFT . We thus arrive at a dual action I DUAL [ dχ , dχ ] with theGoldstones χ s associated with R and no Goldstones associated with Q . Moreover, I EFT and I DUAL are evidently related by the Legendre transformation (117).
C. Non-equilibrium higher-form ’t Hooft anomalies
To kill the conservation of the higher-form current without killing the higher-form sym-metry, it is important to understand what constitutes a p -form symmetry. Recall that all p -form Goldstones enjoy the time-dependent symmetry (77), or equivalently ϕ ps → ϕ ps + α ps , (119)where α ps for s = 1 , are closed p -forms. These transformations contain within them theglobal p -form symmetry transformations, but they also contain gauge redundancies. Inparticular, two different sets of transformation parameters α ps and ˜ α ps correspond to thesame physical transformation if and only if their differences α ps − ˜ α ps are exact for s = 1 , .Since in the distant future, SK boundary conditions impose ϕ p (+ ∞ ) = ϕ p (+ ∞ ) , the s = 1 and s = 2 transformation parameters must contain the same topological features, meaningthat α p − α p must be exact. Therefore, there is only one copy of the global p -form symmetryand it only acts on the retarded p -form Goldstones.We find it convenient to work in synchronous gauge ϕ ps M ...M p ( σ ) = 0 . (120) At leading order in the derivative expansion, they can be solved algebraically. α p = α p = α p ( σ I ) for I = 1 , . . . d , which is nowboth closed and time-independent. If the higher-form symmetry is unbroken, then we alsohave the gauge symmetry (75). Notice that in either case, the set of symmetries acting onthese higher-form fields is time-independent. As a result, the SK boundary conditions—therequirement that the fields be continuous on the closed-time-path —force us to use thesame α for s = 1 , . Thus, the advanced higher-form fields do no transform under anyshift-symmetries, so we expect that there exist advanced building-blocks that contain noderivatives and are invariant under all physical symmetries. It turns out that this is so andwe can compute them in the following way. Define the higher-form symmetry generalizationof (47), denoted by G a , as e R Σ p G a = g − (Σ p ) g (Σ p ) . (121)Notice that G a is an element of the Lie algebra as its exponential is a group element. Wecan therefore express it as a linear combination of symmetry generators G a ⊃ i Φ pa Q p + · · · . (122)If Q p has a non-equilibrium ’t Hooft anomaly, then we may include Φ pa as building-block forthe action.It may seem strange that we are gauge-fixing the fields before constructing the action.Usually, one constructs the action and then gauge-fixes; after all, the components ϕ s M ...M p yield important constraints as their equations of motion. However, since we are aimingto destroy the conservation of the p -form current, these constraint equations are not desir-able. It is therefore appropriate—indeed necessary—to impose the synchronous gauge-fixingconditions before constructing the effective action. D. Quadratic examples
To illustrate some key features of mixed and non-equilibrium ’t Hooft anomalies as appliedto systems with higher-form symmetries, we will investigate some simple examples. Forconcreteness, we will work in dimensions and suppose the existence of two one-formcharges with mixed anomaly, Q el and Q mag , which we interpret as the electric and magneticone-form charges from electromagnetism. Working at finite temperature /β , and ignoringthe conservation of the energy-momentum tensor, we can fix X µs ( σ ) = δ µM σ M . Then we canconstruct our action on the physical spacetime coordinates x µ ≡ X µs . Let A sµ and ϕ sµ bethe Goldstone fields associated with Q el and Q mag , respectively. Since Q el and Q mag enjoya mixed anomaly, we can never have both A s and ϕ s appearing in the same effective action.In the A -picture, let F s = dA s E is = F is , B is = 12 ǫ ijk F jks , (123)and in the ϕ -picture, let f s = dϕ s , B ′ is = f is , E ′ is = 12 ǫ ijk f jks . (124) If φ s are continuous on the closed-time-path, then in the limit that σ → + ∞ , we have φ (+ ∞ ) = φ (+ ∞ ) .
27e will work in the retarded-advanced basis, but for economy of expression we will drop all r -subscripts on retarded fields. All actions we construct will involve only the relevant andmarginal terms and will respect the dynamical KMS symmetry.
1. Hot Maxwell theory
Begin by supposing that Q el and Q mag are both conserved and enjoy a mixed ’t Hooftanomaly; as a result, they are both spontaneously broken. The fact that they are bothconserved means that there are no electric or magnetic monopoles. The resulting theorywill be the source-free Maxwell theory at finite temperature. Then, we can choose to workin the ϕ - or the A -picture. We choose to work in the A -picture as this choice will highlightthe similarities with ordinary electromagnetic theory in vacuum. The action consisting ofrelevant and marginal terms is I = Z d x (cid:16) ~E a · ~E − c s ~B a · ~B (cid:17) . (125)The equations of motion are then ~ ∇ · ~E = 0 , ˙ ~E = c s ~ ∇ × ~B. (126)We additionally have the mathematical identities ~ ∇ · ~B = 0 , ˙ ~B = − ~ ∇ × ~E. (127)Evidently, there are no electric or magnetic monopoles in this theory. Combining the aboveequations, we find that ¨ ~B = c s ~ ∇ ~B, ¨ ~E = c s ~ ∇ ~E. (128)We thus have a wave that propagates at speed c s . Notice that our equations look just like thevacuum Maxwell equations except the speed of wave-propagation is no longer the standardspeed of light. This discrepancy arises because in our example, Lorenz boosts are explicitlybroken.Now let us consider the conserved currents associated with the one-form symmetries. Wehave J µν el and J µν mag , where J i el = E i , J ij el = c s ǫ ijk B k , J i mag = B i , J ij mag = − ǫ ijk E k . (129)Then, we have the conservation equations: ∂ µ J µν el = 0 , which holds on-shell, and ∂ µ J µν mag = 0 ,which holds as an identity. Notice that expressions for the two different currents involvethe same building-blocks, allowing us to express the electric higher-form current entirely interms of components of the magnetic higher-form current, namely J i el = c s ǫ ijk J jk mag , J ij el = ǫ ijk J k mag . (130)28e can therefore express the conservation of the electric higher-form current in terms of thecomponents of the magnetic higher-form current by ǫ ijk (cid:18) ∂ j J k mag + c s ∂ t J jk mag (cid:19) = 0 , ∂ i J i mag = 0 . (131)Notice that if Q el and Q mag were unrelated charges, then we would have two independentconserved current, that is there would be no special relationship between the componentsof the two currents. In this case, we would have two diffusive modes. But in our case, thereis a mixed anomaly, which relates the components of the conserved charges. This specialrelationship (130) along with the conservation equations is sufficient to derive the existenceof a propagating wave. We therefore see the existence of wave solutions is intimately con-nected with the existence of a mixed anomaly. Moreover these findings readily generalizeto arbitrary dimension and arbitrary p - and d − p − -form U (1) symmetries with mixed’t Hooft anomaly.
2. Magnetohydrodynamics I
We will now remove Q el as a physical conserved quantity of the theory by supposingthat Q mag is not spontaneously broken. Physically, the resulting action will correspond to atheory of MHD in which we ignore the fluid degrees of freedom arising from the conservationof the stress-energy tensor. We work in the ϕ -picture and impose the time-independent shiftsymmetries ϕ si → ϕ si + κ i ( ~x ) , (132)for arbitrary κ i ( ~x ) , which ensure Q mag is unbroken and that Q el is not gauge-invariant (seebelow for more details). Thus, Q el is entirely removed from the theory. The action consistingof relevant and marginal terms is I = Z d x (cid:18) i Dβ ~E ′ a + ~B ′ a · ~B ′ − D ~E ′ a · ˙ ~E ′ (cid:19) , (133)and the resulting equations of motion are ~ ∇ · ~B ′ = 0 , ˙ ~B ′ = D ~ ∇ ~B ′ . (134)We therefore find a diffusion equation for the magnetic field. Compare this result with thatof the previous example: when Q mag is spontaneously broken, there exists a propagatingwave; when it is unbroken, there is a diffusive mode. This relationship between SSB patternand dispersion relation in non-equilibrium systems is quite common [32].Now consider the conserved currents j µν el and j µν mag associated with Q el and Q mag , respec-tively. Explicitly, these currents are j i el = E ′ i , j ij el = ǫ ijk B ′ k , j i mag = B ′ i , j ij mag = Dǫ ijk ˙ E ′ k . (135) We used calpital letter J when working in the A -picture and lower-case j when working in the ϕ -picture. ∂ µ j µν mag = 0 on the equations of motion and ∂ µ j µν el = 0 identically. It thereforeseems like there are two conserved one-form symmetries; however j µν el is not invariant underthe gauge transformation (132) and is hence not physical. We therefore only have oneconserved current, namely j mag .
3. Magnetohydrodynamics II
We will now consider a gentler way of removing Q el from the set of conserved charges. Wewill see that we can make the non-conservation of this charge arbitrarily weak by adjustinga time-scale τ . To do this, work in the A -picture and let Q el be both spontaneously brokenand have a non-equilibrium ’t Hooft anomaly. As a result, Q mag will still exist, but it willno longer enjoy a mixed anomaly with another conserved charge, which indicates that it is not spontaneously broken. Thus, if we only consider conserved charges, we have the sameSSB pattern as in the previous example. The resulting theory will therefore describe MHD,but unlike the more standard formalism, this new theory allows for large charge density.In order to give Q el a non-equilibrium anomaly, we must operate in synchronous gauge,that is we fix A s = 0 , which yields the residual symmetry A s → A s + α ( ~x ) , (136)for closed, time-independent one-form α such that α = 0 . Notice that ~A a is now a covariantbuilding-block. The action consisting of relevant and marginal terms is therefore I = Z d x (cid:18) ~E a · ~E − c s ~B a · ~B + 1 τ ~A a · ~E + iτ β ~A a (cid:19) . (137)The terms with a factor of /τ are a result of the non-equilibrium ’t Hooft anomaly. Bychoosing weak anomaly, we can make this relaxation time τ arbitrarily long.The resulting equations of motion are ˙ ~E − c s ~ ∇ × ~B = − τ ~E. (138)Since we have no ~A s -field in the action, the electric Gauss law is not an equation of motion.Taking the curl and divergence, respectively, of the above equation yields ¨ ~B − c s ~ ∇ ~B = − τ ˙ ~B, (cid:18) ∂ t + 1 τ (cid:19) ~ ∇ · ~E = 0 , (139)where we have made use of the identities (127). Using the Maxwell relation ~ ∇ · ~E = ρ , where ρ is the electric charge density, we see that the second equation can be solved by ρ ( t, ~x ) = ρ ( ~x ) e − t/τ , meaning that non-zero charge density can exist, but it decays exponentially fast.Let us now investigate the connection to MHD and free Maxwell theory. To recover theMHD equations of motion, work in the low-frequency limit ∂ t ≪ /τ . Then, the equationsof motion become ˙ ~B = τ c s ~ ∇ ~B, ρ = 0 . (140)30f we identify D = τ c s , then we recover the diffusive MHD equation (134). Further, noticethat ρ = 0 , indicating that there can be no electric charge density. These equations ofmotion along with the identities (127) are exactly what we expect for standard MHD. Tosee the connection with free Maxwell theory, work in the high-frequency limit ∂ t ≫ /τ .Then, we have ¨ ~B = c s ~ ∇ ~B, ∂ t ρ = 0 . (141)The first equation indicates that there now exists a propagating wave and the second equa-tion indicates that we are operating on sufficiently short time scales that charge densitydoes not have time to decay. If we take the initial condition ρ t =0 = 0 , then the the high-frequency equations of motion in conjunction with the identities (127) reproduce the freeMaxwell equations except the speed of light is replaced with c s .We arrive at a fascinating result: on short time-scales, Q mag appears unbroken, while onlong time-scales, it appears spontaneously broken. Moreover, the characteristic time scale, τ at which this crossover occurs is fixed by the strength of the non-equilibrium ’t Hooftanomaly. We expect this scale-dependent symmetry-breaking to be a much more generalphenomenon.Finally, the higher-form currents are given by (129), but now only one of them is con-served. In particular, J mag is conserved as an identity, while ∂ µ J µi el = − τ J i el . (142)We see that on time-scales much longer than τ , the r.h.s. of the above equation dominatesand we only have one conserved current, just as in the standard MHD example. By con-trast, on time-scales much shorter than τ , the l.h.s. of the above equation dominates and J el appears essentially conserved, which resembles the hot Maxwell example. Notice thatthere still exists a relationship among the components of the two higher-form currents (130)even though Q el is no longer conserved. We therefore see that in some sense the mixed ’tHooft anomaly still exists. We thus claim that the correct way to construct a theory ofMHD with non-vanishing charge density is to require that Q el and Q mag exhibit a mixed ’tHooft anomaly and that Q el exhibit a non-equilibrium ’t Hooft anomaly.
4. Gapped diffusion
We now consider the situation in which both Q el and Q mag are not conserved. Workingin the ϕ -picture, we take Q mag to be both unbroken and possess a non-equilibrium ’t Hooftanomaly. Therefore we must impose synchronous gauge ( ϕ s = 0 ) and endow ϕ s with thetime-independent gauge symmetries ϕ si → ϕ si + κ i ( ~x ) , (143) If we included the conservation of the stress-energy tensor, the second equation of (141) would tell us thatthe charge in each fluid element is ‘locked in’ at leading order in the derivative expansion. If higher-ordercorrections are considered, this equation becomes diffusive. κ i ( ~x ) . Notice that ϕ ai is an allowed building-block. The action consisting ofrelevant and marginal terms is then I = Z d x (cid:18) i Dβ ~E ′ a + ~B ′ a · ~B ′ − D ~E ′ a · ˙ ~E ′ + 1 τ ~ϕ a · ~B ′ + iτ β ~ϕ a (cid:19) . (144)The equations of motion are therefore (cid:18) ∂ t + 1 τ (cid:19) ~B ′ = D ~ ∇ ~B ′ , (145)which is a gapped diffusion equation. Thus, there are no gapless excitations.The higher-form currents are given by (135), such that j el is not gauge invariant and istherefore not a physical current. But now, j mag is not conserved, namely ∂ µ j µi mag = − τ j i mag . (146)Thus, we see explicitly that there are no conserved currents in this theory, which is whythere are no gapless excitation.
5. Legendre transform for unbroken symmetries
We have seen that when a p - and d − p − -form current are conserved and enjoy a mixedanomaly, it is possible to perform a Legendre transform to convert the theory involving the p -form Goldstone into a dual theory involving the d − p − -form Goldstone. Because theseactions are related by a Legendre transform, we can confidently say that they are physicallyequivalent. By contrast, we merely provided general arguments that the two distinct actionsfor MHD given by (133) and (137) were in some sense physically equivalent. Here we willshow that there is a duality relating these two actions.Begin by considering (133) and replace ~E ′ s → ~e s and ~B ′ s → ~b s , where ~e s and ~b s arefundamental fields. We will work in the retarded-advanced basis and drop the r -subscripton the retarded fields. Notice that ~e always appears with a time-derivative, meaning thatthere is a gauge symmetry ~e → ~e + ~ξ ( ~x ) , (147)for arbitrary time-independent ~ξ . Following (117), we would like to construct the auxiliaryaction I AUX ? = Z d x (cid:18) i Dβ ~e a + ~b a · ~b − D~e a · ˙ ~e + ~e · ~E a + ~e a · ~E − ~b · ~B a − ~b a · ~B (cid:19) , (148)where ~E s and ~B s are given by (123). Unfortunately this action contains the term ~e · ~E a = ~e · (cid:16) ~ ∇ A a − ∂ t ~A a (cid:17) , (149)32hich is not gauge invariant. If, however, we impose A a = 0 , then integration by partsremoves the gauge non-invariant term. We are then left with the auxiliary action I AUX = Z d x (cid:18) i Dβ ~e a + ~b a · ~b − D~e a · ˙ ~e + ˙ ~e · ~A a + ~e a · ~E − ~b · ~B a − ~b a · ~B (cid:19) . (150)Notice that the equations of motion for ~A s and A yield ~ ∇ × ~b s = − ˙ ~e s , ~ ∇ · ~e a = 0 , (151)which, with appropriate gauge choice for (147), imply that there exist one-forms ϕ s suchthat ~e s = ~E ′ s , ~b s = ~B ′ s , (152)where ~E ′ s and ~B ′ s are given by (124). We thus recover the original action (133).Conversely, to find the dual action, we may instead integrate out ~e s and ~b s . The equationsfor ~b s are very straight-forward, ~b s = ~B s . (153)The equations for ~e s are Diβ ~e a − D ˙ ~e + ~E = 0 , ∂ t (cid:16) D~e a − ~A a (cid:17) = 0 . (154)The second equation can be integrated over time to give D~e a + ~λ ( ~x ) = ~A a , where ~λ is anarbitrary integration function. But the SK boundary condition mandates that all advancedfields vanish in the infinite future, forcing ~λ = 0 , which yields ~e a = ~A a . (155)And finally, the first equation of (154) can be solved to give ˙ ~e = 2 iDβ ~A a + 1 D ~E. (156)Plugging the solutions (153), (155), and (156) into the auxiliary action (150), we obtain thedual action I DUAL = Z d x (cid:18) − c s ~B a · ~B + 1 τ ~A a · ~E + iτ β ~A a (cid:19) , (157)where we have identified D = τ c s and re-scaled the fields by ~A s → c s ~A s . At this stage wemay gauge-fix A = 0 with no consequence as the resulting equations of motion from thiscomponent will be entirely redundant. Notice that this dual action exactly matches (137)except for the term ~E a · ~E . The reason this term is missing has to do with power-counting:in the original action (133), the dynamics are diffusive, so we consider ∂ t ∼ ~ ∇ . If τ is large,that is /τ is of order the energy cutoff for the EFT, then we should expect ∂ t ∼ ~ ∇ inthe dual action (157). In such a case, the term ~E a · ~E must be considered irrelevant, so weought not include it in the leading-order action. By contrast, when we constructed (137)from symmetry considerations, we assumed that the non-equilibrium ’t Hooft anomaly wasweak, meaning that /τ is much less than the cutoff energy. As a result, we may consider ∂ t ∼ ~ ∇ , making the term ~E a · ~E marginal. 33 II. DUAL SUPERFLUIDS
In this section, we will construct the dual two-form EFT for -dimensional superfluidsat zero and finite temperature. For a coset construction of the ordinary superfluid action,consult [32].
A. Zero temperature
In the dual description of the superfluid, we have, in addition to Poincaré symmetry,the U (1) two-form symmetry generator Q and U (1) zero-form generator N , which countsparticle number. Moreover, there is a mixed anomaly between Q and N , meaning that wecan only include one of them in our coset construction.First, suppose we take N to be in our coset. Then, the standard claim is that both Q and P are spontaneously broken, but their diagonal subgroup ¯ P = P + µN is preserved [53, 54].Boosts are also spontaneously broken but all other symmetries are preserved. Thus the mostgeneral coset element takes the form g ( x ) = e ix µ ¯ P µ e iπ ( x ) N e iη i ( x ) K i . (158)Notice that we can rewrite this group element as g ( x ) = e ix µ P µ e iψ ( x ) N e iη i ( x ) K i , ψ ≡ µt + π. (159)Thus, we can forget about the fact that P is broken; instead we may act as though only N is broken and then after the fact give ψ a time-dependent VEV. The coset constructionfor the ordinary superfluid action is already well-studied, so we will now change gears andconsider the dual theory.In the dual picture, we take Lorentz boosts and Q to be spontaneously broken and wesuppose our system exists at finite particle-number density. Because we are constructing anEFT describing a zero-temperature system, we need not bother with the SK formalism, sothere is just one copy of each field. Further, we define our coset on the physical spacetimedirectly and we only include Goldstones associated with broken symmetries. The mostgeneral coset element is g (Σ) = e ix µ P µ e iη i ( x ) K i e i Q R Σ2 A , (160)where Σ = { x, Σ } , for two-dimensional manifold Σ , and where A is a two-form field. Noticethat in the dual picture, we do not treat P as spontaneously broken. Much like (159) wemust instead give A ( x ) a spacetime-dependent VEV. It turns out that requiring A µν = h A µν i + ϕ µν , h A i i = 0 , h A ij i ∝ ǫ ijk x k , (161)is equivalent to having a dual theory at finite particle-number density [55]. Thus, it is thedual-theory equivalent of having P and N broken but their diagonal subgroup preserved inthe original picture (158).The resulting Maurer-Cartan form is Ω = iE ν ( P ν + ∇ ν η i ) + ω i J i + ⋆F Q , (162)34here E νµ = Λ νµ , ∇ µ η i = ( E − ) νµ [Λ − ∂ ν Λ] i ,ω iµ = 12 ǫ ijk [Λ − ∂ µ Λ] jk ,F µ = 12 ǫ µνλρ ∂ ν A λρ . (163)It is convenient to also define F µ = F ν E µν , which is boost-invariant.Now impose IH constraints to remove the boost Goldstones. In particular, fix F i . (164)The quickest way to solve this equation is to note that e ( ν ) µ = Λ µν form a set of orthonormalvectors such that e ( ν ) µ η µµ ′ e ( ν ′ ) µ ′ = η νν ′ . Thus, the IH constraints imply that F µ is orthogonalto e ( i ) µ and hence parallel to e (0) µ = Λ µ . We therefore have that Λ µ = F µ F , (165)where F = p − F µ F µ . In terms of the boost Goldstones η i , we have η i η tanh η = F i F , (166)where η = p ~η and we have used the fact that Λ = cosh η and Λ i = ( η i /η ) sinh η . Noticethat this equation is only non-singular because we supposed that A µν enjoys the VEV (161).With these IH constraints solved, we have just one invariant building-block at leadingorder in the derivative expansion, namely Y ≡ F t = p − F µ F µ . (167)The leading-order effective action is therefore S = Z d xL ( Y ) , (168)for some function L .Notice that the zero-form charge N associated with particle number conservation did notappear anywhere in this coset construction. The reason is that in the dual picture, theconserved current appears as J µU (1) = 12 ǫ µνλρ ∂ ν A λρ , (169)which is conserved off-shell as a mathematical identity. Similarly in the ordinary coset construction, the two-form symmetry does not appear. . Finite temperature We now turn our attention to the finite temperature case. The SSB patter is of coursethe same. The difference is that now we define the action on the fluid worldvolume andparameterize the full symmetry group with Goldstones even if they are associated withunbroken symmetries. At finite temperature, we define our action on the SK contour andas a result must have doubled field content. If, however, we are only interested in theleading-order action, then it turns out that the dynamical KMS symmetries force the non-equilibrium action to factorize as the difference of two ordinary actions, each with a singlecopy of the fields [32]. We will therefore work with just one copy of the fields. Higher-ordercorrections require two copies of the fields.The most general group element is g (Σ) = e iX µ ( σ ) P µ e iη i ( σ ) K i e iθ i ( σ ) J i e i Q R Σ2 A , (170)where Σ = { σ, Σ } , for two-dimensional manifold Σ . Then the Maurer-Cartan form is Ω = iE µ ( P µ + ∇ µ η i K i ) + iω i J i + i ⋆ F Q , (171)where E µM = ∂ M X ν [Λ R ] νµ , ∇ µ η i = ( E − ) Mµ [Λ − ∂ M Λ] j R ji ,ω iM = 12 ǫ ijk [(Λ R ) − ∂ M (Λ R )] jk ,F M = 12 ǫ MNP Q ∂ N A P Q , (172)such that Λ µν = ( e iη i K i ) µν and R ij = ( e iθ i J i ) ij .We can now impose IH constraints to remove the Lorentz Goldstones. First, to removeboost Goldstones, we impose (98), which can be solved to yield η i η tanh η = − ∂ X i ∂ X t , (173)where η ≡ p ~η . Second, to remove the rotation Goldstones, fix the constraint (100). Asmentioned previously, we need not solve this IH constraint for leading-order actions.After imposing these IH constraints, the leading-order building-blocks are as follows. Be-gin by defining the fluid worldvolume metric by (101). Then, we have the local temperature T given by (102) as an invariant building-block. Finally, we have that Y = p F M G MN F N , Z = F M G M , (174)are the remaining invariant building-blocks. Thus, transforming to the phsyical spaceitmecoordinates x µ = X µ , the leading-order action is S = Z d xL ( T, Y, Z ) . (175) We could have alternatively imposed IH constraints F M ( E − ) iM = 0 , in keeping with (164). At least atleading-order these two IH constraints end up yielding the same EFT. III. DUAL SOLIDS
In this section, we will construct the dual two-form EFTs for solids at zero and finitetemperature. The two form charges count the number of crystal defects of the lattice.
A. Zero temperature
At zero temperature, the standard EFT for a solid involves three Goldstone modes φ i for i = 1 , , , corresponding to spontaneously broken translations [2, 32, 56, 57]. At leadingorder in the derivative expansion, we have S = Z d xP ( γ ij ) , γ ij = ∂ µ φ i ∂ µ φ j , (176)for some function P . Note that these Goldstones are invariant under the shift symmetries φ i → φ i + c i , (177)for constant c i . Thus this effective field theory has three internal U (1) symmetries, whichhave the interpretation of lattice momentum. Let P i be the generators of these U (1) sym-metries. We additionally have three distinct two-form symmetries with conserved currents ~ J µνλ = ǫ µνλρ ∂ ρ ~φ. (178)It is the aim of this subsection to take these two-form symmetries as the starting point.Recall that SSB pattern of solids is as follows. Boosts and rotations are spontaneouslybroken. Physical spatial translations and lattice-momentum translations are spontaneouslybroken, but their diagonal subgroup P i + P i are preserved, and temporal translations arepreserved. In the dual picture, we do not make use of the lattice momentum generators P i and hence we cannot treat them or P i as spontaneously broken. We must, however, stillcontinue to treat boosts and rotations as spontaneously broken.In the dual picture, in addition to Poincaré symmetry, we have a three-vector of spon-taneously broken two-form symmetries with generator ~ Q that enjoy mixed anomalies with ~ P . As a result, if we include Goldstones for ~ Q , we cannot have Goldstones for ~ P in thecoset. Since we are working at zero temperature, we only need one copy of the fields, we canformulate our EFT directly on the physical spacetime manifold, and we need only includeGoldstones for spontaneously broken symmetries. The most general coset element is g (Σ) = e ix µ P µ e iη i ( x ) K i e iθ i ( x ) J i e i ~ Q · R Σ2 ~ϕ , (179)where Σ = { x, Σ } , for two-dimensional manifold Σ . Then the resulting Maurer-Cartanform is Ω = iE ν ( P ν + ∇ ν η i + ∇ ν θ i J i + ⋆ ~F · ~ Q ) , (180)37here E νµ = (Λ R ) νµ , ∇ µ η i = ( E − ) νµ [(Λ R ) − ∂ ν (Λ R )] i , ∇ µ θ i = 12 ǫ ijk ( E − ) νµ [(Λ R ) − ∂ ν (Λ R )] jk ,~ F µ = E µν ~F ν , (181)such that ~F µ ≡ ǫ µνλρ ∂ ν ~ϕ λρ , Λ µν = ( e iη i K i ) µν , and R ij = ( e iθ i J i ) ij .We can now impose IH constraints to remove the Lorentz Goldstones. To remove therotation Goldstones, we impose ǫ ijk F jk = 0 , (182)which can be solved to give F ij = ( Y / ) ij , Y ij = F iµ F jµ , (183)where Y / is the matrix square root of Y . Then, to remove the boost Goldstones, we impose F it = 0 , (184)which can be solved to give η i η tanh η = − F jt ( Y − / ) ji . (185)We are thus left with only one set of invariant building-blocks, namely Y ij . The leading-orderdual action is then S DUAL = Z d xL ( Y ij ) , (186)for some function L . Notice how both the standard solid action and the dual solid actiondepend on symmetric × matrices. Finally, as in the superfluid case, we should expectthe two-form fields to have non-trivial background values. In particular, we want F iµ ∝ η iν .We therefore suppose that (cid:10) ϕ i j (cid:11) = 0 , (cid:10) ϕ ijk (cid:11) ∝ ǫ ijk t. (187) B. Finite temperature
At finite temperature, the symmetry-breaking pattern is unchanged; however, lattice mo-mentum generated by P i need not be conserved. If it is conserved, we have the phenomenonof second sound; otherwise there is no second sound [31, 58].Since our EFTs now exist at finite temperature, we have two copies of the fields, inparticular, we have two copies of the solid fields φ is for s = 1 , . Alternatively, we have in The indices i, j = 1 , , are now playing two roles: the first is the role of spatial rotation index and thesecond is to enumerate the components of ~ Q . We can get away with this redundant notation becauserotations are spontaneously broken. φ ir and φ ia . If lattice momentum is conserved, then both φ ir and φ ia must always appear in the action with at least one derivative, namely ∂ µ φ ir and ∂ µ φ ia .If, however, lattice momentum is not conserved, but P i is still a symmetry, then φ ia mayappear without derivatives even as φ ir must always appear with a derivative. Recall that φ ir are the classical fields, while φ ia encode information about fluctuations. Thus, whether ornot lattice momentum is conserved, we have the three-form conserved currents ~ J µνλρ = ǫ µνλρ ∂ ρ ~φ r . (188)If we then switch to the dual picture and work with two-form fields ~ϕ sµν , the currentsassociated with P i are now given by ~J µ = 12 ǫ µνλρ ∂ ν ~ϕ rλρ . (189)But notice that ∂ µ ~J µ = 0 identically because partial derivatives commute. It thereforeseems that the dual picture automatically has conserved lattice momentum; however, thisneed not be true if J µ is not gauge invariant. Notice that if ~ Q are unbroken, then we havethe gauge symmetries of the form (75), which do not leave J µ invariant. Thus, from thedual perspective, we have conserved lattice momentum and hence second sound if and onlyif ~ Q are spontaneously broken.We now proceed to constructing the dual actions for solids with and without second sound.As in the zero-temperature case, we have spontaneously broken boosts and rotations, butwe do not treat translations as broken. We also have the three-form symmetry charges ~ Q .The most general group element is g (Σ) = e iX µ ( σ ) P µ e iη i ( σ ) K i e iθ i ( σ ) J i e i ~ Q · R Σ2 ~ϕ , (190)where Σ = { σ, Σ } , for two-dimensional manifold Σ . And the Maurer-Cartan form is Ω = iE µ ( P µ + ∇ µ η i K i + ∇ µ θ i J i ) + i ⋆ ~F · ~ Q , (191)where E µM = ∂ M X ν [Λ R ] νµ , ∇ µ η i = ( E − ) Mµ [Λ − ∂ M Λ] j R ji , ∇ µ θ i = 12 ǫ ijk ( E − ) Mµ [(Λ R ) − ∂ M (Λ R )] jk ,~F M = 12 ǫ MNP Q ∂ N ~ϕ P Q , (192)such that Λ µν = ( e iη i K i ) µν and R ij = ( e iθ i J i ) ij . It will be convenient to define F iµ = E µM F iM . We can take ~ϕ sµν to be the pullback of ~ϕ sMN to the physical spacetime via the inverse of the map X µs ( σ ) . Instead of treating spatial translations as spontaneously broken, we endow the two-form fields with theVEVs (187) E i = 0 = ⇒ η i η tanh η, (193)and the IH constraints necessary to remove rotation Goldstones are ǫ ijk F jk = 0 = ⇒ F ij = ( y / ) ij , (194)where y ji = F iµ F jµ .The invariant building-blocks are as follows. First, we have the local temperature givenby (102). Next, we have the symmetric matrix y ij as defined above; notice that both ofthese terms are invariant under the gauge symmetry ϕ IJ ( σ ) → ϕ IJ ( σ ) + κ IJ ( σ K ) . (195)Thus these building-blocks are invariant whether or not ~ Q are spontaneously broken. Lastly,we have the building-block z i ≡ F it = u µ F iµ , where u µ ≡ T ∂X µ ∂σ , (196)but it is not invariant under the gauge symmetry (195). Thus it is only an allowed building-block if ~ Q are spontaneously broken, that is, when lattice momentum is conserved.Transforming to physical spacetime, the dual actions for solids are therefore S DUAL = Z d xL ( T, y ij , z i ) , (197)describing a solid with conserved lattice momentum, and S DUAL = Z d xL ( T, y ij ) , (198)describing a solid without conserved lattice momentum. The building-block z i captures therelative motion of the fluid and the lattice when second sound is present. IX. DISCUSSION
In this work, we presented a systematic procedure to formulate non-equilibrium effec-tive actions for hydrodynamic (gapless) and quasi-hydrodynamic (weakly-gapped) excita-tions involving p -form symmetries. To aid in the systematization, we employed the cosetconstruction, generalizing it to account for higher-form symmetries in non-equilibrium sys-tems. In addition, we extended the coset construction to account for theories exhibitingnon-equilibrium ’t Hooft anomalies, which are exact symmetries of the action but haveno conserved Noether current. As the name suggests these anomalies can only occur innon-equilibrium systems.Further, we noticed some interesting features of non-equilibrium EFTs that involve higher-form symmetries: Second sound emerges when a thermal cloud of solid phonons can flow as a fluid independently of thesolid lattice [58, 59]. Then we have both fluid sound waves and solid (i.e. lattice) sound waves.
40 When implementing the coset construction, the Maurer-Cartan form associated witha p -form symmetry, which we denote by Ω p +1 , is a p + 1 -form. If various higher-formsymmetries of different p exist in a system, then the resulting Maurer-Cartan form isa mixed-form in the sense that it can be expressed as a sum of the form Ω = P p Ω p +1 .• Whenever a U (1) p -form symmetry is spontaneously broken, there exists a d − p − -form U (1) symmetry that is also spontaneously broken. Further, the p -form and d − p − -form symmetries exhibit a mixed ’t Hooft anomaly (not to be confusedwith a non-equilibrium ’t Hooft anomaly). Moreover, if there exist p -form and a d − p − -form U (1) symmetries with mixed ’t Hooft anomaly, then both higher-formsymmetries must be spontaneously broken. Such a phenomenon holds both at zeroand finite temperature.• Whenever such a mixed anomaly exists, it is impossible to have the p -form and the d − p − -form Goldstones in the same action. There are therefore dual physically equivalentdescriptions. In particular, one effective action involves the p -form Goldstone and theother involves the d − p − -form Goldstone.• In non-equilibrium systems, Goldstone-like excitations exist even for unbroken p -formsymmetries. The distinguishing feature between spontaneously broken and unbroken p -form Goldstones is the emergence of a gauge symmetry. These findings are a naturalextension of those in [32]. The conserved d − p − -form charge that would exist if the p -form symmetry were spontaneously broken becomes gauge non-invariant and thusfails to be a physical charge.• There is a second way to construct an action with an unbroken U (1) p -form symmetry.We work in the dual picture involving the d − p − -form Goldstone and give it a non-equilibrium ’t Hooft anomaly. This leads to non-conservation of the d − p − -formcurrent but does not affect the conservation of the p -form current. As a result, the p -form symmetry, as it enjoys no mixed ’t Hooft anomaly, cannot be spontaneouslybroken. Further, we can give the d − p − -form current an arbitrarily weak non-equilibrium ’t Hooft anomaly, meaning that we can, in a certain sense, have a weakly unbroken p -form symmetry.• Lastly, using the coset construction, we were able to reproduce the well-known Chern-Simons action. That is, from symmetry principles alone, we were able to construct apurely topological theory.This work admits generalizations and applications in many directions. First, there isan interesting consequence of these findings that warrants further investigation: in non-equilibrium systems, whether or not a given symmetry is spontaneously broken, may dependon the time-scale over which it is observed. Consider the situation of a p - and a d − p − -form symmetry with mixed ’t Hooft anomaly. If we give the d − p − -form symmetry anon-equilibrium ’t Hooft anomaly, the strength of the anomaly is characterized by a time-scale τ in the sense that for frequency ωτ ≪ , the current is not conserved, whereas for ωτ ≫ , the current appears conserved. In this way, the non-equilibrium ’t Hooft anomaly41learly exists in the low-frequency regime, meaning that the p -form symmetry is unbro-ken. However in the high-frequency regime, the non-equilibrium ’t Hooft anomaly is highlysuppressed, meaning that the p -form symmetry will appear broken. In this paper, we sawthat our quadratic models for electromagnetic systems behave in this way, but it appears tobe a more general phenomenon. Second, the existing non-equilibrium effective actions formagnetohydrodynamic systems do not account for situations in which there is a build-up ofelectric charge. By exploiting principles of mixed and non-equilibrium ’t Hooft anomalies, wewere able to construct a quadratic magnetohydrodynamic action that permits large electriccharge density, which exponentially decays to zero density. Such an effective action may ul-timately yield important insights into the nature of plasma instabilities, which are of greatimportance in fusion reactions. Third, higher-form symmetries are often associated withtopological phases of matter. The non-equilibrium coset construction involving higher-formsymmetries may therefore prove to be a powerful tool for constructing theories for topolog-ical phases beyond the Chern-Simons action. Fourth, we have only considered one kind ofmixing between various p -form symmetries, namely we considered mixed ’t Hooft anomalies.However, there are other, more complicated relationships that can exist among higher-formsymmetries, like two-groups [45]. Extending the coset construction to account for objectssuch as two-groups is of significant theoretical interest. Finally, higher-form symmetries havemany applications in a wide variety of areas. For example, they can be used to constructeffective field theories of superfluid vortices [55] and we expect they should play a similarrole in describing the dynamics of disclinations in solids in the non-equilibrium EFT formal-ism [60]. Understanding such topological objects from the perspective of non-equilibriumEFT may have important applications in many areas of physics including understandingvortices in certain dark matter models [61]. Acknowledgments:
I would like to thank Alberto Nicolis and Lam Hui for their wonderfulmentorship and Sašo Grozdanov and Matteo Baggioli for insightful conversations. This workwas partially supported by the US Department of Energy grant DE-SC0011941. [1] S. Dubovsky, L. Hui, A. Nicolis and D. T. Son,
Effective field theory for hydrodynam-ics: thermodynamics, and the derivative expansion,
Phys. Rev. D , 085029 (2012)doi:10.1103/PhysRevD.85.085029 [arXiv:1107.0731 [hep-th]].[2] A. Nicolis, R. Penco, F. Piazza and R. Rattazzi, Zoology of condensed matter: Framids,ordinary stuff, extra-ordinary stuff,
JHEP , 155 (2015) doi:10.1007/JHEP06(2015)155[arXiv:1501.03845 [hep-th]].[3] A. Nicolis, R. Penco, F. Piazza and R. A. Rosen,
More on gapped Goldstones at finitedensity: More gapped Goldstones,
JHEP , 055 (2013) doi:10.1007/JHEP11(2013)055[arXiv:1306.1240 [hep-th]].[4] A. Nicolis, R. Penco and R. A. Rosen,
Relativistic Fluids, Superfluids, Solids andSupersolids from a Coset Construction,
Phys. Rev. D , no. 4, 045002 (2014)doi:10.1103/PhysRevD.89.045002 [arXiv:1307.0517 [hep-th]].
5] A. Nicolis,
Low-energy effective field theory for finite-temperature relativistic superfluids, arXiv:1108.2513 [hep-th].[6] S. Grozdanov and J. Polonyi, “Viscosity and dissipative hydrodynamics from effectivefield theory,” Phys. Rev. D , no.10, 105031 (2015) doi:10.1103/PhysRevD.91.105031[arXiv:1305.3670 [hep-th]].[7] H. Liu and P. Glorioso, Lectures on non-equilibrium effective field theories and fluctuatinghydrodynamics,
PoS TASI , 008 (2018) doi:10.22323/1.305.0008 [arXiv:1805.09331 [hep-th]].[8] P. Glorioso, M. Crossley and H. Liu,
Effective field theory of dissipative fluids (II):classical limit, dynamical KMS symmetry and entropy current,
JHEP , 096 (2017)doi:10.1007/JHEP09(2017)096 [arXiv:1701.07817 [hep-th]].[9] M. Crossley, P. Glorioso and H. Liu,
Effective field theory of dissipative fluids,
JHEP ,095 (2017) doi:10.1007/JHEP09(2017)095 [arXiv:1511.03646 [hep-th]].[10] P. Glorioso, H. Liu and S. Rajagopal,
Global Anomalies, Discrete Symmetries, andHydrodynamic Effective Actions,
JHEP , 043 (2019) doi:10.1007/JHEP01(2019)043[arXiv:1710.03768 [hep-th]].[11] P. Gao, P. Glorioso and H. Liu,
Ghostbusters: Unitarity and Causality of Non-equilibriumEffective Field Theories, arXiv:1803.10778 [hep-th].[12] P. Glorioso and H. Liu,
The second law of thermodynamics from symmetry and unitarity, arXiv:1612.07705 [hep-th].[13] M. Blake, H. Lee and H. Liu,
A quantum hydrodynamical description for scrambling and many-body chaos,
JHEP , 127 (2018) doi:10.1007/JHEP10(2018)127 [arXiv:1801.00010 [hep-th]].[14] M. Harder, P. Kovtun and A. Ritz,
On thermal fluctuations and the generating func-tional in relativistic hydrodynamics,
JHEP , 025 (2015) doi:10.1007/JHEP07(2015)025[arXiv:1502.03076 [hep-th]].[15] N. Banerjee, J. Bhattacharya, S. Bhattacharyya, S. Jain, S. Minwalla and T. Sharma,
Con-straints on Fluid Dynamics from Equilibrium Partition Functions,
JHEP , 046 (2012)doi:10.1007/JHEP09(2012)046 [arXiv:1203.3544 [hep-th]].[16] K. Jensen, N. Pinzani-Fokeeva and A. Yarom,
Dissipative hydrodynamics in superspace,
JHEP , 127 (2018) doi:10.1007/JHEP09(2018)127 [arXiv:1701.07436 [hep-th]].[17] K. Jensen, R. Marjieh, N. Pinzani-Fokeeva and A. Yarom,
A panoply of Schwinger-Keldysh transport,
SciPost Phys. , no. 5, 053 (2018) doi:10.21468/SciPostPhys.5.5.053[arXiv:1804.04654 [hep-th]].[18] K. Jensen, R. Marjieh, N. Pinzani-Fokeeva and A. Yarom, An entropy current in superspace,
JHEP , 061 (2019) doi:10.1007/JHEP01(2019)061 [arXiv:1803.07070 [hep-th]].[19] K. Jensen, M. Kaminski, P. Kovtun, R. Meyer, A. Ritz and A. Yarom,
To-wards hydrodynamics without an entropy current,
Phys. Rev. Lett. , 101601 (2012)doi:10.1103/PhysRevLett.109.101601 [arXiv:1203.3556 [hep-th]].[20] P. Kovtun, G. D. Moore and P. Romatschke,
Towards an effective action for relativistic dissipa-tive hydrodynamics,
JHEP , 123 (2014) doi:10.1007/JHEP07(2014)123 [arXiv:1405.3967[hep-ph]].
21] F. M. Haehl, R. Loganayagam and M. Rangamani,
Two roads to hydrodynamic effective ac-tions: a comparison, arXiv:1701.07896 [hep-th].[22] F. M. Haehl, R. Loganayagam and M. Rangamani,
Effective Action for Relativistic Hy-drodynamics: Fluctuations, Dissipation, and Entropy Inflow,
JHEP , 194 (2018)doi:10.1007/JHEP10(2018)194 [arXiv:1803.11155 [hep-th]].[23] F. M. Haehl, R. Loganayagam and M. Rangamani,
The Fluid Manifesto: Emergent symme-tries, hydrodynamics, and black holes,
JHEP , 184 (2016) doi:10.1007/JHEP01(2016)184[arXiv:1510.02494 [hep-th]].[24] M. Hongo, S. Kim, T. Noumi and A. Ota,
Effective Lagrangian for Nambu-Goldstone modesin nonequilibrium open systems, arXiv:1907.08609 [hep-th].[25] D.V. Volkov,
Phenomenological Lagrangians , Fiz.Elem.Chast.Atom.Yadra 4 (1973) 3–41.[26] T. Hayata, Y. Hidaka, T. Noumi and M. Hongo,
Relativistic hydrodynamics from quantum fieldtheory on the basis of the generalized Gibbs ensemble method,
Phys. Rev. D , no. 6, 065008(2015) doi:10.1103/PhysRevD.92.065008 [arXiv:1503.04535 [hep-ph]].[27] M. Hongo, Path-integral formula for local thermal equilibrium,
Annals Phys. , 1 (2017)doi:10.1016/j.aop.2017.04.004 [arXiv:1611.07074 [hep-th]].[28] M. Hongo,
Nonrelativistic hydrodynamics from quantum field theory: (I) Normal fluid com-posed of spinless Schrödinger fields, doi:10.1007/s10955-019-02224-4 arXiv:1801.06520 [cond-mat.stat-mech].[29] M. J. Landry, “Dynamical chemistry: non-equilibrium effective actions for reactive fluids,”[arXiv:2006.13220 [hep-th]].[30] M. Baggioli and M. Landry, “Effective Field Theory for Quasicrystals and Phasons Dynamics,”SciPost Phys. , no.5, 062 (2020) doi:10.21468/SciPostPhys.9.5.062 [arXiv:2008.05339 [hep-th]].[31] M. J. Landry, “Second sound and non-equilibrium effective field theory,” [arXiv:2008.11725[hep-th]].[32] M. J. Landry, “The coset construction for non-equilibrium systems,” JHEP , 200 (2020)doi:10.1007/JHEP07(2020)200 [arXiv:1912.12301 [hep-th]].[33] D. Gaiotto, A. Kapustin, N. Seiberg and B. Willett, “Generalized Global Symmetries,” JHEP , 172 (2015) doi:10.1007/JHEP02(2015)172 [arXiv:1412.5148 [hep-th]].[34] J. Zhao, Z. Yan, M. Cheng and Z. Y. Meng, “Higher-form symmetry breaking at Ising transi-tions,” [arXiv:2011.12543 [cond-mat.str-el]].[35] E. Lake, “Higher-form symmetries and spontaneous symmetry breaking,” [arXiv:1802.07747[hep-th]].[36] Y. Hidaka, Y. Hirono and R. Yokokura, “Counting Nambu-Goldstone modes of higher-formglobal symmetries,” [arXiv:2007.15901 [hep-th]].[37] D. M. Hofman and N. Iqbal, “Goldstone modes and photonization for higher form symmetries,”SciPost Phys. , no.1, 006 (2019) doi:10.21468/SciPostPhys.6.1.006 [arXiv:1802.09512 [hep-th]].[38] L. V. Delacrétaz, D. M. Hofman and G. Mathys, “Superfluids as Higher-form Anomalies,”SciPost Phys. , 047 (2020) doi:10.21468/SciPostPhys.8.3.047 [arXiv:1908.06977 [hep-th]].[39] D. Harlow and H. Ooguri, “Symmetries in quantum field theory and quantum gravity,” arXiv:1810.05338 [hep-th]].[40] S. Grozdanov, D. M. Hofman and N. Iqbal, “Generalized global symmetriesand dissipative magnetohydrodynamics,” Phys. Rev. D , no.9, 096003 (2017)doi:10.1103/PhysRevD.95.096003 [arXiv:1610.07392 [hep-th]].[41] S. Grozdanov and N. Poovuttikul, “Generalized global symmetries in states with dynamicaldefects: The case of the transverse sound in field theory and holography,” Phys. Rev. D ,no.10, 106005 (2018) doi:10.1103/PhysRevD.97.106005 [arXiv:1801.03199 [hep-th]].[42] P. Glorioso and D. T. Son, “Effective field theory of magnetohydrodynamics from generalizedglobal symmetries,” [arXiv:1811.04879 [hep-th]].[43] S. Grozdanov, A. Lucas and N. Poovuttikul, “Holography and hydrodynamics with weaklybroken symmetries,” Phys. Rev. D , no.8, 086012 (2019) doi:10.1103/PhysRevD.99.086012[arXiv:1810.10016 [hep-th]].[44] S. Grozdanov and N. Poovuttikul, “Generalised global symmetries in holography: mag-netohydrodynamic waves in a strongly interacting plasma,” JHEP , 141 (2019)doi:10.1007/JHEP04(2019)141 [arXiv:1707.04182 [hep-th]].[45] N. Iqbal and N. Poovuttikul, “2-group global symmetries, hydrodynamics and holography,”[arXiv:2010.00320 [hep-th]].[46] V. I. Ogievetsky, Nonlinear realizations of internal and space-time symmetries , in X-th winterschool of theoretical physics in Karpacz, Poland. 1974.[47] E. Ivanov and V. Ogievetsky,
The Inverse Higgs Phenomenon in Nonlinear Realizations ,Teor.Mat.Fiz. 25 no. 2, (1975) 1050–1059.[48] I. Low and A. V. Manohar,
Spontaneously broken space-time symmetries and Goldstone’s theo-rem,
Phys. Rev. Lett. , 101602 (2002) doi:10.1103/PhysRevLett.88.101602 [hep-th/0110285].[49] Steven Weinberg, The quantum theory of fields. Vol. 2: Modern applications , (CambridgeUniversity Press, 1996).[50] L. V. Delacrétaz, S. Endlich, A. Monin, R. Penco and F. Riva, (Re-)Inventing theRelativistic Wheel: Gravity, Cosets, and Spinning Objects,
JHEP , 008 (2014)doi:10.1007/JHEP11(2014)008 [arXiv:1405.7384 [hep-th]].[51] M. J. Landry and G. Sun, “The coset construction for particles of arbitrary spin,”[arXiv:2010.11191 [hep-th]].[52] S. Endlich, A. Nicolis and R. Penco,
Ultraviolet completion without symmetry restoration,
Phys.Rev. D , no. 6, 065006 (2014) doi:10.1103/PhysRevD.89.065006 [arXiv:1311.6491 [hep-th]].[53] M. Greiter, F. Wilczek and E. Witten, Hydrodynamic Relations in Superconductivity,
Mod.Phys. Lett. B , 903 (1989). doi:10.1142/S0217984989001400[54] D. T. Son, Low-energy quantum effective action for relativistic superfluids, hep-ph/0204199.[55] B. Horn, A. Nicolis and R. Penco, “Effective string theory for vortex lines in fluids and super-fluids,” JHEP , 153 (2015) doi:10.1007/JHEP10(2015)153 [arXiv:1507.05635 [hep-th]].[56] J. Armas and A. Jain, “Hydrodynamics for charge density waves and their holographic duals,”Phys. Rev. D , no.12, 121901 (2020) doi:10.1103/PhysRevD.101.121901 [arXiv:2001.07357[hep-th]].[57] J. Armas and A. Jain, “Viscoelastic hydrodynamics and holography,” JHEP , 126 (2020)doi:10.1007/JHEP01(2020)126 [arXiv:1908.01175 [hep-th]].
58] L. P. Pitaevski˘ı,
Second Sound in Solids,
Soviet Physics Uspekhi , 3 1968doi:10.1070/pu1968v011n03abeh003839[59] A. Cepellotti, G. Fugallo, L. Paulatto, M. Lazzeri, F. Mauri, N. Marzari, “Phonon hydrody-namics in two-dimensional materials,” Nature Communications , no. 1, 6400 (2015) DOI:10.1038/ncomms7400[60] A. J. Beekman, J. Nissinen, K. Wu, K. Liu, R. Slager, Z. Nussinov, V. Cvetkovic, J. Zaanen,“Dual gauge field theory of quantum liquid crystals in two dimensions” [arXiv:1603.04254[cond-mat.str-el]].[61] L. Hui, A. Joyce, M. J. Landry and X. Li, “Vortices and waves in light dark matter,”[arXiv:2004.01188 [astro-ph.CO]]., no. 1, 6400 (2015) DOI:10.1038/ncomms7400[60] A. J. Beekman, J. Nissinen, K. Wu, K. Liu, R. Slager, Z. Nussinov, V. Cvetkovic, J. Zaanen,“Dual gauge field theory of quantum liquid crystals in two dimensions” [arXiv:1603.04254[cond-mat.str-el]].[61] L. Hui, A. Joyce, M. J. Landry and X. Li, “Vortices and waves in light dark matter,”[arXiv:2004.01188 [astro-ph.CO]].