G E N E R A L I S E D G L O B A L S Y M M E T R I E S A N D D I S S I PAT I V E M A G N E T O H Y D R O DY N A M I C S

I N S T I T U U T- L O R E N T Z F O R T H E O R E T I C A L P H Y S I C S L E I D E N U N I V E R S I T Y

O X F O R D, 1 4 . 7 . 2 0 1 7

S AŠO G R O Z D A N O V

O U T L I N E

• hydrodynamics

• generalised global symmetries and magnetohydrodynamics

• plasma and holography

• outlook

2

3

with 
D. Hofman and N. Iqbal

with 
N. Poovuttikul

FRAGMENT I FRAGMENT II

H Y D R O DY N A M I C S

H Y D R O DY N A M I C S• QCD and quark-gluon plasma

• low-energy limit of QFTs (effective field theory)

• tensor structures (phenomenological gradient expansions) with transport coefficients (microscopic)

Tµ⌫�u�, T, µ

�= ("+ P )uµu⌫ + Pgµ⌫ � ⌘�µ⌫ � ⇣r · u�µ⌫ + . . .

5

rµJµ = 0rµTµ⌫ = 0

Jµ(u�, T, µ) = nuµ � �T�µ⌫r⌫(µ/T ) + . . .

H Y D R O DY N A M I C S

• conformal (Weyl-covariant) hydrodynamics

• infinite-order asymptotic expansion

• classification of tensors beyond Navier-Stokes

first order: 2 (1 in CFT) - shear and bulk viscosities

second order: 15 (5 in CFT) - relaxation time, … [Israel-Stewart and extensions]

third order: 68 (20 in CFT) - [S. G., Kaplis, PRD 93 (2016) 6, 066012, arXiv:1507.02461] 


Tµµ = 0

gµ⌫

! e�2!(x)gµ⌫ Tµ⌫ ! e6!(x) Tµ⌫

! =OHX

n=0

↵nkn+1Tµ⌫ =

1X

n=0

Tµ⌫(n)

6

H Y D R O DY N A M I C S• diffusion and sound dispersion relations in CFT

• loop corrections break analyticity of the gradient expansion (long-time tails), but are 1/N suppressed [Kovtun, Yaffe (2003)]

• entropy current, constraints on transport and new transport coefficients (anomalies, broken parity)

• non-relativistic hydrodynamics

• hydrodynamics from effective Schwinger-Keldysh field theory with dissipation [Nicolis, et. al.; S. G., Polonyi; Haehl, Loganayagam, Rangamani; de Boer, Heller, Pinzani-Fokeeva; Crossley, Glorioso, Liu]

shear: ! = �i ⌘"+ P

k2 � i"

⌘2⌧⇧

("+ P )2� 1

2

✓1"+ P

#k4 +O �k5�

sound: ! = ±csk � i�ck2 ⌥ �c2cs

��c � 2c2s⌧⇧

�k3 � i

"8

9

⌘2⌧⇧

("+ P )2� 1

3

✓1 + ✓2"+ P

#k4 +O �k5�

7

M A G N E T O H Y D R O DY N A M I C S

S TAT E S O F M AT T E R

• solids

• liquids

• gases

• plasma

9

P L A S M A A N D M H D

• ionised gas, uncharged at large distances (screening)

• the theory: magnetohydrodynamics (MHD)

10

@✏

@t+r · (✏~v) = 0

✏

✓@

@t+ ~v ·r

◆~v = �rp+ ~J ⇥ ~B

r⇥ ~E = �@~B

@t

r⇥ ~B = µ0 ~J

r · ~B = 0

r · ~E = 0

continuity + Euler (or Navier-Stokes)

Maxwell

Equation of state✓

@

@t+ ~v ·r

◆⇣ p✏�

⌘= 0

~E + ~v ⇥ ~B = 0 , (� ! 1) Ohm’s law (ideal)

P L A N A N D W I S H E S• generalise MHD to bring it to the effective field theory level of (charged)

relativistic hydrodynamics: symmetries and field content

• is there a way to avoid assumptions of the matter content?

• what if a plasma is strongly coupled, contains strong magnetic fields, 
 
its equation of state depends on magnetic properties and the matter sector is strongly coupled to the charged sector,

• quantum effects: pair-creation (Euler-Heisenberg), Landau levels, …?

• systematic derivative expansions: transport coefficients

• look at an example of such a plasma and learn something new about real plasmas (fusion, magnetars, quark-gluon plasma, …)?

11

p(T,B)

B/T 2 � 1

G E N E R A L I S E D G L O B A L S Y M M E T R I E S

• symmetry based formulation of MHD 
[S. G., Hofman, Iqbal, PRD (2017) 9, 096003, arXiv:1610.07392]

• gauge vs. global U(1) symmetry in electromagnetism

• 0-form symmetry (1-form conserved current)

• 1-form symmetry (2-form conserved current) 
[Gaiotto, Kapustin, Seiberg, Willet] 


• matter-less electromagnetism has two U(1) 1-form symmetries

• spontaneous breaking of the symmetry gives a photon

12

O(x) ! eiq⇤O(x)

W (C) ! exp✓iq

Z

C⇤µdx

⌫

◆W (C)

1

g2rµFµ⌫ = j⌫el Jµ⌫ =

1

2✏µ⌫⇢�F⇢�

G E N E R A L I S E D G L O B A L S Y M M E T R I E S

• such a symmetry can be source or gauged like any other symmetry

• related to one-form current by a dualisation

• our theory has the following conserved global symmetries: energy-momentum and a U(1) number of magnetic flux lines (in the presence of charged matter)

13

rµTµ⌫ = 0 rµJµ⌫ =1

2rµ ("µ⌫⇢�F⇢�) = 0

rµTµ⌫ = H⌫⇢�J⇢�

S[b] ⌘ S0 +�S[b], �S[b] ⌘Z

d

4x

p�g bµ⌫Jµ⌫

�S[b] =

Zd

4

x

p�gA�j�

ext

, j

�ext

⌘ ✏�⇢µ⌫@⇢bµ⌫

external source

I D E A L M H D A N D W A V E S

• construct a general hydrodynamic gradient expansion with fields: 


• ideal MHD
(string hydro)

• thermodynamics

• Alfvén and (slow and fast) magnetosonic waves from

14

Tµ⌫(0) = ("+ p)uµu⌫ + p gµ⌫ � µ⇢hµh⌫

Jµ⌫(0) = 2⇢u[µh⌫]

v2M =1

2

⇢�V2A + V20

�cos

2 ✓ + V2S sin2 ✓ ±q⇥

(V2A � V20 ) cos2 ✓ + V2S sin2 ✓

⇤2+ 4V4 cos2 ✓ sin2 ✓

�

v2A = V2A cos2 ✓

! = ±vAk ! = ±vMk

uµ, T, hµ, µ u2 = �1, h2 = 1, hµuµ = 0

"+ p = Ts+ µ⇢ dp = sdT + ⇢dµ

rµTµ⌫ = 0rµJµ⌫ = 0

I D E A L M H D A N D W A V E S

• Alfvén waves have been observed in solar corona [McIntosh, et. al (2011)]

15

S P E E D S O F M H D W A V E S• assume weak field limit of the equation of state to reproduce all of standard

MHD results (Alfvén and magnetosonic waves) [Hernandez, Kovtun]

• strong field limit

16

pweak (µ, T ) =a

4T 4 +

g2

2µ2 +

�

4

µ4

T 4+ · · ·

pstrong

(µ, T ) =g02

2µ2 +

a0

4T 4 +

�0

8

T 8

µ2+ · · ·T 2 ⌧ µ :

µ ⌧ T 2 :

D I S S I PAT I O N• construct first-order (anisotropic) dissipative corrections

• 7 terms consistent with positive entropy production and charge-conjugation invariant MHD (2 shear, 3 bulk viscosities, 2 resistivities); 11 in general MHD [Hernandez, Kovtun]

17

Sµ = s uµ � 1TTµ⌫(1)u⌫ �

µ

TJµ⌫(1)h⌫ , rµS

µ � 0

Tµ⌫(1) = �f �µ⌫ + �⌧ hµh⌫ + 2 `(µh⌫) + tµ⌫

Jµ⌫(1) = 2m[µh⌫] + sµ⌫

�f = �⇣?�µ⌫rµu⌫ � ⇣(1)⇥ hµh⌫rµu⌫�⌧ = �⇣(2)⇥ �µ⌫rµu⌫ � 2⇣khµh⌫rµu⌫`µ = �2⌘k�µ�h⌫r(�u⌫)

tµ⌫ = �2⌘?✓�µ⇢�⌫� � 1

2�µ⌫�⇢�

◆r(⇢u�)

mµ = �2r?T�µ�h⌫r[�✓h⌫]µ

T

◆

sµ⌫ = �2rkµ�µ⇢�⌫�r[⇢h�]

⌘? � 0 ⌘k � 0r? � 0 rk � 0⇣? � 0 ⇣?⇣k � ⇣2⇥

D I S S I PAT I O N• corrections to the dispersion relations of MHD waves

• limits of small momentum and large angle do not commute

• expand in small k, e.g. the Alfvén wave is

• for non-infinitesimal k, there is a critical angle at which propagating modes (slow-MS and Alfvén) cease to exist and become diffusive
(weak/strong B)

18

! = ±k cos ✓s

1

1 +

Tsµ⇢

� 12

i

✓sin

2 ✓

Ts+ µ⇢⌘? +

cos

2 ✓

Ts+ µ⇢⌘k +

µ cos2 ✓

⇢r? +

µ sin2 ✓

⇢rk

◆k2

✓c(k)

D I S S I PAT I O N• in the regime of a strong magnetic field, when anisotropic

corrections become large, critical angle decreases from

• Kubo formulae, e.g. resistivities

• normally

• now, new diagrams: e.g.

19

rk = lim!!0

Gxy,xyJJ

(!)

�i! r? = lim!!0Gxz,xz

JJ

(!)

�i!

hEE(!)i ⇠ �(�i!)2 g2G��(!)

1� hjeljel(!)i g2G��(!)r ⇠ (�i!) 1hjeljel(!)i

⇠ 1�

+hjj(!)i

+ · · ·

+g2G��(!)

+hjj(!)i

G�(!)

+ · · ·

+

r 6= 1�

M H D AT Z E R O T E M P E R AT U R E • take the limit of zero temperature

• a sector of the theory without entropy production, and

• symmetry breaking patterns

• degrees of freedom s.t.

• ideal hydrodynamics

(no 1st order correction)

• entropy conservation vs. closure of equations (reduces 8 (7) to 5)

20

�T = 0

finite T : SO(3, 1) ! SO(2)zero T : SO(3, 1) ! SO(1, 1)⇥ SO(2)

uµ⌫uµ⌫ = �2, uµ⌫u⌫⇢u⇢� = uµ�µ, uµ⌫

Tµ⌫(0) = �"⌦µ⌫ + p⇧µ⌫

Jµ⌫(0) = ⇢uµ⌫

⌦µ⌫ ⌘ uµ�u ⌫�

(rµTµ⌫)⌦⌫� + µ (rµJµ⌫)u⌫� = 0

⇧µ⌫ ⌘ gµ⌫ � ⌦µ⌫

M H D AT Z E R O T E M P E R AT U R E • this theory has no slow MS waves due to vanishing fluctuation of the

temperature

• universal dispersion relation for Alfvén waves (to all orders)

• fast MS waves receive corrections from 2nd order hydrodynamics and travel at the

• in absence of any scales beyond B, speed of fast MS waves is the speed of light

21

! = ±k cos ✓

!M = ±�vMk + ↵Mk

3 + · · ·�

vM =

✓cos

2 ✓ +⇢

µ�sin

2 ✓

◆1/2� =

@⇢

@µ

H O L O G R A P H I C P L A S M A

H O L O G R A P H I C M H D• example of a strongly coupled plasma [S. G., Poovuttikul, arXiv:1707.04182]

• fields at the boundary source dual conserved operators

• generating functional

• construct a theory of a metric and a two-form gauge field in 5D

• naively, it is dual to the bulk Einstein-Maxwell theory with 
if we write :

23

W [gµ⌫ , bµ⌫ ] =

⌧exp

i

Zd

4x

p�g

✓1

2

T

µ⌫gµ⌫ + J

µ⌫bµ⌫

◆��

S =1

225

Zd5x

p�G

✓R+

12

L2� 1

3e2HHabcH

abc

◆

FabFab

H = dB

H = ?F F ^ ?F ! H ^ ?H

H O L O G R A P H I C M H D• the connection to Einstein-Maxwell theory is more subtle

• dualise by writing the path integral over F

24

Z �Z

DFab DBab exp⇢i

N

2c

8⇡

2

Zd

5x

p�G

�FabF

ab+ e

�1H Bab✏

abcdercFde��

hJµRi = �N2c2⇡2

limu!0

Fuµ = � N2c

2⇡2eHlimu!0

✏µ⌫⇢�@⌫B⇢�

• the gauge field on the boundary is dynamical (photons) [Hofman, Iqbal]

Dirichlet BC 
(standard quantisation)

von Neumann BC 
(alternative quantisation)

gab , Bab gab , Aab

Zd

4xJ

µ⌫�Bµ⌫

Zd

4xAµ�Jµ

H O L O G R A P H I C M H D• full holographic action

• D’Hoker-Kraus black brane

• on-shell action at a cut-off (FG coordinates)

• source

25

S =N2c8⇡2

"Zd5x

p�G✓R+ 12� 1

3e2HHabcH

abc

◆+

Z

@Md4x

p�g✓2 trK � 6 + 1

e2HHµ⌫Hµ⌫ ln C

◆#

Hµ⌫ = naHaµ⌫

ds2 = Gabdxadxb = r2h

✓�F (u)dt2 + e

2V(u)

v(dx2 + dy2) +

e2W(u)

wdz2

◆+

du2

4u3F (u)

H =Br2he

�2V+W

2u3/2pw

dt ^ dz ^ du

�S

on�shell = �N2c

4⇡2

Zd

4x

p�gHµ⌫⇣�B

(0)µ⌫ + �B

(1)µ⌫ ln C2⇢⇤

⌘

B(0)µ⌫ +B(1)µ⌫ ln C2⇢⇤ =

4⇡2

N2cbµ⌫

H O L O G R A P H I C M H D• expectation values of dual conserved operators

26

hTµ⌫i =� N2c

4⇡2lim✏!0

r2h✏

✓Kµ⌫ � �µ⌫K � 3�µ⌫ + 1

2R[�]µ⌫ � 1

4�µ⌫R[�]

�✓Hµ�H �⌫ �

1

4�µ⌫H↵�H↵�

◆ln(C⇢)

◆����⇢=✏

hJµ⌫i =� lim✏!0

Hµ⌫��⇢=✏

matter

Maxwell

• stress-energy tensor has two contributions, more precisely [Fuini, Yaffe]

hTµ⌫i = lim✏!1/⇤2

N2c2⇡2

✓g(2)µ⌫ � g(0)µ⌫ (g(2))�� +

1

2h̃µ⌫ + h̃µ⌫ ln

�C2⇢�+O(⇢, @2)◆����

⇢=✏

N2c⇡2

h̃µ⌫ ln(⇤/C) =N2c⇡2

h̃µ⌫ ln(⇤/M) + h̃µ⌫

✓2

e2r� N

2c

⇡2ln(⇤/M)

◆

H O L O G R A P H I C M H D

• trace anomaly and N=4 NSVZ beta function of U(1)R (from SU(4)R)

27

hTµ⌫i = hTmatterµ⌫ i+ hTEMµ⌫ i

hTmatterµ⌫ i =N2c2⇡2

✓g(2)µ⌫ � g(0)µ⌫ (g(2))�� +

1

2h̃µ⌫

◆� N

2c

⇡2h̃µ⌫ ln(⇤/M)

hTEMµ⌫ i = �✓

2

e2r� N

2c

⇡2ln (⇤/M)

◆h̃µ⌫

SEM = �1

4e(⇤/M)2

Zd

4x

p�gFµ⌫Fµ⌫

• electromagnetic term comes from the Maxwell action

hTEMµ⌫ i =1

e(⇤/M)2

✓hFµ↵F ↵⌫ i �

1

4⌘µ⌫hF↵�F↵�i

◆=

1

e(⇤/M)2

✓hFµ↵ihF ↵⌫ i �

1

4⌘µ⌫hF↵�ihF↵�i

◆

hTµµi = ��(e)

e3B2 = �N

2c

4⇡2B2

��1/e2

�= µ

de�2

dµ= �N

2c

2⇡2

"1

6

4X

↵=1

(q↵f )2 +

1

12

3X

a=1

(qas )2

#= �N

2c

2⇡2

H O L O G R A P H I C M H D• final renormalised expectation values

• RG-condition-dependent electromagnetic coupling

• Ward identities from MHD are satisfied

• interpretation: not quite gauged N=4 SYM (gauge anomaly of U(1)R, no Chern-Simons term, B-term not in IIB supergravity)

28

hTµ⌫i =N2c2⇡2

✓g(2)µ⌫ � g(0)µ⌫ (g(2))�� +

1

2h̃µ⌫

◆� 2

e2rh̃µ⌫

hJµ⌫i = 2B(1)µ⌫

↵̄ =N2c2⇡2

↵ =N2c2⇡2

e2r4⇡

=1

137

rµTµ⌫ = H⌫⇢�J⇢� rµJµ⌫ = 0

T H E R M O DY N A M I C S29

T R A N S P O R T C O E F F I C I E N T S30

• maximal resistivity

• bulk viscosities

⇣?⇣k = ⇣2⇥

S P E E D S O F S O U N D31

! = ±vk � iDk2

S O U N D AT T E N U AT I O N32

! = ±vk � iDk2

T R A N S M U TAT I O N O F M O D E S33

✓c(k)

transition from propagating to non-propagating modes (from sound to diffusion) at a k-dependent critical angle

E L E C T R I C C H A R G E• behaviour of waves (e.g. Alfvén) strongly dependent on the

renormalisation condition for the boundary electric charge

34

S U M M A R Y• new, symmetry-based formulation of MHD, with new transport

coefficients which agrees with (complete) standard MHD at low magnetic field strength [Hernandez, Kovtun]

• we claim this theory can be used for any plasma

• MHD may survive the zero-T limit and become non-dissipative

• the theory can be studied via a holographic toy model, which includes dynamical electromagnetism (other applications?)

• MHD modes could be examined at any strength of B

• transmutation of propagating to non-propagating modes, least-conductive plasma, saturation of bulk viscosity inequality at strong coupling

35

O U T L O O K

is any of this useful for realistic plasmas?

36

T H A N K Y O U !