††institutetext: Department of Mathematical Sciences, Durham University,
Durham, DH1 3LE, United Kingdom

# A Goldstone theorem for continuous non-invertible symmetries

Iñaki García Etxebarria and  Nabil Iqbal<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We study systems with an Adler-Bell-Jackiw anomaly in terms of non-invertible
symmetry. We present a new kind of non-invertible charge defect where a key
role is played by a local current operator localized on the defect. The charge
defects are now labeled by elements of a continuous $U(1)$. We use this
construction to prove an analogue of Goldstone’s theorem for such non-
invertible symmetries. We comment on possible applications to string theory.

##  1 Introduction

Global symmetries and their associated conservation laws are one of the most
fundamental tools that we have for the quantitative understanding of nature.
One of the many uses of conventional global symmetries is to characterize
phases of systems and their low-energy dynamics. To take one example, whenever
a conventional continuous global symmetry is spontaneously broken, Goldstone’s
theorem guarantees that a gapless mode is present in the spectrum, and the
low-energy dynamics is essentially completely characterized by the pattern of
symmetry breaking.

Recently, our understanding of global symmetries has undergone something of a
renaissance. The idea that the existence of conserved quantities can be recast
in terms of the topological surface operators who count the charge has led to
powerful generalizations of the very concept of “symmetry”. Two such
generalizations which will concern us in this paper are those of higher-form
symmetries – i.e. the symmetries associated with the conservation of extended
objects Gaiotto:2014kfa  – and non-invertible symmetries, i.e. symmetries for
which the charges do not obey a simple group composition law Frohlich:2009gb ;
Carqueville:2012dk ; Brunner:2013xna ; Bhardwaj:2017xup ; Gaiotto:2019xmp ;
Heidenreich:2021xpr ; Choi:2021kmx ; Kaidi:2021xfk ; Roumpedakis:2022aik ;
Bhardwaj:2022yxj ; Arias-Tamargo:2022nlf ; Choi:2022zal ; Choi:2022jqy ;
Cordova:2022ieu ; Kaidi:2022uux ; Antinucci:2022eat ; Bashmakov:2022jtl ;
Damia:2022rxw ; Damia:2022bcd ; Bhardwaj:2022lsg ; Lin:2022xod ;
Bartsch:2022mpm ; Apruzzi:2022rei ; GarciaEtxebarria:2022vzq ; Lu:2022ver ;
Heckman:2022muc ; Niro:2022ctq ; Kaidi:2022cpf ; Mekareeya:2022spm ;
Antinucci:2022vyk ; Giaccari:2022xgs ; Bashmakov:2022uek ; Cordova:2022fhg .
See e.g. McGreevy:2022oyu  for a recent pedagogical review of some of these
developments.

In particular, consider a system (such as conventional QED with a single
massless Dirac fermion) where a current $j^{A}$ is nonconserved due to an
Adler-Bell-Jackiw anomaly Adler:1969gk ; Bell:1969ts , i.e.

$d\star j^{A}=\frac{1}{4\pi^{2}}F\wedge F$ (1)

where the operator $F\wedge F$ on the right-hand side is constructed from a
dynamical $U(1)$ photon. It has recently been shown that though the naive
symmetry is explicitly broken by the anomaly, this is not a “generic” kind of
breaking; instead the system can be understood in terms of a novel kind of
non-invertible symmetry Choi:2022jqy ; Cordova:2022ieu . This precise
characterization opens new doors for a non-perturbative understanding of
systems exhibiting such an anomaly.

In this work, we present a variation of this construction that permits us to
prove a Goldstone theorem for such systems; i.e. we prove using Euclidean
partition function techniques that if an operator ${{\mathcal{O}}}$ charged
under the non-invertible symmetry has a non-zero vacuum expectation value,
there must exist a gapless mode in the spectrum. As an application, we show
how many of the massless fields of string theory can be understood as
Goldstone modes of spontaneously broken non-invertible symmetries; this
provides an alternative viewpoint for their masslessness.

Note added: In the last stages of preparation of this paper, Karasik:2022kkq
appeared, where a charge defect similar to our (11) (involving an extra scalar
field on the defect) was constructed for different motivations.

##  2 Goldstone’s theorem for non-invertible symmetries

###  2.1 Goldstone’s theorem in Euclidean formulation

To orient ourselves, we begin by reviewing a simple reformulation of the usual
Goldstone theorem in the language of Euclidean path integrals. This argument
was first given in Hofman:2018lfz  to prove a Goldstone theorem for higher-
form symmetries.

Consider a Lorentz invariant quantum field theory with a $U(1)$ 0-form
symmetry, with associated 1-form conserved current $j$. For simplicity we will
restrict ourselves to the case of four spacetime dimensions. The Ward identity
for the conserved current $j$ in the presence of a charged operator
${{\mathcal{O}}}(x)$ with charge $q$ is

$d\star j(x){{\mathcal{O}}}(0)=iq{{\mathcal{O}}}(0)\delta^{(4)}(x)$ (2)

i.e. the current is not quite conserved in the presence of the charged
operator. Let us now integrate both sides of this equation over a solid 4-ball
of radius $R$ centered at the origin, as in Figure 1. We find the equation

$\left(\int_{S^{3}(R)}\star j\right){{\mathcal{O}}}(0)=iq{{\mathcal{O}}}(0)$
(3)

where on the left hand side we have used Stokes theorem; the integral is taken
over the boundary of the 4-ball. Finally, take the expectation value of both
sides:

$\bigg{\langle}\left(\int_{S^{3}(R)}\star
j\right){{\mathcal{O}}}(0)\bigg{\rangle}=iq\langle{{\mathcal{O}}}\rangle$ (4)

Figure 1:  Charge operator defined on an $S^{3}$ of radius $R$ wrapping
operator ${{\mathcal{O}}}(0)$ at the origin.

Now, if we are in a phase where the symmetry is spontaneously broken, then
$\langle{{\mathcal{O}}}\rangle$ is nonzero, and the integral on the left-hand
side must be both nonzero and independent of the radius of the 3-sphere $R$.
By spherical symmetry, we see that the correlation of the local operator $j$
on the 3-sphere and ${{\mathcal{O}}}$ must therefore depend on $R$ as

$\langle j^{i}(x){{\mathcal{O}}}(0)\rangle\sim iqn^{i}R^{-3}$ (5)

where $n^{i}$ is an outwardly pointing normal vector on the 3-sphere. The
dependence on $R$ is fixed by the requirement that the integral over the
3-sphere result in an $R$-independent constant. Thus there is a power-law
correlation in the theory.111This argument is of course exactly the same as
the one used to obtain the inverse square electric field of a point charge
using Gauss’s law in elementary electrodynamics. We have shown the existence
of at least one gapless excitation: this is the Goldstone mode.

We note that this is simply a reformulation of the usual Hamiltonian
arguments; if we appropriately deform the $S^{3}$ and cut open the path
integral then we obtain the commutators of $j$ and ${{\mathcal{O}}}$ that are
used in the standard proofs. This Euclidean formulation will be useful for the
generalization that follows.

###  2.2 Axial symmetry defect operators

We now turn to the actual case of interest. In recent work Choi:2022jqy ;
Cordova:2022ieu , it has been shown that models exhibiting the Adler-Bell-
Jackiw anomaly actually are invariant under a novel kind of non-invertible
symmetry. We briefly review some aspects of that discussion here.

Consider massless QED, defined by the action:

$S[\psi,{\bar{\psi}},A]=\int
d^{4}x\left(\frac{1}{4e^{2}}F^{2}+i{\bar{\psi}}\left(\not{\partial}-i\not{A}\right)\psi+\cdots\right)$
(6)

The $U(1)$ gauge redundancy acts as

$\psi\to e^{i\Lambda(x)}\psi\qquad A\to A+d\Lambda\ .$ (7)

In this theory the axial current
$j_{A}^{\mu}={\bar{\psi}}\gamma^{5}\gamma^{\mu}\psi$ is not conserved.
Instead, due to the ABJ anomaly it satisfies the following non-conservation
equation Adler:1969gk ; Bell:1969ts :

$d\star j^{A}=\frac{1}{4\pi^{2}}F\wedge F$ (8)

We stress that the right-hand-side of this expression is a dynamical operator,
and not a fixed external source (as would be the case for a ’t Hooft anomaly).
There will thus be dynamical violation of axial charge conservation, and we
cannot construct a conserved charge in the conventional manner. There is a
temptation to instead consider the following current:

$\star j^{\mathrm{not-gauge-inv}}_{A}=\star j_{A}-\frac{1}{4\pi^{2}}A\wedge
dA\ .$ (9)

This current is conserved, but as the notation suggests it is not gauge-
invariant. Indeed it is not possible to construct a conserved gauge-invariant
local current in this theory. However as the integral of the Chern-Simons term
is gauge-invariant, one may try to construct a charge defect operator as
follows:

$\begin{split}\hat{U}_{{\alpha}}({{\mathcal{M}}}_{3})=\exp\left(i\frac{\alpha}{2}\int_{{{\mathcal{M}}}_{3}}\left(\star
j_{A}-\frac{1}{4\pi^{2}}A\wedge dA\right)\right)\end{split}$ (10)

This is a topological operator under small deformations of
${{\mathcal{M}}}_{3}$, and thus is a candidate operator to define a conserved
axial charge defect, which acts on an operator ${{\mathcal{O}}}$ with integer
charge $q$ as ${{\mathcal{O}}}\to e^{i\alpha q/2}{{\mathcal{O}}}$.

However for general ${{\mathcal{M}}}_{3}$ – in particular those with
nontrivial 1-cycles – the expression (10) is not gauge-invariant under large
gauge transformations. Such an invariance would require the coefficient of the
Chern-Simons term to be quantized as $\alpha\in 2\pi\mathbb{Z}$, thus
resulting in a trivial action on all operators. In Choi:2022jqy ;
Cordova:2022ieu  this deficiency was remedied by adding extra degrees of
freedom – a TQFT – living on ${{\mathcal{M}}}_{3}$. Their construction then
permits the topological charge defect operator to be defined on any
${{\mathcal{M}}}_{3}$, provided that the rotation angle $\alpha$ is in
$\mathbb{Q}/\mathbb{Z}$. Thus rather than having a continuous $U(1)$ symmetry
as in the conventional case the defect operators are now labeled by
$\mathbb{Q}/\mathbb{Z}$.

In this work however, we seek to generalize the Goldstone theorem described
above, and it is important that we have some notion of a local conserved
current. Thus we perform a slightly different construction. Consider the
following charge defect operator, defined on $3$-manifolds $\Sigma$:

$U_{{\alpha}}({{\mathcal{M}}}_{3})=\int[\mathcal{D}\theta]\exp\left(i\frac{{\alpha}}{2}\int_{{{\mathcal{M}}}_{3}}\star\tilde{j}_{A}\right)\equiv\int[\mathcal{D}\theta]\exp\left(i\frac{{\alpha}}{2}\int_{{{\mathcal{M}}}_{3}}\left(\star
j_{A}-\frac{1}{4\pi^{2}}(A-d\theta)\wedge dA\right)\right)$ (11)

Here we have introduced a new degree of freedom; this is a compact scalar
$\theta$ that lives only on the 3-manifold ${{\mathcal{M}}}_{3}$, and which
transforms under the gauge redundancy (7) as ${\theta}\to{\theta}+\Lambda$. In
the above expression we perform a path integral over ${\theta}$. Thus the
integrand $\star\tilde{j}_{A}$ is now locally gauge invariant, and there is no
rationality constraint on the parameter ${\alpha}$, so $e^{i{\alpha}}\in
U(1)$, and we have constructed a continuous $U(1)$ symmetry. We will show
below it is non-invertible.

It should be noted that ${\theta}$ really has almost no dynamics; we will
restrict attention only to the case where ${{\mathcal{M}}}_{3}$ is $S^{3}$,
and in that case this construction can be viewed as a convenient way to
extract the gauge-invariant information that is present in the more naive
construction (10).

We now discuss some properties of the defect operator
$U_{{\alpha}}({{\mathcal{M}}}_{3})$.

  1. 1.

It is topological under small variations of ${{\mathcal{M}}}_{3}$. This can be
seen heuristically by imagining some extension of ${\theta}$ off of the
defect, and then noting that $d\star\tilde{j}^{A}=0$ for any such extension.
In Appendix A we provide a somewhat more detailed explanation of this fact.

  2. 2.

As usual for a 0-form defect operator, when the defect operator is collapsed
onto an appropriately charged local operator it performs an axial rotation by
an angle $\alpha$, i.e. if ${{\mathcal{O}}}(x)$ carries axial charge $q$ then

$U_{{\alpha}}(S^{3}){{\mathcal{O}}}(x)=e^{\frac{i\alpha
q}{2}}{{\mathcal{O}}}(x)$ (12)

if the $S^{3}$ in question is a small sphere wrapping $x$.

  3. 3.

We may immediately imagine interesting phenomena whenever ${\theta}$ is
allowed to have non-trivial winding, i.e. when (a) ${{\mathcal{M}}}_{3}$ is a
manifold with non-trivial 1-cycles, or (b) in the presence of ’t Hooft lines.
We discuss some such properties – which do not affect our Goldstone proof – in
a subsequent section.

###  2.3 Goldstone’s theorem for non-invertible symmetry

We may now use this object to prove a variant of Goldstone’s theorem by
generalizing slightly the arguments used in Section 2.1. We begin by
considering the setup of (12), where $U_{{\alpha}}$ is defined on an $S^{3}$
of radius $R$. Now take a derivative with respect to ${\alpha}$ and set
${\alpha}$ to $0$: we then find

$\bigg{\langle}i\left(\int_{S^{3}(R)}\star\tilde{j}_{A}\right){{\mathcal{O}}}(0)\bigg{\rangle}=iq\langle{{\mathcal{O}}}\rangle$
(13)

Note that the expectation value on the left-hand side now involves the path
integral over ${\theta}$ defined on the defect as well. $\tilde{j}_{A}$ is now
defined only on the defect.

Now imagine that we are in a phase where $\langle{{\mathcal{O}}}\rangle\neq
0$, i.e. an operator charged under the non-invertible symmetry has a non-
vanishing expectation value. Then, just as in (5), by spherical symmetry we
see that the correlation function must depend on $R$ in a very specific way:

$\langle\tilde{j}^{i}_{A}(R){{\mathcal{O}}}(0)\rangle\sim iqn^{i}R^{-3}$ (14)

with $n^{i}$ an outwards pointing normal vector on the $S^{3}$. We thus see
that there is a non-trivial power-law correlation between an operator
${{\mathcal{O}}}$ in the bulk and a current $\tilde{j}_{A}$ defined on the
defect. Thus there must exist at least one massless mode in the bulk, which is
the Goldstone mode.

In this formulation the proof is essentially the same as in the conventional
invertible symmetry case, except that the current $\tilde{j}_{A}$ is now
defined only on the defect. Note that there is no way for the new degree of
freedom ${\theta}$ living only on the defect to create such a correlation
between bulk and defect operators unless the bulk is gapless.222It might be
helpful to note that the expression above coincides with the _connected_
correlator, as the expectation value of the defect-localized current _without_
a local operator inserted is $\langle\tilde{j}_{A}\rangle=0$, by the same
argument as above with $q=0$. Thus the connected correlator displays power-law
behavior, requiring a gapless mode:
$\langle\tilde{j}^{A}(R){{\mathcal{O}}}(0)\rangle-\langle\tilde{j}^{A}(R)\rangle\langle{{\mathcal{O}}}\rangle\sim
iqR^{-3}\,.$ (15) This completes the proof.

The effective theory describing this low energy Goldstone mode (usually called
an axion) is well understood. Denoting the axion by $\phi$, the first few
terms in the derivative expansion are:

$S[A,\phi]=\int_{\mathbb{R}^{4}}\left(\frac{1}{2{{\gamma}}^{2}}d\phi\wedge\star
d\phi+\frac{1}{2e^{2}}F\wedge\star F+ig\phi F\wedge F\right)$ (16)

It was shown in Choi:2022jqy  that this theory does indeed exhibit the non-
invertible symmetry if $g=\frac{1}{4\pi^{2}}$. It is easy to verify that the
$\phi$ field saturates the Ward identity (14), where the charged operator
${{\mathcal{O}}}$ is realized as ${{\mathcal{O}}}\sim e^{iq\phi}$.

An example of a microscopic theory which realizes this phase is if one adds a
complex scalar $\Phi(x)$ and a Yukawa coupling $h\Phi(x){\bar{\psi}}\psi+h.c.$
to the Dirac action (6), and then arranges a potential $V(\Phi^{\dagger}\Phi)$
to condense the scalar $\Phi$; $\phi(x)$ may then be viewed as the phase of
$\Phi(x)$ (or equivalently the phase of the fermion condensate
${\bar{\psi}}\psi$).

Another example (discussed in Choi:2022jqy ) is QCD with massless quarks,
where the (non-invertible) axial symmetry is spontaneously broken, and indeed
the anomaly famously provides the dominant channel for pion decay to two
photons. We now see that the $\pi_{0}$ can be viewed as a Goldstone boson of
this breaking; the effective Lagrangian takes the form above with $\phi$
identified with the $\pi_{0}$ (and with other interactions involving the
photon $A$).

We make two statements about generalizations of this Goldstone theorem:

  1. 1.

It may appear that the construction above requires access to a weakly coupled
photon description of the theory. However the essential information of the ABJ
anomaly can be phrased in a universal way333See e.g. Das:2022auy  where a
similar characterization was used to understand finite-temperature physics
from holography. in terms of the combined symmetry structure of a conserved
2-form current $J$ and the non-conserved axial current $j^{A}$:

$d\star j^{A}=\frac{1}{4\pi^{2}}J\wedge J\qquad d\star J=0$ (17)

We believe the argument can be extended to any theory with this structure. The
second equation implies that we can locally write $J=\frac{1}{2\pi}\star dA$
in terms of an effective photon $A$. $A$ will not have a simple action, but we
can nervertheless use it to construct the operator (11) on a topologically
trivial $S^{3}$ and run the argument above.

  2. 2.

Similar arguments can be made for a higher-form continuous non-invertible
symmetry (e.g. as in Damia:2022rxw ; Damia:2022bcd ) by changing the
dimensionality of the defect manifold ${{\mathcal{M}}}_{3}$ and of the charged
operator ${{\mathcal{O}}}$ (which will generically become an extended object),
as was done in Lake:2018dqm ; Hofman:2018lfz  for (invertible) higher-form
symmetries. The statement then is that if the charged object exhibits a
perimeter law (i.e. its expectation value depends only locally on geometric
data characterising its worldvolume), then there exists a gapless mode in the
spectrum.

###  2.4 Behaviour in the non-zero flux sector

For the proof above we were only required to construct the defect operator
$U_{{\alpha}}$ on an $S^{3}$ embedded inside $\mathbb{R}^{4}$. In the interest
of completeness, we would like to highlight some properties of this defect on
more general manifolds, particularly those with non-trivial homology groups.

Let us consider formulating our bulk theory on $S^{1}\times S^{1}\times
S^{2}$. We are interested in studying a sector of the path integral where
there is $U(1)$ electromagnetic flux $2\pi n$ on the $S^{2}$, i.e.

$\int_{S^{2}}F=2\pi n\,$ (18)

where $n\in\mathbb{Z}$.

Now let us place the defect operator $U_{{\alpha}}({{\mathcal{M}}}_{3})$ on an
${{\mathcal{M}}}_{3}=S^{1}\times S^{2}$, where it fills one of the two
$S^{1}$’s, i.e. we evaluate

$\begin{split}\langle
U_{{\alpha}}({{\mathcal{M}}}_{3})\rangle=\bigg{\langle}\exp\left(i\frac{{\alpha}}{2}\int_{{{\mathcal{M}}}_{3}}\left(\star
j^{A}-\frac{1}{4\pi^{2}}(A-d\theta)\wedge dA\right)\right)\,.\end{split}$ (19)

We consider performing the path integral over $\theta$ first, keeping $A$
fixed. Note that the dependence on ${\theta}$ is only through its
characteristic class (i.e. the winding), and any other ${\theta}$ in the same
class can be obtained by adding a topologically trivial ${\theta}$ to a fixed
representative of its characteristic class. We have

$\begin{split}\int_{\mathcal{M}_{3}}d\theta\wedge dA=4\pi^{2}wn\end{split}$
(20)

with $w\in\mathbb{Z}$ the winding of $\theta$ on the $S^{1}$ factor. Large
gauge transformations act as

$\begin{split}G:(a,\lambda)\sim(a+n,w+n)\end{split}$ (21)

where $a$ is the holonomy of $A$ on the $S^{1}$. In this calculation we treat
$A$ as a background and we do not impose these identifications when summing
over $w$; in a more complete treatment they will be taken into account when
performing the full path integral over $A$ (where they reduce the effective
range of the holonomy).

The result of the path integral over $\theta$, including the winding sectors,
is therefore proportional to

$\begin{split}\sum_{w\in\mathbb{Z}}\exp\left(i\frac{{\alpha}wn}{2}\right)\,.\end{split}$
(22)

Performing the sum over $w$, we find a delta function in ${\alpha}n$ that only
has support if

$\begin{split}\frac{{\alpha}n}{2}\in 2\pi\mathbb{Z}\ .\end{split}$ (23)

We thus see that for irrational $\frac{\alpha}{2\pi}$ our operator annihilates
the path integral unless $n=0$. This is a very non-invertible operator indeed.
It would be interesting to further understand the properties of this defect
and compare to the operator of Choi:2022jqy ; Cordova:2022ieu , which is
defined for rational $\frac{\alpha}{2\pi}$.

##  3 Goldstone bosons and string theory

One of our original motivations for seeking an extension of Goldstone’s
theorem that applied to non-invertible symmetries was the observation that in
string theory there are fields which appear to be exactly massless, even in
non-supersymmetric configurations (as in Alvarez-Gaume:1986ghj ; Dixon:1986iz
; Sagnotti:1995ga ; Sagnotti:1996qj ; Sugimoto:1999tx , for instance). The
masslessness of some of these fields is usually explained from the fact that
they are connections for higher-form abelian gauge symmetries.

This argument is somewhat unsatisfactory on its own; a gauge redundancy is
after all a statement about a description of a system. We should instead seek
an explanation in terms of realizations of (possibly approximate) global
symmetries. Given our lack of understanding of the non-perturbative aspects of
string theory, a systematic argument based on symmetry principles alone, valid
for arbitrary compactifications, seems to be of some value.444A different
argument for the masslessness of the photon, based on swampland
considerations, was given in Reece:2018zvv .

Consider, as an example, the operator measuring the D4 “Page” charge
Page:1983mke 555See Marolf:2000cb  for other notions of charge.,666The
integrand $\tilde{F}_{4}+A_{1}\wedge H_{3}$ can alternatively be written as
$dA_{3}$, but the way we have written it makes it clearer that it is not gauge
invariant under gauge transformations of $A_{1}$.

$\begin{split}\hat{U}_{\alpha}(\Sigma_{4})=\exp\left(2\pi
i\alpha\int_{\Sigma_{4}}\tilde{F}_{4}+A_{1}\wedge H_{3}\right)\,,\end{split}$
(24)

where $\tilde{F}_{4}\equiv dA_{3}-A_{1}\wedge H_{3}$, $A_{1}$ and $A_{3}$ are
RR potentials, and $H\equiv dB_{2}$ is the NSNS field strength. Here
$\Sigma_{4}$ is a 4-surface linking a 5-manifold $X_{5}$ where we have placed
a D4 brane. We include no other background fields, and for avoidance of
tadpoles we take the directions transverse to the D4 to be non-compact.

Importantly, we will first treat the D4 brane as an infinitely heavy object,
so that we may consider it as a defect operator $D(X_{5})$ in the effective
supergravity field theory. We will eventually incorporate the fact that the
tension is finite below.

In this setting – when the D4 branes are infinitely massive –
$\hat{U}_{\alpha}(\Sigma_{4})$ is topological, and would naively define a
5-form $U(1)$ symmetry. This is not so for a number of reasons. First, for
generic $\alpha$ the resulting expression is not gauge invariant under large
gauge transformations of $A_{1}$ if $H^{1}(\Sigma_{4};\mathbb{R})\neq 0$, and
$H_{3}$ generic. We can fix these issues by taking $\Sigma_{4}$ to be $S^{4}$.
We also note that the background we discuss does not source $H_{3}$, so we
might be tempted to simply claim that $U_{\alpha}(\Sigma_{4})$ is gauge
invariant. But our goal is to argue that the presence of Goldstone bosons is
robust, so we will not set $H_{3}=0$. Instead, we introduce an additional
dynamical scalar on the charge defect, as above:

$\begin{split}U_{\alpha}(\Sigma_{4})=\exp\left(2\pi
i\alpha\int_{\Sigma_{4}}\tilde{F}_{4}+(A_{1}-d\theta)\wedge
H_{3}\right)\,,\end{split}$ (25)

with $\theta$ transforming as $\theta\to\theta+\lambda$ under a $A_{1}\to
A_{1}+d\lambda$ gauge transformation. This defines a non-invertible symmetry
analogous to (11).

The D4 brane is charged under $U_{\alpha}(\Sigma_{4})$, so we may write:

$\begin{split}U_{\alpha}(\Sigma_{4})D(X_{5})=e^{i\alpha}D(X_{5})\end{split}$
(26)

Now the insertion of the defect operator $D(X_{5})$, i.e. a D4 wrapping
$X_{5}$, in the string theory partition function will give a contribution
proportional to the volume of $X_{5}$, arising from the Dirac-Born-Infeld
coupling on the brane worldvolume. In terms of standard language, it obeys a
“perimeter law”, so this nonzero expectation value implies a form of
spontaneous symmetry breaking. We can, in particular, run a straightforward
generalisation of the argument given above (see Lake:2018dqm ; Hofman:2018lfz
for details) that implies the existence of a Goldstone mode. The minimal
possibility saturating (26) is that we have a 5-form $C_{5}$ in the spectrum,
with a coupling

$\begin{split}\exp\left(i\int_{X_{5}}C_{5}\right)\end{split}$ (27)

to the D4 brane worldvolume. We identify this massless field as the ordinary
RR 5-form field.

Clearly, this argument generalises with minor modifications to all branes in
string theory, implying the exact masslessness of the fields that they are
charged under.

So far we have worked in a semi-classical approximation to string theory,
where the branes are infinitely massive objects. In actual string theory, this
is not the case: branes are dynamical objects, and this means that there are
no longer any exact global symmetries or completely topological objects
Arkani-Hamed:2006emk ; Harlow:2018tng ; Banks:2010zn . Nevertheless it seems
reasonable that a large mass for these objects should not affect questions in
the extreme infrared, such as the masslessness of putative Goldstone modes. An
argument to this effect for 1-form symmetries was given in Iqbal:2021rkn . In
our context, the main physical effect is expected to be screening from
“virtual” D4-branes. This makes the value of the measured Page charge depend
on the the distance $R$ between the surface where we measure the charge and
the brane, i.e. (26) becomes

$\begin{split}U_{\alpha}(\Sigma_{4})D(X_{5})=e^{i\alpha
f(R)}D(X_{5})\end{split}$ (28)

for some nontrivial function of $R$. We are not aware of a systematic
treatment of this effect for the case at hand, but related cases are analysed
in detail in Cordova:2022rer . Extrapolating the main features of the cases
discussed in that paper, we expect that the operator we have described above
will be topological — i.e. $f(R\to\infty)$ approaches a constant — up to
exponentially suppressed effects, for scales larger than the inverse of the
brane tension.

This is good news: any small mass for the would be Goldstone would
dramatically affect our argument for large enough radii (the relevant
correlator would become exponentially suppressed), but it is precisely in this
regime that the screening effect itself becomes exponentially suppressed. So
we conclude that screening effects, while they do break the symmetry, cannot
give a mass to the Goldstone boson. This is perhaps not much of a surprise:
giving mass to a Goldstone boson for a higher form symmetry, since it is
vector-like, is a discontinuous operation, so these Goldstone modes are much
more robust than ordinary 0-form Goldstone scalars (which have the same
degrees of freedom whether they are massive or not).

A summary of what we have just found is that the massless form fields in
string theory are massless because branes are heavy. This motivates a final
speculative comment: if we try to think what it would take to gap the spectrum
of supergravity, by the arguments above it will necessarily involve making the
branes tensionless. This resonates well with the idea that gapped phases of
gravity should be thought of as phases where the metric vanishes. (We refer
the reader to Witten:1988ze ; Witten:1988sy  and the final comments in
McGreevy:2016myw  for further references and discussion of such ideas.) Indeed
recent work on emergent higher-form symmetries associated with the the
gravitational sector alone Hinterbichler:2022agn ; Benedetti:2021lxj  may
eventually allow us to make such ideas concrete.

###### Acknowledgements.

We thank Mohamed Anber, Ben Heidenreich, Arpit Das, Avner Karasik, Miguel
Montero, Napat Poovuttikul, and Tin Sulejmanpasic for discussions. Both
authors are supported by the STFC consolidated grant ST/T000708/1, and I.G.E
is additionally supported by the Simons Foundation via the Simons
Collaboration on Global Categorical Symmetries.

##  Appendix A Topological invariance of defect operator

Here we discuss the topological properties of the new coupling added to the
action of the defect operator in (11), i.e.

$\exp\left(i\frac{{\alpha}}{8\pi^{2}}\int_{{{\mathcal{M}}}_{3}}d\theta\wedge
F\right)$ (29)

In particular, when placed in the path integral, is this invariant under
deformations of ${{\mathcal{M}}}_{3}$?

Heuristically, if we extend ${\theta}$ off of the defect, then we can consider
deforming ${{\mathcal{M}}}_{3}$ to a nearby surface
${{\mathcal{M}}}_{3}^{\prime}$; comparing the integral over the two regions we
then find we then find

$\int_{{{\mathcal{M}}}_{3}}-\int_{{{\mathcal{M}}}_{3}^{\prime}}\left(d\theta\wedge
F\right)=\int_{{{\mathcal{M}}}_{4}}d(d\theta\wedge F)=0$ (30)

where
$\partial{{\mathcal{M}}}_{4}={{\mathcal{M}}}_{3}\cup{{\mathcal{M}}}_{3}^{\prime}$.

We can present a slightly more abstract formulation of the argument which
shows the relationship between the above relation and the 1-form symmetry.
Consider coupling an external 2-form source $b$ using the 1-form symmetry of
QED:

$S[\psi,{\bar{\psi}},A;b]=\int
d^{4}x\left(\frac{1}{4e^{2}}F^{2}+i{\bar{\psi}}\left(\not{\partial}-i\not{A}\right)\psi+\frac{i}{2\pi}b\wedge
F\right)$ (31)

(Such a coupling is possible for any conserved 2-form current $J$, where in
this particular case we have $J\equiv\frac{1}{2\pi}F$). As $b$ couples to a
conserved 2-form current, it is always true that

$Z[b]=Z[b+d\Lambda]$ (32)

for an arbitrary 1-form $\Lambda$.

Now we turn on the following source for $b$, parametrized by a choice of
3-surface ${{\mathcal{M}}}_{3}$:

$b=b_{{{\mathcal{M}}}_{3}}=\frac{\alpha}{4\pi}d\left(\Theta_{{{\mathcal{M}}}_{3}}(x)d\theta(x)\right)$
(33)

Here ${\theta}(x)$ is a scalar field defined on $\mathbb{R}^{4}$ which will
eventually be restricted to the defect worldvolume ${{\mathcal{M}}}_{3}$.
$\Theta_{{{\mathcal{M}}}_{3}}(x)$ is a scalar function with
$\Theta_{{{\mathcal{M}}}_{3}}(x)=0$ if $x$ is “on one side” of
${{\mathcal{M}}}_{3}$ and $\Theta_{{{\mathcal{M}}}_{3}}(x)=1$ if $x$ is on the
“other side” of ${{\mathcal{M}}}_{3}$. (E.g. if we were studying the simple
case where ${{\mathcal{M}}}_{3}$ is the flat plane $x_{0}=0$, we would have
$\Theta_{{{\mathcal{M}}}_{3}}(x)=\Theta(x_{0})$ with $\Theta$ the usual
Heaviside step function.) To make the manipulations well-defined, let us
imagine that this interpolation from $0$ to $1$ takes place over a small
length scale.

Now as the choice for $b$ in (33) is exact, the invariance (32) means that the
partition function does not change under inserting this source. In particular
it does not care about the choice of ${{\mathcal{M}}}_{3}$, independently of
the value of ${\theta}$, i.e.

$Z[b_{{{\mathcal{M}}}_{3}}]=Z[b_{{{\mathcal{M}}}_{3}^{\prime}}]$ (34)

This invariance also does not care about details on how the Heaviside function
above is regulated. However in the limit where the step function is made
arbitrarily sharp, we find that $d\Theta_{{{\mathcal{M}}}_{3}}$ becomes a
1-form localized on the defect, and we have:

$\frac{i}{2\pi}\int_{\mathbb{R}^{4}}b_{{{\mathcal{M}}}_{3}}\wedge
F=\frac{i\alpha}{8\pi^{2}}\int_{{{\mathcal{M}}}_{3}}d\theta\wedge F$ (35)

Thus we have constructed the coupling desired and confirmed that it is
invariant under shifts of ${{\mathcal{M}}}_{3}$. At the final stage we see
that it depends only on ${\theta}(x)$ on the defect. This argument is
essentially the same as that leading to (30), but this presentation highlights
the relationship between the topological character of the operator and the
unbroken 1-form symmetry.

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