Symmetries and topological operators, on average

Andrea Antinucci, Giovanni Galati, Giovanni Rizi and Marco Serone

SISSA, Via Bonomea 265, I-34136 Trieste, Italy

INFN, Sezione di Trieste, Via Valerio 2, I-34127 Trieste, Italy

###### Abstract

We study Ward identities and selection rules for local correlators in
disordered theories where a 0-form global symmetry of a QFT is explicitly
broken by a random coupling $h$ but it re-emerges after quenched average. We
consider $h$ space-dependent or constant. In both cases we construct the
symmetry operator implementing the group action, topological after average. In
the first case, relevant in statistical systems with random impurities, such
symmetries can be coupled to external backgrounds and can be gauged, like
ordinary symmetries in QFTs. We also determine exotic selection rules arising
when symmetries emerge after average in the IR, explaining the origin of
LogCFTs from symmetry considerations. In the second case, relevant in AdS/CFT
to describe the dual boundary theory of certain bulk gravitational theories,
the charge operator is not purely codimension-1, it can be defined only on
homologically trivial cycles and on connected spaces. Selection rules for
average correlators exist, yet such symmetries cannot be coupled to background
gauge fields in ordinary ways and cannot be gauged. When the space is
disconnected, in each connected component charge violation occurs, as expected
from Euclidean wormholes in the bulk theory. Our findings show the obstruction
to interpret symmetries emergent after average as gauged in the bulk.

###### Contents

  1. 1 Introduction
  2. 2 Symmetries in quenched disorder
    1. 2.1 Pure theories and explicit symmetry breaking
    2. 2.2 Quenched disorder and Ward identities
    3. 2.3 Topological operators for disordered symmetries
    4. 2.4 ’t Hooft anomalies for continuous disordered symmetries
    5. 2.5 Discrete disordered symmetries: ’t Hooft anomalies and gauging
  3. 3 Disordered symmetries and the replica trick
    1. 3.1 The replica trick
    2. 3.2 Disordered symmetries from replica theory
  4. 4 Disordered emergent symmetries and LogCFTs
    1. 4.1 Emergent disordered symmetries
    2. 4.2 LogCFTs
  5. 5 Symmetries in ensemble average
    1. 5.1 Factorization and spurions
    2. 5.2 Ensemble average and Ward identities
    3. 5.3 A gravity discussion
  6. 6 Conclusions
  7. A Toy model examples
    1. A.1 Free scalar theories
      1. A.1.1 Quenched disorder
      2. A.1.2 Ensemble Average
    2. A.2 ’t Hooft anomalies from replicas
  8. B Symmetry operators for averaged symmetries

##  1 Introduction

Global symmetries constitute an indispensable tool for studying physical
systems, especially when the dynamics cannot be analyzed using exact
techniques. The idea of symmetry is sometimes vaguely stated and often
confused with slightly different concepts, such as _selection rules_. While
these two ideas are often connected, they are logically distinct. Moreover,
symmetry must be distinguished from gauge invariance, which is a redundancy of
a certain formalism rather than an actual symmetry of the physical system.

For these reasons, the idea of symmetry has recently been made more precise
and intrinsic through the notion of _topological operators_ [1]. These are
extended operators or defects in Quantum Field Theory (QFT) supported on
submanifolds in space-time. Their dependence is purely topological: small
deformations of the support do not change the correlation functions, but when
they pass a charged operator, it undergoes a symmetry transformation. The
topological nature implies that, in any quantization scheme, if they are
placed on a space-like slice, they become operators on the Hilbert space
commuting with the Hamiltonian, recovering the standard notion of symmetry in
quantum systems [2]. In Euclidean QFTs, which we will focus on, we do not
distinguish between operators and defects, so we will refer to such operators
generically as topological operators.

In finite volume, when a QFT has invertible topological operators implementing
a group of symmetries, they imply selection rules on correlation functions,
and the same remains true in infinite volume if the symmetry is not
spontaneously broken. More importantly, the formalization of symmetry in terms
of topological operators allows several interesting generalizations. These
include higher-form symmetries (starting from [1, 3]), higher groups
(introduced in the physics literature in [4, 5, 6]), non-invertible symmetries
in two-dimensions (see e.g. [7, 8]) and more recently also in higher
dimensions after [9, 10], leading to the notion of higher-categorical
symmetries [11, 12]. These developments are based on the notion of topological
operators introduced in [1] and allowed the extension of the concept of ’t
Hooft anomalies to discrete and generalized symmetries and new constraints on
the dynamics of strongly coupled systems (see for instance [13, 14, 15, 16]).

The aim of this work is to extend the formalization of symmetries of [1] to
QFTs where the interactions are randomly distributed, for the case of 0-form
global symmetries. We believe that a more systematic treatment of symmetries
in QFTs of this kind can be useful, given the notorious difficulties in
treating such systems. There are two relevant possibilities considered here.

  1. 1.

The random couplings $h(x)$ vary in space and are distributed according to a
probability functional $P[h]$.

  2. 2.

The random couplings $h$ are constant and drawn from a probability density
$P(h)$.

Scenario 1. is relevant for statistical mechanical systems with impurities or
disorder (for a review see [17]). There are two main variants of disorder QFT:
quenched if the impurities are treated as external random sources and annealed
if the impurities are taken dynamical. Physically the two situations depend on
the time scale we are looking at. At extremely long time-scales, where the
entire system reaches equilibrium, we should take the impurities dynamical.
Since impurities have very long thermalizations time scales, quenching is
useful for time-scales where the system essentially thermalizes, with the
impurities taken fixed. In the quenched case, the properties of the QFT will
of course depend on the impurities. If we assume that impurities are random,
possible observables are taken by averaging over the impurities with the
chosen distribution. In a lattice formulation an impurity is modelled by an
interaction which is different at any site, and its presence is unpredictable.
In the continuum limit it is often the case that we can describe such systems
as the average over an ensemble of field theories where the coupling constants
are space dependent. Particularly interesting is the case of the Ising model
perturbed with a random magnetic field (dubbed as random field Ising model)
[18] or with a random interaction between nearby spins (dubbed as random bond
Ising model) [19]. See e.g. [20, 21, 22, 23] for recent works on these models.

Scenario 2. is relevant for quantum gravity and has received significant
attention lately. The connection between averaging and euclidean gravity path
integrals dates back to [24, 25] in association to Euclidean wormholes. In the
context of the AdS/CFT correspondence [26, 27, 28], the connection has been
invoked in [29] as a possible way to interpret from a boundary point of view
the origin of interactions between disconnected components of a boundary
theory induced by bulk Euclidean wormholes (factorization puzzle). Further
elaborations with concrete examples appear in [30]. Ensemble averaging
features also in the Sachdev-Ye-Kitaev (SYK) model [31, 32, 33]. A concrete
connection has recently been made in [34], where it has been shown that the
sum over geometries in Jackiw-Teitelboim gravity [35, 36] with $n$
disconnected boundaries is dual to the ensemble average of an $n$-point
correlation function in a matrix model. Other notable examples of ensemble
averaging after [34] include averages over free compact bosons in 2d [37, 38]
(see also [39, 40, 41, 42] for related studies and generalizations), averages
over OPE coefficients in effective 2d CFTs [43, 44], averages over the gauge
coupling in 4d ${\cal N}=4$ super Yang-Mills theory [45].

In both scenarios 1. and 2. we focus on correlation functions of local
operators with _quenched disorder_ averaging. These include averages of
products of correlators, which are effectively independent observables. In
disconnected spaces, when $h$ is constant, also averaged single correlators
can lead to averages of products of correlators, which is the mechanism
leading to the factorization puzzle in the context of AdS/CFT. In order to
distinguish scenario 1. from scenario 2. we dub the first as “quenched
disorder” and the latter as “ensemble average”, but it should be kept in mind
that quenching is involved also in scenario 2.

We start from a _pure theory_ , that is an ordinary QFT with no disorder, and
deform it with a certain local interaction. In the quenched disorder case the
strength of this interaction varies from point to point, while it is constant
in ensemble average. In both cases the interaction can break part of all of
the global symmetries of the pure system, so that each specific realization
generically has less symmetries and less predictive power than the pure
theory. On the other hand it has been noticed in several examples that
symmetries of the pure systems can be recovered after the average on the
coupling is taken into account. These statements are mostly based on the
observation that the averaged system satisfies selection rules which are not
enjoyed by the generic specific realization. Intuitively speaking, even if the
random coupling breaks the symmetry, this re-emerges provided we average over
all the ensemble in a sufficiently symmetric way (in a sense to be clarified).
For simplicity, in what follows we refer to such symmetries as _disordered
symmetries_ and _averaged symmetries_ respectively in the context of quenched
disorder and ensemble average. Note that this is distinct from the notion of
emergent symmetries used in pure QFTs when a symmetry is approximately
conserved in the IR. In the disorder case the symmetry is exact at all energy
scales, but only on average.111We can also have emergent symmetries in both
senses, namely emerging after average and in the IR. We will discuss this case
in section 4.

We will review these kind of arguments from a spurionic point of view at the
beginning of section 2.2 for quenched disorder, and in section 5.1 for
ensemble average, deriving under which condition the selection rules of the
pure theory are satisfied after the average.

This is still an imprecise information since, as we emphasized, having a
global symmetry is stronger than just observing the validity of some selection
rule. This is crucial in order to get stronger dynamical constraints implied
by ’t Hooft anomalies, and eventually gauging the symmetry. Our goal is to
clarify the sense in which these systems recover the symmetry, aiming to
construct the analog of topological operators for both quenched disorder and
ensemble average QFTs.

Sections 2, 3 and 4 focus on quenched disordered systems. We consider theories
defined in the continuum and admitting a description in terms of an action
(Hamiltonian) obtained from that of the pure theory $S_{0}$ by adding a local
operator $\mathcal{O}_{0}(x)$ with a space-time dependent coupling $h(x)$:

$S[h]=S_{0}+\int d^{d}x\,h(x)\mathcal{O}_{0}(x)\ .$ (1.1)

This is what we will call a specific realization. Correlation functions of
local operators ${\mathcal{O}_{i}}$ for a given value of $h(x)$ are computed
by a path integral:

$\big{\langle}\mathcal{O}_{1}(x_{1})\cdots\mathcal{O}_{k}(x_{k})\big{\rangle}=\frac{\int\mathcal{D}\mu\,e^{-S[h]}\,\mathcal{O}_{1}(x_{1})\cdots\mathcal{O}_{k}(x_{k})}{\int\mathcal{D}\mu\,e^{-S[h]}}\
.$ (1.2)

Given a probability functional $P[h]$, a set of observables of the disordered
system are the averaged correlation functions

$\overline{\big{\langle}\mathcal{O}_{1}(x_{1})\cdots\mathcal{O}_{k}(x_{k})\big{\rangle}}=\int\mathcal{D}h\,P[h]\,\big{\langle}\mathcal{O}_{1}(x_{1})\cdots\mathcal{O}_{k}(x_{k})\big{\rangle}\,,$
(1.3)

or more generally the averages of products of correlators

$\overline{\prod_{j=1}^{N}\Big{\langle}\mathcal{O}_{1}^{(j)}(x_{1}^{(j)})\cdots\mathcal{O}_{n_{j}}^{(j)}(x_{k_{j}}^{(j)})\Big{\rangle}}\
.$ (1.4)

The starting point for systematizing global symmetries of the disordered
system which are not enjoyed by the specific realizations is to derive Ward
identities for the averaged correlators. We do this in section 2.2, starting
from the simplest case of continuous 0-form invertible symmetries. The Noether
current $J^{\mu}$ associated to the symmetry of the pure theory is no longer
conserved in the specific realizations if $\mathcal{O}_{0}(x)$ is charged
under the symmetry. However we find that the _shifted current_ in (2.32) leads
to standard Ward identities (2.36) for averaged single correlators and the
less standard identities (2.40) for averages of products of correlators.

In order to generalize our results to discrete symmetries, where a Noether
current is unavailable, in section 2.3 we construct the symmetry operators,
topological on average, which implements the finite group action. This is not
as easy as in pure theories because of the disorder. The topological operator
$\widetilde{U}_{g}$ is a complicated power series of integrated currents which
however can be resummed to give the simple expression

$\widetilde{U}_{g}=U_{g}\langle U_{g}\rangle^{-1}\,,$ (1.5)

where $U_{g}$ is the topological operator of the pure theory. Its action on
averages of simple correlators is given in (2.53). In products of correlators
(1.4) the operator $\widetilde{U}_{g}$ is topological on average only if
inserted in all the correlators involved, as in (2.56). This characterizes
intrinsically the disordered symmetries and implies somewhat exotic selection
rules which are weaker with respect to symmetries not broken by the random
interactions.

The Ward identities satisfied by $\widetilde{U}_{g}$, when the latter is
supported on a compact surface $\Sigma^{(d-1)}$ are valid regardless of how
the symmetry is realized on the vacuum. When the symmetry operator is well
defined also on infinite surfaces the disordered symmetry is not spontaneously
broken and implies selection rules. The same is not true for spontaneously
broken symmetries, we will briefly discuss this situation in the final section
6.

Beyond selection rules, our analysis allows us to show that disordered
symmetries (both continuous and discrete) can be coupled to external
backgrounds, can be gauged, and can have ’t Hooft anomalies, precisely like
ordinary symmetries. We also argue that a symmetry of a pure system with a ’t
Hooft anomaly, when it reappears as disordered symmetry, enjoys the same ’t
Hooft anomaly thus implying the same constraints on the dynamics, and that a
possible higher-group structure of the underlying 0-form symmetry with higher-
form symmetries of the pure theory is recovered after average due to the
topological nature of the higher-group structure. Symmetry Protected
Topological (SPT) phases [46], protected by what we denoted disordered
symmetries, appeared already in condensed matter, see e.g. [47, 48, 49, 50,
51, 52, 53, 54]. Our findings can possibly provide a different theoretical
QFT-based framework for such phases of matter.

In section 3 the above results, derived directly from the disordered theory,
are reproduced using the replica trick, the standard way to deal with theories
of this kind. Disordered symmetries manifest themselves as standard symmetries
in the replica theory, thus offering a conceptual different viewpoint on these
kind of symmetries. Aside from providing a sanity check of the results, the
replica theory allows us to also study another scenario: disordered symmetries
emerging at long distances, discussed in section 4. The effect of the disorder
can now lead to the more exotic selection rules (4.11) and (4.12). The
phenomenon manifests in the replica theory as two irreducible representations
of the replica symmetry transforming in different representations of the
emergent disordered symmetry. As an interesting application of this result we
consider the prime example of an emergent symmetry, conformal invariance, and
we show that as a consequence of these exotic selection rules, a quenched
disordered system can flow in the IR to a fixed point described by a
Logarithmic conformal field theory (LogCFT) [55, 56, 57, 58, 59].

We analyze ensemble average in section 5. While the intuitive idea that the
average restores the symmetry is still true, and a spurionic argument (section
5.1) implies selection rules, the status of the averaged symmetry is
drastically different. A hint already comes from the replica trick: when
applied with constant couplings, the replica theory is non-local, and even if
the symmetry is manifest its Noether current is not a local operator. This is
problematic for constructing a topological operator. Indeed, independently of
the replica trick, we imitate the analysis done for disordered theories, and
we get the exotic topological charge operator (5.14). This is not really a co-
dimension one operator, since it depends both on a closed surface
$\Sigma^{(d-1)}$ and on a filling region $D^{(d)}$ such that $\partial
D^{(d)}=\Sigma^{(d-1)}$. In particular the operator cannot be supported on
homologically non-trivial cycles. Crucially, the operator $\widehat{Q}$
implies selection rules, because the second term in (5.14), when inserted on
average of correlation functions of local operators, vanishes when integrated
over the full space. If the space manifold is connected, there are only two
possible filling regions of a homologically trivial $\Sigma^{(d-1)}$, and
$\widehat{Q}$ is independent of the choice. On the other hand on a
disconnected space there are several choices of filling region $D^{(d)}$, and
the charge operator does depend on these choices. Nevertheless, we do have
selection rules for averages of correlators, if one takes into account all the
connected components of space, and we can construct operators (B.20)
implementing the finite group action. In each connected component the
selection rules can be violated, allowing the charges to escape from one
component to the other. We have then the somewhat exotic situation of a 0-form
symmetry in the sense of selection rules on correlation functions of local
operators, but without having genuine topological operators (even after
average). In contrast to ordinary symmetries and disordered symmetries in the
quenched disordered case above, averaged symmetries cannot be coupled to
background gauge fields in ordinary ways and hence cannot be gauged.

In section 5.3 we comment about the gravity interpretation of these results.
Whenever the average theory admits a gravitational bulk dual, the local charge
violation in presence of disconnected space has the natural interpretation in
the bulk as charge violation induced by Euclidean wormholes configurations, as
pointed out in [60, 61, 62]. The difficulty (impossibility) of gauging
averaged boundary symmetries that we have found clarify why such symmetries
cannot be identified with bulk gauge symmetries.

We conclude in section 6. In appendix A we work out some specific examples for
concreteness, and in appendix B we explicitly construct the operator which
implements the action of the group for averaged symmetries.

##  2 Symmetries in quenched disorder

In this section we study global 0-form symmetries in quenched disorder
theories which arise only after the average. We start in section 2.1 by
reviewing how Ward identities for ordinary 0-form symmetries are recasted in
terms of topological operators in pure QFTs. We generalize the analysis to
theories with quenched disorder in section 2.2 and construct the topological
operator implementing the symmetry group action in section 2.3. We then
discuss ’t Hooft anomalies and gaugings for both continuous and discrete
disordered symmetries in sections 2.4 and 2.5.

###  2.1 Pure theories and explicit symmetry breaking

Consider a standard $d$-dimensional Euclidean QFT described by the action
$S_{0}$. If this theory is invariant under some continuous symmetry group $G$,
correlation functions of local operators must satisfy the usual constraints
imposed by the Ward-Takahashi identities:

$i\big{\langle}\partial_{\mu}J^{\mu}(x){\cal O}_{1}(x_{1})\ldots{\cal
O}_{k}(x_{k})\big{\rangle}=\sum_{l=1}^{k}\delta^{(d)}(x-x_{l})\big{\langle}{\cal
O}_{1}(x_{1})\ldots\delta{\cal O}_{l}(x_{l})\ldots{\cal
O}_{k}(x_{k})\big{\rangle}\,.$ (2.1)

Here $J_{\mu}(x)$ is the Noether current222For convenience we define the
Noether current as $\delta S=i\int\\!\epsilon(x)\partial^{\mu}J_{\mu}$. Notice
that this has an extra factor of $i$ with respect to the one obtained by Wick
rotating the standard Minkowski current. and $\delta\mathcal{O}_{l}(x_{l})$ is
the transformation of the local operator $\mathcal{O}_{l}$ under the action of
the Lie algebra of $G$. For instance if $G=U(1)$ and $\mathcal{O}_{l}$ has
charge $q_{l}$, then $\delta\mathcal{O}_{l}=iq_{l}\mathcal{O}_{l}$.
Integrating over the full space $X^{(d)}$, the left hand side of (2.1)
vanishes if $X^{(d)}$ has no boundary and the symmetry is not spontaneously
broken, and we get selection rules on the correlators.

The modern approach [1] to interpret the same constraints consists in
associating global symmetries to co-dimension one _topological operators_
$U_{g}[\Sigma^{(d-1)}]$, $g\in G$, namely extended operators supported on some
$(d-1)$-dimensional closed surface $\Sigma^{(d-1)},$ which are invariant under
continuous deformations of their support. In the case of continuous symmetries
such topological operators are simply333In the following we suppress the group
and algebra indices. In (2.2) the element $g\in G$ is the exponential of
$\alpha$ valued in the dual of the Lie algebra of $G$.

$U_{g}[\Sigma^{(d-1)}]=e^{i\alpha Q[\Sigma^{(d-1)}]}\,,$ (2.2)

where $Q[\Sigma^{(d-1)}]=\int_{\Sigma^{(d-1)}}J_{\mu}n^{\mu}$ is the Noether
operator which measures the charge enclosed within the region $D^{(d)}$
delimited by $\Sigma^{(d-1)}$ with $\partial D^{(d)}=\Sigma^{(d-1)}$. We can
then write integrated Ward identities. For instance, if $G=U(1)$ we have

$\Big{\langle}Q[\Sigma^{(d-1)}]{\cal O}_{1}(x_{1})\ldots{\cal
O}_{k}(x_{k})\Big{\rangle}=\chi(\Sigma^{(d-1)})\big{\langle}{\cal
O}_{1}(x_{1})\ldots{\cal O}_{k}(x_{k})\big{\rangle}\,,$ (2.3)

with

$\chi(\Sigma^{(d-1)})=\sum_{l,x_{l}\in D^{(d)}}q_{l}\,.$ (2.4)

The integrated Ward identity can be iterated using the fact that $J^{\mu}(x)$
is uncharged with respect to $Q[\Sigma^{(d-1)}]$,444This is not true for non-
abelian $G$. However with simple manipulations one can reach the same
conclusion. Here we focus on the abelian case just for notational simplicity.
resulting in

$\Big{\langle}Q^{n}[\Sigma^{(d-1)}]{\cal O}_{1}(x_{1})\ldots{\cal
O}_{k}(x_{k})\Big{\rangle}=\chi^{n}(\Sigma^{(d-1)})\big{\langle}{\cal
O}_{1}(x_{1})\ldots{\cal O}_{k}(x_{k})\big{\rangle}\ .$ (2.5)

This implies that the exponentiated operators (2.2) satisfy

$\Big{\langle}U_{g}[\Sigma^{(d-1)}]{\cal O}_{1}(x_{1})\ldots{\cal
O}_{k}(x_{k})\Big{\rangle}=e^{i\alpha\chi(\Sigma^{(d-1)})}\big{\langle}{\cal
O}_{1}(x_{1})\ldots{\cal O}_{k}(x_{k})\big{\rangle}\ ,\ \ \ g=e^{i\alpha}\ .$
(2.6)

More generally the integrated Ward identities associated to a finite
transformation $g\in G$ can be written as

$\Big{\langle}U_{g}[\Sigma^{(d-1)}]{\cal O}_{1}(x_{1})\ldots{\cal
O}_{k}(x_{k})\Big{\rangle}=\Big{\langle}{\cal
O^{\prime}}_{1}(x_{1})U_{g}[\Sigma^{\prime}_{d-1}]\mathcal{O}_{2}(x_{2})\ldots{\cal
O}_{k}(x_{k})\Big{\rangle}\,,$ (2.7)

where $\mathcal{O}^{\prime}_{1}(x_{1})=(R_{1}(g)\cdot\mathcal{O}_{1})(x_{1})$
is the transformed operator according to its representation $R_{1}$ under $G$,
$\Sigma^{(d-1)}$ is a surface linking with the point $x_{1}$ and
$\Sigma^{\prime(d-1)}$ is its deformation across the point. The selection
rules on correlation functions now follow from the fact that a topological
operator $U_{g}[\Sigma^{(d-1)}]$ supported on a very big surface at infinity
is trivial, but shrinking it to a point, $U_{g}$ passes and transforms all the
local operators. We then get

$\begin{split}\langle{\cal O}_{1}(x_{1})\ldots{\cal
O}_{n}(x_{n})\rangle=R_{1}(g)\ldots R_{n}(g)\cdot\langle{\cal
O}_{1}(x_{1})\ldots{\cal O}_{n}(x_{n})\rangle\,,\end{split}$ (2.8)

which is the desired selection rule. A correlation function of local operators
can be non-vanishing only if the direct product of representations contains
the singlet representation. While $Q[\Sigma^{(d-1)}]$ and
$U_{g}[\Sigma^{(d-1)}]$ enforce equivalent constraints on the theory, the
advantage of using the exponentiated operator $U_{g}[\Sigma^{(d-1)}]$ is that
in (2.7) we do not need to define the infinitesimal transformation
$\delta\mathcal{O}$ so that the generalization to discrete symmetries is
straightforward.

If we add a deformation of the pure theory which explicitly breaks $G$, the
Ward identities (2.1) acquire a new term and, as expected, the operator
$Q[\Sigma^{(d-1)}]$ (or equivalently $U_{g}[\Sigma^{(d-1)}]$) is no longer
topological. For example, for $G=U(1)$ and a deformation described by the
action (the term $h\mathcal{O}_{0}(x)$ is always paired with its hermitian
conjugate, which we leave implicit)

$S=S_{0}+h\int\\!d^{d}x\,\mathcal{O}_{0}(x)\,,$ (2.9)

where $\mathcal{O}_{0}(x)$ is a local operator with charge $q_{0}$ under
$U(1)$ and $h$ is a coupling, we get

$\begin{split}i\big{\langle}\partial_{\mu}J^{\mu}(x){\cal
O}_{1}(x_{1})\ldots{\cal
O}_{k}(x_{k})\big{\rangle}=&\sum_{l=1}^{k}\delta^{(d)}(x-x_{l})\big{\langle}{\cal
O}_{1}(x_{1})\ldots\delta{\cal O}_{l}(x_{l})\ldots{\cal
O}_{k}(x_{k})\big{\rangle}\\\ &-ihq_{0}\big{\langle}{\cal
O}_{1}(x_{1})\ldots{\cal O}_{k}(x_{k}){\cal
O}_{0}(x)\big{\rangle}\,.\end{split}$ (2.10)

Integrating over an open region $D^{(d)}$ with boundary $\Sigma^{(d-1)}$ we
have

$\Big{\langle}Q[\Sigma^{(d-1)}]{\cal O}_{1}\ldots{\cal
O}_{k}\Big{\rangle}=\chi(\Sigma^{(d-1)})\langle{\cal O}_{1}\ldots{\cal
O}_{k}\rangle-hq_{0}\int_{D^{(d)}}d^{d}x\,\big{\langle}{\cal O}_{1}\ldots{\cal
O}_{k}{\cal O}_{0}(x)\big{\rangle}\,.$ (2.11)

If the coupling $h$ is irrelevant, at large distances and for sufficiently
large surfaces $\Sigma^{(d-1)}$, the second term in the r.h.s. of (2.11) is
suppressed with respect to the first one, and the operators
$Q(\Sigma^{(d-1)})$ become approximately topological.555For a related
discussion on approximate symmetries in the language of topological operators
see [63]. In this case we say that the symmetry $G$ is emergent in the IR.

###  2.2 Quenched disorder and Ward identities

Theories with quenched disorder in the continuum limit can often be described
starting from a pure theory $S_{0}$ and adding a perturbation like in (2.9)
(see e.g. [64, 65]), where $h$ is taken to be space-dependent (again we always
implicitly pair up $h(x)\mathcal{O}_{0}(x)$ with its hermitian conjugate):

$S[h]=S_{0}+\int d^{d}x\,h(x)\mathcal{O}_{0}(x)\,.$ (2.12)

The random coupling is sampled from a distribution $P[h]$ and we should think
of an ensemble of systems, each member being described by the action (2.12).
Note that the considerations above on the explicit breaking are valid, with
minor modifications, for each member of the ensemble.

A relevant example (which we will extensively consider in the sections 3 and
4) is the case of white noise, where $P[h]$ is Gaussian

$P[h]\propto\exp\Big{(}-\frac{1}{2v}\int\\!d^{d}x\,h^{2}(x)\Big{)}\,,$ (2.13)

parametrized by a coupling $v$ which governs the width of the Gaussian
distribution. Dimensional analysis fixes the dimension of $v$ to be

$[v]=d-2\Delta_{{\cal O}_{0}}\,,$ (2.14)

where $\Delta_{{\cal O}_{0}}$ is the classical scaling dimension of the
operator ${{\cal O}_{0}}$. The disorder is classically irrelevant in the RG
sense when

$\Delta_{{\cal O}_{0}}>\frac{d}{2}\,.$ (2.15)

The equation (2.15) is called Harris criterion [66]. If the disorder is
classically relevant or marginal, it has an important effect on the IR
dynamics. For instance, other fixed points could emerge, so called random
fixed points, which can also have logarithmic behavior (see section 4), or we
could have no fixed points at all. When (2.15) is satisfied, the IR behaviour
of the system is unaffected by the impurities.

Like in the pure theory case, if the coupling $h(x)$ breaks a symmetry $G$ and
is irrelevant, then the symmetry $G$ will appear as an emergent symmetry in
the IR theory. On the other hand, in disordered theories symmetries might also
appear on average, but exactly, namely at all energy scales, independently on
the scaling dimension of $h(x)$. It is important to keep into account this
distinction in the considerations that will follow. The latter case is the one
that we will call _disordered symmetries_.

The observables we are interested in are averaged correlation functions of
local operators defined as (we adopt here the notation of [65])

$\overline{\langle{\cal O}_{1}(x_{1})\ldots{\cal
O}_{k}(x_{k})\rangle}=\int\\!{\cal D}h\,P[h]\frac{\int\\!{\cal
D}\mu\,e^{-S[h]}{\cal O}_{1}(x_{1})\ldots{\cal O}_{k}(x_{k})}{\int\\!{\cal
D}\mu\,e^{-S[h]}}\,,$ (2.16)

where $\mu$ is the path integral measure and $P[h]$ is an arbitrary
distribution, not necessarily of the form (2.13). Correlation functions can be
obtained as usual by coupling each local operator ${\cal O}_{i}$ to an
external source $K_{i}$ and by taking functional derivatives with respect to
the $K_{i}$’s of the averaged generating functional $Z_{D}[K_{i}]$ defined as

$Z_{D}[K_{i}]:=\int\mathcal{D}h\,P[h]\frac{\int\\!{\cal D}\mu\,e^{-S[h]+\int
K_{i}\mathcal{O}_{i}}}{\int\\!{\cal
D}\mu\,e^{-S[h]}}=\int\mathcal{D}h\,P[h]\frac{Z[K_{i},h]}{Z[0,h]}\ .$ (2.17)

We can also define the disordered free energy $W_{D}[K_{i}]$ as

$W_{D}[K_{i}]:=\int\\!{\cal D}h\,P[h]\log Z[K_{i},h]=\int\\!{\cal
D}h\,P[h]W[K_{i},h]=\overline{W[K_{i},h]}\,,$ (2.18)

that generates averages of _connected_ correlation functions

$\overline{\langle{\cal O}_{1}(x_{1})\ldots{\cal
O}_{k}(x_{k})\rangle}_{c}=\frac{\delta^{k}W_{D}[K_{i}]}{\delta
K_{1}(x_{1})\ldots\delta K_{k}(x_{k})}\bigg{|}_{K_{i}=0}\,.$ (2.19)

We stress that, unlike standard QFTs, in quenched disorder theories not all
correlators can be determined from the connected ones and in particular

$\overline{\langle\mathcal{O}_{i}(x)\rangle\langle\mathcal{O}_{j}(y)\rangle}\not=\overline{\langle\mathcal{O}_{i}(x)\rangle}\,\,\overline{\langle\mathcal{O}_{j}(y)\rangle}\,.$
(2.20)

This is one of the crucial properties of disordered systems which will play an
important role in the following. This motivates to introduce a more general
generating functional

$Z_{D}^{(N)}[K^{(1)}_{i},...,K^{(N)}_{i}]:=\int\mathcal{D}h\,P[h]\prod_{j=1}^{N}\frac{Z[K^{(j)}_{i},h]}{Z[0,h]}$
(2.21)

whose functional derivatives produce the average of products of correlators.
The generalization of (2.20) is

$Z_{D}^{(N)}[K^{(1)}_{i},...,K^{(N)}_{i}]\neq\prod_{j=1}^{N}Z_{D}[K^{(j)}_{i}]\,.$
(2.22)

Now suppose that the pure theory $S_{0}$ has some global 0-form invertible
symmetry $G$. If the random deformation is $G-$invariant every realization of
the system enjoys the symmetry, therefore $G$ is a symmetry of the full
disordered theory and it will show up in the averaged correlators. Indeed from
the Ward identities of the theory in presence of a random source $h(x)$, by
simply taking the average we immediately get the expected identities. This
applies also to higher-form symmetries which cannot be broken by adding local
operators to the action [3, 1].

If the random deformation breaks some or all of the symmetries of the pure
theory, the story is more interesting. In this case we want to understand if
and under which conditions the disordered theory still enjoys these
symmetries. We start by considering an internal invertible continuous 0-form
symmetry $G$, but our conclusions apply also in more general setups. In order
to gain some intuition it is useful to use a spurionic argument. The path
integral of the theory coupled to a random source $h(x)$ is

$Z[h]=\int\\!{\cal D}\mu\exp\left(-S_{0}-\int h(x){\cal O}_{0}(x)\right)\,.$
(2.23)

Because of the explicit breaking the partition function obeys

$Z[h]=Z[R_{0}^{\vee}(g)\cdot h],\quad\quad g\in G$ (2.24)

where $\mathcal{O}_{0}$ transforms in representation $R_{0}$ of $G$, and
$R_{0}^{\vee}$ is its transpose. Turning on sources $K_{i}$ for operators of
the pure theory we see that the generating functional satisfies

$\begin{split}Z[K_{i},h]&=\int\\!{\cal D}\mu\exp\left(-S_{0}-\int h(x){\cal
O}_{0}(x)+\int\\!K_{i}(x)\mathcal{O}_{i}(x)\right)\\\ &=Z[R_{i}^{\vee}(g)\cdot
K_{i},R_{0}^{\vee}(g)\cdot h].\end{split}$ (2.25)

Thus the correlators before averaging are not $G-$invariant but

$\frac{\delta}{\delta K_{1}\ldots\delta
K_{n}}Z[K_{i},h]\bigg{|}_{K_{i}=0}=R_{1}(g)\ldots
R_{n}(g)\cdot\frac{\delta}{\delta K_{1}\ldots\delta
K_{n}}Z[K_{i},R_{0}^{\vee}(g)\cdot h]\bigg{|}_{K_{i}=0}\\!\quad.$ (2.26)

This implies that

$\displaystyle\overline{\langle{\cal O}_{1}(x_{1})\ldots{\cal
O}_{n}(x_{n})\rangle}$ $\displaystyle=\int\\!{\cal
D}h\,P[h]\frac{1}{Z[h]}\frac{\delta Z[K_{i},h]}{\delta K_{1}\ldots\delta
K_{n}}\bigg{|}_{K_{i}=0}$ (2.27) $\displaystyle=R_{1}(g)\ldots
R_{n}(g)\cdot\int\\!{\cal D}h\,P[h]\frac{1}{Z[h]}\frac{\delta
Z[K_{i},R_{0}(g)^{\vee}\cdot h]}{\delta K_{1}\ldots\delta
K_{n}}\bigg{|}_{K_{i}=0}\,.$

We can now change variable in the $h$-path integral,
$R_{0}(g^{-1})^{\vee}\cdot h(x)\rightarrow h(x)$. Crucially, if the
probability measure $\mathcal{D}h\,P[h]$ is invariant, the averaged correlator
obeys the $G$ selection rules

$\begin{split}\overline{\langle{\cal O}_{1}(x_{1})\ldots{\cal
O}_{n}(x_{n})\rangle}=R_{1}(g)\ldots R_{n}(g)\cdot\overline{\langle{\cal
O}_{1}(x_{1})\ldots{\cal O}_{n}(x_{n})\rangle}\,,\end{split}$ (2.28)

but only on average. For example, a space-dependent coupling breaks
translations, but if $P[h]$ is translation-invariant (like e.g. in (2.13)),
then momentum conservation is recovered on average.

Although the above spurion analysis is enough to determine selection rules, it
does not provide the explicit form of the conserved currents and which Ward
identities are satisfied (and how). The existence of topological operators is
not even guaranteed and the common lore which identifies symmetries with
topological defects needs a more detailed analysis in order to be verified.
Let us then derive the form of Ward identities for disordered symmetries. For
notational simplicity we focus on $G=U(1)$, but the analysis can be extended
to any Lie group. Consider the generating functional $Z_{D}[K_{i}]$ defined in
(2.17). The usual Ward identities are derived by changing variables in the
path integral at the numerator, transforming all the fields with a space-time
dependent $U(1)$ element $e^{i\epsilon(x)}$, so that

$S_{0}\rightarrow S_{0}+i\int\epsilon(x)\partial_{\mu}J^{\mu}(x)\ ,$ (2.29)

$J^{\mu}$ being the Noether current. Here the symmetry is broken by $h(x)$ in
any specific realization, nevertheless we can modify the standard procedure by
changing variable also in the path integral at the denominator. Since $h(x)$
is space dependent, Poincaré invariance is explicitly broken in each specific
realization and generally $\langle J^{\mu}\rangle\neq 0$. This suggests that
even if the symmetry is recovered on average the current must be modified
somehow. The above-mentioned change of variable in both numerator an
denominator, expanding at first order in $\epsilon(x)$ leads to

$\int\\!\\!\mathcal{D}h\,P[h]\left(\big{\langle}-\partial_{\mu}J^{\mu}-q_{0}h\mathcal{O}_{0}+q_{i}K_{i}\mathcal{O}_{i}\big{\rangle}_{K}+\frac{Z[K_{i},h]}{Z[0,h]}\big{\langle}\partial_{\mu}J^{\mu}+q_{0}h\mathcal{O}_{0}\big{\rangle}\right)=0\,.$
(2.30)

By taking functional derivatives with respect to the sources $K_{i}$ and then
setting them to zero we get

$\overline{\big{\langle}\partial_{\mu}\widetilde{J}^{\mu}(x)\mathcal{O}_{1}(x_{1})\cdots\big{\rangle}}+q_{0}\overline{\big{\langle}h(x)\widetilde{\mathcal{O}}_{0}(x)\mathcal{O}_{1}(x_{1})\cdots\big{\rangle}}=\sum_{i}q_{i}\delta^{(d)}(x-x_{i})\overline{\big{\langle}\mathcal{O}_{1}(x_{1})\cdots\big{\rangle}}\
,$ (2.31)

where we introduced the shifted operators

$\widetilde{J}^{\mu}(x):=J^{\mu}(x)-\langle J^{\mu}(x)\rangle\ ,\ \ \ \ \
\widetilde{\mathcal{O}}_{0}(x):=\mathcal{O}_{0}(x)-\langle\mathcal{O}_{0}(x)\rangle\
.$ (2.32)

The vacuum expectation values should be thought of as certain (generally non-
local) functionals of $h(x)$, whose presence is important in the average.

Since $h(x)$ is integrated over all space-dependent configurations, the second
term in (2.31) vanishes identically provided that the probability measure
satisfies certain invariance conditions. Indeed we are allowed to perform the
change of variable $h(x)\rightarrow e^{-iq_{0}\epsilon(x)}h(x)$ in the $h$
path integral of (2.17), and if the probability measure is invariant under
this formal transformation we obtain

$q_{0}\int\mathcal{D}hP[h]\left(\big{\langle}h\mathcal{O}_{0}\big{\rangle}_{K_{i}}-\frac{Z[K_{i},h]}{Z[0,h]}\big{\langle}h\mathcal{O}_{0}\big{\rangle}\right)=0\,.$
(2.33)

Taking arbitrary functional derivatives with respect to the external sources
and setting them to zero we find

$q_{0}\overline{\big{\langle}h(x)\widetilde{\mathcal{O}}_{0}(x)\mathcal{O}_{1}(x_{1})\ldots\big{\rangle}}=0\,,$
(2.34)

which implies the vanishing of the second term in the left hand side of
(2.31). By changing variables in the path integral, we also get the relation

$\Big{\langle}\partial^{\mu}J_{\mu}(x)+q_{0}h(x)\mathcal{O}_{0}(x)\Big{\rangle}=0\,,$
(2.35)

valid before averaging. We are now ready to discuss Ward identities. If
$q_{0}=0$, namely the $U(1)$ symmetry is unbroken in any realization of the
ensemble, plugging (2.35) in (2.31) leads to the averaged version of the
ordinary Ward identities (2.1). This is of course expected, given that (2.1)
holds even before average in this case. More interestingly, for $q_{0}\neq 0$,
thanks to (2.34) we find the disordered Ward identities

$i\overline{\big{\langle}\partial_{\mu}\widetilde{J}^{\mu}(x)\mathcal{O}_{1}(x_{1})\cdots\mathcal{O}_{k}(x_{k})\big{\rangle}}=\sum_{i=1}^{k}iq_{i}\delta^{(d)}(x-x_{i})\overline{\big{\langle}\mathcal{O}_{1}(x_{1})\cdots\mathcal{O}_{k}(x_{k})\big{\rangle}}\
.$ (2.36)

Several comments are in order.

  * •

The relation we obtained has the same form of a standard Ward identity, but
for a modified current $\widetilde{J}^{\mu}=J^{\mu}-\langle J^{\mu}\rangle$.
The modification is proportional to the identity operator in any of the
specific realization of the ensemble, and can be thought of as an
$h-$dependent counterterm which restores the conservation in the disordered
theory. Note that the Ward identities written as in (2.36) apply for arbitrary
correlation functions of local operators which do not contain explicit powers
of $h(x)$.

  * •

Before averaging the current $J^{\mu}$ (as well as its shifted version
$\widetilde{J}^{\mu}$) is sensitive to the UV renormalization of the theory,
i.e. it acquires a non-vanishing anomalous dimension (in contrast to ordinary
conserved currents in pure theories). A proper definition of $J^{\mu}$ would
require a regularization of the theory and a choice of renormalization scheme.
Luckily enough, if we are only interested in averaged correlators, we do not
need to worry about these issues, since (2.36) guarantees that
$\widetilde{J}^{\mu}$ is effectively conserved inside averaged correlators.

  * •

The Ward identities (2.36) are valid independently of the behavior of the
current at infinity. When the integral of $\partial_{\mu}\widetilde{J}^{\mu}$
over the full space diverges (this requires the space to be non compact) the
disordered symmetry is spontaneously broken. We do not discuss spontaneous
disordered symmetry breaking in detail in this paper. We briefly comment on it
in the conclusions. If the symmetry is not spontaneously broken the integral
of $\partial_{\mu}\widetilde{J}^{\mu}$ over the full space vanishes. Thus
(2.36) implies the selection rules we already derived from the spurionic
argument. However (2.36) is a more refined constraint being a local
conservation equation: local currents can be used to discuss ’t Hooft
anomalies and eventually gauging the symmetry, as we will see shortly.
Moreover we will show in the next subsection that, with some modification with
respect to the usual story, the conservation of $\widetilde{J}^{\mu}$ leads to
topological operators as in the pure case.

  * •

Since the random coupling $h(x)$ is space dependent, in every member of the
ensemble translational symmetry is explicitly broken. The analysis above can
be repeated for the stress-energy tensor $T^{\mu\nu}$, showing that also
traslational invariance is recovered in a theory with quenched disorder,
provided $P[h]$ is translational invariant.

With simple modifications we have a similar identity for any Lie group $G$:

$i\overline{\big{\langle}\partial_{\mu}\widetilde{J}_{a}^{\mu}(x)\mathcal{O}_{1}(x_{1})\cdots\big{\rangle}}=\sum_{i}\delta^{(d)}(x-x_{i})\overline{\big{\langle}\mathcal{O}_{1}(x_{1})\cdots
r_{i}(T_{a})\cdot\mathcal{O}_{i}(x_{i})\cdots\big{\rangle}}\ .$ (2.37)

Here $T_{a}$ is a Lie algebra generator and $r_{i}$ is the representation of
the Lie algebra, induced by $R_{i}$, under which $\mathcal{O}_{i}$ transforms.
A more general situation could take place, in which the disorder deformation
does not break the full group, but leaves a subgroup $H\subset G$ unbroken. In
this case any specific realization is $H-$symmetric, and thus the currents
$J_{\alpha}^{\mu}$, with $T_{\alpha}$ generator of
$\mathfrak{h}=\text{Lie}(H)$, satisfy the standard Ward identity without the
necessity of averaging. In particular
$\langle\partial_{\mu}J^{\mu}_{\alpha}\rangle=0$, even if the expectation
value of the current itself is not necessarily vanishing due to the lack of
Poincaré invariance. Even if $G/H$ is generically not a group, the associated
currents, which are not conserved in any specific realization, after the
appropriate shift by their expectation values turn out to satisfy the Ward
identity (2.37) in the disordered theory, and reconstruct the full group $G$.

Sometimes a 0-form symmetry $G$ can form an higher-group structure with
higher-form symmetries of the theory [4, 6, 5]. In this case $G$ is not really
a subgroup of the full symmetry structure, since the product of several
$G-$elements can also produce an element of the higher-form symmetry. This
kind of extension is classified by group-cohomology classes, the Postnikov
classes: for instance in a 2-group, mixing $G$ with a 1-form symmetry
$\Gamma$, the relevant datum is a class $\beta\in H^{3}(BG,\Gamma)$, with $BG$
the classifying space of $G$. The important thing is that this is a discrete
datum and cannot change under continuous deformation. Suppose we add a
disorder breaking $G$, and this re-emerges as a disordered symmetry. A natural
question is whether the higher-group structure is also recovered. The answer
is affirmative as a consequence of the discrete nature of this structure.
Indeed the probability distributions $P[h]$ have some tunable continuous
parameters, like $v$ in the Gaussian case (2.13), such that the pure theory is
recovered in some limit ($v\rightarrow 0$ in the Gaussian case). The
cohomology class characterising the higher-group is discrete and cannot change
with this continuous parameter. Since all these disordered theories are
continuously connected to the pure one, the higher-group structure is
unchanged.

Up to this point disordered symmetries seem to behave like ordinary global
symmetries in pure theories. The difference arises by considering averages of
products of correlators

$\overline{\prod_{j=1}^{N}\Big{\langle}\mathcal{O}_{1}^{(j)}(x_{1}^{(j)})\cdots\mathcal{O}_{k_{j}}^{(j)}(x_{k_{j}}^{(j)})\Big{\rangle}}\
.$ (2.38)

Because of (2.22) these are independent correlators, and we do not expect them
to satisfy Ward identities immediately implied by (2.36), or to be constrained
by the usual selection rules. Let us consider the more general generating
functional $Z_{D}^{(N)}[\\{K^{(j)}_{i}\\}]$ introduced in (2.21). With the
same manipulations which led to (2.33), we get

$q_{0}\sum_{j=1}^{N}\int\mathcal{D}hP[h]\left(\left(\big{\langle}h\mathcal{O}_{0}\big{\rangle}_{K^{(j)}_{i}}-\frac{Z[K^{(j)}_{i},h]}{Z[0,h]}\langle
h\mathcal{O}_{0}\rangle\right)\prod_{l\neq
j}\frac{Z[K^{(l)}_{i},h]}{Z[0,h]}\right)=0\,,$ (2.39)

while the individual terms of the sum are generically non-vanishing. This
implies that the only Ward identity we can prove from
$Z_{D}^{(N)}[\\{K^{(j)}_{i}\\}]$ are obtained by changing variable in _all_
the path integrals involved: if we try to change variables only in a subset of
these path integrals, the extra term arising would be not be of the form
(2.39), but the sum would be over that subset of indices. Repeating the steps
above we obtain the Ward identities for averages of products of correlators:

$\sum_{j=1}^{N}\overline{\big{\langle}\partial_{\mu}\widetilde{J}^{\mu}\mathcal{O}_{1}^{(j)}\cdots\mathcal{O}_{k_{j}}^{(j)}\\!\big{\rangle}\left(\prod_{l\neq
j}\big{\langle}\mathcal{O}_{1}^{(l)}\cdots\mathcal{O}_{k_{l}}^{(l)}\big{\rangle}\right)}=\sum_{j=1}^{N}\sum_{i_{j}=1}^{k_{j}}q_{i_{j}}^{(j)}\delta^{(d)}(x-x_{i_{j}}^{(j)})\overline{\prod_{l=1}^{N}\big{\langle}\mathcal{O}_{1}^{(l)}\cdots\mathcal{O}_{k_{l}}^{(l)}\big{\rangle}}\
.$ (2.40)

These Ward identities imply weaker selection rules. For instance, the
correlator

$\overline{\langle\mathcal{O}_{1}(x_{1})\cdots\mathcal{O}_{k_{1}}(x_{k_{1}})\rangle\langle\mathcal{O}_{k_{1}+1}(x_{k_{1}+1})\cdots\mathcal{O}_{k_{1}+k_{2}}(x_{k_{1}+k_{2}})\rangle}$
(2.41)

can be non zero when $\sum_{i=1}^{k_{1}}q_{i}\neq 0$ and
$\sum_{i=k_{1}+1}^{k_{1}+k_{2}}q_{i}\neq 0$, provided that
$\sum_{i=1}^{k_{1}+k_{2}}q_{i}=0$.

In a theory with quenched disorder ordinary and disordered symmetries can be
present at the same time, and we see that their different action shows up in
looking at averages of products of correlators.

See appendix A.1.1 for an explicit derivation of (2.36) for a two-point
($k=2$) function in a simple solvable model.

###  2.3 Topological operators for disordered symmetries

We now address the question of whether there exist topological symmetry
operators implementing disordered symmetries, placing them in the general
framework of [1]. This is important to e.g. generalize to discrete symmetries,
coupling them to backgrounds and discuss non-perturbative anomalies. For
notational simplicity we again focus on the $G=U(1)$ case, but all the
considerations can be extended to any Lie group. We introduce the modified
charge operator

$\widetilde{Q}[\Sigma^{(d-1)}]=\int_{\Sigma^{(d-1)}}\\!\\!\widetilde{J}_{\mu}n^{\mu}=Q[\Sigma^{(d-1)}]-\Big{\langle}Q[\Sigma^{(d-1)}]\Big{\rangle}$
(2.42)

which satisfies the integrated Ward identity

$\overline{\Big{\langle}\widetilde{Q}[\Sigma^{(d-1)}]{\cal
O}_{1}(x_{1})\ldots{\cal
O}_{k}(x_{k})\Big{\rangle}}=\chi(\Sigma^{(d-1)})\overline{\langle{\cal
O}_{1}(x_{1})\ldots{\cal O}_{k}(x_{k})\rangle}\ ,$ (2.43)

with $\chi(\Sigma^{(d-1)})$ as in (2.4), as well as the generalization to
arbitrary products

$\sum_{j=1}^{N}\overline{\Big{\langle}\widetilde{Q}[\Sigma^{(d-1)}]\mathcal{O}_{1}^{(j)}\cdots\mathcal{O}_{k_{j}}^{(j)}\Big{\rangle}\left(\prod_{l\neq
j}\langle\mathcal{O}_{1}^{(l)}\cdots\mathcal{O}_{k_{l}}^{(l)}\rangle\right)}=\chi(\Sigma^{(d-1)})\overline{\prod_{l=1}^{N}\langle\mathcal{O}_{1}^{(l)}\cdots\mathcal{O}_{k_{l}}^{(l)}\rangle}\
.$ (2.44)

The reason why the naive procedure of constructing the symmetry operator by
exponentiating $\widetilde{Q}[\Sigma^{(d-1)}]$ does not work can be already
understood at the second order: $\widetilde{Q}^{2}[\Sigma^{(d-1)}]$ does not
measure the square of the total charge. Let $\Phi$ be a generic product of
local operators.666In order to avoid cluttering in the formulas, from now on
we will adopt a lighter notation omitting often the support of local operators
or indices. We have

$\overline{\langle\widetilde{Q}^{2}\Phi\rangle}=\overline{\langle\widetilde{Q}\,Q\Phi\rangle}-\overline{\langle
Q\rangle\langle\widetilde{Q}\Phi\rangle}=\chi\overline{\langle\,Q\Phi\rangle}-\chi\overline{\langle
Q\rangle\langle\Phi\rangle}+\overline{\langle\widetilde{Q}Q\rangle\langle\Phi\rangle}=\chi^{2}\overline{\langle\,\Phi\rangle}+\overline{\langle\widetilde{Q}Q\rangle\langle\Phi\rangle}\
.$ (2.45)

In the second step we used both the Ward identity (2.43) and (2.44) with
$N=2$. We deduce that what measures the total charge square is not
$\widetilde{Q}^{2}$ but

$\widetilde{Q}_{2}:=\widetilde{Q}^{2}-\langle\widetilde{Q}Q\rangle=Q^{2}-2\langle
Q\rangle Q+2\langle Q\rangle^{2}-\langle Q^{2}\rangle\ .$ (2.46)

In order to construct the topological symmetry operator we need, for any
$n\in\mathbb{N}$, an operator $\widetilde{Q}_{n}$ such that

$\overline{\langle\widetilde{Q}_{n}\mathcal{O}_{1}\cdots\mathcal{O}_{k}\rangle}=\chi^{n}\overline{\langle\mathcal{O}_{1}\cdots\mathcal{O}_{k}\rangle}$
(2.47)

and then define the symmetry operators as

$\widetilde{U}_{g}=\sum_{n=0}^{\infty}\frac{(i\alpha)^{n}}{n!}\widetilde{Q}_{n}\
,\ \ \ \ \ \ \ \ g=e^{i\alpha}\ .$ (2.48)

To prove that such operators exist, and show how to compute them, we start
from $\overline{\langle Q^{n}\Phi\rangle}$ (again $\Phi$ denotes a generic
product of local operators), and we rewrite one $Q$ as $\widetilde{Q}+\langle
Q\rangle$, so that we can use a linear Ward identity for $\widetilde{Q}$, and
we iterate until we eliminate all the $Q$s:

$\begin{array}[]{rl}\overline{\langle
Q^{n}\Phi\rangle}=&\overline{\langle\widetilde{Q}Q^{n-1}\Phi\rangle}+\overline{\langle
Q\rangle\langle Q^{n-1}\Phi\rangle}=\chi\overline{\langle
Q^{n-1}\Phi\rangle}+\overline{\langle Q\rangle\langle Q^{n-1}\Phi\rangle}\\\
=&\chi^{2}\overline{\langle Q^{n-2}\Phi\rangle}+\chi\overline{\langle
Q\rangle\langle Q^{n-2}\Phi\rangle}+\overline{\langle Q\rangle\langle
Q^{n-1}\Phi\rangle}\\\ \vdots&\\\
=&\displaystyle\chi^{n}\overline{\langle\Phi\rangle}+\sum_{k=0}^{n-1}\chi^{k}\overline{\langle
Q\rangle\langle Q^{n-k-1}\Phi\rangle}\ .\end{array}$ (2.49)

The terms $\chi^{k}\overline{\langle Q\rangle\langle Q^{n-k-1}\Phi\rangle}$
can be managed as follows. We eliminate one $\chi$ by using the linear Ward
identity for the averaged product of two correlators for $\widetilde{Q}$,
which we then re-expand as $Q-\langle Q\rangle$:

$\begin{split}\chi^{k}\overline{\langle Q\rangle\langle
Q^{n-k-1}\Phi\rangle}=\,&\chi^{k-1}\Big{(}\,\overline{\langle\widetilde{Q}Q\rangle\langle
Q^{n-k-1}\Phi\rangle}+\overline{\langle
Q\rangle\langle\widetilde{Q}Q^{n-k-1}\Phi\rangle}\,\Big{)}\\\
=\,&\chi^{k-1}\Big{(}\,\overline{\langle Q^{2}\rangle\langle
Q^{n-k-1}\Phi\rangle}-2\overline{\langle Q\rangle^{2}\langle
Q^{n-k-1}\Phi\rangle}+\overline{\langle Q\rangle\langle
Q^{n-k}\Phi\rangle}\,\Big{)}\,.\end{split}$ (2.50)

Then we eliminate an other $\chi$ from each term, again using the linear Ward
identity, in some terms with the product of two correlators, in others with
the product of three correlators. We continue in this way until we eliminate
all the $\chi$s, and remain with a sum of averages of products of expectation
values of $\langle Q^{a}\rangle$ for various $a$, and $\langle
Q^{b}\Phi\rangle$ for a certain $b$, generally different for each term. This
defines the operator $\widetilde{Q}_{n}$. For instance

$\widetilde{Q}_{3}=Q^{3}-3\langle Q\rangle Q^{2}-3\langle Q^{2}\rangle
Q+6\langle Q\rangle^{2}Q-\langle Q^{3}\rangle+6\langle Q\rangle\langle
Q^{2}\rangle-6\langle Q\rangle^{3}\,.$ (2.51)

While this seems very complicated, one can check until arbitrarily high order
that the expansion can be beautifully resummed as

$\widetilde{U}_{g}=\sum_{n=0}^{\infty}\frac{(i\alpha)^{n}}{n!}\widetilde{Q}_{n}=e^{i\alpha
Q}\Big{\langle}e^{i\alpha Q}\Big{\rangle}^{-1}\,,$ (2.52)

where $\widetilde{Q}_{0}:=1$. Note that this is the only result consistent
with $\big{\langle}\widetilde{U}_{g}\big{\rangle}=1$, which must be true by
construction since $\langle\widetilde{Q}_{n}\rangle=0$ as a direct consequence
of the Ward identities (2.47) satisfied by $\widetilde{Q}_{n}$ in absence of
local operators.

The operator $\widetilde{U}_{g}$ in averaged correlators behaves as

$\overline{\langle\widetilde{U}_{g}[\Sigma^{(d-1)}]\mathcal{O}_{1}\cdots\mathcal{O}_{k}\rangle}=e^{i\alpha\chi(\Sigma^{(d-1)})}\overline{\langle\mathcal{O}_{1}\cdots\mathcal{O}_{k}\rangle}$
(2.53)

and is hence a topological symmetry operator, on average. It satisfes the
group law

$\overline{\langle\widetilde{U}_{g}\widetilde{U}_{h}\Phi\rangle}=\overline{\langle\widetilde{U}_{gh}\Phi\rangle}\,,$
(2.54)

$\Phi$ being an arbitrary product of local operators. As a consequence, the
naive expectation that $e^{i\alpha Q}e^{i\beta Q}=e^{i(\alpha+\beta)Q}$ is
wrong because of the disorder. Note that before averaging the operator
$\widetilde{U}$ is subject to renormalization and its proper definition
requires a choice of renormalization scheme. We do not need to keep track of
these subtleties, however, because they are washed away after the average is
taken.

We now consider how $\widetilde{U}_{g}$ behaves inside averages of products of
correlators (2.38), extending (2.44) to finite symmetry actions. This is
important because, as we mentioned, products of correlators is what really
characterizes disordered symmetries with respect to ordinary ones, and we need
the symmetry operator version of the criterion we discussed at the end of
section 2.2. In principle one could explicitly construct the correct
combination of charges $\widetilde{Q}_{n}$ entering the Ward identities using
the results above. For example, in the average of products of two correlators,
at quadratic order in the charges we have

$\overline{\langle\widetilde{Q}_{2}\Phi_{1}\rangle\langle\Phi_{2}\rangle}+\overline{\langle\Phi_{1}\rangle\langle\widetilde{Q}_{2}\Phi_{2}\rangle}+2\overline{\langle\widetilde{Q}_{1}\Phi_{1}\rangle\langle\widetilde{Q}_{1}\Phi_{2}\rangle}=\chi^{2}\overline{\langle\Phi_{1}\rangle\langle\Phi_{2}\rangle}\,,$
(2.55)

$\Phi_{1,2}$ being two distinct generic products of local operators. Similarly
for multiple products.

We claim that the correct Ward identities consist in inserting
$\widetilde{U}_{g}$ in all the (un)factorized correlators under average:

$\overline{\prod_{j=1}^{N}\Big{\langle}\widetilde{U}_{g}[\Sigma^{(d-1)}]\mathcal{O}^{(j)}_{1}(x_{1}^{(j)})\cdots\mathcal{O}^{(j)}_{k_{j}}(x_{k_{j}}^{(j)})\Big{\rangle}}=e^{i\alpha\chi(\Sigma^{(d-1)})}\overline{\prod_{j=1}^{N}\Big{\langle}\mathcal{O}^{(j)}_{1}(x_{1}^{(j)})\cdots\mathcal{O}^{(j)}_{k_{j}}(x_{k_{j}}^{(j)})\Big{\rangle}}\
.$ (2.56)

This can be checked by expanding both members in powers of $\alpha$, which
gives a series of Ward identities for the $\widetilde{Q}_{n}$’s. For example,
for two correlators ($N=2$) we have

$\sum_{l=0}^{k}{k\choose
l}\overline{\Big{\langle}\widetilde{Q}_{l}\Phi_{1}\Big{\rangle}\Big{\langle}\widetilde{Q}_{k-l}\Phi_{2}\Big{\rangle}}=\chi^{k}\overline{\big{\langle}\Phi_{1}\big{\rangle}\big{\langle}\Phi_{2}\big{\rangle}}\,,$
(2.57)

where $\chi=\chi_{1}+\chi_{2}$ are the sum of charges of the local operators
in the product $\Phi_{1,2}$ which are inside the support of the charge
operators. Checking (2.57) directly is cumbersome, but we can proceed as
follows. We rewrite the last term appearing in (2.49) using (2.57) (assuming
its validity) with $\Phi_{1}=Q$ and $\Phi_{2}=Q^{n-k-1}\Phi$.777Note that
$\Phi$ can include integrated current operators $J_{\mu}$, hence powers of
charges $Q$, but not powers of $\widetilde{Q}$. The latter is still the
integral of a local operator, but with an explicit dependence on $h(x)$, in
which case the analysis does not apply. In this way we get

$\widetilde{Q}_{n}=Q^{n}-\sum_{k=0}^{n-1}\sum_{l=0}^{k}{{k}\choose{l}}\big{\langle}\widetilde{Q}_{l}Q\big{\rangle}\widetilde{Q}_{k-l}Q^{n-k-1}=Q\widetilde{Q}_{n-1}-\sum_{l=0}^{n-1}{{n-1}\choose{l}}\langle\widetilde{Q}_{l}Q\rangle\widetilde{Q}_{n-l-1}\
.$ (2.58)

This is a recursion formula which determines $\widetilde{Q}_{n}$ in terms of
all the $\widetilde{Q}_{m}$ for $m<n$, and it is equivalent to (2.57). It can
be checked that computing the topological charges with this formula gives the
same result as computing them directly from the linear Ward identities,
proving in this way the validity of (2.56) and (2.57).

For averages of multiple correlators the group law (2.54) generalizes to

$\overline{\prod_{j=1}^{N}\Big{\langle}\widetilde{U}_{g}\widetilde{U}_{h}\Phi_{j}\Big{\rangle}}=\overline{\prod_{j=1}^{N}\Big{\langle}\widetilde{U}_{gh}\Phi_{j}\Big{\rangle}}\
.$ (2.59)

We are finally able to characterize disordered symmetries in full generality.
These are symmetries of theories with quenched disorder implemented by
symmetry operators $\widetilde{U}_{g}$, $g\in G$, which become topological
after quenched average. They satisfy the identity (2.53) and the group law
(2.54) as operator equations valid in any averaged correlator. Differently
from ordinary global symmetries, in averages of products of correlators like
(2.38) they are topological only if inserted in each factor of the product,
and satisfy the generalized group law (2.59) inside averaged correlators.
Disordered symmetries are symmetries of the pure system broken by the disorder
but with a symmetric probability measure. It is then not surprising that
$\widetilde{U}_{g}$ can be written in terms of the corresponding topological
operator $U_{g}$ of the pure system as

$\widetilde{U}_{g}[\Sigma^{(d-1)}]=U_{g}[\Sigma^{(d-1)}]\Big{\langle}U_{g}[\Sigma^{(d-1)}]\Big{\rangle}^{-1}\
.$ (2.60)

However the characterization above is intrinsic and does not require to know
the pure system. The resummation of the series (2.48) into the compact
expression (2.60) allows us to immediately generalize the analysis to more
general groups $G$, including discrete ones where there is no current or
charge operator available.

###  2.4 ’t Hooft anomalies for continuous disordered symmetries

We examine in this and the next subsections some general properties of
disordered symmetries. We will argue that the concept of ’t Hooft anomalies,
for both continuous and discrete symmetries, extends to this context. In
particular we show that disordered symmetries inherit the anomaly of their
pure counterpart. This is important because we can use anomalies to constraint
the IR dynamics of quenched disordered theories, whose flow is generally
extremely complicated. We start discussing ’t Hooft anomalies for continuous
disordered symmetries, postponing to section 2.5 the case of discrete
symmetries.

A theory with a global symmetry can be coupled to a background gauge field $A$
which acts as an external source for the conserved current $J$, and results in
a partition function $Z[A]$. A ’t Hooft anomaly arises whenever $Z[A]$ is not
invariant under gauge transformations of the background (see e.g. [67] for a
modern review). Denoting by $A^{\lambda}$ the gauge transformed background, we
have

$Z[A^{\lambda}]=e^{i\int_{X^{(d)}}\alpha(\lambda,A)}Z[A]\,,$ (2.61)

where the phase in the exponent is the t’Hooft anomaly, a functional depending
on $\lambda$ and $A$, which cannot be cancelled by local counterterms.
Coupling to backgrounds for disordered symmetries is more subtle, because the
symmetry is explicitly broken in any specific realization of the ensemble. If
the symmetry is restored on average, however, a coupling to an external
background becomes possible via the shifted current $\widetilde{J}$ defined in
(2.32), namely we define

$\overline{Z[A]}=\int\\!{\cal D}h\,P[h]\int\\!{\cal D}\mu e^{-S_{0}-\int
h(x){\cal O}_{0}(x)+\int A_{\mu}\widetilde{J}^{\mu}}\,.$ (2.62)

A ’t Hooft anomaly for a disordered symmetry $G$ can be defined in close
analogy with the ordinary case (2.61):

$\overline{Z[A^{\lambda}]}=e^{i\int_{X^{(d)}}\alpha(\lambda,A)}\overline{Z[A]}\,.$
(2.63)

Anomalies (both continuous and discrete) are known to be invariant under RG
flow thanks to their topological nature (typically associated to a Chern-
Simons level taking value in a cohomology group, see e.g. [68, 69]). In
particular, the value of the anomaly cannot depend on possible continuous
parameters entering in the disorder distribution $P[h]$, such as $v$ in the
Gaussian example (2.13). By adiabatically changing such parameters, we can
make the distribution arbitrarily peaked around $h=0$, in which case we
effectively recover the pure theory.888For the gaussian case this is achieved
by taking $v\rightarrow 0$. We then expect that a ’t Hooft anomaly (2.63)
associated to a disordered symmetry $G$ can only appear if the associated pure
theory (before adding the disorder perturbation) had a ’t Hooft anomaly for
the same symmetry $G$. Moreover, the two anomalies must coincide. This can be
easily verified for all anomalies which, from a path integral point of view,
can be seen to derive from the non-invariance of the path integral measure
[70]. Starting from the left hand side of (2.63) when $\lambda$ is
infinitesimal, we perform a change of variable in the path integral in
$Z[A^{\lambda}]$, which corresponds to an $x$-dependent transformation under
$G$ such that $A^{\lambda}\rightarrow A$. As in pure theories, the non-
invariance of the measure leads to the anomaly term. The derivative of the
current coming from the action variation is cancelled by the explicit symmetry
breaking term and we are left with the anomalous term only. Crucially, the
latter does not depend on the disorder $h$ and hence we immediately get the
infinitesimal version of the right hand side of (2.63), where $\alpha$ is
exactly the same as in the underlying pure theory. If the anomaly vanishes,
the disordered symmetry can be gauged by making the gauge field $A_{\mu}$ in
(2.62) dynamical.

We report in appendix A.2 an example of matching of ’t Hooft anomalies between
the pure and the disorder theories using the replica trick, which will be
introduced in section 3, for the case of the $U(1)$ chiral anomaly in four
dimensions.

###  2.5 Discrete disordered symmetries: ’t Hooft anomalies and gauging

The topological operators $U_{g}[\Sigma^{(d-1)}]$ are crucial to handle
discrete symmetries for which there is no current. In pure theories the
coupling to background gauge fields associated to a discrete symmetry group
$G$ can be achieved by modifying the path integral with the topological
symmetry operators [1]. There are several equivalent ways to introduce a
background gauge field for a discrete symmetry group $G$. One of these (see
e.g. [5] for further details) consists in taking an atlas $\\{U_{i}\\}$ of the
$d$-dimensional space $X^{(d)}$ and assigning group-valued connections
$A_{ij}\in G$ on $U_{i}\cap U_{j}$ such that $A_{ij}=A_{ji}^{-1}$ and
$A_{ij}A_{jk}A_{ki}=1$ on triple intersections $U_{i}\cap U_{j}\cap U_{k}$. A
codimension one symmetry operator $U_{g_{p}}[\Sigma^{(d-1)}_{p}]$ assigns
$A_{ij}=g_{p}$ (or $g_{p}^{-1}$ depending on its orientation) if
$\Sigma^{(d-1)}_{p}$ has a non trivial intersection number with the line dual
to $U_{i}\cap U_{j}$ and $A_{ij}=1$ otherwise.999In the dual triangulation the
charts $U_{i}$ are points, the intersections $U_{i}\cap U_{j}$ are lines, and
so on. The resulting sets of connections $A_{ij}$ defines a background gauge
field for $G$ and can be represented by a cohomology class $A\in
H^{1}(X^{(d)},G)$. The operators $U_{g_{p}}[\Sigma^{(d-1)}_{p}]$ can intersect
in three-valent junctions of codimension two provided that

$g_{i}g_{j}g_{k}=1\,,$ (2.64)

or also in higher multi-valued junctions. The configuration described above
requires few choices, and one must check independence on those. Since the
operators are topological local changes in their support are immaterial. We
could also change the mesh locally near the junctions, which corresponds to
resolve a multi-valent junction in three-valent ones in different ways. This
corresponds to background gauge transformations and a non-invariance under
them signals a ’t Hooft anomaly for discrete symmetries. In $d$ dimensions a
’t Hooft anomaly is classified by a class $\alpha\in
H^{d+1}(BG,U(1))$.101010Strictly speaking, this is the case for bosonic
theories in $d<3$ dimensions. More in general, anomalies are classified by a
cobordism group [71].

Consider now a theory $T$ with quenched disorder, obtained by deforming a pure
theory $T_{0}$, and denote by $T_{h}$ the member of the ensemble with coupling
$h(x)$. Suppose $T$ has a discrete disordered symmetry $G$. As we have seen
this is implemented by the operators

$\widetilde{U}_{g}[\Sigma^{(d-1)}]=U_{g}[\Sigma^{(d-1)}]\Big{\langle}U_{g}[\Sigma^{(d-1)}]\Big{\rangle}^{-1}\
.$ (2.65)

We introduce a fine-enough mesh of topological operators
$\widetilde{U}_{g_{i}}[\Sigma_{i}^{(d-1)}]$ satisfying (on average) the
cocycle condition (2.64) in the three-valent junctions. Since
$\widetilde{U}_{g_{i}}[\Sigma_{i}^{(d-1)}]$ is not topological in $T_{h}$, the
junctions (as well as the operators $\widetilde{U}$ themselves) are not really
well-defined because of UV divergences. However we can employ an arbitrary
regularization scheme for these divergences, without the need of specifying a
renormalization scheme to try to define the junctions and the operators
$\widetilde{U}$ (recall the second comment after (2.36)). This is because we
know that the operators become topological after the average and hence such
divergences are expected to be washed away from the integration over $h(x)$.
We define

$Z_{T_{h}}\big{[}\left\\{g_{i}\right\\},h\big{]}=\int\\!\\!\mathcal{D}\mu\,e^{-S[\phi]-\int
h(x)\mathcal{O}_{0}(x)}\prod_{i}\widetilde{U}_{g_{i}}[\Sigma^{(d-1)}_{i}]=\Big{\langle}\prod_{i}\widetilde{U}_{g_{i}}[\Sigma_{i}^{(d-1)}]\Big{\rangle}$
(2.66)

which, contrary to the pure case, does depend on the specific location of the
planes $\Sigma_{i}^{(d-1)}$. At this point there is no notion of background
gauge fields. However, as a consequence of the Ward identity discussed in
section 2.3,

$Z_{T}\big{[}\left\\{g_{i}\right\\}\big{]}=\int\\!\\!\mathcal{D}h\,P[h]\,Z_{T_{h}}\big{[}\left\\{g_{i}\right\\},h\big{]}$
(2.67)

is independent of the choice of location for $\Sigma_{i}$ and hence the set of
operators $\widetilde{U}_{g_{i}}$ inserted in (2.67) corresponds to a well-
defined discrete gauge field $A\in H^{1}(X^{(d)},G)$. It is important to
emphasize here that the gauge field $A$ arises only after the average over
$h(x)$ is performed. Differently said, if a pure system has a symmetry $G$,
perturbing it with quenched disorder and coupling it to a background are non-
commutative operations. In what follows we denote the above partition function
by $Z_{T}[A]$.

Local modifications of the three-valent junctions change the gauge field by an
exact 1-cocycle $A\rightarrow A^{\lambda}=A+\delta\lambda$. This can change
the partition function $Z_{T}[A]$ by a phase, which represents a class
$\alpha\in H^{d+1}(BG,U(1))$: this is the diagnostic for an ’t Hooft anomaly
for a discrete disordered symmetry. Since the topological operator
$\widetilde{U}_{g}[\Sigma^{(d+1)}]$ is different from the one in the pure
theory by the stacking of an $h(x)-$dependent functional, it is not a priori
obvious that the contact terms arising in the local moves are the same as
those in the pure theory, precisely as it occurred in the continuous case
discussed in section 2.4. However, the fact that anomalies are classified by
classes in $H^{d+1}(BG,U(1))$, which are discrete, immediately proves that
they cannot depend on the strength of the disorder and must be equal to those
of the pure theory. As a result, a system with a disordered symmetry with a ’t
Hooft anomaly cannot be trivially gapped. This is in agreement with previous
works in condensed matter where – mostly in the context of topological
insulators [47, 48, 49, 50, 51, 53] but not only (see e.g. [52]) – SPT phases
of matter where the symmetry is disordered were found. We see that in general
disordered symmetries can lead to protected non-trivial topological phases
(see [54] for a recent analysis).111111In [54] it is considered a Lorentizan
theory with a disorder coupling depending on space but not in time. In this
set-up it is found that purely disordered symmetries, i.e. in absence of pure
symmetries, necessarily have a trivial t’ Hooft anomaly. This is not in
contradiction with our findings, based on Euclidean theories.

Now suppose that the ’t Hooft anomaly vanishes. Then $Z_{T}[A]$ is well
defined and is possible to gauge the symmetry by summing over all consistent
insertions of symmetry operators, or equivalently over cohomology classes
$A\in H^{1}(X^{(d)},G)$. We denote the resulting theory by $T/G$, whose
partition function is121212In the pure case it is possible to modify this sum
weighting the terms with phases. Consistency conditions related with
associativity constraint these phases to be of the form
$\int_{X^{(d)}}A^{*}\nu$, where $\nu\in H^{d}(BG,U(1))$ is a _discrete torsion
class_ and we think $A$ as a homotopy class of maps $X^{(d)}\rightarrow BG$,
so that $A^{*}\nu\in H^{d}(X^{(d)},U(1))$. Since the same kind of constraints
are valid also in the disordered theories, we expect the very same
modification of the gauging procedure to be possible also in this context.

$Z_{T/G}=\sum_{A\in H^{1}(X^{(d)},G)}Z_{T}[A]\ .$ (2.68)

At this point everything is essentially the same as in the pure case (see e.g.
[1, 7]). The operators of $T$ with a counterpart in $T/G$ are the gauge-
invariant ones, while we also add the $(d-2)$ dimensional operators in the
twisted sector of $G$. Indeed the topological operators
$\widetilde{U}_{g}[\Sigma^{(d-1)}]$ become trivial in $T/G$, and their
boundary operators turn into genuine operators (on average). Finally, since
$A\in H^{1}(X^{(d)},G)$ is dynamical, $T/G$ has a dual symmetry generated by
the Wilson lines of the $G$ gauge field. This is a $(d-2)$-form symmetry whose
charged objects are the operators coming from the twisted sectors of $G$. For
$G$ abelian the symmetry is the Pontryagin dual $G^{\vee}$, while it is a non-
invertible symmetry in the non-abelian case [7].

##  3 Disordered symmetries and the replica trick

Disordered systems are often treated by means of the _replica trick_ , which
expresses the averaged correlation functions as certain limits of correlation
functions of a standard QFT, the replica theory. In this section we interpret
the disordered symmetries from the point of view of the replica theory. In
addition to provide a sanity check of the results found in section 2, the
method of replicas allows us to consider emergent symmetries in the disordered
theory for which the results in the previous section do not apply. We will
discuss emergent symmetries in section 4. For the rest of this section and the
next section we assume a Gaussian probability distribution like (2.13) (and
its generalization for complex $h$) with variance $v$.131313Normalization
factors of $P[h]$, which ensure that probabilities add to one, will not play a
role in our considerations and are then left implicit.

###  3.1 The replica trick

To fix our notation we briefly review the _replica trick_. This is a useful
tool that allows to compute connected and full (i.e. both its connected and
disconnected parts) correlators of the disordered theory as limits of
correlators of a pure theory. The starting point of the replica trick is the
identity

$W=\log Z=\lim_{n\rightarrow 0}\left(\frac{\partial Z^{n}}{\partial
n}\right)\,.$ (3.1)

The idea is to replicate the pure system $n$ times, indexing each copy with a
label $a$

$Z^{n}[h,K_{i}]=\int\\!\prod_{a=1}^{n}{\cal
D}\mu_{a}\,\exp\left(-\sum_{a}S_{0,a}-\sum_{a}\int h(x){\cal
O}_{0,a}(x)+\sum_{i,a}\int K_{i}(x){\cal O}_{i,a}(x)\right)\,,$ (3.2)

with the same random field coupling $h$ and external sources $K_{i}$ for all
replicas. When $P[h]$ is Gaussian the average over $h(x)$ can be performed
explicitly and we get

$W_{n}[K_{i}]:=\int\\!{\cal
D}h\,P[h]Z^{n}[h,K^{i}]=\int\\!\prod_{a=1}^{n}{\cal D}\mu_{a}\,e^{-S_{{\rm
rep}}+\sum_{i,a}\int K_{i}{\cal O}_{i,a}}\,,$ (3.3)

where

$S_{{\rm rep}}=\sum_{a}S_{0,a}-\frac{v}{2}\sum_{a,b}\int\\!d^{d}x\,{\cal
O}_{0,a}(x){\cal O}_{0,b}(x)\,$ (3.4)

is the replica action. We see how a coupling between the replica theories has
been generated after the average. Renormalization will possibly induce other
couplings in the replica theory, all compatible with the symmetries of the
system. Among these, importantly the replica theory enjoys an $S_{n}$ replica
symmetry that permutes the various copies of the pure theory. We now assume
that $W_{n}$ can be analytically continued for arbitrary values of $n$
including the origin in the complex $n$-plane.141414This is a notoriously
subtle limit. In particular we can have the phenomenon of spontaneous replica
symmetry breaking (see [72] and references therein). We assume in what follows
that the replica symmetry is not spontaneously broken. Using (3.1) we find

$W_{D}=\lim_{n\rightarrow 0}\Big{(}\frac{\partial W_{n}}{\partial
n}\Big{)}\,,$ (3.5)

where $W_{D}$ is defined in (2.18), and thus

$\begin{split}\overline{\big{\langle}{\cal O}_{1}(x_{1}){\cal
O}_{2}(x_{2})\ldots\big{\rangle}}_{c}&=\lim_{n\rightarrow
0}\partial_{n}\,\left(\Big{\langle}\sum_{a}{\mathcal{O}}_{1,a}(x_{1})\sum_{b}{\mathcal{O}}_{2,b}(x_{2})\ldots\Big{\rangle}^{{\rm
rep}}\right)\,,\end{split}$ (3.6)

where we used the fact that

$\lim_{n\rightarrow 0}W_{n}[K_{i}]=1\,.$ (3.7)

Note that the in the left hand side of (3.6) we have the connected part of the
correlator (indicated with the subscript $c$) which is computed in the replica
theory by a suitable limit of a full correlator. Moreover, we see from (3.6)
that a local operator ${\cal O}$ inside connected correlators of the
disordered theory is mapped in the replica theory to its $S_{n}$-singlet
component $\sum_{a}{\cal O}_{a}$.

The replica trick is also useful to compute general correlation functions in
the disordered theory. Denoting by

$S_{a}[h]=S_{0,a}+\int\,h(x)\mathcal{O}_{0,a}(x)\,,$ (3.8)

we have

$\begin{split}\overline{\langle{\cal O}_{1}(x_{1})\ldots{\cal
O}_{k}(x_{k})\rangle}&=\int\\!{\cal D}h\,P[h]\frac{\int\\!{\cal
D}\mu\,e^{-S[h]}{\cal O}_{1}(x_{1})\ldots{\cal O}_{k}(x_{k})}{Z[h]}\\\
&=\int\\!{\cal D}h\,P[h]\frac{\int\\!\prod_{a}{\cal
D}\mu_{a}\,e^{-\sum_{a}S_{a}[h]}{\cal O}_{1,1}(x_{1})\ldots{\cal
O}_{k,1}(x_{k})}{Z[h]^{n}}\,,\end{split}$ (3.9)

which is an identity for any positive integer $n$. Assuming again that it can
be analytically continued for $n\rightarrow 0$ we get151515Note that we have
actually taken the limit $n\rightarrow 0$ in the denominator of (3.9)
($Z^{n}[h]\rightarrow 1$) before integrating over $h$, while in the numerator
it is kept after the integration over $h$.

$\overline{\langle{\cal O}_{1}(x_{1})\ldots{\cal
O}_{k}(x_{k})\rangle}=\lim_{n\rightarrow 0}\langle{\cal
O}_{1,1}(x_{1})\ldots{\cal O}_{k,1}(x_{k})\rangle^{{\rm rep}}\,.$ (3.10)

In general correlators, in contrast to connected correlators, local operators
are mapped to a specific copy (the same for all operators in the correlation
function) in the replica theory. Equation (3.10) can easily be generalized to
averages of products of general correlation functions. For example, omitting
for simplicity the $x$-dependence of the local operators, we have

$\begin{split}\overline{\Big{\langle}\prod_{i=1}^{k}{\cal
O}_{i}^{(1)}\Big{\rangle}\Big{\langle}\prod_{j=1}^{l}{\cal
O}_{j}^{(2)}\Big{\rangle}}&=\lim_{n\rightarrow 0}\int\\!{\cal
D}h\,P[h]\frac{\\!\int\\!\\!\prod_{a}\ {\cal
D}\mu_{a}\,e^{-\sum_{a}S_{a}[h]}\prod_{i=1}^{k}{\cal
O}_{i,1}^{(1)}\prod_{j=1}^{l}{\cal O}_{j,2}^{(2)}}{Z^{n}[h]}\\\
&=\lim_{n\rightarrow 0}\Big{\langle}\prod_{i=1}^{k}{\cal
O}_{i,1}^{(1)}\prod_{j=1}^{l}{\cal O}_{j,2}^{(2)}\Big{\rangle}^{{\rm
rep}}\,,\end{split}$ (3.11)

and similarly for more than two products. The last observables which we need
to evaluate are averages of products of $N$ connected correlators. Before
averaging, these correlators are obtained by taking functional derivatives of
the product $W[K_{i}^{(1)}]\cdots W[K_{i}^{(N)}]$. For each of them we can use
the replica trick to express this product as a unique path integral. We then
have

$\overline{\prod_{l=1}^{N}\Big{\langle}\prod_{j_{l}=1}^{k_{l}}\mathcal{O}_{j_{l}}^{(l)}\Big{\rangle}_{c}}=\bigg{(}\prod_{k=1}^{N}\lim_{n_{k}\rightarrow
0}\frac{\partial}{\partial
n_{k}}\bigg{)}\Big{\langle}\prod_{l=1}^{N}\prod_{j_{l}=1}^{k_{l}}\sum_{a_{j_{l}}^{(l)}=1}^{n_{l}}\mathcal{O}^{(l)}_{j_{l},a_{j_{l}}^{(l)}}\Big{\rangle}^{\text{rep}}\,,$
(3.12)

where $S_{\rm rep}$ is the replica theory for $n=\sum_{i=1}^{N}n_{i}$
replicas. Note that averages of products of general or connected correlators
in the disordered theory are always expressed in the replica theory as
suitable limits of a single general correlator. Since any correlator can be
expanded in its connected components, (3.12) is actually sufficient to compute
generic correlation functions of the disordered theory. Any operator of the
disordered theory gives rise to a multiplet transforming in the
$n$-dimensional (natural) representation of $S_{n}$. Averages of connected
correlators of operators of the disordered theory are given by the $S_{n}$
singlet operators inside the natural representation in the replica theory.
More general correlation functions of the disoredered theory are instead given
by considering operators singlets under subgroups $S_{n_{i}}\subset S_{n}$
induced by the natural representation in the replica theory.

###  3.2 Disordered symmetries from replica theory

Our first task is to understand how disordered symmetries manifest themselves
in the replica theory. For concreteness we consider again the case of a
$G=U(1)$ symmetry, the replica action reads

$S_{{\rm rep}}=\sum_{a}S_{0,a}-\frac{v}{2}\sum_{a,b}\int\\!d^{d}x\,{\cal
O}_{0,a}(x)\overline{{\cal O}}_{0,b}(x)\,.$ (3.13)

The $U(1)^{n}$ symmetry of the replicated pure part is broken by the disorder
coupling to its diagonal $U(1)$ subgroup, which is then a symmetry of the
replica theory. In particular there is a conserved current

$J_{D}^{\mu}=\sum_{a}J^{\mu}_{a}$ (3.14)

constructed as the $S_{n}$ singlet out of the multiplet induced by the current
$J^{\mu}$ of the disordered theory.

We can recover the Ward identities of the disordered symmetry from those
produced by $J^{\mu}_{D}$ in the replica theory by using (3.12) for averages
of products of connected correlators. The general key idea is to write a sum
of averages of products of connected correlators with current insertions that,
once mapped to correlators of the replica theory, reconstruct the complete
diagonal current $J_{D}^{\mu}$. Then we can use the Ward identity in the
replica theory and finally we rewrite the results back in terms of the
disordered theory.

Determining the Ward identities for averages of single connected correlators
is simple, because the diagonal current $J_{D}$ appears directly in the
replica theory and we can immediately use the ordinary Ward identities there.
We have

$\displaystyle\overline{\big{\langle}\partial_{\mu}J^{\mu}(x)\mathcal{O}_{1}(x_{1})\mathcal{O}_{2}(x_{2})\cdots\big{\rangle}}_{c}$
$\displaystyle=\lim_{n\rightarrow
0}\partial_{n}\Big{(}\Big{\langle}J^{\mu}_{D}(x)\sum_{a}{\mathcal{O}}_{1,a}(x_{1})\sum_{b}{\mathcal{O}}_{2,b}(x_{2})\ldots\Big{\rangle}^{{\rm
rep}}\Big{)}$
$\displaystyle=\sum_{i}q_{i}\delta^{(d)}(x-x_{i})\lim_{n\rightarrow
0}\partial_{n}\big{\langle}\sum_{a}{\mathcal{O}}_{1,a}(x_{1})\sum_{b}{\mathcal{O}}_{2,b}(x_{2})\ldots\big{\rangle}^{{\rm
rep}}$ $\displaystyle=\sum_{i}q_{i}\delta^{(d)}(x-x_{i})\overline{\langle{\cal
O}_{1}(x_{1}){\mathcal{O}}_{2}(x_{2})\ldots\rangle}_{c}\,,$ (3.15)

which reproduces the connected version of (2.36). Averages of products of
connected correlators are also easy to treat, because it is enough to consider
a sum of correlators where the current is inserted in each term to reconstruct
$J_{D}$ in the replica theory and then use the Ward identities there. Skipping
obvious steps, we get

$\sum_{j=1}^{N}\overline{\langle\partial_{\mu}J^{\mu}(x)\mathcal{O}_{1}^{(j)}\ldots\mathcal{O}_{k_{j}}^{(j)}\rangle_{c}\bigg{(}\prod_{l\neq
j}\langle\mathcal{O}_{1}^{(l)}\dots\mathcal{O}_{k_{l}}^{(l)}\Big{\rangle}_{c}\bigg{)}}=\sum_{j=1}^{N}\sum_{i_{j}=1}^{k_{j}}\delta_{i_{j}^{(j)}}q_{i_{j}}^{(j)}\overline{\prod_{l=1}^{N}\langle\mathcal{O}_{1}^{(l)}\ldots\mathcal{O}_{k_{l}}^{(l)}\rangle_{c}}\,,$
(3.16)

which is similar to (2.40), but expressed in terms of connected correlators
and the unshifted current.

Due to the different way the replica trick handles connected and general
correlators, determining the Ward identities for the latter will produce the
improved current $\widetilde{J}_{\mu}$. We use (3.10) to write

$\begin{split}&\overline{\langle\partial^{\mu}J_{\mu}\mathcal{O}_{1}\cdots\mathcal{O}_{n}\rangle}=\lim_{n\rightarrow
0}\langle\partial^{\mu}J_{\mu,1}{\cal O}_{1,1}\ldots{\cal
O}_{k,1}\rangle^{{\rm rep}}\\\ &=\lim_{n\rightarrow
0}\langle\partial^{\mu}J_{\mu,1}{\cal O}_{1,1}\ldots{\cal
O}_{k,1}\rangle^{{\rm rep}}-\lim_{n\rightarrow
0}\frac{1}{n-1}\langle\sum\limits_{a=2}^{n}\partial^{\mu}J_{\mu,a}{\cal
O}_{1,1}\ldots{\cal O}_{k,1}\rangle^{{\rm rep}}\\\ &+\lim_{n\rightarrow
0}\langle\partial^{\mu}J_{\mu,2}{\cal O}_{1,1}\ldots{\cal
O}_{k,1}\rangle^{{\rm rep}}\,.\end{split}$ (3.17)

In the last step, the last two terms add to zero due to the $S_{n}$ symmetry
enjoyed by the replica theory. In the limit $n\rightarrow 0$ we have

$\begin{split}&\lim_{n\rightarrow 0}\langle\partial^{\mu}J_{\mu,1}{\cal
O}_{1,1}\ldots{\cal O}_{k,1}\rangle^{{\rm rep}}-\lim_{n\rightarrow
0}\frac{1}{n-1}\langle\sum\limits_{a=2}^{n}\partial^{\mu}J_{\mu,a}{\cal
O}_{1,1}\ldots{\cal O}_{k,1}\rangle^{{\rm rep}}\\\ &=\lim_{n\rightarrow
0}\langle\partial^{\mu}J_{\mu,D}{\cal O}_{1,1}\ldots{\cal
O}_{k,1}\rangle^{{\rm rep}}\,\end{split}$ (3.18)

and

$\lim_{n\rightarrow 0}\langle\partial^{\mu}J_{\mu,2}{\cal O}_{1,1}\ldots{\cal
O}_{k,1}\rangle^{{\rm
rep}}=\overline{\langle\partial^{\mu}J_{\mu}\rangle\langle\mathcal{O}_{1}\cdots\mathcal{O}_{k}\rangle\,.}$
(3.19)

Therefore, by using the standard Ward identities of the replica theory, from
(3.17) we get (2.36), as expected. The Ward identities (2.40) for products of
generic correlators can be derived using a similar treatment:

$\displaystyle\sum_{j=1}^{N}\overline{\big{\langle}\partial_{\mu}J^{\mu}\mathcal{O}_{1}^{(j)}\cdots\mathcal{O}_{k_{j}}^{(j)}\\!\big{\rangle}\left(\prod_{l\neq
j}\big{\langle}\mathcal{O}_{1}^{(l)}\cdots\mathcal{O}_{k_{l}}^{(l)}\big{\rangle}\right)}=\lim_{n\rightarrow
0}\sum_{j=1}^{N}\langle\partial^{\mu}J_{\mu,j}\prod\limits_{j=1}^{N}(\mathcal{O}_{1,j}^{(j)}\cdots\mathcal{O}_{k_{j},j}^{(j)})\rangle^{{\rm
rep}}$ $\displaystyle=\lim_{n\rightarrow
0}\sum_{j=1}^{N}\langle\partial^{\mu}J_{\mu,j}\prod\limits_{j=1}^{N}(\mathcal{O}_{1,j}^{(j)}\cdots\mathcal{O}_{k_{j},j}^{(j)})\rangle^{{\rm
rep}}-\lim_{n\rightarrow
0}\frac{N}{n-N}\langle\sum\limits_{a=N+1}^{n}\partial^{\mu}J_{\mu,a}\prod\limits_{j=1}^{N}(\mathcal{O}_{1,j}^{(j)}\cdots\mathcal{O}_{k_{j},j}^{(j)})\rangle^{{\rm
rep}}$ $\displaystyle+\lim_{n\rightarrow
0}N\langle\partial^{\mu}J_{\mu,N+1}\prod\limits_{j=1}^{N}(\mathcal{O}_{1,j}^{(j)}\cdots\mathcal{O}_{k_{j},j}^{(j)})\rangle^{{\rm
rep}}$ (3.20)
$\displaystyle=\sum_{j=1}^{N}\sum_{i_{j}=1}^{k_{j}}q_{i_{j}}^{(j)}\delta^{(d)}(x-x_{i_{j}}^{(j)})\overline{\prod_{l=1}^{N}\big{\langle}\mathcal{O}_{1}^{(l)}\cdots\mathcal{O}_{k_{l}}^{(l)}\big{\rangle}}+N\overline{\langle\partial^{\mu}J_{\mu}\rangle\prod\limits_{i}\langle\mathcal{O}_{1}^{(i)}\cdots
O_{k_{i}}^{(i)}}\rangle\,.$

The last term in the right-hand-side in the third row of (3.20) precisely
combines with the left-hand-side to reproduce the shifted current
$\widetilde{J}_{\mu}$ and hence the Ward identities (2.40).

The above analysis shows that the replica counterpart of the disordered
symmetry is an ordinary symmetry generated by the diagonal current
$J^{\mu}_{D}$ and all the Ward identities of the disordered theory reduce to
Ward identities involving $J^{\mu}_{D}$ in the replica theory. The exotic
selection rules (see discussion around (2.41)) of the disordered symmetry are
a consequence of the non-trivial map between the observables of the replica
theory and those in the theory with quenched disorder.

##  4 Disordered emergent symmetries and LogCFTs

Our analysis of Ward identities in section 2.2 applies for disordered
symmetries, namely symmetries which are present in the underlying UV theory,
are broken by the disorder, and get restored after disorder average. On the
other hand, as in pure theories, we can have genuinely emergent symmetries in
the IR, namely symmetries which are not present in the UV theory even before
adding the disorder coupling. If the symmetry emerges for each theory in the
ensemble, then we expect that it gives rise to approximate selection rules of
the same kind as in pure theories with emergent symmetries in the IR. However,
we could also have symmetries that emerge in the IR only after disorder
average. By definition, this implies the existence of additional selection
rules which are valid on average in the IR of the theory. For non-emergent,
actual disordered symmetries such selection rules arise from a conserved
current which is a shifted version of the current operator $J^{\mu}$ of the UV
theory $\widetilde{J}^{\mu}=J^{\mu}-\langle J^{\mu}\rangle$. For emergent
symmetries we cannot determine its explicit form, as the description in terms
of the UV action is useless, and the analysis in section 2.2 does not hold.
However, as we will see, we can deduce which are the selection rules that the
emergent disordered symmetry imposes on averaged correlation functions using
the replica theory.

From a symmetry point of view, the key qualitative feature of the replica
theory (for any finite $n$) is the presence of a $S_{n}$ global permutation
symmetry not present in the original theory with disorder. In the analysis in
section 3.2 the internal symmetry $G$ generated by the current $J_{D}^{\mu}$
commutes with $S_{n}$, namely the infinitesimal transformations
$\delta\mathcal{O}_{j,a}$ of the fields do not mix different replicas. This is
guaranteed by the fact that $G$ in the replica theory is the diagonal subgroup
of the $G^{n}$ global symmetry of the replica theories when $v=0$. On the
other hand, in the case of an emergent symmetry this is not necessarily the
case: each irreducible representation of $S_{n}$ can sit in a different
$G$-representation, or even more generally, the local operators could sit in
representations of the semi-direct product $G\rtimes S_{n}$. We expect that
emergent symmetries in the replica theory of this kind correspond to
disordered emergent symmetries in the theory with disorder. As we will see
below, even in the deep IR the resulting selection rules will be modified with
respect to those coming from (2.36) and its generalizations. As an application
we will show how these modified Ward Identities allow for logarithmic
conformal field theories (LogCFTs) as IR fixed points of disordered systems.

###  4.1 Emergent disordered symmetries

Let us analyze in some detail the Ward Identities for emergent symmetries in
the replica theory. We study theories in which the total symmetry is a direct
product $G\times S_{n}$, since this particular case already exhibits
interesting features. For further simplification, we consider $G=U(1)$ and
correlators where only the singlet and the standard representations of $S_{n}$
are involved. Generalizations to other representations of $S_{n}$ or more
general groups $G$ should be straightforward.

Consider the average of a single correlation function of $k$ local operators
in the disordered theory. We consider both the general and the connected part
of the correlator. Using (3.10) and (3.6), they are mapped in the replica to
the $n\rightarrow 0$ limit of respectively $\langle{\cal O}_{1,1}\ldots{\cal
O}_{k,1}\rangle^{\text{rep}}$ and $\partial_{n}\langle\sum_{a_{1}}{\cal
O}_{1,a_{1}}\ldots\sum_{a_{k}}{\cal O}_{k,a_{k}}\rangle^{\text{rep}}$,
omitting the space dependence of the operators in the correlators for
simplicity. The replica theory is an ordinary pure theory and the emergent
symmetry should manifest with the existence of a vector local operator
$J_{D}^{\mu}$, which becomes conserved in the IR. The operator $J_{D}^{\mu}$
is necessarily a singlet of $S_{n}$, since $U(1)$ commutes with $S_{n}$ by
definition. Note that we do not need to assume the knowledge of the full
multiplet $J^{\mu}_{a}$ for which $J_{D}^{\mu}=\sum_{a=1}^{n}J_{a}^{\mu}$.
Indeed, while in the UV, for weak disorder, the existence of vector operators
in the natural representation of $S_{n}$ is guaranteed, we do not need to keep
track of the IR fate of the non-singlet components. Assuming that
$J_{D}^{\mu}$ is conserved in the IR also at finite $n$, the following
standard selection rules on $k$-point correlators apply:

$\displaystyle\sum_{j=1}^{k}\langle\sum_{a_{j}=1}^{n}\delta{\cal
O}_{j,a_{j}}\prod_{j\neq i=1}^{k}\sum_{a_{i}=1}^{n}{\cal
O}_{i,a_{i}}\rangle^{\text{rep}}$ $\displaystyle=0\,,$ (4.1)
$\displaystyle\sum_{j=1}^{k}\langle\delta{\cal O}_{j,1}\prod_{j\neq
i=1}^{k}{\cal O}_{i,1}\rangle^{\text{rep}}$ $\displaystyle=0\,.$ (4.2)

The key point is now to look more closely to the variations $\delta{\cal
O}_{j,a_{j}}$. Indeed, the natural representation of $S_{n}$ is reducible and
the ${\cal O}_{i}$’s split in

$\mathcal{O}_{i}^{(S)}=\sum_{a=1}^{n}\mathcal{O}_{i,a}\ ,\ \ \ \ \ \
\mathcal{O}_{i,a}^{(F)}=\mathcal{O}_{i,a}-\frac{1}{n}\mathcal{O}^{(S)}_{i}\,,$
(4.3)

which transform in the singlet and in the standard, or fundamental,
representation respectively.161616More general representations arise for
composite operators of the disordered theory which, once replicated,
correspond to multiplets of $S_{n}$ transforming in a (reducible) tensor
product of two or more natural representations. The $U(1)$ symmetry acts as

$\delta\mathcal{O}_{i}^{(S)}=q_{S,i}\mathcal{O}_{i}^{(S)},\quad\quad\quad\delta\mathcal{O}_{i,a}^{(F)}=q_{F,i}\mathcal{O}_{i,a}^{(F)}\,,$
(4.4)

where the charges are generically different, $q_{S,i}\neq q_{F,i}$, and can
possibly depend on $n$. The variations entering the Ward identities of the
replica theory are then

$\delta\mathcal{O}_{i,a}=\delta\mathcal{O}_{i,a}^{(F)}+\frac{1}{n}\delta\mathcal{O}_{i}^{(S)}=q_{F,i}\mathcal{O}_{i,a}+\frac{\Delta
q_{i}}{n}\sum_{a=1}^{n}\mathcal{O}_{i,a}\,,$ (4.5)

where

$\Delta q_{i}:=q_{i,S}-q_{i,F}\,.$ (4.6)

Since in connected correlators we only have singlet components, plugging (4.5)
in (4.1) gives simply

$\sum_{j=1}^{k}q_{S,j}\langle\prod_{i=1}^{k}\sum_{a_{i}=1}^{n}{\cal
O}_{i,a_{i}}\rangle^{\text{rep}}=0\,.$ (4.7)

On the other hand, plugging (4.5) in (4.2) equals

$\displaystyle 0$
$\displaystyle=\sum_{j=1}^{k}q_{F,j}\langle\prod_{i=1}^{k}{\cal
O}_{i,1}\rangle^{\text{rep}}+\sum_{j=1}^{k}\frac{\Delta
q_{j}}{n}\langle\sum_{b=1}^{n}{\cal O}_{j,b}\prod_{j\neq i=1}^{k}{\cal
O}_{i,1}\rangle^{\text{rep}}$
$\displaystyle=\sum_{j=1}^{k}\Big{(}q_{F,j}+\frac{\Delta
q_{j}}{n}\Big{)}\langle\prod_{i=1}^{k}{\cal
O}_{i,1}\rangle^{\text{rep}}+\sum_{j=1}^{k}\frac{\Delta
q_{j}}{n}\langle\sum_{b=2}^{n}{\cal O}_{j,b}\prod_{j\neq i=1}^{k}{\cal
O}_{i,1}\rangle^{\text{rep}}$
$\displaystyle=\sum_{j=1}^{k}q_{F,j}\langle\prod_{i=1}^{k}{\cal
O}_{i,1}\rangle^{\text{rep}}+\sum_{j=1}^{k}\Delta q_{j}\langle{\cal
O}_{j,2}\prod_{j\neq i=1}^{k}{\cal O}_{i,1}\rangle^{\text{rep}}$
$\displaystyle+\frac{1}{n}\sum_{j=1}^{k}\Delta
q_{j}\bigg{(}\langle\prod_{j\neq i=1}^{k}{\cal
O}_{i,1}\rangle^{\text{rep}}-\langle{\cal O}_{j,2}\prod_{j\neq i=1}^{k}{\cal
O}_{i,1}\rangle^{\text{rep}}\bigg{)}\,.$ (4.8)

The existence of the limit $n\rightarrow 0$ requires that

$\Delta q_{j}(n)=nK_{j}+O(n^{2})\,,\qquad\text{as}\;\;\;n\rightarrow 0\,,$
(4.9)

where

$K_{j}=\frac{\partial\Delta q_{j}}{\partial n}\bigg{|}_{n=0}\,.$ (4.10)

We can use (4.9) to go back to the averaged correlators of the disordered
theory and obtain the desired selection rules

$\displaystyle\sum_{j=1}^{k}q_{j}\overline{\langle\prod_{i=1}^{k}{\cal
O}_{i}\rangle}+\sum_{j=1}^{k}K_{j}\Big{(}\overline{\langle\prod_{i=1}^{k}{\cal
O}_{i}\rangle}-\overline{\langle{\cal O}_{j}\rangle\langle\prod_{j\neq
i=1}^{k}{\cal O}_{i}\rangle}\Big{)}$ $\displaystyle=0\,,$ (4.11)
$\displaystyle\sum_{j=1}^{k}q_{j}\overline{\langle\prod_{i=1}^{k}{\cal
O}_{i}\rangle_{c}}$ $\displaystyle=0\,,$ (4.12)

where

$q_{j}=q_{F,j}|_{n=0}=q_{S,j}|_{n=0}\,,\quad j=1,\ldots,k\,.$ (4.13)

A similar analysis can be repeated for averages of products of correlation
functions of the kind (2.38). We report here only the final result:

$\displaystyle\sum_{m=1}^{N}\sum_{j=1}^{k_{m}}\Bigg{[}\left(q_{j}^{(m)}+K_{j}^{(m)}\right)\overline{\prod_{l=1}^{N}\langle\Upsilon^{(l)}\rangle}$
$\displaystyle+K_{j}^{(m)}\left(\sum_{a\neq
m}\overline{\langle\Upsilon^{(m)}_{j}\rangle\langle\mathcal{O}_{j}^{(m)}\Upsilon^{(a)}\rangle\prod_{l\neq
m,a}\langle\Upsilon^{(l)}\rangle}-N\overline{\langle\mathcal{O}_{j}^{(m)}\rangle\langle\Upsilon^{(m)}_{j}\rangle\prod_{l\neq
m}\langle\Upsilon^{(l)}\rangle}\right)\Bigg{]}=0$ (4.14)

where we introduced the notations

$\Upsilon^{(l)}=\prod_{i=1}^{k_{l}}\mathcal{O}_{i}^{(l)}\,,\,\,\,\,\Upsilon^{(l)}_{j}=\prod_{i=1,i\neq
j}^{k_{l}}\mathcal{O}_{i}^{(l)}\,.$ (4.15)

When $K_{j}=0$, the selection rules (4.11) are the standard ones associated to
a $U(1)$ conserved symmetry, while for $K_{j}\neq 0$ we get additional terms
which affect the disconnected component of the correlator only, given that the
connected part satisfies the ordinary selection rule (4.12). The fact that
(4.12) holds implies that in the disordered theory we have a notion of
operators $\mathcal{O}_{i}$ carrying a definite $U(1)$ charge $q_{i}$, yet in
disconnected correlators some effect is responsible for the appearance of the
extra terms proportional to $K_{j}$. It would be interesting to understand the
origin of these extra factors directly from the disordered theory.

For $k=2$, (4.11) and (4.12) simplify and can be rewritten as

$\begin{split}(q_{1}+q_{2})\overline{\langle\mathcal{O}_{1}\rangle_{c}\langle\mathcal{O}_{2}\rangle_{c}}+(K_{1}+K_{2})\overline{\langle\mathcal{O}_{1}\mathcal{O}_{2}\rangle_{c}}&=0\,,\\\
(q_{1}+q_{2})\overline{\langle\mathcal{O}_{1}\mathcal{O}_{2}\rangle_{c}}&=0\,.\end{split}$
(4.16)

If $K_{1}+K_{2}\neq 0$, independently of the value of $q_{1}+q_{2}$, the
connected part of the 2-point function has to vanish and only a disconnected
component is allowed. We are not aware of disordered theories with $K_{j}\neq
0$ for an internal global symmetry. On the other hand, we will show in the
next section that the exotic selection rules derived above, applied to the
case of emergent conformal symmetry, are at the origin of the possible
appearance of logarithmic CFTs in the IR of disordered theories.

###  4.2 LogCFTs

Infrared fixed points of theories with quenched disorder can be described by
non-unitary LogCFTs, first discussed in 2d [56, 57]. See e.g. [73] for a
review of 2d LogCFTs or [58] for an introduction to LogCFTs in $d$ dimensions
from an axiomatic point of view. It was recognized in [57] that LogCFTs are
intrinsically associated in having primary operators that are highest weight
of indecomposable but not irreducible representations of the conformal group.
A derivation of how LogCFTs can arise as random fixed points was given in [59]
and more recently in [65] by means of (suitable generalizations of) Callan-
Symanzik equations, in both cases using replica methods. We provide here an
alternative derivation, working out the generalization of (4.16) when the
emergent group is assumed to be the conformal one.

In the IR fixed point of the replica theory we have a dilatation current
$J_{d}^{\mu}$ which yields the topological dilatation operator

$D\left[\Sigma^{(d-1)}\right]=\int_{\Sigma^{(d-1)}}\\!\\!J_{d}^{\mu}n_{\mu}\
.$ (4.17)

The conformal Ward identities applied to a primary operator ${\cal O}$ imply

$D\left[\Sigma^{(d-1)}_{x}\right]\mathcal{O}(x)=\delta_{D}\mathcal{O}(x)+\mathcal{O}(x)D\left[\Sigma^{(d-1)}_{\text{no}\,x}\right]\,,$
(4.18)

where

$\delta_{D}\mathcal{O}=\left(\Delta+x^{\mu}\partial_{\mu}\right)\mathcal{O}(x)\,,$
(4.19)

($\Sigma_{\text{no}\,x}^{(d-1)}$) $\Sigma_{x}^{(d-1)}$ is a closed codimension
1 surface (not) encircling $x$. The dilatation operator acts diagonally only
on the irreducible representations (4.3):

$\delta_{D}\mathcal{O}_{i}^{(S)}(x)=\left(\Delta_{S,i}(n)+x^{\mu}\partial_{\mu}\right)\mathcal{O}_{i}^{(S)}(x)\
,\ \
\delta_{D}\mathcal{O}_{i,a}^{(F)}(x)=\left(\Delta_{F,i}(n)+x^{\mu}\partial_{\mu}\right)\mathcal{O}_{i,a}^{(F)}(x)\
.$ (4.20)

Thus on $\mathcal{O}_{i,a}(x)$ we have

$\delta_{D}\mathcal{O}_{i,a}(x)=\left(\Delta_{F,i}+x^{\mu}\partial_{\mu}\right)\mathcal{O}_{i,a}(x)+\frac{\Delta_{S,i}-\Delta_{F,i}}{n}\sum_{\alpha=1}^{n}\mathcal{O}_{i,\alpha}(x)\,,$
(4.21)

where in general $\Delta_{S,i}(n)\neq\Delta_{F,i}(n)$ for finite $n$. We plug
the above transformations in (4.2) with $k=2$ and equal operators. In this way
we find the analogues of (4.16) for scaling transformations:

$\begin{split}\left(x^{\mu}\partial_{\mu}+2\Delta\right)\overline{\langle\mathcal{O}(x)\rangle_{c}\langle\mathcal{O}(0)\rangle_{c}}+2K\overline{\langle\mathcal{O}(x)\mathcal{O}(0)\rangle_{c}}&=0\,,\\\
\left(x^{\mu}\partial_{\mu}+2\Delta\right)\overline{\langle\mathcal{O}(x)\mathcal{O}(0)\rangle_{c}}&=0\,,\end{split}$
(4.22)

where

$\Delta:=\Delta_{F}|_{n=0}=\Delta_{S}|_{n=0}\,,\qquad\qquad
K=\partial_{n}(\Delta_{S}-\Delta_{F})|_{n=0}\,.$ (4.23)

The general solution of (4.22) reads

$\begin{split}\overline{\langle\mathcal{O}(x)\mathcal{O}(0)\rangle}_{c}&=\frac{c_{1}}{|x|^{2\Delta}}\\\
\overline{\langle\mathcal{O}(x)\mathcal{O}(0)\rangle}&=\frac{c_{2}}{|x|^{2\Delta}}-\frac{c_{1}\log(\mu|x|)}{|x|^{2\Delta}}\,,\end{split}$
(4.24)

where $c_{1,2}$ are two integration constants with mass dimension $-2\Delta$
and $\mu$ is an arbitrary mass scale. Note that in a LogCFT, due to the
peculiar way dilatations act on operators, the presence of a mass scale is
actually compatible with conformal symmetry (see e.g. [58] for a more detailed
explanation). We see that the log term arises when $K\neq 0$, which acts as a
source term in the second equation in (4.22).

Whenever the LogCFT has some internal global symmetry $G$ which is not
emergent in the IR but is an exact symmetry present along the whole RG flow
(i.e. present for each member of the ensemble and not broken by the disorder),
the derivation above shows that logarithms can only appear in two-point
functions of operators singlets under $G$. Indeed, in the replica theory the
symmetry $G$ gets replicated in $n$ (unbroken) copies $G_{a}$, while the
conformal symmetry generally is not, being only emergent at the fixed point. A
representation $\rho$ of $G$ acting on a primary operator $\mathcal{O}$ is
then replicated into $n$ copies $\rho_{a}$, each acting only on
$\mathcal{O}_{a}$. Let $g\in G_{a}$, by simple manipulations we get

$\begin{split}\rho_{a}(g)\cdot\mathcal{O}^{(S)}&=\rho_{a}(g)\cdot\mathcal{O}_{a}-\mathcal{O}_{a}+\mathcal{O}^{(S)}\\\
&=\left(\rho_{a}(g)-\mathbbm{1}\right)\cdot\mathcal{O}^{(F)}_{a}+\frac{1}{n}\left(\rho_{a}(g)+(n-1)\mathbbm{1}\right)\cdot\mathcal{O}^{(S)}\,.\end{split}$
(4.25)

Since $G_{a}$ are internal symmetries, which necessarily commute with the
dilatation operator $D$, we have

$\begin{split}0=\left[D,\rho_{a}(g)\right]\cdot\mathcal{O}^{(S)}=\left(\Delta_{F}-\Delta_{S}\right)\left(\rho_{a}(g)-\mathbbm{1}\right)\cdot\mathcal{O}^{(F)}_{a}\,.\end{split}$
(4.26)

Unless $\rho$ is in the trivial representation, the only solution of (4.26) is

$\Delta_{S}(n)=\Delta_{F}(n)\,,$ (4.27)

which implies that the factor $K$ defined in (4.23) vanishes, and thus
logharithms cannot appear in the two-point function of ${\cal O}$ at the IR
fixed point.

##  5 Symmetries in ensemble average

We discuss in this section the case in which the random coupling is taken to
be constant:

$h(x)\rightarrow h\,.$ (5.1)

Such set-up, which does not physically describe impurities as in quenched
disorder, is particularly interesting in the light of the recent understanding
of the role of average QFTs in the AdS/CFT correspondence [34]. As in the case
of quenched disorder, we are interested in the situation where a symmetry is
explicitly broken in any element of the ensemble and we want to see when and
under which conditions it can emerge after the average. To distinguish them
from the case of disordered systems, we will call these symmetries _averaged
symmetries_. A notable example of this kind is the $O(N)$ symmetry in the SYK
model [31, 32, 33] which rotates the $N$ Majorana fermions, broken by the
random fermion coupling, and restored after average (provided the average is
taken with an $O(N)$-invariant distribution, as is often the case).

We will see that the simple replacement (5.1) leads to crucial differences
with respect to the quenched disorder case. We discuss the importance of
connectedness of the full space in section 5.1, we derive the Ward identities
and the topological operators emerging after ensemble average in section 5.2,
and finally in section 5.3 we comment on the implications of our results in
the context of the AdS/CFT correspondence where the ensemble average is
supposed to be the dual theory of a bulk theory of gravity in $d+1$
dimensions.

###  5.1 Factorization and spurions

The presence of a constant random coupling $h$ over the entire space $X^{(d)}$
leads to a new effect, not present in the quenched disorder, which is the lack
of factorization of correlation functions in disconnected spaces. For
definiteness, consider a theory deformed by a random coupling $h$ in a space
$X^{(d)}$ which is the union of two spaces $X^{(d)}=X^{(d)}_{1}\sqcup
X_{2}^{(d)}$, with $X^{(d)}_{1}\cap X^{(d)}_{2}=\emptyset$. At this stage we
are not specifying whether the coupling is a constant or not, we only assume
that it breaks a global 0-form symmetry $G$ of the pure theory. For each
element of the ensemble we can define a generating functional introducing
sources $K_{i}$ for the local operators $\mathcal{O}_{i}$. Since the space
manifold is disconnected, for each local operator ${\cal O}$ we effectively
need two sources, $K_{1}$ and $K_{2}$, defined in $X^{(d)}_{1}$ and
$X^{(d)}_{2}$. For any $h$, constant or not, the total functional
factorizes171717This follows from the observation that any map whose domain is
disconnected can be written uniquely as a sum of maps each supported in a
connected component.

$Z[X^{(d)},K,h]=Z[X^{(d)}_{1},K_{1},h]\;Z[X^{(d)}_{2},K_{2},h]\,,$ (5.2)

and so will do arbitrary correlation functions of local operators $\Phi$:

$\langle\Phi\rangle_{X}=\langle\Phi_{1}\rangle_{X_{1}}\langle\Phi_{2}\rangle_{X_{2}}\,,$
(5.3)

with obvious notation. When $h$ is space dependent (quenched disorder), its
support and its probability measure splits into $X_{1}$ and $X_{2}$. Hence
quenched averaged correlators factorize in the two distinct
components:181818It should not be confused this factorization of correlators
in disconnected space with the non-factorization of products of averaged
correlators due to quenched disorder considered in section 2 and present in
any space $X^{(d)}$, connected or not.

$\overline{\langle\Phi_{1}\rangle_{X_{1}}\langle\Phi_{2}\rangle_{X_{2}}}=\overline{\langle\Phi_{1}\rangle}_{X_{1}}\,\overline{\langle\Phi_{2}\rangle}_{X_{2}}\,.$
(5.4)

Thanks to this factorization, the selection rules of the disordered theory are
realized independently on each connected component:

$\overline{\langle\Phi_{i}\rangle}_{X_{i}}=R_{i}\;\overline{\langle\Phi_{i}\rangle}_{X_{i}}\,,\qquad
i=1,2,\qquad\text{(quenched disorder)}\,,$ (5.5)

where $R_{i}$ are the direct products of the representations of the local
operators in $X_{i}^{(d)}$, which should each contain a singlet to get a non-
vanishing correlator.

Crucially, in the ensemble average case (5.4) cannot hold, because a constant
$h$ does not split on the connected components and the average correlates the
operators across $X_{1}^{(d)}$ and $X_{2}^{(d)}$. In particular, we now get
the selection rules

$\overline{\langle\Phi_{1}\rangle_{X_{1}}\langle\Phi_{2}\rangle}_{X_{2}}=R_{1}\cdot
R_{2}\;\overline{\langle\Phi_{1}\rangle_{X_{1}}\langle\Phi_{2}\rangle}_{X_{2}}\,,\qquad\text{(ensemble
average)}\,.$ (5.6)

In contrast to the quenched disorder case, averages of single correlators in
the ensemble average effectively turn into averages of products of correlators
when the space is disconnected. The constraint (5.6) is weaker than (5.5),
obtained in the quenched average theory. In (5.6) we need the singlet to
appear only in the product $R_{1}\cdot R_{2}$, in (5.5) separately for $R_{1}$
and $R_{2}$. For symmetries that emerge after ensemble average, which we dub
_average_ symmetries, the charge is then not conserved on a single connected
component of the manifold, but can “escape” to the other connected components
(see the end of appendix A.1.2 for an explicit computation in a free scalar
model). We will see how this relates to the violation of global symmetries by
Euclidean wormholes in section 5.3. The above analysis is trivally generalized
to a space with an arbitrary number of disconnected components and to
arbitrary products of correlation functions of local operators.

###  5.2 Ensemble average and Ward identities

The analysis presented in section 2.2 can be repeated in the case of constant
$h$. For concreteness we consider again the case in which the pure theory has
a $U(1)$ global symmetry under which $\mathcal{O}_{0}$ has charge $q_{0}$. We
have one complex parameter $h$ and the average generating functional is

$\overline{Z[K_{i}]}=\int\\!dhd\bar{h}\,P[\bar{h}h]\,\frac{\int\mathcal{D}\mu
e^{-S_{0}-(h\int\mathcal{O}_{0}+c.c.)+\int
K_{i}\mathcal{O}_{i}}}{\int\mathcal{D}\mu
e^{-S_{0}-(h\int\mathcal{O}_{0}+c.c.)}}\,.$ (5.7)

We derive identities between correlators by changing variables inside the
various integrals in (5.7). By changing variable in the numerator with an
infinitesimal space-dependent symmetry transformation of parameter
$\epsilon(x)$, we get

$\langle\partial_{\mu}J^{\mu}(x)\Phi\rangle=\sum_{i}\delta^{(d)}(x-x_{i})q_{i}\langle\Phi\rangle+q_{0}\langle\mathcal{D}(x)\Phi\rangle\,,$
(5.8)

where the sum runs over all the local operators defining $\Phi$ and we have
defined

$\mathcal{D}(x):=-h\mathcal{O}_{0}(x)+\bar{h}\,\overline{\mathcal{O}}_{0}(x)\,.$
(5.9)

Note that (5.8) holds before taking the average. Indeed, this is nothing else
than the Ward identities one obtains in a pure theory for an explicitly broken
symmetry. We are now not allowed to do a change of variable in the $h$
integral to possibly prove the vanishing on average of the last term in (5.8).
However, we can perform a _global_ transformation $h\rightarrow
e^{-iq_{0}\epsilon}h$, with $\epsilon$ constant, inside (5.7). In this way, we
get

$\int_{X^{(d)}}\\!\overline{\big{\langle}\mathcal{D}(x)\Phi\big{\rangle}}=\int_{X^{(d)}}\\!\overline{\big{\langle}\mathcal{D}(x)\big{\rangle}\big{\langle}\Phi\big{\rangle}}\,,$
(5.10)

where $X^{(d)}$ is the _full_ space manifold. Finally we can perform a space
dependent $U(1)$ transformation only in the path integral in the denominator
of (5.7), getting

$\langle\partial_{\mu}J^{\mu}\rangle=q_{0}\langle\mathcal{D}\rangle\,,$ (5.11)

valid before ensemble average. From now on we will assume that
$\mathcal{O}_{0}$ is a scalar under spatial rotations,191919This assumption is
not crucial. For non-scalar deformations, rotational invariance is broken
before the average and we need to keep track of all the vacuum expectation
values induced by the random variable, as done in the quenched disorder case.
This can be repeated in the ensemble average case, but makes the analysis more
involved. so that every element of the ensemble is $\mathfrak{so}(d)$
invariant. We then have $\langle J_{\mu}\rangle=0$ and thanks to (5.11) the
relation (5.10) simplifies to

$\int_{X^{(d)}}\\!\overline{\big{\langle}\mathcal{D}(x)\Phi\big{\rangle}}=0\,.$
(5.12)

See appendix A.1.2 for an explicit derivation of (5.12) for a two-point
function in a simple solvable model. The combination
$\partial^{\mu}J_{\mu}-q_{0}\mathcal{D}(x)$ satisfies the condition

$\int_{X^{(d)}}\\!\\!d^{d}x\,\overline{\big{\langle}\left(\partial^{\mu}J_{\mu}(x)-q_{0}\mathcal{D}(x)\right)\Phi\big{\rangle}}=0\,,$
(5.13)

which ensures that the Ward identities (5.8), when integrated over the full
space and after ensemble average, imply charge conservation. As expected from
a spurionic argument, the symmetry is restored after average.202020In a pure
theory the identities (5.8) apply but ${\cal{D}}(x)$ does not integrate to
zero when inserted in arbitrary correlators. As a consequence no selection
rules are implied, as expected for an explicitly broken symmetry!

Let us now see if we can define more general operators
$\widehat{Q}[\Sigma^{(d-1)},D^{(d)}]$, topological after ensemble average. The
natural choice from (5.13) is

$\widehat{Q}[\Sigma^{(d-1)},D^{(d)}]=Q[\Sigma^{(d-1)}]-q_{0}\int_{D^{(d)}}\\!\\!d^{d}x\,\mathcal{D}(x)\,,\qquad
Q[\Sigma^{(d-1)}]:=\int_{\Sigma^{(d-1)}}\\!\\!n_{\mu}J^{\mu}(x)\,,$ (5.14)

where $D^{(d)}$ is an arbitrary region such that $\partial
D^{(d)}=\Sigma^{(d-1)}$. Note that this requires $\Sigma^{(d-1)}$ to be
homologically trivial otherwise, by definition, the surface $D^{(d)}$ does not
exist. In the terminology of [74], the operator (5.14) is a _non-genuine_ co-
dimension one operator, since it requires a topological surface attached to
it.212121The requirement is however of different nature. In [74] (and
subsequent works) the surface is required to have a well-defined gauge-
invariant operator, here the surface is required to make the operator
topological (on average).

We can discuss the dependence of $\widehat{Q}$ in (5.14) on the choice of the
filling region $D^{(d)}$. Given another such manifold $D^{\prime(d)}$ we can
glue it along $\Sigma^{(d-1)}$ with the orientation reversal of $D^{(d)}$ to
form a closed manifold $Y^{(d)}=D^{\prime(d)}\sqcup\overline{D^{(d)}}$, and
$\widehat{Q}[\Sigma^{(d-1)},D^{(d)}]$ is independent on $D^{(d)}$ if and only
if

$\int_{Y^{(d)}}\\!\overline{\big{\langle}\mathcal{D}(x)\Phi\big{\rangle}}=0\
.$ (5.15)

We see that (5.15) is not satisfied unless the space-time $X^{(d)}$ is
connected, and we will generically refer to $\widehat{Q}$ as a non-genuine
operator. On the other hand, if $X^{(d)}$ is connected any homologically
trivial co-dimension one submanifold $\Sigma^{(d-1)}$ of $X^{(d)}$ divides
$X^{(d)}-\Sigma^{(d-1)}$ in two disjoint connected components glued along
$\Sigma^{(d-1)}$, hence necessarily $Y^{(d)}=X^{(d)}$ and (5.15) reduces to
(5.12), showing the independence of $\widehat{Q}[\Sigma^{(d-1)},D^{(d)}]$ on
the filling region. $\widehat{Q}$ is still expressed with an integral over
$D^{(d)}$, but the dependence of the non-genuine symmetry operator on the
filling region is only apparent, and for all practical purposes this can be
regarded as independent on the filling region. We refer to this situation as a
quasi-genuine co-dimension one operator.

If $X^{(d)}$ has several connected components, $Y^{(d)}$ can be a proper sub-
region, since adding or removing from it an entire connected component which
does not intersect $\Sigma^{(d-1)}$ preserves the property that $Y^{(d)}$ is
the union of regions glued along $\Sigma^{(d-1)}$. For instance if $X^{(d)}$
has two connected components $X_{1}^{(d)}$ and $X_{2}^{(d)}$, and suppose
$\Sigma^{(d-1)}$ is entirely contained in $X_{1}^{(d)}$, the latter is divided
by $\Sigma^{(d-1)}$ into two regions $D^{(d)}$ and $D^{{}^{\prime}(d)}$, and
choosing one or the other leads to different operators
$\widehat{Q}[\Sigma^{(d-1)}]$, since (5.12) holds only in the entire space and
not to each connected component:

$\overline{\Big{\langle}\int_{D^{(d)}}\\!\\!\mathcal{D}(x)\Phi\Big{\rangle}}=\overline{\Big{\langle}\Big{(}\int_{D^{{}^{\prime}(d)}}+\int_{X_{2}^{(d)}}\Big{)}\mathcal{D}(x)\Phi\Big{\rangle}}\neq\overline{\Big{\langle}\int_{D^{{}^{\prime}(d)}}\\!\\!\mathcal{D}(x)\Phi\Big{\rangle}}\,.$
(5.16)

In this case we cannot define a quasi-genuine co-dimension one topological
operator and therefore, even if the total charge is conserved thanks to
(5.13), we cannot measure it locally in a subregion of the entire
(disconnected) space.

$\mathcal{O}_{1}$

$\mathcal{O}_{2}$

$\mathcal{O}_{3}$

$\mathcal{O}_{4}$

$\Sigma^{(d-1)}$

$D^{(d)}$

$=$

$\mathcal{O}_{1}$

$\mathcal{O}_{2}$

$\mathcal{O}_{3}$

$\mathcal{O}_{4}$

$\Sigma^{(d-1)}$

$D^{{}^{\prime}(d)}$

Figure 1: Selection rules (5.18) for correlators when $X^{(d)}$ is connected.
The integral over the region $D^{(d)}$ in the left panel equals the integral
over the region $D^{{}^{\prime}(d)}$ in the right panel thanks to (5.12). When
$D^{(d)}$ is shrunk to a point the region $D^{{}^{\prime}(d)}$ extends to the
whole $X^{(d)}$.

In order to measure the charge of operators in the whole space, we can
consider $\widehat{Q}$ on a codimension 1 closed surface
$\Sigma^{(d-1)}=\Sigma_{1}^{(d-1)}\sqcup\Sigma_{2}^{(d-1)}$, with
$\Sigma_{i}^{(d-1)}\subset X_{i}^{(d)}$ ($i=1,2$), and two regions
$D_{i}^{(d)}$ such that $\partial D_{i}^{(d)}=\Sigma_{i}^{(d-1)}$. In each
given connected component, the charge cannot be conserved, as we have seen,
but if we simultaneously consider the two regions, then the Ward identities
still apply. In the schematic notation of section 5.1 we have

$\begin{split}\overline{\langle\widehat{Q}[\Sigma,D]\Phi\rangle}_{X}&=\overline{\langle\widehat{Q}[\Sigma_{1},D_{1}]\Phi_{1}\rangle_{X_{1}}\langle\widehat{\Phi}_{2}\rangle}_{X_{2}}+\overline{\langle\Phi_{1}\rangle_{X_{1}}\langle\widehat{Q}[\Sigma_{2},D_{2}]\Phi_{2}\rangle}_{X_{2}}\\\
&=\Big{(}\chi_{1}(\Sigma_{1})+\chi_{2}(\Sigma_{2})\Big{)}\overline{\langle\widehat{\Phi}_{1}\rangle_{X_{1}}\langle\widehat{\Phi}_{2}\rangle}_{X_{2}}=\Big{(}\chi_{1}(\Sigma_{1})+\chi_{2}(\Sigma_{2})\Big{)}\overline{\langle\widehat{\Phi}\rangle}_{X}\,,\end{split}$
(5.17)

where $\chi_{1,2}(\Sigma_{1,2})$ denotes the sum of the charges of the local
operators $\Phi_{1,2}$ which are inside the surface $\Sigma^{(d-1)}_{1,2}$.
Since $\Sigma^{(d-1)}$ depends now on $D^{(d)}$, it is crucial to consider the
complement space in both connected spaces at the same time. The generalization
to spaces $X^{(d)}$ with more than two connected components is obvious.

We refer the reader to appendix B for a proof of the existence of the operator
$\widehat{U}_{g}$ which implements the action of the group rather than the
action of the corresponding Lie algebra. By definition, the operator
$\widehat{U}_{g}$, given in (B.20), satisfies

$\overline{\langle\widehat{U}_{g}[\Sigma^{(d-1)},D^{(d)}]\mathcal{O}_{1}\cdots\mathcal{O}_{n}\rangle}=e^{i\alpha\chi(\Sigma^{(d-1)})}\overline{\langle\mathcal{O}_{1}\cdots\mathcal{O}_{n}\rangle}\,.$
(5.18)

Since $\widehat{U}_{g}[\emptyset,X^{(d)}]=1$, (5.18) implies the selection
rules we derived from the spurion argument (see figure 1). The equivalent of
(5.17) for a finite group action precisely reproduces the selection rule
(5.6). With $\Sigma^{(d-1)}$ as in figure 2, we have

$\overline{\langle\Phi\rangle}_{X}=\overline{\langle\widehat{U}_{g}[\Sigma,D]\Phi\rangle}_{X}=\overline{\langle\widehat{U}_{g}[\Sigma_{1},D_{1}]\Phi_{1}\rangle_{X_{1}}\langle\widehat{U}_{g}[\Sigma_{2},D_{2}]\Phi_{2}\rangle}_{X_{2}}=e^{i\alpha(\chi_{1}(\Sigma_{1})+\chi_{2}(\Sigma_{2})}\overline{\langle\Phi\rangle}_{X}$
(5.19)

while, say,

$\overline{\langle\Phi\rangle}_{X}=\overline{\langle\widehat{U}_{g}[\Sigma_{1},D_{1}]\Phi_{1}\rangle_{X_{1}}\langle\Phi_{2}\rangle}_{X_{2}}\neq
e^{i\alpha\chi_{1}(\Sigma_{1})}\overline{\langle\Phi\rangle}_{X}\,.$ (5.20)

We have then found an instance of a theory with a global zero-form symmetry in
the sense of giving rise to selection rules for correlation functions of local
operators, but with no genuine co-dimension one topological operator. Aside of
being topological only on average, the operator $\widehat{U}_{g}[\Sigma,D]$ is
not genuine and it can be defined only on homologically trivial cycles.

$\mathcal{O}_{1}$

$\mathcal{O}_{2}$

$\Sigma_{1}^{(d-1)}$

$D_{1}^{(d)}$

$\mathcal{O}_{4}$

$\mathcal{O}_{3}$

$\Sigma_{2}^{(d-1)}$

$D_{2}^{(d)}$

$\not=$

$\mathcal{O}_{1}$

$\mathcal{O}_{2}$

$\Sigma_{1}^{(d-1)}$

$D^{{}^{\prime}(d-1)}_{1}$

$\mathcal{O}_{4}$

$\mathcal{O}_{3}$

$\Sigma_{2}^{(d-1)}$

$D_{2}^{(d)}$

Figure 2: Violation of the selection rules (5.20) when $X^{(d)}$ is
disconnected. The integral over the region $D_{1}^{(d)}$ in $X_{1}^{(d)}$
(left) is not equal to the integral over the region $D_{1}^{{}^{\prime}(d)}$
in $X_{1}^{(d)}$ (right) because of the presence of the component
$X_{2}^{(d)}$. An equality sign would require to reverse the region of
integration also in $X_{2}^{(d)}$ (right) from $D_{2}^{(d)}$ to its
complement.

The local charge violation (5.20) in a single connected component of space
when $X^{(d)}$ is an union of several connected components indicate the
presence of non-local interactions in the theory. Their presence is manifest
by using the replica trick. Consider a Gaussian random distribution
$P[\bar{h}h]\propto\exp(-\bar{h}h/v)$ (e.g. as in the SYK model). Repeating
the steps described in section 3 we find _non-local_ interactions among
replicas

$S_{\text{rep}}=\sum\limits_{a=1}^{n}S_{0,a}-v\int\\!d^{d}x\\!\int\\!d^{d}y\,\sum_{a,b=1}^{n}\overline{\mathcal{O}}_{0,a}(x)\mathcal{O}_{0,b}(y)\,.$
(5.21)

The replica theory enjoys a diagonal $U(1)_{D}$ global symmetry, but the naive
diagonal current $J^{\mu}_{D}=\sum_{a}J^{\mu}_{a}$ does not satisfy standard
Ward identities. By performing an infinitesimal $U(1)$ transformation with a
local parameter $\alpha(x)$ we get

$\displaystyle\delta S_{\text{rep}}$ $\displaystyle=\int
dx\,\alpha(x)\partial_{\mu}J^{\mu}_{D}(x)-q_{0}v\sum_{a,b}\int
dx\,dy\,\Big{(}\alpha(y)-\alpha(x)\Big{)}\overline{\mathcal{O}}_{0,a}(x)\mathcal{O}_{0,b}(y)$
(5.22)
$\displaystyle=\int_{X^{(d)}}dx\,\alpha(x)\bigg{(}\partial_{\mu}J^{\mu}_{D}(x)+q_{0}v\sum_{a,b}\int_{X^{(d)}}dy\,\Big{(}\overline{\mathcal{O}}_{0,a}(x)\mathcal{O}_{0,b}(y)-\mathcal{O}_{0,a}(x)\overline{\mathcal{O}}_{0,b}(y)\Big{)}\bigg{)}\,.$

Thus the Ward identities for the diagonal symmetry are modified by a non-local
term and read

$\begin{array}[]{l}\displaystyle\Big{\langle}\Big{(}\partial_{\mu}J^{\mu}_{D}(x)+q_{0}v\sum_{a,b}\int_{{X^{(d)}}}\\!\\!d^{d}y\left(\overline{\mathcal{O}}_{0,a}(x)\mathcal{O}_{0,b}(y)-\overline{\mathcal{O}}_{0,a}(y)\mathcal{O}_{0,b}(x)\right)\Big{)}\Phi\Big{\rangle}^{{\text{rep}}}\\\
\displaystyle\qquad=\sum_{i}\delta^{(d)}(x-x_{i})q_{i}\big{\langle}\Phi\big{\rangle}^{\text{rep}}\,.\end{array}$
(5.23)

In the replica theory the operator

$\partial_{\mu}J^{\mu}_{D}(x)+q_{0}v\sum_{a,b}\int_{X^{(d)}}\\!\\!d^{d}y\,\left(\overline{\mathcal{O}}_{0,a}(x)\mathcal{O}_{0,b}(y)-\overline{\mathcal{O}}_{0,a}(y)\mathcal{O}_{0,b}(x)\right)$
(5.24)

satisfies the Ward identities and its integral over the full space evidently
vanishes (inside arbitrary correlators), implying the $U(1)_{D}$ selection
rules. This is how the properties of the averaged symmetry show up in the
replica theory, where the non-local nature of the symmetry is manifest for
Gaussian distributions. The property (5.12) of the operator $\mathcal{D}(x)$
defined in (5.9) is mapped to the property of the extra term in (5.24) of
integrating to zero exactly as an operator equation. This is consistent with
the dictionary between correlators of the averaged theory and the replica one.

###  5.3 A gravity discussion

We have found that averaged global symmetries are intrinsically different from
ordinary global symmetries. They imply selection rules as dictated by the
global symmetry but, in contrast to ordinary global symmetries, they do not
admit genuine co-dimension one operators, topological after average. Even in a
connected space such operators cannot be defined in homologically non-trivial
cycles. As a result, these symmetries cannot consistently be coupled to an
external background field, at least not in a natural way.222222For discrete
symmetries, for example, coupling to an external background field corresponds
to insert a mesh of symmetry defects on homologically non-trivial cycles of
space. Note that this is different from the concept of ’t Hooft anomalies. In
the latter the obstruction is in making the gauge fields dynamical but there
is a well defined notion of coupling the theory to backgrounds gauge fields.
The difficulty of coupling the symmetry to an external background is clear in
the replica theory from the presence of the second term in (5.24), which is
non-local and not manifestly the divergence of a current.

The results have interesting consequences when applied to averaged theories
which are assumed to have an holographic dual bulk gravitational theory in
asymptotically AdS space-times.

$X_{1}^{(d)}$

$X_{2}^{(d)}$

$M^{(d+1)}$

$\mathcal{O}$

$\mathcal{O}$

$\mathcal{O}^{\dagger}$

$\mathcal{O}^{\dagger}$

Figure 3: Example of a wormhole bulk geometry contributing to the average
correlator $\overline{\langle{\cal O}{\cal O}\rangle\langle{\cal
O}^{\dagger}{\cal O}^{\dagger}\rangle}$, with ${\cal O}$ a charged boundary
operator.

In the ordinary AdS/CFT correspondence a given theory of gravity in
asymptotically AdS space-time is dual to a given CFT. Ordinary global
symmetries of the CFT become gauge symmetries in the bulk. This correspondence
fits nicely with the widely accepted common lore that in quantum gravity
unbroken global symmetries cannot exist [75, 76, 77, 78]. A natural question
then arises: when the dual theory is given by an ensemble average, what is the
bulk interpretation of the symmetries emerging after average? In [60] (see
also [61, 62]) it has been conjectured that boundary emergent symmetries
correspond in the bulk to global, and not gauge, symmetries which are broken
non-perturbatively by Euclidean wormhole configurations, which allow the
global symmetry charge to flow from one connected component to another one,
see figure 3. From the boundary point of view, this charge violation induced
by bulk wormholes correspond to the lack of selection rules in the average
theory that we have discussed before, when the space is not connected, in
agreement with the findings in [60, 61, 62]. Since averaged symmetries simply
cannot be gauged, our results clarify why they cannot be interpreted as gauge
symmetries in the bulk, at least in the case where the average is of the form
(2.9).232323In particular, our results do not straightforwardly apply when the
average is over OPE coefficients, as e.g. discussed in [43, 44].

Note that boundary emergent symmetries are compatible with recent works where,
motivated by the connection with the lore of spectrum completeness in
gravitational theories [79], “absence of global symmetries in gravitational
theories” is replaced by “absence of topological operators”, including those
related to non-invertible symmetries [80, 81].

##  6 Conclusions

In this paper we have studied disordered QFTs where an ordinary symmetry of a
pure QFT is explicitly broken by a random coupling, but the symmetry re-
emerges after quenched average. We focused our attention to understand if and
under what conditions we can have operators, topological on average, in
analogy to ordinary QFTs [1]. We considered _quenched disorder theories_ ,
where the pure theory is deformed with a space dependent coupling, and
_ensemble average theories_ , where the latter is kept constant.

In the quenched disordered case, we can write Ward identities for averages of
products of correlators and construct the symmetry operator implementing the
finite group action, topological after average. Such disordered symmetries can
be coupled to external background, can be gauged, and can have ’t Hooft
anomalies (i.e. can exclude a trivially gapped phase at long distances),
precisely like ordinary symmetries. Using the replica trick, we also discussed
genuinely emergent symmetries in the IR after average, namely symmetries which
are not present in the UV theory even before adding the disorder coupling. We
pointed out that whenever a symmetry $G$ is emergent in the IR, exotic
selection rules can explain the origin of LogCFTs.

In ensemble average theories the analogy to pure QFTs is more loose. We still
have selection rules for averages of correlators and we can construct
operators implementing the finite group action, but the charge operator is not
purely codimension-1 and cannot be defined if $\Sigma^{(d-1)}$ is
homologically non-trivial. When the space is disconnected, the selection rules
apply only globally and in each connected component charge violation can
occur. Such averaged symmetries cannot be coupled to background gauge fields
in ordinary ways. The difficulty (impossibility) of gauging emergent boundary
symmetries clarify why such symmetries cannot be identified with bulk gauge
symmetries when the average theory admits a gravitational bulk dual.

It would be interesting to analyze spontaneous breaking of disordered
symmetries in more detail.242424We thank O. Aharony for a question that
prompted the paragraph that follows. There are essentially two ways in which
the disordered symmetry could spontaneously break: i) the symmetry is
spontaneously broken in the pure theory before adding the random interaction,
ii) the symmetry is unbroken in the pure theory and the random interaction
induces a spontaneous breaking of the disordered symmetry. Let us consider the
case of continuous symmetries. From the replica theory point of view, i) and
ii) are distinguished by which components of the replica currents
$J^{\mu}_{a}$ are subject to spontaneous breaking, all components in case i)
and only the singlet $\sum_{a}J^{\mu}_{a}$ in case ii). Assuming the existence
of the analytic continuation in $n$ and of a smooth $n\rightarrow 0$ limit, we
expect for $d>2$ gapless excitations (Goldstone boosons) in the replica
theory, giving rise to power-like correlators. From the disordered theory
point of view, in case i) there is a Goldstone mode in the pure theory which
acquires a mass in each specific realization of the ensemble, turning into a
pseudo Goldstone boson. In contrast, no Goldstone boson is present in the pure
theory in the more exotic case ii). In both cases it would be nice to identify
which correlators (if any) exhibit power like-behavior on average as a result
of the spontaneous breaking of the disordered theory.

It would be also interesting to generalize our findings to quantum disorder,
namely to Lorentzian theories where the random coupling depends only on space.
The natural extension of our analysis beyond 0-form symmetries does not seem
straightforward. Higher-form symmetries can be broken only by non-local
deformations, which should be also taken random. It is possibly easier to
consider a set-up in $d=2$ where non-invertible symmetries can be obtained by
0-form symmetries only, and see if and in what sense we can have a non-
invertible symmetry re-emerging after average.

An important remark about the ensemble average case is that, in comparing our
findings with the existing literature on the factorization problem in AdS/CFT,
one should keep in mind that we only considered averaging over couplings.
There are other setups, like averaging over OPE coefficients [43, 44] or over