Published for SISSA by Springer
Received: July 11, 2016 Revised: January 11, 2017 Accepted: February 16, 2017 Published: February 23, 2017
A new approach to non-Abelian hydrodynamics
Jose J. Fern andez-Melgarejo,a Soo-Jong Reyb;c;d and Piotr Sur owkaa;e
aCenter for the Fundamental Laws of Nature, Harvard University,
Cambridge, MA 02138, U.S.A.
bSchool of Physics & Astronomy and Center for Theoretical Physics, Seoul National University, Seoul, 08826 Korea
cDepartment of Fundamental Sciences, University of Science and Technology, Daejeon, 34113 Korea
dCenter for Gauge, Gravity & Strings, Institute for Basic Sciences, Daejeon, 34047 Korea
eMax-Planck-Institut fur Physik (Werner-Heisenberg-Institut), Fohringer Ring 6, D-80805 Munich, Germany
E-mail: mailto:[email protected]
Web End [email protected] , mailto:[email protected]
Web End [email protected] , mailto:[email protected]
Web End [email protected]
Abstract: We present a new approach to describe hydrodynamics carrying non-Abelian macroscopic degrees of freedom. Based on the Kaluza-Klein compacti cation of a higher-dimensional neutral dissipative uid on a manifold of non-Abelian isometry, we obtain a four-dimensional colored dissipative uid coupled to Yang-Mills gauge eld. We derive transport coe cients of resulting colored uid, which feature non-Abelian character of color charges. In particular, we obtain color-speci c terms in the gradient expansions and response quantities such as the conductivity matrix and the chemical potentials. We argue that our Kaluza-Klein approach provides a robust description of non-Abelian hydrodynamics, and discuss some links between this system and quark-gluon plasma and uid/gravity duality.
Keywords: Gauge-gravity correspondence, Holography and condensed matter physics (AdS/CMT), Holography and quark-gluon plasmas
ArXiv ePrint: 1605.06080 Denn die Menschen glauben an die Wahrheit dessen, was ersichtlich stark geglaubt wird.
All truthful things are subject to interpretation. Which interpretation prevails at a given time is a function of power, not truth.
| Friedrich Nietzsche | The Will to Power
Open Access, c
[circlecopyrt] The Authors.
Article funded by SCOAP3. doi:http://dx.doi.org/10.1007/JHEP02(2017)122
Web End =10.1007/JHEP02(2017)122
JHEP02(2017)122
Contents
1 Introduction 1
2 Dissipative uid dynamics with Yang-Mills charge 5
3 Kaluza-Klein approach 73.1 Self-gravitating dissipative uid 73.2 Non-Abelian Kaluza-Klein reduction 9
4 Charged uid coupled to Maxwell theory 104.1 Reduction on Abelian group manifold 114.2 Abelian reduction of energy-momentum tensor 13
5 Colored uid coupled to Yang-Mills theory 145.1 Compacti cation on SU(2) group manifold 145.2 Field equations for Yang-Mills plasma 165.3 Conservation laws 17
6 Colored uid from non-Abelian reduction 196.1 Non-Abelian reduction of uid 196.2 Perfect colored uid 196.3 Entropy current 206.4 Non-Abelian dissipative uid 21
7 Outlooks 23
A Einstein equations on a group manifold 25A.1 General ansatz 25A.2 SU(2) group manifold 26A.3 Equations of motion 27
B Conservation laws 28B.1 Current conservation 29B.2 Lorentz force 29
1 Introduction
Hydrodynamics has been an e cient approach for the description of strongly interacting state of matter. This boosted the research and application of hydrodynamics models, such as transport phenomena or hydrodynamic instabilities. One aspect in hydrodynamics that
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has not been explored in detail yet is the dynamics of a colored uid charged under non-Abelian Yang-Mills gauge elds, where the constituents of the uid carry non-Abelian color charges and interact with non-Abelian vectors. Due to its non-Abelian nature, we expect that this system gives rise to a variety of physical phenomena richer than its Abelian counterpart, viz. Maxwell plasma. Nevertheless, the level of rigor in formulating the theoretical foundations of this model and the understanding of its ensuing physical properties are far lesser.
A robust description will contribute to the characterization of some important physical systems. For example, the quark-gluon plasma behaves as an almost perfect dense uid carrying SU(3) color charge. However, the detailed microscopic understanding of the equilibration mechanisms after the heavy-ion collisions is still left to be an outstanding problem. A transient phase in the equilibration process is reached when the system is at local thermal equilibrium with yet non-equilibrated colored quark and gluon degrees of freedom (DOFs). Most of the analysis done so far is based on kinetic theory [1{8] and on the single-particle approach [9]. Integrating out momentum, one obtains a covariant color continuity equation which, together with the mechanical conservation laws of the uid, constitute the main equations of the system. Still, the construction of the required collision terms which enter the Boltzmann equation is highly non-trivial and, except at weak coupling regime, there is no rst-principles derivation. In addition, the applicability of kinetic theory is valid for not-so-far from equilibration situations. Consequently, we conclude that kinetic theory is a useful complementary tool, requiring prior knowledge of the structure of the hydrodynamic equations.
Alternative approaches include the Poisson bracket formulation [10] and the action principle [11, 12] of ideal uid dynamics. In contrast, the study of dissipative e ects, which constitute an integral part of hydrodynamics is well understood only at the level of the equations of motion (EOMs). The description of these e ects at the level of an action requires placing the uid on the Schwinger-Keldysh contour [13], which leads to certain additional supersymmetric DOFs [14, 15].
Another aspect that sheds light on the understanding of hydrodynamic structure is the duality between uids and black holes [16{19]. This allowed us to discover previously neglected parity-breaking terms that were originated by quantum anomalies [20{22]. To study non-Abelian DOFs coupled to uids, we need a new background of black hole with non-Abelian Yang-Mills hair [23{25]. However, in AdS/CFT correspondence, local symmetry in the bulk gravity is mapped to global symmetry in the boundary theory. Therefore, as the background eld in the boundary theory is usually external and non-dynamical, we have no way of promoting non-Abelian global symmetries to gauge symmetries in the boundary theory. We note that some proposals have been put forward to modify the boundary conditions in such a way the the resulting boundary has dynamical elds [26]. However, these ideas have not been consistently embedded into the uid/gravity duality and may present additional di culties in the hydrodynamic formulation of non-Abelian uids.
For these reasons, we view this state of a air at odds: self-gravitating hydrodynamics, whose gravitational interaction is also intrinsically nonlinear, has been rigorously investigated in various contexts of relativistic astrophysics of compact objects [27] and cosmology
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of large-scale structures [28, 29]. We thus expect that non-Abelian hydrodynamics, at least at classical level, can also be rigorously formulated and investigated as much as the self-gravitating hydrodynamics. Such study would have a direct application to wider phenomena featuring non-Abelian DOFs such as the quark-gluon plasma [30] and the spintronics with strong spin-orbit coupling [31, 32].
In this work we propose a completely new approach to bypass all the above conceptual and technical di culties. We start from a neutral and dissipative uid coupled to Einstein gravity in D dimensions, which we assume is completely characterized. The idea is to perform a Kaluza-Klein (KK) compacti cation [33, 34] of this system and obtain a uid in d = D n dimensions whose constituents are charged under non-Abelian Yang-Mills elds,
where n is the dimension of the internal manifold. That is to say, we use KK dimensional reduction as a method to construct an ab initio description of non-Abelian hydrodynamics. The KK compacti cation mechanism endows the lower-dimensional system with a set of gauge elds, the so-called KK gauge elds. The compacti cation ansatz of internal manifold elucidate the resulting gauge symmetry of d-dimensional system. As we are interested in non-Abelian hydrodynamics, we will compactify on an SU(2) group manifold [35, 36]. Therefore, we will take n = dim(G) = 3, where G is the gauge group. We will perform this procedure on the EOMs of the starting higher-dimensional neutral uid, which include dissipative terms.1
Our approach is based on the non-Abelian Kaluza-Klein compacti cation on a SU(2) group manifold, which we interpret as an internal manifold whose isometries generate the non-Abelian color symmetry in the physical system. Since we start with a uid from the outset, the resulting theory is valid in the long-wavelength limit, coupled to new non-Abelian DOFs that the compacti cation generates.
KK compacti cation provided a robust tool for the understanding of the (hidden) structure and the dynamics of gravity-matter systems, which descends from a more fundamental theory such as string/M-theories. If we start with a fundamental theory in D dimensions de ned on a manifold MD, we can nd a stable solution of its equations of
motion of the form MD = Md [notdef] Xn, where d = (D n), Md is non-compact, reduced
spacetime, and Xn is a compact manifold of characteristic size R. At low energies, the compact space Xn is not accessible by direct observations: it would take excitations of energy E 1/R to probe spacetime structures of a scale of order R. If R is su ciently
small, this energy scale is gapped from the low-energy dynamics on Md. Nevertheless, the
properties of Xn will have important e ects on the reduced theory. As emphasized, if Xn is a manifold with isometry group G, then metric uctuations along the Killing directions of Xn generate Yang-Mills gauge elds with gauge group G, which will be present in the dynamics of the lower-dimensional theory.
From the viewpoint of KK theory, a novelty of our work is that we include energy-momentum tensor of dissipative uid, sourcing the Einstein eld equations. The procedure, however, must be self-consistent. A KK compacti cation is said to be consistent if all the
1Heavy-ion collisions and other phenomenologically relevant phenomena occur o the equilibrium and consequently, dissipative e ects are very important in their descriptions.
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Figure 1. Our starting system is a D-dimensional dissipative uid coupled to gravity (left). After KK compacti cation on a n-dimensional internal manifold with non-Abelian isometries, we obtain a d-dimensional dissipative uid that, apart from being coupled to gravity, is charged under dynamic non-Abelian Yang-Mills gauge elds (right).
solutions of the d-dimensional theory satisfy the D-dimensional EOMs. In this work, we also present the necessary conditions to achieve a consistent reduction of uid energy-momentum tensor.
Summarizing, the salient features of our approach are the following:
We apply the KK method to a neutral uid at the outset coupled to gravity, thus
bypassing kinetic theory.
The approach applies to dissipative uids, for the compacti cation is at the level of
equations of motion rather than action.
The proposed KK method \generates" dynamical (non-)Abelian gauge elds which
are self-consistently coupled to a charged/colored uid.
This mechanism provides an ab initio approach to (non-)Abelian hydrodynamics,
distinct from gauge-gravity duality or uid/gravity duality.
This paper is organized as follows. In section 2 we present the main results of our work: the dynamics of the system, its symmetries and its properties. In the following sections we explain the KK dimensional reduction and the method for obtaining our results. In particular, section 3 reviews the basics of relativistic hydrodynamics and provides the necessary set-up and notations for our calculations. In section 4, we review the dimensional reduction of the system Einstein-perfect uid on a circle. This results in a uid charged under a U(1) gauge eld. In section 5 we do the KK compacti cation on an SU(2) group manifold of the Einstein-dissipative uid system and study the conservation laws of the system. In section
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6 we evaluate our energy-momentum tensor and identify all the dissipative coe cients of the d-dimensional uid. In section 7 we explain the main properties of our system and discuss future directions we are currently investigating. Appendices provide the details of our computations.
2 Dissipative uid dynamics with Yang-Mills charge
In this section, we recapitulate the dynamics and the main properties of a d-dimensional dissipative uid that carries charges of non-Abelian Yang-Mills gauge group G.
We denote space-time indices by [notdef], , = 1, . . . , d, and the adjoint representation of the Yang-Mills group G indices by , , . . . = 1, . . . , dim(G).2
The energy-momentum tensor consists of two contributions by dissipative uid and non-Abelian Yang-Mills gauge elds:
T total =T uid + 12Q2c(x)
, (2.1)
where F is the non-Abelian eld strength of the gauge eld A , Qc refers to the coupling constant3 and repeated color indices are summed over. The uid energy-momentum tensor is further split to perfect uid and dissipative parts,
T uid = T perfect + T diss . (2.2)
The perfect uid contribution is given by
T perfect (x) = [[epsilon1](x) + p(x)] u u + p(x) , (2.3)
where p is pressure, [epsilon1] is energy density, u is the velocity eld and is Minkowski metric.
As for the dissipative part T diss , in this description we will not choose any speci c frame and will consider a generic energy-momentum tensor. Though we will make further explicit assumptions in section 6.4, we can generalize our results to any frame independent prescription.
The thermodynamic relation for the SU(2) charged perfect uid after the KK compacti cation accounts for the chemical potentials color associated to the color charges Q ,
[epsilon1] + p = T s + Q color . (2.4)
T is the temperature and s is the entropy density.
Let us specify the dynamics of the system. The rst EOM corresponds to the uid dynamics evolution. Inspired by the KK compacti cation of a uid coupled to gravity (in particular Bianchi identities of Einstein equations, cf. section 5), we obtain the conservation of the total energy-momentum tensor,
r T total = 0 . (2.5)
2dim(G) will correspond to the dimension of the internal group manifold in the KK compacti cation.
3From the KK perspective Qc = Qc(x) is a scalar quantity that corresponds to the dilaton-dependent gauge coupling, Qc(x) / e(x). When we set constant-valued, then Qc plays the role of a Yang-Mills
coupling constant and it disappears from the EOMs of the gauge elds.
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F F
1
2 (F )2
The second EOM describes the dynamics of the non-Abelian Yang-Mills gauge elds and introduces a non-Abelian colored current.
(D F ) (x) = J (x) . (2.6)
The quantity Jcolor (x)
J ;color (x) = Qc Q (x) u (x) + J ;diss (x) , (2.7)
allows us to de ne a covariantly conserved current4
J = Q2c J D
Q2c
F
, (D J )m = 0 , (2.8)
where Q (x) is the color charge density attached to the uid.
Although the dissipative part contribution in Jdiss (x) is frame-dependent, the color current Jcolor (x) is always covariantly conserved independent of the choice of frame. Further details for speci c frame choices can be found in section 6.4.
The colored uid must interact with the Yang-Mills gauge elds through the Lorentz force. In our formulation, for a the uid characterized by the energy-momentum tensor T uid , the Lorentz force naturally arises from the conservation of energy-momentum tensor current,
r T uid (x) = Qc F (x) Jcolor (x) . (2.9)
As the expression of energy-momentum tensor current is frame-dependent, departure from the Landau frame does not permit to read the transport coe cients associated with the dissipative e ects from T diss . To correctly identify these coe cients, we need a frame-invariant formulation of the dissipative terms which is in agreement with the second law of thermodynamics, r Js 0, where Js refers to the covariant entropy current. We adopt
the following generalizations:
P (x)P (x) 1d n
P (x)P (x)
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T diss (x) = 2 (x) (x) ,
@p(x)
@Qm u (x)Jdiss m(x) +
1d n
P ab
@p
@[epsilon1] u u
(x)T diss (x) = (x) (x) ,
P (x)
Jdiss m+ Qm[epsilon1]+puT diss mn(x)
P D n
= 0 , (2.10)
where P = + u u is the projector to the hypersurface orthogonal to the uid, mn is the non-Abelian conductivity tensor, and r u , , , are various dissipative
coe cients. Covariant derivatives are de ned in section 5.
This completes the summary of the equations that govern our system. It now remains to establish this set of EOMs and conservation laws. In this paper, we established them by starting from a higher-dimensional neutral uid and then making a KK dimensional reduction. The idea was that we used the KK compacti cation as a guiding principle to
4If Qc plays the role of a coupling constant, then the second term vanishes. We will explore the details of non-constant Qc and in section 5.
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T
+ 1T Fn u
obtain expressions that preserve SU(2) covariance and the conservation laws, which arise upon recasting the higher-dimensional ones.
In the following sections, we explicitly show the calculations that lead to these equations.
3 Kaluza-Klein approach
Our goal is to construct non-Abelian hydrodynamics. It consists of two components: the colored matter uid and the Yang-Mills gauge eld. Constructing its hydrodynamics starting from a microscopic Yang-Mills-matter theory (such as QCD) is just a theoretical idea: it is not feasible nor shedding light on physics. As such, we look for a mesoscopic approach. The idea is to utilize the Kaluza-Klein compacti cation to construct both components of non-Abelian hydrodynamics simultaneously. Our starting point is a self-gravitating, dissipative and neutral uid in a dynamic D-dimensional spacetime MD(
bgMN), viz. a dissipative and neutral uid coupled to the Einstein gravity, all in D dimensions.5 Our working assumption is that the D-dimensional matter is strongly interacting at the outset. While gravity is fundamentally weak, e ective strength for the uid depends on macroscopic conditions such as density and temperature.
3.1 Self-gravitating dissipative uid
We will rst characterize strongly interacting dissipative, neutral uid in curved D-dimensional spacetime MD(
bgMN). The hydrodynamic eld variables of uid consist of the velocity vector eld
buM( bx) and various other scalar elds. The velocity eld is time-
like, normalized6
bx) = 1 , (3.1) such that it carries (D 1) independent components. On the other hand, the number of
independent scalar elds is set by the number of equations of state that we consider. For a perfect uid, we will consider temperature
bT (
bx), pressure bp( bx), and energy density b[epsilon1]( bx) to be independent scalar variables. Likewise, for the dissipative coe cients, we take shear viscosity
b , bulk viscosity
b , shear tensor
bAB, and expansion scalar
bx) = 0. (3.2) In the long-wavelength limit, the energy-momentum tensor is given by a derivative expansion of hydrodynamic elds, which in our case consists of parity-even terms up to the rst-order in gradients. It is given by two terms:
bT uidMN(
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buM(
buN(
bx)
bx)
bgMN(
b as the independent response variables associated with the D-dimensional neutral uid.
The conservation laws and EOMs of the D-dimensional dissipative, neutral uid follow from the conservation of energy-momentum tensor
brN
bT uidNM(
bx) , (3.3) 5We denote all D-dimensional variables as hatted quantities.6We use the mostly plus signature.
bx) =
bx) +
bT dissMN(
bT perfectMN(
where
bT perfectMN is the perfect uid part and
bT dissMN contains the dissipative e ects. In this work, we do not assume a priori an equation of state for the uid, so we treat all hydrodynamic elds as being independent. For later treatment, we nd it convenient to use the vielbein formalism. The vielbein EMA is related to the metric as7
bgMN( bx) = EMA( bx)ENB( bx) AB , AB = ( + . . . +) . (3.4)
Thus,
bT uidMN(
bx) = EMA(
bx)ENB(
bx)
b
T perfectAB(
bx) +
bT dissAB(
bx)
. (3.5)
At zeroth-order in the gradient expansion, the uid is perfect, so
bT perfectMN(
bx) = [
b[epsilon1](
bx) . (3.6)
To study the dissipative part of energy-momentum tensor, it is necessary to specify the hydrodynamic frame. This dependence on the hydrodynamic frame arises as a consequence that the macroscopic variables that characterize the uid do not have unique microscopic de nitions. This permits us to have some freedom to select a convenient frame and therefore rede ne them in a simple manner. A convenient choice to x this arbitrariness utilizes the projection of the dissipative part in the energy-momentum tensor to the hypersurface orthogonal to the velocity vector,
buM
bT dissMN = 0 . (3.7)
This is referred to as the Landau frame. In this frame, the most general form of dissipative part of energy-momentum tensor is given by
bT dissAB(
bx) = 2
b (
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bx) +
bp(
bx)]
buM(
bx)
buN(
bx) +
bp(
bx)
bgMN(
bx)
bAB(
bx)
b (
bx)
bPAB(
bx)
b (
bx) , (3.8)
where
b is the shear viscosity and
b is the bulk viscosity of the D-dimensional neutral uid.
We also denote the projection tensor to the hypersurface orthogonal to the velocity vector as
bPAB, the shear tensor as
bAB, and the expansion scalar as
b . They are de ned as follows:
bPAB(
bx) =
b AB +
buA(
bx)
buB(
bx) ,
bAB(
bx) =
bP(AC(
bx)
bPB)D(
bx)
bDC
buD(
bx)
1
bp(x)
b (
bx)
bPAB(
bx) , (3.9)
bx) , where DA = EAM@M +
b!A(
bx) is the Lorentz covariant derivative, and
b (
bx) =
bDA
buA(
b!A is the spin
connection acting on the tangent frame.
We minimally couple this D-dimensional neutral, dissipative uid to the D-dimensional gravity, whose metric eld is given by
bgMN. The system is described by the D-dimensional Einstein eld equations sourced by the uid,
bRMN(
bx)
1
2
. (3.10)
7We can straightforwardly incorporate fermionic DOFs, such as a supersymmetric uid interacting with supergravity, as our scheme utilizes the vielbein formalism approach.
bgMN(
bx)
bR(
bx) = 8GD
h b
T perfectMN(
bx) +
bT dissMN(
bx)
i
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Two remarks are in order. First, it is important to stress that the energy-momentum tensor sourcing the Einsteins equation includes both ideal and dissipative parts. Second, our approach admits straightforward extension to any higher orders in the gradient expansion. This is an interesting program we leave for future development.
Before we dwell into details of computations, in the next section, we overview the main aspects of the KK compacti cation of this system.
3.2 Non-Abelian Kaluza-Klein reduction
Our goal is to construct self-consistent non-Abelian hydrodynamics using the approach of the dimensional reduction in KK theory. In this section, we sketch the main aspects of the KK compacti cation approach and the guidelines of our developments.
We start with the D-dimensional Einstein-neutral uid system given by eq. (3.10) and dimensionally reduce it on n-dimensional compact space Xn. We can e ectively split the gravitational DOFs in D dimensions into gravitational and additional DOFs in the d = (D n)-dimensional reduced spacetime. The additional DOFs are scalar elds that
characterize the size and shape of Xn and, if the manifold admits Killing symmetries, vector elds with gauge symmetries. Likewise, we can split the uid energy-momentum tensor in D dimension into uids energy-momentum tensor and some vector currents in d-dimensional, reduced spacetime. Depending on the properties of Killing vectors on Xn, these vector currents can be either Abelian or non-Abelian. In this treatment, one must only keep a consistent truncation of light modes, setting the massive modes to zero. Consistency requires that heavy modes that are dropped are not sourced by the light modes one keeps.
Note that we are performing the KK dimensional reduction for both the gravity in the left-hand side of (3.10) and the uid in the right-hand side. As for the gravity, it is known that the KK dimensional reductions that involve Abelian isometries are always guaranteed to be consistent, as the heavy and light modes do not mix each other. It is also known that, for some internal spaces (maximally symmetric spaces and group manifolds), dimensional reductions that involve non-Abelian isometries are consistent as well. As for the matter, KK compacti cation of a uid without gravity (and hence, without dynamical gauge elds coupled to the uid) on n-dimensional torus Xn = Tn is straightforward, as was recently studied in [37]. The reduction leads to a uid carrying
U(1)n \global" charges, and to relations between D-dimensional heat transport coe cients and d-dimensional, reduced charge transport coe cients. The results are in agreement with results known independently, so it suggests that the KK reduction that involves Abelian isometries is consistent for the uid as well.
Consider next the KK dimensional reduction of Einstein- uid system on a group manifold Xn = G [35] of dimension n = dim(G) and of curvature scale R. The group manifold G is describable in terms of the Maurer-Cartan one-forms m. These one-forms are invariant under left multiplications by a group element g 2 G. Thus, this left multiplication is an
isometry of metric g(G) of the n-dimensional internal space. So, in d-dimensional reduced spacetime, the gauge symmetries include the di eomorphisms of spacetime and the massless elds of the d-dimensional, system will be the metric g and the non-Abelian Yang-Mills gauge elds with gauge group G. Likewise, in d-dimensional reduced spacetime, the neu-
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tral uid we started with becomes a uid carrying G global charges. The Einstein- uid equation then gauges this global charges to G color charges so that the uid is minimally coupled to the non-Abelian gauge eld. This is the main reason why we reduce the higher-dimensional Einstein- uid system on group manifolds: the reduction naturally lead to color charges and couples the G-colored uid to dynamical G-color Yang-Mills elds. The reduction will translate the D-dimensional conservation laws into the d-dimensional, reduced conservation of both energy-momentum tensor and non-Abelian vector currents.
From the KK compacti cation, we obtain the system of colored uid interacting with Yang-Mills theory. Nevertheless, the reduction also will bring in additional DOFs. Depending on the physical situations we are interested in, one may keep them as part of the system or truncate them out. For the formulation of non-Abelian hydrodynamics, we will only keep the non-Abelian gauge eld dynamics but none others such as the gravitational dynamics. That is, we will decouple the gravitational DOFs and consider non-Abelian hydrodynamics on d-dimensional Ricci- at spacetime. Such decoupling can be achieved if, for instance, one takes in D dimensions nontrivial cosmological constant and n-form eld strength and the Freund-Rubin ansatz. With ne-tuning of the cosmological constant and taking GD to zero while keeping Rn+2/GD held xed, one can decouple the gravity while keeping nontrivial Yang-Mills gauge dynamics in Ricci- at d-dimensional spacetime. We will also need to truncate the dilaton (that parametrizes the volume of G) and other scalar elds that emerge by setting them to be constant-valued. Varying them, however, would result in change of the d-dimensional equations of state.
Let us stress that the above approach we propose relies on neither kinetic theory nor Lagrangian formulations. In this regard, our approach o ers an ab initio derivation of the non-Abelian hydrodynamics modulo well-motivated assumption that a neutral uid coupled to Einstein eld equations is self-consistent in D dimensions.
Finally, let us comment on a technical caveat related to the Yang-Mills gauge group. In our approach, the KK dimensional reduction is done on the EOMs. This bears some consequences in the possible choices of the group manifold Xn = G. In particular, dimensional reduction of the EOMs allows for gauge groups whose structure constants are traceful, i.e., fmnn [negationslash]= 0 (cf. [38]).
4 Charged uid coupled to Maxwell theory
As a step to introduce the technicalities that KK theory requires and build intuitions therein, we rst consider the KK reduction of Einstein- uid system on a group manifold with Abelian isometries. Thus, we choose the internal manifold to be a n-torus, Xn = U(1)n. For simplicity, we will take the internal manifold isotropic,
R1 = R2 = [notdef] [notdef] [notdef] = Rn = R, and we will restrict ourselves to a perfect uid, leaving
incorporation of the dissipative e ects to next section.
Consider the KK reduction of a perfect uid given by eq. (3.10) on a S1 internal circle of radius R, where
bT uidMN = bT perfectMN. We will show that the KK reduction gives rise to a
charged perfect uid interacting with Maxwell electromagnetism.
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4.1 Reduction on Abelian group manifold
For the KK reduction on a circle, let us assume the following ansatz for the vielbein EMA(
in eq. (3.4) as
. (4.1)
Curved indices of the D-dimensional spacetime will be split as M = [notdef][notdef], z[notdef] whereas we
will denote at indices as A = [notdef]a, z[notdef]. We will also assume that all the elds that appear
in the ansatz only depend on the d-dimensional coordinates x of Md.8 The dilaton (x),
which measures the size of Xn, is weighed by the reduction-speci c coe cients
2 = n2(d + n 2)(d 2)
Though in this section we evaluate n = 1, we will keep n generic.
Let us start by substituting the compacti cation ansatz into the D = (d + 1)-dimensional Einstein eld equations
bGMN(x) bRMN(x)
and recast the di erential equations. The components
analyzing the component equations.
The
= Q2e(x)Je (x) , (4.4)
where Qe is the dilaton-dependent gauge coupling,
Qe(x) e(d1) (x), (4.5)
and the current is given by
Je (x) 2e( +2 )(x)Qe(x) e a(x)
Hence, the
bG z components of Einstein equations automatically de ne the electromagnetic dynamics of the system, including the current Je (x) of the uid. Thus, the uid becomes charged whenever it has non-vanishing ow around S1. Being proportional to T uidaz(x), the electric current Je (x) will be proportional to the reduced velocity eld u (x). The dilaton eld that measures the size of S1 has the e ect of spacetime-dependent unit of electric charge, Qe(x). As discussed in the previous section, we take the KK reduction as an ab initio approach for deriving consistent hydrodynamic equations. As such, we will eventually set the dilaton to be constant-valued.
8In section 5, we will assume some dependence on the internal coordinates, which will yield to non-Abelian gauge symmetry upon reduction.
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bx)
EMA(
bx) = e (x)e a(x) e (x)A (x)0 e (x)
!
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and =
(d 2)
n . (4.2)
1
2
bgMN(x)
bR(x) =
bT uidMN(x) , (4.3)
bG z, and bGzz give the d-dimensional gravitational, gauge, and dilaton eld equations, respectively. Though we do not specify the structure of uid energy-momentum tensor
bT uidMN, we will return to it after
bG ,
bG z components imply the Maxwell equations coupled to a current.
r Q2e(x)F
bT uidaz(x) . (4.6)
This same pattern to the other components of (3.10). From the
bG components, we obtain the d-dimensional Einstein equations sourced by the charged uid, the U(1) gauge eld and the dilaton:
G (x) R (x)
1
2g (x)R(x) = T total (x) , (4.7)
where the right-hand side de nes the total energy-momentum tensor of the d-dimensional system
T total (x) = T uid (x) +
12@ @ 14g (@)2 (x) + 12Q2e(x)
F 2 14F 2g (x) .
(4.8)
The last two terms are contributions of dilaton eld and Maxwell eld, while the rst term is the energy-momentum tensor of charged uid, de ned by
T uid (x) e2 (x)e a(x)e b(x)
bT uidab(x) . (4.9)
Finally, let us consider the
bGzz component. We obtain the d-dimensional dilation eld equation, sourced by both the uid and the Maxwell gauge eld,
(x) = 2 D(x) , (4.10)
Again, the right-hand side of the equation de nes the dilatation current,
D(x)
d 1
JHEP02(2017)122
4 Q2e(x)F 2 (x) + (d 1) e2 (x)
bT uidzz(x) e2 (x)
bT uid(x) . (4.11)
where
bT uid :=
bT uidMM is the trace of the D = (d + 1)-dimensional energy-momentum
tensor.
The Einstein tensor in the de ning equation eq. (4.3) obeys the Bianchi identity, from which conservation laws of various currents we identi ed above are derived. The conservation laws on current Jel and total energy-momentum tensor T total result relevant for the
Maxwell-plasma system. For the charge current, covariant divergence of eq. (4.4) gives
r
r
Q2e(x)
F = 0 , (4.12)
where we have used the torsion-free condition for d-dimensional spacetime. This implies
r (e( (d3) )e a
bTaz)(x) = 0 ! r (Qe(x)Je (x)) = 0 , (4.13) which results the to the conservation law of electric current Je , generalized by the dilaton eld.
From the Bianchi identity of Einstein tensor in eq. (4.7) we obtain
r T total (x) = 0 . (4.14)
This implies that variations in the uid energy-momentum tensor are balanced by the change of the Maxwell energy-momentum tensor and the dilaton.
r T uid (x) = r
Q2eF = r[ r ]
Q2e
F 2 14F 2g (x). (4.15)
{ 12 {
On-shell, this conservation is equivalent to
r T uid + (e[ (d3) ]e a
bT uidaz)F + e2 [(d 1)
bT uidzz
bT uid]r = 0 . (4.16)
We interpret this as the generalization of the Lorentz force equation of Maxwell-plasma under the presence of the dilaton eld. Once again, the role of the KK approach is just a tool to facilitate the ab initio derivation of charged uid interacting with Maxwell theory. Therefore, setting the dilaton to be constant-valued we obtain the standard form of the Lorentz force equation:
r T uid (x) = Qe(x)F (x)Jel (x) . (4.17)
4.2 Abelian reduction of energy-momentum tensor
So far, we have not made any assumption on the energy-momentum tensor
bT uidMN of the
neutral uid we started from. We now study
bT uidMN under a well-motivated ansatz for
the higher-dimensional velocity eld
bu(
bx) and the other scalar quantities. To gain better intuition about physics, we will restrict the D-dimensional neutral uid to a perfect uid. In section 5, we will consider the dissipative contributions.
The D-dimensional velocity eld
buM has (D 1) independent components, as it is
conveniently normalized by eq. (3.1):
buM( bx) buN( bx) bgMN( bx) = 1 . (4.18)
The ansatz that we will assume for the velocity eld is:
bua = ua(x) cosh '(x) ,
buz = sinh '(x) ,
(4.19)
where ua(x) is the velocity eld of charged uid in d dimensions, which is normalized as ua(x)ub(x) ab = 1. The scalar eld '(x) parametrizes the degree of freedom associated
with the internal component of the velocity,
buz. Substituting the ansatze for the vielbein eq. (4.1) and the velocity elds eq. (4.19) into the energy-momentum tensor, we will obtain the de ning variables of the d-dimensional uid in terms of the D-dimensional ones. That is to say, we nd that the energy-momentum tensor in d dimensions is
T perfect (x) = e a(x)e b(x)
bTab(x)= ([epsilon1] + p) u (x)u (x) + p g (x) , (4.20)
where the energy density [epsilon1](x) and the pressure p(x) are given by
[epsilon1](x) = e2 (x)
b[epsilon1](x) cosh2 '(x) + bp(x) sinh2 '(x)
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, p(x) = e2 (x)
bp(x) . (4.21)
By substituting the velocity ansatz eq. (4.19) into eq. (4.6) we obtain the electric current
Je (x) 2 ([epsilon1](x) + p(x)) e[ +(d+1) ](x) tanh '(x) u (x) . (4.22)
{ 13 {
As anticipated, the charge current is proportional to the velocity eld u . Again, let us analyze the case for which the dilaton eld is constant. Then, the energy-momentum conservation, eq. (4.20) leads to
r T perfect (x) = Qe(x)F (x)Je (x) . (4.23)
This is precisely the Lorentz force equation we have directly derived from the reduction of the Einstein- uid system in the last section.
One can straightforwardly generalize the above construction by taking the internal space Xn to be an n-torus Tn. It will give rise to a uid charged under n independent
Abelian electromagnetic elds with U(1)n gauge symmetry.
After analyzing the system of a uid charged under Abelian gauge elds, we will address the case for which the gauge symmetry is non-Abelian. To carry out this problem, the internal manifold will be a group manifold whose isometry group is non-Abelian. We will choose SU(2) for simplicity but the procedure applies to any other gauge group.
5 Colored uid coupled to Yang-Mills theory
We now construct non-Abelian hydrodynamics of Yang-Mills plasma. Here, our goal is to derive ab initio the EOMs of a dissipative uid carrying non-Abelian SU(2) charges and interacting with Yang-Mills theory. To do so, our idea is again to start with an Einstein- uid system in D dimensions eq. (3.10) and perform a KK dimensional reduction on a SU(2) group manifold [35] (for a review, cf. [38{40]). After the reduction, we will nd an SU(2) colored uid interacting with SU(2) Yang-Mills theory in d dimensions. As SU(2) group manifold is three-dimensional, our setup corresponds to n = 3 and hence D = d + 3. Nevertheless, this method can be applied to any group manifold G, having thus a colored uid interacting with Yang-Mills theory of gauge group G.
5.1 Compacti cation on SU(2) group manifold
Let us consider the following KK ansatz for the D-dimensional vielbein:
EMA = e
where
bx) = (u1)nm(y)A n(x) .and g is a coupling constant g. As in the Abelian reduction, we split the curved manifold indices as M = ([notdef], m) where [notdef] = 1 . . . d and m = 1 . . . n, tangent space indices as A =
{a, [notdef], where a = 1 . . . d and = 1 . . . n, and local coordinates as
bxM = (x , ym). For the
SU(2) case, n = 3. These algebra-valued indices can be freely lowered and raised without loss of generality.
{ 14 {
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bep 0 g1e
be a e
bA p
!
, (5.1)
bem
be a(
bem (
bA m(
bx) = e a(x) ,
bx) = umn(y)Vn (x) , (5.2)
In various Weyl factors, the dilaton eld (x) is weighed with the coe cients
2 = n2(d + n 2)(d 2)
and =
(d 2)
n . (5.3)
The matrix umn(y) in eq. (5.2) is a twist eld that carries the information of the SU(2) group manifold. After the reduction, this information is encoded in the d-dimensional system through the structure constants,
fmnp := 2(u1)ms(y)(u1)nt(y)@[sut]p(y) . (5.4)
Though the twist matrix eld umn(y) varies over the group manifold (hence depends on the internal coordinates y), the combination on the r.h.s. of this equation needs to be constant-valued in order for them to be the structure constants of the Lie algebra associated with the group manifold.
The ansatz can be explicitly expressed in terms of the Maurer-Cartan one-forms m of the SU(2) group manifold by combining the elds as
bEa(x) = e (x)ea(x) ,
bE (x) = g1e (x)Vm (x)(m gAm(x)) , (5.5) where m unmdxn is the left-invariant one-form of G, satisfying the Maurer-Cartan
equation
dm + 12fnpmn ^ p = 0 , (5.6)
and thus fnpm are the structure constants of the isometry group G of the internal manifold.
Before carrying out the non-Abelian reduction on the group manifold G, we introduce new notations for the physical variables in d dimensions. We shall build from the scalar vielbein V two scalar metrics
Mmn = Vm Vn and M Vm Vn mn, (5.7)
which are SU(2) invariant and SU(2) covariant, respectively. We denote the trace as M M . We de ne the covariant derivatives D (A) and D (V) as
D Vm @ Vm gA m
n
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Vn ,
D
Vm D Vm + Q Vm ,
(5.8)
where the elementary gauge eld used in D is given by
A mn(x) Ap (x)fpmn (5.9)
and the composite gauge elds used in D are built from the scalar vielbein as
Pa (x) ea P =
1
2
V m Da(A)Vm + V m Da(A)Vm
,
V m Da(A)Vm V m Da(A)Vm
.
(5.10)
Qa (x) ea Q =
1
2
{ 15 {
The distinction is that, while D is the ordinary gauge covariant derivative, D accounts
for quantities that are adjoined by the scalar vielbein Vm . Finally, the Yang-Mills eld
strength two-form Fm of Am is de ned as
Fm dAm +
1
2 g fnpm An ^ Ap . (5.11)
This eld strength typically appears dressed up by the scalar elds, so we also denote the tangent space (both in internal and spacetime manifolds) eld strength two-form as F ab Vm Fmab.
5.2 Field equations for Yang-Mills plasma
To obtain the EOMs of the d-dimensional system we will substitute the ansatz eq. (5.1) into Einstein equations and recast the resulting expressions.9
Let us start with the EOMs for the SU(2) gauge elds. They descend from the
bG n
components in eq. (3.10). Working in the tangent space we obtain
Db(Q2c(x)F ab(x)) + Q2c(x)Pb (x) F ab(x) = Q2d(x)J a(x) , (5.12) where
Qc(x) := e
13 (d+1)(x) (5.13)
is the dilaton-dependent gauge coupling, and
(5.14)
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Ja (x) = 2
hg Q2c(x) [epsilon1] Pa (x) M (x) Qc(x) e2 (x)
bT uida (x)
i
.
is the color current. For covariantly constant scalars, eq. (5.12) is reduced to
Db(F ab)(x) = J a(x) , (5.15) which is the standard form of the Yang-Mills eld equations coupled to color current.
The Einstein eld equations descend from the
bG components:
G (x) = R (x)
1
2g (x)R(x) = T total (x) , (5.16)
where T total := e ae bT totalab is the total energy-momentum tensor, with
T totalab(x) = e2 (x)
bT uidab(x) + 12Q2c(x) cdF ac(x)F bd(x)
1
2 ab(F )2(x)
+ 12
@a(x)@b(x)
1
2 ab(@)2(x)
+ Pa (x)Pb (x) 1 2P2(x) ab
g2 Q2c(x)
M (x)M (x)
1
2M2(x)
ab . (5.17)
From the rst line, we read o the energy-momentum tensor T uidab of the colored uid:
T uidab(x) = e2 (x)
bT uidab(x) . (5.18) 9Details of the calculations for extracting the equations of motion are relegated to appendix A.
{ 16 {
Other eld equations also yield relevant information on currents and their conservation laws. The equation of motion for dilaton eld is obtained from the trace of eq. (3.10),
bGmm:
(x) = 1
2 D(x) , (5.19)
where D(x) is the dilation current
(5.20)
D(x) =
1
2(d 2)
Q2c(x)(F )2(x) + 2g2
d 2
Q2c(x)
M (x)M (x) 1 2M2(x)
2d 2
e2 (x)
3d + 1
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bT uid(x)
bT uid (x)
.
The rst line is the contribution of SU(2) gauge elds and scalar elds, whereas the second one is the contribution of colored uid. As we can check, there is no non-linear contribution of the dilaton eld itself apart from the Weyl factors.
The equation of motion for the algebra-valued scalar elds Vm (x) is given by a linear
combination of the
bGmn components and the trace bGmm:
Da(Q)Pa = J , (5.21)
where
J (x) = 12Q2c(x)
F ab(x)F cd(x) ac bd
1 3(F )2
+ 2e2
1 3
bT uid
bT uid
M M 12M2
. (5.22)
The rst line of this expression is the contribution of SU(2) gauge elds and colored uid, while the last line corresponds to the contribution of algebra-valued scalar elds.
5.3 Conservation laws
The non-Abelian reduction of the Einstein- uid system has led to a Yang-Mills plasma, consisting of colored uid interacting with non-Abelian gauge elds (and also coupled to gravity, dilaton and algebra-valued scalar elds). In this section, we will further investigate the conservation laws of the system.
Likewise in section 4 for Maxwell plasma, we have not made any assumption on gravity and scalar elds so far. Nevertheless, in order to study the conservation of the simplest model for Yang-Mills plasma, we will truncate the system so that the d-dimensional metric is at and scalar elds are covariantly constant. Such truncations will impose some constraints on the corresponding eld equations of and Vm , namely, eqs. (5.20) and (5.22).
For this truncation to be consistent, we would need to solve these constraints. They will in turn impose some conditions on the d-dimensional Einstein equations10 and the Yang-Mills eld equations through Weyl factors and scalar potentials. In this section, we will
10As for gravity, we can decouple the DOFs associated to the metric by taking the limit GD ! 0 for the
Einstein equations.
{ 17 {
+4g2 Q2c(x)
M M 12M M 1 3
simply consider the simplest consistent solution of these scalar elds, but will not explore the arena of possible non-trivial solutions. Nevertheless, it should be interesting to look into the implications of such nontrivial solutions (and their stability) in the context of uid/gravity duality. It will also be important to understand to what extent these solutions constrain the values of the transport coe cients and other quantities that characterize the lower-dimensional uid.
Firstly, let us analyze the color currents of the system and their conservation laws. The SU(2) Yang-Mills eld equation eq. (5.12) can be recast:
Db Q2cMmnFnab
bT uida Vm . (5.24)
This allows to de ne a current J m which is covariantly conserved, D J m = 0. Its
expression is given by eq. (5.14) (see appendices for calculation):
J ma = Q2c Mmn h
. (5.25)
The interpretation is clear: the rst term is the color current sourced by the algebra-valued scalar elds, while the second term is the color current sourced by the colored uid itself. Being the non-Abelian counterpart of the U(1) charged current, the second term is proportional to the o -diagonal block of the energy-momentum tensor,
bT uida . This block is non-zero if the D-dimensional uid ows on the group manifold, so Jcolorma is proportional to the internal velocity elds ua.
Secondly, let us analyze the heat current of the Yang-Mills plasma and their conservation laws. We already discussed that the Bianchi identity r G = 0 of the d-dimensional
Einstein equation, eq. (5.16) leads to the conservation of the total energy-momentum tensor
r T total = 0 . (5.26)
We would like to obtain the relations that this condition imposes among the d-dimensional degrees of freedom. Applying a covariant divergence on the total energy-momentum tensor eq. (5.17) and substituting the eld equations of the Yang-Mills elds and scalar elds, we are left with an expression that involves rst derivatives of the scalar elds and components of the energy-momentum tensor
bT uidMN.11 This expression is the non-Abelian generalization of the Lorentz force, which involves not only the Yang-Mills eld strength but also the algebra-valued scalar elds. Nevertheless, if we set these scalar elds to be covariantly constant, DaVm = Da' = 0, we obtain
DaT uidab + 2Q1c(x)e2 (x)
bTc Vn Fnbc = e2 Da
I.e., we get the standard expression of Lorentz force for Yang-Mills plasma:
DaT uidab(x) = Qc(x)F ab(x)Jcolor a(x). (5.28)
After doing the KK reduction of gravity sourced by a generic uid
bT uidMN, we are going to
bT perfect + bT diss)MN and study in detail the resulting d-dimensional uid. 11We relegate details of the calculation to appendix A.
{ 18 {
= Jma , (5.23)
13 (d5)
JHEP02(2017)122
where
Jma = 2g[epsilon1] M Pa Vm 2e
Jna Db Q2cMnp
i
F pab
bT uidab + 2Q1c(x)
bTc F bc
= 0 . (5.27)
evaluate
bT uidMN = (
6 Colored uid from non-Abelian reduction
In this section, we will implement the KK compacti cation of the uid energy-momentum tensor to construct the colored uid and read o its de ning variables.
6.1 Non-Abelian reduction of uid
The energy-momentum tensor and the de ning variable of the d-dimensional uid will be read o after inserting the compacti cation ansatze for the vielbein and the rest of the expressions into the EOMs of the D-dimensional system.
For the non-Abelian reduction of the velocity elds
buA, we will assume an ansatz such that none of its components depend on the coordinates of the internal group manifold G. We can parametrize them as follows
bua = ua(x) cosh '(x) ,
bu = n (x) sinh '(x) , (6.1)
where
The d-dimensional velocity has (d 1) independent components, and the n-dimensional
unit vector n has (n 1) independent components. In total, along with ', there are
(d 1) + (n 1) + 1 = D 1 independent components. The angular variable ' measures
the relative magnitude between the external and \internal" velocity elds. The unit vector ua is the boost in external spacetime, while the unit vector n is the boost in the internal group manifold. They all uctuate in external spacetime.
With this ansatz, we will now study the d-dimensional energy-momentum tensor of the uid, eq. (3.3).
6.2 Perfect colored uid
Firstly, we are going to characterize the colored perfect uid in d dimensions. This will allow us to identify its thermodynamic and scalar quantities in terms of quantities in D dimensions.
The energy-momentum tensor of the d-dimensional perfect colored uid is given by
T perfectab(x) = [[epsilon1](x) + p(x)] ua(x)ub(x) + p(x) ab , (6.3)
where, using eq. (5.18), the quantities are related to the D-dimensional ones as
p(x) = e2 (x)
bp(x) , [epsilon1](x) = e2 (x)
cosh2 '(x)
1
bc2s 1) + 1
{ 19 {
JHEP02(2017)122
uaub ab = 1 and n n = 1. (6.2)
. (6.4)
From this, we nd the speed of sound, cs, in the perfect colored uid as
c2s
@p
@[epsilon1] =
b[epsilon1](x) + sinh2 '(x)
bp(x)
, where
bc2s = @
bp @
cosh2 '(x)(
b[epsilon1] . (6.5)
The faster the uid is boosted inside the group manifold, the slower the sound speed of the colored uid.
The boost inside the group manifold generates the color current. From the current Jcolorma, eq. (5.25), we have
Jcolorma(x) = Qc(x)Qm(x) ua(x) . (6.6)
Here, Qm(x) is the color charge density attached to the uid, which is de ned as
Qm(x) = 2([epsilon1](x) + p(x))Vm (x)n (x) tanh '(x) . (6.7)6.3 Entropy current
The D-dimensional neutral uid has entropy density
bs, so the entropy current is given by
bJ[hatwide]sA=
bs buA, (6.8) In the perfect uid limit, the entropy current is covariantly conserved
brM
bJ[hatwide]sM= 0 . (6.9) From the ansatz eq. (4.19), the entropy in d dimensions is given by
s = e2
bs cosh ' , (6.10) and the entropy current in d dimensions is given by
Js = s(x) n (x) tanh '(x), Js (x) = s(x) u (x). (6.11)
The conservation law eq. (6.9) is reduced to
r Js = 0 . (6.12) where we have used the spin connection components of appendix A.
The neutral perfect uid in D dimensions satis es the thermodynamic relation
b[epsilon1] + bp =
bT
bT is the temperature. After the reduction, the d-dimensional uid is colored, so its thermodynamic relation must account for the chemical potentials colorm associated to the charges Qm in the form
[epsilon1] + p = T s + Qmcolorm . (6.14) Requiring this Euler relation to hold in d dimensions, we obtain that the d-dimensional temperature and chemical potentials are given by
T (x) =
bT (x)
1cosh '(x) ,colorm(x) = n (x)V m(x) tanh '(x) . (6.15)
So far, we have described the d-dimensional perfect uid carrying non-Abelian SU(2) charges and given all its de ning quantities in terms of the D-dimensional neutral uid parameters. These results are in full agreement with the ones obtained for the Abelian case in section 4. Built upon these consistency checks, we are going to consider dissipative e ects of the uid in the next section.
{ 20 {
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bs , (6.13)
where
6.4 Non-Abelian dissipative uid
We are going to extend our previous analysis by considering the dissipative part of energy-momentum tensor,
bT dissMN. This piece is given by
bT dissAB = 2
b
bAB
b
bPAB
b . (6.16)
The correction of rst-order in derivatives in
bT dissAB will generate terms of rst-order derivatives of the components of velocity elds
buA. Being velocity elds, these terms play the same role as second-order derivative of ordinary elds. Therefore, we will eliminate the derivatives by using their equations of motions, namely, the conservation laws.
In particular, if we consider eqs. (5.23) and (5.27), we obtain
u (x)
br '(x) = c2s(x) (x) tanh '(x) , (6.17) where (x) r u (x). Moreover, (x) is related to
b
brM
buM(x) by
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b (x) = cosh '(x)
+ n u
br n
(x) , (6.18)
so that when substituting, we have
b (x) = cosh3 '(x)
c2s(x)
bc2s(x)
(x) . (6.19)
In addition, the d-dimensional acceleration a u
br u is given by
a = sech2'(x)
e2 (x)
b[epsilon1](x) + p(x)
br
1 p(x)
+ n (x)c2s(x)u (x) , (6.20)
where
br
b[epsilon1](x) + p(x)2p2(x) sinh 2'(x)
br '(x) . (6.21)
With these results, we can estimate the d-dimensional coe cients associated with the dissipative terms. For the D-dimensional neutral uid, the shear and bulk viscosities can be read o from
bT dissAB. This occurs due to the fact that the uid is described in the Landau frame, i.e.,
buA
bT dissAB = 0 . (6.22)
Upon the non-Abelian KK dimensional reduction, the rearrangement of DOFs into d-dimensional Lorentz covariant representations implies that the reduced ones do not satisfy the Landau frame condition. In particular, we obtain
ua(x)
bT dissab(x) + 1
cosh '(x)
bu (x)
1 p(x)
= e2 (x)
bT diss b(x) = 0 , (6.23)
which straightforwardly leads to uaT dissab [negationslash]= 0.
On account of the frame-dependent structure of the energy-momentum tensor, departure from the Landau frame means that we cannot read o the d-dimensional transport coe cients associated with the dissipative terms from T diss . To correctly identify these
{ 21 {
coe cients, we need a frame-invariant formulation of the dissipative terms. In addition, according to the second law of thermodynamics, it has to be guaranteed that the entropy current Jsa satis es r Js 0. Such frame-invariant description was developed in [41] for
a uid charged under an Abelian gauge eld A . Here, we generalize this result to account for non-Abelian symmetry.
Using the frame-invariant approach as a guiding principle and also based on the gauge covariance of SU(2) algebra-valued quantities, we formulate the following expressions for the transport coe cients in the presence of non-Abelian gauge elds Am :
Pac(x)Pbd(x) 1d n
Pab(x)P cd(x)
T disscd(x) = 2 (x) ab(x) ,
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@p(x)
@Qm ua(x)Jdissam(x) +
1d n
P ab
@p
@[epsilon1] uaub
(x)T dissab(x) = (x) (x) ,
Pab(x)
Jdissbm+ Qm[epsilon1] + pucT disscb mn(x)
PabDb n
T
+ 1T Fnabub
= 0 , (6.24)
where Jdissam follows from eq. (5.25) using
bTMN = bT dissMN, mn is the non-Abelian conductivity tensor, and , , are the d-dimensional dissipative coe cients.
At this stage, in order to obtain the e ective dissipative coe cients, we need to substitute the expressions that we obtained for Jdissam and
bT dissab and work out these three equations.12 From them, we read o the following expressions:
(x) = e2
b (x) cosh '(x) , mn(x) = e2
b (x)T (x) cosh '(x)
mn sinh4 'cosh2 ' Vm n Vn n (x) . (6.25)
(x) = 2e2 (x)
b (x) cosh '
"
1 dn
2
+c4s 1cosh4 '
1 p
@ b[epsilon1] +e2 (x) cosh5 '
b
@
bp @
b[epsilon1]
!#
.
It is important to stress that when getting rid of any dependence on the scalar elds ', we recover the d-dimensional quantities multiplied by the dilaton factor e2 , which parametrizes the volume of the internal manifold. On the other hand, it is worth to mention that the non-Abelian behavior of the conductivity matrix arises from the dependence of the scalar vielbein Vm .
The analysis in this section demonstrates that the non-Abelian KK dimensional reduction is an ab initio and e cient method for deriving the structure and dynamics of Yang-Mills plasma. Moreover, the construction that leads to eq. (6.25) gives a hydrodynamic frame-independent transport. We see from eq. (6.25) that, apart from viscosities, we have the non-Abelian conductivity matrix mn, which is directly connected to the non-
Abelian degrees of freedom in the system. We remark that a similar quantity was obtained in the context of the uid/gravity duality [24].
Now that we have clearly formulated non-Abelian hydrodynamics, we can study various related issues. Understanding conductivity is a major challenge in recent approaches to holographic super uids. One can show that, at the phase transition, a set of SU(2) currents
12In ref. [37], this calculation was performed for a neutral uid compacti ed on a torus.
{ 22 {
can be used as an order parameter [42]. Moreover, it was observed in [25] that employing a non-Abelian gauge transformation allows one to obtain a nite conductivity without breaking translational symmetry.
On the other hand, this theory results in a very suitable and robust framework where to study the quark-gluon plasma. In this respect, one important phenomenon of this system is the study of the relaxation time. This is the time at which the non-Abelian character of the plasma is relaxed, thus becoming purely Abelian. This is a known property that has not been theoretically understood neither for quark-gluon plasma nor for spintronics systems.13
Since our construction can describe the dissipative part of non-Abelian hydrodynamics, we expect it to be useful in elucidating the relaxation mechanism of the color current.
7 Outlooks
In this work, we have proposed a new approach for constructing non-Abelian hydrodynamics, consisting of colored uid interacting with Yang-Mills theory. Based on non-Abelian KK dimensional reduction, the geometric systematics of proposed approach enables one to understand the properties of Yang-Mills plasma even in strongly coupled, non-perturbative regime.
We presented an ab initio approach for constructing hydrodynamics charged under both Maxwell and Yang-Mills plasma. With the non-Abelian KK reduction, we compacti- ed the Einstein- uid equations on a group manifold. The only working assumption is that we started with the most general dissipative, neutral uid coupled to Einstein equation. After the reduction, we obtained Yang-Mills plasma equations for a dissipative, colored uid interacting non-Abelian gauge elds. Though having done the reduction on S1 and SU(2)
group manifold, this procedure can be applied to any type of group manifold. Our approach is not restricted by symmetries that are only symmetries of the Lagrangian. Hence, the KK reduction approach seems to be a robust and covariant method to naturally obtain hydrodynamics coupled to (non-)Abelian gauge elds. The method straightforwardly extends to dissipative hydrodynamics coupled to gravity and a speci c form of dilaton scalar eld, which would also bear applications to early universe cosmology, formation of large-scale structure or compact objects, and colored turbulence.
We studied the conservation laws of colored uid and obtained a non-Abelian covariantly conserved current Jam, which is proportional to the uid velocity eld, as predicted by [30]. In addition, truncating the scalar elds coming from the gravity sector to constant values, we obtained the equation for non-Abelian Lorentz force.
We showed that the reduction procedure does not preserve the hydrodynamic frames. As a consequence, the e ective transport coe cients could not be straightforwardly read o from the reduced system. We proposed a frame-independent formulation of dissipative uids for the non-Abelian gauge elds that is thermodynamically valid and generalizes the one given in [41]. With this construction, we identi ed the d-dimensional dissipative susceptibilities that characterize the e ective uid in terms of the D-dimensional ones. In
13It is worth to mention that our system can be coupled to additional fermionic degrees of freedom, as we are using the vielbein formalism.
{ 23 {
JHEP02(2017)122
particular, we have obtained a conductivity matrix whose non-Abelian nature is given by the scalar vielbein Vm .
The Yang-Mills plasma equations we obtained were in complete agreement with the equations of Maxwell plasma derived in section 4. If we set the structure constants fmnp = 0, we could check that these equations were reduced to the equations for charged uid coupled to U(1)3 Abelian gauge elds. The results of this section could also be straightforwardly extended to other, higher-dimensional group manifold G. We claimed that, for xed d, the large-D limit should be taken seriously as it corresponds to the limit for which rank(G) gets large, revealing a new perspective to the planar limit of Yang-Mills plasma. Results on this aspect will be relegated to a separate publication.
We believe the proposed approach marks signi cant advances toward the understanding of the evolution of nuclear matter after a heavy-ion collision. Hydrodynamics with non-Abelian degrees of freedom that have not thermalized is a transient phase and the lack of a rst-principle derivation of the equations that govern its evolution has been a major obstacle for further developments.
Having now the ab initio construction of uid and eld equations, we can utilize complementary methods such as kinetic theory or gauge/gravity duality to shed more light of this regime. Gravitational solutions with Abelian gauge elds have recently been studied [19, 43, 44]. Therefore, we provide a robust formulation of non-Abelian hydrodynamics where to test uid/gravity duality beyond Abelian uids.
In addition to a phenomenological description of quark-gluon plasma, recent formulation of uid dynamics in terms of uid/gravity duality has increased the interest in the analysis of uids coupled to Yang-Mills elds. In this picture, uid is a eld theory dual to a black hole in higher-dimensional, asymptotically anti-de Sitter spacetime (see [45] for a review). It would be interesting to further explore the physics of black holes with non-Abelian and dilatonic hairs using the non-Abelian Kaluza-Klein reduction [46].
Acknowledgments
We thank Yong-Min Cho, Richard Davison, Sa so Grozdanov, Seungho Gwak, Dima Kharzeev, Jaewon Kim, Andy Lucas, Jeong-Hyuck Park, Malcolm Perry, Chris Pope, Yuho Sakatani, and Jan Zaanen for useful discussions. SJR and PS acknowledge hospitality of NORDITA program \Holography and Dualities 2016 " during the nal stage. The work of JJFM was supported by the Fundaci on S eneca - Talento Investigador Program. The work of SJR was supported in part by the National Research Foundation Grants 2005-0093843, 2010-220-C00003 and 2012K2A1A9055280. SJR was also supported in part by the Munich Institute for Astro- and Particle Physics (MIAPP) of the DFG cluster of excellence \Origin and Structure of the Universe". The work of PS was supported by a Marie Curie International Outgoing Fellowship, grant number PIOF-GA-2011-300528. JJFM acknowledges Alejandro R. Vicente for comments on the new version.
{ 24 {
JHEP02(2017)122
A Einstein equations on a group manifold
In this appendix, we elaborate technical details of the non-Abelian Kaluza-Klein compact-i cation on a group manifold. We also explain the convention used in this work.
We will consider that our starting system is de ned on a D dimensional manifold
MD(
bg) with coordinates bxM, for M = 1, . . . , D. For the tangent spacetime description we introduce a vielbein EMA, where A = 1, . . . , D, which satis es
bgMN( bx) = EMA( bx)ENB( bx) AB , AB = ( + . . . +) . (A.1)
bg) =
bg) is the d-dimensional external spacetime manifold on which our resulting system will live whereas Xn(M) is the n-dimensional internal manifold. The coordinates are split as
bxM = [notdef]x , ym[notdef], where [notdef] = 1, . . . , d and m = 1, . . . , n. Despite the scalar matrix Mmn will parametrize the uctuations of the internal manifold, the nal d-dimensional system cannot have any functional dependence on Xn. The
We start with the reduction ansatz for the vielbein expressed in terms of the Maurer-Cartan one-forms:
bEa(x, y) = e (x)ea(x) ,
bE (x, y) = g1e (x)Vm (x)(m gAm(x)) ,
Md(g) [notdef] Xn(M). Md(
(A.2)
where m unm(y)dyn are the twist matrices, which will depend on the group manifold
coordinates y. Here, g is a gauge coupling parameter.We will compute various geometric quantities. The spin-connection is de ned as
b!C;AB =
where
@NEMD
b DC . (A.4)
Substituting the vielbein ansatz, we obtain the following expressions:
b!c;ab = e
!c;ab + 2 c[a@b]
,
b!c;a =
1
2e(2 + )F nec ea Vn ,
b!c; = +12e( + )
hV mec D (e Vm ) V mec D (e Vm )i,
b! ;ab = +1 2e(2 + )F mea eb Vm ,
b! ;a =
1
2e( + )ea V mV nD (e Mmn) ,
b! ; = +g2e fmnp [V nV mVp + V mV nVp V mV nVp ] ,
{ 25 {
JHEP02(2017)122
A.1 General ansatz
We will perform a KK dimensional reduction. To do so, we will assume that MD(
b CA;B +
b AB;C
b BC;A (A.3)
b AB;C = 12 EAMENB EBMEAN
(A.5)
where
Mmn = Vm Vn ,
Fm @ Am @ Am + gfnpmAn Ap ,
D Vm @ Vm gfnmpAn Vp
(A.6)
and
fmnp = (u1)ms(u1)nt(@sutp @tusp) . (A.7) We will calculate the components of the Ricci tensor
bRAB = bRACBD
b CD and the scalar
curvature
bR =
bRAB
b AB by substituting the components of the spin connection
b!MAB into
the expression for the Riemann tensor,
JHEP02(2017)122
bRMNAB = @M
b!NAB @N
b!MAB +
b!MAE
b!NEB
b!NAE
b!MEB . (A.8)
A.2 SU(2) group manifold
In what follows, we restrict to the SU(2) group manifold, so that fmnp will be the SU(2) structure constants, fmnp = [epsilon1]mnp. In this case, the components of the spin connection are given by [47]
b!ab = !ab + 2 e c[aDb] bec + 12e(2 + )F ab
be ,
b!a = e Pa
be e Da
be + 12e(2 + )F ab
beb ,
(A.9)
b! = e Qa
bea + g2e (M [epsilon1] + M [epsilon1] M [epsilon1] )
be .
where
F ab Vm Fmab , (A.10)
M is the SU(2) covariant scalar matrix
M Vm Vn mn , (A.11)
and
Pa
1
2[V m DaVm + V m DaVm ] , Qa
1
2[V m DaVm V m DaVm ] . (A.12)
The Ricci tensor components are
bRab = e2 Rab 12@a@b Pa Pb ab 12e23 (n+1)F acF bd cd
,
bRa =
1
2e
13 (n5)
hDb(e23 (n+1)F ab)+e23 (n+1)F abPb 2g[epsilon1] M Pa i,
bR =
1
2e2 Da
Pa
2
3 (d 2)
1
2e
2
3 (d+1)F abF cd ac bd (A.13)
4g2e
2
3 (d+1)
M M 12M M +2g2e23 (d+1)
M M 12M2
,
where
M M and DaVm = DaVm + Qa Vm . (A.14)
{ 26 {
A.3 Equations of motion
Our starting point is the D-dimensional Einstein- uid equation
GMN
bRMN
bTMN + 1D 2 gMN
bT
= 0 . (A.15)
We will analyze the tensor GAB = EAMEBNGMN:
GAB
bRAB
bTAB + 1D 2 AB
bT
= 0 . (A.16)
Here, we analyze each components of the Einstein- uid equation. We begin with the internal components, Gmn:
Gmn = g2e2 umpunqVp Vq G . (A.17)
On one hand, this equation has to be satis ed for any scalar elds Vm . As the twist matrices umn depend on internal coordinates, we have that G = 0, where G is given by
G =
1
2e2 Da
JHEP02(2017)122
Pa
2
3 (d 2)
1
2e
2
3 (d+1)F abF cd ac bd
4g2e
2
3 (d+1)
M M 12M M + 2g2e23 (d+1)
M M 12M2
+ 1
d + 1
bT uid
bT uid ,
(A.18)
Solving the trace part, G = 0, where
G = e2 (d 2)
+ 14(d 2) e
2
3 (d+1)(F )2
g2d 2
e
2
3 (d+1)
M M 12M2 + 1d 2e2 3d + 1
bT uid
bT uid
,
(A.19)
we solve for and substitute back to eq. (A.18). We then obtain from G = 0 that
DaPa =
1
2e
2
3 (d+1)
F abF cd ac bd 1 3(F )2
+ 4g2e
2
3 (d+1)
M M 12M M 1 3
M M 12M2
+ 2e2
1d + 1
bT uid
bT uid
1 3
3d + 1
bT uid
bT uid
. (A.20)
Let us consider now the eld equations G n = 0. As
G n = E AEn GA = E aEn Ga = g1e( + )umpe aVp Ga , (A.21)
it follows that G n = 0 is equivalent to Ga = 0.
{ 27 {
We now study the eld equations Ga = 0. We have
Db(e
2
3 (d+1)F ab) + e
bT uida = 0 .(A.22)
Finally, let us consider the d-dimensional components of the Einstein eld equations,
G = 0. Using the equations for other components, this equation implies
G = E AE BGAB = e2 e ae bGab = 0 . (A.23)
Therefore the resulting equation is Gab = 0, where
(A.24)
Gab = Rab
2
3 (d+1)F abPb 2g[epsilon1] M Pa + 2e
13 (d5)
JHEP02(2017)122
1
2@a@b Pa Pb
1
2e
2
3 (d+1)
F acF bd cd 12(d 2) (F )2 ab
g2d 2
e
2
3 (d+1)
M M 12M2
ab e2
bT uidab
1d 2
bT uidcd cd ab
.
These are the Einstein equations for the d-dimensional system, which can be equivalently rewritten as
R
1
2g R = T total , (A.25)
where T total = e ae bT totalab is
T totalab = 12
@a@b
1
2(@)2 ab
+ Pa Pb
1
2P2 ab
+ e
2
3 (d+1)
12F acF bd cd 1 4(F )2 ab
(A.26)
g2e
2
3 (d+1)
M M 12M2
ab + e2
bT uidab .
From this expression, we also see that the energy-momentum tensor of non-Abelian hydrodynamics T uidab is given by
T uidab = e2
bT uidab . (A.27)
B Conservation laws
In this section we will calculate the conservation laws of the d-dimensional theory, namely the current conservation and the Lorentz force. Despite of not making any assumption on the scalar elds, after obtaining the most general expressions we will study the cases for which scalar elds are covariantly constant,
D Vma = D = 0 ,
in order to make contact with the conservation laws considered in hydrodynamics, where no degrees of freedom associated to scalar elds take place.
{ 28 {
B.1 Current conservation
Current conservation follows from the consistency condition of the EOMs. Before applying a covariant derivative Da on (A.22), we rst rewrite the equation of motion for gauge eld as
hDb e23 (d+1)MnmFmab 2g[epsilon1] M Pa Vn + 2e13 (d5)
bT uida Vn i V n= 0 .(B.1)
As V n is non-degenerate in general, without loss of generality, we can assume the equation
of motion to beDb
e23 (d+1)MnmFmab = Jma , (B.2)
whereJma = 2g[epsilon1] M Pa Vm 2e
JHEP02(2017)122
13 (d5)
bT uida Vm . (B.3)
We can expand the covariant derivative and multiply by (Q2cM)1,
DbF mab = e
2
3 (d+1)Mmn
hJna Db
e23 (d+1)Mnp
F pabi. (B.4)
Then if we apply another covariant derivative Da, the l.h.s. vanishes and we nd that the current Jm that is covariantly conserved, DaJma = 0, is given by
J ma = e
2
3 (d+1)Mmn
hJna Db
e23 (d+1)Mnp
F pabi. (B.5)
If we set the scalar elds DaVm = 0, then Pa = 0 and the color current will be purely associated to the o -diagonal components of the D-dimensional uid energy-momentum tensor.
B.2 Lorentz force
To study the Lorentz force, we will make use of the Bianchi identity of the Einstein tensor
r
R
1
2g R
= r T total = 0 . (B.6)
Upon vielbein compatibility, this is equivalent to
DaT totalab = 0 , (B.7)
where Da = ea (@ + ! ), where ! is the d-dimensional spin connection. Explicitly,
DaT totalab = Da
e2
bT uidab
+ 1
2 (Da@a@b + @aDa@b @cDa@c ab) + DaPa Pb + Pa DaPb DaPc Pc ab +
1
2Da
e
2
3 (d+1)
F acF bd cd 1 2(F )2 ab
+ 12e
2
3 (d+1)
Da
F acF bd cd
1
2(F )2 ab
g2Da
e
2
3 (d+1)
M M 12M2
ab
.
(B.8)
{ 29 {
Let us analyze various terms separately.
1
2 (Da@a@b + @aDa@b @cDa@c ab) = @b
1 8(d 2)
e
2
3 (d+1)(F )2
+ 1
2(d 2)
g2e
2
3 (d+1)
1
2(d 2)
e2
3d + 1
bT
bT
,
(B.9)
and
DaPa Pb +
JHEP02(2017)122
1
2D[aPb] Pa =
1
2e
2
3 (d+1)
F F 1 3(F)2
Pb
+ 4g2e
2
3 (d+1)
M M 12M M 13 V (M)
Pb
2P[a| V mD|b]Vm Pa
g 2V mfnmpFnabVp Pa .
(B.10)
Using the Bianchi identity DFm = 0 and the above equations of motion, we have
1
2Da
e
2
3 (d+1)
F acF bd cd 1 2(F )2 ab
= 12Da
e
2
3 (d+1)Mmn
FmacFnbd cd
1
2FmFn ab
1
2Da
e
2
3 (d+1)
F acF bc
+ e
2
3 (d+1)F cdF bcPd g[epsilon1] M Pc F bc + 2e
13 (d5)
bTc Vn Fnbc . (B.11)
Summing up all the terms, we have
DaT totalab = DaT uidab + @b 18(d 2)
e23 (d+1)(F )2 + 12(d 2)
g2e23 (d+1)
1
2(d 2)
e2
3d + 1
bT uid
bT uid
+ 12e
2
3 (d+1)
F F 1 3(F)2
Pb
+ 4g2e
2
3 (d+1)
M M 12M M 1 3 V
Pb
2P[a| V mD|b]Vm Pa
g 2V mfnmpFnabVp Pa
+ 12Da
e
2
3 (d+1)Mmn
FmacFnbd cd
1
2FmFn ab
1
2Da
e
2
3 (d+1)
F acF bc + e
2
3 (d+1)F cdF bcPd
g[epsilon1] M Pc F bc + 2e
13 (d5)
bTc Vn F nbc g2Da h e
2
3 (d+1)V (M) ab
i
.
(B.12)
{ 30 {
If we assume at Minkowski in d dimensions and that scalar elds are covariantly constant,
DaVm = Da = 0 , (B.13)
this expression reduces to
DaT uidab + 2e
13 (d5)
bTc Vn Fnbc = e2 Da
bT uidab + 2e
= 0 . (B.14)
Open Access. This article is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/
Web End =CC-BY 4.0 ), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.
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Abstract
Abstract
We present a new approach to describe hydrodynamics carrying non-Abelian macroscopic degrees of freedom. Based on the Kaluza-Klein compactification of a higher-dimensional neutral dissipative fluid on a manifold of non-Abelian isometry, we obtain a four-dimensional colored dissipative fluid coupled to Yang-Mills gauge field. We derive transport coefficients of resulting colored fluid, which feature non-Abelian character of color charges. In particular, we obtain color-specific terms in the gradient expansions and response quantities such as the conductivity matrix and the chemical potentials. We argue that our Kaluza-Klein approach provides a robust description of non-Abelian hydrodynamics, and discuss some links between this system and quark-gluon plasma and fluid/gravity duality.
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