Holographic dualities for Magnetohydrodynamics

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MSc PHYSICS AND ASTRONOMY

Theoretical Physics

Master Thesis

Holographic dualities for Magnetohydrodynamics

Basil B. van Wijk

10772314

February 10

th

, 2021

Supervisor/Examiner dr. Jacom´e Armas

Institute of Theoretical Physics, ITP University of Amsterdam

Examiner dr. Ben W. Freivogel Institute of Theoretical Physics, ITP University of Amsterdam

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Abstract

This thesis aims to establish a duality between holographic theories for magnetohydrodynamics (MHD). In recent work, a duality between the “traditional” formulation of MHD, which contains a dynamical one-form gauge field, and the “dual” formulation, which has U (1) one-form global symmetry was estab-lished. This duality is exploited within the holographic context to obtain a duality between holographic theories. We show that the “dual” theory with an external two-form gauge field is equivalent to the Einstein-Maxwell theory with a dynamical one-form gauge field in five-dimensional Anti-de Sitter (AdS5)

spacetime. This difference in external and dynamical gauge fields is due to the exchange of boundary conditions inherent to a well-defined duality procedure. Using holographic renormalisation methods, we obtain the Ward identities that describe the dynamics of MHD in both “traditional” and “dual” formulations. Furthermore, we explicitly demonstrate the duality using a specific asymptotically AdS solution describing magnetic black branes. Finally, we provide a general analysis of holographic dualities in D-dimensions containing a (p − 1)-form gauge field.

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Acknowledgements

First and foremost, I am grateful to my supervisor Jay; for the academic guidance, genuine help and interest, and no-nonsense conversations and feedback. I truly enjoyed every meeting or call, which helped me further in the work, as well as motivated me.

Moreover, I would like to thank my fellow students for the interesting discussions and support that helped me put everything in perspective. Especially when it was still possible to meet in the Masters Room, and all those countless coffee breaks both were inspirational and helped to unwind.

Lastly, I want to thank all the friends and family that helped me through this long and covid-induced-lonely process. Especially those that showed any genuine interest in my abstract and seemingly irrelevant work, that interest meant a lot to me.

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5 Generalisation 41 5.1 Original theory . . . 41 5.1.1 Near-boundary expansion . . . 42 5.1.2 Boundary conditions . . . 43 5.2 Dual theory . . . 44 5.2.1 The duality . . . 44 5.2.2 Boundary conditions . . . 46 Discussion 48 A Duality and boundary conditions for the gauge fields 50 A.1 Explicit duality . . . 50

A.2 Boundary conditions . . . 51

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Introduction

Magnetohydrodynamics (MHD) is the theory defined by the combination of the Navier-Stokes equations for fluid dynamics and the Maxwell’s equations for electromagnetism, describing e.g. plasmas. MHD is the regime of hot electromagnetism where the electric fields are Debye screened and the plasma is electrically neutral. Alfv`en was the first to research MHD in his 1942 paper [1], for which he later received a Nobel prize. Later, this theory was further developed and is a dynamical gauge theory. (See [2] for an overview of relativistic MHD.) In general, this traditional approach has proven to be successful in describing many phenomena in e.g. astrophysics, nuclear and solar physics [3–6], which are usually numerical descriptions. However, this traditional approach lacks any fundamental principle in describing MHD, as it has a dynamical gauge field. To address this shortcoming, this thesis aims to establish a duality for MHD using 1) a framework of generalised global (higher-form) symmetries [7] and 2) the holographic principle, which both are modern and fundamental approaches to MHD.

Firstly, the higher-form global symmetry was only recently proposed as an approach to MHD [8,

9]. This is a more intuitive way of thinking about MHD as it includes a non-scalar conserved quantity. These conserved charges and currents, associated with the generalised symmetry, form the basis of - what in this thesis is called - the “dual” approach to MHD. Specifically, in this case, the dual theory has a one-form U (1) global symmetry resulting in a two-form gauge field Bµν. This alternative approach has

been proven to be equivalent to the traditional approach at full non-linear level using effective field theory and hydrodynamic expansions [10]. This so-called string fluid formulation of MHD is a consequence of the partial breaking of the one-form U (1) global symmetry. The duality at the heart of this equivalence relation is J(2)= ?4F(2), ⇔ Jµν = 1 2 µνσλ_F σλ,

where J(2) _{is the two-form current in the dual theory. The current is conserved through the Bianchi}

identity for the field strength F(2) _{of the traditional theory. Hence, the MHD traditional theory can}

alternatively be based on this partially broken one-form U (1) global symmetry1_{. This duality is}

schemat-ically shown between the top two blocks in fig.1.

Secondly, the seminal AdS/CF T correspondence [11] is a powerful way to describe quantum field theories outside the weak-coupling limit through a holographic duality between gravity and gauge theories [12–14]. In recent work, such a holographic duality has been proposed for dual MHD [15,16], where the dual MHD theory “lives” at the boundary of the bulk spacetime. However, as there are divergences at the boundary in such a holographic theory, the method of holographic renormalisation must be applied [17]. In this procedure, a near-boundary hypersurface is introduced on which the counterterms can be added that cancel the divergent terms in the boundary-limit. To do so, the expanded equations of motion can be solved, and the counterterms in terms of the bulk fields must be added to the on-shell action on this hypersurface. These counterterms directly give rise to the correct four-dimensional MHD description and are double-trace operators [18]. In fig.1, this holographic duality is shown in the vertical relations between the bulk and boundary theories. Therefore, the connection between the holographic dual theory, with a two-form gauge field Bab(bottom right) and the traditional MHD theory with a dynamical gauge

field Aµ (top left) can be made, following the blue arrow.

In this thesis, a different connection between these theories is examined. This is done through a duality directly at the bulk, between a two-form and one-form U (1) gauge field. As shown in the bottom

1_{The nuance of the partially breaking of this symmetry is briefly mentioned in section}_1.2_{based on [}₁₀_]. 1

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Figure 1: An overview of the theories discussed in this thesis.

A schematic overview of the “different” MHD theories and their dual relations. The horizontal and vertical relations are (Hodge) dualities and holographic respectively. The bulk is in both theories

5-dimensional AdS with a 4-dimensional boundary.

relation of fig.1, the resulting duality between their field strengths is

H(3)= ?5F(2), ⇐⇒ Habc=

1 2abcdeF

de_,

where H(3) _{= dB}(2) _{is the three-form field strength of the dual theory. Although this type of bulk}

duality has been studied in various contexts, see e.g. [19] for an holographic analysis of viscoelastic hydrodynamics, it has yet to be applied to MHD.

The bulk duality is formally established by adding a Lagrange multiplier [20] to the dual bulk ac-tion, which can be identified as a gauge field sourced by a current. Inherent to this bulk duality is the exchange of boundary conditions, which is a result of the double-trace operators. As mentioned above, the counterterms needed to renormalise the theory are precisely these double-trace operators. Therefore, Dirichlet boundary conditions for the dual theory result in Neumann boundary conditions for the tradi-tional theory. This exchange affects the boundary gauge fields; the Neumann boundary conditions for the traditional bulk theory correspond to a dynamical rather than an external gauge field. Hence, the initial choice for the boundary conditions for the dual theory should indeed be of the Dirichlet type, for the traditional theory to describe MHD. DeWolf and Higginbotham, in their recent article [21] examined the same boundary conditions and explicit bulk duality, and drew similar conclusions to those of this thesis.

Lastly, the traditional MHD theory can be recovered by repeating the holographic renormalisation procedure for this traditional bulk theory, as shown in the left relation of fig.1. In this thesis, it will be explicitly shown for the dual theory and briefly repeated for the traditional theory. In conclusion, a connection can be made from dual bulk theory to the traditional boundary theory following the red arrow.

The duality will be explicitly demonstrated using a specific AAdS solution describing magnetic black branes, which is separately done for the dual and for the traditional theory. It is shown that the stress-energy tensors are identical and that the boundary duality between the two-form current and field strength holds explicitly.

This thesis concludes with an analysis of a more general case of the holographic duality for MHD. Holographic dualities for D-dimensions and (p − 1)-gauge fields will be examined including the exchange of boundary conditions.

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Outline

This thesis is structured as follows: in chapter1a brief introduction to MHD is presented and the duality at the boundary (top duality in fig. 1) is discussed. That chapter first describes the traditional and dual MHD approaches separately and is based on [2, 10]. In chapter 2, dual MHD is holographically renormalised following [17] (right relation in fig. 1), and the dual MHD equations are retrieved at the boundary starting with a bulk action as stated in [15]. Thereafter, the bulk duality (bottom duality in fig.1) is derived in chapter 3, following the methods in [19,20]. The obtained traditional bulk theory is also renormalised at the boundary in that same chapter (left relation in fig.1), to retrieve the traditional MHD equations, in parallel with chapter2. Subsequently, chapter4further strengthens the result for the bulk duality and an explicit solution to the bulk theories is considered. For both dual and traditional MHD the conserved quantities are explicitly calculated and compared for the magnetic black brane solution. In chapter5, a separate analysis of holographic dualities in D-dimensions containing a (p − 1)-form gauge field is presented. Finally, the discussion and future outlook conclude this thesis.

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Chapter 1

Relativistic Magnetohydrodyanmics

In this chapter, an introductory discussion of magnetohydrodynamics will be given. Historically, MHD would be seen as fluid dynamics and electromagnetism minimally coupled to each other, also known as weakly-coupled MHD. However, as is shown in [10] (abbreviated below as AJ), this weakly-coupled minimal MHD can, without loss of generality, be described by full MHD, without any weak-coupling assumptions. In section 1.1, the fluid part is completely standard (continuity equation, Navier-Stokes equation etc.), and the electromagnetism part is considered with a dynamical gauge field rather than external, following textbook MHD (e.g. [2,22–24]). However, the constitutive relation for the full MHD is also derived using the same approach as for the higher-form theory following AJ. In section1.2, the dual approach is discussed, which comes from higher-form global symmetries first described by [8, 9] (abbreviated below as GHI) and further complemented by AJ. Lastly, in section1.3, the duality between the theories are further discussed: presenting the differences and similarities of the theories.

In both approaches of MHD, the specific regime is considered where the electric field is of subleading order compared to the magnetic field. Effectively, the electric field is Debye screened and short-range compared to the long-range excitations of the magnetic field. MHD can be seen as a specific regime of the more general theory of “hot electromagnetism”, where the electric field is not of subleading order [10]. Similarly, electrohydrodynamics is defined where the magnetic field would be subleading to the electric field. The fields are expanded in derivatives of the dynamical variables [25, 26], which for MHD yields Bµ= O(1) and Eµ= O(∂), and is the main assumption of this thesis. Moreover, this thesis is limited to

the leading order, or ideal MHD.

1.1 Traditional magnetohydrodynamics

In this section, a short overview of non-relativistic, relativistic fluid dynamics with additional electro-magnetism is presented.

Starting with non-relativistic fluid dynamics, which are governed by the ideal fluid dynamics equations

∂tρ + ∇ · (ρ~v) = 0, (1.1)

∂t(ρ~v) + ∇(ρ~v2) + ∇P = ~F , (1.2)

where the density ρ, velocity ~v, pressure P are the 5 variables. The next step in constructing non-relativistic MHD is the electromagnetic part that gives 3 more variables for ~B (as the electric field vanishes for ideal MHD). The force in the momentum/Euler equation for MHD is the Lorentz force. In the usual MHD regime this is ~j × ~B, which in turn, when using Amp`ere’s law, becomes1

~

F = (∇ × ~B) × ~B. The electromagnetic dynamical equations are

∇ · ~B = 0, (1.3)

∂tB = ∇ × (~~ v × ~B), (1.4)

1_{This thesis works with unitary constants µ}

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1.1. TRADITIONAL MAGNETOHYDRODYNAMICS

which directly come from the Maxwell’s equations and uses the assumption of ideal Ohm’s law. That this the ideal law can be justified due to the fact that in the MHD regime, the conductivity is near-infinity. The last equation of ideal MHD is the equation of state relating pressure and density

(∂t+ ~v · ∇)(

P

ργ) = 0, (1.5)

where, usually, the heat capacity ratio γ = 5/3 (for monoatomic gas).

These 5 equations give 9 dynamical equations for 8 degrees of freedom, P , ρ, ~v and ~B. However, eq. (1.3) is seen as a separate constraint rather than a dynamical equation, leaving these equations a closed and solvable system.

1.1.1 Relativistic magnetohydrodynamics

Just like non-relativistic MHD, the relativistic MHD equations are conservation laws expressed in terms of the MHD-variables. These conservation laws are a result of continuous symmetries by Noether’s theorem. From a traditional hydrodynamical approach, the Lorentz symmetries lead to the conservation of the symmetric stress-energy tensor Tµν_{. Furthermore, any presence of a global U (1) symmetry, allows}

a conserved current Jµ. As usual, this is best seen from the action point of view, where Seff[g, A] =

Z

d4x√−gF , (1.6)

and F [g, A] is known as the free energy density. It is this quantity that can be expanded in derivatives of the hydrodynamical variables [25,26], which is the pressure to leading order, and defines the conservation laws. Varying the action with respect to the sources gives the hydrodynamical conservation laws

∇νTµν = 0,

∇µJµ= 0.

Note that this is a set of 5 equations, with 10 degrees of freedom for Tµν _{and J}µ_{, making it an open,}

insolvable system a priori. Hence, further assumptions are needed. The hydrodynamical assumption is that both Tµν _{and J}µ _{can be expressed as functions of the temperature T (x}µ_{), the chemical potential}

µ(xµ_{) and the four-velocity u}µ_(xµ_{). Where the velocity is timelike normalised u}µ_u

µ = −1, leaving 5

degrees of freedom, making it a closed solvable system.

Subsequently, adding the electromagnetic potential Aµ, with its associated field strength Fµν =

2∂[µAν], to these hydrodynamic equations, makes it a “hot electromagnetic” system. To make it a

magnetohydrodynamical system, the gauge field is treated as a dynamical rather than external field (this would change a sign in eq. (1.9) [2]). That is, in the correct MHD regime mentioned before. The field strength can be decomposed as

Fµν = 2u[µEν]− µνσλuσBλ, (1.7)

where Eµ = Fµνuν and Bµ = ₂1µνσλuνFσλ, both transverse to uµ. Where in the MHD regime the

electric field is screened Eµ = O(∂). The conservation laws also change when there is a dynamical U (1)

gauge field. To best see this we consider the action S[g, A], as a function of the metric gµν and gauge

field Aµ. However, now that the gauge field is dynamical, it can be coupled to an external current as

follows S[g, A] = Seff[g, A] + Z d4x√−gAµJ µ ext. (1.8)

Then diffeomorphism- and gauge-invariance gives us ∇µTµν = JµextF

µν_, _(1.9)

∇µJµ= 0, (1.10)

respectively.

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Finally, the Maxwell’s equations and Bianchi identity must be discussed. We write F = Fmat−

1 4F

µν_F

µν, (1.11)

for the free energy density, where there is a matter contribution. The equations of motion then become ∇µFµν+ Jextµ + J

µ

mat= 0, (1.12)

J_extµ + Jµ= 0, (1.13)

∇[µFνσ]= 0, (1.14)

where J_matµ is the current associated with the “matter” part of the theory. Firstly, the Bianchi identitiy in equilibrium explicitly becomes

∇µBµ= Bµaµ− EµΩµ, (1.15)

uµµνσλ∇σEλ= uµµνσλEσaλ, (1.16)

where the vorticity and acceleration are respectively defined as Ωµ = µνσλuν∇σuλ and aµ = uν∇νuµ.

These are the first “half” of the well-known Maxwell’s equations, the second “half” is given by eqs. (1.12) and (1.14).

The MHD equations are generally taken to be the combination of eqs. (1.9), (1.13) and (1.14). The addition of the electromagnetic fields clearly brings more degrees of freedom, which are constrained by the MHD equations. Thus, in MHD there are 4 additional degrees of freedom coming from Bµ_(xµ_{), to}

the 5 fluid variables.

1.1.2 Constitutive relations

The constitutive relations are the expressions of Tµν _{and J}µ _{in terms of the free variables T , µ, u}µ

and Bµ_{. In a historical textbook approach to MHD, the theory is treated as a weakly-coupled effective}

field theory, i.e. the stress-energy tensor can simply be expressed as the sum of the hydrodynamical and electromagnetic stress-energy tensor [22, 23]

T_MHDµν = T_fluidµν + T_EMµν, (1.17) T_fluidµν = (ε(T, µ) + P0(T, µ)) uµuν+ P0(T, µ)gµν, (1.18) T_EMµν = FµσFνσ− 1 4g µν_F2_, _(1.19)

where the prime in P0(T, µ) denotes the weakly-coupled assumption. For the fluid part of the constitutive relations follow the usual thermodynamic relations

ε + P0= sT + qµ, dP0= sdT + qdµ. (1.20)

This defines s = ∂TP0 and q = ∂µP0 as the entropy density and particle number density, respectively. In

the MHD regime, this becomes

T_MHDµν = (ε + P0+ B2)uµuν+ (P0+1 2B 2_)gµν_{− B}µ_Bν_{+ O(∂)} _(1.21) = (ε + P0)uµuν+ (P0−1 2B 2_)gµν_{+ B}2 Bµν+ O(∂), (1.22)

Where the Bµν_{= ∆}µν_{− ˆ}_Bµ_Bˆν _{and ∆}µν _{= g}µν_{+ u}µ_uν _{are the projectors for the unit vectors u}µ_{and ˆ}_Bµ_.

Even though this textbook approach has proven to be effective, it describes minimal MHD and is not quite adequate enough. We will now show how this minimally-coupled assumption can be relaxed, and the theory can be generalised to full MHD following AJ.

To derive the constitutive relations for the full MHD, a different method is used. This method will also be used for dual MHD, and it will be shown that the full MHD from the traditional point of view, is equivalent to the dual MHD.

Starting from the effective action in eq. (1.6), the stress-energy tensor can be found by taking the variation. The free energy density can be split in a matter contribution and the Maxwells contribution

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through eq. (1.11). This free energy density, in its most general form, is a function of all the MHD variables {T, µ, uµ_{, B}µ_{}. More specifically, of the scalar variables F = F (T, µ, B}2_{), where the unit vectors u}µ _and

ˆ

Bµ _{will arise in the stress-energy tensor.}

Finding the constitutive relations we use the definition hTµν_{i =}_√2

−g δS δgµν

. (1.23)

This in turn becomes

hTµνi = 2 ∂F ∂TδgT + ∂F ∂µδgµ + ∂F ∂B2δgB 2 + F (T, µ, B)gµν, (1.24) where δg is the functional derivative with respect to the gµν field.

Generally, the stress-energy tensor is identified as the pressure in hydrodynamical theories, such that eq. (1.11) becomes F = P (T, µ, B2) = Pm(T, µ, B2) − 1 2B 2 + O(∂), (1.25)

where Pmis the “matter” part of the pressure. At this point no (weak-coupling) assumptions have been

made, as the matter pressure is still a function of the magnetic field. Next, the partial derivatives of the pressure P (T, µ, B2) need to be defined through2

s =∂P ∂T, q = ∂P ∂µ, $ = ∂P ∂B2, (1.26)

which imply the thermodynamic expressions

ε + P = sT + qµ, dP = sdT + qdµ + $dB2. (1.27) Before explicitly writing the stress-energy tensor, the variation of the free variables with respect to the metric will be taken to be of the form in AJ eq. (7.12) [10]

δgT = T 2u µ_uν_δg µν, δgµ = µ 2u µ_uν_δg µν, δgB2= −B2Bµνδgµν. (1.28)

Finally, the stress-energy tensor becomes

hTµν_{i = (sT + qµ)u}µ_uν_{+ P g}µν_{− 2$B}2

Bµν+ O(∂) (1.29)

= (sT + qµ − 2$B2)uµuν+ (P − 2$B2)gµν+ 2$B2BˆµBˆν+ O(∂). (1.30) This equation is one of the constitutive relations for the full MHD theory from the traditional point of view, i.e. with a one-form gauge field.

At this point, the new variable $ should be reexamined which will be crucial for the mapping from full to minimal MHD. From eq. (1.25) and its definition we find

$ = ∂P ∂B2 = ∂Pm ∂B2 − 1 2 = $m− 1 2, (1.31)

where the “matter” $m has a nonzero contribution if we do not assume weak-coupling and Pm is a

function of the magnetic field. Including eq. (1.25) in the stress-energy tensor we find hTµν_{i =(sT + qµ)u}µ_uν_{+ P g}µν_{+ B}2 Bµν− 2$mB2Bµν+ O(∂) (1.32) =(sT + qµ)uµuν+ (Pm− 1 2B 2_)gµν_{+ B}2 Bµν − 2$mB2Bµν+ O(∂). (1.33)

2_{The definition of $ here is different than in [}₁₀_{], which is suggestively written in a from directly related to the}

higher-form chemical potential. In this work, this relation is made later, see eq. (1.72).

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Having set out the same stress-energy tensor explicitly in different forms eqs. (1.29) and (1.33), the link to the minimal MHD, in eq. (1.22) can be made. From eq. (1.33), it is clear that the first line in that equation is the same as eq. (1.22) upon the identification of Pm(T, µ, B2) = P0(T, µ), which implies that

$m= 0 and minimal MHD can be seen as a specific region of full MHD. Thus, the difference between

minimal and full MHD can be summarised as follows:

Full MHD : Pm= Pm(T, µ, B2) ⇐⇒ $m6= 0,

Minimal MHD : Pm= Pm(T, µ) ⇐⇒ $m= 0.

Furthermore, the energy densities and the other variables stay the same in the mapping from the minimal to the full MHD, once the thermodynamical identities are defined for the matter contribution that match eq. (1.20) for minimal MHD

ε + Pm= sT + qµ, dPm= sdT + qdµ. (1.34)

Important to remark here, the generalisation that is shown here only holds to ideal order in derivative. However, AJ has shown that this generalisation is exact to all orders in derivative.

Moving on to the current: for both the minimal and full MHD, a similar approach leads to

Jµ= q(T, µ, B2)uµ+ O(∂). (1.35) As this thesis only works with ideal MHD (i.e. to leading order) these expressions will be sufficient, further details of subleading terms are discussed in [2,10], to which this is in agreement to leading order.

With these constitutive relations at hand, eqs. (1.13) and (1.35) can, at ideal order, be written as uµJ_µext = q(T, µ, B2). (1.36) This is one of the 8 dynamical MHD equations, and through the definition of q this can be written as

∂P (µ, T, B2₎

∂µ = u

µ_Jext

µ + O(∂). (1.37)

It was shown by AJ that, when assuming uµJ_µext= O(∂), this can be solved for µ

µ = µ0(T, B2) + O(∂). (1.38)

Implying that the chemical potential is not a true free variable, but instead a function of the other variables: the magnetic field and temperature. The fact that this is not a true degree of freedom will be implicitly exploited in section1.3.

In conclusion, we have the MHD equations (eqs. (1.9), (1.13) and (1.14)), their conservation relations (eqs. (1.22) and (1.35)) and the explicit MHD constitutive relations (eqs. (1.29) and (1.35)) that form traditional MHD. In doing so, Aµwas treated as a dynamical field with source Jextµ , rather than vice versa,

which is crucial and related to the alternative boundary conditions that will be discussed in chapter3

[20,21,27].

Moreover, as will be discussed in more detail in the next section, the traditional MHD description is not based on any fundamental principle of physics, there is no symmetry underlying the above description. The gauge symmetry that leads to electromagnetism, is merely a redundancy of the theory and not a real symmetry. In section1.3, an overview of this, and the Lorentz symmetry of this theory, will be given.

Lastly, the constitutive relation for the stress-energy tensor was derived formally and without any assumptions. For the dual theory in the next section, this approach is used as well. However, as that has a one-form global symmetry, a brief introduction to higher-form global symmetries needs to be discussed first.

1.1.3 Zero-form U (1) global symmetry

Electromagnetism is a well-known example of a U (1) global (gauge) symmetry. To be more precise, this is a U (1) zero-form global symmetry. It was shown that there are also higher-form global symmetries [7]. A

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1.2. DUAL MAGNETOHYDRODYNAMICS

brief discussion of the zero-form U (1) symmetry is presented in form-notation, which will be generalised to the one-form global symmetry case in the next section.

As usual, electromagnetism has a conserved current related to the zero-form U (1) gauge symmetry. This conservation equation is given by eq. (1.10), and its form language equivalent is

d ?4J(1)= 0. (1.39)

Then, when integrated over the full 4-dimensional spacetime M4 (the left hand side ofeq. (1.39) is a

four-form object), Stoke’s theorem implies 0 = Z M4 d ?4J(1)= Z ∂M4 ?4J(1). (1.40)

In getting to the conserved current we integrate over the “boundary” of our open spacetime. As we have the freedom to choose our space, the boundary can be an arbitrary Cauchy slice and could be two spatial codimension-3 volumes V3. This gives us the conserved charge through3

0 = Z V₃(1) ?4J(1)− Z V₃(2) ?4J(1) −→ Q(0)= Z V3 ?4J(1), (1.41)

where Q(0) is the conserved quantity under the zero-form U (1) symmetry. This is widely interpreted

as the conserved electric charge in a volume. Lastly, the transformation to which the zero-form global symmetry owes its name to is the transformation of the one-form gauge field under which the theory remains invariant

Aµ→ Aµ+ ∂µΛ. (1.42)

The zero-form parameter Λ is identical to the constant phase in the transformation of any operator O(x) that is charged under the symmetry

O(x) → eiqΛ_O(x), _(1.43)

from which it is clear that this indeed concerns a U (1) symmetry. AJ has explicitly related this parameter to the chemical potential. Assuming a background manifold with a timelike isometry K = (kµ, Λk_{), the}

dynamical variables {uµ_{, T, µ} are expressed in the dynamical fields B = (β}µ_{, Λ}β_{) through}

β = u µ T , Λ β_{+ β}µ_A µ= µ T. (1.44)

Through this identification, it is clear how the chemical potential is associated with the gauge field. Formally, it is this set B that parametrises the infinitesimal symmetry transformation used in the formal derivation of the constitutive relations, where βµ and Λβare associated with diffeomorphisms and gauge transformations respectively. B has 5 degrees of freedom, as any zero-form hydrodynamical theory. Note that the magnetic field Bµ still needs to be added to obtain MHD, this will not change eq. (1.44) but will enter in the free density F (T, µ, B2_).

At this point, the zero-form U (1) global symmetry has been discussed, associated with a conserved one-form current, a zero-form Λ parameter and a zero-form chemical potential µ. In the next section, this is generalised to the one-form U (1) global symmetry that belongs to the dual theory.

1.2 Dual magnetohydrodynamics

In this section we will discuss an alternative approach to MHD, we call “dual” MHD, using generalised global symmetries [7]. This was extensively described in [8], which exploited the fact that the dual of the two-form field strength is conserved by construction and the Bianchi identity. Using this symmetric principle leading to MHD makes this a more fundamental approach. We also show that this is based on a one-form U (1) global symmetry, which will be discussed in analogy with the zero-form U (1) global

3_{Notice that this is identical to the usual Q}

(0)=R d3xJt=R SµJµ, where the spacial vector Sµis pointing in the time

direction. This can help us to intuitively generalise the conservation charges to higher-form symmetries.

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symmetry in the previous section. Lastly, the constitutive relations are derived from this higher-form point of view.

The duality at the base of this formulation is

J(2)= ?4F(2) ⇔ Jµν=

1 2

µνσλ_F

σλ. (1.45)

From this formulation, and the Biachi identity eq. (1.14), it is immediately clear that this two-form current is trivially conserved ∇µJµν = 0.

It is this conserved two-form current that induces the global symmetry on which this dual approach is based; the conserved quantity will be given in section1.2.1.

In contrast to the traditional theory, where the one-form gauge field sources a one-form conserved current, here we have a conserved two-form current. To find the dynamics of this theory, we will couple this two-form current to an external two-form gauge field source Bµν. This can be done through the

action by adding a term to the effective action4 S∗[g, B] = S_eff∗ [g, B] +

Z

d4x√−gBext µνJ

µν_. _(1.46)

Then, in analogy with eqs. (1.9) and (1.10) diffeomorphism- and gauge-invariance dictate that

∇µT∗µν= HνσλJσλ, (1.47)

∇µJµν= 0, (1.48)

where H(3) _{= dB}(2)_{. The duality J}(2) _{= ?}

4dA(1), after integrating by parts, allows us to rewrite the

action in terms of the one-form gauge field and its associated current S∗[g, B] = S_eff∗ [g, B] −

Z

d4x√−g∇σ(µνσλBµνext)Aλ, (1.49)

such that we can identify the external current J_extµ = ∇ν(Bσλext

µνσλ_{) =} 1

3

µνσλ_H νσλ.

At this point, it is clear that the one-form gauge field is dynamical and its current external. That this higher-form theory has different boundary conditions will play an important role in chapter 3 and is explicitly discussed in appendixA. With this duality, we can now see that eqs. (1.9) and (1.47) are both the same manifestation of the Lorentz force5

H_extνσλJσλ = JµextF

µν_. _(1.50)

Depending on the point of view the Lorentz force is generated by Jext or Bext.

The origins of free variables will be discussed later, and there will also be 7 free variables as in the traditional theory: {˜µ, T, uµ_{, h}µ_{}, where ˜}_{µ is the chemical potential associated with the two-form gauge}

field and hµ _{will be defined momintarily. For now it is sufficient to know that it spans the plane of}

the gauge field together with uµ_{. These hydrodynamical variables are subject to the eight equations}

in eqs. (1.47) and (1.48), which makes this a seemingly over-constrained system. However, due to the antisymmetric property of the current, the time component of eq. (1.48) does not have a time derivative, classifying it as a separate constraint, just like eq. (1.3). More specifically, eq. (1.47) governs the dynamics for {T, uµ}, where eq. (1.47) those of {˜µ, hµ} and uµ∇µJµν = 06 is the constraint [28]. Hence, this is a

closed system as well.

To sum up, all the dynamics of the system is captured in eqs. (1.47) and (1.48) which is a consequence of the one-form U (1) global symmetry. Next, a more detailed description of this higher-form global symmetries is given, but it is clear that this is a more fundamental approach to MHD than traditionally.

4_{Throughout this thesis, the}∗_{label denotes that this is a dual quantity rather than a traditional quantity: e.g. T}∗ µν vs

Tµν.

5_{Note that this holds for all orders in derivative.}

6_{This is a more general statement of the constraint, where the absence of a time derivative is more conceptual. The}

usual choice uµ_{∼ δ}µ

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1.2.1 One-form U (1) global symmetry

For the one-form U (1) symmetry, a similar approach as for the zero-form U (1) symmetry (see section1.1.3) leads to a similar result but with a different interpretation. As mentioned, the dual approach starts with the Bianchi identity that leads directly to the conservation equation for the two-form current, which will give the one-form U (1) global symmetry

d ?4J(2)= 0, (1.51)

which is the form-notation equivalent of eq. (1.48). In contrast to the zero-form symmetry, this now is a three-form object that needs to be integrated over a 3-dimensional spacetime V3

0 = Z V3 d ?4J(2)= Z ∂V3 ?4J(2). (1.52)

Now again, we are free to choose this boundary to be in the spatial dimensions, but this time this is a codimension-2 surface. This gives us the conserved quantity

Q(1) = Z Σ2 ?4J(2)= Z Σ2 F(2). (1.53)

The interpretation of this conserved quantity is different. It can be seen as the conservation of the magnetic field lines crossing a codimension-2 surface Σ, which agrees with the well-known Gauss’ law for magnetism that has been known for centuries. For this reason, Bµν is sometimes called the magnetic

gauge field, and Aµthe electric gauge field [21]. Lastly, the 1-dimensional objects that are charged under

this one-form symmetry transform as

W (C) → eiqRCdxµΛ µ

W (C), (1.54)

where Λµ is a one-form constant phase, and C is a contour. In analogy with the zero-form symmetry,

this is the same Λµused in the transformation Bµν → Bµν+ ∂[µΛν], under which the field strength Hµνσ

remains invariant.

For this unbroken symmetry, it can be related to the work by AJ, the dynamical fields now are B = (βµ_{, Λ}β µ), where β = u µ T , Λ β µ+ β ν_B νµ= µµ T . (1.55)

It is this one-form chemical potential µµ, associated with the two-form gauge field. However, the main

difference with the zero-form is that µµis not gauge-invariant, which is exploited to find the constitutive

relations and takes away one degree of freedom. From B, we would naively count eight degrees of freedom otherwise. This can be seen as the analogy of the fact that ∇µJµt = 0 is a constraint rather

than a separate equation of motion.

1.2.2 Spontaneously partially broken one-form symmetry

Before the constitutive relations are explicitly derived, some of the frameworks need to be discussed based on AJ. The global one-form symmetry, discussed before, can be spontaneously broken, leading to a vector Goldstone mode ϕµ resulting in the theory of one-form superfluids. Furthermore, for a dual

theory of MHD, a specific limit of the one-form superfluids is needed: string fluids. In this regime, the one-form symmetry is partially broken along the velocity uµ, which would lead to a scalar Goldstone mode ϕ = βµϕµ. The breaking of the symmetry allows us to write a gauge-invariant term of the chemical

potential in eq. (1.55)

ζµ= T ∂µ(ϕ) − µµ, (1.56)

in this string fluid limit of the one-form superfluids. It is also this one-form that we can relate to the hydrodynamical variables of the dual theory by setting: ζµ = −˜µhµ, where ˜µ is the (positive) norm

and hµhµ = 1 the unit vector. Moreover, it is necessary to introduce the one-form equivalent of the

“superfluid velocity”

ξµν = 2∂[µϕν]+ Bµν, (1.57)

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which is gauge-invariant. By using the one-form equivalent of the Josephson equation AJ showed that these two quantities are related through

ζµ= ξµνuν, (1.58)

such that ζµ, and thereby hµ, are transverse to uµ. In proving this, AJ has showed that δBϕµ = 0

is an exact all-order on-shell statement, and, as a consequence, the superfluid velocity transforms as δBξµν= δBµν.

1.2.3 Constitutive relations

The constitutive relations can be derived similarly as in section 1.1.2, following [2, 25, 26], or, more rigorously, as is done in [10, 29] based on the spontaneous partial breaking of the symmetry discussed before. In this subsection the latter method is used, in parallel to the derivation of the full MHD for the zero-form U (1) symmetry leading to eq. (1.29).

With the current framework, the variation on of the dynamical variables {˜µ, T, uµ, hµ} with respect to the background fields gµν and Bµν can be calculated. δB(uµ) and δB(hµ) are found using the vanishing

variation of their norm δB(u2) = δB(h2) = 0, δB(T ) is found using the norm of βµ: δB(β2) = δB(−T−2),

while for δB(˜µ) the identity δB(ζ2) = 2˜µδB(˜µ) is used. The results are in agreement with AJ:

δB(uµ) = − 1 2u ν_δg µν, (1.59) δB(hµ) = − 1 2h ν_δg µν, (1.60) δBT = T 2u µ_uν_δg µν, (1.61) δBµ =˜ ˜ µ 2(u µ_uν_{− h}µ_hν_)δg µν+ u[µhν]δBµν. (1.62)

Finally, using these variations, the constitutive relations can be derived explicitly. In MHD the most generic free density is a function of the scalar variables

S∗= Z

d4x√−gF (T, ˜µ). (1.63)

Finding the constitutive relations we use the definition hTµν ∗ i = 2 √ −g δS∗ δgµν , hJµν_{i =} _√2 −g δS∗ δBµν . (1.64)

This in turn becomes

hTµν ∗ i = 2 ∂F ∂TδgT + 2 ∂F ∂ ˜µδgµ + F (T, µ)g˜ µν_, _(1.65) hJµν_{i = 2}∂F ∂TδBT + 2 ∂F ∂ ˜µδBµ,˜ (1.66)

where δg/B is the functional derivative with respect to the gµν/Bµν field. Now, the variation of the

individual fields T and ˜µ eqs. (1.61) and (1.62) can be used to find the final constitutive relations7 T_∗µν =( + p)uµuν+ pgµν− ˜q ˜µhµhν+ O(∂), (1.67)

Jµν =2˜qu[µhν]+ O(∂), (1.68)

that are in agreement with [9,10,29]. The thermodynamic quantities are related to F (T, ˜µ) through

F (T, ˜µ) = p(T, ˜µ), s = ∂F ∂T, q =˜ ∂F ∂ ˜µ, = ∂F ∂TT + ∂F ∂ ˜µµ − F ,˜ (1.69) 7_{Notice that, a priori, these thermodynamic variables are not the same as the for the traditional theory and will be}

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1.3. TRADITIONAL-DUAL THEORY

leading to the usual thermodynamic relation,

+ p = sT + ˜q ˜µ, dp = sdT + ˜qd˜µ. (1.70) Here the extensivity assumption was used, in which the Landau potential scales linearly with the volume. Generally, ˜q can be interpreted as the magnetic flux density and s is the entropy current. Intuitively, eqs. (1.67) and (1.68) are the most general (anti)symmetric 2-rank tensors at leading order, where the u(µhν) terms can be removed by a boost in the (u, h) plane without the loss of generality. Interestingly, the coefficient for hµ_hν _{is not independent, which can be related to the tension in the field lines through}

thermodynamics [9].

In conclusion, we have the dual MHD equations (eqs. (1.47) and (1.48)) and the explicit constitutive relations (eqs. (1.67) and (1.68)) that form the dynamics of dual MHD. It is this dual theory that forms the basis of the next chapter: holographically renormalising this theory. In contrast to section 1.1, we now have a two-form gauge field that is external; this difference in boundary conditions is essential to obtaining this duality and will be further discussed in section 1.3, based on [21], and chapter 3, where this duality will be further developed in a holographic context. A solution to this theory will be discussed in the magnetic black brane discussed in section4.1. The biggest advantage of this dual approach is that it is a direct consequence of the (spontaneously breaking of) one-form U (1) symmetry.

1.3 Traditional-Dual theory

Having discussed the traditional and dual theory separately, the connection between them can be discussed further. In chapter3, this duality is explored in great detail in the holographic context. In this section, on the other hand, the duality between the traditional and dual theory is examined at MHD-level, and the similarities and differences are discussed.

1.3.1 Identification map

To further examine the connection between the two dual theories, an identification map is presented such that the conserved quantities are identical. The stress-energy tensor in eqs. (1.29) and (1.67) must be identical. Also, the two-from currents must be identical; the “two-form current” in the traditional convention is given by the dual of Fµν linked through eq. (1.45)

Jµν = 2u[µBν]+ O(∂), (1.71)

which must be compared to eq. (1.68). As mentioned in section1.1.2the zero-form chemical potential is not a true degree of freedom, allowing us to set µ = 0 such that we can make the following identification:

hµ= B

µ

|B|, q = |B|,˜ µ = −2$|B|,˜ p = P − 2$B

2_, _{= ε,} _(1.72)

all up to leading order in O(∂), and in agreement with [2]. Hence, it is clear that the thermodynamical parameters used in the two dual theories are not the same, but related through this mapping. Especially, the pressure p seems to be shifted by 2$B2, and $ is related to the chemical potential through ˜µ.

1.3.2 Symmetry groups

As is clear from sections1.1and1.2, the traditional formulation of MHD is a gauge theory, whereas the dual formulation is based on a one-form global symmetry. Here, we will briefly discuss a separate aspect of the symmetry within MHD: Lorentz symmetry, and present some key differences between the two dual theories.

Regarding the Lorentz symmetry of the theories, we will discuss the finite temperature case, which will prove to be relatively elementary. Although it is still uncertain of what will happen exactly in the zero-temperature limit, we will present a brief summary of what GHI discusses in that limit.

From the traditional point of view, the Lorentz symmetry SO(3, 1) breaks into SO(3). Intuitively, this can be explained due to the presence of the velocity uµ_{: having a preferred direction, only the rotating}

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1.3. TRADITIONAL-DUAL THEORY

symmetry is left. Although, in section VI of GHI, an alternative interpretation is presented. Here, the authors make the connection to the conserved Q(0), which are point-like. Due to the presence of the

zero-form chemical potential µ, associated with Aµ through eq. (1.44), only SO(3) is left. Intuitively, this

can now be seen as a conserved-in-time point-like Q(0), which still has rotational symmetry. Interestingly,

the temperature does not affect this symmetry further, as the boost symmetry is already broken. This can schematically be seen the following table1.1.

From the dual point of view, the breaking down of the Lorentz symmetry must be interpreted similarly. However, this is slightly more complex as Q(1)conserves the flux of (magnetic) field lines, which are

one-dimensional objects, and is schematically presented in table 1.2. For finite temperature, the systems boost symmetry breaks into SO(3). In the presence of the one-form chemical potential, associated with Bµν through eq. (1.55), there is still a rotating symmetry around the magnetic fields (SO(2)). In the

zero-temperature limit, the symmetry might actually be enhanced, as the boosting-symmetry would not be broken. Hence, boosting along the magnetic fields enhances to SO(3, 1) and SO(2) × SO(1, 1) for ˜

µ = 0 and ˜µ 6= 0, respectively.

Symmetry group T 6= 0, T = 0 µ = 0 SO(3, 1)

µ 6= 0 SO(3)

Table 1.1: Traditional theory

Symmetry group T 6= 0 T = 0 ˜

µ = 0 SO(3) SO(3, 1) ˜

µ 6= 0 SO(2) SO(2) × SO(1, 1) Table 1.2: Dual theory

To conclude, illustrating further what has been discussed in this chapter, the following table is pre-sented summarising some of the (subtle) differences between the two theories.

Essential to the duality is that the one-form gauge field Aµ is dynamical that couples to an external

current Jext

µ , in contrast to traditional electromagnetism. On the other hand, the one-form global

sym-metry approach has an external two-form gauge field Bext

µν, and through the duality in eq. (1.45) this is

related to the current through J(1) _{= ?}

4dB(2), which arise in the presence of double-trace deformations

[18], see appendix A. An external gauge field is the result of regular boundary conditions, whereas, a dynamical rather gauge field is resulting from alternative boundary conditions. In chapter2 the holo-graphic dual theory with external gauge field will be discussed with regular boundary conditions and in chapter3the traditional MHD with dynamical gauge field constraint to alternative boundary conditions. As this thesis was being completed, an interesting article [21] came out which presented these boundary conditions schematically. They also explored the opposite with regular boundary conditions for Aµ and

alternative boundary conditions for Bµν and showed that these are also identical, see appendix A for

further details.

Dual theory Global symmetry

Traditional theory Gauge symmetry one-form U (1) global symmetry

intrinsic property of the theory

zero-form U (1) gauge symmetry redundancy of the description couples to external gauge field gauge field is dynamical

magnetic gauge field Bext

µν electric gauge field Aµ

is spontaneously broken can not be spontaneously broken regular boundary conditions alternative boundary conditions

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Chapter 2

Higher-form Magnetohydrodynamics

The dual MHD approach can be derived, using the holographic principle, from an effective bulk action. As we are interested in a four-dimensional field theory describing dual MHD, a five-dimensional bulk action in Anti-de Sitter space is considered (essentially a form of AdS5/CF T4). At the basis, there is a two-form

gauge field Bext ab

1 _{associated with the one-form global symmetry which is external at the boundary and}

therefore will lead to the dual MHD formulation we are looking for as described in chapter1.

In this chapter, a full derivation will be given of the equations of motion and the conserved quantities that are related to the action. The holographic principle will be considered with a Fefferman-Graham bulk-metric that needs to be holographically renormalised, and the boundary terms of the source fields will be used to derive the stress-energy tensor and two-form current. A simple holographic action will be considered described by the Einstein-Hilbert action with a negative cosmological constant a one-form U (1) global symmetry2_: S_bulk∗ = 1 2κ2 5 Z d5x√−G R − 2Λ − 1 3e2HabcH abc , (2.1)

where Gabis the bulk-metric and Habcis the three-form field strength associated with the dynamical gauge

field Bab. For bulk dimension D = 5, the cosmological constant simplifies to Λ =

−(D−1)(D−2) 2L2 = −6

when we set the AdS radius L = 1 for convenience, as is done throughout the whole thesis3_.

Before the equations of motion can be calculated, the correct boundary terms need to be added to ensure that the action principle is adhered, δS = 0. The manifold M has a well-defined boundary ∂M where the metric does not vanish, i.e. Gµν = γµν 6= 0, where γµν is the induced metric of the boundary

∂M. The total boundary term for the whole action is merely the usual Gibbons-Hawking-York term, as the gauge part does not give rise to a similar boundary term,

S_bndry∗ = 1 κ2 5 Z ∂M d4x√−γK. (2.2)

This is the usual Gibbons-Hawking boundary term for AdS, with K = gµν_K

µν the trace of the extrinsic

curvature of the boundary [30]. Then, the total action becomes

S∗_tot= 1 2κ2 5 Z d5x√−G R + 12 − 1 3e2HabcH abc + 2 Z ∂M d4x√−γK ! , (2.3)

which can be varied. The bulk equations of motion are considered for each dynamical field, creating a 1_{Latin and Greek letters will be used for bulk and boundary indices, respectively.}

2_{This is the simplest one-form U (1) symmetry, with its field strength minimally coupled.} 3_{The coupling constant is sometimes also set to e}2_{= 1.}

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2.1. FEFFERMAN-GRAHAM HOLOGRAPHIC RENORMALISATION

separate set of tensor equations. Varying with respect to the metric and gauge field gives us Rab− (6 + 1 2R)Gab= 1 e2HacdH cd b − 1 6e2GabHcdeH cde_, 0 = ∇aHabc. (2.4)

The equations of motion for the gauge field are thus given by eq. (2.4), and those of the metric in AdS5

are given by R = −20 − 1 9e2HabcH abc_, _(2.5) Rab+ 4Gab= 1 e2HacdH cd b − 2 9e2GabHcdeH cde_. _(2.6)

The left-hand side of eqs. (2.4) and (2.6) is due to the gravity part of the action, whereas the right-hand side is due to the gauge part of the action. To clarify the following holographic derivation these two sides will be approached separately, where the left-hand side signifies the vacuum case, after setting up the holographic renormalisation context in Fefferman-Graham coordinates.

This chapter is structured as follows. In section2.1, an overview of the existing literature on holo-graphic renormalisation is presented, following [31]. In section 2.2, this is applied to the gravity part of the action in eq. (2.3), which is effectively pure AdS5. Based on [17], this is done including the

anomaly and up to higher-order in derivatives. Lastly in section 2.3, the gauge section of the action is also renormalised. As a result of the counterterms, the correct equations for dual MHD are recovered.

2.1 Fefferman-Graham holographic renormalisation

To solve eqs. (2.4) and (2.6) given a conformal structure at the boundary4_{, the Fefferman-Graham}

co-ordinate system can be used. This is a coco-ordinate system where the radial coco-ordinate is ρ, with the boundary at ρ = 0, that allows for an expansion in ρ to solve the equations of motion perturbatively. The Fefferman-Graham expansion is defined for any boundary dimension (d) and a logarithmic term appears in even d, which results in the following5

ds2= Gabdxadxb = dρ2 4ρ2+ γµν(x, ρ)dx µ_dxν ₌dρ2 4ρ2 + 1 ρgµν(x, ρ)dx µ_dxν_, gµν(x, ρ) = gµν(0)(x) + ρg (1) µν(x) + ρ 2_g(2) µν(x) + ρ 2_{log ρ˜}_h µν(x) + O(ρ3). (2.7)

In these coordinates, the background metric g(0)µν at the boundary is used to raise and lower the

4-dimensional boundary indices.

Following Skenderis [31] renormalising this holographic theory in this section, we start from the Fefferman-Graham metric that is the asymptotic solution to the bulk equations. As one can check that eq. (2.7) is asymptotically AdS (AAdS) as it satisfies:

R(G)_µνσλ= (GµσGνλ− GµλGνσ) + O(ρ).

This is one of the requirements to the metric for renormalising the theory.

The bulk equations are second-order differential equations that are constraint to Dirichlet boundary conditions and determine the subleading terms of the dynamical metric. This means that in a near-boundary analysis, ρ = → 0, the bulk field equations uniquely dictate all the subleading terms of the expansion in terms of the source metric and its derivatives. Thus, there are algebraic relations that write g(1)µν, g(2)µν and ˜hµν in terms of g

(0)

µν, which is the reason that the leading term is generally interpreted as the

source of the metric. Furthermore, the logarithmic term of the field ˜hµν is necessary to obtain a solution

4_{Strictly speaking the bulk metric cannot induce a metric at the boundary due to its second-order poles [}₁₇_].

5_{These particular coordinates are chosen to simplify the calculation for even dimensions [}₁₇_,₃₂_{]. In the literature, one}

can find related FG-coordinates related by ρ = r2_{, see e.g.[}₃₃_,₃₄_{] which also include further steps of this calculation, but}

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2.1. FEFFERMAN-GRAHAM HOLOGRAPHIC RENORMALISATION

in even dimensions and is related to conformal anomalies, which was first noted by [17]. The previous analysis of the metric holds for any dynamical field, generally

F (x, ρ) = ρm_f

(0)(x) + ρf(2)(x) + · · · + ρn(f(2n)(x) + log(ρ) ˜f(2n)(x)

, (2.8)

where 2m and n are non-negative integers. In particular, the two-form gauge field will be discussed in section2.3, and the expansion of Aµ is discussed in chapter3. Moreover, f(0)(x) is the source field of the

generic bulk field F (x, ρ), which is important for defining the counterterms.

2.1.1 The renormalisation scheme

Once we have the generic asymptotic solutions to the field equations, the on-shell bulk action can be expressed in terms of the perturbation terms. To regularise the divergent terms, we first have to introduce a cut-off hypersurface at ρ = (see fig. 2.1) where the boundary terms can be evaluated, such that the divergent terms of the action can be identified before introducing counterterms. Performing the ρ-integral the regulated on-shell boundary action can always be written as

S_regon−shell(f(0), ) =

Z

ρ=

d4xp−g(0) −ν_a

(0)+ −ν−1a(2)+ . . . − log a(2ν) , (2.9)

Figure 2.1: A visualisation of the bound-ary M, and the dotted hypersurface at ρ = within the bulk M.

where ν is a positive constant exclusively depending on the scale dimensions of the dual operator, and the coefficients a(i)are direct functions of the source fields. All the divergent

coefficients in the regularised action determine the countert-erms such that the subtracted action does not diverge, as becomes imminent from eq. (2.11). However, the divergent terms in the regulated on-shell action Sregare found in terms

of f(0)(x) and , whereas the general counterterms SCT are

expressed as “living” on the hypersurface ρ = , and are therefore in terms of the bulk field F (x, ). This means that the asymptotic Fefferman-Graham expansion for the metric in eq. (2.7) needs to be inverted up to the right order to find f(0) = f(0)(F (x, ), ). Specifically, in order to do this the

in-duced metric, γµν = 1ρgµν(x, ρ), needs to be inverted. Only

then we can properly define the counterterms:

SCT(F (x, ), ) = − divergent terms of Sregon−shell(f(0), ),

(2.10) such that we can define the non-divergent subtracted action6

Ssub(F (x, ), ) = Son−shellreg (f(0), ) + SCT(F (x, ), ). (2.11)

This subtracted action is defined on the hypersurface, and the → 0 limit defines the renormalised action on the boundary that we are interested in (see fig.2.1)

Srenon−shell(f(0)) = lim

→0Ssub(F (x, ), ). (2.12)

Notice that the difference between the two are seemingly trivial but subtly crucial for calculating any further quantities from the action. Specifically in calculating stress-energy tensor of the boundary theory. When calculating the variation of the renormalised on-shell action, the difference between the bulk field and its boundary source needs to be taken into account. Varying the action with respect to the dynamical boundary metric, we find expectation value of the stress-energy tensor:

hTµνi = 2 p−g(0) δSrenon−shell δgµν₍₀₎ = lim→0 2 p−g(x, ) δSsub δgµν_{(x, )} (2.13) = lim →0 1 2 √ −γ δSsub δγµν_{(x, )} . (2.14)

6_{This suggestively called term has a minus sign in [}₃₁_{] due to the different sign in eq. (}_2.10_).

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2.2. THE ADS GRAVITY SECTION

This was historically described by Brown and York in [35], for a spacetime with boundary and a positive energy density using the Hamilton Jacobi theory. This will be calculated in the next section for pure gravity. In section2.3the stress-energy tensor will include a gauge field, and there will be an additional conserved two-form current that will be derived slightly differently, but defined identically.

Before analysing the AdS gravity and gauge section of the action in eq. (2.3) separately, the inverse metric is needed. This will also be an perturbation in ρ and can be found analytically by requiring that gµν _{= g}µλ_gµσ_g

λσ at each order in ρ, and was checked numerically , resulting in

gµν= g₍₀₎µν − ρg₍₁₎µν − ρ2(g₍₂₎µν− gµλ₍₁₎gν_(1)λ) − ρ2log(ρ)˜hµν+ O(ρ3). (2.15) The indices of the subleading terms have been raised with the source term g₍₀₎µν, e.g. gµν₍₁₎= gµσ₍₀₎gνλ₍₀₎g_σλ(1).

2.2 The AdS gravity section

In this section, the renormalisation scheme that is laid out in the previous section will be performed for the gravity part of the theory. This effectively implies setting Bab= 0 and the equations of motion

simplify to a pure gravity theory in AdS5 in vacuum

Rab+ 4Gab= 0. (2.16)

In Fefferman-Graham coordinates, this equation needs to be written more explicitly in terms of the radial coordinate ρ and the derivatives of the field to it. As the bulk metric is in block diagonal form, this can be done for three separate “channels”: scalar, vector and tensor, referring to the number of free (Greek) indices of the equation. Computing the Christoffel symbols explicitly to find the Ricci tensor in the FG-coordinates, the channels result in

−1 2tr[g −1_g00_{] +}1 4tr[g −1_g0_g−1_g0_{] = 0,} _(2.17) −1 2(∇ (g) µ tr[g −1_g0_{] − ∇}ν (g)g 0 µν) = 0, (2.18) R(g)_µν − ρ 2g00 µν− 2(g0g−1g0)µν+ tr[g−1g0]g0µν + 2gµν0 + tr[g−1g0]gµν = 0, (2.19)

where the prime denotes differentiating with respect to ρ. The Ricci tensor and covariant derivatives are labelled by the metric and are constructed from g and g−1_{, denoting its inverse (in components g}µν₎7_.

Each of the channel equations of motions can now be expanded around small ρ, using the perturbed metrics in eqs. (2.7) and (2.15). The following derivation has been performed numerically using the Mathematica package xAct [36], and uses the fact that eqs. (2.17) to (2.19) need to hold for every order of ρ. To this end, the vanishing leading order of eq. (2.19) leads to the following

R(0)_µν + 2g_µν(1)+ g_µν(0)g_(1)λλ = 0. (2.20) This, in turn, gives the statement that the first order of the metric expansion can be expressed in terms of the source metric and its derivatives

g_µν(1)= −1 2(R (0) µν − 1 6g (0) µνR (0)_), _gλ (1)λ= − 1 6R (0)_. _(2.21)

Interestingly, gµν(1)is proportional to the second derivative of the boundary metric, g (1)

µν = O(∂2), through

the Ricci tensor. For many applications, this term is subleading [15], as it will only give rise to a higher-order expansion for MHD. We will include it here for now, and disregard it later in section2.3.

7_{Notice that eq. (}_2.19_{) has an opposite sign for the Ricci tensor compared to [}₁₇_{] which is due to a difference in}

convention of the Ricci tensor. This leads to different intermediate results, but the same physical conclusion compared to [15]. Similarly, this difference in convention gives an overall sign difference in eq. (2.18) that will give different intermediate results in the next section compared to the gauge part, see eq. (2.49). Here the following definition used: Rµν = ∂σΓσµν−

∂µΓσσν+ ΓσσλΓ λ

µν− ΓσµλΓ λ σν.

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The leading order of eq. (2.17) gives the identities for gµν(2) and ˜hµν, which are separated by the

logarithm. Starting with the logarithmic term we directly find ˜

hλ_λ= 0, (2.22)

hence will not contribute to the action or have a physical contribution and can be neglected later on [33]. The other identity is also given in trace-form, and in the derivation the tracelessness of ˜h is used, leading to gλ_(2)λ=1 4g µν (1)g (1) µν. (2.23)

Lastly, the vector channel eq. (2.18) does not contribute to any interesting expressions.

2.2.1 The holographic renormalisation

Now that we have generic asymptotic solutions to the bulk field equations on the boundary, we can explicitly calculate the regularised on-shell action on the boundary in terms of these perturbation terms, according to eq. (2.9), which includes performing the radial integral. As the on-shell action is considered, eq. (2.6) allows us to fill in R(G) _{= −20. Introducing the cut-off hyperplane at ρ = without any}

divergences, according to the renormalisation scheme in the previous section. Still focusing on the pure gravity part (Bab= 0) eq. (2.3) becomes8

S_regon-shell= 1 2κ2 5 Z ρ= d4x Z ∞ dρ√−G(−8) + 2√−γK _ρ= . (2.24)

There are several aspects that needs to be calculated here: the volume elements of the manifold and its boundary, and the extrinsic curvature. The volume element of the boundary changes according to γµν =1_ρgµν, and det(cA) = cndet(A) for all n × n matrix A, resulting in |γ| = _ρ14|g|. The bulk metric is

block diagonal, hence |G| = 1 4ρ2

1

ρ4|g|. Imposing K = ∇ana with a normalised normal vector in the radial

direction na _{= −2ρδ}a

ρ, the trace of the extrinsic curvature becomes

K = ∇ana= −2∇ρρ = − 2 √ −G∂ρ( √ −Gρ). (2.25)

Using the chain rule this implies S_regon-shell(g, ) = 1 2κ2 5 Z ρ= d4x Z ρΛ dρ−4 ρ3 √ −g −4 ρ∂ρ √ −g _ρ=+ 8 ρ2 √ −g _ρ= . (2.26) The boundaries of the ρ-integral has two cut-off values and ρΛ, where → 0 corresponds to the

hypersurface that is renormalised below. However, ρΛ→ ∞ can be seen as the horizon after which there

might be divergences, such that the integral is defined smoothly in the whole bulk. In the magnetised black brane described in chapter4, this is the black hole horizon9. Moreover, the metric expansion used in deriving the counterterms only holds for small ρ, and this bulk boundary does not contribute to any divergences. Therefore, the radial integral only has divergent terms on the lower boundary ρ = , that will be renormalised below.

Before we can identify the divergent terms in the form of the a(i)coefficients in eq. (2.9), the

determi-nant√−g needs to be expanded and written in terms of the source field g(0)µν. This is done perturbatively,

as the FG expansion could, for example, be written as a small perturbation hµν that is identified as

follows gµν = gµν(0)+ hµν, hµν = ρgµν(1)+ ρ 2_g(2) µν + ρ 2_{log ρ˜}_h µν+ O(ρ3), hλλ= ρg_(1)λλ + ρ2g_(2)λλ + ρ2log ρ˜hλλ+ O(ρ3), hλ_λhν_ν = ρ2gλ (1)λg ν (1)ν+ O(ρ 3_), hµνhµν = ρ2gµν₍₁₎gµν(1)+ O(ρ 3_).

8_{Even though this is the dual action S}* on-shell

reg , as this only concerns the gravity part of the action, the star∗is omitted.

9_{This is not explicitly discussed in this thesis, see e.g. [}₃₇_{] for such an analysis.}

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Subsequently, using log det g = tr[log g]10and the Taylor series, these expressions lead to √ −g =p−g(0)_exp1 2tr(1 − 1 2g −1 (0)h + O(h 2_)), =p−g(0) r 1 + hλ_λ+1 2(h λ λhνν− hµνhµν) + O(ρ3), =p−g(0)(1 + ρ 2g (1)λ λ+ ρ2 8(g λ (1)λg ν (1)ν− g µν (1)g (1) µν) + O(ρ 3 )), (2.27) ∂ρ √ −g =p−g(0)( 1 2g (1)λ λ+ ρ 4(g λ (1)λg ν (1)ν− g µν (1)g (1) µν) + O(ρ 2 )). (2.28)

Where the series expansion for both the logarithm, exponent and square root is used for 1 + x, x 1, and in eq. (2.27) the relations in eqs. (2.22) and (2.23) are used. Now the radial integration in eq. (2.26) can be performed and, together with the relations in eqs. (2.27) and (2.28), the divergent terms are collected as in eq. (2.9), which gives the following

Son-shell_reg (g(0), ) = 1 2κ2 5 Z ρ= d4xp−g(0) 6 2 + 0 + 1 2log (g λ (1)λg ν (1)ν− g µν (1)g (1) µν) + O( 0₎ . (2.29)

Hence the coefficients for the divergent terms, a(i), are

a(0)= 6, a(1)= 0, a(2)= − 1 2(g λ (1)λg ν (1)ν− g µν (1)g (1) µν), (2.30)

which are in agreement with [17], up to an overall sign. This sign can be accounted for by a sign difference in at three parts. Firstly, the orientation of the normal vector can be inward or outward pointing. Secondly, a related quantity is the sign in front of the extrinsic curvature K in the boundary action and the orientation of the radial integral boundaries - which are different in [17]. Lastly, the convention of including the sign eq. (2.10) and then adding the counterterms, or alternatively not including the minus sign, but subtracting. Ultimately, the sign difference at this point, will be overcome in the next section.

2.2.2 The counterterms

The next step in renormalising this theory is determining the counterterms. In order to do so, the regularised action in eq. (2.29) needs to be rewritten in terms of the bulk field γµν = gµν/ rather than

the boundary field on the hypersurface ρ = . To this end, eq. (2.27) needs to be inverted and written as p−g(0)= √ −γ2_{(1 −} 2g λ (1)λ+ 2 8(g λ (1)λg ν (1)ν+ g µν (1)g (1) µν) + O(3)). (2.31)

Moreover, as the subleading terms of the metric are still given in terms of the source term on the boundary and its curvature R(0) as in eq. (2.21), these quantities need to be rewritten as living on the hypersurface R(γ). This is done by writing R(γ) as a series in ρ [30]:

R(γ)µν = Rµν(0)+ ρ

∂R(0)µν

∂ρ |ρ=+ O(ρ

2_), _(2.32)

= R_µν(0)+ ρ(traceless terms) + O(ρ2), (2.33) such that its inverse results in

Rµν_(γ)= γµλγσνR_λσ(γ)= ρ2Rµν₍₀₎+ O(ρ3). (2.34) With these expansions, the Ricci scalar on the hypersurface can be found when contracted with γµν ₌

ρgµν _{given by eq. (}_2.15₎ R(γ)= γµνR(γ)_µν = ρR(0)− ρ2_gµν (1)R (0) µν + O(ρ 3_), _(2.35) = ρR(0)− ρ2₍1 12R 2 (0)− 1 2R µν (0)R (0) µν) + O(ρ3), (2.36)

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where the first equation of eq. (2.21) has been used. Subsequently, this leads to the following identities that are used in the next derivation:

R(0) =1 R(γ)+ 2( 1 12R 2 (0)− 1 2R µν (0)R (0) µν) + O( 3₎ , R₍₀₎2 = 1 2R 2 (γ)+ O( 0_), Rµν₍₀₎R(0)_µν = (Rµν_(γ)− O())1 2(R µν (γ)+ O()) = 1 2R µν (γ)R µν (γ)+ O().

These expressions now can be used to evaluate the second-order terms in ρ of eqs. (2.29) and (2.31) at the hypersurface as g(1)µν is related to the Ricci tensor at this surface

g_(1)λλ = −1 6R (0) = 1 6 R(γ)+ ( 1 12R 2 (γ)− 1 2R µν (γ)R (γ) µν) + O(R 3 (γ)) , (2.37) g_(1)λλ g_(1)σσ = 1 362R 2 (γ)+ O(R 3 (γ)), (2.38) g₍₁₎µνg_µν(1)= −1 2(R µν (0)− 1 6g µν (0)R (0)₎ −1 2(R (0) µν − 1 6g (0) µνR (0)₎ = 1 2 1 4R µν (γ)R (γ) µν − 1 18R 2 (γ)+ O(R 3 (γ)) , (2.39)

which are in agreement with [17] at d = 4.

With eqs. (2.31) and (2.37) to (2.39) at hand, the divergent terms of the regularised action in eq. (2.29) can be isolated and identified as the counterterms, through eq. (2.10). Putting everything together, the counterterms are given by

SCT∗ = 1 2κ2 5 Z ρ= d4x√−γ −6 + 1 2R (γ) − log 1 24R 2 (γ)− 1 8R µν (γ)R (γ) µν + O(R3_(γ)) , (2.40)

which is in agreement with [17,30,38, 39] for AdS5, up to the anomaly. Aside from the anomaly, there

is no separate term of R2 _{for boundary dimensions 4. This renormalisation is done completely general}

for AdS5, and up to second order in Ricci tensor. However, in the context of (dual) MHD, the derivative

expansion and eq. (2.21) dictate Ricci tensors are in higher order in derivatives. In the MHD limit, this eq. (2.40) simply reduces to the leading term −3. Nonetheless, this section focuses on general holographic renormalisation for pure gravity, and therefore uses the full counterterms in eq. (2.40).

2.2.3 The stress-energy tensor

Once we have the counterterms, the action can be renormalised to find the stress-energy tensor. The subtracted action on the cut-off hypersurface as specified in eq. (2.11) is as follows

Ssub(g, ) = 1 2κ2 5 Z d5x√−G (R + 12) + Z ∂M d4x√−γ 2K − 6 +1 2R (γ)₋1 2log A ! , (2.41)

where the conformal anomaly A is given by A = 1 12R 2 (γ)− 1 4R µν (γ)R (γ) µν (2.42)

Holographic dualities for Magnetohydrodynamics

MSc PHYSICS AND ASTRONOMY

Theoretical Physics

Master Thesis

Holographic dualities for Magnetohydrodynamics

Basil B. van Wijk

10772314

February 10

, 2021

Abstract

Acknowledgements

Contents

Introduction

Chapter 1

Relativistic Magnetohydrodyanmics

1.1

Traditional magnetohydrodynamics

1.1.1

Relativistic magnetohydrodynamics

1.1.2

Constitutive relations

1.1.3

Zero-form U (1) global symmetry

1.2

Dual magnetohydrodynamics

1.2.1

One-form U (1) global symmetry

1.2.2

Spontaneously partially broken one-form symmetry

1.2.3

Constitutive relations

1.3

Traditional-Dual theory

1.3.1

Identification map

1.3.2

Symmetry groups

Chapter 2

Higher-form Magnetohydrodynamics

2.1

Fefferman-Graham holographic renormalisation

2.1.1

The renormalisation scheme

2.2

The AdS gravity section

2.2.1

The holographic renormalisation

2.2.2

The counterterms

2.2.3

The stress-energy tensor