# Higher-form symmetry breaking and holographic flavour

Nabil Iqbal<EMAIL_ADDRESS>Centre for Particle Theory, Department
of Mathematical Sciences, Durham University, South Road, Durham DH1 3LE, UK
Kieran Macfarlane<EMAIL_ADDRESS>Centre for Particle
Theory, Department of Mathematical Sciences, Durham University, South Road,
Durham DH1 3LE, UK

###### Abstract

We study the realisation of higher-form symmetries in the holographic dual of
gauge theories coupled to probe matter in the fundamental. We particularly
focus on the dual of $U(N)$ gauge theory coupled to fundamental matter. We
demonstrate the existence of a continuous 1-form symmetry associated with the
conservation of magnetic flux and show that this symmetry is spontaneously
broken in the IR when the flavour degrees of freedom are gapped. We
numerically compute the spectral function of the 2-form current and
demonstrate the existence of the associated Goldstone mode. We compare to
expectations at weak-coupling.

###### Contents

  1. I Introduction
    1. I.1 Higher-form symmetries in non-Abelian gauge theory
  2. II Symmetries of holographic flavour
    1. II.1 Bulk holographic action
    2. II.2 Examples in lower dimensions
  3. III $SU(N)$ gauge theory
    1. III.1 Bulk action
    2. III.2 Charged operators
      1. III.2.1 Baryon vertices in pure $SU(N)$ gauge theory
      2. III.2.2 Baryon operator in theory with dynamical flavour
  4. IV $U(N)$ gauge theory
    1. IV.1 Bulk action
    2. IV.2 Charged line operator
  5. V Fluctuation spectrum
    1. V.1 Goldstone modes and numerics
      1. V.1.1 1-form
      2. V.1.2 2-form
      3. V.1.3 Results
    2. V.2 Comparison to weak coupling
  6. A Conventions and differential form identities
  7. B Normalisations
    1. B.1 Kinetic Terms
    2. B.2 Chern-Simons Term
    3. B.3 DBI term
    4. B.4 Couplings of other branes
      1. B.4.1 Wrapped $D5$-brane: baryon vertex
      2. B.4.2  Wrapped $D5$-brane with boundary: DBI monopole
      3. B.4.3 $D1$ string
  8. C Numerical Solution
    1. C.1 Equations of motion
    2. C.2 Asymptotic analysis
    3. C.3 Source-response method
  9. D Index of symbols

##  I Introduction

In recent years, higher-form symmetries, also called generalised global
symmetries Gaiotto:2014kfa , have emerged as a powerful tool for understanding
quantum field theory. Just as conventional global symmetries are associated
with the conservation of densities of particles, higher-form symmetries are
associated with the conservation of extended objects such as strings or flux
tubes. They are on conceptually the same footing as ordinary global
symmetries; they can spontaneously break (resulting in Goldstone modes
Hofman:2018lfz ; Lake:2018dqm ), they can have anomalies (see e.g.
Gaiotto:2017yup  for an early example), and they can be used to build theories
of hydrodynamics Grozdanov:2016tdf ; Glorioso:2018kcp ; Armas:2018atq .

In general, a $U(1)$ $p$-form symmetry is associated with a conserved
$(p+1)$-form current $J$.

$d\star J=0$ (1)

The conventional case is $p=0$; in this case we have a 1-form current that
counts a density of particles. Here we will focus on the case $p=1$; in this
case we have a conserved $2$-form current satisfying
$\partial_{\mu}J^{\mu\nu}=0$.

In this work we will study some aspects of the realisation of higher-form
symmetries in quantum field theories with holographic duals. We will focus on
the study of gauge theory coupled to probe matter in the fundamental
representation. As we will review below, in the case where the gauge group is
$SU(N)$, this theory has no 1-form symmetries, though it does have a 0-form
symmetry associated with baryon number; we will clarify some aspects of how
the holographic representation of this baryon number intertwines with the
(explicitly broken) putative center symmetry of the pure gauge theory. In the
case where the gauge group is $U(N)$ however, the theory has an unbroken
1-form symmetry associated with the conservation of magnetic flux of the “U(1)
factor” in the gauge group. We will study the realisation of this symmetry,
identifying the charged line operators and studying the correlation function
of its currents. As we will elaborate on below, we note that this is perhaps
the simplest holographic model in which such a continuous 1-form symmetry can
be spontaneously broken, motivating our study.

In this leisurely introduction we begin by reviewing how higher-form
symmetries are realised in various types of non-Abelian gauge theory with and
without matter couplings.

###  I.1 Higher-form symmetries in non-Abelian gauge theory

$SU(N)$ gauge theory: Let us begin our study by reviewing the higher-form
symmetry structure of $SU(N)$ gauge theory with only adjoint matter. If we
have access to a Lagrangian description of the theory, the action is

$S=\int d^{4}x\left(-\frac{1}{g_{YM}^{2}}\mathop{\rm tr}|F|^{2}+\cdots\right)$
(2)

where $F$ is the non-Abelian field strength, and the $\cdots$ refers to
possible supersymmetrisations or other terms in the action. This action
depends only on the Lie algebra of the group $su(N)$. As it turns out, though
the Lie algebra specifies the action, it does not actually fully define the
theory itself. This is because the full theory contains line operators, and
the spectrum of line operators depends on the global form of the gauge group
Aharony:2013hda . Let us first consider the case where the global form of the
gauge group is $SU(N)$; we then have the usual Wilson lines in the fundamental
representation of the gauge group in the theory:

$W(C)\equiv\mathop{\rm Tr}\mathrm{P}\exp\left(\oint_{C}A\right)$ (3)

In a modern understanding, these Wilson lines are charged under 1-form
symmetries. To be more precise, there is a $\mathbb{Z}_{N}$ valued surface
operator $U_{q}({{\mathcal{M}}}_{2})$ that is defined on closed 2-manifolds
and is topological in that it is invariant under small modifications of the
2-manifold ${{\mathcal{M}}}_{2}$. This surface operator has a non-trivial
braiding algebra with the Wilson line, i.e. we have inside the path integral,

$U({{\mathcal{M}}}_{2})\;W(C)=\exp\left(\frac{2\pi iq}{N}\right)W(C)\qquad
q\in\\{0,\ldots,N-1\\}$ (4)

if ${{\mathcal{M}}}_{2}$ wraps the curve $C$. This 1-form symmetry is a
refinement of the usual “center” symmetry of non-Abelian gauge theory, under
which adjoint matter fields $\Phi^{a}_{b}$ are invariant:

$\Phi^{a}_{b}\to\exp\left(\frac{2\pi
iq}{N}\right)\delta^{a}_{c}\;\Phi^{c}_{d}\;\exp\left(-\frac{2\pi
iq}{N}\right)\delta^{d}_{b}=\Phi^{a}_{b}$ (5)

(More explicitly, the insertion of a surface operator $U({{\mathcal{M}}}_{2})$
induces a gauge transformation that is not single-valued as one winds around
${{\mathcal{M}}}_{2}$; instead this gauge transformation returns to itself
only up to an element of the center of the gauge group tHooft:1977nqb .
However if all fields transform in the adjoint, this operation is non-singular
from the point of view of the gauge fields).

Let us now consider what happens if we instead couple this theory to $N_{f}$
flavours of bosons and fermions charged in the fundamental under $SU(N)$, i.e.
if the action is taken to be:

$\displaystyle S^{\prime}=\int d^{4}x\;\left[-\frac{1}{g_{YM}^{2}}\mathop{\rm
tr}|F|^{2}+\sum_{i=1}^{N_{f}}\left(-|\partial\phi_{i}-iA\phi_{i}|^{2}-m_{\phi}^{2}\;|\phi_{i}|^{2}+\bar{\psi}_{i}\left(i{{\gamma}}^{\mu}\left(\partial_{\mu}-iA_{\mu}\right)-m_{\psi}\right)\psi_{i}\right)+\cdots\right]$
(6)

The 1-form symmetry above is now explicitly broken – though the line operator
$W(C)$ can still be defined, the operator $U_{q}$ is no longer topological and
thus there is no longer a 1-form symmetry.111In familiar $0$-form language, a
fundamental matter field transforms as
$\phi^{a}\to\exp\left(\frac{2\pi
iq}{N}\right)\delta^{a}_{b}\;\phi^{b}=\exp\left(\frac{2\pi
iq}{N}\right)\phi^{a}\neq\phi^{a}$.

However there is a new $0$-form symmetry: the baryon number current, which
acts as a diagonal phase rotation on both $\phi_{i}$ and $\psi_{i}$. The
associated conserved current is defined in the usual way as:

$j^{\mu}_{B}=\sum_{i=1}^{N_{f}}\left(\bar{\psi}_{i}{{\gamma}}^{\mu}\psi_{i}+2\;\textrm{Im}\,\phi_{i}^{\dagger}D^{\mu}\phi_{i}\right)$
(7)

It is worth noting that the local gauge-invariant operators that are charged
under this baryon number symmetry are fully antisymmetrized products of
$N_{c}$ fundamental fields, and so will have charge $N_{c}$ in the appropriate
units.

Symmetry structure of $U(N)$ gauge theory: Let us now change the theory under
consideration by studying instead the $U(N)$ gauge theory with only adjoint
matter. It is convenient to write the gauge group as

$U(N)=\frac{U(1)\times SU(N)}{\mathbb{Z}_{N}}$ (8)

If we have access to a Lagrangian description of the $U(N)$ gauge theory, it
is straightforward to see that the gauge field corresponding to the $U(1)$
factor separates off, and the action (2) can now be written as

$S=\int
d^{4}x\left(-\frac{1}{2g_{1}^{2}}|f|^{2}-\frac{1}{g_{YM}^{2}}\mathop{\rm
tr}|F|^{2}+\cdots\right)$ (9)

where $f$ corresponds to the field strength of the new $U(1)$ gauge field
$f=da$. As all matter is in the adjoint, nothing couples to the $U(1)$ gauge
field, which has a free Maxwell action. There is thus a precisely marginal
$U(1)$ gauge coupling which we have named $g_{1}$.

Unlike above, where we had only a single discrete $\mathbb{Z}_{N}$ 1-form
symmetry, this theory has two continuous $U(1)$ 1-form symmetries
corresponding to the simultaneous conservation of electric $U(1)_{e}$ and
magnetic $U(1)_{b}$ flux. Their respective conserved currents
$J^{\mu\nu}_{e,b}$ are:

$J^{\mu\nu}_{e}=\frac{1}{g_{1}^{2}}f^{\mu\nu}\qquad
J^{\mu\nu}_{b}={{\frac{1}{2}}}\epsilon^{\mu\nu\rho\sigma}f_{\rho\sigma}$ (10)

In the phase described by a free $U(1)$ gauge theory action both of these
symmetries are spontaneously broken, and the usual 4d photon is the Goldstone
mode of this breaking. The line operator that is charged under the electric
1-form symmetry is the usual $U(1)$ Wilson line222This can be easily
understood by applying the usual Noether procedure to the action (9), where
the symmetry in question is realised by a shift of the $U(1)$ gauge potential
by a closed 1-form $\Lambda$: $a\to a+\Lambda$., and that charged under the
magnetic 1-form symmetry is the t’Hooft line. The diagnosis of this symmetry
breaking in terms of these line operators is discussed in Hofman:2018lfz ;
Lake:2018dqm .

Following along the lines of the discussion above, we now add the same flavour
degrees of freedom to the $U(N)$ gauge theory. The $U(1)$ gauge field $a$ now
couples minimally to both $\phi$ and $\psi$:

$\displaystyle S^{\prime}=\int d^{4}x\;\Bigg{[}-$
$\displaystyle\frac{1}{2g_{1}^{2}}|f|^{2}-\frac{1}{g_{YM}^{2}}\mathop{\rm
tr}|F|^{2}+$
$\displaystyle\sum_{i=1}^{N_{f}}\left(-|\partial\phi_{i}-ia\phi_{i}-iA\phi_{i}|^{2}-m_{\phi}^{2}\;|\phi_{i}|^{2}+\bar{\psi}_{i}\left(i{{\gamma}}^{\mu}\left(\partial_{\mu}-ia_{\mu}-iA_{\mu}\right)-m_{\psi}\right)\psi_{i}\right)+\cdots\Bigg{]}$
(11)

where $a$ and $A$ are the $U(1)$ and $SU(N)$ gauge potentials respectively.
This changes the dynamics of the $U(1)$ 1-form symmetries described above.
Importantly, the symmetry corresponding to conservation of electric flux is
now generically explicitly broken by the presence of the electrically charged
matter; the simplest way to see this is to note that the conserved current
identified in (10) is now no longer conserved; indeed the $U(1)$ Maxwell
equations are simply:

$\partial_{\nu}J^{\mu\nu}_{e}=j_{B}^{\mu}[\phi,\psi],$ (12)

where $j_{B}^{\mu}[\phi,\psi]$ is precisely the baryon number current (7).
Physically, this simply captures the idea that electric field lines can now
end on the electric charges that are carried by $\phi$ and $\psi$.

The magnetic flux current $J^{\mu\nu}_{b}$ is still conserved, as is clear
from its definition:

$\partial_{\nu}J^{\mu\nu}_{b}=0$ (13)

Thus, this theory has a single $U(1)$ 1-form symmetry.

However, the realisation of this symmetry now depends on the dynamics of the
$\phi,\psi$ fields. Let us now consider how the energy scale of interest $E$
compares to the masses $m_{\phi},m_{\psi}$:

  1. 1.

$E\ll m_{\phi,\psi}$: In this case the matter fields are gapped and can
essentially be ignored. We are then in the same situation as when there were
no flavour fields at all; $j^{\mu}[\phi,\psi]$ is effectively zero, and both
the electric and magnetic flux currents are conserved. The associated
symmetries are again both spontaneously broken, as described around (10). In
particular, the relevant line operators should display a perimeter law in this
phase.

  2. 2.

$E\gg m_{\phi,\psi}$: In this case we probe electric charge fluctuations in
the vacuum, and the electric flux symmetry is explicitly broken. The magnetic
flux symmetry is now realised differently; in particular, it is no longer
spontaneously broken. Relatedly, in this regime the $U(1)$ gauge coupling
$g_{1}$ runs logarithmically with the energy scale $E$.

Summary: In this work, we will study the manifestation of the higher-form
symmetry structures described above in a strongly coupled model, given by the
holographic realisation of maximally supersymmetric ${{\mathcal{N}}}=4$ Super-
Yang-Mills coupled to matter in the fundamental. We will primarily focus on
the case of the $U(N)$ gauge theory where we have a continuous 1-form
symmetry, but along the way we will clarify some aspects of the $SU(N)$ case
as we proceed.

An outline of the rest of the paper follows. In Section II we introduce the
(well-known) bulk action and discuss its symmetry structure, also discussing
some lower-dimensional examples to build some intuition for the extensive
dualisations that follow. In Sections III and IV we discuss the bulk dynamics
and appropriate charged operators in the duality frames that are appropriate
for the $SU(N)$ and $U(N)$ gauge theories respectively. Finally in Section V
we numerically compute the spectral function for the 2-form current in the
$U(N)$ theory and compare with expectations at weak coupling.

##  II Symmetries of holographic flavour

In this section we describe the holographic dual of the system described
above; in particular we study the maximal supersymmetrisation of the gauge
theory, i.e. ${{\mathcal{N}}}=4$ SYM with holographic flavour added. In most
discussions of holography it is implicitly assumed that the gauge group is
$SU(N)$; for us however the precise distinction between the $U(N)$ and $SU(N)$
gauge theories will be of considerable importance. This issue has been
clarified recently (see Hofman:2017vwr  for a perspective from higher-form
symmetry) and we briefly review it here, taking special care with the issues
that will be relevant for our construction.

###  II.1 Bulk holographic action

The holographic dual of the above is Type IIB string theory on
$\text{AdS}_{5}\times S^{5}$, giving rise to kinetic terms for the NS-NS
$2$-form $B_{2}$ and the R-R $2$-form $C_{2}$, as well as a Chern-Simons term.
After compactifying on the $S^{5}$ we obtain an action which is an integral
over all of $\text{AdS}_{5}$. To add fundamental matter, we wrap $N_{f}\ll
N_{c}$ probe $D7$-branes around the $S^{5}$ Karch:2002sh . See e.g.
CasalderreySolana:2011us  for a review of holographic flavour.

The final form of the dimensionally-reduced action on $\text{AdS}_{5}$ is

$\begin{split}S_{\text{bulk}}&=S_{\text{kin}}+S_{\text{CS}}+N_{f}\;S_{\text{DBI}}\\\
&=\frac{N_{c}^{2}}{8\pi^{2}R^{3}}\int{\left[-\frac{1}{2}H_{3}^{2}-\frac{1}{2}\left(\frac{\lambda}{4\pi
N_{c}}G_{3}\right)^{2}+\kappa B_{2}\wedge\left(\frac{\lambda}{4\pi
N_{c}}G_{3}\right)-\frac{1}{2}\kappa^{2}\mu f(z)\left(B_{2}+\frac{2\pi
R^{2}}{\sqrt{\lambda}}F_{2}\right)^{2}\right]}\end{split}$ (14)

Note our conventions for writing differential forms. For a $p$-form
$\Omega_{p}$ in $n$-dimensions we can define a corresponding $n$-form by

$\Omega_{p}^{2}\equiv\Omega_{p}\wedge\star\Omega_{p}$ (15)

Sometimes it is preferable to work in components, in which case we borrow from
Polchinski:1998rr  and write

$|\Omega_{p}|^{2}\equiv\frac{1}{p!}\;\Omega_{\mu_{1}\mu_{2}\ldots\mu_{p}}\Omega^{\mu_{1}\mu_{2}\ldots\mu_{p}}$
(16)

It is straightforward to translate between these descriptions using the
identity

$\Omega_{p}^{2}=|\Omega_{p}|^{2}\sqrt{|g|}\;d^{n}x$ (17)

Further conventions and notation are explained in Appendix A.

The forms appearing in the action are the field strengths $H_{3}=dB_{2}$,
$G_{3}=dC_{2}$ and $F_{2}=dA_{1}$. Note that we have used the unusual name
$G_{3}$ for the field strength of the R-R form to avoid confusion with the
field strength of the $D7$-brane Maxwell field. Our zoo of higher-form fields
is extensive – and will become even more so as we dualise fields in the bulk –
so we have provided an index in Appendix D. The constants in the action are
given by

$\displaystyle\kappa$ $\displaystyle=\frac{4}{R}$ (18a) $\displaystyle\mu$
$\displaystyle=\frac{N_{f}}{N_{c}}\frac{\lambda}{32\pi^{2}}$ (18b)

$\mu$ denotes the relative dynamical importance of the flavour and colour
degrees of freedom. The function $f$ is given by

$f(z)=\begin{cases}1-\left(z/z_{c}\right)^{2}&z\leq z_{c}\\\
0&z>z_{c}\end{cases}$ (19)

We will work with $\text{AdS}_{5}$ in Poincaré coordinates:

$ds^{2}=\frac{R^{2}}{z^{2}}\left(dx^{\mu}dx_{\mu}+dz^{2}\right)$ (20)

where $R$ is the AdS radius.

Some words about the probe limit are in order here. Usually one considers the
limit where $\mu\to 0$, and thus where the backreaction of the flavour degrees
of freedom on the colour degrees of freedom can be ignored. We will work with
the simple quadratic action above, but we will allow $\mu$ to take on finite
values. This corresponds to studying some subset of the interplay between
flavour and colour degrees of freedom, in particular those associated with the
realisation of the symmetries. As explained below, this results in novel
effects associated with the Higgsing of the 2-form $B$ field by the DBI gauge
field $A_{1}$; these effects qualitatively affect the physics but are
invisible in the strict probe limit. Strictly speaking, however, we are not
studying all aspects of this interplay, because we neglect the gravitational
backreaction of the flavour branes; thus the approximation we take should be
considered as an illustrative one that is designed to highlight the physics of
interest.

Let us now note the gauge symmetry of the action above. We have two
independent 1-form gauge transformations shifting $B_{2}$ and $C_{2}$
respectively; we also have a 0-form gauge transformation shifting the DBI
worldvolume field. The full transformation of the fields is

$\displaystyle\delta B_{2}$ $\displaystyle=d\Xi^{(B)}_{1}$ (21a)
$\displaystyle\delta C_{2}$ $\displaystyle=d\Xi^{(C)}_{1}$ (21b)
$\displaystyle\delta A_{1}$ $\displaystyle=-\frac{\sqrt{\lambda}}{2\pi
R^{2}}\;\left(\Xi^{(B)}_{1}+d\xi^{(A)}\right)$ (21c)

Note the simultaneous transformation of $A_{1}$ and $B_{2}$ under a shift by
$\Xi^{(B)}_{1}$; this encodes the fact that string worldsheets can end on the
$D$-brane, and will be of considerable importance to us.

We turn now to the Chern-Simons term $B_{2}\wedge G_{3}$; this well-known term
Aharony:1998qu ; Maldacena:2001ss ; Witten:1998xy  is closely related to the
physics of higher-form symmetry in holography Hofman:2017vwr  and will play a
key role in our analysis. Obtaining the precise prefactor can be somewhat
subtle; it can naively be thought of as arising from a dimensional reduction
of a 10d Chern-Simons term $B_{2}\wedge dc_{4}\wedge G_{3}$ involving the R-R
$4$-form $c_{4}$. Integrating over the $S^{5}$ we pick up a factor of the flux
$dc_{4}\sim N_{c}$, giving a term with qualitatively the correct form, as
first noted in Witten:1998xy .

However, this is not quite consistent: in fact, a covariant action for the
(self-dual) RR 4-form does not actually exist, and pursuing the above route
results in an inconsistent normalisation for the Chern-Simons term, as noted
in Belov:2004ht . In this work we take a different approach. Consistency of
the theory in the presence of magnetic charges actually requires the
coefficient of this term to be quantized; we review this Dirac quantisation
condition in Appendix B.2 and identify the integer coefficient of the term
with $N_{c}$, as we expect on symmetry grounds.

Finally, the last term arises from the dimensional reduction of the DBI
action. We can embed $N_{f}$ probe $D7$-branes into the target space by means
of the DBI action:

$N_{f}\;S_{\text{DBI}}=-N_{f}\;\tau_{7}\int{d^{8}\xi\;\sqrt{-\det\left(g_{\alpha\beta}+B_{\alpha\beta}+2\pi
l_{s}^{2}\;F_{\alpha\beta}\right)}}$ (22)

where $\xi^{\alpha}$ are the brane worldvolume coordinates, $g_{\alpha\beta}$
are the components of the induced worldvolume metric on the $D7$-brane,
$B_{\alpha\beta}$ are the components of $B_{2}$ and $F_{\alpha\beta}$ are the
components of $F_{2}=dA_{1}$, the Maxwell field strength living on the brane
worldvolume. $\tau_{p}$ is the effective $Dp$-brane tension after absorbing
the effect of the dilaton $e^{\Phi}=g_{s}$ and is given by
$\tau_{7}=\frac{1}{g_{s}}\frac{1}{(2\pi)^{7}l_{s}^{8}}$.

The $S^{5}$ factor in the metric may be written as

$d\Omega_{5}^{2}=d\theta^{2}+\cos^{2}{\theta}\;d\psi^{2}+\sin^{2}{\theta}\;d\Omega_{3}^{2},$
(23)

where we have chosen coordinates that make manifest an $S^{3}\subset S^{5}$.
As usual for a D3/D7 embedding, the desired brane configuration fills all of
AdS and wraps the $3$-sphere around the $S^{5}$. Thus the embedding is
parametrized by the transverse coordinates $\psi,{\theta}$; $\psi$ is a
Killing direction and may be taken to be constant, and the appropriate
solution for ${\theta}(z)$ corresponding to massive holographic flavour is
Karch:2002sh :

$\theta(z)=\theta_{c}\equiv\begin{cases}\arccos(z/z_{c})&z\leq z_{c}\\\
0&z>z_{c}\end{cases}$ (24)

A careful matching to the field theory shows that

$z_{c}=\frac{\sqrt{\lambda}}{2\pi}\frac{1}{m_{F}}$ (25)

where $m_{F}$ is the mass of the flavour degrees of freedom Karch:2007pd ;
Karch:2006bv .

Geometrically, this embedding means that the $S^{3}$ wrapping the $S^{5}$ is
of maximal size ($\theta=\pi/2$) at $z=0$ on the boundary, and the $D7$-brane
vanishes ($\theta=0$) at the critical value $z=z_{c}$. For $z>z_{c}$ the
$D7$-brane has no effect; this is dual to the fact that at energies smaller
than the mass gap the flavour degrees of freedom can no longer be excited. If
we set the mass to zero $\theta$ is constant and the theory is conformal.

If we now substitute the on-shell angle $\theta=\theta_{c}$ back into the DBI
action and expand to quadratic order in $B_{2}$ and $F_{2}$ we obtain the
following quadratic action for the fluctuations of the DBI gauge field:

$N_{f}\;S_{\text{DBI}}=-\frac{N_{c}^{2}}{8\pi^{2}R^{3}}\int{\frac{1}{2}\kappa^{2}\mu
f(z)\left(B_{2}+\frac{2\pi R^{2}}{\sqrt{\lambda}}F_{2}\right)^{2}}$ (26)

Some details about this computation are given in Appendix B.3.

###  II.2 Examples in lower dimensions

The action (14) has several interesting features, arising from the interplay
of the higher form symmetry with the baryon number symmetry. To the best of
our knowledge these have not yet been fully explained in the literature, and
we will unpack these below. It is first helpful to orient ourselves with some
more familiar examples in lower dimension.

Let us begin with the following action for a Goldstone mode in three
dimensions:

$S_{1}=-\int d^{3}x\;\frac{v^{2}}{2}|d\theta|^{2}$ (27)

Clearly this has a single degree of freedom, which is gapless. The situation
is however very different if we consider gauging this scalar Goldstone with a
1-form gauge field $a_{\mu}$, resulting in the following action

$S_{2}=\int
d^{3}x\left(-\frac{1}{2g^{2}}\left|da\right|^{2}-\frac{v^{2}}{2}\left|a-d\theta\right|^{2}\right)$
(28)

This theory is now massive; the Goldstone mode is eaten by the photon,
resulting in a gapped theory with mass gap $g^{2}v^{2}$. As turning on a small
gauge coupling $g$ results in a mass gap, the weak coupling limit and the
infrared limit don’t commute.

Higgsing a gauge field is one way to obtain a mass gap. Another way to give a
gauge field a mass is through a Chern-Simons term Deser:1981wh . Let us thus
imagine removing the Goldstone mode and adding a second gauge field $b$ to
obtain the following theory:

$S_{3}=\int
d^{3}x\left(-\frac{1}{2g^{2}}\left|da\right|^{2}-\frac{1}{2v^{2}}\left|db\right|^{2}+a\wedge
db\right)$ (29)

What is the spectrum of this theory? An illuminating way to understand this is
to dualise the gauge field $b$ to a scalar $\psi$. Following the standard
algorithm, we find

$S_{3}^{\prime}=\int
d^{3}x\left(-\frac{1}{2g^{2}}\left|da\right|^{2}-\frac{v^{2}}{2}\left|d\psi-a\right|^{2}\right)$
(30)

where in terms of the original degree of freedom $db=g^{2}\star(d\psi-a)$.
This is essentially the same as the Higgs-ed theory studied above in (28), and
is also gapped. Thus we see that adding a Chern-Simons term and Higgsing a
gauge field are the same mechanism, just written in different duality frames.

Finally, let us imagine both adding a Chern-Simons term and Higgsing, i.e. we
study the following action:

$S_{4}=\int
d^{3}x\left(-\frac{1}{2g^{2}}|da|^{2}-\frac{1}{2v^{2}}|db|^{2}+a\wedge
db+\frac{v^{2}}{2}|a-d\theta|^{2}\right)$ (31)

What is the spectrum now? It is again helpful to dualise the $b$ field, after
which we obtain:

$S_{4}^{\prime}=\int
d^{3}x\left(-\frac{1}{2g^{2}}|da|^{2}-\frac{v^{2}}{2}|d\psi-a|^{2}-\frac{v^{2}}{2}|d\theta-a|^{2}\right)$
(32)

i.e. after a duality this is like “Higgsing twice”. But we only have one
photon to eat a putative Goldstone; thus there should remain one Goldstone
left uneaten, which can be seen by rewriting the action to be:

$S_{4}^{\prime\prime}=\int
d^{3}x\left(-\frac{1}{2g^{2}}|da|^{2}-\frac{v^{2}}{4}|d\psi+d{\theta}-2a|^{2}-\frac{v^{2}}{4}|d\psi-d{\theta}|^{2}\right)$
(33)

Thus the gauge-charged combination $\psi+{\theta}$ is eaten, and forms part of
a massive photon; however the gauge-invariant combination $\psi-{\theta}$
remains uneaten, and is a gapless mode in the spectrum. The general lesson is
that if we try to give a gauge field a mass twice, both by Higgsing and by
adding a Chern-Simons term, then we find that a gapless mode survives. We
could also have chosen to dualise the scalar ${\theta}$ in (31) into a 1-form
gauge field; in this case the gapless mode would have appeared to be a gauge
field and not a scalar, but this gauge field would be related to
$\psi-{\theta}$ by the usual Abelian duality.

What does all of this have to do with AdS/CFT? In most discussions of
holographic flavour in AdS/CFT, one works in the probe limit, considering the
limit $\mu\sim\frac{N_{f}}{N_{c}}$ to $0$ and studying the fluctuations of the
DBI worldvolume gauge field $A_{1}$, which then decouples from the other
fields, and whose action takes the form $S\sim\int(dA_{1})^{2}$. This is the
action of a massless photon and is analogous to a higher-form version of (27).
The field theory dual of this massless gauge field is the baryon number
current.

However, if we consider the complete action (14), we see that this is actually
somewhat dangerous; in fact the gauge field does not appear by itself but
rather in the gauge-invariant combination $B_{2}+\frac{2\pi
R^{2}}{\sqrt{\lambda}}F_{2}$. At first glance, this appears somewhat
problematic, as the action contains the following terms:

$\frac{N_{c}^{2}}{8\pi^{2}R^{3}}\int\left[-\frac{1}{2}(dB_{2})^{2}-\frac{1}{2}\kappa^{2}\mu\left(B_{2}+\frac{2\pi
R^{2}}{\sqrt{\lambda}}F_{2}\right)^{2}+\cdots\right]$ (34)

(where for simplicity we consider a case where $f(z)$ is constant). Compare
this to (28); it actually now appears that the field $A_{1}$ has been eaten by
the higher form gauge field $B_{2}$, in a higher-form analogue of the Higgs
mechanism. This suggests that the theory should have only massive modes, with
the mass scaling like ${\kappa}^{2}\mu$. In a sense, the infrared limit no
longer commutes with the probe limit. However, this is clearly a nonsensical
result: the dual field theory should still have a baryon number current, which
should be dual to a bulk gauge field that is exactly massless to all orders in
$\frac{N_{f}}{N_{c}}$.

The resolution comes in studying the full action, which contains an extra
field $C_{2}$ which has its own kinetic term as well as a mixed Chern-Simons
term coupling it to $B_{2}$. The action now appears more like a higher form
version of (31), which indeed does support a gapless mode, though it is not
apparent at first glance. In this work we will unpack the analogous mechanism
in the AdS/CFT context; we will indeed find that the bulk action always
supports a gapless mode. For a certain set of boundary conditions (those which
are dual to the $SU(N)$ gauge theory), this mode is dual to the baryon number
current. For a different set of boundary conditions (those which are dual to
the $U(N)$ gauge theory), this mode is dual to the 2-form current for magnetic
flux.

The fact that the existence of this gapless mode depends crucially on the
interplay between the Chern-Simons and DBI terms is dual to the fact that in
the field theory the baryon number current is intertwined in some sense with
the 1-form center $\mathbb{Z}_{N}$ symmetry of the pure gauge theory.

##  III $SU(N)$ gauge theory

The bulk may be understood in various duality frames. We begin by studying it
in a frame which is useful when the dual field theory is the $SU(N)$ gauge
theory coupled to fundamental flavour.

###  III.1 Bulk action

To begin, it is helpful to Poincaré dualise the R-R form $C_{2}$ to a $1$-form
$\tilde{C}_{1}$ in the usual way. The procedure is explained in for example
Appendix B.4 of Polchinski:1998rr ; applying the algorithm we find:

$\tilde{G}_{2}=\frac{\lambda}{4\pi N_{c}}\star G_{3}-\kappa B_{2}$ (35)

where $\tilde{G}_{2}=d\tilde{C}_{1}$. Substituting into the action now gives a
different presentation of the same theory.

$S=-\frac{N_{c}^{2}}{8\pi^{2}R^{3}}\int{\left[\frac{1}{2}H_{3}^{2}+\frac{1}{2}\kappa^{2}\left(\left(B_{2}+\frac{1}{\kappa}\tilde{G}_{2}\right)^{2}+\mu
f(z)\left(B_{2}+\frac{2\pi
R^{2}}{\sqrt{\lambda}}F_{2}\right)^{2}\right)\right]}$ (36)

Observe now that from the duality relation (35), $\tilde{C}_{1}$ inherits the
gauge-transformation under the 1-form gauge symmetry:

$\delta\tilde{C}_{1}=-\kappa\left(\Xi^{(B)}_{1}+d\xi^{(C)}\right)$ (37)

which ensures that the action is indeed still gauge-invariant. Here
$\Xi^{(B)}$ is the original 1-form gauge transformation of $B_{2}$, whereas
$\xi^{(C)}$ is an emergent 0-form gauge transformation that exists only in
this duality frame. In a sense it is the magnetic dual of the 1-form gauge
transformation of $C_{2}$, which has been dualised away.

We can diagonalise the action in this duality frame to better understand its
spectrum. After diagonalising, it will also be easier to fix a gauge and solve
the equations of motion. To this end, it is convenient to define the function
$h(z)$ by

$h(z)=\frac{1}{1+\mu f(z)}$ (38)

so that we can define the new $1$-form fields

$\displaystyle\eta_{1}$
$\displaystyle=\frac{1}{\kappa}\;\tilde{C}_{1}-\frac{2\pi
R^{2}}{\sqrt{\lambda}}\;A_{1}$ (39a) $\displaystyle\tau_{1}$
$\displaystyle=\frac{2\pi R^{2}}{\sqrt{\lambda}}\;A_{1}+h\;\eta_{1}$ (39b)

and their respective field strengths

$\displaystyle Y_{2}$ $\displaystyle=d\eta_{1}$ (40a) $\displaystyle T_{2}$
$\displaystyle=d\tau_{1}$ (40b)

These linear combinations inherit the following gauge-transformations

$\displaystyle\delta\eta_{1}$ $\displaystyle=d\xi$ (41a)
$\displaystyle\delta\tau_{1}$ $\displaystyle=-\Xi_{1}^{(B)}+h\;d\xi$ (41b)

where $\xi=\xi^{(A)}-\xi^{(C)}$. Observe that $\eta_{1}$ has the same gauge
transformation as an ordinary free Maxwell field. When the field theory is
gapless, $\eta_{1}$ is precisely a massless gauge field in the bulk, so is the
holographic dual of a $0$-form symmetry in the field theory. This $0$-form
symmetry corresponds to baryon number conservation.

The diagonalised action can now be written cleanly as

$S=-\frac{N_{c}^{2}}{8\pi^{2}R^{3}}\int\left\\{\frac{1}{2}H_{3}^{2}+\frac{1}{2}\kappa^{2}\left[(1-h)Y_{2}^{2}+h^{-1}\left(B_{2}+T_{2}+\eta_{1}\wedge
dh\right)^{2}\right]\right\\}$ (42)

Gauge-invariance is easy to check: $H_{3}$, $Y_{2}$ and
$B_{2}+T_{2}+\eta_{1}\wedge dh$ are each individually gauge-invariant
quantities.

This action is somewhat complicated, as it is dual to an RG flow captured
holographically by the profile of the function $f(z)$. To understand the
symmetry structure, it is helpful to consider the limit of zero fermion mass
$m_{F}\to 0$. We now have $z_{c}\to\infty$, and so $f(z)=1$ for all $z$. This
gives $h(z)=(1+\mu)^{-1}$.

We then find:

$S\to-\frac{N_{c}^{2}}{8\pi^{2}R^{3}}\int{\left[\frac{1}{2}H_{3}^{2}+\frac{1}{2}\kappa^{2}(1+\mu)(B_{2}+d\tau_{1})^{2}+\frac{1}{2}\kappa^{2}\left(\frac{\mu}{1+\mu}Y_{2}^{2}\right)\right]}$
(43)

We can partially gauge-fix to an analogue of unitary gauge in which we set
$T_{2}=0$. This describes a 2-form gauge field $B_{2}$ which has been Higgs-ed
by the 1-form $\tau_{1}$; the resulting dynamical bulk field is massive. It
also has a precisely massless 1-form gauge field $\eta_{1}$, as anticipated
above. This is dual to the baryon number current. Note that this structure
arose out of the interplay between the Chern-Simons term and the flavour
terms; this is dual to the interplay between the $U(1)$ baryon number current
and the $\mathbb{Z}_{N}$ center symmetry of the field theory. In the remainder
of this section we further describe some universal aspects of this interplay.

Up to boundary terms, the variation of the action is

$\begin{split}\delta S=-\frac{N_{c}^{2}}{8\pi^{2}R^{3}}\int\bigg{\\{}&\delta
B_{2}\wedge\star\left[\star d\star
dB_{2}+\kappa^{2}h^{-1}(B_{2}+T_{2}+\eta_{1}\wedge dh)\right]\\\
&-\kappa^{2}\;\delta\eta_{1}\wedge\star\left[\star d((1-h)\star
d\eta_{1})-h^{-1}\star(dh\wedge\star(B_{2}+T_{2}+\eta_{1}\wedge dh))\right]\\\
&+\kappa^{2}\;\delta\tau_{1}\wedge
d\left[h^{-1}\star(B_{2}+T_{2}+\eta_{1}\wedge dh)\right]\bigg{\\}}\end{split}$
(44)

Hence the equations of motion are333The $\tau_{1}$ equation is redundant since
it follows by taking the exterior derivative of the $B_{2}$ equation.

$\displaystyle\star\;d\star
dB_{2}+\kappa^{2}h^{-1}(B_{2}+T_{2}+h^{\prime}\eta_{1}\wedge dz)$
$\displaystyle=0$ (45a) $\displaystyle(1-h)\star d\star
d\eta_{1}+h^{-1}h^{\prime}\star[dz\wedge\star(-h\;d\eta_{1}+B_{2}+T_{2}+h^{\prime}\eta_{1}\wedge
dz)]$ $\displaystyle=0$ (45b)

where $h^{\prime}\equiv\frac{dh}{dz}$.

The spectrum of fields thus consists of a massive $2$-form gauge field
$B_{2}$, a massless $1$-form gauge field $\eta_{1}$ which is dual to the
baryon number current, and a $1$-form gauge field $\tau_{1}$ which appears
only through its field strength $T_{2}$. $T_{2}$ is of less physical
importance since it can be easily gauged away; in a sense it simply provides
the longitudinal degrees of freedom of the massive tensor $B_{2}$.

If we are studying the $SU(N)$ gauge theory coupled to fundamental flavour, it
is important to note that the appropriate boundary conditions at the AdS
boundary are those where we hold fixed the boundary value of $\eta_{1}$; this
guarantees that we obtain a conserved $0$-form baryon number current
$j^{\mu}_{b}$ in the boundary theory. As we will see, this will be different
when we study the $U(N)$ gauge theory.

###  III.2 Charged operators

We now describe the bulk object that is charged under the baryon number
symmetry.

####  III.2.1 Baryon vertices in pure $SU(N)$ gauge theory

Let us first review the conventional case with no flavour branes Witten:1998xy
. There we set $N_{f}\to 0$, and we find simply:

$S=-\frac{N_{c}^{2}}{8\pi^{2}R^{3}}\int{\left[\frac{1}{2}H_{3}^{2}+\frac{1}{2}\kappa^{2}(B_{2}+T_{2})^{2}\right]}$
(46)

With no flavour branes the DBI gauge field $A_{1}$ does not exist, and from
(38) and (39b) we see that when $h=1$ we have simply
$\tau_{1}=\kappa^{-1}\tilde{C}_{1}$, i.e. $\tau_{1}$ is directly the magnetic
dual of the RR 2-form.

We will now revisit this action from the point of view of symmetry. Note that
under the 1-form gauge transformation of $B_{2}$, $\tau_{1}$ must also
transform:

$B_{2}\to B_{2}+d\Xi_{1}\qquad\tau_{1}\to\tau_{1}-\Xi_{1}$ (47)

We now study the bulk objects that are charged under these gauge symmetries.
We have fundamental string worldsheets, which couple minimally to $B_{2}$ as

$\frac{1}{2\pi l_{s}^{2}}\int_{{{\mathcal{M}}}}B_{2}$ (48)

We turn now to $\tau_{1}$; as $\tau_{1}$ is a usual 1-form gauge field in the
bulk, it couples naturally to one-dimensional particle worldlines in AdS5.
What are these objects?

From the definition of the duality relationship (35), we can see that these
objects couple magnetically to the RR 2-form $C_{2}$. In the ten-dimensional
picture, these are thus $D5$-branes. Of their six dimensional worldvolume,
five of them are wrapped on the $S^{5}$, and the remaining one dimension
traces a worldline on AdS5. By using the normalisations in Appendix B.2 one
can verify that a single such $D5$-brane couples to $\tau_{1}$ as

$\frac{N_{c}}{2\pi l_{s}^{2}}\int_{L}\tau_{1}$ (49)

Note however that this coupling alone is not invariant under the 1-form gauge
transformation (47): indeed we see that the the one-dimensional worldline $L$
can exist only as the boundary of $N_{c}$ string worldsheets, i.e. the
combined coupling

$N_{c}\left(\frac{1}{2\pi
l_{s}^{2}}\int_{{{\mathcal{M}}}}B_{2}\right)+\frac{N_{c}}{2\pi
l_{s}^{2}}\int_{\partial{{\mathcal{M}}}}\tau_{1}$ (50)

is invariant. This fact – that the wrapped $D5$-brane is the endpoint of
$N_{c}$ fundamental strings and thus acts as a baryon vertex in the dual field
theory – can also be understood directly from the original Chern-Simons
coupling Witten:1998xy ; Aharony:1998qu ; Maldacena:2001ss ; Witten:1998wy .
Here we simply restate it in an alternative (bulk) duality frame.

It is now instructive to imagine the bulk worldline intersecting the AdS
boundary at a point. Each of the $N_{c}$ string worldsheets will also
intersect the boundary as a series of curves ending at the same point.
Holographically, the combined object is a non-dynamical baryon vertex serving
as the endpoint of $N_{c}$ Wilson lines in the fundamental representation. It
is clear that the baryon vertex, being tied to $N_{c}$ Wilson lines, is not a
local operator in the field theory; the bulk dual of this statement is that it
does not correspond to a free particle worldline but rather only as the
boundary of $N_{c}$ F-strings.

Figure 1:  The hanging $D5$-brane forms 1-dimensional worldline in the bulk;
it is the boundary of $N_{c}$ F-string worldsheets which also intersect the
AdS boundary. At their intersection with the boundary they define the
insertion of fundamental Wilson lines in the dual CFT.

####  III.2.2 Baryon operator in theory with dynamical flavour

We now restore the flavour branes, i.e. we return to (43). In the bulk, we now
have a new massless field $\eta_{1}$, which we understand is dual to the
baryon number current in the field theory $U(1)_{B}$. In the dual field
theory, we now expect the existence of local baryon operators that carry
charge $N_{c}$ (in units of the baryon charge of the fundamental gauge-charged
fields).

What is the bulk dual of this operator? Consider a general particle-like
object in the bulk that couples to both $\eta_{1}$ and $\tau_{1}$, i.e.

$\int_{L}\left(q_{\eta}\eta_{1}+q_{\tau}\tau_{1}\right)$ (51)

As argued above, any coupling to $\tau_{1}$ will necessarily mean that the
particle has strings attached, in order to ensure gauge-invariance. Let us
consider an object which has $q_{\tau}=0$. As $\eta_{1}$ does not transform
under the 1-form gauge transformation (47), this coupling is entirely gauge-
invariant on its own. Thus a particle in the bulk that couples in this way is
dual to a local boundary operator that carries baryon charge. From field
theory considerations, we understand that this object should be related to a
bound state of a $D5$-brane and F-strings in some manner.

The presence of the new field $\eta_{1}$ lets the $D5$-brane exist as an
independent object that is untethered to any strings. To see this more
explicitly, we can express this coupling in terms of the original fields
$\tilde{C}_{1}$ and $A_{1}$; we find that the unit quantized $D5$-brane
couples as

$q_{\eta}\int_{L}\eta_{1}=\frac{N_{c}}{2\pi l_{s}^{2}}\int_{L}\left(\frac{2\pi
R^{2}}{\sqrt{\lambda}}A_{1}-\kappa\;\tilde{C}_{1}\right)$ (52)

where we have used the quantized coupling to $\tilde{C}_{1}$ worked out above,
and where the coupling to $A_{1}$ is correlated with that of $\tilde{C}_{1}$
by the condition that $q_{\tau}=0$. This can be compared to the coupling of a
single F-string ending on the $D7$-brane:

$\frac{1}{2\pi l_{s}^{2}}\left(\int_{ws}B_{2}+\int_{\partial ws}\frac{2\pi
R_{2}}{\sqrt{\lambda}}A_{1}\right)$ (53)

In other words, the coupling to $A_{1}$ is as $N_{c}$ F-strings.
Microscopically one can actually imagine that the $D5$-brane is connected by
very small strings to the flavour $D7$-brane, where the string charge is now
carried by the $A_{1}$ field living on the brane. The resulting composite
object is the particle-like excitation that we describe above. Related work in
different holographic models to directly construct bulk objects carrying
baryon number can be found in Hata:2007mb ; Seki:2008mu ; Sfetsos:2008yr ; see
in particular Sfetsos:2008yr . We stress that our construction makes no real
statements about the dynamics of the internal structure, and simply shows how
their charges are captured in the low-energy description.

##  IV $U(N)$ gauge theory

We would now like to understand the theory with the boundary conditions that
are appropriate to having a $U(N)$ gauge theory dual. We now expect to obtain
a 2-form conserved current on the boundary; it is thus appropriate to study
the bulk in a different duality frame.

###  IV.1 Bulk action

We can Poincaré dualise the field $\tau_{1}$ by integrating out its field
strength $T_{2}$ in the usual way. This yields a $2$-form $\mathcal{A}_{2}$
with field strength $\mathcal{F}_{3}=d\mathcal{A}_{2}$ given by:

$\mathcal{F}_{3}=\kappa^{2}h^{-1}\star\left(B_{2}+T_{2}+\eta_{1}\wedge
dh\right)$ (54)

Substituting into the action and integrating by parts gives

$S=-\frac{N_{c}^{2}}{8\pi^{2}R^{3}}\int{\left[\frac{1}{2}H_{3}^{2}+\frac{1}{2}\kappa^{2}(1-h)Y_{2}^{2}+\frac{1}{2}\kappa^{-2}h\;\mathcal{F}_{3}^{2}+B_{2}\wedge\mathcal{F}_{3}-Y_{2}\wedge\mathcal{A}_{2}\wedge
dh\right]}$ (55)

Note the last term in the action where ${{\mathcal{A}}}_{2}$ appears
explicitly; this arises from an integration by parts so that the action
depends on $Y_{2}=d\eta_{1}$ and not $\eta_{1}$ explicitly. As a result we can
now dualise $\eta_{1}$ using exactly the same procedure to give a $2$-form
$\mathcal{P}_{2}$ whose field strength $\mathcal{Q}_{3}=d\mathcal{P}_{2}$ is
given by

$\mathcal{Q}_{3}+\mathcal{A}_{2}\wedge dh=\kappa^{2}(1-h)\star Y_{2}$ (56)

(Note that ${\mathcal{P}}_{2}$ can be thought of as – modulo mixing with other
fields – the electric-magnetic dual of $\eta_{1}$, i.e. the bulk field dual to
the baryon number current). Substituting this back into the action then gives:

$S=-\frac{N_{c}^{2}}{8\pi^{2}R^{3}}\int{\left[\frac{1}{2}H_{3}^{2}+B_{2}\wedge\mathcal{F}_{3}+\frac{1}{2}\kappa^{-2}\left((1-h)^{-1}(\mathcal{Q}_{3}+\mathcal{A}_{2}\wedge
dh)^{2}+h\mathcal{F}_{3}^{2}\right)\right]}$ (57)

Note that we have a gauge freedom given by

$\displaystyle\delta\mathcal{A}_{2}$ $\displaystyle=d\Xi_{1}$ (58a)
$\displaystyle\delta\mathcal{P}_{2}$ $\displaystyle=-\Xi_{1}\wedge
dh+d\Lambda_{1}$ (58b)

under which the action is invariant, where $\Lambda_{1}$ is a new free
$1$-form gauge parameter. From the perspective of the $2$-form picture, two of
the equations of motion in the $1$-form picture are simply the Bianchi
identities $d\mathcal{F}_{3}=0$ and $d\mathcal{Q}_{3}=0$.

The spectrum is easiest to understand in the case where the flavour mass is
zero so that $dh=0$. We then have two coupled 2-forms $B_{2}$ and
$\mathcal{A}_{2}$ which constitute a massive degree of freedom. We also have a
single massless 2-form $\mathcal{P}_{2}$ whose dependence is only through its
field strength $\mathcal{Q}_{3}$; this massless bulk is dual to the only
conserved $2$-form current $J^{\mu\nu}=J_{b}^{\mu\nu}$, identified in (10).

The equations of motion in the $2$-form picture are

$\displaystyle d\star[(1-h)^{-1}(\mathcal{Q}_{3}+\mathcal{A}_{2}\wedge dh)]$
$\displaystyle=0$ (59a) $\displaystyle d\star H_{3}-\mathcal{F}_{3}$
$\displaystyle=0$ (59b) $\displaystyle
d\star(h\mathcal{F}_{3})+\kappa^{2}H_{3}-(1-h)^{-1}dh\wedge\star(\mathcal{Q}_{3}+\mathcal{A}_{2}\wedge
dh)$ $\displaystyle=0$ (59c)

From the perspective of the $1$-form picture, two of these equations give the
Bianchi identities $dY_{2}=0$ and $dT_{2}=0$. The third equation is the same
in both pictures.

Thus, to obtain the bulk dual to the $U(N)$ gauge theory coupled to flavour,
we should use AdS/CFT boundary conditions where we hold fixed the boundary
value of the 2-form field ${\mathcal{P}}$. The usual rules of AdS/CFT will
then guarantee that in the dual field theory, we will obtain a 2-form
conserved current $J$, as expected.

We note that the $U(N)$ theory seems to contains one extra parameter as
compared to the $SU(N)$ theory; as explained around (11), the coupling
constant $g_{1}$ associated to the “$U(1)$ factor” seems to be an extra
parameter that can be tuned. In a universal sense this can be understood as a
double-trace coupling associated to the 2-form current $J$. When there are
flavor degrees of freedom present this coupling is expected to run
logarithmically, becoming strong in the UV. Thus, due to dimensional
transmutation the extra data that needs to be provided is not a dimensionless
coupling but rather the energy scale for the Landau pole at which this
coupling becomes strong. As explained in Hofman:2017vwr ; Faulkner:2012gt ,
the boundary conditions for a massless 2-form field such as ${\mathcal{P}}$ in
AdS5 indeed require one to specify such a scale. We will see this explicitly
when solving the bulk equations of motion in later sections.

We provide a few more details; as usual, $J$ is obtained by taking a
functional derivative of the bulk on-shell action with respect to the boundary
value of $\mathcal{P}_{2}$. If we set

$\lim_{z\to 0}{\mathcal{P}_{2}}=p_{2}$ (60)

and use the equation of motion then we obtain

$\displaystyle J^{\mu\nu}=\frac{\delta S_{\text{on-shell}}}{\delta
p_{\mu\nu}}$
$\displaystyle=\frac{\mathcal{N}}{2\kappa^{2}}\frac{\delta}{\delta
p_{\mu\nu}}\int{d\left[(1-h)^{-1}\mathcal{P}_{2}\wedge\star(\mathcal{Q}_{3}+\mathcal{A}_{2}\wedge
dh)\right]}$ $\displaystyle=\frac{\mathcal{N}}{2}\frac{\delta}{\delta
p_{\mu\nu}}\int{d(\mathcal{P}_{2}\wedge Y_{2})}$

From here we can conclude that the 2-form symmetry current is:

$J^{\mu\nu}=\lim_{z\to
0}\frac{{{\mathcal{N}}}}{2}\left(\star_{4}Y_{2}\right)^{z\mu\nu}$ (61)

where the normalisation is given by
$\mathcal{N}=\frac{N_{c}^{2}}{8\pi^{2}R^{3}}$.

###  IV.2 Charged line operator

We would now like to understand the bulk operators that are charged under the
$2$-form gauge field ${\mathcal{P}}_{2}$. In the field theory, these are dual
to line operators that are charged under the corresponding $1$-form symmetry.
In this subsection only we will work only to first order in $\mu$ to simplify
the formulas. We begin by tracing back through the chain of dualities; from
(39a) to (39b), in the small $\mu$ limit we find:

$\tau_{1}=\frac{1}{{\kappa}}\tilde{C}_{1}+2\pi\ell_{s}^{2}\mu
A_{1}\qquad\eta_{1}=\frac{1}{{\kappa}}\tilde{C}_{1}-2\pi\ell_{s}^{2}A_{1}$
(62)

Furthermore, in the same limit we find

$d{\mathcal{P}}_{2}={\kappa}^{2}\mu\star d\eta_{1}\qquad
d{{\mathcal{A}}}_{2}={\kappa}^{2}\left(B_{2}+d\tau_{1}\right)$ (63)

An object which couples minimally to ${\mathcal{P}}_{2}$ is one that appears
on the right hand side of the equation of motion $d\star
d{\mathcal{P}}_{2}=0$; we thus need to find bulk objects that couple
magnetically to the fields $\tilde{C}_{1}$ and $A_{1}$. As $\tilde{C}_{1}$ is
the magnetic dual of the RR 2-form $C_{2}$, the object coupling magnetically
to it is simply a D1-string. In the Appendix we work out the normalisation of
this coupling in our conventions to show that

$\frac{1}{{\kappa}}\int_{S^{2}}d\tilde{C}_{1}=\frac{(2\pi\ell_{s})^{2}}{N_{c}}$
(64)

where here the $S_{2}$ wraps a $D1$-string that is hanging down into AdS5.

The object which couples magnetically to the DBI worldvolume gauge field
$A_{1}$ is somewhat more interesting. We will call this object the DBI
monopole. In this section we will work in the case where $N_{f}=1$; the
situation for generic $N_{f}$ is more interesting still and we will touch on
it briefly later. A similar problem was discussed in Iqbal:2014cga  in a lower
dimensional construction, and we may take over the same ideas. The desired
magnetically charged object turns out to be a wrapped $D5$-brane that ends on
the $D7$ flavour brane. To be more precise, recall from the earlier sections
that the $D7$ flavour brane wraps an $S^{3}\subset S^{5}$:

$d\Omega_{5}^{2}=d\theta^{2}+\cos^{2}{\theta}\;d\psi^{2}+\sin^{2}{\theta}\;d\Omega_{3}^{2}$
(65)

where the $S^{3}$ is spanned by the coordinates $\Omega_{3}$. The $D7$-brane
does not extend in $\theta$: more precisely, for each value of the radial
coordinate $z$, the $D7$-brane sits at a single $\theta(z)$. In the conformal
case, it lives at $\theta_{D7}=\frac{\pi}{2}$ for all $z$, whereas in the non-
conformal case $\theta_{D7}$ interpolates from $\frac{\pi}{2}$ at the UV
boundary to $0$ in the interior.

In contrast, consider a $D5$-brane that wraps this $S^{3}$ and ends on the
$D7$-brane. The $D5$-brane extends in $\theta$ from $\theta=0$ to the
$\theta_{D7}$ coordinate of the $D7$-brane, as shown in Figure 2. It sits at a
particular value of $\psi$; as $\psi$ is a Killing direction this choice is
arbitrary.

Figure 2:  The $D5$-brane wrapping half of the $S^{4}\subset S^{5}$ formed by
$({\theta},\Omega_{3})$, ending on the $D7$-brane which lives at
$\theta=\theta_{D7}$. The remaining 2 coordinates of the $D5$-brane
worldvolume form a two-dimensional string worldsheet in the bulk.

The boundary of the $D5$-brane is a five-dimensional manifold; three of these
dimensions are compact and form the $S^{3}$, and the remaining two dimensions
define a two dimensional manifold ${{\mathcal{M}}}_{2}$ which extends into the
bulk of AdS5. As is well known Strominger:1995ac , the boundary of this
$D5$-brane appears magnetically charged to the DBI gauge field $A_{1}$ living
on the $D7$-brane worldvolume. Hence the wrapped $D5$-brane is the DBI
monopole that we seek.

In the Appendix, we work out the coupling of this brane and show that the
coupling to one such wrapped brane is

$2\pi\ell_{s}^{2}\int_{S^{2}}F_{2}=(2\pi\ell_{s})^{2}$ (66)

where $F_{2}=dA_{1}$ and the $S^{2}$ surrounds ${{\mathcal{M}}}_{2}$ in AdS5.
By comparing this to (64) and (62), we see that the $D1$-brane couples to
${\mathcal{P}}$ with $1/N_{c}$ the charge of the DBI monopole. We may write an
effective coupling to the ${\mathcal{P}}$ field for both of these objects:

$S=q_{D1}\int_{D1}{\mathcal{P}}+q_{D5}\int_{D5}{\mathcal{P}}$ (67)

The overall normalisation of $q_{D1}$ and $q_{D5}$ depends on the (arbitrary)
convention chosen to normalize ${\mathcal{P}}$ in our action, but their ratio
is fixed on topological grounds to be $N_{c}^{-1}$.

$\frac{q_{D1}}{q_{D5}}=\frac{1}{N_{c}}$ (68)

Let us now turn to an understanding of this from the dual field theory. The
intersection of the $D1$ string with the AdS boundary defines a t’Hooft line
in the $SU(N)$ gauge theory sector. Similarly the wrapped $D5$-brane defines a
t’Hooft line for the $U(1)$ gauge theory sector; the simplest way to see this
is to note that when evaluated at the boundary, (66) is precisely the
definition of a t’Hooft line. It has been previously noted (see e.g. Appendix
C of Hofman:2017vwr ) that from the point of view of the $U(1)$ magnetic
1-form current, the charge of a non-Abelian t’Hooft line has $U(1)$ charge of
1/$N_{c}$-th the Dirac monopole, consistent with (68).

Let us now understand the dynamics of symmetry breaking. Consider the wrapped
DBI monopole such that it intersects the AdS boundary on a 1d curve $C$. This
defines the insertion of a line operator into the field theory $\langle
W(C)\rangle$, and as usual from the rules of AdS/CFT we should compute:

$\langle W(C)\rangle\sim\exp\left(-S_{D5}[C]\right)$ (69)

with $S_{D5}[C]$ the on-shell action of the wrapped $D5$-brane. We now seek to
understand the dependence of this answer on the curve $C$; if it depends only
locally on the data of the curve $C$ (e.g. as a perimeter law) then the
symmetry is spontaneously broken. If it depends non-locally – e.g. as an area
law, or more generally in any way that cannot be locally determined by the
curve, then the symmetry is unbroken.

The precise holographic arguments are a higher-form analogue of the arguments
presented in Iqbal:2014cga . Consider first the case where the mass of the
flavour degrees of freedom is zero, i.e. $z_{c}\to\infty$. In this case the
$S^{3}$ factor of the $D7$-brane remains the same size everywhere in the bulk,
i.e. it is independent of $z$. As the brane always hangs down into the bulk,
this defines a minimal area problem, essentially the same as in the usual
studies of Wilson lines from holography Maldacena:1998im . It is clear from
the geometry that the on-shell action will always depend more strongly on the
curve itself than its perimeter. Thus by the previous paragraph, the symmetry
is unbroken. See Figure 3 for a visualisation of this geometry.

Let us now consider the case where the mass of the flavour degrees of freedom
is nonzero. Then there is a value of $z_{c}$ at which the $D7$-brane caps off.
At this value of $z_{c}$ the wrapped $D5$-brane also pinches off and is
allowed to smoothly end. There are now two disconnected possibilities for the
topology of the hanging DBI monopole; it can form topologically a disc, or it
can be topologically a cylinder which hangs straight down and ends where the
brane caps off. For sufficiently large sizes of the curve, the cylinder
solution will dominate. Such topologically non-trivial phase transitions are
common in holography Brandhuber:1998bs ; Rey:1998bq ; Klebanov:2007ws . As the
surface now hangs straight down, the action will depend only on the perimeter
of the curve (multiplied by a constant distance in the holographic direction),
and in this phase the $U(1)$ symmetry is spontaneously broken, as expected.

Finally, one could attempt to generalize the construction of defect operators
to the case $N_{F}>1$; in this case there is presumably an extra quantum
number associated with which of the $N_{F}$ D7 branes the D5 brane ends on. It
seems that a careful study of the braiding algebra of bulk operators should
allow the holographic identification of the mixed symmetry of rank
$\mbox{gcd}(N,N_{F})$ identified by Cherman:2017tey . We leave this for later
study.

Figure 3:  Two distinct topologies that are possible for the DBI monopole. On
the left is the situation when the flavour sector is gapless; the $D7$-brane
then has no boundary, and the DBI monopoles hangs down into the bulk with a
disc topology, whose action depends non-locally on the data describing the
boundary curve. On the right, when the flavour brane ends, the $D5$-brane is
also allowed to end, permitting a cylinder topology. The action of this
configuration depends only on the perimeter of the boundary curve.

##  V Fluctuation spectrum

In the remainder of this paper we study only the $U(N)$ theory, with its
associated 1-form symmetry associated with magnetic flux. We have argued above
that if the flavour degrees of freedom are gapped, then the 1-form symmetry is
spontaneously broken, as can be seen from the fact that the charged line
operator exhibits a perimeter law. On general grounds, we thus expect that
there exists a gapless Goldstone mode in the spectrum Hofman:2018lfz ;
Lake:2018dqm . In this section we will explicitly solve the equations of
motion to show the existence of this Goldstone mode. We first digress slightly
to explain precisely what a Goldstone mode means in this context.

Consider a completely general Lorentz-invariant four-dimensional quantum field
theory with a conserved 2-form current $J^{\mu\nu}$. For simplicity, let us
study the theory in Euclidean signature; as explained in (e.g.) Hofman:2017vwr
, the two-point function of the current in momentum space then takes the
general form

$\langle
J^{\mu\nu}(k)J^{\rho\sigma}(-k)\rangle=\left(-\frac{1}{k^{2}}\left(k^{\mu}k^{\rho}g^{\nu\sigma}-k^{\nu}k^{\rho}g^{\mu\sigma}-k^{\mu}k^{\sigma}g^{\nu\rho}+k^{\nu}k^{\sigma}g^{\mu\rho}\right)+\left(g^{\mu\rho}g^{\nu\sigma}-g^{\mu\sigma}g^{\nu\rho}\right)\right)f_{JJ}\left(\frac{|k|}{m}\right)$
(70)

where here $f_{JJ}$ is a dimensionless function and $m$ is a scale. The
correlation is completely determined by the function $f$. In this context,
spontaneous breaking of the symmetry means that
$f_{JJ}\left(\frac{|k|}{\Lambda}\right)$ approaches a constant as $k\to 0$;
the $k\to 0$ limit then results in a gapless mode from the inverse factors of
$k^{-2}$ arising from the tensor structure.

A simple example is given by pure 4d electrodynamics; here the 1-form symmetry
is broken, and the correlator of the magnetic flux $J=\star F$ takes precisely
this form with

$f_{JJ}(k)=\frac{1}{g^{2}}$ (71)

where $g^{2}$ is the electromagnetic coupling.

An example where the symmetry is not spontaneously broken is given by the
holographic example studied in Hofman:2017vwr . Here the theory in question
was a simple bottom-up holographic realisation of a 1-form symmetry, and the
function $f_{JJ}$ was given by

$f_{JJ}(k)=\frac{1}{g^{2}\log\left(\frac{|k|}{\Lambda}\right)}$ (72)

where $\Lambda$ is a Landau pole, i.e. a UV scale where the theory breaks
down, as described in Hofman:2017vwr . We note here that $f_{JJ}$ vanishes at
$k\to 0$, and the symmetry is not spontaneously broken. A similar result is
found whenever there are electrically charged degrees of freedom present that
are massless.

In this section we will explicitly solve the bulk equations of motion and
compute the function $f_{JJ}$ in our theory, showing that the low-frequency
limit does not vanish. We will then compare it to expectations at weak
coupling.

###  V.1 Goldstone modes and numerics

We will proceed by computing the correlation function of spatial components of
$J^{ij}$ with the (Euclidean) momentum purely in the time direction. Although
the Green’s function of interest is better extracted in the “$2$-form” duality
frame with the fields $\mathcal{P}_{2}$ and $\mathcal{A}_{2}$, it is easier to
solve the equations of motion in the “$1$-form” duality frame consisting of
the fields $\eta_{1}$ and $\tau_{1}$. Our strategy will be to solve the bulk
equations of motion in the $1$-form frame and then exploit a simple
correspondence between the frames at the UV boundary to extract the Green’s
function.

For numerical convenience, we will set $\mu=1$. As explained below (20), we
are working in an illustrative approximation where we capture some aspect of
the backreaction of the flavor degrees of freedom on the color dynamics, while
neglecting gravitational backreaction. The results below do not depend
qualitatively on this choice of $\mu$, but this $\mathcal{O}(1)$ choice allows
us to conveniently find numerical solutions to the equations of motion.

####  V.1.1 1-form

We solve the equations of motion given in (45) by partially fixing the gauge
so that $T_{2}=0$. Next we Fourier transform the fields in the field theory
directions and exploit Lorentz invariance to choose the momentum
$k^{\mu}=(\omega,0)$. This allows us to expand some expressions involving
differential forms in terms of their components as

$\star d\star
d\eta_{1}=\frac{z}{R^{2}}\left\\{\left(z\eta_{i}^{\prime\prime}-\eta_{i}^{\prime}+z\omega^{2}\eta_{i}\right)dx^{i}+i\omega
z(\eta_{t}^{\prime}-i\omega\eta_{z})\;dz+\left[z(\eta_{t}^{\prime\prime}-i\omega\eta_{z}^{\prime})-\eta_{t}^{\prime}+i\omega\eta_{z}\right]\;dt\right\\}$
(73)

and

$H_{3}=\frac{1}{2}B_{ij}^{\prime}\;dx^{i}\wedge dx^{j}\wedge
dz+\left(B_{it}^{\prime}+\frac{1}{2}i\omega B_{ij}dx^{i}\wedge dx^{j}\wedge
dt-i\omega B_{iz}\right)dx^{i}\wedge dt\wedge dz$ (74)

We next note that for a general $2$-form $\Omega_{2}$, we have

$\star(dz\wedge\star\Omega_{2})=-\frac{z^{2}}{R^{2}}(\Omega_{iz}\;dx^{i}+\Omega_{tz}\;dt)$
(75)

Similarly, for a general $3$-form $\Omega_{3}$ we have

$\star
d\star\Omega_{3}=-\frac{z}{R^{2}}\left\\{\frac{1}{2}\left((z\Omega_{ijz})^{\prime}+i\omega
z\Omega_{ijt}\right)dx^{i}\wedge dx^{j}+(z\Omega_{itz})^{\prime}dx^{i}\wedge
dt+i\omega z\Omega_{itz}dx^{i}\wedge dz\right\\}$ (76)

In pure AdS we can also show that

$\partial_{S}\;\epsilon^{MNP}{}_{QR}=\frac{1}{z}\;\delta_{S}^{z}\;\epsilon^{MNP}{}_{QR}$
(77)

which allows us to write

$\star d\star
H_{3}=-\frac{z}{R^{2}}\left\\{\frac{1}{2}((zB_{ij}^{\prime})^{\prime}-\omega^{2}zB_{ij})dx^{i}\wedge
dx^{j}+(zH_{zit})^{\prime}dx^{i}\wedge dt+i\omega zH_{zit}dx^{i}\wedge
dz\right\\}$ (78)

where

$H_{zit}=B_{it}^{\prime}-i\omega B_{iz}$ (79)

Now we can write the equations of motion more explicitly in components. The
$B_{2}$ equation of motion is

$\begin{split}&zh\left[\frac{1}{2}\left(zB_{ij}^{\prime}\right)^{\prime}dx^{i}\wedge
dx^{j}+(zH_{zit})^{\prime}dx^{i}\wedge dt+i\omega zH_{zit}dx^{i}\wedge
dz\right]\\\ &=16\left[(B_{iz}+h^{\prime}\eta_{i})\;dx^{i}\wedge
dz+(B_{tz}+h^{\prime}\eta_{t})dt\wedge dz+\frac{1}{2}B_{ij}\;dx^{i}\wedge
dx^{j}+B_{it}\;dx^{i}\wedge dt\right]\end{split}$ (80)

We are interested in the vector channel, namely the components with a single
spatial index; after imposing the duality relation at the boundary this
contains the information of the transverse channel of the $J^{ij}$ correlation
function.

$\displaystyle zh(zH_{zit})^{\prime}$ $\displaystyle=16B_{it}$ (81a)
$\displaystyle zh(i\omega zH_{zit})$
$\displaystyle=16(B_{iz}+h^{\prime}\eta_{i})$ (81b)

These can be combined to give

$zh\left[\frac{z(B_{it}^{\prime}+i\omega
h^{\prime}\eta_{i})}{16-\omega^{2}z^{2}h}\right]^{\prime}-B_{it}=0$ (82)

where we eliminated $B_{iz}$ using

$B_{iz}=\frac{i\omega
z^{2}hB_{it}^{\prime}-16h^{\prime}\eta_{i}}{16-\omega^{2}z^{2}h}$ (83)

We also have the $\eta_{1}$ equation

$\displaystyle(1-h)\left[\left(z\eta_{i}^{\prime\prime}-\eta_{i}^{\prime}+z\omega^{2}\eta_{i}\right)dx^{i}+i\omega
z(\eta_{t}^{\prime}-i\omega\eta_{z})dz+(z(\eta_{t}^{\prime\prime}-i\omega\eta_{z}^{\prime})-\eta_{t}^{\prime}+i\omega\eta_{z})dt\right]$
(84) $\displaystyle-
zh^{-1}h^{\prime}[(h\eta_{i}^{\prime}+h^{\prime}\eta_{i}+B_{iz})dx^{i}+(h\eta_{t}^{\prime}-i\omega
h\eta_{z}+B_{tz}+h^{\prime}\eta_{t})dt]=0$ (85)

We are most interested in the vector channel equation

$(1-h)\left(\eta_{i}^{\prime\prime}-\frac{1}{z}\eta_{i}^{\prime}+\omega^{2}\eta_{i}\right)-h^{-1}h^{\prime}\left(h\eta_{i}^{\prime}+h^{\prime}\eta_{i}+\frac{i\omega
z^{2}hB_{it}^{\prime}-16h^{\prime}\eta_{i}}{16-\omega^{2}z^{2}h}\right)=0$
(86)

Observe that equation 86 contains no information for $z>z_{c}$, since $h(z)=1$
and $h^{\prime}(z)=0$ in that region. Tracing this back to Eq 45, we conclude
that $\eta_{1}$ simply does not exist for $z>z_{c}$. In this interpretation,
the field $B_{2}$ starts out life in the deep interior of the bulk and evolves
continuously through the $D$-brane cap until it reaches the UV boundary.
However, the field $\eta_{1}$ does not exist on the IR side of the $D$-brane
cap - it begins its life at $z=z_{c}$ and evolves to the UV boundary. As
$\eta_{1}$ started its life as the DBI worldvolume gauge field (which was then
mixed together with other bulk fields to obtain the physical spectrum), it
makes sense that it only exists where the $D$-brane is present.

Figure 4: Brane caps off at $z=z_{c}$ Figure 5: $B_{2}$ (in green) evolves
continuously from the IR to the UV boundary; $\eta_{1}$ (in red) is “born” at
the brane cap and evolves to the UV

However, we now need to understand how to evolve the existing fields through
the transition at $z=z_{c}$. Imposing continuity of the $\eta_{1}$ equation of
motion yields a useful boundary condition. As $z\to z_{c}$ from below we have
$h(z)\to 1$ and $h^{\prime}(z)\to 2\mu/z_{c}$, so we obtain

$dz\wedge\star\left(-d\eta_{1}+B_{2}+T_{2}+\frac{2\mu}{z_{c}}\eta_{1}\wedge
dz\right)=0$

i.e.

$B_{\mu z}+T_{\mu
z}-\partial_{\mu}\eta_{z}+\left(\partial_{z}+\frac{2\mu}{z_{c}}\right)\eta_{\mu}=0$
(87)

With our gauge choice, the relevant boundary condition at the cap is given by

$\eta_{i}^{\prime}=-\left(\frac{i\omega
B_{it}^{\prime}-2\omega^{2}\mu\eta_{i}}{16-\omega^{2}}\right);\quad z=z_{c}\
.$ (88)

Finally, if we expand the dynamical equations of motion in the UV, from the
asymptotic behavior of the fields we can read off the dual conformal
dimensions (using e.g. CasalderreySolana:2011us )

$\displaystyle\Delta_{\eta}$ $\displaystyle=3$ (89a) $\displaystyle\Delta_{b}$
$\displaystyle=2+4\sqrt{1+\mu}$ (89b)

As $\eta$ is dual to the conserved baryon current, its dimension is fixed at
$3$ as expected; $B$ is dual to a massive tensor mode that does not have a
simple universal interpretation.

To solve the equations of motion, we now Wick-rotate to Euclidean signature by
setting $\omega=i\tilde{\omega}$. The equations of motion become

$\displaystyle(1-h)\left(\eta_{i}^{\prime\prime}-\frac{1}{z}\eta_{i}^{\prime}-\tilde{\omega}^{2}\eta_{i}\right)+h^{-1}h^{\prime}\left(-h\eta_{i}^{\prime}-h^{\prime}\eta_{i}+\frac{\tilde{\omega}z^{2}hB_{it}^{\prime}+16h^{\prime}\eta_{i}}{16+\tilde{\omega}^{2}z^{2}h}\right)$
$\displaystyle=0$ (90a) $\displaystyle
zh\left[\frac{z(B_{it}^{\prime}-\tilde{\omega}h^{\prime}\eta_{i})}{16+\tilde{\omega}^{2}z^{2}h}\right]^{\prime}-B_{it}$
$\displaystyle=0$ (90b)

We can also rewrite the equations in terms of a dimensionless holographic
radial coordinate and frequency by defining

$\displaystyle\zeta$ $\displaystyle=z/z_{c}$ (91a) $\displaystyle w$
$\displaystyle=\tilde{\omega}\;z_{c}$ (91b)

Dropping the subscripts $i,t$ and exploiting the fact that
$z\partial_{z}=\zeta\partial_{\zeta}$, we have

$\displaystyle(1-h)\left(\frac{d^{2}\eta}{d\zeta^{2}}-\frac{1}{\zeta}\frac{d\eta}{d\zeta}-w^{2}\eta\right)+h^{-1}\frac{dh}{d\zeta}\left(-h\frac{d\eta}{d\zeta}-\frac{dh}{d\zeta}\eta+\frac{z_{c}w\zeta^{2}h\frac{dB}{d\zeta}+16\frac{dh}{d\zeta}\eta}{16+w^{2}\zeta^{2}h}\right)$
$\displaystyle=0$ (92a) $\displaystyle\zeta
h\frac{d}{d\zeta}\left[\frac{\zeta}{16+w^{2}\zeta^{2}h}\left(z_{c}\frac{dB}{d\zeta}-w\frac{dh}{d\zeta}\eta\right)\right]-z_{c}B$
$\displaystyle=0$ (92b)

Note that instances of $z_{c}$ remain - this is to be expected since it is
precisely the mass scale $m_{\text{meson}}=z_{c}^{-1}$ which breaks conformal
invariance of the dual field theory. However, the factors of $z_{c}$ appear
only when multiplied by $B$. Hence we can define $b=z_{c}B$, so that

$\displaystyle(1-h)\left(\frac{d^{2}\eta}{d\zeta^{2}}-\frac{1}{\zeta}\frac{d\eta}{d\zeta}-w^{2}\eta\right)+h^{-1}\frac{dh}{d\zeta}\left(-h\frac{d\eta}{d\zeta}-\frac{dh}{d\zeta}\eta+\frac{w\zeta^{2}h\frac{db}{d\zeta}+16\frac{dh}{d\zeta}\eta}{16+w^{2}\zeta^{2}h}\right)$
$\displaystyle=0$ (93a) $\displaystyle\zeta
h\frac{d}{d\zeta}\left[\frac{\zeta}{16+w^{2}\zeta^{2}h}\left(\frac{db}{d\zeta}-w\frac{dh}{d\zeta}\eta\right)\right]-b$
$\displaystyle=0$ (93b)

The boundary condition at the cap is given in Euclidean signature by

$\frac{d\eta}{d\zeta}=\frac{w\frac{db}{d\zeta}-2\mu w^{2}\eta}{16+w^{2}}$ (94)

####  V.1.2 2-form

The above set of equations is a closed system that can be conveniently
numerically solved. However we are ultimately interested in studying the
behavior of the system in the $U(N)$ duality frame, in which the physics is
encoded in the fields ${\mathcal{P}}_{2}$ and ${{\mathcal{A}}}_{2}$ rather
than $\eta_{1}$ and $\tau_{1}$. To relate them, we note that in the UV ($z\to
0,\;dh=0$), we can match the fields using (56) to get
$d\mathcal{P}_{2}=\frac{\kappa^{2}\mu}{1+\mu}\star d\eta_{1}$. After a Wick
rotation we can fix some UV scale $z_{\Lambda}$ to get

$\displaystyle\frac{w}{z_{c}}\;\mathcal{P}_{12}(z_{\Lambda})$
$\displaystyle=\alpha\;\frac{\eta_{3}^{\prime}(z_{\Lambda})}{z_{\Lambda}}$
(95a) $\displaystyle z\;\mathcal{P}_{12}^{\prime}(z_{\Lambda})$
$\displaystyle=\alpha\;\frac{w}{z_{c}}\;\eta_{3}(z_{\Lambda})$ (95b)

where $\alpha=\frac{16\mu}{(1+\mu)R}$. As in Hofman:2017vwr , in the UV we
have

$\mathcal{P}_{jk}\sim p_{jk}+J_{jk}\log z,\quad z\to 0$ (96)

We may also directly verify that the leading order asymptotic behaviour of
$\eta_{3}(z)$ is given by

$\eta_{3}(z)\sim\eta_{0}+\eta_{2}z^{2}+\bar{\eta}_{2}z^{2}\log z,\quad z\to 0$
(97)

Hence matching these components at the cutoff we find that

$\displaystyle J_{12}$ $\displaystyle=\alpha\;\frac{w}{z_{c}}\;\eta_{0}$ (98a)
$\displaystyle\frac{w}{z_{c}}(p_{12}+J_{12}\log z_{\Lambda})$
$\displaystyle=\alpha\left(2\eta_{2}+\bar{\eta}_{2}+2\bar{\eta}_{2}\log
z_{\Lambda}\right)$ (98b)

Consistency of the unambiguous coefficients of $\log z_{\Lambda}$ fixes the
coefficient $\bar{\eta}_{2}$ to be

$\bar{\eta}_{2}=\frac{1}{2}\left(\frac{w}{z_{c}}\right)^{2}\eta_{0}$ (99)

We now turn to the interpretation of the logarithm in Eq (96). As explained in
detail in Hofman:2017vwr , this logarithm arises from the fact that the
double-trace coupling $J^{2}$ is marginally irrelevant. This marginal
irrelevance breaks conformality, and more information must be given to specify
the theory. (Indeed, the only conformal theory with a continuous 1-form
symmetry in four dimensions is free Maxwell electrodynamics Hofman:2018lfz ;
Cordova:2018cvg ). This information can be given in the form of the value of
the double-trace coupling $\frac{1}{\theta}J^{2}$ at a particular scale. (Note
that in this strongly coupled model one can now identify $\theta$ with the
gauge coupling of the $U(1)$ sector $g_{1}$ in (11)).

Following the algorithm in Hofman:2017vwr , we can now determine the source
$p_{12}$ by

$p_{12}=\mathcal{P}_{12}(z_{\Lambda})-\frac{J_{12}}{\theta}=2\alpha\;\frac{z_{c}}{w}\;\eta_{2}+J_{12}\log\bar{z}_{*}$
(100)

Here the scale $\bar{z}_{*}$ is given by

$\bar{z}_{*}\equiv e^{1/2}\;z_{*}\equiv e^{1/2}\;z_{\Lambda}\;e^{-1/\theta}$
(101)

The value of this scale should be understood as the Landau pole where the
theory breaks down; as ${\theta}>0$, we note that it is an extremely small
scale, much smaller than the cutoff. Concretely, we can numerically extract
the 2-point function content $f_{JJ}(\omega)$ by solving the equations of
motion for $B_{3t}$ and $\eta_{3}$. See Appendix C for further details of this
method.

####  V.1.3 Results

Here we present a plot of the numerically calculated Green’s function as a
function of $w=\omega z_{c}$ for various values of the dimensionless number
$\gamma\equiv z_{c}/\bar{z}_{*}$, i.e. the meson mass in units of the Landau
pole scale.

Note that at weak coupling the mass gap is given by the bare flavor mass
$m_{F}$. However at strong coupling the mass gap is the mass of the lightest
meson which is given by
$\frac{1}{z_{c}}=\frac{1}{2\pi}\frac{\sqrt{\lambda}}{m_{F}}$, where here
$m_{F}$ should be understood as the coefficient of the relevant mass
deformation in the UV. We have thus chosen to plot the result in units of the
physical meson mass $z_{c}$.

Figure 6: The symmetry current correlator at strong coupling as a function of
$w=\omega z_{c}$ for various masses, computed numerically using holography.
For this plot we set $\bar{z}_{*}=10^{-8}$.

We observe that the asymptotic behaviour is as expected: for small $w$ the
leading order contribution is a constant which depends on $\gamma$. For large
$w$ we expect logarithmic behaviour, but this is difficult to see explicitly
because we cannot numerically access the solution for an exponential range of
values of $w$.

###  V.2 Comparison to weak coupling

Here we will try to compare the functional form of the results above to a
weak-coupling computation. By weak-coupling, we mean that we will take the
non-Abelian t’Hooft coupling $\lambda$ to zero; however we will keep fixed the
Landau pole associated with the $U(1)$ factor. Note that in the $\lambda$ to
zero limit, the $U(1)$ sector of the field theory is effectively super QED
with $N_{f}$ flavours, i.e. a $U(1)$ gauge field $a_{1}$ coupled to $N_{f}$
Dirac fermions and $N_{f}$ complex scalars of mass $m$ with coupling constant
$g_{1}$. Up to a normalisation, the current associated with the $1$-form
global symmetry is $J_{b}=\star f$.

The current-current correlator can be shown to be

$\braket{\tilde{J}_{b}^{\mu\nu}(k)\tilde{J}_{b}^{\rho\sigma}(-k)}=\epsilon^{\mu\nu\alpha\beta}\epsilon^{\rho\sigma\gamma\delta}k_{\alpha}k_{\gamma}\;\tilde{\Delta}_{\beta\delta}(k)$
(102)

where $\tilde{\Delta}_{\mu\nu}(k)$ is the loop-corrected photon propagator in
momentum space. We are interested in the purely spatial components
$\braket{\tilde{J}_{b}^{xy}(k)\tilde{J}_{b}^{xy}(-k)}$.

The contributions to the photon propagator $\Delta_{\mu\nu}$ arise from
resumming scalar and fermion loops as in Figure 7 and Figure 8:

Figure 7: Scalar loop diagram contributing to correction of photon propagator
Figure 8: Fermion loop diagram contributing to correction of photon propagator

This is a textbook calculation - see e.g. srednicki2007quantum  for a
reference which matches our conventions. We use dimensional regularisation in
the $\overline{MS}$ renormalisation scheme and put momentum purely in the time
direction. This allows us to write

$\braket{\tilde{J}_{b}^{xy}(\omega)\tilde{J}_{b}^{xy}(-\omega)}=\left\\{1-\frac{N_{f}\;g_{1}^{2}(\mu)}{4\pi^{2}}\int_{0}^{1/2}dy\;(1-2y^{2})\log\left[\frac{1+(1/4-y^{2})\;\hat{\omega}^{2}}{(\mu/m)^{2}}\right]+\mathcal{O}(g_{1}^{4})\right\\}^{-1}$
(103)

where $\mu$ is an arbitrary mass scale and we define a dimensionless number by
$\hat{\omega}\equiv\omega/m$. The coupling $g_{1}$ runs logarithmically with
the energy scale. Let’s fix the coupling $g_{1}$ at some UV scale
$\mu_{\Lambda}$ to be $g_{R}$. Then the Landau pole scale $\mu^{*}$ at which
the renormalized coupling $g_{1}$ diverges is related to $\mu_{\Lambda}$ by

$\mu^{*}=\mu_{\Lambda}\;e^{1/\chi}$ (104)

where here $\chi$ is given by

$\chi=\frac{5N_{f}\;g_{R}^{2}}{24\pi^{2}}$ (105)

Note that we have combined the fermion contribution of
$N_{f}\;g_{R}^{2}/(6\pi^{2})$ with the scalar contribution of
$N_{f}\;g_{R}^{2}/(24\pi^{2})$. Here $\mu^{*}$ is the physical scale which we
should identify with the holographic Landau pole $z^{*}$ when comparing the
two theories.

This gives an expression for the current-current correlator in terms of the
Landau pole scale and the double-trace coupling as

$\braket{\tilde{J}_{b}^{xy}(\omega)\tilde{J}_{b}^{xy}(-\omega)}^{-1}=\frac{6}{5}\chi\int_{-1/2}^{1/2}dy\;(1-2y^{2})\;\frac{1}{2}\log\left[\frac{1+(1/4-y^{2})\;\hat{\omega}^{2}}{(\mu^{*}/m)^{2}}\right]$
(106)

See Figure 9 for a plot of the correlator at weak coupling. As we can see, the
weak-coupling and strong-coupling plots are extremely similar: they approach a
constant for small $\omega$ in relation to the relevant mass scale (as
dictated by the spontaneous symmetry breaking), and diverge logarithmically
for large $\omega$ (as dictated by the running of the coupling at large
frequencies). It is curious to note that these two properties appear to be
sufficient to control the correlator at all scales, giving the same dynamical
behavior from strongly coupled gravity and from weakly coupled Feynman
diagrams.

Figure 9: The symmetry current correlator at weak coupling as a function of
$\hat{\omega}=\omega/m_{F}$, computed analytically in perturbation theory

Conclusion:  In this work, we have studied the realisation of 1-form
symmetries in perhaps the simplest holographic model in which such a symmetry
could be spontaneously broken; along the way we have clarified some aspects of
the interplay between $0$-form baryon number symmetry and the $1$-form
$\mathbb{Z}_{N}$ symmetry in $SU(N)$ gauge theory. We identified the charged
line operator and verified the expected behavior of the current-current
correlation function, demonstrating the existence of the expected Goldstone
mode. We can identify various directions for future research. It would be very
interesting to extend this study to finite temperatures, where we could expect
to make contact with recent symmetry-based formulations of
magnetohydrodynamics Grozdanov:2016tdf . In a more formal direction, it would
also be very interesting to understand the bulk holographic dual of the
colour-flavour-center symmetry identified in Cherman:2017tey .

Acknowledgements

We thank N. Poovuttikul for discussions. NI and KM are supported in part by
the STFC under consolidated grant ST/L000407/1.

##  Appendix A Conventions and differential form identities

In this work we normally use $M,N$ to refer to 5d bulk indices, $\mu,\nu$ to
refer to 4d field theory bulk indices, and $i,j$ to refer to 3d spatial
indices. $\alpha,\beta$ refer to $D7$-brane worldvolume coordinates and $A,B$
refer to 10d target space indices.

Our our conventions for differential forms are those of carroll2004spacetime ,
and we record some useful identities below:

$\displaystyle d(\omega_{p}\wedge\eta_{q})$
$\displaystyle=d\omega_{p}\wedge\eta_{q}+(-1)^{p}\omega_{p}\wedge d\eta_{q}$
(107) $\displaystyle\omega_{p}\wedge\eta_{q}$
$\displaystyle=(-1)^{pq}\eta_{q}\wedge\omega_{p}$ (108)
$\displaystyle\omega_{p}\wedge\star\eta_{p}$
$\displaystyle=\eta_{p}\wedge\star\omega_{p}$ (109)

The square of the Hodge star acting on a $p$ form in $n$ dimensions on a
metric with $s$ minus signs in its eigenvalues is

$\star^{2}=(-1)^{s+p(n-p)}\ .$ (110)

In particular, in Lorentzian signature in 4d acting on a 2-form, we have
$\star_{4}^{2}=-1$.

To translate between our expressions involving forms and our expressions
involving components, we can use the identity

$A_{p}^{2}=|A_{p}|^{2}\;\epsilon$ (111)

where $\epsilon$ is the volume form associated with the metric determinant $g$

$\epsilon=\star 1=\sqrt{|g|}\;d^{n}x$ (112)

As in Polchinski:1998rr , we define

$|A_{p}|^{2}=\frac{1}{p!}A_{\mu_{1}\ldots\mu_{p}}A^{\mu_{1}\ldots\mu_{p}}$
(113)

and we use the shorthand

$A_{p}^{2}\equiv A_{p}\wedge\star A_{p}$ (114)

The integral of an $n$-form $\Omega$ over an $n$-dimensional manifold
$\mathcal{M}$ of signature $s$ is defined by

$\int_{\mathcal{M}}\Omega\equiv\int_{\mathbb{R}^{n}}(-1)^{s}(\star\Omega)\;\epsilon=\int_{\mathbb{R}^{n}}d^{n}x\;\sqrt{|g|}\;(-1)^{s}(\star\Omega)$
(115)

So in particular, for a $p$-form we have

$\int_{\mathcal{M}}A_{p}^{2}=\int_{\mathbb{R}^{n}}d^{n}x\;\sqrt{|g|}\;|A_{p}|^{2}$
(116)

##  Appendix B Normalisations

To translate between field theory quantities and bulk quantities we use the
holographic dictionary Polchinski:2010hw

$\frac{R^{4}}{l_{s}^{4}}=\lambda\equiv g_{YM}^{2}N_{c}=4\pi g_{s}N_{c}$ (117)

###  B.1 Kinetic Terms

The kinetic terms for $B_{2}$ and $C_{2}$ are of the form

$S_{\text{kin}}=-\int_{\text{AdS}_{5}}{\left(\frac{1}{2}\mathcal{N}_{B}^{2}\;H_{3}^{2}+\frac{1}{2}\mathcal{N}_{C}^{2}\;G_{3}^{2}\right)}$
(118)

and our task is to find the factors $\mathcal{N}_{B}$ and $\mathcal{N}_{C}$.

Consider the type IIB low energy supergravity action in the NS-NS sector444To
match the conventions of Polchinski:1998rr  with ours we have
$\int{d^{10}x\sqrt{-G}\;|F_{p+1}|^{2}}=\int{F_{p+1}^{2}}$. (see e.g.
Polchinski:1998rr ):

$S_{\text{NS}}=\frac{1}{2\kappa_{10}^{2}}\int{d^{10}x\sqrt{-G}\;e^{-2\Phi}\left(R+4\partial_{\mu}\Phi\partial^{\mu}\Phi-\frac{1}{2}|H_{3}|^{2}\right)}$
(119)

where $G$ is the $10$-dimensional metric in the string frame, $\Phi$ is the
dilaton field, $R$ is the Ricci scalar and $\kappa_{10}$ is the gravitational
coupling in $10$ spacetime dimensions given by
$2\kappa_{10}^{2}=(2\pi)^{7}l_{s}^{8}$.

If we choose the dilaton to be constant with $e^{\Phi}=g_{s}$ and choose the
string frame metric to be the usual metric on $\text{AdS}_{5}\times S^{5}$,
the relevant term is

$S_{\text{NS}}=\frac{1}{(2\pi)^{7}l_{s}^{8}g_{s}^{2}}\int_{\text{AdS}_{5}\times
S^{5}}{\left(-\frac{1}{2}H_{3}^{2}\right)}$ (120)

We dimensionally reduce on the $S^{5}$ which yields a factor of
$V_{5}=\pi^{3}R^{5}$

$S_{\text{eff}}=-\frac{R^{5}}{128\pi^{4}l_{s}^{8}g_{s}^{2}}\int_{\text{AdS}_{5}}{\left(\frac{1}{2}H_{3}^{2}\right)}$
(121)

where now the integral is taken only over the $\text{AdS}_{5}$ directions. We
thus conclude that

$\mathcal{N}_{B}^{2}=\frac{R^{5}}{128\pi^{4}g_{s}^{2}l_{s}^{8}}=\frac{N_{c}^{2}}{8\pi^{2}R^{3}}$
(122)

The analysis for the R-R kinetc term is similar. The supergravity action in
the R sector is

$S_{\text{R}}=-\frac{1}{4\kappa_{10}^{2}}\int{d^{10}x\sqrt{-G}\left(|F_{1}|^{2}+|\hat{G}_{3}|^{2}+\frac{1}{2}|\tilde{F}_{5}|^{2}\right)}$
(123)

where the relevant quantity for us is $\hat{G}_{3}\equiv G_{3}-C_{0}\wedge
H_{3}$ and $G_{3}=dC_{2}$ is the R-R field strength. Setting $C_{0}=0$ we have
the term

$S_{\text{R}}=-\frac{R^{5}}{128\pi^{4}l_{s}^{8}}\int_{AdS_{5}}{\left(\frac{1}{2}G_{3}^{2}\right)}$
(124)

after compactifying on the $S^{5}$. By comparing with the original action we
can identify

$\mathcal{N}_{C}^{2}=\frac{R^{5}}{128\pi^{4}l_{s}^{8}}=\frac{\lambda^{2}}{128\pi^{4}R^{3}}$
(125)

as promised.

###  B.2 Chern-Simons Term

Suppose we have a Chern-Simons term in the action of the form

$S_{\text{CS}}=\frac{k}{2\pi}\int{B_{2}\wedge
G_{3}}=\kappa\;\mathcal{N}_{B}\;\mathcal{N}_{C}\int{B_{2}\wedge G_{3}}$ (126)

The coupling to $D1$-branes and $F1$-strings respectively is
$S_{\text{D1}}=\mu_{1}\int{C_{2}}$ and $S_{\text{F1}}=\frac{1}{2\pi
l_{s}^{2}}\int{B_{2}}$, where $\mu_{1}^{-1}=2\pi l_{s}^{2}$ is the tension of
a $D1$-brane. A higher-form Dirac quantisation condition gives

$\frac{\mu_{1}}{2\pi}\int_{S_{3}}F_{3}\in\mathbb{Z}$ (127)

For a magnetic monopole of unit charge we have (in $d=10$)

$dG_{3}+\frac{2\pi}{\mu_{1}}\delta_{4}(W)=0$ (128)

where $W=S^{5}\times L$ is the 6d worldvolume of the $D5$-brane sourcing the
monopole and $L$ is the worldline of the monopole, i.e.a timelike curve in
$\text{AdS}_{5}$.

By taking the wedge with $d\Omega_{5}\wedge\Xi_{1}$ and integrating over all
$10$ dimensions we obtain

$\int_{\text{AdS}}{dG_{3}\wedge\Xi_{1}}+\frac{2\pi}{\mu_{1}}\int_{L}\Xi_{1}=0$
(129)

which allows us to write the gauge variation of the Chern-Simons term as

$\delta S_{\text{CS}}=\frac{k}{\mu_{1}}\int_{L}{\Xi_{1}}$ (130)

This must be cancelled by the gauge variation of $M$ $F_{1}$ strings which end
on the worldline $L$,

$M\delta S_{F1}=\frac{M}{2\pi l_{s}^{2}}\int_{F1}{\delta B_{2}}=\frac{M}{2\pi
l_{s}^{2}}\int_{L}{\Xi_{1}}$ (131)

We identify the integer $M$ with the number of colours in the field theory
$N_{c}$, as in Witten:1998xy .

Hence

$k=\frac{\mu_{1}N_{c}}{2\pi
l_{s}^{2}}=\frac{N_{c}}{4\pi^{2}l_{s}^{4}}=\frac{N_{c}\lambda}{4\pi^{2}R^{4}}$
(132)

This gives

$\kappa=\frac{k}{2\pi\mathcal{N}_{B}\mathcal{N}_{C}}=\frac{4}{R}$ (133)

###  B.3 DBI term

Here we will describe the basic setup to add a $D7$-brane to
$\text{AdS}_{5}\times S^{5}$ by wrapping an $S^{3}$ around the $S^{5}$. We
will follow a similar approach to Karch:2007pd . The final result will be a
contribution to the action of

$-\mathcal{N}_{B}^{2}\int{\left[\frac{1}{2}\kappa^{2}\mu
f(z)\left(B_{2}+\frac{2\pi R^{2}}{\sqrt{\lambda}}F_{2}\right)^{2}\right]}$
(134)

where the factor $\mu$ and the function $f(z)$ will be determined.

The $10d$ string frame metric $G_{AB}$ is given by

$ds^{2}=G_{AB}dX^{A}dX^{B}=\frac{R^{2}}{z^{2}}(-dt^{2}+dz^{2}+dx^{i}dx^{j}\delta_{ij})+R^{2}d\Omega_{5}^{2}$
(135)

Here $i,j\in\\{1,2,3\\}$ are spatial indices and $A,B$ index the coordinates
$z,t,x^{i}$ and all the angles of $S^{5}$. $R$ is the AdS radius and we
parametrise the $5$-sphere as

$d\Omega_{5}^{2}=d\theta^{2}+\cos^{2}{\theta}d\psi^{2}+\sin^{2}{\theta}d\Omega_{3}^{2},$
(136)

where $d\Omega_{3}^{2}$ is the standard metric for a $3$-sphere, the angle
$\psi\in[0,2\pi]$ is azimuthal and the angle $\theta$ takes values in
$[0,\frac{\pi}{2}]$. This coordinate choice is analogous to the so-called Hopf
coordinates on $S^{3}$. For our purposes, these coordinates provide a simpler
way to embed a $3$-sphere inside a $5$-sphere than the usual hyperspherical
coordinates.

We can embed a probe $D7$-brane into the target space by means of the DBI
action:

$S_{\text{DBI}}=-\tau_{7}\int{d^{8}\xi\sqrt{-\det\left(g_{\alpha\beta}+B_{\alpha\beta}+2\pi
l_{s}^{2}F_{\alpha\beta}\right)}}$ (137)

where $\xi^{\alpha}$ are the brane worldvolume coordinates, $g_{\alpha\beta}$
is the induced worldvolume metric on the $D7$-brane, $B_{\alpha\beta}$ are the
components of the NS-NS $2$-form and $F_{2}=dA_{1}$ is the Maxwell field
strength living on the brane.

$\tau_{p}$ is the effective $Dp$-brane tension after absorbing the effect of
the dilaton $e^{\Phi}=g_{s}$ and is given by equation (13.3.23) of
Polchinski:1998rr  as

$\tau_{7}=\frac{1}{g_{s}}\frac{1}{(2\pi)^{7}l_{s}^{8}}$ (138)

The desired brane configurations fills all of AdS and wraps a $3$-sphere
around the $S^{5}$, so the transverse fluctuations will be the $\theta$ and
$\psi$ angles. Since $G_{AB}$ is independent of $\psi$, i.e.$\partial_{\psi}$
is a Killing vector of the target space metric, we will take $\psi$ to be a
constant. Crucially, the $\theta$ angle will be a function of the AdS radial
coordinate: $\theta=\theta(z)$.

The worldvolume metric is the pull-back of the target space metric onto the
worldvolume

$g_{\alpha\beta}=\frac{\partial X^{A}}{\partial\xi^{\alpha}}\frac{\partial
X^{B}}{\partial\xi^{\beta}}G_{AB}$ (139)

If we choose static gauge $\xi^{\alpha}=X^{\alpha}$ then we simply have

$\displaystyle g_{zz}$
$\displaystyle=G_{zz}+\left(\frac{d\theta}{dz}\right)^{2}G_{\theta\theta}=G_{zz}(1+(z\theta^{\prime})^{2})$
(140a) $\displaystyle g_{\alpha\beta}$ $\displaystyle=G_{\alpha\beta},\hskip
200.0pt\alpha\neq z$ (140b)

Let’s write $B_{\alpha\beta}+2\pi
l_{s}^{2}F_{\alpha\beta}\equiv\tilde{B}_{\alpha\beta}$. We can expand the
determinant in the DBI Lagrangian as

$\displaystyle\det(g_{\alpha\beta}+\tilde{B}_{\alpha\beta})$
$\displaystyle=\det(g_{\alpha\gamma}\delta^{\gamma}_{\beta}+g_{\alpha\gamma}g_{\beta\delta}\tilde{B}^{\gamma\delta})$
$\displaystyle=\det(g_{\alpha\gamma})\det(\delta^{\gamma}_{\beta}+g_{\beta\delta}\tilde{B}^{\gamma\delta})$
$\displaystyle=\det(G_{MN})(1+(z\theta^{\prime})^{2})(R^{2}\sin^{2}{\theta})^{3}\det(\delta^{\gamma}_{\beta}+g_{\beta\delta}\tilde{B}^{\gamma\delta})$

For a traceless matrix $A$ we have
$\det(\mathbb{1}+A)=1-\frac{1}{2}\mathop{\rm Tr}(A^{2})+\mathcal{O}(A^{3})$.

Since $g_{\beta\delta}\tilde{B}^{\gamma\delta}$ is traceless, the leading
order behaviour of
$\det(\delta^{\gamma}_{\beta}+g_{\beta\delta}\tilde{B}^{\gamma\delta})$ is
given by

$\det(\delta^{\gamma}_{\beta}+g_{\beta\delta}\tilde{B}^{\gamma\delta})=1+\frac{1}{2}\tilde{B}^{\alpha\beta}\tilde{B}_{\alpha\beta}+\cdots$
(141)

Putting this into the DBI action and performing the integral over the unit
$3$-sphere to obtain a factor of $2\pi^{2}$ gives

$S_{\text{DBI}}=-\frac{1}{g_{s}}\frac{2\pi^{2}R^{3}}{(2\pi)^{7}l_{s}^{8}}\int{\sqrt{-G_{MN}}\;d^{5}x\left[\sqrt{1+(z\theta^{\prime})^{2}}\;\sin^{3}{\theta}\left(1+\frac{1}{4}\tilde{B}^{\alpha\beta}\tilde{B}_{\alpha\beta}\right)\right]}$
(142)

to quadratic order in $\tilde{B}_{\alpha\beta}$.

Let’s turn off the Kalb-Ramond field and the Maxwell field. Now we have an
effective action of the form

$S=\mathcal{N}\int{dzL[\theta(z),\theta^{\prime}(z)]}$ (143)

where

$L=\frac{\sin^{3}{\theta}}{z^{5}}\sqrt{1+(z\theta^{\prime})^{2}}$ (144)

and we absorbed the integral over field theory directions into the overall
normalisation factor $\mathcal{N}$.

Solving the Euler-Lagrange equation gives the single-valued on-shell angle

$\theta(z)=\theta_{c}\equiv\begin{cases}\arccos(z/z_{c})&z\leq z_{c}\\\
0&z>z_{c}\end{cases}$ (145)

Geometrically, this means that the $S^{3}$ wrapping the $S^{5}$ is of maximal
size ($\theta=\pi/2$) at $z=0$ on the boundary, and the $D7$-brane vanishes
($\theta=0$) at the critical value $z=z_{c}$. For $z>z_{c}$ the $D7$-brane has
no effect.

It is straightforward to show that

$\sqrt{1+(z\theta_{c}^{\prime})^{2}}\;\sin^{3}{\theta_{c}}=f(z)$ (146)

where $f$ is the dimensionless scalar function given by

$f(z)=\begin{cases}1-(z/z_{c})^{2}&z\leq z_{c}\\\ 0&z>z_{c}\end{cases}$ (147)

Using the holographic dictionary we find that $2\pi l_{s}^{2}=\frac{2\pi
R^{2}}{\sqrt{\lambda}}$. In general to add $N_{f}$ flavours we simply add
$N_{f}$ probe $D7$-branes. After holographic renormalisation this gives an
overall contribution to the action of

$N_{f}S_{\text{DBI}}=-\frac{N_{c}^{2}}{8\pi^{2}R^{3}}\int{\left[\frac{1}{2}\left(\frac{4}{R}\right)^{2}\frac{N_{f}}{N_{c}}\frac{\lambda}{32\pi^{2}}f(z)\left(B_{2}+\frac{2\pi
R^{2}}{\sqrt{\lambda}}F_{2}\right)^{2}\right]}$ (148)

from which we can read off that

$\mu=\frac{N_{f}}{N_{c}}\frac{\lambda}{32\pi^{2}}$ (149)

as expected.

###  B.4 Couplings of other branes

Here we work out the couplings to various other bulk objects in our
normalisation. We will sometimes make use of the form delta function
$\delta_{{{\mathcal{M}}}_{p}}(x)$. This is a delta function that is nonzero
only if $x$ is on the submanifold ${{\mathcal{M}}}_{p}$; more precisely it is
a $d-p$-form such that $\delta_{{{\mathcal{M}}}_{p}}(x)=0$ if
$x\notin{{\mathcal{M}}}_{p}$ and the integral over any $p$-form $C_{p}$
satisfies:

$\int_{\mathbb{R}^{d}}\delta_{{{\mathcal{M}}}_{p}}\wedge
C_{p}=\int_{{{\mathcal{M}}}_{p}}C_{p}\ .$ (150)

####  B.4.1 Wrapped $D5$-brane: baryon vertex

We recall the action:

$S=-\frac{N_{c}^{2}}{8\pi^{2}R^{3}}\int\left[\frac{1}{2}H_{3}^{2}+\frac{1}{2}\kappa^{2}(B_{2}+T_{2})^{2}\right]$
(151)

where $T_{2}=d\tau_{1}$. Recall from (35) also that $\tau_{1}$ is related to
the original RR and NS 2-forms as

$d\tau_{1}=\frac{\lambda}{4\pi N_{c}{\kappa}}\star dC_{2}-B_{2}$ (152)

We now start with a $D5$-brane wrapped on the $S^{5}$, and would like to
determine its coupling to the field $\tau_{1}$; in other words we add to the
action a term

$q_{\tau}\int_{L}\tau_{1}$ (153)

and would like to determine the coefficient $q_{\tau}$. For simplicity, study
a configuration with $B_{2}=0$; varying the action with respect to $\tau_{1}$
we have

$\frac{N_{c}^{2}{\kappa}^{2}}{8\pi^{2}R^{3}}d\star
d\tau_{1}=q_{\tau}\delta_{L}(x)$ (154)

Integrating both sides of this over a ball with boundary $S^{3}$ that
intersects the worldline $L$, we have

$\frac{N_{c}^{2}{\kappa}^{2}}{8\pi^{2}R^{3}}\int_{S^{3}}\star
d\tau_{1}=q_{\tau}$ (155)

Now use $G_{3}=dC_{2}$ and insert (152) to find that for a minimally charged
$D5$-brane as in (127) we have

$q_{\tau}=\frac{N_{c}}{2\pi l_{s}^{2}}$ (156)

This is $N_{c}$ times the “unit charge” of a single $F$-string in the
appropriate units, as we discuss in the bulk of the text.

####  B.4.2  Wrapped $D5$-brane with boundary: DBI monopole

We now discuss a different bulk object, though also arising from a wrapped
$D5$-brane, here wrapping a half $S^{4}$ and ending on the D7 flavour branes.
The geometry is described around (65). Here we work out the precise charges;
the computation outlined here is a higher-dimensional analogue of the
calculations in Iqbal:2014cga . The relevant parts of the bulk action are

$N_{F}T_{7}\int_{D7}2\pi l_{s}^{2}\;F_{2}\wedge
C_{6}+T_{5}\int_{D5}C_{6}+\cdots$ (157)

We study the case with $N_{f}=1$. We study the configuration where the
$D5$-brane has a boundary $\partial D5$ ending on the $D7$-brane. This
boundary means that the coupling to $C_{6}$ alone is no longer gauge
invariant; indeed if we now do a 5-form gauge transformation of the RR 6-form
$C_{6}$, $C_{6}\to C_{6}+d\Lambda_{5}$, we find that gauge-invariance requires
that

$T_{7}\;2\pi l_{s}^{2}\;dF_{2}=-T_{5}\;\delta_{\partial D5}(x)$ (158)

and thus that if we consider an $S^{2}$ that surrounds $\partial D5$ on the
$D7$-brane worldvolume, we have that

$\int_{S^{2}}F_{2}=2\pi$ (159)

where we have used that $\frac{T_{5}}{T_{7}(2\pi l_{s}^{2})}=2\pi$. Thus the
edge of the wrapped brane couples magnetically to the DBI worldvolume field.
As expected, this is the magnetic flux that saturates the Dirac quantisation
condition, where the conjugate electric charge is viewed as the endpoint of an
F-string ending on the $D7$-brane.

####  B.4.3 $D1$ string

Here we work out the coupling of the $D1$ string to $\tilde{C}_{1}$, which is
the magnetic dual of the RR 2-form $C_{1}$. We begin with the relevant part of
the effective 5d kinetic term for $C_{2}$ from (14), which is

$S=-\frac{\lambda^{2}}{128\pi^{3}R^{3}}\int_{AdS_{5}}{{\frac{1}{2}}}(dC_{2})^{2}+\cdots$
(160)

From here we and the coupling to a $D1$ string used in B.2 we find that the
equation of motion in the presence of a $D1$ source is

$\int_{S^{2}}\star
dC_{2}=\frac{64\pi^{3}}{l_{s}^{2}}\frac{R^{3}}{\lambda^{2}}$ (161)

where the integral is taken over an $S^{2}$ that surrounds the $D1$ brane.
Next, using the relation between $C_{2}$ and $\tilde{C}_{1}$ in (35) in a
configuration where $B=0$, we find that

$\int_{S^{2}}dC_{1}=\frac{16\pi^{2}l_{s}^{2}}{N_{C}R}$ (162)

Restoring the factor of ${\kappa}^{-1}$ this reduces to (64) quoted in the
main text.

##  Appendix C Numerical Solution

Here we give a brief explanation of the numerical procedure used to obtain the
spectral function in Figure 6.

An overview: the equations of motion for $B_{2}$ and $\eta_{1}$ are
numerically solved twice, each time with different boundary conditions. The
solution for $\eta_{1}$ is dualised to a solution for $\mathcal{P}_{2}$ in the
UV. The solutions for $B_{2}$ and $\mathcal{P}_{2}$ are then used to construct
the Green’s function $f_{JJ}$. For concreteness, we work with the components
$B_{3t}$ and $\eta_{3}$.

###  C.1 Equations of motion

The boundary conditions used are the values of the fields $B$ and $\eta$ at
the $D7$-brane cap $\zeta=1$. This then fixes the derivatives of the fields as
follows. The derivative of $B$ is determined by solving the equation of motion
in the range $1<\zeta<\infty$ and the derivative of $\eta$ is determined by
imposing continuity as $\zeta\to 1$ from below.

In the region $1<\zeta<\infty$, the equation of motion for $B$ is considerably
simpler. In fact, it can be reduced to a first-order nonlinear differential
equation for the new field $\Sigma$ defined by

$\Sigma(\zeta)\equiv\frac{1}{B(\zeta)}\frac{dB}{d\zeta}$ (163)

The appropriate asymptotic boundary condition in the IR is constrained by
regularity to be

$\Sigma(\zeta)\sim-w;\quad\zeta\to\infty$ (164)

This is the consequence of the asymptotics of $B$ itself:

$B(\zeta)\sim e^{-w\zeta}\;\quad\zeta\to\infty$ (165)

After solving for $\Sigma$ we can read off the value of the derivative of $B$
at the cap as

$\frac{dB}{d\zeta}(1)=\Sigma(1)B(1)$ (166)

The coupled equations of motion are solved up to some UV-cutoff scale (a
minimum value for $z_{c}$), at which $\eta_{1}$ can be straightforwardly
dualised to $\mathcal{P}_{2}$. $\mathcal{P}_{2}$ corresponds to a $1$-form
global symmetry, so its asymptotic form in the UV is well-understood.

###  C.2 Asymptotic analysis

To extract the data needed for the Green’s functions, we need a careful
understanding of the asymptotic falloffs of various fields. In the 1-form
picture, one finds the following form for the fields:

$B(z\to 0)\sim
z^{-\nu}\left(b_{0,-}+z^{2}b_{2,-}+\cdots\right)+z^{\nu}\left(b_{0,+}+z^{2}b_{2,+}+\cdots\right)$
(167)

Here, by the usual rules of AdS/CFT, $b_{0,-}$ is the source and $b_{0,+}$ is
the response. Similarly, we may expand the field $\eta(z)$ at infinity: we
find

$\eta(z\to 0)\sim\eta_{0}+\eta_{2}z^{2}+\bar{\eta}_{2}z^{2}\log
z+\cdots+z^{4-\nu}\left(\eta_{-,0}+z^{2}\eta_{-,2}+\cdots\right)$ (168)

Here a somewhat unfamiliar role is played by the terms in $z^{4-\nu}$; these
arise from the mixing between the two bulk fields. The coefficients
$\eta_{-,0},\eta_{-,2}$ are all proportional to $b_{0,-}$ and may be
explicitly calculated from the asymptotic analysis of the equations of motion.

Numerically it is more practical to fit the solutions of the equations of
motion to the known asymptotic form using linear regression. This “numerical
holographic renormalisation” allows us to pick out the coefficients we need.
Crucially however, we implemented the numerics using the $\zeta$ coordinate
defined by $\zeta=\frac{z}{z_{c}}$. We can write the above asymptotic
expansions in this coordinate system as

$\displaystyle B(\zeta\to 0)$ $\displaystyle\sim
z_{c}^{-\nu}\;\zeta^{-\nu}\left(b_{0,-}+z_{c}^{2}\;\zeta^{2}\;b_{2,-}+\cdots\right)+z_{c}^{\nu}\;\zeta^{\nu}\left(b_{0,+}+z_{c}^{2}\;\zeta^{2}\;b_{2,+}+\cdots\right)$
(169)
$\displaystyle=\zeta^{-\nu}\left(\hat{b}_{0,-}+\cdots\right)+\zeta^{\nu}\left(\hat{b}_{0,+}+\cdots\right)$

A linear regression in the $\zeta$ coordinate system will thus fit the
coefficients

$\displaystyle\hat{b}_{0,-}$ $\displaystyle\equiv z_{c}^{-\nu}b_{0,-}$ (170a)
$\displaystyle\hat{b}_{0,+}$ $\displaystyle\equiv z_{c}^{\nu}\;b_{0,+}$ (170b)

This scaling can be accounted for, but will anyway cancel out at the end of
our calculation.

However for $\eta$ the presence of the logarithmic term in the asymptotic
expansion produces a more subtle transformation. We have (after holographic
renormalisation)

$\displaystyle\eta(\zeta\to 0)$
$\displaystyle\sim\eta_{0}+\eta_{2}z_{c}^{2}\zeta^{2}+\bar{\eta}_{2}z_{c}^{2}\zeta^{2}(\log
z_{c}+\log\zeta)+\cdots$ (171)
$\displaystyle=\eta_{0}+z_{c}^{2}\left(\eta_{2}+\bar{\eta}_{2}\log
z_{c}\right)\zeta^{2}+\bar{\eta}_{2}\;z_{c}^{2}\;\zeta^{2}\log\zeta+\cdots$
$\displaystyle=\hat{\eta}_{0}+\hat{\eta}_{2}\zeta^{2}+\hat{\bar{\eta}}_{2}\zeta^{2}\log\zeta+\cdots$

Hence for $\eta$ a naive linear regression in the $\zeta$ coordinates will fit
the coefficients

$\displaystyle\hat{\eta}_{0}$ $\displaystyle\equiv\eta_{0}$ (172a)
$\displaystyle\hat{\eta}_{2}$ $\displaystyle\equiv
z_{c}^{2}\left(\eta_{2}+\bar{\eta}_{2}\log z_{c}\right)$ (172b)

$\bar{\eta}_{2}$ is given in terms of $\eta_{0}$ by consistency, so we can
invert these transformations to obtain $\eta_{0}$ and $\eta_{2}$, i.e.the
expansion coefficients in the $z$ coordinate system. These are the physically
useful constituents for computing the Green’s function.

We finally map these coefficients to the source and response in the $2$-form
picture via

$\displaystyle p$
$\displaystyle=2\alpha\;\frac{z_{c}}{w}\;\eta_{2}+J_{12}\log\bar{z}_{*}$
(173a) $\displaystyle J$ $\displaystyle=\alpha\;\frac{w}{z_{c}}\;\eta_{0}$
(173b)

###  C.3 Source-response method

To construct the Green’s function we refer to the source-response picture, in
which the Green’s function is understood as acting on the source to produce a
response. Labelling the fields as $I,J$, the sources as $S_{I}$ and the
responses as $R_{I}$, the components $G_{IJ}$ of the Green’s function are thus
given by

$G_{IJ}S_{J}=R_{I}$ (174)

Hence for example,

$G_{PB}S_{B}+G_{PP}S_{P}=R_{P}$ (175)

To extract $G_{PP}$, we need to obtain two sets of the source and response
data, which we label as $S_{I}^{(1)}$ and $S_{I}^{(2)}$, etc. We thus obtain a
straightforward matrix equation

$\begin{pmatrix}S_{B}^{(1)}&S_{P}^{(1)}\\\
S_{B}^{(2)}&S_{P}^{(2)}\end{pmatrix}\begin{pmatrix}G_{PB}\\\
G_{PP}\end{pmatrix}=\begin{pmatrix}R_{P}^{(1)}\\\ R_{P}^{(2)}\end{pmatrix}$
(176)

which we can trivially invert to find

$G_{PP}=\frac{S_{B}^{(1)}R_{P}^{(2)}-S_{B}^{(2)}R_{P}^{(1)}}{S_{B}^{(1)}S_{P}^{(2)}-S_{B}^{(2)}S_{P}^{(1)}}$
(177)

In our earlier notation, we have $S_{B}=b_{0,-}$, $R_{B}=b_{0,+}$, $S_{P}=p$,
$R_{P}=J$ and $f_{JJ}=G_{PP}$. Hence running the numerical algorithm twice
provides all the data we need to input into equation 177 to compute the
Green’s function of interest.

##  Appendix D Index of symbols

For the convenience of the reader, here we present a roughly alphabetical list
of the symbols in this paper, a brief description, and where it is first
defined. As a rule, the subscript on a form indicates the degree of the form.

  1. 1.

$A_{1}$: the usual DBI worldvolume gauge field living on the flavour brane.
First appears in (14), where $F_{2}=dA_{1}$.

  2. 2.

$\mathcal{A}_{2}$: the magnetic dual of the $1$-form field $\tau_{1}$.

  3. 3.

$B_{2}$: the NS-NS 2-form. First appears in (14).

  4. 4.

$C_{2}$: the R-R 2-form. First appears in (14).

  5. 5.

$\tilde{C}_{1}$: the magnetic dual of the R-R 2-form $C_{2}$. Defined in (35).

  6. 6.

$f(z)$: the function describing how the brane caps off in the bulk. Defined in
(19).

  7. 7.

$f_{JJ}(w)$: the scalar function capturing the dependence of the symmetry
current two-point function on $w$. Defined in (70).

  8. 8.

$F_{2}$: the field strength of $A_{1}$. First appears in (14).

  9. 9.

$\mathcal{F}_{3}$: the field strength of $\mathcal{A}_{2}$. Defined in (54).

  10. 10.

$G_{3}$: the field strength of $C_{2}$. First appears in (14).

  11. 11.

$\tilde{G}_{2}$: the field strength of $\tilde{C}_{1}$. Defined in (35).

  12. 12.

$h(z)$: a function of $f$ first defined in (38)

  13. 13.

$H_{3}$: the field strength of $B_{2}$. First appears in (14).

  14. 14.

$m_{F}$: the fermion mass, i.e.the mass gap at weak coupling. First appears in
(25).

  15. 15.

$m_{\text{meson}}$: the lightest meson mass, $z_{c}^{-1}$, i.e.the mass gap at
strong coupling.

  16. 16.

$N_{c}$: the number of colours of the gauge group.

  17. 17.

$N_{f}$: the number of flavours of fundamental matter; the number of
$D7$-branes in the bulk.

  18. 18.

$\mathcal{P}_{2}$: the magnetic dual of $\eta_{1}$. Defined in (56).

  19. 19.

$\mathcal{Q}_{3}$: the field strength of $\mathcal{P}_{2}$. Defined in (56).

  20. 20.

$R$: the AdS radius. First appears in (20).

  21. 21.

$T_{2}$: the field strength of $\tau_{1}$. Defined in (40b).

  22. 22.

$w$: the dimensionless number $\omega z_{c}$. Defined in (91b).

  23. 23.

$Y_{2}$: the field strength of $\eta_{1}$. First defined in (40a).

  24. 24.

$z$: the radial AdS coordinate.

  25. 25.

$z_{c}$: the value of $z$ where the brane caps off. First appears in (19).

  26. 26.

$\eta_{1}$: a combination of $A_{1}$ and $\tilde{C}_{1}$. Defined in (39a).

  27. 27.

$\kappa$: Factor appearing in (14) equal to $\frac{4}{R}$. Defined in (18a).

  28. 28.

$\mu$: the ratio of mass contributions from $A_{1}$ and $\tilde{C}_{1}$.
Defined in (18b).

  29. 29.

$\tau_{1}$: a combination of $A_{1}$ and $\tilde{C}_{1}$. Defined in (39b).

  30. 30.

$\zeta$: the dimensionless number $z/z_{c}$. Defined in (91a).

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