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JHEP05(2018)192

Published for SISSA by Springer

Received: April 14, 2018

Revised: May 15, 2018

Accepted: May 25, 2018

Published: May 30, 2018

Dissipative hydrodynamics with higher-form symmetry

Jay Armas,

a,b

Jakob Gath,

Akash Jain

and Andreas Vigand Pedersen

Institute for Theoretical Physics, Univer s it y of Amsterdam,

1090 GL Amsterdam, The Netherlands

Dutch Institute for Emergent Phenomena, University o f Amsterdam,

1090 GL Amsterdam, The Netherlands

Department of Physics, Technical University of Denmark,

DK-2800 Kgs. Lyngby, Denmark

Centre for Particle Theory & Department of Mathematical Sciences, Durham University,

South Road, Durham DH1 3LE, United Kingdom

Seaborg Technologies,

Bredgade 36B, 1260 Copenhagen, Denmark

E-mail:

j.armas@uva.nl, jagath@fysik.dtu.dk, akash.jain@durham.ac.uk,

andreas@seaborg.co

Abstract: A theory of parity-invariant dissipative ﬂuids with q-form symmetry is for-

mulate d to ﬁrst order in a derivative ex p ansi on. The ﬂuid is anisotropic with symmetry

SO(D −1 −q) ×SO(q) and carries dissolved q-dimensional charged objects that couple to

a (q + 1)-form background gauge ﬁeld. The case q = 1 for which the ﬂui d carries string

charge is related to magnetohydrodynamics in D = 4 spacetime dimensions. We ide ntify

q+7 parity-even independent transport c oeﬃcients at ﬁrst order in derivatives for q > 1.

In particular, compared to the q = 1 case under the assumption of parity and charge con-

jugation invariance, ﬂuids with q > 1 are charact er i s ed by q extra transport coeﬃcients

with the physical interpretation of shear viscosity in the SO(q) sector and current resistiv-

ities. We discuss certain issues related to the existence of a hydrostati c sector for ﬂuids

with higher-form symmetry for any q ≥ 1. We extend these results in order to include

an int er f ace s ep ar ati ng diﬀerent ﬂuid phases and study the dispersion relation of capillary

waves ﬁnding clear signatures of anisotropy. The formalism developed here can be easily

adapted to study hydrodynamics with multiple higher-form symmetries.

Keywords: Eﬀective Field Theories, Holography and condensed matter physics

(AdS/CMT), Holography and quark-gluon plasmas

ArXiv ePrint:

1803.00991

Open Access,

 The Authors.

Article funded by SCOAP

https://doi.org/10.1007/JHEP05(2018)192

JHEP05(2018)192

Despite hydrodynamics being a well established research subject, there has been a

substant i al progress in its structural foundations in recent years. This includes an oﬀ-shell

formul ati on of hydrodynamics and the development of cl assi ﬁc at i on schemes of its trans-

port properties [

1–3]; the construction of eﬀective ﬁeld theories and the formul ati on of

action principles for dissipative ﬂuid dynamics [

4–9]; the establishment of a framework for

describing interfaces between diﬀerent ﬂuid phases [

10–12]; a new formalism for studying

non-relativistic ﬂuids [

13–16]; and the development of hydrodynamic theories with gener-

alised global 1-form symmetries and their connec t i ons to magnetohydrodynami c s [

17–20]

as well as their role in the understanding of eﬀective theories with translational symmetry

breaking and state s with dynamical defects [

21].

This paper introduces a framework for building eﬀective hydrodynamic theories of dis-

sipative ﬂuids with q-form symmetries, general i si n g previous work for q = 0, 1. These eﬀec-

tive t he or i es correspond to the hydrodynamic limit of microscopi c descriptions whose un-

derlying fundamental charged objects are q-dimensional (i.e. q-dimensional branes). These

q-dimensional objects couple to a background gauge ﬁeld A

q+1

. In the l anguage of [

22],

these ﬂuids describe microscopic systems with a generalised q-form global symmetry. Asso-

ciated wi th the q-form symmetry is a (q+1)-form current J whose int egr al over a (D−q−1)

dimensional hypersurface M

yields a conserved dipole charge

⋆J , (1.1)

where the operator ⋆ is the Hodge dual operator in D-dimensional spacetime. This dipole

charge counts the number of q-dimensional objec ts that cross the (D −q − 1)-dime nsional

hypersurface M

The hydr odynamic theories constructed here capture the collective

excitations of the se charged q-dimensional objects around a state of thermal equilibrium.

This work has been highly mot i vated by th e structure of long-wavelength perturbations

of black branes in supergravity, wher e dissip ati ve ﬂuids wi t h higher -f orm symmet r i e s are

naturally realised [

23–27]. As the fundamental ﬁelds in supergravity include sever al higher-

form gauge ﬁelds, generic black brane bound states can carry multiple higher-form (dipole)

charge s. For instance , the D3-F1 bound state in type IIB str i ng theory carries two 2-

form currents and one 4-form current [

27]. Within this context, the ﬂuid stress tensor

and charge currents appear as the eﬀective currents sourcing the gravitational and el e c tr i c

ﬁelds far away from the black brane horizon [

27], while their conservation laws are reali se d

as constraint equations when solving Einstein equations in a long- wavelength expansion.

Tackling the general problem of establishing a hydr odynamic theory of dissipative ﬂuids

carrying multiple higher-form charges is certainly of interest and in this wor k we take the

ﬁrst step towards thi s goal by const r uc ti n g the eﬀective hydrodynam i c the or y of ﬂuids

with one single q-f orm symmetry. These ﬂuids are anisotropic with SO(D −q −1) ×SO(q)

symmetry and the source of this anisotropy is t he existence of a higher-form charge current.

Grav i tat i onal duals to these ﬂuids are encountered in black brane geometries in gravity

In the case q = 1 in D = 4 an d in the context of magnetohydrodynamics where J

µν

= ⋆F

µν

, the

fundamental objects are strings and Q

counts the number of magnetic ﬁeld lines crossing a codimension-2

hypersurface. See [

18] for a detailed discussion.

– 2 –

JHEP05(2018)192

theories with a metric g

µν

, one single (q + 1)-form gauge ﬁeld A

q+1

and possibly a dilaton

ﬁeld. Several examples of s uch gravitational duals with arbitrary q were found in [

24].

In this work, we formulate the theory of dissipative ﬂuids with q-form symmetry to

ﬁrst order in a long-wavelength expansion (to ﬁrst order in derivatives of the ﬂuid ﬁelds)

focusing on the parity-even sector of the theory.

We also study equilibrium conﬁ gur ati ons

and highlight some of the technical complications that arise while descr i bi ng the hydrostatic

sector of ﬂuids with higher-form symm et r i e s. In addition, we generalise our results t o

include the presence of interfaces separating diﬀerent ﬂuid phases and study speciﬁc cases

of surface waves, ﬁnding signatures of anisotropy in their dispersion relation.

1.1 Summary of the results and organisation of the material

In section

2, the ideal order d y nami c s of ﬂui ds with q-form symmetry living in a background

spacetime with metric g

µν

and a (q+1)-form gauge ﬁeld A

q+1

is introduced. These ﬂuids are

anisotropic with SO(D−q−1)×SO(q) sy mm et r y and carry a dipole charge de ns i ty Q. They

are characterised by a stress tensor, charge (q + 1)-form current and an entropy current.

The existence of this (q + 1)-for m current is responsible for the mic r osc opi c anisotropic

properties of the ﬂuid. When the charge Q vanishes, the current vani s he s as well and the

usual uncharged isotropic ﬂuid is recovered. We establish their t he r modynamic properties

and conservation laws. As far as we are aware, this is the ﬁrst time that the ideal order

dynamics of these ﬂuids is formalised in the literature.

In section

3 we formulate the dissi p ati ve sector of the theory up to ﬁrst order in

derivatives. We focus on the parity-even sector for q ≥ 0 while for q = 1 we in addition

require charge conjugation invariance. First, we describe how frame t r ansf orm ati ons act

on the stress tensor and currents and comment on diﬀerent choices of frames. Picking a

higher-form analogue of the Landau frame, we proceed and require the divergence of the

entr opy current to be positive semi-deﬁnite. This leads to the existence of q+8 transport

coeﬃcients for q > 1 and 8 for q = 1, all of which are dissipative, thereby cont r i but i ng to

entr opy product i on. Once Kubo formulae are obtained, we note that Onsager’s relation for

mixed correlation functions sets a constr ai nt among these transport coe ﬃc i e nts, thereby

leading to q+7 indepe nde nt transport coeﬃcients at ﬁrst order in derivatives for q > 1 and

7 for q = 1. Compared to the q = 1 case studied in [

18, 19], for q > 1 there i s one additional

transport coeﬃcient with the physical interpretation of shear viscosity in the SO(q) sector

and q −1 extra current resistivities. At t he end of this section, we study the constraints on

these transport coeﬃcients in isotropic limits. We observe that some of these constraints

are satisﬁed by gravitational duals even away from the isotr opi c limits.

Section

4 contains a detailed analysis of equilibrium conﬁgurations in theories wi t h a

q-form symmetry. We begin by noting that the equilibrium partition funct i on presented

in [

18] for q = 1 does not describe the hydrostatic se c tor of the theory completely. As

such, the hydrostatic solution as deﬁned in [

18], which as sum es hydros tat i c backgr ounds

to admit not j ust a timelike isometry but also q spacelike isometries, causes production of

The q = 1 case deserves considerably more attention and will be the focus of a later pub lic a tio n [

28].

In this paper we further restrict the q = 1 case by requiring charge conjugation invariance in addition to

parity invariance. In this case, our analysis for q = 1 is the same as that inp ro [18 ].

– 3 –

JHEP05(2018)192

entr opy, which is incompatible wi t h equilibrium. We show that to avoid this, an additional

constraint must be imposed on the hydrostatic backgrounds. Furthermore, [

18] assumed

that spacelike and timelike isometries admitted by the hydrost at i c backgrounds must have

vanish i ng mutual inner products, which further restricts t he class of backgrounds on which

equilibrium can be realised. We explicitl y derive the most general partition function where

these inner products are not assumed to be zero and show that the resulting solution is a

q-form generalisation of the q = 1 solution presented in [

24]. We comment on other issues

regarding the hydrostatic sector of these theories, wherein the requirement of an equilib-

rium parti ti on function seems to impose more constrai nts than requiring the second law

of thermodynamics alone to hold. This is in striking contrast with q = 0 hydrody nam i cs ,

where the second law guarantees the existence of an equilibrium partition function at all

orders in the derivative expansion [

29, 30].

In section

5, following [10, 12], we generalise our results in order to include the presence

of an interface/surface separating two diﬀerent ﬂuid phases. Similar to section

4, we write

down a partition function for ﬂuids with q-form symmetry in the presence of the interface.

We then analyse the divergence of the surface entropy current and obtain the surface

thermodynamics as well as a constraint on the normal component of the bulk ﬂuid velocity.

We observe that th i s matches partition function ex pectations. Having es tabl i s he d a notion

of equilibrium i n this setting, we obtain the dispersion relation for capillary waves and

ripples on the interface, ﬁnding clear signals of anisotropy.

Finally, in section 6 we comment on some open issues and future research directions.

We also provide appendix

A with some of our results written in another ﬂuid frame in

order to ease comparison with earlier literature.

2 Ideal order ﬂuids with q-form symmetry

In this sec ti on we introduce the ide al order currents and conservation e q uat i ons for the

propagation of an anisot r opi c ﬂuid with q-form symmetry carrying q- br ane charge in

a D-dimensional background geomet r y (M, g

µν

, A

q+1

) with p-spatial direct i ons so that

D = (p + 1) with p ≥ q. The manifold M is endowed with the Levi-Civita connection ∇

built out from the background metric g

µν

with coordinate s x

. These ﬂuids are charac-

terised by a (q + 1)-form current J that couples to the b ackground gauge ﬁeld A

q+1

. In

general, introducing a conserved higher-form current breaks the SO (p) sy m me tr y enjoyed

by the ordinary relativistic “point charged” (or neutral) ﬂuid to a SO(p −q) ×SO(q) sym-

metry.

As usual, at each point of M there exists a rest-frame in which the ﬂuid is static.

At ideal order, this frame is unambiguously deﬁned and is char act er i s ed by a timelike vec-

tor u

(the ﬂuid velocity) normalised such that u

= −1. For later use, we introduce

the projector transverse to the ﬂuid veloci ty ∆

µν

= g

µν

+ u

. The local thermodynamic

ﬁelds of the ﬂuid are then unambiguously deﬁned as their local values in the rest-frame.

We now proceed to write now the ideal order hydrodynamics descri bi n g this system.

If multiple q

-form conserved currents with i = 1, . . . , ℓ are introduced, the ﬂuid is expec ted to have the

symmetry SO(p −

ℓ

i=1

) × SO(q

) × · · · × SO(q

ℓ

– 4 –

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