aainstitutetext: Center for Cosmology and Particle Physics, Department of
Physics, New York University, New York, NY 10003, USAbbinstitutetext:
Department of Physics, Princeton University, Princeton, NJ 08544,
USAccinstitutetext: C. N. Yang Institute for Theoretical Physics, Stony Brook
University, Stony Brook, NY 11794, USAddinstitutetext: Simons Center for
Geometry and Physics, SUNY, Stony Brook, NY 11794, USA

# Moduli Spaces in CFT: Large Charge Operators

Gabriel Cuomo c Leonardo Rastelli d Adar Sharon<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Using the large-charge expansion, we prove a necessary condition for a CFT to
exhibit conformal symmetry breaking, under the assumption that a continuous
global symmetry is also broken on the moduli space: there must be a tower of
charged local operators whose scaling dimensions are asymptotically linear in
the charge. In supersymmetric theories with a continuous R-symmetry and a
holomorphic moduli space, the existence of such a tower of operators follows
trivially from a BPS condition: their scaling dimensions are then exactly
linear in the R-charge. We illustrate the more general statement in several
examples of three-dimensional ${\cal N}=1$ CFTs, where the leading linear
behavior receives nontrivial corrections. By considering a suitable scaling
limit, we also relate the spectrum of states with large charge on the cylinder
(isomorphic to local operators) to the spectrum of massive particles on the
moduli space.

##  1 Introduction

Conformal field theories that exhibit spontaneous breaking of conformal
symmetry (a moduli space of vacua) are highly non-generic. Their key property,
the absence of a potential for the dilaton, is a special feature that requires
fine tuning. It seems however difficult to fully characterize this feature in
terms of abstract CFT data. In principle, one can set up an abstract bootstrap
problem, involving the usual data (scaling dimensions and OPE coefficients) as
well as new data such as the spectrum of asymptotic states in the broken
vacuum and form factors. In a recent paper Cuomo:2024vfk , we illustrated the
simplest bootstrap equation Karananas:2017zrg  in a concrete perturbative
example.111See Ivanovskiy:2024vel  for a similar analysis in ${\cal N}=4$ SYM
theory. Our findings confirmed the general intuition that theory data must
satisfy some very special relations for conformal symmetry breaking to occur,
but a simple general criterion is still elusive.

In this work we address the problem from a different angle. We prove a simple
necessary condition for spontaneous conformal symmetry breaking to occur,
under the (crucial) assumption that a continuous global symmetry is also
broken on the moduli space. Let $\Delta_{\rm min}(Q)$ be the minimum scaling
dimension among all local operators of charge222Here for ease of notation we
take the global symmetry to be $U(1)$. $Q$. The necessary condition is that
$\Delta_{\rm min}(Q)$ must be asymptotically linear,

$\text{CFT with moduli space s.t. }\cancel{Q}\;\implies\;\Delta_{\rm
min}(Q)=\alpha_{0}Q+\dots\quad{\rm for}\,Q\to\infty\,,$ (1.1)

where the dots indicate possible subleading corrections.

It will come as no big surprise that our basic tool is the large-charge
expansion initiated in Hellerman:2015nra ,333 This is by now a large subject,
see e.g. Gaume:2020bmp  for a review. a beautiful and surprising application
of effective field theory (EFT) ideas. Even in a strongly-coupled CFT, certain
observables of large charge $Q\gg 1$ are captured by a weakly-coupled EFT, for
which $1/Q$ is the natural small parameter. One such observable is precisely
$\Delta_{\text{min}}(Q)$, which by the state operator correspondence is given
by the ground-state energy on the cylinder of the large-charge EFT at fixed
$Q$. Our claim (1.1) follows by a straightforward extension of the analysis of
Hellerman:2017veg , enforcing the sole additional assumption that the theory
has a moduli space where the charge is spontaneously broken. The form of the
subleading corrections is dimension dependent. In $d=3$ we find the behavior

$\Delta_{\text{min}}(Q)=\alpha_{0}Q+\alpha_{1}+\alpha_{2}/Q+O(1/Q^{2})\;,$
(1.2)

while in $d=4$ the expansion takes the form

$\Delta_{\text{min}}(Q)=\alpha_{0}Q+\alpha_{1}\log Q+\alpha_{2}+O(1/Q)\;.$
(1.3)

The necessary condition (1.1) is established in full generality, with no
recourse to supersymmetry. However, when it comes to check it in concrete
examples, we must face the predicament that all known interacting, local444In
most of the paper we assume the existence of a local stress tensor with a
finite two-point function. It is not difficult to find examples of CFTs with
an approximate moduli space at leading in a large $N$ expansion, as we briefly
discuss in section 3.5. CFTs with a moduli space of vacua are in fact
supersymmetric. The most familiar class of examples have at least four real
supercharges in $d=3$ and $d=4$ spacetime dimensions. Their moduli spaces are
holomorphic, parametrized by the vacuum expectation values of chiral
operators. Chiral operators are charged under a continuous R-symmetry, which
is thus also spontaneously broken. In these theories, holomorphy goes hand in
hand with the existence of infinite towers of protected chiral operators with
exactly linearly-spaced scaling dimensions, proportional to the R-charge,
$\Delta(Q)=\alpha Q$. Mathematically, the ring of chiral operators coincides
with the holomorphic coordinate ring on the moduli space.555The more precise
statement is that the coordinate ring is identified with reduced chiral ring,
i.e. with the ring of chiral operators where nilpotents elements have been
quotiented out. This can be established rigorously in Lagrangian models
Luty:1995sd  and it is believed to be true (general lore!) in any 3d and 4d
SCFT with at least four real supercharges. Physically, the chiral operators
arise (at least at some intuitive level) from quantization of the holomorphic
moduli. In this whole class of examples, (1.1) follows trivially from a BPS
condition. From the viewpoint of the large-charge EFT, this amount of
supersymmetry imposes selection rules that set to zero the higher coefficients
$\alpha_{i\geq 1}$.

Fortunately, there is another class of supersymmetric theories for which (1.1)
is nontrivial: the three-dimensional SCFTs with only ${\cal N}=1$ SUSY (two
real supercharges). These theories have no continuous R-symmetry, no
holomorphy and no protected operators. Nevertheless, they can still admit
exact moduli spaces thanks a discrete $\mathbb{Z}_{2}$ R-symmetry
Gaiotto:2018yjh , as we review in Appendix B. They are thus the perfect
playground to study the interplay between moduli spaces and global symmetries.

We consider several explicit examples of $\mathcal{N}=1$ SUSY theories with
moduli spaces, and check that (1.1) holds with non-vanishing corrections to
the leading linear behavior. In selecting the matter content and interactions
of these examples, some care had to be taken to ensure that they have a moduli
space with a broken global charge while at the same time no emergent ${\cal
N}=2$ supersymmetry in the IR. Most of our examples are Wess-Zumino theories,
which can be studied perturbatively in $4-\epsilon$ dimensions by analytically
continuing in the number of fermions Fei:2016sgs ; ThomasSeminar , as we
review in appendix B. We go beyond the EFT analysis and compute explicitly the
coefficients $\alpha_{i}$ in the $\epsilon$-expansion, following Badel:2019oxl
.

In our concrete examples, the leading correction to the linear behavior of
$\Delta_{\text{min}}(Q)$ turns out to be negative or zero. This is perhaps
related to the charge convexity conjecture Aharony:2021mpc , which
schematically claims that $\Delta_{\text{min}}(Q)$ is a convex function of
$Q$. While a counterexample to the original conjecture appeared in
Sharon:2023drx , our findings might support a modification of the conjecture
(“convexity only at large enough charge”) that bypasses the counterexample.
However we have not been able to prove this weaker conjecture and leave
further exploration to future work.

The large-charge EFT allows us compute additional observables as well,
including OPE coefficients and dimensions of operators just above
$\Delta_{\text{min}}(Q)$. Further predictions can be obtained by assuming the
existence of a macroscopic limit Jafferis:2017zna . This limit relates
correlation functions in the large-charge state on the cylinder to flat-space
correlators on the moduli space by taking $Q\to\infty$ as well as taking the
cylinder radius $R\to\infty$ by keeping a specific ratio fixed. Physically,
this limit corresponds to “zooming in” to distances much smaller than the
sphere radius but keeping the charge large, so that we expect the result to be
a flat-space correlator in a nontrivial state. By applying this procedure to
e.g. four-point functions of large-charge operators, we can relate the
spectrum of large charge operators to the spectrum of massive particles on the
moduli space.

We can summarize the main conceptual lesson as follows. In CFTs with a moduli
space where a global charge is also broken, large charge operators create, in
radial quantization, semiclassical states that closely resemble the vacua
where both the conformal and the internal symmetry are broken. This implies
the existence of a tower of operators whose scaling dimension grows linearly
with the charge. The existence of a conserved charge is essential, as it
allows to focus on a precise tower of states, those corresponding to the
lowest dimensional charged operators. However one might wonder whether more
generally, in CFTs with a moduli space but no global charge, there could still
be some simple semiclassical signature of spontaneous conformal symmetry
breaking in the high energy spectrum. In section 5 we offer some musings on
this question, starting from the EFT of a real dilaton. We speculate that in
the general case the moduli space is reflected in the existence of certain
special resonant states on the cylinder, whose width becomes parametrically
narrower than their energy in the high energy limit.

The remainder of the paper is organized as follows. In section 2 we review the
construction of the moduli space EFT and we compute $\Delta_{\text{min}}(Q)$
as well as of some additional observables. We explain in detail how to extract
the leading-order result and the form of the corrections to it. Next in
section 3 we discuss several explicit examples of moduli spaces at large
charge in $d=3$ ${\cal N}=1$ theories. We compute the first correction to the
leading-order result and discuss some applications. We also briefly consider
examples of non-supersymmetric large $N$ theories with an approximate moduli
space. In section 4 we apply the macroscopic limit to large-charge correlators
and relate them to flat-space correlators on the moduli space. This limit
allows us to write down precise equations relating the spectrum of large
charge local operators to the massive spectrum on the moduli space. In section
5 we offer some speculations on the general case of moduli spaces with no
broken global symmetry. Several appendices complement the text with further
technical details.

##  2 EFT and CFT data at large charge

In this section we argue that if a CFT admits a moduli space of vacua in which
a continuous internal symmetry group $G$ is (either partially or completely)
spontaneously broken, then the lowest dimensional operator with large charge
under a broken Cartan generator $Q$ has scaling dimension linearly
proportional to the charge:

$\text{CFT with moduli space s.t. }\cancel{Q}\;\implies\;\Delta_{\rm
min}(Q)\propto Q\quad{\rm for}\,Q\to\infty\,.$ (2.1)

For non-abelian groups, $\Delta_{\rm min}(Q)$ is the scaling dimension of an
operator in a large representation of $G$.

The argument leading to this result uses the EFT for the massless modes on the
moduli space and the state-operator correspondence. This approach is a simple
generalization of that of Hellerman:2017veg , where the authors studied a
specific supersymmetric theory, and the lowest dimensional charged operators
have protected dimensions $\Delta(Q)=\alpha Q$ due to a BPS shortening
condition. As it will become clear below, the conclusion that $\Delta_{\rm
min}(Q)\propto Q$ for large $Q$ is independent of the details of the theory
and of supersymmetry, as long as the CFT admits a moduli space. The
proportionality coefficient in (2.1), as well as the subleading corrections
and the spectrum of large-charge operators with $\Delta\gtrsim\Delta_{\rm
min}$ are also calculable from the EFT.

###  2.1 Moduli effective theory

Consider a CFT with an $n$-dimensional moduli space in $d>2$. We denote its
internal symmetry group $G$, and assume that along the moduli space the
conformal symmetry is spontaneously broken along with the internal symmetry
group as $G\to H$, for some (possibly trivial) $H\subset G$. We can use an EFT
to describe states on a moduli space at energies much smaller than the gap $m$
of the massive states. The massless degrees of freedom always include at least
$n$ scalar fields whose expectation values locally parametrize the moduli
space; among these, we always find a dilaton $\Phi$ together with
$\text{dim}(G/H)$ Goldstone bosons. In supersymmetric theories the massless
degrees of freedom also include the superpartners. At a generic point on the
moduli space it is always possible to make a change of coordinates such that
there is one noncompact direction (which we associate with the dilaton $\Phi$)
and $n-1$ compact directions $\phi_{A}$, $A=1,...,n-1$ corresponding to the
additional moduli, including the Goldstones for the internal symmetry.

The EFT is constrained by the nonlinear realization of the conformal symmetry
and the spontaneously broken internal symmetries Salam:1969bwb ; Isham:1970gz
; Ellis:1971sa . In practice, the EFT is most easily constructed by imposing
Weyl invariance and introducing a Weyl invariant combination of the metric and
the dilaton as (see e.g. Luty:2012ww )

$\hat{g}_{\mu\nu}=g_{\mu\nu}|\Phi|^{\frac{4}{d-2}}\,,$ (2.2)

where we parametrized the dilaton $\Phi$ so that it transforms as a real
canonically normalized scalar field in $d$-dimensions under the conformal
group, and we take the additional moduli $\phi^{A}$ dimensionless, and thus
with zero Weyl weight. In this section $g_{\mu\nu}$ can be taken to be the
flat metric $g_{\mu\nu}=\text{diag}(1,-1,-1,-1)$, but note that we could also
work on any other Weyl equivalent manifold by Weyl invariance; we will make
use of this observation in the next section.

Denoting with a hat geometric invariants constructed from the rescaled metric
(2.2), it is easy to write the effective action to leading order in
derivatives

$S_{EFT}=\int
d^{d}x\sqrt{\hat{g}}\left[\frac{1}{2}\hat{G}_{AB}(\phi)\hat{g}^{\mu\nu}\partial_{\mu}\phi^{A}\partial_{\nu}\phi^{B}+\hat{G}_{\Phi\Phi}(\phi)\frac{d-2}{8(d-1)}\hat{\mathcal{R}}\right]\\\
+S_{\rm CFT_{\rm IR}}+\text{irrelevant couplings}\,,$ (2.3)

where $\hat{G}_{AB}(\phi)$ and $\hat{G}_{\Phi\Phi}(\phi)$ specify the metric
of the non-linear sigma model (NLSM) and are constrained by the nonlinear
realization of the internal symmetry or supersymmetry when present. The
coefficient multiplying $\hat{G}_{\Phi\Phi}$ was chosen for future convenience
and the rescaled Ricci scalar contains the dilaton kinetic term

$\hat{\mathcal{R}}=|\Phi|^{-\frac{4}{d-2}}\left[\mathcal{R}-\frac{4(d-1)\nabla^{2}\Phi}{(d-2)\Phi}\right]\,.$
(2.4)

Note indeed that in flat space the first line of the action (2.3) can be
written as a standard NLSM

$\int
d^{d}x\left[\frac{1}{2}G_{AB}(\Phi,\phi)\partial_{\mu}\phi^{A}\partial^{\mu}\phi^{B}+\frac{1}{2}G_{\Phi\Phi}(\phi)(\partial\Phi)^{2}+G_{A\Phi}(\Phi,\phi)\partial_{\mu}\phi^{A}\partial^{\mu}\Phi\right]\,,$
(2.5)

where conformal invariance determines the dependence of the metric on $\Phi$
and relates $G_{\Phi\Phi}$ and $G_{A\Phi}$.

$G_{AB}(\Phi,\phi)=\Phi^{2}\hat{G}_{AB}(\phi)\,,\quad
G_{\Phi\Phi}(\phi)=\hat{G}_{\Phi\Phi}(\phi)\,,\quad
G_{A\Phi}(\Phi,\phi)=\frac{1}{2}\Phi\frac{\partial}{\partial\phi^{A}}\hat{G}_{\Phi\Phi}(\phi)\,.$
(2.6)

In (2.3) $S_{\rm CFT_{\rm IR}}$ schematically denotes all the other massless
IR fields, including the superpartners and/or accidental strongly coupled
sectors. The Poincaré invariant vacua with broken conformal symmetry
correspond to constant solutions of the form

$\langle\Phi\rangle=v^{\frac{d-2}{2}}\,,\qquad\langle\phi^{A}\rangle=\text{const}\,.$
(2.7)

Expanding in fluctuations around such solutions we find that the moduli
correspond to $n$ weakly coupled massless scalar particles. Note that by
dimensional analysis the gap of the massive modes, and hence the cutoff of the
EFT, scales as $m\propto v$.

All the interactions of the moduli among themselves and/or with the additional
fields present in $S_{\rm CFT_{\rm IR}}$ are irrelevant. From the expansion of
the action (2.5) we see that the moduli are derivatively coupled. In the
simplest case where no strongly coupled sector is present, additional
interactions arise from the couplings to the superpartners (if present) and
from higher derivative terms. The latter are suppressed by inverse powers of
$|\Phi|^{\frac{2}{d-2}}\sim v$ by Weyl invariance. For instance, based solely
on conformal invariance the first higher derivative operators arise at fourth
order in derivatives and are given by

$\begin{split}S_{4}=&\int
d^{d}x\sqrt{\hat{g}}\left[c_{1}(\phi)(\hat{\mathcal{R}}_{\mu\nu})^{2}+c_{2}(\phi)(\hat{\mathcal{R}}_{\mu\nu\rho\sigma})^{2}+c_{3,AB}(\phi)\hat{\mathcal{R}}^{\mu\nu}\partial_{\mu}\phi^{A}\partial_{\nu}\phi^{B}\right.\\\
+&\,\left.c_{4,ABCD}(\phi)\hat{\mathcal{R}}^{\mu\nu\rho\sigma}\partial_{\mu}\phi^{A}\partial_{\nu}\phi^{B}\partial_{\rho}\phi^{C}\partial_{\sigma}\phi^{D}+c_{5,ABCD}(\phi)(\partial_{\mu}\phi^{A}\hat{\nabla}^{\mu}\phi^{B})(\partial_{\nu}\phi^{C}\hat{\nabla}^{\nu}\phi^{D})\right]\,,\end{split}$
(2.8)

where we used the leading order equations of motion to eliminate operators
proportional to $\hat{\mathcal{R}}$ and $\hat{\nabla}^{2}\phi^{A}$. The
components $\hat{G}_{AB}$ and $\hat{G}_{\Phi\Phi}$ of the NLSM metric in
(2.3), as well as the functions $c_{i}$’s appearing in (2.8) are Wilson
coefficients, whose value depends on the microscopic dynamics of the specific
underlying CFT. For an underlying strongly coupled theory, these scale as
inverse powers of $(4\pi)$ according to generalized dimensional analysis
Georgi:1992dw  (at a generic point of the moduli space), while weakly coupled
theories correspond instead to non-generic sizes for these coefficients
(generically in the form of a large $\sim 1/g^{2}$ prefactor).

In even dimensions, higher derivative operators include also Wess-Zumino terms
required to match the conformal anomaly and other ’t Hooft anomalies of the
internal symmetries. In particular, in $d=4$ conformal anomaly matching
implies that the EFT should include the following four-derivatives term
Schwimmer:2010za ; Komargodski:2011vj :

$S_{\rm WZ}=-\int
d^{d}x\log\left(|\Phi|^{\frac{2}{d-2}}/\mu\right)\left(\Delta a\,E_{4}-\Delta
c\,W^{2}_{\mu\nu\rho\sigma}\right)+\ldots\,,$ (2.9)

where $\mu$ is an arbitrary mass scale, $\Delta a$ and $\Delta c$ denote the
difference between the UV and IR anomaly coefficients, $E_{4}$ and
$W_{\mu\nu\rho\sigma}$ are, respectively, the Euler density and the Weyl
tensor, and the dots stand for terms with at least two derivatives acting on
$\Phi$, which are required by the Wess-Zumino consistency conditions and whose
specific form will not be important for our purposes. In general in
$2n$-dimensions, the Wess-Zumino anomaly matching term is of order $2n$ in
derivatives Baume:2013ika .

It is important to remark that the absence of relevant couplings for the
moduli is a very special property. Indeed, based solely on conformal
invariance, the dilaton always admits at least a relevant coupling, the
cosmological constant term:

$\int d^{d}x\sqrt{\hat{g}}f(\phi)=\int d^{d}x\sqrt{g}\,|\Phi|^{d}f(\phi)\,,$
(2.10)

where the function $f$ is constrained by the internal symmetry and may just be
a constant. This term, as well as other potential couplings to relevant
operators of $S_{\rm CFT_{\rm IR}}$, would create a potential for $\Phi$ which
would lift the flat direction. Therefore, for the CFT to admit a moduli space,
its selection rules must set all the relevant couplings of the dilaton
including (2.10) to zero. This is a highly nontrivial condition, and typically
requires some amount of supersymmetry.666An alternative scenario, considered
in Coradeschi:2013gda , would be to have a family of theories labeled by some
marginal couplings, such that a flat direction opens up at special points of
the conformal manifold. As elaborated in Coradeschi:2013gda  (see also
Bellazzini:2013fga ), the problem of constructing interacting non-
supersymmetric CFTs with a moduli space of vacua can be thought as a scalar
analogue of the cosmological constant problem. In our analysis, we will simply
use self-consistently the general EFT (2.3), remaining agnostic about the
mechanism that produces the moduli space and tunes the cosmological constant
term to zero.

The EFT breaks down when the metric of the sigma model $\sim
v^{d-2}\hat{G}(\bar{\phi})$ becomes singular. This obviously happens at $v=0$,
but it may also happen at other isolated points or surfaces of dimension
$p<n$, at which additional degrees of freedom become light. At those points we
may still use an EFT of the form (2.3) for the $n-p$ moduli _parallel_ to the
surface, formally including the contribution of the additional light sector in
$S_{\rm CFT_{\rm IR}}$. A familiar illustration of this behaviour is in
$\mathcal{N}=4$ SYM with gauge group $SU(N)$. At a generic point of the moduli
space the gauge group is fully Higgsed and the low energy theory consists of
$N$ $\mathcal{N}=4$ $U(1)$ gauge theories, but at specific points the gauge
group is partially restored and the massless modes include nontrivial sectors
with gauge group $SU(p<N)$.777 A less trivial example is a $3d$
$\mathcal{N}=1$ theory of $11$ real superfields with superpotential
$W=g_{1}AX_{i}Y_{i}+g_{2}BX_{i}Z_{i}+g_{2}CY_{i}U_{i}\,,$ (2.11) where
$i=1,2$. This model admits a fixed-point in the $\epsilon$-expansion, and its
symmetries protect a moduli space which includes an $O(2)$ breaking branch
$\langle X_{i}\rangle=\hat{n}_{i}v_{x}\,,\quad\langle
Y_{i}\rangle=\varepsilon_{ij}\hat{n}_{j}v_{y}\,,\quad\langle
U_{i}\rangle=\hat{n}_{i}v_{u}\,,\quad\langle
Z_{i}\rangle=\varepsilon_{ij}\hat{n}_{j}v_{z}\,,$ (2.12) where $\hat{n}_{i}$
is a unit vector. The moduli space (2.12) is isomorphic to
$\mathds{R}^{4}\times S^{1}$ and the low energy EFT at a generic point
consists of $5$ derivatively coupled real superfields. This manifold is
singular when $v_{x}=0$ or $v_{y}=0$, where the low energy EFT includes 8
moduli as well as an interacting sector. Within this surface we encounter an
additional singularity when three of the vevs in (2.12) are zero, where there
are nine light modes, an axio-dilaton and an interacting SCFT.

###  2.2 Derivation of the leading order result

Let us suppose that in some region of the moduli space the internal symmetry
$G$ is spontaneously broken to $H$. Denoting with $Q_{a}$ the broken Cartan
charges, without loss of generality we can parametrize the coset $G/H$ such
that the action of $Q_{a}$ simply amounts to a shift of the associated
Goldstone field $\pi^{a}$. This implies that $\hat{G}_{AB}(\phi)$ and
$\hat{G}_{\Phi\Phi}(\phi)$ in (2.3) can only depend on the $\pi^{a}$’s through
derivative terms. From now on we shall distinguish the $\pi^{a}$’s from the
other moduli, that will be denoted $\phi^{A}$ as before with a slight abuse of
notation.

In the following it will be convenient to work on the Lorentzian cylinder
$\mathds{R}\times S^{d-1}$, for which the moduli action reads

$\begin{split}S_{EFT}=\int
d^{d}x\sqrt{g}&\left[\frac{\Phi^{2}}{2}\partial_{\mu}\pi^{a}\partial^{\mu}\pi^{b}\hat{G}_{ab}(\phi)+\Phi^{2}\partial_{\mu}\pi^{a}\partial^{\mu}\phi^{B}\hat{G}_{aB}(\phi)\right.\\\
&\left.+\frac{\Phi^{2}}{2}\partial_{\mu}\phi^{A}\partial^{\mu}\phi^{B}\hat{G}_{AB}(\phi)+\frac{\hat{G}_{\Phi\Phi}(\phi)}{2}\left(-\Phi\partial^{2}\Phi-
m_{d}^{2}\Phi^{2}\right)\right]+\ldots\,,\end{split}$ (2.13)

where we separated the $\pi^{a}$’s explicitly from the other moduli and

$m_{d}^{2}=\frac{d-2}{4(d-1)}\mathcal{R}=\frac{(d-2)^{2}}{4R^{2}}\,.$ (2.14)

is the conformal mass coupling on the cylinder and $R$ denotes the radius of
$S^{d-1}$.

While the EFT is derived around the background (2.7), the low energy effective
action may be reliably used around arbitrary backgrounds as long as the
derivative expansion is under control. We shall therefore search for a
solution of the equations of motion on the cylinder with the ansatz

$\Phi=v^{\frac{d-2}{2}}\,,\qquad\pi^{a}=m^{a}t/R\,,\qquad\phi^{A}=\bar{\phi}^{A}\,,$
(2.15)

where $t$ denotes the time coordinate. The ansatz (2.15) ensures that the
expectation values of the current and the stress-tensor agree with the
structure of a three-point function of primary operators Cuomo:2021ygt ;
therefore (2.15) describes a charged primary state.888This can also be checked
pedestrianly by computing the expression for the special conformal generators
in terms of the fluctuations around the background. Physically, the ansatz
(2.15) describes a stationary solution where both time translations $D/R$
(corresponding to dilations in flat space) and the charges $Q_{a}$ are
spontaneously broken,999Of course, strictly speaking there is no spontaneous
symmetry breaking at finite volume and integrating over the zero-modes of the
solution (2.15) the symmetry is restored at a quantum level (see e.g. the
discussion in Monin:2016jmo ). leaving invariant a diagonal combination
$D-m_{a}Q_{a}$: this is the spontaneous symmetry breaking pattern defining a
superfluid phase Son:2002zn ; Nicolis:2015sra , that recently attracted much
attention in the study of large charge operators in CFTs, see e.g.
Hellerman:2015nra ; Monin:2016jmo ; Gaume:2020bmp . In generic CFTs however
the _chemical potentials_ $m^{a}$ grow with the charge and the scaling
dimension scales as $\Delta_{\rm min}(Q)\propto Q^{\frac{d}{d-1}}$. The
existence of the moduli space (or in other words the absence of a potential
for the dilaton (2.10)) makes the models at hand non-generic, and we will find
below that the $m^{a}$’s do not scale with $Q$, resulting in a linear relation
between the charge and the scaling dimension.

The equation of motion (EOM) for $\Phi$ imposes the following condition (for
$v\neq 0$)

$\displaystyle
m^{a}m^{b}\hat{G}_{ab}(\bar{\phi})/R^{2}-\hat{G}_{\Phi\Phi}(\bar{\phi})m_{d}^{2}=0\,,$
(2.16)

while the $\bar{\phi}_{A}$ satisfy the following condition

$\left[m_{d}^{2}\frac{\partial\hat{G}_{\Phi\Phi}(\phi)}{\partial\phi^{A}}-\frac{m^{a}m^{b}}{R^{2}}\frac{\partial\hat{G}_{ab}(\phi)}{\partial\phi^{A}}\right]_{\phi=\bar{\phi}}=0\,.$
(2.17)

Finally, computing the Noether currents $j^{\mu}_{a}$, we find the values of
the Cartan charges on the solution (2.15)

$j^{0}_{a}=\frac{Q_{a}}{R^{d-1}\Omega_{d-1}}=\hat{G}_{ab}(\bar{\phi})m^{b}v^{d-2}/R\,,$
(2.18)

where $\Omega_{d-1}=2\pi^{d/2}/\Gamma(d/2)$ is the volume of the
$d-1$-dimensional sphere. Equations (2.16), (2.17) and (2.18) completely
specify the solution (2.15) in terms of the charges $Q_{a}$.

Let us consider first the case in which only one charge
$Q_{a}=\delta_{a}^{1}Q$ is non-zero (in some basis). We can thus solve (2.18)
for $v$ as

$v=\frac{1}{R}\left(\frac{Q}{\Omega_{d-1}\hat{G}_{1b}m^{b}}\right)^{\frac{1}{d-2}}\,.$
(2.19)

The remaining equations do not depend on $v$ nor $Q$. For fixed values of the
$\bar{\phi}^{A}$ such that $\hat{G}_{AB}$ and $\hat{G}_{\Phi\Phi}$ are non-
singular, it is always possible to find $m^{a}$’s that solve (2.16) and (2.18)
as a function of the $O(1)$ ratio $\hat{G}_{AB}/\hat{G}_{\Phi\Phi}$.101010The
explicit solution for arbitrary charges reads  $\displaystyle
v^{2d-4}=\frac{j_{a}^{0}(\hat{G}^{-1})^{ab}j_{b}^{0}}{\hat{G}_{\Phi\Phi}m^{2}_{d}}\,,\qquad\frac{m^{a}}{R}=\frac{(\hat{G}^{-1})^{ab}j_{b}^{0}}{v^{d-2}}=\frac{(\hat{G}^{-1})^{ab}j_{b}^{0}}{\sqrt{j_{a}^{0}(\hat{G}^{-1})^{ab}j_{b}^{0}/(\hat{G}_{\Phi\Phi}m^{2}_{d})}}\,.$
(2.20) Our only assumption is that (2.17) also admits a solution at a point
$\bar{\phi}^{A}$ that lies within EFT - we shall comment further about this
hypothesis later. The energy of the state is obtained from the energy momentum
tensor:

$\Delta_{\rm min}(Q)=R\int d^{d-1}x\sqrt{g}\,T_{00}=R\int
d^{d-1}x\sqrt{g}\left(j^{0}_{a}m_{a}/R-\mathcal{L}\right)=m_{1}Q\,,$ (2.21)

where we used that (2.16) implies that the Lagrangian vanishes,
$\mathcal{L}=0$, and $m_{1}\sim O(1)$ depends upon the Wilson coefficients of
the EFT through the equations of motion. By the state-operator correspondence,
(2.21) is the scaling dimension of the lowest dimensional charged operator of
the theory. When the solution to (2.17) is not unique, the minimal energy one
has to be taken (i.e. the one that gives the smallest $m_{1}$) and in some
cases there might be several degenerate operators with the same charge and
scaling dimension at this order.

The generalization of the argument to operators in different representations,
whose highest weight state has multiple non-zero Cartan charges, is trivial
and yields

$\Delta_{\rm min}(\\{Q_{a}\\})=m_{a}Q_{a}=Qf\left(Q_{a}/Q_{b}\right)\,,\qquad
Q=\left(\sum_{a}Q_{a}^{2}\right)^{1/2}\,,$ (2.22)

where the function $f$ is again determined by the EFT Wilson coefficients.
Similarly to (2.21), (2.22) holds as long as $Q\gg 1$. The function $f$ is
expectedly analytic except at isolated points in the space span by the ratios
$Q_{a}/Q_{b}$, corresponding to singularities in the moduli space.

Our analysis is reliable as long as the dilaton vev is much larger than the
derivatives of the Goldstone fields $v\sim
Q^{\frac{1}{d-2}}/R\gg|\partial\pi_{a}|\sim m_{a}/R$; we conclude that the
result (2.21) holds for $Q\gg 1$. At the classical level subleading
contributions arise from higher derivative terms and are suppressed by powers
of $R^{2}v^{2}\sim Q^{\frac{2}{d-2}}$. For instance, it is simple to check
that the operators in (2.8) yield a contribution of order
$Q^{\frac{d-4}{d-2}}$ to (2.21). Note also that the Wess-Zumino term (2.9)
evaluates to zero on the cylinder and therefore does not contribute to the
scaling dimension.

For a purely weakly-coupled EFT $Q$ is the loop counting parameter and thus
quantum corrections to the result (2.21) (generically) scale as $Q^{0}$. These
are more important than classical higher derivative contributions for $d\leq
4$. We will discuss such contributions and the spectrum of operators
corresponding to excitations on top of the large charge ground state in the
next subsection.

It is perhaps useful at this point to discuss a trivial example: the free
theory of a charged scalar $\varphi$. In this case the moduli space EFT
coincides with the full action:

$\mathcal{L}=|\partial\varphi|^{2}-m_{d}^{2}|\varphi|^{2}\,.$ (2.23)

The parallel with (2.13) becomes obvious in cylindrical coordinates
$\varphi=\Phi e^{-i\pi}/\sqrt{2}$. In this case the solution (2.15) is

$\varphi=e^{-im_{d}t}v^{\frac{d-2}{2}}\,.$ (2.24)

The charge and energy densities yield the scaling dimension

$j^{0}=2m_{d}v^{d-2}\,,\quad T_{00}=m_{d}j^{0}\quad\implies\quad\Delta_{\rm
min}(Q)=\frac{d-2}{2}Q\,,$ (2.25)

which is obviously exact in this case.

Besides free theories, there are many other instances for which the relation
(2.21) is exact. This is generally the case for theories with BPS operators,
where $Q$ is one of the $R$-charges. In these cases supersymmetry ensures the
absence of subleading corrections even if the EFT is generically nontrivial.
In some case, e.g. for Coulomb branch operators in theories with rank larger
than one, there are many degenerate solutions, corresponding to different
directions in the chiral ring. We will discuss examples in which the relation
(2.21) is not related to a BPS shortening condition in the next section.

Note that, while the cylinder makes the physical properties of the solution
(2.15) manifest, it is also possible to work directly in flat space or any
other conformally flat manifold by Weyl invariance; this is for instance
illustrated by the semiclassical saddle-point calculation of the flat space
two-point function $\langle\varphi^{n}\bar{\varphi}^{n}\rangle$ in free
theory, presented, e.g., in Badel:2019oxl .

Let us finally come back to the solution of (2.17). Suppose that no solution
for the $\bar{\phi}^{A}$’s exist at nonsingular points of the moduli space. In
this case we can still consider the configuration (2.15) for some
$\bar{\phi}^{A}$ such that $G_{AB}$ and $c$ are non-singular. We therefore
find a non-minimal energy state with $\Delta(Q)=m_{1}Q$ for some $m_{1}$. We
conclude that the real ground state, even if it does not lie within EFT, must
have lower energy: $\Delta_{\rm min}(Q)<m_{1}Q$.

In practice we expect that, when the solution to (2.17) lies at a singular
submanifold of the moduli space, we can simply use the low energy theory along
that surface for the $n-p$ parallel moduli to compute the scaling dimension.
Such EFT is formulated in terms of different Wilson coefficients than those at
a generic point of the moduli space, but we nonetheless recover the linearity
of $\Delta_{\rm min}(Q)$ at large charge.

###  2.3 Nearby operators

We now use the EFT to compute the spectrum of excited states on top of the
large charge ground states as well as quantum corrections on top of the result
(2.21).

Let us begin analyzing the spectrum of excited states. To compute the scaling
dimension of operators associated with excitations of the moduli we need to
compute the action for the fluctuations around the solution (2.15). We do this
in appendix A. We find that the fluctuations always include two modes with
dispersion relation

$\omega_{B,1}(\ell)=\ell/R\,,\qquad\omega_{B,2}(R)=(\ell+d-2)/R\,,$ (2.26)

as a function of the angular momentum $\ell$. Note that (2.26) coincides with
the spectrum of fluctuations around the solution (2.24) in free theory. The
dispersion relations for the additional $n-2$ fields are also approximately
linear in the limit $\ell\gg 1$:

$\omega_{B,k}(\ell)=\ell/R+O(1/R)\,,\qquad k=3,\ldots,n\,.$ (2.27)

The general form of the dispersion relations (2.27) at finite $\ell$ may be
quite involved due to the spontaneous breaking of time translations in (2.15);
we were not able to obtain general closed form expressions. A simplification
occurs when there is a unique Cartan charge and a symmetry acting as
$\pi\rightarrow-\pi$ on the corresponding Goldstone boson. This is for
instance the case in theories with a $U(1)$ charge invariant under charge
conjugation. In this case we find a fully relativistic result

$\omega_{B,k}(\ell)=\sqrt{n_{k}^{2}/R^{2}+J_{\ell}^{2}}\,,$ (2.28)

where $J^{2}_{\ell}=\ell(\ell+d-2)/R^{2}$ and the $n_{k}$’s are $O(1)$ numbers
that depend upon the coefficients in (A.9) of the appendix.

The Fock space of these quasi-particles describes operators in the same
representation of the internal symmetry with higher scaling dimension, except
for the zero modes of $\omega_{B,1}$ and the other $n_{G/H}-1$ Goldstone
bosons, that relate different charge sectors or create states in the same
representation of the ground state. States with one or more spin $1$
$\omega_{B,1}$ particles, whose gap is $\omega_{B,1}(1)=1/R$, are descendants.

At a generic point of the moduli space, in most cases, there are additional
massless modes. In the simplest and most generic scenario, these consist of
derivatively coupled massless fermions and vector fields (with Abelian gauge
group in $d\leq 4$), these often being the superpartners of the
moduli.111111Note that, unless the solution (2.15) describes a BPS state, the
dispersion relation of these modes is not directly fixed by the supersymmetry.
The excitations of these fields describe additional operators. Below, we
briefly analyze these additional states.

Assuming four-component Dirac fermions in $2<d\leq 4$ for concreteness (the
generalization to chiral spinors and other dimensions is straightforward), the
most general fermionic action compatible with conformal invariance at first
order in derivatives reads

$S_{\rm fermions}\simeq\int
d^{d}x\sqrt{\hat{g}}\left\\{\frac{i}{2}\bar{\psi}^{i}G_{\psi,ij}(\phi)\left[\hat{\not{\nabla}}\psi^{j}+\Gamma^{j}_{ka}(\phi)\gamma^{\mu}\psi^{k}\partial_{\mu}\pi^{a}+\Gamma^{j}_{kA}(\phi)\gamma^{\mu}\psi^{k}\partial_{\mu}\phi^{A}\right]+c.c.\right\\}\,,$
(2.29)

where the hatted metric is defined in (2.2), the indices $i,j$ run from $1$ to
the number of fermions $N_{F}$ and the _metric_ and _connections_ in field
space $G_{\psi}$ and $\Gamma$ are model-dependent. Note that terms of the form
$\sqrt{\hat{g}}\bar{\psi}\psi$ would yield a large mass $\sim v\bar{\psi}\psi$
in the broken vacua (2.7), as well as a potentially relevant Yukawa coupling.
We therefore assumed that such operators are ruled out by the selection rules
of the model similarly to (2.10). Around the solution (2.15) the quadratic
action reads

$S_{\rm fermions}\simeq v\int
d^{d}x\sqrt{g}\left\\{\frac{i}{2}\bar{\psi}^{i}G_{\psi,ij}(\bar{\phi})\left[\not{\nabla}\psi^{j}+m^{a}\Gamma^{j}_{ka}(\bar{\phi})\gamma^{0}\psi^{k}\right]+c.c.\right\\}\,.$
(2.30)

We see that the coupling to the Goldstones $\pi^{a}$ creates a chemical
potential. Therefore, canonically normalizing the fields and decomposing them
in spinor harmonics (see e.g. Camporesi:1995fb ), we find that the dispersion
relations take the form

$\omega_{F,i}^{(\pm)}(\ell)=p_{F}(\ell)\pm\tilde{\mu}_{i}\,,$ (2.31)

where $p_{F}(\ell)=\left[\ell+(d-1)/2\right]/R$ and the $\tilde{\mu}_{i}$ are
$O(1/R)$ chemical potentials that depend on the Wilson coefficients $G_{\psi}$
and $\Gamma_{\psi}$ in (2.30). Each mode in (2.30) has spin $\ell+1/2$ and
must be counted twice for a four component Dirac field (besides the group-
theoretical multiplicity). Interestingly, (2.31) implies that when one or more
of the $\tilde{\mu}_{i}$’s becomes larger than some of the $p_{F}(\ell)$ we
have negative energy levels. The fermions must therefore occupy such states,
forming a (small) Fermi sphere that contributes an $O(1)$ amount to the
charges $Q_{a}$. The gap of the Fock states around the Fermi sphere is given
by the absolute value of (2.31): $|\omega_{F,i}(\ell)|$.

In practical examples the fermionic metric and connection $G_{\psi},\Gamma$ in
(2.29) are related to the bosonic NLSM (2.13) by supersymmetry. A relevant
example for us is $3d$ $\mathcal{N}=1$ SUSY theories. In this case the generic
EFT consists of $n$ real superfields, with bosonic action as in (2.3) and
fermionic action given by:121212The second term in (2.32) can also be written
in terms of the connection associated with the NLSM (2.6) (using that the
$\psi$’s are Grassmanian).

$S_{\rm fermions}=-\frac{i}{2}\int
d^{3}x\,\psi^{i}\sigma^{\mu}\left[G_{ij}(\varphi)\partial_{\mu}+\partial_{\mu}\varphi^{k}\frac{\partial
G_{kj}(\phi)}{\partial\varphi^{i}}\right]\psi^{j}+\ldots$ (2.32)

where the $\psi^{i}$’s are Majorana, $i=1,\ldots,n$,
$\varphi^{i}=\\{\Phi,\pi^{a},\phi^{A}\\}$ collectively denotes all the moduli,
the NLSM metric is the one given in (2.6) and the ellipsis denote 4-fermion
interactions. The bosonic 2-derivative terms thus completely determine the
fermionic two-derivative terms through SUSY.

Finally, the two derivative action for the vector fields at leading order is
simply the free theory action accounting for the conformal coupling to the
dilaton131313An apparent exception in $d=3$ is the following coupling
$F_{\mu\nu}\varepsilon^{\mu\nu\rho}\partial_{\rho}\phi^{A}f_{V,A}(\phi)$ for
an Abelian vector field, but this is not a new term compared to (2.5) since an
Abelian vector is dual to a shift invariant scalar.

$S_{\rm vectors}\simeq-\frac{1}{4}\int
d^{d}x\sqrt{\hat{g}}\,G_{V,ij}(\phi)\hat{g}^{\mu\rho}\hat{g}^{\nu\sigma}F_{\mu\nu}^{i}F_{\rho\sigma}^{j}\,,$
(2.33)

where the field strength is defined as usual and the function $G_{V,ij}(\phi)$
is constrained by gauge-invariance for non-Abelian gauge groups (that can
occur in $d>4$). Expanding around the solution (2.15) we find that each vector
field behaves as a free Maxwell field and thus we find $N_{V}$ modes with
dispersion relation Giombi:2015haa :

$\omega_{V,i}(\ell)=\sqrt{(\ell+1)(\ell+d-3)}/R\,,$ (2.34)

where $\ell\geq 1$.

###  2.4 Quantum corrections

Let us now discuss the quantum corrections to the result (2.21). In the
generic case where all the fields in the EFT are derivatvely coupled, these
arise from the Casimir energy due the quasi-particles analyzed above:

$\frac{R}{2}\left[\sum_{k=1}^{n}\sum_{\ell=0}^{\infty}n_{B}(\ell)\omega_{B,k}(\ell)-2\sum_{k=1}^{N_{F}}\sum_{\ell=0}^{\infty}\sum_{\pm}n_{F}(\ell)|\omega_{F,k}^{(\pm)}(\ell)|+\sum_{k=1}^{N_{V}}\sum_{\ell=1}^{\infty}n_{V}(\ell)\omega_{V,k}(\ell)\right]\,,$
(2.35)

where the multiplicities are given by

$\begin{gathered}n_{B}(\ell)=\frac{(d+2\ell-2)\Gamma(d+\ell-2)}{\Gamma(d-1)\Gamma(\ell+1)},\quad
n_{F}(\ell)=\frac{2\Gamma(d+\ell-1)}{\Gamma(d-1)\Gamma(\ell+1)}\,,\\\\[5.0pt]
n_{V}(\ell)=\frac{\ell(\ell+d-2)(2\ell+d-2)\Gamma(\ell+d-3)}{\Gamma(d-2)\Gamma(\ell+2)}\,.\end{gathered}$
(2.36)

The sum in (2.35) of course must be properly regularized compatibly with the
symmetries of the model. (2.35) naively provides a $Q^{0}$ contribution to the
scaling dimension. For odd spacetime dimensions, the result must be finite
since it cannot be renormalized by any counterterm. For even $d$ however we
find higher derivative operators that contribute to the scaling dimension at
order $Q^{0}$ compatibly with conformal invariance, such as the operators in
(2.8) for $d=4$. Therefore in even dimension we expect that, barring special
symmetries of the solution, the sum (2.35) displays a logarithmic divergence
and thus results in a logarithm of the cutoff $v\sim Q^{\frac{1}{d-2}}$
similarly to the discussion in Cuomo:2020rgt . In summary:

$\delta\Delta^{(\rm 1-loop)}_{\rm
min}(Q)=\begin{cases}\alpha_{0}&d\;\text{odd}\\\ \alpha_{0}\log
Q&d\;\text{even}\,,\end{cases}$ (2.37)

where $\alpha_{0}$ is a constant that depends upon the dispersion relations.
(2.37) is the leading correction to (2.21) in $d=3,4$. When multiple Cartan
charges are non-zero as in (2.22), the coefficient $\alpha_{0}$ in (2.37) is a
function of the ratios $Q_{a}/Q_{b}$.

Of course, in many cases supersymmetry enforces cancellations. E.g. for BPS
states we always have $\delta\Delta^{(\rm 1-loop)}_{\rm min}(Q)=0$. We will
discuss in the next sections examples for which the Casimir energy is
nontrivial.

In generic schemes, such as dimensional regularization141414Note that in the
presence of a dilaton there exist many perturbative schemes that explicitly
preserve conformal symmetry Gretsch:2013ooa , including a version of
dimensional regularization., (2.35) can be further simplified. Indeed, the
first two scalar modes in (2.26) do not contribute to the sum (as it can be
easily checked in dimensional regularization) because their dispersion
relations coincide with those for the fluctuations of a free scalar field. For
the same reason, the contribution of the fermions in (2.31) vanishes unless
one or more of the $\tilde{\mu}_{i}$ are larger than some of the
$p_{F}(\ell)$, in which case the fermions’ contribution equals the (negative)
energy of the Fermi sphere.151515Note that the Casimir energy (2.35) is the
first correction to $\Delta_{\rm min}(Q_{a})-m^{a}Q_{a}$, where the $Q_{a}$’s
include the $O(1)$ contributions from the Fermi surface. Finally the vector
fields contribution to (2.35) is trivial in $d=4$ since the cylinder is a
conformally flat manifold and the four dimensional Maxwell action is Weyl
invariant on its own, without the need of a dilaton.

Finally, we estimate the subleading corrections to $\Delta_{\rm min}(Q)$ when
$S_{\rm CFT_{\rm IR}}$ in (2.5) includes some arbitrary interacting
sector.161616This discussion might apply in the example discussed in footnote
7 if the saddle describing the lowest dimensional charged operator lies in one
of the singular submanifolds discussed below (2.12). Recall that by assumption
this sector couples to the moduli only via irrelevant terms. This means that
when expanding around (2.7) all operators are irrelevant. As we show below,
this does not ensure that around a different background, such as (2.15), no
relevant perturbation arises. We will nonetheless argue that such terms only
yield subleading contribution to (2.21), therefore proving the robustness of
our result.

If $S_{\rm CFT_{\rm IR}}$ contains a primary scalar operator
$\mathcal{O}_{\Delta}$ of dimension $\Delta<d-2$, excluding total derivatives
the leading couplings around the background (2.15) are given by

$S\supset\int
d^{d}x\sqrt{\hat{g}}\left[\gamma_{ab}(\phi)\partial_{\mu}\pi^{a}\hat{\partial}^{\mu}\pi^{b}+\beta(\phi)\hat{\mathcal{R}}\right]\mathcal{O}_{\Delta}|\Phi|^{-2\frac{\Delta}{d-2}}\,,$
(2.38)

where the powers of the dilaton are dictated as usual by conformal invariance.
Recalling (2.2) and using (2.15), (2.38) results in a perturbation of the form

$\sim M^{d-\Delta}\int d^{d}x\sqrt{g}\mathcal{O}_{\Delta}\,,$ (2.39)

where by (2.19) the mass scale upfront is given by

$M=\left[\frac{v^{d-2-\Delta}}{R^{2}}\right]^{\frac{1}{d-\Delta}}\sim\frac{1}{R}Q^{\frac{d-2-\Delta}{(d-\Delta)(d-2)}}\,.$
(2.40)

As announced, (2.39) is a relevant perturbation for $\Delta<d$, and the
coupling (2.40) is larger than the geometric scale $1/R$ for $\Delta<d-2$.
Therefore the coupling (2.40) triggers a nontrivial RG flow. This results (at
most) in a contribution $\sim M^{d}$ to the potential. Using (2.40) we
conclude that (2.38) implies the existence of the following contribution to
the scaling dimension:

$\delta\Delta_{\rm min}(Q)\sim Q^{1-\frac{2\Delta}{(d-\Delta)(d-2)}}\,,$
(2.41)

which is larger than the Casmir energy (2.37) for $\Delta<d-2$, but is smaller
than the leading term in (2.21) for $\Delta>0$.171717Of course $\Delta>0$ is
always realized in unitary theories, but we must have $\Delta>0$ also in non-
unitary theories for the term (2.39) to be irrelevant in the Poincaré’
invariant state (2.7). In other words, the contribution of the coupling (2.38)
to the dilaton potential is always subleading compared to the conformal
coupling to the Ricci, as expected.181818As a concrete illustration of this
mechanism we can imagine that $S_{\rm CFT_{\rm IR}}$ consists of a four-
dimensional scalar $\varphi$ with potential $\lambda\varphi^{4}$ (this
coupling can be made conformal compensating the dependence on the RG scale
with the dilaton Komargodski:2011xv ). Identifying therefore
$\mathcal{O}_{\Delta}=\varphi$ in (2.38), with the appropriate sign the scalar
acquires a vev $\langle\varphi\rangle\sim M$ and at its minimum the potential
contributes to the energy as $\sim Q^{2/3}$, up to logarithms of $Q$
associated with the running of $\lambda$, in agreement with (2.41).

When $S_{\rm CFT_{\rm IR}}$ does not include scalars of dimension
$\Delta<d-2$, assuming unitarity the only non-irrelevant coupling between the
interacting sector and the moduli can take the form

$S\supset\int
d^{d}x\sqrt{\hat{g}}\lambda_{a}(\phi)\partial_{\mu}\pi^{a}J^{\mu}|\Phi|^{-\frac{2d}{d-2}}\simeq\int
d^{d}x\sqrt{g}\lambda_{a}(\bar{\phi})m^{a}J^{0}/R\,,$ (2.42)

where $J^{\mu}$ is a conserved current of $S_{\rm CFT_{\rm IR}}$. As for the
chemical potential term in the fermionic action (2.30), this operator may
result in a $\sim Q^{0}$ contribution to $\Delta_{\rm min}(Q)$. Similarly, a
marginally irrelevant operator would couple to the dilaton through the
coupling beta-function Komargodski:2011xv , and might result in a contribution
to the scaling dimension proportional to some inverse power of $\log Q$.

In conclusion, accidental strongly coupled sector might yield corrections to
the scaling dimension $\Delta_{\rm min}(Q)$ which are non-polynomial in $Q$,
but are nonetheless suppressed with respect to the leading order result
(2.21).

###  2.5 Correlation functions

Let us conclude the general discussion by mentioning that the EFT can also be
used to compute correlation functions with two or more large charge operators
and an arbitrary number of light operators as in the non supersymmetric case
Monin:2016jmo . For instance, we can represent a neutral operator of scaling
dimension $\Delta\sim O(1)$ in the EFT in terms of the dilaton as

$\mathcal{O}_{\Delta}=c_{\mathcal{O}}(\phi)\Phi^{\frac{2\Delta}{d-2}}+\ldots\,,$
(2.43)

where $c_{\mathcal{O}}$ is some function of the moduli and the dots stand for
higher derivative terms. Evaluating its expectation value in the large charge
ground state we obtain the OPE coefficient

$\lambda_{Q\mathcal{O}Q}=\langle Q|\mathcal{O}_{\Delta}|Q\rangle\simeq
c_{\mathcal{O}}(\bar{\phi})v^{\Delta}\quad\implies\quad\lambda_{Q\mathcal{O}Q}\propto
Q^{\frac{\Delta}{d-2}}\,.$ (2.44)

Similar results hold for charged operators. For spinning operators one has to
introduce derivatives of the Goldstones, and thus multiply the formula (2.43)
by factors of $\partial_{\mu}\pi^{a}/\Phi^{\frac{2}{d-2}}$. One obtains

$\lambda_{Q\mathcal{O}Q}|_{\text{spin }J}\propto Q^{\frac{\Delta-J}{d-2}}\,,$
(2.45)

barring additional selection rules. Note that (2.45) is obviously satisfied by
the conserved currents. One may also consider correlation functions with
several insertions of the large charge ground-state operator, as in
Cuomo:2021ygt .

Note that the result (2.44) holds in free theory, for which the OPE for
$\langle\bar{\varphi}^{Q+q}\varphi^{q}\varphi^{Q}\rangle$ is
$\sqrt{(Q+q)!}/\sqrt{Q!q!}\sim Q^{q/2}$ and
$\Delta_{\phi^{q}}=q\frac{d-2}{2}$. Less trivially, the same results holds in
strongly interacting $4d$ $\mathcal{N}=2$ SCFTs for Coulomb branch operators,
as it follows from localization Grassi:2019txd ; Beccaria:2020azj , and for
Higgs branch operators, whose correlators can be computed using the chiral
algebra Beem:2013sza ; Lemos:2020pqv . The EFT calculation can also be
extended to obtain subleading orders, see Hellerman:2017sur ;
Hellerman:2018xpi ; Hellerman:2020sqj ; Hellerman:2021duh ; Hellerman:2021yqz
. One can also check that (2.45) agrees with the known chiral algebra result
for the OPE between two Higgs branch operators and the spin $1$ Schür operator
that is obtained fusing the $SU(2)_{R}$ current $J^{J_{1}J_{2}}_{\mu}$ and a
Higgs branch operator $\mathcal{H}^{I_{1}\ldots I_{q}}$ in the $2q+1$
$SU(2)_{R}$ representation with $q\sim O(1)$.191919This OPE is fixed because,
within the chiral algebra, the _product_ operator $\sim
J^{J_{1}J_{2}}_{\mu}\mathcal{H}^{I_{1}\ldots I_{q}}$ is a Virasoro descendant
of $\mathcal{H}^{I_{1}\ldots I_{q}}$.

##  3 Moduli spaces at large charge: examples

In this section we test our basic claim (1.1) in several nontrivial examples.
To this end we focus on $3d$ $\mathcal{N}=1$ SCFTs, which have no continuous
R-symmetry nor protected chiral operators (see appendix B for a brief review).
We will see explicitly how the asymptotic linear behavior of $\Delta_{\rm
min}(Q)$ receives subleading corrections.

We first review how to compute the large charge scaling dimension
$\Delta_{\text{min}}(Q)$ in the presence of a small coupling $g$, in a double-
scaling limit where the mass of the single-particle states on the moduli space
is held fixed. We then study two specific Wess-Zumino models, where we can
achieve perturbative control by working in $d=4-\epsilon$ dimensions, such
that $g^{2}\sim\epsilon$. We are of course interested in models with a moduli
space and at least a global $U(1)$ symmetry. The concrete Wess-Zumino examples
that we consider may look somewhat baroque, but some care needs to be taken in
engineering models that do not exhibit an accidental supersymmetry enhancement
in the IR, with the global $U(1)$ becoming the continuous R-symmetry of an
${\cal N}=2$ SCFT.

We then consider the example of ${\cal N}=1$ SQED, going beyond perturbation
theory using the effective action formalism discussed above. Finally we
discuss some non-supersymmetric examples that exhibit approximate moduli
spaces to leading order in a large $N$ expansion.

###  3.1 Large charge double-scaling limit

We quickly review the method for extracting large-charge data using the
double-scaling limit discussed in Badel:2019oxl  (see also Watanabe:2019pdh ).
Assuming a weak (cubic) coupling $g$ and a $U(1)$ symmetry, the method allows
for the computation of $\Delta_{\text{min}}(Q)$ in the double-scaling limit
$g^{2}\rightarrow 0$ with $g^{2}Q=\text{fixed}$ and yields a result of the
form

$\Delta_{\text{min}}(Q)=\frac{1}{g^{2}}\hat{\Delta}_{-1}\left(g^{2}Q\right)+\hat{\Delta}_{0}\left(g^{2}Q\right)+g^{2}\hat{\Delta}_{1}\left(g^{2}Q\right)+\ldots\;,$
(3.1)

where the function $\hat{\Delta}_{-1}$, $\hat{\Delta}_{0}$, $\dots$ are,
respectively, the tree-level, one-loop, and higher loop contributions. When
multiple charges are present, the functions $\hat{\Delta}_{i}$ also depend on
the ratios $Q_{a}/Q_{b}$. In order to match the standard perturbative results,
for small $g^{2}Q$ the $\hat{\Delta}_{i}(g^{2}Q)$ admit a Taylor expansion in
$g^{2}Q$ which starts at order $g^{2}Q$. In all the examples we work in the
$\epsilon$-expansion, such that $g^{2}\sim\epsilon$.

We use the state-operator correspondence, and compute the ground state energy
at fixed charge $Q$ on the cylinder $\mathbb{R}\times S^{d-1}$. At small $g$
this can be done using a saddle-point approximation. These saddles take the
form of the “helical” superfluid solution similar to (2.24), schematically

$X=\rho e^{i\mu t}$ (3.2)

where $X$ is a complex field on which the charge acts by phase rotations and
$t$ is time. The constants $\rho$ and $\mu$ are fixed by the equations of
motion and the charge-fixing condition. Then the leading-order term
$\hat{\Delta}_{-1}$ is just the value of the energy computed on this saddle.

The next-order contribution $\hat{\Delta}_{0}$ comes from the Casimir energy
evaluated around the saddle. We must therefore quantize the theory around the
saddle (3.2) and compute the contribtions from the zero-point energies of the
various modes. The regularization of the final infinite sum is straightforward
within dimensional regularization, see Badel:2019oxl  for details. In
principle one can use this formalism to compute higher orders in
$\Delta_{\text{min}}(Q)$ as well, although we will be content with focusing on
these two leading contributions.

###  3.2 7-field WZ model

We study the model defined by the superpotential Gaiotto:2018yjh

$W=\frac{g}{2}\overline{u}\sigma^{i}u\,R_{i}\equiv\frac{g}{2}\overline{u}Ru\;,$
(3.3)

where $u$ is a complex superfield which is a doublet under $SU(2)$ and $R$ is
a real superfield and a triplet under $SU(2)$. The global symmetry group is
$SU(2)\times U(1)$ where the $U(1)$ rotates the phase of $u$, and there is an
additional $\mathbb{Z}_{2}^{R}$ symmetry taking $R\to-R$. The classical moduli
space is generated by $R$ and is protected from quantum corrections. The model
is IR dual to SQED with two charge-one particles Gaiotto:2018yjh ;
Benini:2018bhk .

In the $\epsilon$-expansion one finds the two-loop beta function

$\beta_{g}=-\frac{g\epsilon}{2}+\frac{3g^{3}}{8\pi^{2}}-\frac{g^{5}}{8\pi^{4}}\;,$
(3.4)

leading to the two-loop anomalous dimensions202020This was computed in
Benini:2018bhk  apart from the $\epsilon^{2}$ term in $\Delta_{R^{2}}$, which
is new.

$\begin{split}\Delta_{R}&=1-\frac{\epsilon}{3}-\frac{\epsilon^{2}}{108}\;,\\\
\Delta_{R^{2}}&=2-\frac{2}{3}\epsilon-\frac{\epsilon^{2}}{54}\;,\end{split}$
(3.5)

where by $R^{2}$ we mean the symmetric product of two $R$’s. Surprisingly we
see that $\Delta_{R^{2}}=2\Delta_{R}$ to two-loop order, which seems to hint
at higher SUSY. However, this model is not compatible with emergent
$\mathcal{N}=2$ SUSY, since the moduli space is 3-real-dimensional and so
cannot be Kähler. Therefore the relation $\Delta_{R^{2}}=2\Delta_{R}$ cannot
be exact and we should expect higher order to violate it. Below we will use
the double-scaling limit analysis to analyze large charge operators. We will
also provide a structural explanation for why the relation
$\Delta_{R^{Q}}=Q\Delta_{R}$ holds at two-loop, but not at higher orders.

To perform the analysis in the double-scaling limit with $g^{2}Q$ fixed, we
first rewrite the superpotential in a more convenient form. Writing the fields
in components as

$u=\begin{pmatrix}u_{1}\\\ u_{2}\end{pmatrix}\;,\qquad
R=\begin{pmatrix}R_{3}&R_{1}+iR_{2}\\\ R_{1}-iR_{2}&-R_{3}\end{pmatrix}\;,$
(3.6)

the calculation consists of expanding around the saddle point
$R_{1}+iR_{2}=ve^{-im_{d}t}$. To compute the leading terms
$\hat{\Delta}_{-1},\,\hat{\Delta}_{0}$ in the expansion (3.1), it is enough to
consider quadratic fluctuations around the saddle point (since
$\hat{\Delta}_{-1}$ can be read off from the classical charges and
$\hat{\Delta}_{0}$ is the Casimir energy around this background). The
superpotential takes the form

$W=\frac{g}{2}(R_{1}+iR_{2})u_{1}^{*}u_{2}+c.c.+\frac{g}{2}R_{3}(|u_{1}|^{2}-|u_{2}|^{2})\;,$
(3.7)

and so in computing $\hat{\Delta}_{-1}$ and $\hat{\Delta}_{0}$ we can ignore
the last term which is cubic around the saddle point. However, removing this
term, the superpotential reduces to that of the $\mathcal{N}=2$ supersymmetric
$XYZ$ model, which consists of three chiral superfields and superpotential

$W=\frac{g}{2}XYZ\;.$ (3.8)

The theory thus exhibits enhanced SUSY at this order, which leads to some
operator dimensions being protected. The $XYZ$ model was studied in the
double-scaling limit in Sharon:2020mjs , and the results are

$\frac{\hat{\Delta}_{-1}}{g_{*}^{2}}=\frac{d-2}{2}Q,\quad\hat{\Delta}_{0}=\frac{g_{*}^{2}v^{2}}{8}=\frac{g_{*}^{2}Q}{32\pi^{2}}=\frac{\varepsilon}{6}Q\;.$
(3.9)

Summing up we find

$\frac{\hat{\Delta}_{-1}}{g_{*}^{2}}+\hat{\Delta}_{0}=\frac{d-1}{3}Q\;,$
(3.10)

as expected from the BPS bound in the $XYZ$ model.

Equation (3.10) implies that $\Delta_{\text{min}}(Q)=Q\Delta_{\text{min}}(1)$
up to two-loop order in standard perturbation theory. This is because, as
formerly mentioned, the functions $\hat{\Delta}_{k}(g^{2}Q)$ in (3.1) are
polynomials starting at linear order, and the deviation from
$\Delta_{\text{min}}(Q)=Q\Delta_{\text{min}}(1)$ arises from the term
$g^{2}\hat{\Delta}_{1}$, which admits a Taylor expansion of the form
$g^{2}\hat{\Delta}_{1}=\\#g^{4}Q+\\#g^{6}Q^{2}+\ldots$ for small $g^{2}Q$.

Finally notice that the mass of the gapped mode $u$ is proportional to
$g^{2}v^{2}\sim g^{2}Q/R$. Therefore for $g^{2}Q\gg 1$ we can integrate it out
and obtain the moduli EFT. We will analyze in detail such EFT in section 3.4.

In summary, we found that the large charge scaling dimension in the model
(3.3) is linear in $Q$ as expected. Techinically, to the order we are working
this result arises as a trivial consequence of the similarity with the
$\mathcal{N}=2$ model studied in Sharon:2020mjs , as we explained below (3.7)
using the double-scaling limit. We will provide a less trivial check of our
predictions in the next section.

###  3.3 5-field WZ model

We study the WZ model defined by the superpotential

$W=\frac{g}{2}A(|X|^{2}-|Y|^{2})\;,$ (3.11)

with $A$ a real superfield and $X,Y$ complex superfields. Its continuous
symmetries are $U(1)\times U(1)$, and in addition it has a discerete symmetry
that flips the sign of $A$ and interchanges either $X$ and $Y$, or $X$ and
$\bar{Y}$. Finally, the $\mathbb{Z}_{2}^{R}$ symmetry acts as $A\to-A$. The
symmetries protect the classical moduli space (where either $\langle
A\rangle\neq 0$ or $\langle|X|^{2}\rangle=\langle|Y|^{2}\rangle\neq 0$) from
quantum corrections. According to our discussion, we expect operators with
large charge under either of the $U(1)$’s to be well described by the
effective action on the region of the moduli space where $X\neq 0$ and/or
$Y\neq 0$, where the internal symmetry group is either partially or fully
broken. Below we perform a detailed study of the charged operators’ spectrum
of this theory within the $\epsilon$-expansion. This will provide a nontrivial
check of the general predictions in section 2, concretely illustrating the
relation between the moduli EFT and the large charge operators’ spectrum.

In the $\epsilon$-expansion from $4d$ one finds the two-loop beta function

$\beta_{g}=-\frac{\epsilon}{2}g+\frac{3g^{3}}{32\pi^{2}}-\frac{3g^{5}}{512\pi^{4}}\,,$
(3.12)

and the scaling dimensions

$\displaystyle\Delta_{X}=\Delta_{Y}$
$\displaystyle=1-\frac{5\epsilon}{12}+\frac{\epsilon^{2}}{144}\,,\qquad$
$\displaystyle\Delta_{A}$
$\displaystyle=1-\frac{\epsilon}{3}+\frac{\epsilon^{2}}{36}\;,$ (3.13)
$\displaystyle\Delta_{X^{2}}=\Delta_{Y^{2}}$
$\displaystyle=2-\frac{2\epsilon}{3}-\frac{\epsilon^{2}}{36}\;,\qquad$
$\displaystyle\Delta_{XY}=\Delta_{X\bar{Y}}$
$\displaystyle=2-\epsilon+\frac{\epsilon^{2}}{36}\;,$
$\displaystyle\Delta_{AX}=\Delta_{AY}$
$\displaystyle=2-\frac{5\epsilon}{12}+\frac{\epsilon^{2}}{144}\;,\qquad$
$\displaystyle\Delta_{|X|^{2}-|Y|^{2}}$
$\displaystyle=2-\frac{\epsilon}{3}+\frac{\epsilon^{2}}{36}\;,$
$\displaystyle\Delta_{A^{2}-|X|^{2}-|Y|^{2}}$
$\displaystyle=2-\epsilon+\frac{\epsilon^{2}}{24}\;,\qquad$
$\displaystyle\Delta_{A^{2}+|X|^{2}+|Y|^{2}}$
$\displaystyle=2-\frac{\epsilon}{3}-\frac{\epsilon^{2}}{8}\;.$

Using that to one-loop order the scaling dimension of the operator
$X^{Q_{x}}Y^{Q_{y}}$ is a quadratic polynomial in $Q_{x}$ and $Q_{y}$, from
(3.13) we infer that the dimension of the lowest dimensional operator for
arbitrary values of the charges is

$\Delta_{X^{Q_{x}}Y^{Q_{y}}}=\Delta_{X^{Q_{x}}\bar{Y}^{Q_{y}}}=\left(Q_{x}+Q_{y}\right)\left(1-\frac{\epsilon}{2}\right)+\frac{\epsilon}{12}\left(Q_{x}-Q_{y}\right)^{2}+O\left(\epsilon^{2}\right)\,.$
(3.14)

This equation shows that to to one-loop order
$\Delta_{X^{Q}Y^{Q}}=Q\Delta_{XY}$. We will find below that this relation is
exact at leading order in the large charge double-scaling limit, but it
receives corrections at higher orders.

We now study the theory in the large-charge double-scaling limit. We start by
computing the leading term $\hat{\Delta}_{-1}$ in the expansion (3.1) for
general charges $Q_{x},\,Q_{y}$ (corresponding to the operators
$X^{Q_{x}}Y^{Q_{y}}$). Without loss of generality we assume $Q_{x}\geq
Q_{y}\geq 0$. The leading contribution is given by the semiclassical energy of
the saddle of the form

$X=\frac{v_{x}}{\sqrt{2}}e^{-i\mu_{x}t}\;,\quad
Y=\frac{v_{y}}{\sqrt{2}}e^{-i\mu_{y}t}\;,$ (3.15)

with $A=0$. The EOMs and charge-fixing conditions fix

$\displaystyle\mu_{x}$
$\displaystyle=\sqrt{m_{d}^{2}+g^{2}v_{x}^{2}-g^{2}v_{y}^{2}}\;,\qquad$
$\displaystyle\frac{Q_{x}}{R^{d-1}\Omega_{d-1}}$
$\displaystyle=|v_{x}|^{2}\mu_{x}\;,$ (3.16) $\displaystyle\mu_{y}$
$\displaystyle=\sqrt{m_{d}^{2}+g^{2}v_{y}^{2}-g^{2}v_{x}^{2}}\;,$
$\displaystyle\frac{Q_{y}}{R^{d-1}\Omega_{d-1}}$
$\displaystyle=|v_{y}|^{2}\mu_{y}\;.$

(3.16) admit a unique solution for real values of $\mu_{x/y}$ and $|v_{x/y}|$.
Interestingly, we find that the second derivative of the scaling dimension is
discontinuous at $g^{2}Q_{x}/(R^{d-1}\Omega_{d-1})=8\sqrt{2}m_{d}$ and
$Q_{y}=0$; the result it’s otherwise analytic for $Q_{x},\,Q_{y}>0$. The
scaling dimension admits simple expansions for small and large values of
$g^{2}\sqrt{Q_{x}^{2}+Q_{y}^{2}}$

$\frac{\hat{\Delta}_{-1}}{g^{2}R}=\begin{cases}\displaystyle
m_{d}(Q_{x}+Q_{y})+\frac{g^{2}(Q_{x}-Q_{y})^{2}}{32R^{d-1}\Omega_{d-1}m_{d}^{2}}+\ldots&g^{2}\sqrt{Q_{x}^{2}+Q_{y}^{2}}\ll(4\pi)^{2}\\\\[10.00002pt]
\displaystyle\sqrt{2}m_{d}\sqrt{Q_{x}^{2}+Q_{y}^{2}}\left[1-\frac{\sqrt{2}m_{d}^{4}R^{d-1}\Omega_{d-1}(Q_{x}^{2}-Q_{y}^{2})^{2}}{g^{2}(Q_{x}^{2}+Q_{y}^{2})^{5/2}}+\ldots\right]&g^{2}\sqrt{Q_{x}^{2}+Q_{y}^{2}}\gg(4\pi)^{2},\end{cases}$
(3.17)

which agrees with (3.14) for small $g^{2}Q_{x/y}$. We were not able to obtain
a closed form expression for $\hat{\Delta}_{-1}$ for arbitrary values of the
charges, but we found that the result simplifies drastically for
$Q_{x}=Q_{y}$, for which one surprisingly find that

$\frac{\hat{\Delta}_{-1}}{g^{2}}\stackrel{{\scriptstyle
Q_{x}=Q_{y}=Q}}{{=}}2Rm_{d}Q=(d-2)Q\,,$ (3.18)

coinciding with the free theory result.

We now turn to the one-loop correction. We focus on the operators with
$Q_{x}=Q_{y}=Q$ for which the leading order result (3.17) is particularly
simple. These states are special since they are invariant under the discrete
symmetry exchanging $X$ and $Y$, and the solution (3.15) explicitly reads

$\mu_{x}=\mu_{y}=m_{d}\,,\qquad|v_{x}|^{2}=|v_{y}|^{2}=\frac{Q}{m_{d}R^{d-1}\Omega_{d-1}}\,.$
(3.19)

Expanding the action to quadratic order around the saddle point, we find the
dispersion relations given in appendix C. Since the leading order result is
independent of the coupling, we can neglect the renormalization of the
coupling and the one-loop correction is just the Casimir energy

$\hat{\Delta}_{0}=\frac{R}{2}\sum_{k=1}^{5}\sum_{\ell=0}^{\infty}\left[n_{B}(\ell)\omega_{B,k}(\ell)-2N_{f}\sum_{\pm}n_{F}(\ell)\omega^{(\pm)}_{F,k}(\ell)\right]\;,$
(3.20)

where $\omega_{B,k}$ and $\omega_{F,k}$ are the bosonic and fermionic
dispersion relations respectively, $N_{f}=1/4$ is the number of Dirac Fermions
and the multiplicities are given in (2.36). We checked that the sum (3.20) is
regular (i.e. it contains no $1/\epsilon$ pole) in the limit
$\epsilon\rightarrow 0$ when evaluated within dimensional regularization.

First, as a crosscheck, we compute the sum for small $g^{2}Q$. We find that
the Casimir energy admits an expansion starting at order $\epsilon^{2}Q^{2}$

$\hat{\Delta}_{0}=\frac{g^{4}Q^{2}}{4(4\pi)^{4}}+O\left(\frac{g^{6}Q^{3}}{(4\pi)^{6}}\right)=\frac{\epsilon^{2}Q^{2}}{36}+O\left(\epsilon^{3}Q^{3}\right)\,.$
(3.21)

The absence of an $O\left(\epsilon Q\right)$ correction is in agreement with
the fact that both the double-scaling limit leading order result (3.18) and
the one-loop diagrammatic calculation (3.14) coincide with the free theory
answer.212121Incidentally, comparing (3.21) with the diagrammatic result
(3.13) and using that the two-loop anomalous dimension of $X^{Q}Y^{Q}$ is a
cubic polynomial with no $Q^{0}$ term, we obtain that the two-loop anomalous
dimension (at fixed $Q$) is
$\Delta_{X^{Q}Y^{Q}}=\left(2-\epsilon\right)Q+\frac{\epsilon^{2}Q^{2}}{36}+O\left(\epsilon^{3}\right)\,.$
(3.22)

For large $(g^{2}Q)\gg(4\pi)^{2}$ we can evaluate the sum using the method of
matched asymptotic expansion, as explained in appendix D of Cuomo:2021cnb .
The leading terms of the result are

$\hat{\Delta}_{0}=\frac{g^{2}Q}{64\pi^{2}}-\frac{1}{32}\log\left(\frac{g^{2}Q}{4\pi^{2}}\right)+O\left(g^{0}Q^{0}\right)\,,$
(3.23)

and thus, summing the leading order contribution (3.18), we arrive at

$\frac{\hat{\Delta}_{-1}}{g^{2}}+\hat{\Delta}_{0}=\left[2-\frac{11\epsilon}{12}+O\left(\epsilon^{2}\right)\right]Q-\frac{1}{32}\left[1+O\left(\epsilon\right)\right]\log\left(\epsilon
Q\right)+O\left(\epsilon^{0}Q^{0}\right),.$ (3.24)

(3.24) agrees with the general prediction of section 2 that the scaling
dimension grows linearly with the charge. This is a very nontrivial result at
the technical level, and it is due to nontrivial cancellations between the
bosonic and fermionic contributions (whose individual sums are divergent and
scale as $\sim(g^{2}Q)^{2}$ for large $g^{2}Q$) in (3.20). Note also that the
coefficient of the linear term in (3.24) deviates from the the free-theory
result at this order.

Interestingly, the fist subleading contribution in (3.24) at large $Q$ is
logarithmic. This also agrees with the EFT prediction (2.37), since to leading
order in $\epsilon$ the theory lives in four dimensions. The coefficient of
the logarithmic term is also predicted by the EFT. It is instructive to check
this explicitly.

For large charge $g^{2}\sqrt{Q_{x}^{2}+Q_{y}^{2}}\gg 1$ the fluctuations
around the solution (3.15) include two massive modes with gap $M^{2}\sim
g^{2}(|v_{x}|^{2}+|v_{y}|^{2})\sim g^{2}\sqrt{Q_{x}^{2}+Q_{y}^{2}}/R^{2}\gg
1/R^{2}$. The moduli space is $\mathds{R}\times S^{1}\times S^{1}$, and
therefore the low energy dynamics is well described by the EFT for the dilaton
$\Phi$ and two $U(1)$ Goldstones $\pi_{x}$ and $\pi_{y}$ (and the
superpartners). Using that the discrete symmetries act on the Goldstones as
$\pi_{x}\leftrightarrow\pm\pi_{y}$, the most general bosonic effective action
to leading order in derivatives reads

$\mathcal{L}_{bos}=\frac{c}{2}\Phi^{2}\left[(\partial\pi_{x})^{2}+(\partial\pi_{y})^{2}\right]+\frac{1}{2}\left[(\partial\Phi)^{2}-m_{d}^{2}\Phi^{2}\right]\,,$
(3.25)

which depends on a unique Wilson coefficient $c$ to this order (we work in
conventions such that the Goldstones are $2\pi$-periodic). The saddle-point
describing an operator with charges $(Q_{x},Q_{y})$ is

$\Phi=v\,,\quad\pi_{x}=\mu_{x}t\,,\quad\pi_{y}=\mu_{y}t\,,$ (3.26)

where the EOMs and the charge-fixing condition give

$\mu_{x/y}=\frac{Q_{x/y}m_{d}}{\sqrt{c(Q_{x}^{2}+Q_{y}^{2})}}\,,\qquad
v^{2}=\frac{\sqrt{Q_{x}^{2}+Q_{y}^{2}}}{\sqrt{c}\,m_{d}R^{d-1}\Omega_{d-1}}\,.$
(3.27)

Using (2.22), we obtain

$\Delta_{\rm
min}(Q_{x},Q_{y})=\frac{m_{d}}{\sqrt{c}}\sqrt{Q_{x}^{2}+Q_{y}^{2}}+\ldots\,.$
(3.28)

We can extract $c$ near four dimensions matching this result for $Q_{x}=Q_{y}$
with (3.24). We obtain

$c=\frac{1}{2}-\frac{\epsilon}{24}+O\left(\epsilon^{2}\right)\,.$ (3.29)

To evaluate the first correction to (3.28), we compute the Casimir energy of
the fluctuations. Fermions do not contribute at order $\log(Q)$ and thus we
neglect them. Expanding in fluctuations around the solution (3.26) we find the
following three bosonic modes

$R\,\omega_{B,k=1,2,3}(\ell)=\begin{cases}\ell\,,\\\ \ell+d-2\,,\\\
\sqrt{\ell(\ell+d-2)}\,.\end{cases}$ (3.30)

Notice that the dispersion relations do not depend on $c$ nor the ratio
$Q_{x}/Q_{y}$. From now on we set $d=4$ as appropriate to this order. The
first two modes in (3.30) are as in (2.26) and do not contribute to the
Casimir energy in dimensional regularization. The last mode is more
interesting. Its contribution to the Casimir energy is logarithmically
divergent in $d=4$

$\delta\Delta_{\rm min}^{(one-
loop)}\supset\frac{R}{2}\sum_{modes}\sum_{\ell}n_{B}(\ell)\omega_{B,3}(\ell)\sim-\frac{1}{16}\sum_{\ell}\frac{1}{\ell}+\text{power-
divergences}+\text{finite}\,,$ (3.31)

where we used $n_{B,\ell}=(1+\ell)^{2}$ for the multiplicity in $d=4$. Cutting
off the sum at $M\sim\sqrt{\epsilon(Q_{x}^{2}+Q_{y}^{2})^{1/2}}/R$ and
introducing counterterms to cancel the power-divergences as dictated by
conformal invariance, we infer

$\delta\Delta_{\rm min}^{(one-
loop)}=-\frac{1}{32}\log\left(\epsilon\sqrt{Q_{x}^{2}+Q_{y}^{2}}\right)+O\left(Q_{x/y}^{0}\right)\,.$
(3.32)

The same result can be obtained evaluating the sum in dimensional
regularization in a way that manifestly preserves the Weyl invariance of the
EFT as in Cuomo:2020rgt . Note that the precise function of $Q_{x}/Q_{y}$
appearing in the logarithm in (3.32) depends on the $O(Q^{0})$ higher
derivative terms that we did not study. For $Q_{x}=Q_{y}$, (3.32) agrees with
the result (3.24), therefore providing a nontrivial check of the EFT
methodology. Additionally, the EFT result in (3.32) predicts that the
coefficient multiplying the logarithm is the same for all values of the ratio
$Q_{x}/Q_{y}$.222222We can also use the EFT to predict the $Q^{0}$ Casimir
energy directly in $d=3$. Since the fermionic dispersion relations for the
light modes are independent of $c$,
$\omega_{F,1}^{(\pm)}(\ell)=\omega_{F,2}^{(\pm)}(\ell)=p_{F}(\ell)\pm
m_{d}\,,\qquad\omega_{F,3}^{(\pm)}(\ell)=p_{F}(\ell)\,,$ (3.33) fermions do
not form a Fermi sphere and thus do not contribute the Casimir energy. We
conclude  $\delta\Delta_{\rm min}^{(one-loop)}|_{d=3}\simeq-0.13255\,.$ (3.34)
Note that the Casimir energy is negative for any value of $c$ and
$Q_{x}/Q_{y}$.

###  3.4 SQED

$3d$ $\mathcal{N}=1$ SQED consists of a $U(1)$ gauge multiplet with $N_{f}$
fundamental matter multiplets and vanishing Chern-Simons level. As discussed
in appendix B, the theory possesses a $\mathbb{Z}_{2}$ R-symmetry which
protects its moduli space. This moduli space is $2N_{f}-1$-dimensional moduli
space, and is parametrized by the matter fields with one combination gauged
away. The effective theory on the moduli space is a $\mathbb{CP}^{N_{f}-1}$
NLSM, consisting of $2N_{f}-2$ Goldstone bosons which together with the
dilaton give the $2N_{f}-1$ massless bosonic degrees of freedom that were
expected. As formerly mentioned, SQED with $N_{f}=2$ is thought to be dual to
the 7-field WZ model discussed in section 3.2. We will study the theory
directly in $3d$ using the EFT methods discussed in section 2.

Since the effective theory consists of a dilaton coupled to a
$\mathbb{CP}^{N_{f}-1}$ NLSM, the leading 2-derivative term in the bosonic
part of the EFT takes the form (momentarily working in arbitrary spacetime
dimensions)

$\mathcal{L}=\frac{1}{2}(\partial_{\mu}\Phi)^{2}+\frac{1}{2}m^{2}_{d}\Phi^{2}+2c\Phi^{2}\gamma_{ij}\partial_{\mu}z_{i}\partial^{\mu}\bar{z}_{j}\;.$
(3.35)

Here, $\Phi$ is the dilaton and the complex coordinates $z_{i}$,
$i=1,...,N_{f}-1$ parametrize $\mathbb{CP}^{N_{f}-1}$. $\gamma_{ij}$ is the
Fubini-Studi metric

$\gamma_{i{j}}=\frac{(1+|z|^{2})\delta_{ij}-\bar{z}_{i}z_{j}}{(1+|z|^{2})^{2}}\;.$
(3.36)

Finally, $c$ is some constant which cannot be fixed from symmetry
considerations. We have fixed its normalization such that at $c=1$ and
$N_{f}=2$, the model reduces to three real free fields.

We can focus on the specific $U(1)$ isometry rotating $z_{1}$ by a phase and
compute $\Delta_{\text{min}}(Q)$. Following the analysis in section 2, we find
a solution to the EOMs on the cylinder of the form

$\Phi=v\;,\qquad z_{1}=e^{i\mu t}\;,$ (3.37)

with all other fields vanishing. The equations of motion set
$\mu^{2}=\frac{m^{2}_{d}}{c}$, and the solution has charge and energy density

$J_{0}=cv^{2}\mu\;,\qquad T_{00}=m^{2}_{d}v^{2}\;.$ (3.38)

We thus find at the two-derivative level

$\Delta_{\text{min}}(Q)=\frac{m_{d}Q}{\sqrt{c}}+O(Q^{0})\stackrel{{\scriptstyle
d=3}}{{=}}\frac{Q}{2\sqrt{c}}+O(Q^{0})\;.$ (3.39)

As a consistency check, for $c=1$ we obtain the result for a free theory. We
can also set $d=4-\epsilon$ and compare this calculation with the result for
the 7 field WZ model. Matching with equation (3.10), we extract $c$ near four
dimensions:

$c=1-\frac{1}{3}\epsilon+O\left(\epsilon^{2}\right)\,.$ (3.40)

The subleading term in $\Delta_{\text{min}}(Q)$ will be the Casimir energy
$E_{0}$, which will be a $c$-dependent constant. We compute it in Appendix D
for the physical number of dimensions, $d=3$. For the case $N_{f}=2$ where
$\mathbb{CP}^{1}$ is just the 2-sphere, the result is displayed in figure 1.

Figure 1: Casimir energy $E_{0}$ for the $\mathbb{CP}^{1}$ model.

At $c=1$ the theory becomes free and the Casimir energy vanishes, as expected,
while for all other values of $c$ it is negative. For the case $N_{f}>2$, the
result is $E_{0}(N_{f})=E_{0}(N_{f}=2)+2(N_{f}-2)e_{0}$ where
$e_{0}\approx-0.132$. We thus find that fixing a $U(1)$ symmetry, the minimal-
dimension operators with a fixed charge $Q$ should have dimension

$\Delta_{\text{min}}(Q)=\frac{1}{2\sqrt{c}}Q+E_{0}(N_{f})+O(1/Q)\;.$ (3.41)

###  3.5 Approximate moduli in non-supersymmetric theories

While exact moduli spaces are hard to construct in interacting theories
without supersymmetry, there are several models that admit flat directions in
certain parametric limit $\delta\rightarrow 0$, where generically
$\delta^{2}\sim 1/N$, while remaining interacting – see e.g. Rabinovici:1987tf
; PhysRevLett.52.1188 ; Chai:2020zgq ; Chai:2020onq . From the viewpoint of
the EFT (2.3), this means that the coefficient of the dilaton potential (2.10)
is parametrically suppressed with respect to its natural value.232323For a
theory with a unique quartic coupling $\sim\lambda$, which may possibly be
$O(1)$, on a solution (2.7) a generic potential contributes as $M^{d}/\lambda$
to the energy, where $M$ is the generic gap of the massive modes on the moduli
space. Equivalently, the parameter $\delta$ induces a large separation between
the (tachyonic) mass of the dilaton and the gap of the other massive modes in
the moduli space, ensuring that states with spontaneously broken conformal
symmetry are parametrically long lived.

It is interesting to repeat the EFT analysis of section 2 in this setup. To
this aim we simply include the potential (2.10) with a small coefficient
$\sim\delta^{\frac{4}{d-2}}$ (where the powers have been chosen for
convenience, such that $\delta$ would scale as a cubic coupling around a
conformal vacuum). Assuming that the gap of the massive modes in the would-be
moduli space scales as $\sim g^{2}v^{2}$ for some coupling $g$ (that in the
most convenient normalization appears as an overall $1/g^{2}$ in front of the
action and is possibly $O(1)$ for a strongly coupled model) as in the former
examples, we find that the dilaton is light and the large charge operator
spectrum obeys a linear relation as in (2.21) for $1\ll g^{2}Q\ll
1/\delta^{2}$. However for $g^{2}Q\gg 1/\delta^{2}$ eventually the dilaton and
the other moduli acquire a gap of order $\delta^{2}g^{2}Q$, the only remaining
light modes being the Goldstones for the internal symmetry; consequently, the
scaling dimension grows nonlinearly with the charge $\Delta_{\rm
min}(Q)\sim(\delta^{2}g^{2}Q)^{\frac{d}{d-1}}/(\delta^{2}g^{2})$, in agreement
with the prediction of the large charge expansion Hellerman:2015nra ;
Monin:2016jmo . In summary

$\Delta_{\rm min}(Q)\propto\begin{cases}\displaystyle Q&\text{for }1\ll
g^{2}Q\ll 1/\delta^{2}\\\
\displaystyle\frac{(\delta^{2}g^{2}Q)^{\frac{d}{d-1}}}{\delta^{2}g^{2}}&\text{for
}g^{2}Q\gg 1/\delta^{2}\,.\end{cases}$ (3.42)

The simplest model displaying this behaviour Chai:2020zgq  is a scalar
$O(N)\times O(N)$ CFT in $2<d<4$, with potential given by

$V(\vec{\phi}_{1}^{\,2},\vec{\phi}_{2}^{\,2})=\frac{\lambda_{11}}{8N}(\vec{\phi}_{1}^{\,2})^{2}+\frac{\lambda_{22}}{8N}(\vec{\phi}_{2}^{\,2})^{2}+\frac{\lambda_{12}}{4N}\vec{\phi}_{1}^{\,2}\,\vec{\phi}_{2}^{\,2}\,,$
(3.43)

where we normalized the couplings so that they admit a natural large
$N$-limit. To leading order in $N$, the beta functions of the coupling admits
a fixed-point (for any $d$) such that

$\lambda_{11}=\lambda_{22}=-\lambda_{12}\equiv\lambda>0\quad\text{for}\;N\rightarrow\infty\,,$
(3.44)

where $\lambda=16\pi^{2}\epsilon$ for $d=4-\epsilon$. The potential (3.43)
becomes a perfect square at such fixed points and the theory admits a flat
direction for

$\langle\vec{\phi}_{1}^{\,2}\rangle=\langle\vec{\phi}_{2}^{\,2}\rangle\,.$
(3.45)

In the vacua (3.45) the light modes consist of a dilaton and a $O(N)\times
O(N)/O(N)$ NLSM, while the radial modes acquire a mass
$\sim\lambda\langle\vec{\phi}^{\,2}_{1/2}\rangle/N$. Higher order corrections
destroy the moduli space and contribute to a potential of the form

$\frac{\lambda^{\prime}}{4N^{2}}\vec{\phi}_{1}^{\,2}\vec{\phi}_{2}^{\,2}\,.$
(3.46)

where $\lambda^{\prime}=8\lambda$ at leading order in the
$\epsilon$-expansion. This term induces a tachyonic mass $M_{dilaton}^{2}\sim
M^{2}/N\ll M^{2}$ for the dilaton.

We may then apply the former discussion to this example, with
$g^{2}\sim\epsilon/N$ and $\delta^{2}\sim 1/N$ (where $\epsilon$ could be
$O(1)$). We conclude that operators in large traceless symmetric
representations with $Q$ boxes under both the $O(N)$’s have approximately
linear in $Q$ scaling dimension for $1\lesssim\epsilon Q/N\ll N$. Note that
this prediction is not a trivial consequence of large $N$ factorization, since
we are considering charges $Q\gtrsim N$. The behavior eventually changes for
$\epsilon Q/N\gtrsim N$, for which we recover the nonlinear superfluid
behavior. These predictions are simple to verify in the large charge double-
scaling limit in the $\epsilon$-expansion (and it should be possible to
compute the scaling dimension at large $N$ with $Q/N$ fixed). We spare the
details to the reader.

Charged operators in this model should be contrasted to the spectrum of large
charge operators in the $O(N)$ model, studied in the $\epsilon$-expansion in
Antipin:2020abu  and at large $N$ for any $d$ in Alvarez-Gaume:2019biu ;
Giombi:2020enj . In that case there is no approximate moduli space, and the
only relevant parameter is $\epsilon Q/N$, whose value controls the transition
from the perturbative diagrammatic regime to the superfluid phase where
$\Delta_{\rm min}(Q)\sim\frac{N}{\epsilon}(\epsilon Q/N)^{\frac{d}{d-1}}$,
with no intermediate linear regime.

Another potentially interesting testing ground for our predictions could be
the conformal Fishnet model Gurdogan:2015csr , a non-supersymmetric and non-
unitary deformation of $\mathcal{N}=4$ SYM with gauge group $SU(N_{c})$ which
at large $N_{c}$ becomes integrable. It was recently suggested that in the
planar limit this model admits flat directions with a spontaneously broken
$U(1)$ symmetry Karananas:2019fox .242424This was verified to one-loop order
in the ’t Hooft coupling. Our EFT analysis of large charge operators does not
rely in any essential form on unitarity and should therefore apply to
operators with charge $Q/N_{c}\gg 1$ if the moduli space is indeed robust in
the planar limit.252525Encouragingly, charged operators in the planar limit
have been studied in Basso:2018agi , where it was found that $\Delta_{\rm
min}(Q)\simeq f(g)Q$ where $f$ is some nontrivial function of the strong
coupling. It remains unclear at this stage if this predictions survives for
$Q\gtrsim N_{c}$. It would also be interesting to explore connections between
the large charge double-scaling limit and integrability in that model, along
the lines of Caetano:2023zwe .

We finally mention that our discussion provides a simple necessary condition
for the existence of a dilaton at the endpoint of the conformal window of QCD
Ellis:1970yd ; Appelquist:2010gy . The existence of a flat direction in the
conformal window of QCD would be a rather finely tuned phenomenon, as
emphasized in Gorbenko:2018ncu ; Zan:2019idm ; Benini:2019dfy , but it is
nonetheless conceivable at large $N_{c}$, as in the example above. If such a
scenario is indeed realized, it is natural to expect that also the internal
$SU(N_{f})\times SU(N_{f})$ symmetry is spontaneously broken to the diagonal
in the moduli space, as in the gapped phase. It would then follow from our
discussion that at the endpoint of the conformal window the lowest dimensional
operators with charge $Q$ under a Cartan generator in the diagonal of
$SU(N_{f})\times SU(N_{f})$ (corresponding to traceless symmetric
representations with $2Q$ boxes in both Young diagrams) would have scaling
dimension $\Delta_{min}(Q)\propto Q$ for $1\ll Q\ll N_{c}$. We hope that in
the future this prediction can be used as a test of the existence of a flat
direction at the endpoint of the QCD conformal window.

###  3.6 Convexity in CFTs

Motivated by the weak gravity conjecture, the paper Aharony:2021mpc
conjectured a convexity relation for $\Delta_{\text{min}}(Q)$ in unitary $d>2$
CFTs. Specifically, the charge convexity conjecture (CCC) states that there
exists some $q_{0}$ of order 1 for which

$\Delta_{\text{min}}(n_{1}q_{0}+n_{2}q_{0})\geq\Delta_{\text{min}}(n_{1}q_{0})+\Delta_{\text{min}}(n_{2}q_{0})\;,\qquad\forall\;n_{1},n_{2}\,\in\mathbb{N}\;.$
(3.47)

Although a counterexample was found in Sharon:2023drx , the conjecture still
seems to apply to a large range of CFTs, and an obvious question is whether
some alternative version of it still holds.262626One such version is studied
in Aharony:2023ike . It is natural to start from the weakest version, where we
allow any (arbitrarily large) $q_{0}$. We call this version of the conjecture
the weak CCC.

Originally the weak CCC was dismissed as trivial, since all known phases at
asymptotically large charge were manifestly convex in the sense of (3.47).
However, in the models that we discussed this is no longer true. Specifically,
in our examples $\Delta_{\text{min}}(Q)$ is given in an expansion of the form

$\Delta_{\text{min}}(Q)=\alpha_{1}Q+\alpha_{0}+O(1/Q)$ (3.48)

in $3d$, or

$\Delta_{\text{min}}(Q)=\alpha_{1}Q+\alpha_{0}\log Q+O(Q^{0})$ (3.49)

in the $\epsilon$-expansion from $4d$. In both cases convexity is not
manifest, and instead the sign of the Casimir energy $\alpha_{0}$ determines
whether the weak CCC holds (if $\alpha_{0}$ vanishes, one needs to compute
higher-order terms to check the weak CCC, although we have not encountered
such examples). It is then interesting to ask whether the weak CCC is indeed
obeyed in these examples.

Happily, in all of the examples discussed above $\alpha_{0}$ is non-positive
(and is only zero in free theories or for BPS operators in theories with
extended supersymmetry), and so the weak CCC holds. In particular, in the
$5$-field WZ model and the SQED example discussed in sections 3.3 (see
footnote 22) and 3.4, the Casimir energy is always negative, and so even
though we cannot fix the Wilson coefficient $c$ we are still assured that
convexity holds.

We comment that convexity is a misnomer in the $3d$ case, since convexity for
a function $f(x)$ requires $f^{\prime\prime}(x)\geq 0$, while the condition
(3.47) instead corresponds to superadditivity. This distinction was not
important in previous discussions of convexity since these two definitions
were equivalent in all of the examples that were considered (see
Aharony:2021mpc ; Antipin:2021rsh ; Ishii:2024pnh ). However, in the expansion
(3.48) $\alpha_{0}$ determines superadditivity while higher-order terms
determine convexity, and so the two concepts are distinct. We will keep
calling the condition (3.47) “convexity” to avoid confusion.

One might wonder if the Casimir energy (2.35) is always non-positive in $d=3$,
perhaps assuming $\mathcal{N}=1$ supersymmetry. We were not able to establish
this result in full generality. However we found that the contributions of
scalars with _relativistic_ dispersion relations (2.28) to the Casimir energy
is always negative for $d=3$ and $d=4$.272727This is not true anymore in $d=5$
and $d=6$, where the sign depends on the mass $n_{k}$ in (2.28). Additionally,
as already mentioned in section 2, fermions always contribute negatively to
the Casimir energy. These observations are enough to establish that
$\alpha_{0}\leq 0$ in a large class models, including all the ones that we
studied here. Although this is a limited set of examples, it is reassuring
that they all obey the weak CCC. It would be nice to prove the weak CCC in
general.282828The weak CCC can be proven in two dimensions Palti:2022unw .

Note however that it is easy to construct moduli EFTs that violate the weak
CCC, e.g. the theory of an axio-dilaton (or its $3d$ $\mathcal{N}=1$ version)
with the appropriate sign of the higher derivative terms in (2.8) or a purely
bosonic $\mathds{R}\times\mathbb{CP}^{1}$ model (with the dilaton potential
(2.10) finely tuned to zero) as in (3.35), with $c\gtrsim 4.63$ in $d=5$ or
with $0<c<2$ in $d=6$, so that the Casimir energy is negative. Therefore a
proof of the weak CCC, if possible, must go beyond EFT arguments (e.g. of the
form appearing in Orlando:2023ljh ) and must rely nontrivially on the
requirement that the theory admits a well behaved UV completion.

##  4 Large charge macroscopic limit for theories with moduli spaces

Given the relation between moduli spaces and large charge operators that we
discussed so far, we expect that at distances much smaller than the sphere
radius, correlation functions in a large charge state reduce to correlators in
the moduli space. This statement in fact follows from EFT at distances
$R\gg|x|\gg 1/m$, where $m\propto Q^{\frac{1}{d-2}}/R$ is the EFT cutoff, in
this section we further assume that it remains true also at distances $|x|\ll
1/m$.292929More precisely, correlation functions are expected to be well
approximated by a nontrivial flat space limit up to corrections of order
$|x|^{2}/R^{2}$ and $1/(mR)^{2}\sim Q^{-\frac{2}{d-2}}$; in practice we often
do not know how to compute these corrections beyond the EFT regime, and
therefore we do not discuss them here. These considerations relate the scaling
dimension spectrum of large charge operators and the mass spectrum of
particles on the moduli space. This set of ideas is formalized by the
existence of a macroscopic limit Lashkari:2016vgj , first introduced in the
context of charged operators in Jafferis:2017zna . In the context of SCFTs
with moduli spaces, this limit was further explored in Caetano:2023zwe . For
the benefit of the reader, here we provide a self-contained discussion.

###  4.1 Macroscopic limits

Let us begin by reviewing the main ideas behind the concept of macroscopic
limit, following Jafferis:2017zna . Let us consider a CFT with a $U(1)$
symmetry (the generalization to more complicated groups being
straightforward), but not necessarily a moduli space for the moment. Let us
denote with $\mathcal{O}_{Q}$ the lowest dimensional operator as a function of
the charge for each allowed value of $Q$; additional specifications might be
needed when this is not unique. The macroscopic limit conists in sending
$Q\rightarrow\infty$ and $R\rightarrow\infty$ with the ratio
$Q^{\frac{1}{\gamma}}/R=\text{fixed}$, where the power $\gamma$ depends on the
theory and the state, such that correlation functions of light operators
(whose dimensions are held fixed in the limit) between two large charge states
remain finite. The limit must be taken so that the dimensionful distances
between the light operators remain finite. For instance, consider the
expectation value of a light scalar operator $\mathcal{O}_{\delta,q}$ of
dimension $\delta\sim O(1)$ and charge $q\sim O(1)$ in a large charge state

$\langle
Q|\mathcal{O}_{\delta,q}|Q+q\rangle=\frac{\lambda_{Q,\mathcal{O},Q+q}}{R^{\delta}}\,,$
(4.1)

In the macroscopic limit this matrix element should reduce to an appropriate
expectation value in a nontrivial flat-space state:

$\langle
Q|\mathcal{O}_{\delta,q}|Q+q\rangle\xrightarrow{\text{macro.}}\langle\text{flat}|\mathcal{O}_{\delta,q}|\text{flat}\rangle_{\mathds{R}^{d}}$
(4.2)

where we indicated with $|\text{flat}\rangle$ the resulting state, that is
characterized by the dimensionful scale $\nu\equiv Q^{\frac{1}{\gamma}}/R$ \-
therefore dimensional analysis implies
$\langle\text{flat}|\mathcal{O}_{\delta,q}|\text{flat}\rangle_{\mathds{R}^{d}}\propto\nu^{\delta}$.
Using that the macroscopic limit leaves the ratio
$Q^{\frac{1}{\gamma}}/R=\text{fixed}$, (4.2) can be stated purely in terms of
CFT data as:

$\lim_{Q\rightarrow\infty}Q^{-\delta/\gamma}\lambda_{Q,\mathcal{O},Q+q}=\nu^{-\delta}\langle\text{flat}|\mathcal{O}_{\delta,q}|\text{flat}\rangle_{\mathds{R}^{d}}=\text{finite}\,.$
(4.3)

As a less trivial example, consider the correlator of two light operators in
the large charge ground-state

$\langle
Q|\mathcal{O}_{\delta_{1},q_{1}}(\tau,\hat{n}_{1})\mathcal{O}_{\delta_{2},q_{2}}(0,\hat{n}_{2})|Q+q_{1}+q_{2}\rangle\xrightarrow{\text{macro.}}\underbrace{\langle\text{flat}|\mathcal{O}_{\delta_{1},q_{1}}(\tau,\vec{x})\mathcal{O}_{\delta_{2},q_{2}}(0,\vec{0})|\text{flat}\rangle_{\mathds{R}^{d}}}_{\equiv
g_{\mathcal{O}\mathcal{O}}(\tau,|\vec{x}|)}\,,$ (4.4)

where we work on the Euclidean cylinder with metric
$ds^{2}=d\tau^{2}+R^{2}d\hat{n}^{2}$ and the limit is taken keeping fixed
$\tau$ and $\vec{x}^{\,2}=2R^{2}(1-\hat{n}_{1}\cdot\hat{n}_{2})$. Introducing
cross ratios in standard notation

$\langle\mathcal{O}_{Q}(0)\mathcal{O}_{\delta_{1},q_{1}}(z,\bar{z})\mathcal{O}_{\delta_{2},q_{2}}(1)\mathcal{O}_{-Q}(\infty)\rangle\equiv
G_{Q}(z,\bar{z})\,,$ (4.5)

(4.4) can be stated purely in terms of the flat space four-point function as

$\lim_{Q\rightarrow\infty}Q^{-\frac{\delta_{1}+\delta_{2}}{\gamma}}G_{Q}\left(z=1-\nu\frac{w}{Q^{\frac{1}{\gamma}}},\bar{z}=1-\nu\frac{\bar{w}}{Q^{\frac{1}{\gamma}}}\right)=\nu^{-\frac{\delta_{1}+\delta_{2}}{\gamma}}\times
g_{\mathcal{O}\mathcal{O}}(\text{Re}w,|\text{Im}w|)\,,$ (4.6)

where the factors of $\nu$ ensure that $w$ has the dimensions of a length.
Note that for the limit to be nontrivial it is crucial that the result is
nonzero at least for some correlators.

The existence of a macroscopic limit is a natural requirement; physically it
corresponds to the statement that zooming in at distances much shorter than
the sphere radius in a very large charge state we recover the dynamics of a
nontrivial flat space state. However, to identify such state and the power
$\gamma$ in the fixed ratio $Q^{1/\gamma}/R$ we need to make some assumptions.
Below we discuss some possibilities.

Suppose that the scaling dimensions of the operators $\mathcal{O}_{Q}$ grows
as $\Delta(Q)\sim Q^{\alpha}$ for $Q\rightarrow\infty$. We additionally assume
that the corresponding state on the sphere is approximately homogeneous, and
therefore the spin carried by these operators does not scale with $Q$. The
energy density $\varepsilon$ and the charge density $\rho$ of the
corresponding state therefore are

$\varepsilon\sim\frac{Q^{\alpha}}{R^{d}}\,,\qquad\rho\sim\frac{Q}{R^{d-1}}\,.$
(4.7)

In the simplest option, the macroscopic limit is defined so that the energy
density remains fixed in the limit $R\rightarrow\infty$ Lashkari:2016vgj .
This implies that the ratio $Q/R^{\frac{d}{\alpha}}$ must be held fixed, i.e.
$\gamma=\frac{d}{\alpha}$. We have three possibilities:

  1. 1.

for $\alpha>\frac{d}{d-1}$ the charge density vanishes; we expect that in this
case the flat limit corresponds to a generic high energy state, of the kind
analyzed in Komargodski:2021zzy , and that the operators $\mathcal{O}_{Q}$ do
not provide the large charge ground state.

  2. 2.

for $\alpha=\frac{d}{d-1}$ we obtain the relation
$\varepsilon\sim\rho^{\frac{d}{d-1}}$; this is the only possible equation of
state for a CFT at finite density in flat space and corresponds to the
expected behaviour of large charge operators in generic CFTs.

  3. 3.

for $\alpha<\frac{d}{d-1}$ we find that $\rho$ diverges, contradicting the
assumption that correlation functions remain finite in the limit. We thus
conclude that we cannot take the macroscopic limit while keeping the energy
density fixed.

This is a simple but important conclusion: if $\Delta_{\rm min}(Q)\sim
Q^{\alpha}$ with $\alpha<\frac{d}{d-1}$, then the existence of a nontrivial
macroscopic limit necessarily results in a nontrivial state with vanishing
energy density, and therefore the theory must have a flat direction. More in
general, we can drop the assumption that the scaling dimensions grows as a
power law in $Q$ and we conclude:

$\lim_{Q\rightarrow\infty}\Delta_{\rm
min}(Q)/Q^{\frac{d}{d-1}}=0\;\cup\;\exists\text{ macro. limit
}Q\rightarrow\infty\quad\implies\quad\exists\text{ moduli space}\,.$ (4.8)

When the energy density vanishes in the macroscopic limit, we expect that
$\rho$ vanishes as well - otherwise we would have vacua that break
spontaneously the Lorentz symmetry. This implies that $\gamma<d-1$.

###  4.2 Applications

The conclusion (4.8) is a powerful one, and it essentially argues for the
opposite direction of the arrow in the argument (2.1). However at this stage
it does not tell us what is the right value of $\gamma$. We expect that, when
there exist a family operators satisfying the property (4.8), the $U(1)$
symmetry is spontaneously broken in the flat space state. In this case we can
resort to the EFT analysis of the previous sections to identify the proper way
to take the macroscopic limit. From the solution (2.15) we see that the
physical limit corresponds to holding the dilaton’s vev $v$ fixed. Therefore,
using the expression for the charge density (2.18) we consider the limit

$Q\rightarrow\infty\,,\quad
R\rightarrow\infty\,,\quad\text{with}\quad\frac{Q}{R^{d-2}}=\text{fixed}\,,$
(4.9)

which amounts at $\gamma=d-2$. To be more precise, the fixed ratio $Q/R^{d-2}$
is related to the dilaton’s vev in the broken vacuum by (2.19). In this
section we discuss some consequences of the existence of the limit (4.9) for
correlation functions. For simplicity, we also assume that the lowest
dimensional charged operators $\mathcal{O}_{Q}$ are scalars.

Let us warm-up by considering OPE coefficients. Setting $\gamma=d-2$, (4.3)
implies that OPE coefficients of scalar operators scale as

$\lambda_{Q,\mathcal{O},Q+q}\sim Q^{\Delta_{\mathcal{O}}/(d-2)}\,,$ (4.10)

for large $Q$. This is indeed the result that we obtained within EFT (2.44).
Similarly, the requirement that the macroscopic limit reproduces the moduli
space theory implies that OPEs of spinning operators should vanish faster than
in (4.10) for $Q\rightarrow\infty$ \- this is again trivially realized in the
EFT, see (2.45).

In some cases, the existence of a finite macroscopic limit follows from
supersymmetry. As a trivial example, consider a rank $1$ $4d$ $\mathcal{N}=2$
SCFT, and denote with $\Phi_{r}$ the $U(1)_{r}$ charge $r$ Coulomb branch
operator, whose scaling dimension is $\Delta_{r}=r$. Working in the standard
normalization where the chiral ring’s OPE is unit normalized, supersymmetry
fixes the form of the so called extremal correlators, consisting of $n+1$
chiral operators and one anti-chiral operator (see e.g. Eberhardt:2020cxo ):

$\langle\Phi_{Q}(0)\Phi_{r_{1}}(x_{1})\Phi_{r_{2}}(x_{2})\ldots\Phi_{r_{n}}(x_{n})\bar{\Phi}_{Q+\sum_{i}r_{i}}(x)\rangle=\frac{w(Q+\sum_{i}r_{i})}{|x|^{2\Delta_{Q}}}\prod_{i}\frac{1}{|x_{i}-x|^{2\Delta_{i}}}\,,$
(4.11)

where

$w(r)=\langle\Phi_{r}(0)\bar{\Phi}_{r}(1)\rangle\,.$ (4.12)

Mapping the correlator (4.11) to the cylinder we obtain

$\langle
Q|\Phi_{r_{1}}(\tau_{1},\hat{n}_{1})\Phi_{r_{2}}(\tau_{2},\hat{n}_{2})\ldots\Phi_{r_{n}}(\tau_{n},\hat{n}_{n})|Q\rangle=\sqrt{\frac{w(Q+\sum_{i}r_{i})}{w(Q)}}\prod_{i}\frac{e^{\tau_{i}\Delta_{i}/R}}{R^{\Delta_{i}}}\,.$
(4.13)

Taking the macroscopic limit and using the localization result Grassi:2019txd
; Beccaria:2020azj  $\sqrt{w(Q+\sum_{i}r_{i})/w(Q)}\simeq
Q^{\sum_{i}r_{i}/2}$, we obtain

$\langle
Q|\Phi_{r_{1}}(\tau_{1},\hat{n}_{1})\Phi_{r_{2}}(\tau_{2},\hat{n}_{2})\ldots\Phi_{r_{n}}(\tau_{n},\hat{n}_{n})|Q\rangle\xrightarrow{macro}v^{\sum_{i}\Delta_{i}}\,,$
(4.14)

where we identified the fixed-ratio $v\equiv\sqrt{Q}/R$ with the local
coordinate $v$ on the Coulomb branch. We see that taking the macroscopic limit
in (4.14) we obtained a coordinate-indepent result. This matches the well
known holomorphic properties of chiral operators in the Coulomb branch.

The macroscopic limit implies a relation between the particles’ mass spectrum
in the moduli space and the dimensions of charged primary operators that are
much heavier than the large charge ground state. The exact correspondence can
be established relating the conformal block decomposition of a four-point
function with two insertions of the large charge operator $\mathcal{O}_{Q}$
and two light operators, with the Källen Lehman decomposition in the moduli
space. We discuss this relation below. It is important that the macroscopic
limit applies at arbitrary short distances, even if the EFT breaks for
$\vec{x}^{\,2},\tau^{2}\ll m^{-2}\sim R^{2}/Q^{\frac{2}{d-2}}$. For technical
reasons that we mention below, our discussion will be limited to $d\geq 4$.

Consider the two point function of a light neutral operator $\mathcal{O}$
(whose $O(1)$ quantum numbers will be irrelevant for our purposes) in the
large charge ground state:

$F_{Q,\mathcal{O}}(\tau,\hat{n}_{1}\cdot\hat{n}_{2})\equiv\langle
Q|\mathcal{O}(\tau,\hat{n}_{1})\mathcal{O}(0,\hat{n}_{2})|Q\rangle\,.$ (4.15)

It is convenient to evaluate this correlator inserting a complete set of
states between the two light operators. Grouping states into conformal
families, this results into the conformal block decompostion. For $\tau>0$ we
use the s-channel conformal block decomposition

$F_{Q,\mathcal{O}}(\tau,\hat{n}_{1}\cdot\hat{n}_{2})=\frac{1}{R^{2\Delta_{\mathcal{O}}}}\left[\lambda_{Q,\mathcal{O},Q}^{2}g_{\Delta_{Q},0}^{\mathcal{O}_{Q},\mathcal{O}}(\tau,\hat{n}_{2}\cdot\hat{n}_{1})+\sum_{\Delta,\ell}\lambda_{Q,\mathcal{O},(\Delta,\ell)}^{2}g_{\Delta,\ell}^{\mathcal{O}_{Q},\mathcal{O}}(\tau,\hat{n}_{2}\cdot\hat{n}_{1})\right]\,,$
(4.16)

where we isolated the contribution of the operator $\mathcal{O}_{Q}$ itself
and $g_{\Delta,\ell}^{\mathcal{O}_{Q},\mathcal{O}}$ is the conformal block
corresponding to the exchange of an operator of dimension $\Delta$ and spin
$\ell$ in the $\mathcal{O}_{Q}$-$\mathcal{O}$ OPE. We normalized the s-channel
conformal blocks so that the primary’s contribution reads:

$g_{\Delta,\ell}^{\mathcal{O}_{Q},\mathcal{O}}(\tau,\hat{n}_{2}\cdot\hat{n}_{1})=e^{-\tau(\Delta-\Delta_{Q})/R}C_{\ell}^{\left(\frac{d}{2}-1\right)}(\hat{n}_{1}\cdot\hat{n}_{2})+\text{descendants}\,.$
(4.17)

The s-channel decomposition in these coordinates converges for $\tau>0$. For
$\tau<0$ we may use the t-channel expansion, which in the case of identical
neutral operators $\mathcal{O}$ is obtained from the s-channel decomposition
with the replacement $\tau\rightarrow-\tau$ in the blocks.

In the macroscopic limit (4.4), the correlation function (4.15) reduces to the
moduli space correlator. This in turn can be expressed using the spectral
decomposition, i.e.

$\begin{split}F_{Q,\mathcal{O}}(\tau,\hat{n}_{1}\cdot\hat{n}_{2})&\xrightarrow{\text{macro.}}\langle\mathcal{O}(\tau,\vec{x})\mathcal{O}(0,\vec{0})\rangle_{v,\vec{\phi}}\\\
&=\langle\mathcal{O}\rangle_{v,\vec{\phi}}^{2}+\int_{0}^{\infty}dm^{2}\rho_{\mathcal{O}\mathcal{O}}(m^{2})G_{m^{2}}(\tau^{2}+\vec{x}^{\,2})\,,\end{split}$
(4.18)

where $\rho_{ij}$ is the spectral density

$\rho_{ij}(p^{2})=(2\pi)^{d-1}\sum_{n}\langle
0|\mathcal{O}_{i}|n\rangle\langle
n|\mathcal{O}_{j}|0\rangle\delta^{d}(q_{n}-p)\,.$ (4.19)

We would like to relate (4.18) with the conformal block decomposition
described above. To this aim we need to compute the form of the conformal
blocks in the macroscopic limit.

The macroscopic limit of the conformal block depends on whether the gap
$\Delta-\Delta_{Q}$ and the angular momentum $\ell$ of the exchanged operator
scale or not with $R$. When $\Delta-\Delta_{Q}\sim\ell\sim O(1)$, the
conformal blocks are trivial in the macroscopic limit

$g_{\Delta,\ell}^{\mathcal{O}_{Q},\mathcal{O}}(\tau,\hat{n}_{2}\cdot\hat{n}_{1})\xrightarrow{\text{macro.}}1\qquad\text{for
}\;\Delta-\Delta_{Q}\sim\ell\sim O(1)\,.$ (4.20)

This result arises because the contributions of descendants in (4.17) is
suppressed by powers of $\Delta_{Q}$, see e.g. Komargodski:2021zzy . Therefore
operators with $O(1)$ gap and angular momentum can only contribute to the
disconnected part of the correlator in the macroscopic limit. The contribution
is nontrivial only when the corresponding OPE coefficient scales as
$R^{\Delta_{\mathcal{O}}}\sim Q^{\frac{\Delta_{\mathcal{O}}}{d-2}}$. Because
of (4.2), we find immediately that the exchange of $\mathcal{O}_{Q}$ in (4.16)
reproduces exactly the disconnected term in (4.18) in the macroscopic limit.
Therefore we conclude that the OPE coefficients of all the other exchanged
operators vanish in the macroscopic limit

$\lim_{Q\rightarrow\infty}Q^{-\frac{\Delta_{\mathcal{O}}}{d-2}}\lambda_{Q,\mathcal{O},(\Delta,\ell)}=0\,.$
(4.21)

The connected term in (4.18) arises from the exchange of operators with
$\Delta-\Delta_{Q}\sim R$ and/or angular momentum $\ell\sim R$. We therefore
set

$\omega\equiv\frac{\Delta-\Delta_{Q}}{R}\,,\qquad
k\equiv\frac{\ell}{R}\,,\qquad\bar{v}\equiv\frac{\Delta_{Q}}{R^{d-2}}\,.$
(4.22)

The discussion of the macroscopic limit of the conformal blocks at fixed
$\omega$ and $k$ is similar to the one in Komargodski:2021zzy , where the
authors considered the macroscopic limit at fixed energy density. The
contribution of the _n_ th level descendants in (4.17) scales as
$\sim\left[(\omega^{2}+k^{2})R^{2}/\Delta_{Q}\right]^{n}\sim R^{n[2-(d-2)]}$.
Descendants may therefore be neglected for $d>4$. In $d=4$ we may instead
resort to the exact expression of the conformal block Dolan:2000ut ;
Dolan:2003hv , and we find that descendants only contribute to an overall
prefactor. We spare the details to the reader, and only report the final
result:

$g_{\Delta,\ell}^{\mathcal{O}_{Q},\mathcal{O}}(\tau,\hat{n}_{2}\cdot\hat{n}_{1})\xrightarrow{\text{macro.}}\mathcal{N}_{d,\bar{v}}(\omega,k)\frac{R^{d-3}}{k}e^{-\omega\tau}\int
d^{d-1}p\,\delta(|\vec{p}|-k)e^{i\vec{k}\cdot\vec{x}}\,,$ (4.23)

where the prefactor reads

$\mathcal{N}_{d,\bar{v}}(\omega,k)=\frac{R^{d-3}}{2^{d-2}\pi^{\frac{d-2}{2}}\Gamma\left(\frac{d-2}{2}\right)}\times\begin{cases}\displaystyle
1&\text{for }d>4\\\
\displaystyle\exp\left(\frac{\omega^{2}+k^{2}}{2\bar{v}^{2}}\right)&\text{for
}d=4\,.\end{cases}$ (4.24)

We expect that a result analogous to (4.23) holds also in $d<4$ with a
different prefactor, but we did not prove it.

Using (4.23), we recast the macroscopic limit of (4.16) as

$F_{Q,\mathcal{O}}(\tau,\hat{n}_{1}\cdot\hat{n}_{2})\xrightarrow{\text{macro}.}\frac{\lambda_{Q,\mathcal{O},Q}^{2}}{R^{2\Delta_{\mathcal{O}}}}+\int_{0}^{\infty}d\omega\int\frac{d^{d-1}k}{(2\pi)^{d-1}}e^{-\omega|\tau|}e^{i\vec{k}\cdot\vec{x}}K(\omega,|\vec{k}|)\,,$
(4.25)

where we defined

$K(\omega,k)=(2\pi)^{d-1}\sum_{\Delta,\ell}\delta\left(\omega-\frac{\Delta-\Delta_{Q}}{R}\right)\delta\left(k-\frac{\ell}{R}\right)\frac{\mathcal{N}_{d,\bar{v}}(\omega,k)\lambda^{2}_{Q,\mathcal{O},(\Delta,\ell)}}{k\,R^{2\Delta_{\mathcal{O}}+3-d}}\,.$
(4.26)

The existence of the macroscopic limit therefore requires that the combination
in (4.26) becomes $R$-independent in the appropriate sense.303030To this aim,
it might be necessary to smoothen the delta functions in (4.26) with
normalized averages over intervals of infinitesimal size $\delta$ around
$\omega$ and $k$ \- see e.g. footnote $8$ in Komargodski:2021zzy .

Finally, comparing (4.25) with (4.18), we arrive at

$\theta(\omega)\rho_{\mathcal{O}\mathcal{O}}(\omega^{2}-k^{2})=K(\omega,k)\,.$
(4.27)

(4.27) establishes a direct connection between the flat space spectral density
and the operators exchanged in the s- and t- channel of the correlation
function (4.15).

As a simple application, consider the contribution of a mass $m$ particle in
the spectral density in $d=4$:

$\theta(\omega)\rho_{\mathcal{O}\mathcal{O}}(\omega^{2}-k^{2})=c_{m^{2}}\bar{v}^{2\Delta_{\mathcal{O}}-2}\theta(\omega)\delta(\omega^{2}-k^{2}-m^{2})\,,$
(4.28)

where the powers of $\bar{v}$ are dictated by dimensional analysis and
$c_{m^{2}}$ is an arbitrary dimensionless coefficient. Because of (4.27),
(4.28) implies that there exists one or more operators with scaling dimension

$\Delta-\Delta_{Q}\simeq\sqrt{\ell^{2}+R^{2}m^{2}}\,,$ (4.29)

and with OPE coefficient

$\begin{split}\sum\lambda^{2}_{Q,\mathcal{O},(\Delta,\ell)}&\simeq
c_{m^{2}}(R\bar{v})^{2\Delta_{\mathcal{O}}-2}\frac{4\pi\ell}{\sqrt{\ell^{2}+R^{2}m^{2}}}e^{-\frac{R^{2}m^{2}+2\ell^{2}}{2\Delta_{Q}}}\\\
&\propto
Q^{\Delta_{\mathcal{O}}-1}\frac{\ell}{\sqrt{\ell^{2}+R^{2}m^{2}}}e^{-\frac{R^{2}m^{2}+2\ell^{2}}{2\Delta_{Q}}}\,,\end{split}$
(4.30)

where the sum runs over all the degenerate operators with the same scaling
dimension (4.29). Note that it is important that the mass remains finite in
the macroscopic limit $Rm\sim Q^{1/2}$.

(4.29) and (4.30) are in agreement with the explicit results of
Caetano:2023zwe , where the authors studied correlation functions with two
large charge Coulomb branch operators in $\mathcal{N}=4$ SYM with gauge group
$SU(2)$ in the weak coupling double-scaling limit and conjectured a similar
formula.313131To compare the results, note that
$e^{-\frac{\omega^{2}+k^{2}}{2\bar{v}^{2}}}\simeq 1+O(g^{2})$ at weak coupling
in the double-scaling limit. We also expect the prediction (4.29) to be
relevant in large $N$ SCFTs with moduli spaces.

Let us conclude this dicussion with a comment about the macroscopic limit for
$\mathcal{N}=4$ SYM in the holographic limit as explained in
Ivanovskiy:2024vel . We focus for concreteness on the Higgsing pattern
$SU(N)\rightarrow U(1)\times SU(N-1)$. Holographically this is realized by
displacing a brane at some distance from the center of the bulk of AdS5
Klebanov:1999tb . The state which reduces to this vacuum in the macroscopic
limit is the one created by a fully symmetric chiral operator
$\sim\text{Tr}[X^{Q}]$ with $Q\sim N$, which is dual to a giant graviton
spherical $D3$ brane in AdS5 Hashimoto:2000zp  (see also McGreevy:2000cw ;
Balasubramanian:2001nh ; Corley:2001zk ; Lin:2004nb ). For large charge $Q\gg
N$ (but still $Q\ll N^{2}$ in order to neglect the gravitational backreaction)
the brane extends towards the boundary of AdS similarly to the moduli space
one. Zooming in at distances much smaller than the $S^{3}$ radius the two
branes are locally identical. The mapping between moduli space particles and
charged CFT operators then follows straightforwardly. In particular, for both
branes we can consider open strings ending on them. For the moduli space brane
the string state could be e.g. a W-boson particle, whose mass $T\Delta X\sim
gv$ is proportional to the distance between the displaced brane and the bulk
of AdS. For the giant graviton with $Q\gg N$ the string corresponds to a
W-boson like primary operator, and the semiclassical energy of the string
provides the scaling dimensions in agreement with (4.29).323232We thank Shota
Komatsu for clarifying discussions about this point. It would be interesting
to further explore the connection between large charge operators and moduli
spaces in holography.

Let us conclude by mentioning that it might be interesting to apply the
macroscopic limit to the study of partition functions at finite chemical
potential, and in particular to the supersymmetric index. For instance, in
$4d$ $\mathcal{N}=2$ SCFTs certain limits of the supersymmetric index are
related to the physics of the Coulomb and Higgs branches of the theory (see
e.g. Rastelli:2016tbz ; Rastelli:2014jja  for reviews); we expect that many of
these relations can be understood by a generalization of the arguments
presented in this section.

##  5 Comments on moduli spaces with no broken charges

The main conceptual message of this work could be summarized as follows. In
CFTs with moduli spaces where a global charge is also broken, large charge
operators create, in radial quantization, _semiclassical_ states that closely
resemble the vacua where both the conformal and the internal symmetry are
broken. The connection is clear in the EFT saddle-point (2.15), since the
dilaton acquires a large value. This is what allows to prove the existence of
a tower of operators whose scaling dimension grows linearly with the charge.
The existence of a conserved charge is essential, as one can identify a
precise tower of states, those corresponding to the lowest dimensional charged
operators.

One might still wonder whether in the general case (where no internal symmetry
is spontaneously broken) the existence of a moduli space is still reflected in
some semiclassical feature of the spectrum. Below we address this question
starting from the EFT of a real dilaton. Not surprisingly, we conclude that in
the absence of a conserved charge it is not possible to identify a tower of
special operators that look like the spontaneously broken vacua. However, we
speculate that the moduli space is reflected in the existence of certain
resonant states on the cylinder, whose width becomes parametrically narrower
than their energy in the high energy limit.

The basic idea is simple. Consider for concreteness the EFT for a real
dilaton, which is relevant in the $ABC$ model Gaiotto:2018yjh  and other $3d$
$\mathcal{N}=1$ SCFTs. The leading term is simply the free action

$S=\int
d^{d}x\left[\frac{1}{2}(\partial\Phi)^{2}-\frac{m_{d}^{2}}{2}\Phi^{2}+\ldots\right]\,,$
(5.1)

where the dots stand for higher derivative terms and additional fields, such
as the superpartners. On the cylinder, the free theory admits a homogeneous
solution of the form

$\Phi=\frac{v^{\frac{d-2}{2}}}{\sqrt{2}}\cos(m_{d}t+t_{0})\,.$ (5.2)

The parameter $t_{0}\sim t_{0}+2\pi$ is a zero-mode of the solution and should
be integrated over in the path-integral. In appendix 5.2 we show that the
solution (5.2) arises as the saddle-point describing the two-point function
$\langle\Phi^{n}(x_{i})\Phi^{n}(x_{f})\rangle$ for $n\gg 1$ in free theory,
with

$v^{d-2}=\frac{n}{(d-2)\Omega_{d-1}R^{d-2}}\,.$ (5.3)

Therefore the solution (5.2) describes a primary state with dimension
$\Delta_{n}=n$ in free theory. Note that from the cylinder viewpoint, the
quantization condition $n\in\mathds{N}$ follows from the semiclassical
quantization of the field around the saddle-point.333333The simplest way to
see this is to consider the Bohr-Sommerfeld condition  $\oint\Pi
d\Phi=2\pi\,,$ (5.4) where $\Pi=R^{d-1}\int d^{d-1}\hat{n}\,\dot{\Phi}$ is the
field momentum and the integration is over an oscillation period
$T=2\pi/m_{d}$ of the solution.

One might therefore naively conclude from the existence of the solution (5.2)
that CFTs with moduli spaces always admit states with energy $\Delta_{n}\simeq
n$ for $n\gg 1$.343434Note that this would be a stronger claim than the one
for charged operators, since the ratio $\Delta_{n}/n$ is fixed by the EFT.
This is because there is no free parameter in the EFT action (up to rescalings
of the dilaton) unlike the charged case. There is an important subtlety
however. Even when $n\gg 1$ the dilaton vev does not stay large for the full
oscillation period $T=2\pi/m_{d}$ of the solution. This is to be contrasted
with the large charge profile (2.15), where the dilaton retains a constant
value and the solution never exits the EFT valdity regime.

Within EFT, we can trust the profile (5.2) only as long as higher derivative
terms are suppressed, i.e. when

$(\partial\Phi)^{2}\ll|\Phi|^{\frac{2d}{d-2}}\,.$ (5.5)

From this condition we infer that for each oscillation period the solution
(5.2) exits the EFT validity regime for a time $\delta t$ of the order

$\frac{\delta t}{T}\sim\left(\frac{m_{d}}{v}\right)^{\frac{d-2}{d}}\sim
n^{-\frac{1}{d}}\,.$ (5.6)

Physically, during the time in which the dilaton vev is small, all the degrees
freedom of the CFT become again massless and interact nontrivially with the
dilaton mode. The breakdown of EFT therefore signifies that the EFT
Hamiltonian is not complete, and there exist additional off-diagonal terms
that mix the dilaton with the other fields of the theory. These terms are
suppressed by $\delta t/T\sim n^{-1/d}$ with respect to the leading
contribution $\Delta_{n}\simeq n$ at large energy, but they imply nonetheless
that we cannot diagonalize the Hamiltonian restricting ourselves only to the
EFT states. Therefore the energy eigenstates are complicated linear
combinations of dilaton quasi-particles and other strongly coupled modes. This
was in some sense a foreordained conclusion: the spectrum of a general CFT is
dense at high energies, and it is a priori not clear how to identify special
eigenstates.

This discussion is again to be contrasted with the case of charged operators -
in which case the saddle-point (2.15) describes the ground state in a certain
fixed charge sector. In that case the EFT is obtained integrating out only
massive states in a standard fashion, and we can self-consistently focus on
the EFT Hilbert space. In the neutral case instead, when using the EFT around
the solution (5.2) we are neglecting other degenerate states with the same
quantum numbers, that are not separated by a large gap. There is no selection
rule that prevents the semiclassical dilaton state (2.15) from mixing with
these other states made of strongly coupled degrees of freedom.

Yet, there remain the question of which is the right interpretation of the
semiclassical state described by (5.2). After all, this is certainly a well
defined state for short enough times, such that the dilaton did not yet have
time to interact significantly with other modes. Our speculation is that the
saddle-point (5.2) describes a resonant state. Indeed, in general the spectrum
of states of energy $\Delta\gg 1$ is expected to be exponentially dense
Bhattacharyya:2007vs ; Shaghoulian:2015lcn ; Benjamin:2023qsc  and may
therefore be approximated by a continuum.353535For a detailed discussions of
the validity of this approximations, see e.g. Giudice:2000av ; Cuomo:2020fsb .
On general grounds, the mixing of an isolated state with a continuum results
in the existence of a resonant pole in the Green’s function of the theory
di1992lezioni . Therefore we expect that the interactions between the dilaton
and other strongly coupled modes will result into a width that controls the
lifetime of the state (5.2). We can estimate the width on dimensional grounds
from the ratio (5.6)

$\Gamma_{n}\sim\Delta_{n}\times\frac{\delta t}{T}\sim n^{\frac{d-1}{d}}\,.$
(5.7)

We can also reach this conclusion using EFT. Indeed, it is sometimes the case
that in EFT one can integrate out a continuum of states at the price of losing
unitarity. The most famous example is perhaps the EFT for $CP$ violation in
Kaon physics, in which one describes the physics in a small energy window
around the Kaon mass and neglects the pions and other light states, whose
effect is described via the inclusion of small imaginary terms in the
Hamiltonian. In that case the EFT does not describe the full evolution of the
system, but only the probability that the Kaon evolves into another quasi-
degenerate particle rather than decaying into ligher ones.

Assuming that is is possible to integrate out the quasi-continuum of
degenerate states around the solution (5.2), we need to parametrize in EFT our
inability to describe the interactions of the dilaton with the other modes
during the small interval of times in which the dilaton vev becomes small.
Similarly to Hellerman:2020eff , as long as $\delta t/T\ll 1$ this can be done
treating the interaction as instantaneous, introducing effective operators
which are localized in field space $\propto\delta(\Phi)$. The leading such
operator is

$\delta S=c\int
d^{d}x\,\delta(\Phi)\left[(\partial\Phi)^{4}+\ldots\right]^{\frac{1}{2}+\frac{d-2}{4d}}\,,$
(5.8)

where the powers are dictated by Weyl invariance and the dots stand for terms
which are needed to Weyl complete the parenthesis.363636In the notation of
section 2.1, the term in parenthesis can be written as
$\frac{(d-2)^{2}}{4d(d-1)}\hat{\mathcal{R}}_{\mu\nu}^{2}\Phi^{4\left(1+\frac{2}{d-2}\right)}=(\partial\Phi)^{4}+\ldots\,.$
(5.9) The general idea is that
$(\partial\Phi)^{\frac{2}{d}}\sim\left[m_{d}^{2}v^{d-2}\right]^{\frac{1}{d}}\gg
1/R$ controls the derivative expansion at the singular points where $\Phi=0$;
on physical grounds, the reason why we can still use EFT is that the
interactions with the strongly coupled modes is almost instantaneous, and
should therefore be describable by effective _source terms_ localized at
$m_{d}t+t_{0}\approx\frac{\pi}{2}+\pi\mathds{Z}$. (5.8) realizes the leading
such term in a manifestly Weyl invariant fashion.

On the saddle-point, the operator (5.8) results into a contribution to the
stress tensor of the form

$T_{00}\propto
c\,\delta(m_{d}t+t_{0})|\dot{\Phi}|^{\frac{1}{2}+\frac{d-2}{2d}}\simeq
c\times\delta(m_{d}t+t_{0})\left[m_{d}v^{\frac{d-2}{2}}\right]^{\frac{1}{2}+\frac{d-2}{2d}}\,.$
(5.10)

Thus, upon averaging over the zero-mode $t_{0}$, we obtain a correction $\sim
c\,n^{\frac{d-1}{d}}$ to the energy. Since the operator (5.8) has zero support
in the usual moduli space, the coefficient $c$ is unknown and may be taken
complex according to our discussion. This is therefore the leading
contribution to the width in agreement with the estimate (5.7).

Matrix elements in such resonant states behave very differently than in
generic high energy states. Indeed, we can compute the expectation value of an
arbitrary scalar operator $\mathcal{O}_{\delta}$ of dimension $\delta\sim
O(1)$ via operator matching $\mathcal{O}_{\delta}\propto|\Phi|^{\delta}$ as in
section 2.5. We therefore find that $t=0$ matrix elements take the form

$\langle
v|\mathcal{O}_{\delta}(t=0)|v\rangle\propto\Delta^{\frac{\delta}{d-2}}\,.$
(5.11)

Which is very different than the expected result in a generic (thermalized)
high energy state
$\langle\Delta|\mathcal{O}_{\delta}(t=0)|\Delta\rangle\propto\Delta^{\frac{\delta}{d}}$.
These considerations suggest these resonances should be more properly
interpreted as _scar states_ , similarly to Milekhin:2023was .

To summarize, in the absence of a conserved charge it does not seem possible
to find a specific tower of primary operators that carries over the
information of the moduli space physics. It might however still be possible to
construct long-lived resonances which, in a sense, can be thought as
homogeneous excitations of the dilaton.

## Acknowledgments

We thank Zohar Komargodski and Shota Komatsu for useful discussions. The work
of GC was supported by the Simons Foundation grant 994296 (Simons
Collaboration on Confinement and QCD Strings). The work of LR was supported in
part by the National Science Foundation under Grant NSF PHY-2210533 and by the
Simons Foundation under Grant 681267 (Simons Investigator Award).

##  Appendix A Fluctuations around the large charge ground state

We expand the moduli around the solution (2.15)

$\Phi=v^{\frac{d-2}{2}}+\delta\Phi\,,\quad\pi^{a}=m^{a}t/R+\delta\pi^{a}/v^{\frac{d-2}{2}}\,,\quad\phi^{A}=\bar{\phi}^{A}+\delta\phi^{A}/v^{\frac{d-2}{2}}\,.$
(A.1)

Expanding (2.13), we find the following quadratic action

$\begin{split}\mathcal{L}_{fluct}&=\overline{G}_{aB}\partial_{\mu}\delta\pi^{a}\partial^{\mu}\delta\phi^{B}+\frac{m^{a}}{R}\dot{\delta\phi^{B}}\left(\overline{G}_{aB,C}\delta\phi^{C}+2\overline{G}_{aB}\delta\Phi\right)+\frac{m^{a}}{R}\dot{\delta\pi^{b}}\left(\overline{G}_{ab,A}\delta\phi^{A}+2\overline{G}_{ab}\delta\Phi\right)\\\
&+\frac{1}{2}\overline{G}_{ab}\partial^{\mu}\delta\pi^{a}\partial_{\mu}\delta\pi^{b}+\frac{1}{2R^{2}}m^{a}m^{b}\left(\overline{G}_{ab}\delta\Phi^{2}+2\overline{G}_{ab,C}\delta\Phi\delta\phi^{C}+\frac{1}{2}\overline{G}_{ab,CD}\delta\phi^{C}\delta\phi^{D}\right)\\\
&+\frac{1}{2}\overline{G}_{AB}\partial^{\mu}\delta\phi^{A}\partial_{\mu}\delta\phi^{B}-\frac{1}{2}\left(\overline{G}_{\Phi\Phi}\,\delta\Phi+\overline{G}_{\Phi\Phi,A}\delta\phi^{A}\right)\partial^{2}\delta\Phi\\\
&-\frac{m_{d}^{2}}{2}\left(\overline{G}_{\Phi\Phi}\,\delta\Phi^{2}+2\overline{G}_{\Phi\Phi,A}\delta\Phi\delta\phi^{A}+\frac{1}{2}\overline{G}_{\Phi\Phi,AB}\delta\phi^{A}\delta\phi^{B}\right)\,,\end{split}$
(A.2)

where $\overline{G}=\hat{G}(\bar{\phi})$ and the additional subscripts denote
derivatives with respect to the $\phi^{A}$’s, e.g.
$\overline{G}_{ab,C}=\partial\hat{G}_{ab}/\partial\phi^{C}|_{\phi=\bar{\phi}}$,
$\overline{G}_{\Phi\Phi,A}=\partial\hat{G}_{\Phi\Phi}/\partial\phi^{A}|_{\phi=\bar{\phi}}$,
etc. . Using the EOMs (2.16) and (2.17), we see that the terms proportional to
$\delta\Phi^{2}$ and $\delta\Phi\delta\phi^{C}$ cancel; therefore, integrating
by parts the term proportional to $\partial^{2}\delta\Phi$, we arrive at

$\begin{split}\mathcal{L}_{fluct}&=\frac{1}{2}\overline{G}_{ab}\partial^{\mu}\delta\pi^{a}\partial_{\mu}\delta\pi^{b}+\frac{m^{a}}{R}\dot{\delta\pi^{b}}\left(\overline{G}_{ab,A}\delta\phi^{A}+2\overline{G}_{ab}\delta\Phi\right)\\\
&+\frac{1}{2}\bar{G}_{\Phi\Phi}\,\partial_{\mu}\delta\Phi\partial^{\mu}\delta\Phi+\frac{1}{2}\bar{G}_{\Phi\Phi,A}\partial_{\mu}\delta\phi^{A}\partial^{\mu}\delta\Phi+\frac{1}{2}\overline{G}_{AB}\partial^{\mu}\delta\phi^{A}\partial_{\mu}\delta\phi^{B}-\frac{1}{2}M^{2}_{AB}\delta\phi^{A}\delta\phi^{B}\\\
&+\overline{G}_{aB}\partial_{\mu}\delta\pi^{a}\partial^{\mu}\delta\phi^{B}+\frac{m^{a}}{R}\dot{\delta\phi^{B}}\left(\overline{G}_{aB,C}\delta\phi^{C}+2\overline{G}_{aB}\delta\Phi\right)\,,\end{split}$
(A.3)

where we defined

$M^{2}_{AB}=\left(\frac{m_{d}^{2}}{2}\bar{G}_{\Phi\Phi,AB}-\frac{m^{c}m^{d}}{2R^{2}}\,\overline{G}_{cd,AB}\right)\,.$
(A.4)

At this point it is convenient to isolate in the $\delta\pi$’s the component
parallel to the chemical potential vector, according to the scalar product
defined by $\overline{G}_{ab}$. We thus decompose

$\delta\pi^{a}=m^{a}\delta\pi_{0}+\delta\pi^{a}_{\bot}\quad\text{such
that}\quad m^{a}\bar{G}_{ab}\delta\pi^{b}_{\bot}=0\,.$ (A.5)

Note that this decomposition is trivial when there is a unique spontaneously
broken Cartan generator. Using (2.16) and (2.17), we recast the action as

$\begin{split}\mathcal{L}_{fluct}&=\frac{1}{2}\overline{G}_{\Phi\Phi}\,\left(\partial^{\mu}\delta\Phi+\frac{1}{2}\frac{\overline{G}_{\Phi\Phi,A}}{\overline{G}_{\Phi\Phi}}\partial^{\mu}\delta\phi^{A}\right)^{2}+R^{2}m_{d}^{2}\frac{\overline{G}_{\Phi\Phi}}{2}\left(\partial_{\mu}\delta\pi_{0}+\frac{m^{a}\overline{G}_{aB}}{R^{2}m_{d}^{2}\overline{G}_{\Phi\Phi}}\partial^{\mu}\delta\phi^{B}\right)^{2}\\\
&+2\overline{G}_{\Phi\Phi}\,Rm_{d}^{2}\,\left(\dot{\delta\pi_{0}}+\frac{m^{a}\overline{G}_{aB}}{R^{2}m_{d}^{2}\overline{G}_{\Phi\Phi}}\dot{\delta\phi^{B}}\right)\left(\delta\Phi+\frac{1}{2}\frac{\overline{G}_{\Phi\Phi,A}}{\overline{G}_{\Phi\Phi}}\delta\phi^{A}\right)\\\
&+\frac{1}{2}\overline{G}_{ab}\partial^{\mu}\delta\pi^{a}_{\bot}\partial_{\mu}\delta\pi^{b}_{\bot}+\overline{G}_{aB}\partial_{\mu}\delta\pi^{a}_{\bot}\partial^{\mu}\delta\phi^{B}+\frac{m^{a}}{R}\dot{\delta\pi^{b}_{\bot}}\overline{G}_{ab,A}\delta\phi^{A}\\\
&+\frac{1}{2}\left(\overline{G}_{AB}-\frac{\overline{G}_{\Phi\Phi,A}\overline{G}_{\Phi\Phi,B}}{4\overline{G}_{\Phi\Phi}}-\frac{m^{c}\overline{G}_{cA}m^{d}\overline{G}_{dB}}{R^{2}m_{d}^{2}\overline{G}_{\Phi\Phi}}\right)\partial^{\mu}\delta\phi^{A}\partial_{\mu}\delta\phi^{B}\\\
&+\left(\frac{m^{a}}{R}\overline{G}_{aB,C}-\frac{m^{a}\overline{G}_{aB}\bar{c}_{C}}{R\bar{c}}\right)\dot{\delta\phi^{B}}\delta\phi^{C}-\frac{1}{2}M^{2}_{AB}\delta\phi^{A}\delta\phi^{B}\,.\end{split}$
(A.6)

We can rewrite (A.6) compactly by defining

$K^{2}_{AB}=\overline{G}_{AB}-\frac{\overline{G}_{\Phi\Phi,A}\overline{G}_{\Phi\Phi,B}}{4\overline{G}_{\Phi\Phi}}-\frac{m^{c}\overline{G}_{cA}m^{d}\overline{G}_{dB}}{R^{2}m_{d}^{2}\overline{G}_{\Phi\Phi}}\,,\qquad
d_{BC}=\frac{m^{a}}{R}\overline{G}_{aB,C}-\frac{m^{a}\overline{G}_{aB}\overline{G}_{\Phi\Phi,C}}{R\overline{G}_{\Phi\Phi}}\,.$
(A.7)

Then, shifting $\delta\Phi$ and $\delta\pi_{0}$ as

$\delta\Phi\rightarrow\delta\Phi-\frac{1}{2}\frac{\overline{G}_{\Phi\Phi,A}}{\overline{G}_{\Phi\Phi}}\delta\phi^{A}\,,\qquad\delta\pi_{0}\rightarrow\delta\pi_{0}-\frac{m^{a}\overline{G}_{aB}}{R^{2}m_{d}^{2}\overline{G}_{\Phi\Phi}}\delta\phi^{B}\,,$
(A.8)

we arrive at

$\begin{split}\mathcal{L}_{fluct}&=\frac{1}{2}\overline{G}_{\Phi\Phi}\left[\left(\partial\delta\Phi\right)^{2}+R^{2}m_{d}^{2}(\partial\delta\pi_{0})^{2}+4\,Rm_{d}^{2}\,\dot{\delta\pi_{0}}\delta\Phi\right]\\\
&+\frac{1}{2}\overline{G}_{ab}\partial^{\mu}\delta\pi^{a}_{\bot}\partial_{\mu}\delta\pi^{b}_{\bot}+\overline{G}_{aB}\partial_{\mu}\delta\pi^{a}_{\bot}\partial^{\mu}\delta\phi^{B}+\frac{1}{2}K^{2}_{AB}\partial^{\mu}\delta\phi^{A}\partial_{\mu}\delta\phi^{B}\\\
&+\frac{m^{a}}{R}\overline{G}_{ab,A}\dot{\delta\pi^{b}_{\bot}}\delta\phi^{A}+d_{AB}\dot{\delta\phi^{A}}\delta\phi^{B}-\frac{1}{2}M^{2}_{AB}\delta\phi^{A}\delta\phi^{B}\,.\end{split}$
(A.9)

Upon canonically normalizing the fields, we recognize that the first line of
(A.9) coincides with the action for the fluctuations around the helical
solution (2.24) in free theory. We therefore immediately conclude that the
corresponding dispersion relations are given by (2.26) in the main text.