Sharp Boundaries for the Swampland
East Asian String Webinar
Mar 26 2021

Dalimil Mazáč
IASfiarXiv:2102.08591, with S. Caron-Huot, L. Rastelli, D. Simmons-Duf nGoal: Constrain Theories of Quantum Gravity
Our universe contains gravity
What is the set of consistent universes?

• Only a few options or a vast landscape?
• Do all quantum gravity theories come from string theory?

• How does the answer depend on the spacetime asymptotics?
AdS / at space / dSfl• Which low-energy EFTs admit a UV completion?

this talkWhich EFTs Admit a UV Completion?
Powerful constraints on IRfrom UV consistency:• no global symmetries• weak gravity conjecture• distance conjecture

Drawbacks:

Would prefer:• often conjectural, based on examples• parametric bounds: A . B• rigorous, based on general principles• precise bounds: A  3B

a success story: CFT bootstrap

quantum gravity

Ising: Scaling Dimensions
ΔϵMonte Carlo

1.4130?

1.41265

1.41291.41264?

1.41263
1.41262

1.41281.41261
1.41260

0.518146 0.518148 0.518150 0.518152

1.41271.4126

Bootstrap

1.41250.51808?

0.518100.518120.518140.518160.51818

ffi[Kos, Poland, Simmons-Du n, Vichi ‘16]

ΔσRoadmap
1. What is the space of a priori parameters?
irrelevant couplings in the IR e ective eld theory
2. What are the fundamental principles / assumptions?

causality and unitarity of the S-matrix
3. How do we use 2 to put bounds on 1?

fi

ff

use dispersion relations in a novel way

Low-Energy Effective Field Theory of Gravity
1S=16⇡GNZdx

Dpg R + ↵2 R 2 + ↵3 R 3 + . . .

GN = Mp2 D

all interactions irrelevant!
EFT works for energies

E ⌧ M < Mp
UV cutoexpectation:↵2 = # · M

2↵3 = # · M

Planck scale

4[Camanho, Edelstein, Maldacena, Zhiboedov ’14]

Natural task: prove rigorous bounds on the O(1) coe cients|↵2 |  # · M

2|↵3 |  # · M

4ffi

flf

Measurable in in ationary correlators if Hubble scale ⇠ M

First Step: Weakly Coupled Theories
1
S=16⇡GNZdx

Dpg R + ↵2 R 2 + ↵3 R 3 + . . .

overly ambitious goal: show that |↵2 |  # · Mp

2counterexample: weakly-coupled string theory g ⌧ 1gµ⌫0M =g
↵2 ⇠ M

22D2MpMp

Mp 2a natural rst goal: prove bounds when coupling is weak|↵2 |  # · M

2forM ⌧ Mpfffi) can ignore loop e ects at scale M and below

EString Universality?
More generally, classify weakly-coupled theories of gravity

Only 3 known answers in at-space:Is this the complete list?

• Einstein• heterotic string• type II string[Arkani-Hamed, Huang][Chowdhury, Gadde,Gopalka, Halder, Janagal,Minwalla ’19]
[Chakraborty, Chowdhury,Gopalka, Kundu, Minwalla,Mishra ’20]

Related question without gravity: classify theories of weaklycoupled massive higher-spin particles.
Large N gauge theories

fl[Caron-Huot, Komargodski, Sever, Zhiboedov ’14]

Bounds from Causality [Camanho, Edelstein, Maldacena, Zhiboedov ’14]
Idea: too large irrelevant couplings lead to superluminal propagationx+xx

Superluminal propagation unless↵2↵ 3 ⌘ GN Ex = 1± 2 ± 4bbbD 4|↵2 | . b2min|↵3 | . b4min

Want to replace.

!

⇣

wherebmin ⇠ M↵3 M 4?
↵2 M 21

Our Main Result: Bounds for Gravity + Scalar[Caron-Huot, DM, Rastelli, Simmons-Du n ’21]1
S=16⇡GNZp ⇥d x g R + (r )2 +D

04+

2r4 4+

assume the UV cuto exactly at M
and that the theory is weakly coupled, i.e. M ⌧ Mp
g3 M 4483πMG

D=5150

D=6D=7100

D=8D=950- 15- 10-55ffi- 50ff

D=101015M2g22M8πG

2

D=11D=123r

6 4+ ...

⇤Roadmap
1. What is the space of a priori parameters?
irrelevant couplings in the IR e ective eld theory
2. What are the fundamental principles / assumptions?

causality and unitarity
3. How do we use 2 to put bounds on 1?

fiff

use dispersion relations in a novel way

The S-Matrix Axioms
genuinely Lorentzian observable= A(s, t)

1. causality) analyticity in the complex s planesM2u0

3. unitarity

2. weak coupling in the IRA(s, t) = 8⇡GNtu us st+++stu+2 (s2) positivity of the UV contributions

0++ t2 + u 2 ) +4. bound on Regge growth

M23 (stu) + . . .

A(s, t)!02s

Im[M(s,t)t)] =A(s,1X

J=0

as

s!1

⇢J (s)PJ (cos ✓)0Bound on the Regge GrowthA(s, t)
!02
s

as

s!1

su

M20

Satis ed in weakly coupled string theory

M2A(s, t) = O(s2+↵0 u)

Classical Regge growth conjecture A(s, t) = O(s2 )[Camanho, Edelstein, Maldacena, Zhiboedov ’14]
[Maldacena, Shenker, Stanford ’15]
[Chandorkar, Chowdhury, Kundu, Minwalla, ’21]

We need a little more!
Rigorously provable for scattering in AdS

fi[Caron-Huot, ’17]

|G |  |G|Roadmap
1. What is the space of a priori parameters?
irrelevant couplings in the IR e ective eld theory
2. What are the fundamental principles / assumptions?

causality and unitarity
3. How do we use 2 to put bounds on 1?

fi

ff

use dispersion relations in a novel wayDispersion Relations[Adams, Arkani-Hamed, Dubovsky, Nicolis, Rattazzi ’06]

[Arkani-Hamed, Huang, Huang ’20] ...

Causality requires IR and UV to talk to each other!
Idsf (s) = 0su

M20M=02su

M20IR

M2=0UV)IR = UV
positive!

[Arkani-Hamed, Huang, Huang ’20]
[Caron-Huot, DM, Rastelli, Simmons-Du n ’21]

Dispersive Sum Rulessu

M20IR

st(st)

k2 +1

integrate A(s, t) against=0M2withUV

k = 2, 4, . . .

IRCk,u = Ress0 =0002s + u
A(s , u)s0 (s0 + u) ⇥s0 (s0 + u)⇤k/2#*2J even

2u1+ m22m + u PJ
m2 + u ⇥m2 (m2 + u)⇤k/2Z 1X
1dm2 4 D(D)2h(· · · )i ⌘nJm⇢(m) (· · · )J2⇡
M2 mffiwhere"

UV+=0

Dispersive Sum Rules
only sensitive to terms growing in the UV: gravity, 2 , 3 , ...

8⇡GN

✓

1+2 2t3t + 8 4t)

GN > 0t!02+ ...

◆=*2t
1+ m22(2m + t)PJm2 (m2 + t)

+UVgravity is attractive

Suppose no gravity ) can expand around the forward limit t = 02 =⌧1m2>0UV3 =⌧3#m43 2 2MUV

[Adams, Arkani-Hamed, Dubovsky, Nicolis, Rattazzi ’06]
[de Rham, Melville, Tolley, Zhou ’17]
[Arkani-Hamed, Huang, Huang ’20]
[Caron-Huot, Van Duong ’20], [Bellazzini, Miro, Rattazzi, Riembau, Riva ’20], [Tolley, Wang, Zhou ’20],[Sinha, Zahed ’20], [Gopakumar, Sinha, Zahed ’20]

Problem: Would like to isolate individual couplings in the presenceof gravity but can not expand around the forward limit.

Solution of the Problem [Caron-Huot, DM, Rastelli, Simmons-Du n ’21]
8⇡GN✓

1+2 2t3t + 8 4t

2+ ...

◆=*Part 1: Use crossing symmetry to eliminate

8⇡GN

✓

1+2 2t3t◆=*2t
1+ m2(2m2 + t)PJ
m2 (m2 + t)4,+UV5, . . .

2t(2m2 + t)PJ 1 + m
2+ corrections
m2 (m2 + t)

+UVPart 2: Do not use wavepackets localized around the forward limit.
Instead, localize around small impact parameter!

|b| . M1requires going far from the forward limit |t| ⇠ M

2[Adams, Arkani-Hamed, Dubovsky, Nicolis, Rattazzi ’06]

e ectively combines+ffiff[Camanho, Edelstein, Maldacena, Zhiboedov ’14]

A Comparison with the Forward Limit

EFT coecients without gravity (D=6)
0.5

g4 M 4 /g2

0.4
0.3
forward limit
small impact parameter0.2
0.1
0.0-8-6-4-2

g3 M 2 /g2

024

Results with Gravity
spectrum at the corner

g3 M 4483πMG

D=5150

D=654

D=892

4

D=950- 106D=7100- 15

spectrum minimizing g2 (D=6)

m2 /M 28

D=10-551015M2g22M8πG

2

D=112

D=120- 50

5Allowed (g4,g5,g6) in D=6, for any g3, for g82 πMG ∈{-9,-5,0}
21015

2

bounds on higher dimension couplings forg5 M 8 8M
85πG2- 40- 60=g6 M 1010
68M
πG20- 202M

46g4 M 6 6M
84πG9,5, 0g6 M 1010
πG68 M200

200

150

150

100

100

5050- 50

246g4 M 6 6M
84πG- 60- 40- 20- 50

20g5 M 8 8M85πG20J

A Bound in Maximal Supergravity
Asugra (s, t) = 8⇡GN0

1+stu0+222(s+t+u) + ...
22M
 3.000 . . .
0consistent with type II string theory:0M

2= 2⇣(3) ⇡ 2.40.

[Guerrieri, Penedones, Vieira ’21] : a lower bound on

2M
0 pl

saturated by strongly-coupled string theory?

Summary
Derived rigorous bounds on EFT couplings in consistent weakly coupledtheories of gravity.
The bounds follow from unitarity and causality, implemented by dispersion
relations.
The key new idea is to localize the dispersion relation in the transverse
impact parameter space.

This resolves the notorius di culties presented by the t-channel graviton pole.

Note: only used weak coupling of the EFT, not the UV.

ffi

can incorporate EFT loops

Looking Ahead
The technique generalizes to arbitrary S-matrix elementsa systematic bootstrap program for classifying
weakly-coupled gravity theories from rst principles
uniqueness of string theoryor new possibilities?
non-gravitational S-matrices withmassive higher-spin particles

fiexplore the space of large Ngauge theoriesMore Fascinating Questions

Relationship to at space holography

flCosmology: sum rules for de Sitter gravity?

AdS/CFTConjecture: [Heemskerk, Penedones, Polchinski, Sully ’09]
Large N and large gap implies local description in the bulk.

ffi[Hartman, Jain, Kundu]
Caron-Huot, Alday, Bissi, Kravchuk, Meltzer, Simmons-Du n, Penedones, Perlmutter, Zhiboedov, ...

Dispersive Sum Rules in AdS/CFT[Caron-Huot, DM, Rastelli, Simmons-Du n]

1. Structure of the S-matrix $ OPEu⇥=1+|X[M2]n,` +n,`0sM2X

Clight + [composites] +⌧<gap{z2. Im[A(s, t)] $ h⌦|[ (x4 ), (x1 )][ (x3 ), (x2 )]|⌦i}

X|{z⌧>

3. Causality $ crossing equationh⌦| (x4 )[ (x1 ), (x3 )] (x2 )|⌦i = 0

4. Dispersive sum rules (= analytic functionals) come from
integrating along null rays.
ffiCheavygap}

Dispersive Sum Rules in AdS/CFT
Can essentially repeat the at space argument.

Conjecture: [Heemskerk, Penedones, Polchinski, Sully ’09]
Large N and large gap implies local description in the bulk.

for 4pt functions of scalar primaries

flShow that Einstein gravity in AdS
emerges from these assumptions.