axion inflation and the lattice weak gravity …kias-yitp2017/rudelius.pdflandscape the weak gravity...
TRANSCRIPT
![Page 1: Axion Inflation and the Lattice Weak Gravity …kias-yitp2017/Rudelius.pdfLandscape The Weak Gravity Conjecture The WGC: In any U (1) gauge theory that admits a UV completion with](https://reader033.vdocuments.us/reader033/viewer/2022042921/5f684b1c7d50d30998459921/html5/thumbnails/1.jpg)
Axion Inflation and the Lattice Weak Gravity Conjecture
Tom Rudelius IAS
![Page 2: Axion Inflation and the Lattice Weak Gravity …kias-yitp2017/Rudelius.pdfLandscape The Weak Gravity Conjecture The WGC: In any U (1) gauge theory that admits a UV completion with](https://reader033.vdocuments.us/reader033/viewer/2022042921/5f684b1c7d50d30998459921/html5/thumbnails/2.jpg)
Based On
• 1409.5793/hep-th, 1503.00795/hep-th • 1506.03447/hep-th, 1509.06374/hep-th
• with Ben Heidenreich, Matthew Reece • To appear
• with Ben Heidenreich, Cody Long, Liam McAllister, Matthew Reece, John Stout
![Page 3: Axion Inflation and the Lattice Weak Gravity …kias-yitp2017/Rudelius.pdfLandscape The Weak Gravity Conjecture The WGC: In any U (1) gauge theory that admits a UV completion with](https://reader033.vdocuments.us/reader033/viewer/2022042921/5f684b1c7d50d30998459921/html5/thumbnails/3.jpg)
Outline
I. The Weak Gravity Conjecture
II. The WGC and Inflation
III. Strong Forms of the WGC
IV. Strong Forms of the WGC and Inflation
![Page 4: Axion Inflation and the Lattice Weak Gravity …kias-yitp2017/Rudelius.pdfLandscape The Weak Gravity Conjecture The WGC: In any U (1) gauge theory that admits a UV completion with](https://reader033.vdocuments.us/reader033/viewer/2022042921/5f684b1c7d50d30998459921/html5/thumbnails/4.jpg)
The Weak Gravity Conjecture
![Page 5: Axion Inflation and the Lattice Weak Gravity …kias-yitp2017/Rudelius.pdfLandscape The Weak Gravity Conjecture The WGC: In any U (1) gauge theory that admits a UV completion with](https://reader033.vdocuments.us/reader033/viewer/2022042921/5f684b1c7d50d30998459921/html5/thumbnails/5.jpg)
The Big Picture
Vafa ’06, Ooguri, Vafa ‘06
Swampland
Landscape
![Page 6: Axion Inflation and the Lattice Weak Gravity …kias-yitp2017/Rudelius.pdfLandscape The Weak Gravity Conjecture The WGC: In any U (1) gauge theory that admits a UV completion with](https://reader033.vdocuments.us/reader033/viewer/2022042921/5f684b1c7d50d30998459921/html5/thumbnails/6.jpg)
The Weak Gravity Conjecture
The WGC: In any U(1) gauge theory that admits a
UV completion with gravity, there must exist a state
of charge q, mass m, such that
q
m� Q
M
��extremal
Arkani-Hamed, Motl, Nicolis, Vafa, ’06
![Page 7: Axion Inflation and the Lattice Weak Gravity …kias-yitp2017/Rudelius.pdfLandscape The Weak Gravity Conjecture The WGC: In any U (1) gauge theory that admits a UV completion with](https://reader033.vdocuments.us/reader033/viewer/2022042921/5f684b1c7d50d30998459921/html5/thumbnails/7.jpg)
The Generalized WGC
The generalized WGC: In any p-form theory in d dimensions,
there must exist an electrically charged object of dimension p� 1
and a magnetically charged object of dimension d� p� 1 with
Tel .✓
g2
GN
◆1/2
, Tmag .✓
1
g2GN
◆1/2
![Page 8: Axion Inflation and the Lattice Weak Gravity …kias-yitp2017/Rudelius.pdfLandscape The Weak Gravity Conjecture The WGC: In any U (1) gauge theory that admits a UV completion with](https://reader033.vdocuments.us/reader033/viewer/2022042921/5f684b1c7d50d30998459921/html5/thumbnails/8.jpg)
Strong Forms of the WGC
q
q/m � Q/M |ext.
q/m � Q/M |ext.
q/m � Q/M |ext.
• The lightest charged particle must satisfy !
• The particle of smallest charge must satisfy !
• “Most” charge sites must contain a particle with
![Page 9: Axion Inflation and the Lattice Weak Gravity …kias-yitp2017/Rudelius.pdfLandscape The Weak Gravity Conjecture The WGC: In any U (1) gauge theory that admits a UV completion with](https://reader033.vdocuments.us/reader033/viewer/2022042921/5f684b1c7d50d30998459921/html5/thumbnails/9.jpg)
Why should the WGC be true?
• If not, extremal BH will be unable to decay. • If not, near-extremal BH will move towards
extremality
M = Q
M < Q
M > Q
M = Q
M < Q
M > Q
![Page 10: Axion Inflation and the Lattice Weak Gravity …kias-yitp2017/Rudelius.pdfLandscape The Weak Gravity Conjecture The WGC: In any U (1) gauge theory that admits a UV completion with](https://reader033.vdocuments.us/reader033/viewer/2022042921/5f684b1c7d50d30998459921/html5/thumbnails/10.jpg)
Why should the WGC be true?
• Many examples from string theory, KK theory:
Q = M
Q
M
![Page 11: Axion Inflation and the Lattice Weak Gravity …kias-yitp2017/Rudelius.pdfLandscape The Weak Gravity Conjecture The WGC: In any U (1) gauge theory that admits a UV completion with](https://reader033.vdocuments.us/reader033/viewer/2022042921/5f684b1c7d50d30998459921/html5/thumbnails/11.jpg)
The WGC for Multiple U(1)s1. Take theory with N 1-form U(1)s.
2. Consider charge-to-mass vectors ~zi =~qimi
Mp.
3. WGC: Convex Hull of {~zi} must contain unitball in RN .
Cheung, Remmen ’14
Z1
Z2
BH region
![Page 12: Axion Inflation and the Lattice Weak Gravity …kias-yitp2017/Rudelius.pdfLandscape The Weak Gravity Conjecture The WGC: In any U (1) gauge theory that admits a UV completion with](https://reader033.vdocuments.us/reader033/viewer/2022042921/5f684b1c7d50d30998459921/html5/thumbnails/12.jpg)
The WGC and Inflation
![Page 13: Axion Inflation and the Lattice Weak Gravity …kias-yitp2017/Rudelius.pdfLandscape The Weak Gravity Conjecture The WGC: In any U (1) gauge theory that admits a UV completion with](https://reader033.vdocuments.us/reader033/viewer/2022042921/5f684b1c7d50d30998459921/html5/thumbnails/13.jpg)
Inflation
Problem: Why is the universe so flat and so homogeneous?
!Solution: Inflation—postulated period of
exponential expansion in the early universe
![Page 14: Axion Inflation and the Lattice Weak Gravity …kias-yitp2017/Rudelius.pdfLandscape The Weak Gravity Conjecture The WGC: In any U (1) gauge theory that admits a UV completion with](https://reader033.vdocuments.us/reader033/viewer/2022042921/5f684b1c7d50d30998459921/html5/thumbnails/14.jpg)
Inflation in Field Theory
• Inflation can be thought of as the theory of a ball rolling down a hill with friction
V(�)
�
✏V =M2
p
2
✓V 0(�)
V (�)
◆2
, ⌘V = M2pV 00(�)
V (�).
![Page 15: Axion Inflation and the Lattice Weak Gravity …kias-yitp2017/Rudelius.pdfLandscape The Weak Gravity Conjecture The WGC: In any U (1) gauge theory that admits a UV completion with](https://reader033.vdocuments.us/reader033/viewer/2022042921/5f684b1c7d50d30998459921/html5/thumbnails/15.jpg)
Inflation in Field Theory
• Large-field inflation runs afoul of QG: !!!!!!
• Solution: impose shift symmetry
(�� & Mp)
� ! �+ c
�
V (�) Mp
![Page 16: Axion Inflation and the Lattice Weak Gravity …kias-yitp2017/Rudelius.pdfLandscape The Weak Gravity Conjecture The WGC: In any U (1) gauge theory that admits a UV completion with](https://reader033.vdocuments.us/reader033/viewer/2022042921/5f684b1c7d50d30998459921/html5/thumbnails/16.jpg)
Axion Inflation• Axions (scalars with perturbative shift symmetry)
acquire periodic potential via instanton effects: !!!!!!
•
V (�) = ⇤
4(1� cos
�
f) + ...
V(�)
�
Need f > Mp for inflation
“axion decay constant”
![Page 17: Axion Inflation and the Lattice Weak Gravity …kias-yitp2017/Rudelius.pdfLandscape The Weak Gravity Conjecture The WGC: In any U (1) gauge theory that admits a UV completion with](https://reader033.vdocuments.us/reader033/viewer/2022042921/5f684b1c7d50d30998459921/html5/thumbnails/17.jpg)
WGC and Axion Inflation
Consider 0-form � with decay constant f .
WGC ) 1/(fS) & 1/Mp.
Charged object is an instanton of action S.
S > 1 + WGC ) f . Mp
Banks, Dine, Fox, Gorbatov ’03
Agrees with string theory
Incompatible with inflation!
![Page 18: Axion Inflation and the Lattice Weak Gravity …kias-yitp2017/Rudelius.pdfLandscape The Weak Gravity Conjecture The WGC: In any U (1) gauge theory that admits a UV completion with](https://reader033.vdocuments.us/reader033/viewer/2022042921/5f684b1c7d50d30998459921/html5/thumbnails/18.jpg)
• Recruit additional axions:
Multi-Axion Inflation
Decay Constant Alignment N -flation
Kim, Nilles, Peloso ’04
Dimopoulos, Kachru,
McGreevy, Wacker ’05
![Page 19: Axion Inflation and the Lattice Weak Gravity …kias-yitp2017/Rudelius.pdfLandscape The Weak Gravity Conjecture The WGC: In any U (1) gauge theory that admits a UV completion with](https://reader033.vdocuments.us/reader033/viewer/2022042921/5f684b1c7d50d30998459921/html5/thumbnails/19.jpg)
• Simplest models violate convex hull condition
Multi-Axion Inflation
Decay Constant Alignment N -flation
![Page 20: Axion Inflation and the Lattice Weak Gravity …kias-yitp2017/Rudelius.pdfLandscape The Weak Gravity Conjecture The WGC: In any U (1) gauge theory that admits a UV completion with](https://reader033.vdocuments.us/reader033/viewer/2022042921/5f684b1c7d50d30998459921/html5/thumbnails/20.jpg)
• Similar constraints apply to simplest multi-axion models
Multi-Axion Inflation
Decay Constant Alignment N -flation
Kim, Nilles, Peloso ’04
Dimopoulos, Kachru,
McGreevy, Wacker ’05
TR ’14, ’15
Brown, Cottrell, Shiu, Soler ’15
Montero, Uranga, Valenzuela ’15
![Page 21: Axion Inflation and the Lattice Weak Gravity …kias-yitp2017/Rudelius.pdfLandscape The Weak Gravity Conjecture The WGC: In any U (1) gauge theory that admits a UV completion with](https://reader033.vdocuments.us/reader033/viewer/2022042921/5f684b1c7d50d30998459921/html5/thumbnails/21.jpg)
• Take a model with two instantons:
A Loophole
1
Decay constant
N
Instanton Action
S1 = 1
V ⇠ e�S1(1� cos
�
f1) + e�SN
(1� cos
�
fN)
Instanton charge
Satisfies WGC!
ZN = 1 & 1
f1 = N4 Mp fN = 1
4Mp
SN = 4
Z1 = 4/NZWGC = Mp/(fS)
![Page 22: Axion Inflation and the Lattice Weak Gravity …kias-yitp2017/Rudelius.pdfLandscape The Weak Gravity Conjecture The WGC: In any U (1) gauge theory that admits a UV completion with](https://reader033.vdocuments.us/reader033/viewer/2022042921/5f684b1c7d50d30998459921/html5/thumbnails/22.jpg)
A Loophole
V ⇠ e�S1(1� cos
�
f1) + e�SN
(1� cos
�
fN)
V (�)
�
⇠ NMp
) super-Planckian f1 ) successful inflation!
![Page 23: Axion Inflation and the Lattice Weak Gravity …kias-yitp2017/Rudelius.pdfLandscape The Weak Gravity Conjecture The WGC: In any U (1) gauge theory that admits a UV completion with](https://reader033.vdocuments.us/reader033/viewer/2022042921/5f684b1c7d50d30998459921/html5/thumbnails/23.jpg)
Invoking a Strong Form
• In this model: • The instanton of smallest action does not
satisfy the WGC bound • The instanton of smallest charge does not
satisfy the WGC bound • “Most” charge sites are not occupied by
instantons satisfying the WGC bound • Similar considerations apply to multi-axion
models of inflation
![Page 24: Axion Inflation and the Lattice Weak Gravity …kias-yitp2017/Rudelius.pdfLandscape The Weak Gravity Conjecture The WGC: In any U (1) gauge theory that admits a UV completion with](https://reader033.vdocuments.us/reader033/viewer/2022042921/5f684b1c7d50d30998459921/html5/thumbnails/24.jpg)
Strong Forms of the WGC
![Page 25: Axion Inflation and the Lattice Weak Gravity …kias-yitp2017/Rudelius.pdfLandscape The Weak Gravity Conjecture The WGC: In any U (1) gauge theory that admits a UV completion with](https://reader033.vdocuments.us/reader033/viewer/2022042921/5f684b1c7d50d30998459921/html5/thumbnails/25.jpg)
Methodology
UV-Complete Counterexample
Internal Inconsistency
Need Stronger Conjecture
Need Weaker Conjecture
![Page 26: Axion Inflation and the Lattice Weak Gravity …kias-yitp2017/Rudelius.pdfLandscape The Weak Gravity Conjecture The WGC: In any U (1) gauge theory that admits a UV completion with](https://reader033.vdocuments.us/reader033/viewer/2022042921/5f684b1c7d50d30998459921/html5/thumbnails/26.jpg)
Methodology
Internal Inconsistency
Need Stronger Conjecture
![Page 27: Axion Inflation and the Lattice Weak Gravity …kias-yitp2017/Rudelius.pdfLandscape The Weak Gravity Conjecture The WGC: In any U (1) gauge theory that admits a UV completion with](https://reader033.vdocuments.us/reader033/viewer/2022042921/5f684b1c7d50d30998459921/html5/thumbnails/27.jpg)
The WGC and Dim. Reduction• Consider (d+1)-dimensional U(1) theory
satisfying WGC • KK reduce on small circle • Find violation of WGC in d dimensions!
ZKK
Z
…
![Page 28: Axion Inflation and the Lattice Weak Gravity …kias-yitp2017/Rudelius.pdfLandscape The Weak Gravity Conjecture The WGC: In any U (1) gauge theory that admits a UV completion with](https://reader033.vdocuments.us/reader033/viewer/2022042921/5f684b1c7d50d30998459921/html5/thumbnails/28.jpg)
The sLWGC
• Leads us to suspect strong form:
Z
ZKK
Solves inconsistency!
Sublattice Weak Gravity Conjecture: There
exists some sublattice of the full charge lattice
with particles satisfying |~q|/m � | ~Q|/M |ext
.
![Page 29: Axion Inflation and the Lattice Weak Gravity …kias-yitp2017/Rudelius.pdfLandscape The Weak Gravity Conjecture The WGC: In any U (1) gauge theory that admits a UV completion with](https://reader033.vdocuments.us/reader033/viewer/2022042921/5f684b1c7d50d30998459921/html5/thumbnails/29.jpg)
Top-Down Evidence for the sLWGC
• SO(32)/E8 ⇥ E8 heterotic string states
• T 4,6/� orbifolds of type II/heterotic
string theory
• Higgs branch of T 4/Z3 heterotic orbifold
![Page 30: Axion Inflation and the Lattice Weak Gravity …kias-yitp2017/Rudelius.pdfLandscape The Weak Gravity Conjecture The WGC: In any U (1) gauge theory that admits a UV completion with](https://reader033.vdocuments.us/reader033/viewer/2022042921/5f684b1c7d50d30998459921/html5/thumbnails/30.jpg)
Top-Down Evidence for the sLWGC
• Perturbative string partition function:
Z(µ, µ, ⌧, ⌧) := Tr(q�q�yQyQ)
q = e2⇡i⌧ y = e2⇡iµ
�, ˜� = conformal weights
Q, ˜Q = left, right-moving charges
![Page 31: Axion Inflation and the Lattice Weak Gravity …kias-yitp2017/Rudelius.pdfLandscape The Weak Gravity Conjecture The WGC: In any U (1) gauge theory that admits a UV completion with](https://reader033.vdocuments.us/reader033/viewer/2022042921/5f684b1c7d50d30998459921/html5/thumbnails/31.jpg)
Top-Down Evidence for the sLWGC
• Modular invariance implies spectrum is invariant under shifts !!!
• Existence of graviton, then implies sublattice of charged particles satisfying the WGC bound
Q ! Q+ ⇢, Q ! Q+ ⇢
with �� 12Q
2,
˜
�� 12˜Q2
held fixed,
⇢ 2 �⇤Q, ⇢ 2 �⇤
Q
� = � = Q = Q = 0
see also Shiu et al. ’16
![Page 32: Axion Inflation and the Lattice Weak Gravity …kias-yitp2017/Rudelius.pdfLandscape The Weak Gravity Conjecture The WGC: In any U (1) gauge theory that admits a UV completion with](https://reader033.vdocuments.us/reader033/viewer/2022042921/5f684b1c7d50d30998459921/html5/thumbnails/32.jpg)
Methodology
UV-Complete Counterexample
Need Weaker Conjecture
![Page 33: Axion Inflation and the Lattice Weak Gravity …kias-yitp2017/Rudelius.pdfLandscape The Weak Gravity Conjecture The WGC: In any U (1) gauge theory that admits a UV completion with](https://reader033.vdocuments.us/reader033/viewer/2022042921/5f684b1c7d50d30998459921/html5/thumbnails/33.jpg)
WGC Strong Form Counterexamples
R
R0
m2 ⇠ R2 +R02
q2 ⇠ R02
• Type II T 6/Z2 ⇥ Z2 orbifold with non-
trivial space group
![Page 34: Axion Inflation and the Lattice Weak Gravity …kias-yitp2017/Rudelius.pdfLandscape The Weak Gravity Conjecture The WGC: In any U (1) gauge theory that admits a UV completion with](https://reader033.vdocuments.us/reader033/viewer/2022042921/5f684b1c7d50d30998459921/html5/thumbnails/34.jpg)
WGC Strong Form Counterexamples
(0, 1)
(1, 0)
(1
2,1
2)
/Z2
Strong WGCs Violated!
Lightest charged state
0 1/2 1
![Page 35: Axion Inflation and the Lattice Weak Gravity …kias-yitp2017/Rudelius.pdfLandscape The Weak Gravity Conjecture The WGC: In any U (1) gauge theory that admits a UV completion with](https://reader033.vdocuments.us/reader033/viewer/2022042921/5f684b1c7d50d30998459921/html5/thumbnails/35.jpg)
Strong Forms of the WGC and Inflation
![Page 36: Axion Inflation and the Lattice Weak Gravity …kias-yitp2017/Rudelius.pdfLandscape The Weak Gravity Conjecture The WGC: In any U (1) gauge theory that admits a UV completion with](https://reader033.vdocuments.us/reader033/viewer/2022042921/5f684b1c7d50d30998459921/html5/thumbnails/36.jpg)
(s)LWGC and Inflation• Model with P instantons, N axions, WGC
bounds assume !!
• (s)LWGC says !!
• Need to revisit WGC constraints in light of this new conjecture
P = N
P = 1
![Page 37: Axion Inflation and the Lattice Weak Gravity …kias-yitp2017/Rudelius.pdfLandscape The Weak Gravity Conjecture The WGC: In any U (1) gauge theory that admits a UV completion with](https://reader033.vdocuments.us/reader033/viewer/2022042921/5f684b1c7d50d30998459921/html5/thumbnails/37.jpg)
The Setup• Consider square lattice of instantons
saturating WGC bound:
L � �1
2
Kab@�a@�b �
X
~Q2⇤
e�|~Q|cos(
~Q · ~�+ �~Q)
⇤
| ~Q| :=q
~Q ·K�1 · ~Q
![Page 38: Axion Inflation and the Lattice Weak Gravity …kias-yitp2017/Rudelius.pdfLandscape The Weak Gravity Conjecture The WGC: In any U (1) gauge theory that admits a UV completion with](https://reader033.vdocuments.us/reader033/viewer/2022042921/5f684b1c7d50d30998459921/html5/thumbnails/38.jpg)
A Loophole• Set , N = 1,
= V1 + V0cos(�/f0 + �)� e1/f0 cos �
cosh 1/f0 � cos�/f0.
V (�) =X
n2Ze�n/f0
cos
n�
f0+ �
�~Q = � = const.
-5 5 10 15 20 �
V (�)
![Page 39: Axion Inflation and the Lattice Weak Gravity …kias-yitp2017/Rudelius.pdfLandscape The Weak Gravity Conjecture The WGC: In any U (1) gauge theory that admits a UV completion with](https://reader033.vdocuments.us/reader033/viewer/2022042921/5f684b1c7d50d30998459921/html5/thumbnails/39.jpg)
A Second Loophole• Scrunch lattice along one diagonal: !!!!!
• Effect on potential is subleading in in scrunched direction, find
N�1
fe↵ ⇠pNMp
![Page 40: Axion Inflation and the Lattice Weak Gravity …kias-yitp2017/Rudelius.pdfLandscape The Weak Gravity Conjecture The WGC: In any U (1) gauge theory that admits a UV completion with](https://reader033.vdocuments.us/reader033/viewer/2022042921/5f684b1c7d50d30998459921/html5/thumbnails/40.jpg)
A Third Loophole• Set , take two aligned instantons to
have and dominate potential !!!!!!
• Gives rise to standard KNP alignment
N = 2
Kim, Nilles, Peloso ’04
S~Q ⌧ | ~Q|
![Page 41: Axion Inflation and the Lattice Weak Gravity …kias-yitp2017/Rudelius.pdfLandscape The Weak Gravity Conjecture The WGC: In any U (1) gauge theory that admits a UV completion with](https://reader033.vdocuments.us/reader033/viewer/2022042921/5f684b1c7d50d30998459921/html5/thumbnails/41.jpg)
Generic Constraints• More generally, take random phases • Given direction in charge lattice,
decompose potential into harmonics
�~Q
~e0
~e0
![Page 42: Axion Inflation and the Lattice Weak Gravity …kias-yitp2017/Rudelius.pdfLandscape The Weak Gravity Conjecture The WGC: In any U (1) gauge theory that admits a UV completion with](https://reader033.vdocuments.us/reader033/viewer/2022042921/5f684b1c7d50d30998459921/html5/thumbnails/42.jpg)
Phasor Notation• Introduce phasor notation, !!!!
• Total contribution to nth harmonic is then
L � �1
2Kab@�
a@�b �X
~Q2⇤
Z~Qei~Q·~�
Z~Q = e�|~Q|+i�~Q
Zn =X
~Q,P
i Qi=|n|
Z~Q
![Page 43: Axion Inflation and the Lattice Weak Gravity …kias-yitp2017/Rudelius.pdfLandscape The Weak Gravity Conjecture The WGC: In any U (1) gauge theory that admits a UV completion with](https://reader033.vdocuments.us/reader033/viewer/2022042921/5f684b1c7d50d30998459921/html5/thumbnails/43.jpg)
Gaussian Approximation
• For large number of instantons, can use CLT to write: !!!!!
Zn ⇠ N (0,�2n)
�2n =
X
~Q,P
i Qi=|n|
e�2|~Q|
Higher harmonics suppressed , �2n ⌧ �2
1
![Page 44: Axion Inflation and the Lattice Weak Gravity …kias-yitp2017/Rudelius.pdfLandscape The Weak Gravity Conjecture The WGC: In any U (1) gauge theory that admits a UV completion with](https://reader033.vdocuments.us/reader033/viewer/2022042921/5f684b1c7d50d30998459921/html5/thumbnails/44.jpg)
Estimating
Want to estimate sum:
�2n
�2n =
X
~Q,P
i Qi=|n|
e�2|~Q|
![Page 45: Axion Inflation and the Lattice Weak Gravity …kias-yitp2017/Rudelius.pdfLandscape The Weak Gravity Conjecture The WGC: In any U (1) gauge theory that admits a UV completion with](https://reader033.vdocuments.us/reader033/viewer/2022042921/5f684b1c7d50d30998459921/html5/thumbnails/45.jpg)
Estimating
Method 1: use instantons of small charges and ignore the rest (valid for small decay constants)
�2n
= sum up= ignore
![Page 46: Axion Inflation and the Lattice Weak Gravity …kias-yitp2017/Rudelius.pdfLandscape The Weak Gravity Conjecture The WGC: In any U (1) gauge theory that admits a UV completion with](https://reader033.vdocuments.us/reader033/viewer/2022042921/5f684b1c7d50d30998459921/html5/thumbnails/46.jpg)
Estimating
Method 2: use Poisson resummation (valid for large decay constants)
�2n
= sum up= ignore
re-sum
⇤ ⇤⇤
![Page 47: Axion Inflation and the Lattice Weak Gravity …kias-yitp2017/Rudelius.pdfLandscape The Weak Gravity Conjecture The WGC: In any U (1) gauge theory that admits a UV completion with](https://reader033.vdocuments.us/reader033/viewer/2022042921/5f684b1c7d50d30998459921/html5/thumbnails/47.jpg)
Volume Bound• For general , define !!
with the volume of the unit cell, the fundamental domain of axion moduli space:
Kab
|⇤|vol � =
MNp (2⇡)N
|⇤|
�
Decay Constant Alignment N -flation
![Page 48: Axion Inflation and the Lattice Weak Gravity …kias-yitp2017/Rudelius.pdfLandscape The Weak Gravity Conjecture The WGC: In any U (1) gauge theory that admits a UV completion with](https://reader033.vdocuments.us/reader033/viewer/2022042921/5f684b1c7d50d30998459921/html5/thumbnails/48.jpg)
Volume Bound (cont.)
• Demanding and approximating, get !!!!!
with the N-ball of radius R.
�2n ⌧ 1
DN (R)
log[vol �] . log[vol DN (2Mp)] +O(logN)
![Page 49: Axion Inflation and the Lattice Weak Gravity …kias-yitp2017/Rudelius.pdfLandscape The Weak Gravity Conjecture The WGC: In any U (1) gauge theory that admits a UV completion with](https://reader033.vdocuments.us/reader033/viewer/2022042921/5f684b1c7d50d30998459921/html5/thumbnails/49.jpg)
Volume Bound and N-flation
• Isotropic N-flation ( ) violates the volume bound !!!!!!
• But, decay constant alignment survives
Mp
pNMp
Kab / I
![Page 50: Axion Inflation and the Lattice Weak Gravity …kias-yitp2017/Rudelius.pdfLandscape The Weak Gravity Conjecture The WGC: In any U (1) gauge theory that admits a UV completion with](https://reader033.vdocuments.us/reader033/viewer/2022042921/5f684b1c7d50d30998459921/html5/thumbnails/50.jpg)
Volume Bound and Alignment
• Set !!!!
• Set
) fe↵ . O(1Mp)
) fe↵ . O(NMp)
Kab 2 WishN (�2, N)
K�1ab 2 WishN (�2, N)
![Page 51: Axion Inflation and the Lattice Weak Gravity …kias-yitp2017/Rudelius.pdfLandscape The Weak Gravity Conjecture The WGC: In any U (1) gauge theory that admits a UV completion with](https://reader033.vdocuments.us/reader033/viewer/2022042921/5f684b1c7d50d30998459921/html5/thumbnails/51.jpg)
Summary• The (s)LWGC emerges naturally from the
WGC via dimensional reduction • The WGC is incompatible with some models of
axion inflation • If the (s)LWGC is true:
• Isotropic N-flation would be ruled out • Some decay constant alignment models
would be ruled out • But, some would be allowed
![Page 52: Axion Inflation and the Lattice Weak Gravity …kias-yitp2017/Rudelius.pdfLandscape The Weak Gravity Conjecture The WGC: In any U (1) gauge theory that admits a UV completion with](https://reader033.vdocuments.us/reader033/viewer/2022042921/5f684b1c7d50d30998459921/html5/thumbnails/52.jpg)
Open Questions• Can one prove the WGC? The sLWGC? !
• If not, can one find a counterexample? • Can one close the aforementioned loopholes? • If not, can one exploit one of these
opportunities in a stringy model of inflation? • What more can the WGC tell us about low-
energy physics?
c.f. Harlow ’15, Shiu et al. ’16, Hod ’17, Fisher, Mogni ’17