Convexity of Charged Operators in CFTs and theWeak Gravity Conjecture
Eran Palti
Ben-Gurion University
Ofer Aharony, EP 2108.04594

Quantum Mechanical Systems at Large Quantum Number,
IPMU (Zoom), September 2021

String theory provides a large number of different low-energy effective
theories which have an ultraviolet completion to quantum gravity
In general, the different theories have different properties, like number ofparticles, gauge groups, etc..
But there are some (rare) features which, as far as we can tell, arecommon to all of them
The Swampland program is motivated by the existence of such apparently
universal features, with an aim to understand if they are always requiredby any theory of quantum gravity[Vafa ’05]

An old idea is that in quantum gravity there are no U(1) global symmetries

The simplest argument is that a black hole does not reflect any global
symmetry charge on its horizon

Therefore, there are an infinite number of microstates associated to a
black hole – in contradiction with the expected finite entropyIt is possible to rephrase this argument in terms of an infinite number ofstable black hole remnants+
Black hole
+
+
+

+

+

+

+
+

N copies of 1
charged particle

+
+

+

+

+
+

N+1 copies of 1
charged particle

Hawking radiation!"
Remnant ofcharge N

Remnant ofcharge N+1
Infinite number of remnants

Figure 10: Figure illustrating how in the presence of a U (1) global symmetry, black holeformation and decay would lead to an infinite number of stable remnant states below a fixedAnd we can rephrase it again in terms of an infinite number of stablebound states

m
𝑚
𝑚
𝑚
<
<
𝑞 !"#$%
𝑞 &'()*+,𝑞 .$/)0*$1 -,2-Gravity

Gauging the U(1) modifies the story
Gauge

2𝑔𝑞
m

Gravity𝑚 ≥ 2 𝑔 𝑞 𝑀3

But we recover the same physics in the 𝑔 → 0 limit

The Weak Gravity Conjecture proposes that we still should not have thestable bound states / stable black holes (not clear that should be true)[Arkani-Hamed, Motl, Nicolis, Vafa ’06 ]If so, then we must have a charged particle with mass smaller than charge
2 𝑔 𝑞 𝑀3 ≥ 𝑚

The logic naturally leads to the expectation that in the case of masslessscalar fields we should demand the particle is self-repulsive under all forces

2𝑔𝑞
m

Gauge

µ

ScalarGravity

So should formulate the WGC as

2𝑔! 𝑞 ! 𝑀"! ≥ 𝑚! + 𝜇! 𝑀"!

[EP ’17]

Argument receives support from string theory
2𝑔!𝑞!𝑀"!

𝑚!

𝜇!𝑀"!
[Lee, Lerche, Weigand ‘18][…, Lee, Lerche, Weigand ’18,…]

No violation found to date (but difficult to test in non-supersymmetric settings)˜ (n1 + n2 )˜ (n1 ) + ˜ (n2 ) ,(1.2)

e displaced from its copy. In particular there is no obvious relation between the loInanti-de Sitter space (AdS), the separation of a particle from its copy isfor any positive integers n1 , n2 .1
force
and thebybinding
energy. This makes the binding energy formulation of the W
boundedthe radius
Our proposal is motivated by, but is much more general than, the Weak Gravity Conwhat di↵erent from both the statements about unstable Reissner-Nordstrom black ho[1] (WGC)
in Anti-de Sitter
space. Indeed,
we proposeas:that the WGC should be
Itjecture
is morenatural/accurate
to formulate
the conjecture
andformulated
about repulsive
long-range(2.2).
In theAdS
space there
is of
a acontribution
fr(also in flatspace) as aforces
statement
about
self-binding
energyparticle:

Positive Binding
Conjecture:
For awhich(weaklycannot
coupled)begravitational
theory with a U (1)ct interactions
to the binding
energy,dismissed.

field,
there
shouldprecise
exist at forleasttheone charged
in the theory,
withspace,charge offollowingordert usgauge
make
this
more
case ofparticle
5-dimensional
AdS
one, which has a non-negative self-binding energy.

ation in [9]. We consider a 5-dimensional gravitational theory with a U (1) gauge fi
By a self-binding energy here we mean the di↵erence of energies between the lowest two-scalarfieldstate' charged
it withchargeq, thathas anAdS
5 solution:
For
example,consider5-dimensional
AdS
withchargedscalarparticleandtwiceundertheenergyof theone-particle
state.
This
formulation
is closely
related to previous formulations,most✓ closely to self-repulsive statements [2, 3], but has
Z
p
1
1
1 2
2
5
some importantdi↵erences
which
becomemore
pronouncedin
anti-deS = 2 dxg
R+6
Fµ⌫ + |Dµ '| Sitter (AdS) space.
22
4g
Specifically, in AdS space the binding energy receives important contributions from contact2m |'|terms, and not only from long-range forces.

2

4

2

a |'| + b |'| |Dµ '|2.

The organization of this paper is as follows. In section 2 we discuss the formulation of(2

up to two-derivative order). Let us denote the self-binding energy[Fitzpatrick, Shih ‘11]
Binding energy is calculated (at tree level) as
culated at leading order in perturbation
theory in [9], which foundwith⇡2N 4 g2q2
photongravitonquartic
=
,
2
=
+
+
,
(2.4)
2
1
'
'2
'2
'2
2⇡ 2 N 4 2 (2)
graviton
=
,
'2
3(
(2
1)
ue for four non-compact space-time dimensions. In higher dimensions
it1)is2 4
4 2 2⇡
N (a + b (2))⇡ 2 N butg qstill there willFig.not
quartic
ance force couldphoton
be attractive
be
any
bound
state.
2: Witten Diagrams for the quartic contact interaction (left) and photon,
=
.
=
,
with:
'2 right).
graviton exchange (middle,
'2
(
1) (2
1)
2
1
photon'2

graviton'2
quartic'2

2⇡ 2 N 4 2 (2)4.2. AdS/CFT calculation of four-point functions=,
Here we have introduced(the dimension of the CFT operator dual t
3(1) (21) The O(κ ) correction to the four-point function is calculated using the standard techniques of AdS/CFT. For the model (2.2), the relevant Witten diagramsr are shown in fig.
⇡ 2 N 4 (a + b (2 2. We))see that there are three types of contributions to the four-point function in1general2=. scalar quarticminteractions,= (photon exchange,4) (2.5), N= exchange.
.
–
those
from
and graviton
(
1) (2
1)
2⇡ 2 We will2

separate out these contributions and write

introducedC(xBinding, x , x , x ) =Conjecture, x , x implies(x , x in, x ,allx )+CUV-complete(x , x , x , x th)
The Positive(the dimension of the CFT
operator
dual toC ') (xandN, x )+Cas that
(4.10)
O1 O2

1

2

3

4

(quartic)
O1 O2

1

2

3

4

(photon)
O1 O2

1

2

3

4

(graviton)
O1 O2

1

2

3

4

4
there
are no counter-examples
as farwe know.
InAdS
thePoincarécase whe! ≥toPositive Binding Conjecture
demands:
𝛾4calculations
0arethis
Witten diagram
generally
moreastractablein Euclidean
r
m2 =

(

patch coordinatescompleted into
0 2 is taken as a BPS state, w
1a supersymmetric one,dw
" 2and+ (dw')
24) , N =

.

ds =

0 2

.
(2.6)

(4.11)

The binding energy is dual to the anomalous dimension of the operator 𝜙 5
𝛾4! = Δ 𝜙 5 − 2Δ(𝜙)Where 𝜙 5 is the leading operator in the Operator Product Expansion of 𝜙×𝜙
So, if we write the dimension of operators as a function of charge 𝑞 we havethe conjecture:
Δ 2𝑞 − 2Δ 𝑞 ≥ 0

1Introduction

In thiswe formulateconsider propertiesof localoperatorsas:
in unitary Conformal Field Theories
We
cannotethenthis moregenerally(CFTs) with continuous global symmetries (in d

2 space-time dimensions). We proposethat such operators should satisfy a certain convexity-like property:
Abelian Convex Charge Conjecture: Consider any CFT with a U (1) global symmetry. Denote by(q) the dimension of the lowest dimension operator of charge q. Then thismust satisfy a convex-like constraint(n1 q0 + n2 q0 )(n1 q0 ) +(n2 q0 ) ,(1.1)

for any positive integers n1 , n2 , for some q0 of order one.
Equivalently, the conjecture states that the operator product expansion (OPE) of thelowest-dimension operators with positive charges has no singular terms, whenever the charges

1What is 𝑞6 ? From the bound-states intuition we expect it to be the chargeof the operator with the smallest dimension-to-charge ratio in the theory
It is clear that if we define it that way, the conjecture follows almost trivially:Δ(𝑛 𝑞6 ) Δ(𝑞6 )
>
𝑛 𝑞6
𝑞6
So, the non-trivial statement is: 𝒒𝟎 is of order one
This is capturing the WGC: if the operator could form ‘bound states’ then theminimum dimension-to-charge ratio would occur at large q

Now we make a ‘wild leap’:
Since the formulation seems unrelated to the existence of a weakly-curved
gravity dual, we propose that the conjecture holds for any CFT

For example, the formulation has no problem handling a gravity dual with an
infinite number of massless higher spin fields (it makes no reference togravitons or gauge fields)

is that q0 is not parameterically large in any parameter of the CFT. We propose that this isan exact property of any CFT whose continuous global symmetry is precisely U (1).

Before testing the conjecture, we formulate a more general version:

There is a natural generalization of the proposal also to theories with larger symmetrygroups :Convex Charge Conjecture: Consider any CFT with a continuous global symmetrygroup G, and consider a simple factor G0 ⇢ G. Denote by(r) the dimension of the lowest

dimension operator in the representation r of G. Then, there is always some representation
r0 , which is non-trivial in G0 and has weights of order one, such that the dimensions ˜ (q) ⌘(Symq (r0 )) satisfy a convex-like constraint
˜ (n1 + n2 )˜ (n1 ) + ˜ (n2 ) ,(1.2)

for any positive integers n1 , n2 .1
Our proposal is motivated by, but is much more general than, the Weak Gravity Conjecture [1] (WGC) in Anti-de Sitter space. Indeed, we propose that the WGC should beformulated (also in flat space) as a statement about the self-binding energy of a particle:
Positive Binding Conjecture: For a (weakly coupled) gravitational theory with a U (1)Tests of the ConjectureAt large charge the spectrum is convex:
Δ 𝑞

8∼ 𝐴 𝑞 89:

[Hellerman, Orlando, Reffert, Watanabe ‘15]
[Monin, Pirtskhalava, Rattazi, Seibold ‘16]
[Alvarez-Gaume, Loukas, Orlando, Reffert ‘16]

For BPS states in supersymmetric theories, or for free scalar theories, thespectrum is marginally convex:
Δ 𝑞 ∼𝐴𝑞
In 2-dimensional CFTs a U(1) symmetry implies 2 symmetries, each one ofwhich can be gauged to yield a coset CFT, with spectrum
Δ 𝑞 ∼ Δ ;<= + 𝑎 𝑞 5> :For free fermionic theories, the spectrum is not even marginally convex if
we take 𝑞6 equal to the number of components of the fermion
This is because Pauli’s exclusion principle requires derivatives to beinserted in operators with multiple fermions
For example, in 3 dimensions we have the spectrum:
Δ 1 =1Δ 2 =2Δ 3 =4Δ 4 =6Δ 5 =8Δ 6 = 10

[Komargodski, Mezei, Pal, Raviv-Moshe ‘21]

Therefore, any small perturbative interaction will maintain convexity
Way out: Large number of fermions

the caseof weakly-coupledtheories,
canformulatethe
theconjecture
conjectureininterms
terms of
of sp
sp
case
of weakly-coupledtheories,wewe
canformulatetors
relatedtothefieldsappearingtheaction.
Forexample,
example,consider
considerprimary s
relatedtointhefieldsappearingin intheaction.
For
Testing
explicit
theories
requiressome
expansion
parameter
which aaprimaryntorchargedunderaglobalsymmetry,withcharge
of
order
one.
Denote
by
leads
to
some
sense
of
weak-coupling.
n thcharged under a global symmetry, with charge of order one. Denote bythest dimension) primary operator appearing in the (symmetrised) OPE of n ’s. T

imension) primary operator appearing in the (symmetrised) OPE of n ’s. Tnimensionofshould specificsatisfy fields(1.1) associated(or its generalization(1.2)).
We can introd
We
can
then
identify
totheoperators,
andnnsion ofshould satisfy (1.1) (or its generalization (1.2)). We can introd
measure notation
convexityasbycondensed

densed notation as

n1 ,n2 ⌘n1 ,n2 ⌘hen write (1.1) or (1.2) as
Requiring:

write (1.1) or (1.2) as

n1 +n2n1 +n2⇣

n1 ,n2⇣( n1 ) +( n1 ) +⌘( n2 ⌘) ,( n2 )

,

0.

0
.
n
,n
1
2
he last statement depends on the boundary conditions, but it is always true for d

3. In any ca
eststatement
always true;
a continuous
symmetry
CFT trueis related
field
statementisdepends
on theboundary global
conditions,but itinisthealwaysfor d to a3.gauge
In anyca

4point [38]. We follow here theanalysisoftheanomalous
dimension2µ ††L = @ @µ fixed+ point (when.
(5.1)
mensionsthere
isa
Wilson-Fisher
the
mass
is
fine-tuned
to
4
tion
is
given
by
The simplest explicit
theoryto test it inthere
is Wilson-Fisher
U(1) in4 −point
𝜖 dimensions
In d =
4 ✏ dimensions
is a Wilson-Fisher
fixed(when the m
pling
zero)
a coupling fixed point (when the mass is fine-tuned to
4 ✏ dimensions there
is aatWilson-Fisher
✏
3 22
?
3
✏
3 2
?
3 (5.2)
µ †
†
=
+
✏
+
O
✏
.
=
+
✏
+
O
✏
.
t a coupling
L = @ (4⇡)@µ 2+ 5 25 .
(5.1)
2
5 25
(4⇡)

4

✏
3 2
3
=
+
✏
+
O
✏
.
n
2
n
5 of 25 in this model is [39]n ofin this model[39]
Theis(4⇡)
dimension
?

(5.2)

here
Wilson-Fisher
fixed the
point
(when
the mass
is fine-tuned
toNotisaarigorous
test, because
theorycontains
non-unitary
operatorsnmension ofin this model is [39]

d

✏
n
( )=
n + n (nd ✏ 210 3 ✏2
?n2
(

2

)2==

(4⇡)

1)

2 Rychkov, van Rees ‘15]
[Hogervorst,
d 22
✏
✏
✏n2
( )n
= 2n n 8n+ +n5(n+ O1) ✏3 . n 2n2 (5.3)
8n + 5 + O
100
100✏2 2 2 10
3
3+n + ✏ n+(nO 1)✏
10
52 This25gives

.

100

n 2n

8n + 5 + O ✏. (5.2) (5.3)

n1 n2
n1 n2
n
n
n
n
✏
(3n1 + 3n2
1 2
3
n1 ,n2 = 2
ves model is=[39]1 2 ✏his(3n8) ✏ 5+ O ✏ 50 ,1 + 3n2✓ n1 ,n2
n
n
n
n
1
2
1
2
5=
✏50
(3n + 3n8) ✏2 + O ✏3 ,n1 ,n28) ✏2 + O ✏3 ,

(5.4)

1
2
8[Badel, Cuomo, Monin,(5.4)
Rattazzi ‘19]
which
satisfies(3.3)
at
leadingorderin✏.
5 2 8 50

(3.3)
in✏ ✏.
2 at ✏leading order3
8 2
8 order insatisfies(3.3)
at
leading
✏.
n
+
n
(n
1)
n
2n
8n
+
5
+
O
✏
. 𝑛a small(5.3)
Note
that
the
expression(5.3)
isderivedexpansion parameter
Convex, and remains convex for any value of (𝜖𝑛) forassuminglarge

2e expression
10 (5.3) is derived
100
assuming
a small
expansion✏ntheory⌧ 1, goesand overthentothe
first termin (5.4) always
dominates.
Forparameter
large ✏n thethe large ch[Cuomo,(Privatee) thatthedominates.
expression (5.3)
isderivedassuming
smallover
expansion✏nKomargodski⌧regime1, and discussed
then
theComm.)]in theprevious
section,is convex.
The
spectrumremains
convex[40] throu
alwaysForlarge✏n thetheorya whichgoes
to theparameterlarge
charge✓◆223(4⇡ )

18(4⇡ )

3 (2), we find?
5.11) to compute= ✏ + O ✏2 .
28

iven by

✓

✓

2n n

◆2

1 2 2)
n
(n
1)
(n
=
✏
n1 ,n2n+N9 + 82
8⇡
onsistent
with(3.3).
◆4n (n 1) h 2?
The O(N ) sextic
39⇡ (nmodel2) (nin 3)
8⇡
72

2

(5.10)𝑂 𝑁therefore(quartic)− 𝜖◆dimensions:We
havemodel in 4 ✓48⇡?

1,1 = 2

?

n n8⇡
1 2

(N + 8)+ O( 6? ) ,⇣(5.13)[Jack, Jones ‘21]
(5.8)

2
✏
48n(5.11)
1 + 48n2
32✏ dimensions was also
in
[46].
We
can
consider+6nanalysed
N
+
6n
N
16N
+
N
132
1
2

i
he dimension of an3 operatorof total chargen 6(as in section 5.2), with r0 the fundamental
2+8n56n + 272n

456 + O.

?
𝑈and1 soand
𝑂
𝑁
model
in
3
−
𝜖
dimensions:epresentationof O(N). 9 (sextic)
They found(3.3) is satisfied at leading order in ✏.

✓

◆2

n1 n2
n1 n2 (n2 1 + n2
?
=
(n
+
n
2)
✏
+O ✏ .
n1 ,n22n128,n2 = 18⇡
3

2)

(5.12)✓
5.3 The sextic
and O(N ) models in 3
✓ takes◆4 a Uh(1)rder perturbative correction
morecomplicated form, related⌘+ O ✏3 ,(5.14)✏ dimensions

n1 n2
2
3
8
182
+
123n
32n
+
5n
1
1
1
t two-loop order. In particular
we72
see that for n1 , n2 > 1 we have a8⇡
2
4
For the sextic U (1) model in d2 = 3 2✏ (where2 both
the
|
|
and
the
|
|
operators are3nsion at leading order, +123nwhich is the48n
2-looplevel.
But
for
n
=
n
=
1,
1
2
32n2 + 10n1 n2 + 5n2
2
1 n2 + 10n1 n2i The Euclidean Lagrangian isfine-tuned
will4-loop
follow
the analysis
46].
n vanishes,
and we tohavezero),
to lookwe
at thecontribution.
Using of [45,222
+ (16 + N ) 11 9n1 + 2n1 9n2 + 3n1 n2 + 2n2 ⇡ + O( 6? ) .
?

we find

✓ ◆

3
✓ ◆4
µ
¯
¯
[Badel, Cuomo, Monin, Rattazzi ‘19]
L
=
@
@
+
(5.9)
This is consistent with
(3.3)
for
all
valuesofnandn(including
n
=
n
1
2µ
1
2 = .1).
?
6
+
O(
)
,
(5.13)
=
2
✓ 1,1
3!
?
[Jack, Jones ‘21]
8⇡
9

In [46] certain contributions to the dimension ofn

at four loops were not included, specifically self-✓⇣ ⌘ ◆⇣⌘
2the large
N limit,forfundamental
representationof O(N )). in
In thisd(inconsidera generalchargeq, with(q). Expanding
q operator
4 r0qoftheq 3dimension= Nanalyse+ theories+ O large 𝑁 expansion.
(5.16)Can(q)alsousing
2⇡ N of charge q,
N with dimension (q). Expanding inone
obtains a[47,48]2N operator
d consider
general

𝑂 𝑁 (quartic)✓⇣ ⌘ ◆
ne obtains[47, 48]  model in⇣ 3 dimensions:⌘20, consistent with theq conjecture.
4 qq 3(q) = N+ 2
+O.
(5.16)
✓◆
2N dimension⇡ NN⇣= 5⌘(where⇣⌘o formally calculate the
also
in
d
the non-trivial
2[Alvarez-Gaume, Orlando, Reffert ‘19]
q
4 q
q 3
= N than an+ IR2 fixed point),+ O and this gives. [48] [Giombi, Hyman(5.16)✓point(q)‘20]
V fixedrather2N⇡N
N
1 ,n2 > 0, consistent with the conjecture.
✓also◆d = 5 (where the non-trivial
ible to formallycalculatethedimension
in
⇣
⌘
⇣
⌘
2
3
3q with32the conjecture.
qq0,
consistent,n2 >N ratherO point), (non-unitary):. this gives [48] (5.17)
𝑂 (q)𝑁 =(quartic)
model
in
5IR+
dimensions
a UV
fixed
point
than
an
fixed
and
2
3⇡ theN dimension N
ble to formally 2N
calculatealso in d = 5 (where the non-trivial

32 ⇣ q ⌘2

✓⇣

◆
⌘
3
q

[Giombi, Hyman ‘20]

3qconjecture.
a UV 0,fixed
point rather
than an IR
fixed point),
and calculate
this gives
violatingthe
Similarly,
one canin[48]
6Klebanov,✏[Giombi,Huang,
Pufu, Tarnopolsky ‘19]n2 <✗(q) = N+O.
(5.17)
2
2N
3⇡ N
N[Arias-Tamargo, Rodriguez-Gomez, Russo ‘20]2

 believed to✓⇣fixed◆| theories (which are
flow
to IR
pointsrelated to the
⇣
⌘
⌘
2
3
3q
32 q
q
= in
N the
+this
O givesone. calculate in 6 ✏(5.17)
𝑁 (quartic)
model
in
6
−
𝜖
dimensions(non-unitary):✏
expansion,
and
[49,
50]
oentioned0,(q)violating
theconjecture.
Similarly,
cann𝑂
2
1 ,n2 <above)
2N
3⇡ N
N
or | |2 theories (which are believed✓⇣
to flow◆to IR fixed points related to the
⌘
2
n
n
✏n
1
2
[Arias-Tamargo,
Rodriguez-Gomez,
Russo ‘20]<
0,violatingtheconjecture.
Similarly,one
cancalculatein6✏
ntsn1mentioned
above)
inthe✏
expansion,
and
this
gives
[49,
50]
,n
=
264✏
+
O
,
(5.18)
2
n
,n
✗
1 2
2
[Antipin,Bersini,Sannino,Wang,Zhang ’21]NN2
r | | theories (which are believed ✓to⇣ flow
◆to IR fixed points related to the
⌘
2
n1 n2
✏n

we have scalars, we must also consider 4 couplings that are generated by t
Gauge theories, with perturbative approach:ation group flow, and we need to consider fixed points for these couplings
4-dimensional 𝑆𝑈(𝑁? ) gauge theory with 𝑁@ massless fermions and 𝑁A
o the gauge coupling. This question was analyzed in [66] (see also [67]), whemassless scalars has a perturbative (Banks-Zaks type) fixed point at large 𝑁
Ns scalars and a scalar potential of the form
This needs to coincide with the fixed point of the quartic scalar terms
V ( ) = h̃Tr( ††) + f˜(Tr( † ))2 ,(5.2

global symmetry[Hansen,
is SU(Ns )Langaeble,⇥ U (1)
We canconsiderB . Sannino,
Janowski,Mann,
Steele, Wang ’17]

Iossa, Serone ‘19]
viewed as an Nc ⇥ Ns matrix. They[Benini,
foundthat for small enough values of Ns /N⇤ nScalar mesons
chargedunder
𝑆𝑈unitary,
𝑁A global
symmetry
perturbative
fixedarepoints
which
are
with
real valuesOforthecouplings
) ,
n = (

a positive-definite scalarpotential.
They
points of this type, wi
4
4
4 foundE 4 fixedG
Find (at 1-loop): 𝛾:,: = Δ 𝒪5 − 2Δ 𝒪: = 𝐴(ℎ + 𝑓)
𝐴>0mbers of relevantwhereoperators;
ourchooseanalysisthem
below
apply toinallof these fix
now wetowilltransform
symmetricp
points tohaveℎE + 𝑓G perturbative> 0, so spectrum
convex of this✓ type require havice At
Nsfixed/Nc needs
be small,fixedis pointsof SU (Ns ). In particular we can choose, for instance, aGauge theories with semi-classical methods:
3-dimensional 𝑈(𝑁? ) gauge theory with 𝑁@ fermions flows to CFT in the IR
Has global symmetry 𝑈 1 )"C with monopole operators charged under it
At large 𝑁@ the dimensions of these can be computed as

✓

q
1
2
3
4
5

q

0.265 Nf 0.0383 + O(1/Nf )0.673 Nf 0.194 + O(1/Nf )1.186 Nf 0.422 + O(1/Nf )1.786 Nf 0.706 + O(1/Nf )2.462 Nf 1.04 + O(1/Nf )[Pufu ‘13]
[Dyer, Mezei, Pufu ’13]
[Chester, Iliesiu, Mezei, Pufu ‘17]

first few monopole operator dimensions q for monopole charges q in the U (Nc )in 3 dimensions with Nf fermions in the fundamental, taken from [54].

Can do similar computations for large 𝑁@ scalars (with or without quartic
terms), and also for fermions with quartic couplings (Gross-Neveu type)

✓

q
1
2
3
4
5

CPNf
0.125
0.311
0.544
0.816
1.121

tri-critical
0.097
0.226
0.384
0.567
0.771

✓

[Dyer, Mezei,’13] term in the 1/Nf expansion, namely
the coefficientof thePufu
leading[Dyer,
Pufu, Sachdev[Dupuis,Paranjape,(q)/Nf , fortheMezei,
dimensionof the ’15]
first few monopoles,
for theCPNf Witczak-Krempa ’19]
[Chester – Private Communication]
ritical model[62].

Can also add Chern-Simons term with level 𝑘, and find that monopole
spectrum 11becomes more convex monotonically with increasing 𝑘tal of SU (Nc ).

[Chester, Iliesiu, Mezei, Pufo ’17]
[Chester ‘21] [Chester – Private Communication]
4

ars, we must also consider

couplings that are generated by the✓A simple supersymmetric counter-example to 𝑞6 = 1 ∶Can consider single chiral superfield:

𝑊 = 𝑔ΦD

In 3 dimensions flows to IR interacting fixed point (in 4 dimensions IR free):Global 𝑈 1 E symmetry with charges and dimensions:55Scalar: 𝑞4 = D , Δ 2 = D:

GFermion: 𝑞F = D , Δ 1 = H

Requires taking 𝑞6 = 2, and then convexity seems natural (but not proven)from BPS bound:Δ 2𝑛 > 𝑛 Δ(2)All our examples are consistent with strong versions of conjecture:1) 𝑞6 is charge of lightest charged operator2) 𝑞6 is smallest scalar operator charge

Have proposed counter examples on gravity sidein flat space for supersymmetric theories withmultiple U(1)s: [Heidenreich, Reece, Rudelius ‘16]

Can test in completely strongly-coupled theories: O(2) model in 3D
Q
1
2
3
4

✏5
0.518(1)1.234(3)2.10(1)
3.114(4)6

1.23(2)2.10(1)3.103(8)MC
0.5190(1)1.236(1)2.108(2)3.108(6)bootstrap
0.5190(1)1.236(3)-TableI.from:[Banerjee,
Chandrasekharan,
Orlandopreviously
‘18]
TABLE
Conformal
dimensions
D(Q) obtainedby other methods for Q  4: Field theory results in 4 ✏dimensions at five loops are in column two, six loop results at
Theory describing thed = 3 are in column three ([32] for Q = 2, in [33] for Q = 3
I
inMC
𝐻𝑒
and superfluidin [34] for Q transition= 4), previous
results are in column
four [35], and bootstrap results are in column five [36].

In order to overcome this problem we have dalgorithm to efficiently compute the ratio
CQ (r = L/2)
R(L/2) =CQ 1 (r = L/2)

on cubic lattices of side L for 8  L/a 
criticalpoint(the detailsof our algorit
CFT data method c value
ref [Chester et al. ‘20]
found in theEXP
supplementary
1.50946(22) [85] material). Using
MC
1.51122(15) [86]easy to extractthedi↵erence (Q) = D(Q)CB
1.51136(22)using the relation
R(L) ⇠[86]1/L2 (Q) . The ac
MC
0.519050(40)which we areCBable0.519088(22)to compute the ratio R(L/2MC
1.2361(11)[87]values of Q CB
can beseen in Fig. 2. Once the1.23629(11)stSummary
Proposed that the natural formulation of the WGC is in terms of the selfbinding energy of a particle
This leads to a CFT dual statement which is that the spectrum of charged
operators should be convex
Seems to hold in all the examples we tested so far
In the absence of a general argument/proof, we need to keep testing it,
searching for counter-examples (and learning what is special about them
if they exist…)Experimental predictions/tests?

Thank You

A simple supersymmetric counter-example to 𝑞6 = 1 ∶Can consider single chiral superfield:

𝑊 = 𝑔ΦD

In 3 dimensions flows to IR interacting fixed point, in 4 dimensions IR free
5

:

D

D

Global 𝑈 1 E symmetry with charges: Scalar 𝑞4 = , fermion 𝑞F = −Consider 4-dimensional free theory, then have
Δ 2 = 1,

Δ 1 =

D
5

Requires taking 𝑞6 = 2, and then convexity seems natural (but not proven)from BPS bound:Δ 2𝑛 > 𝑛 Δ(2)