Interpolation and Exponentially Suppressed
Cosmological Constantin Non-Supersymmetric Heterotic Strings
with General Z2 Twists
古賀 勇⼀（⼤阪公⽴⼤）
YK. arXiv:2212.14572

2023/03/17 素粒⼦現象論研究会2022 @ ⼤阪公⽴⼤学

1Introduction• 加速器実験⟹ 到達可能なエネルギースケールではSUSYの証拠なし• String/Planck scale 程度の⾼エネルギー領域で既にSUSYが破れている可能性
⾮超対称な弦理論 (Non-SUSY string) に注⽬
（概略図）
⾼

エネルギー
スケール

低

超弦理論

超対称な有効理論
⾮超対称な世界

⾼

エネルギー
スケール

低

⾮超対称な弦理論

⾮超対称有効理論？
⾮超対称な世界2

Introduction• 10 次元の弦理論 (with modular invariance)：• Type IIB• Type IIA• Type I• Heterotic 𝑆𝑂(32)• Heterotic 𝐸! ×𝐸!

例外群• Type 0B• Type 0A• Heterotic 𝑆𝑂(32)• Heterotic 𝑆𝑂(16)×𝐸!
• Heterotic 𝑆𝑂 16 ×𝑆𝑂 16• Heterotic 𝐸"# ×𝑆𝑈(2)#• Heterotic 𝑆𝑂 24 ×𝑆𝑂 8• …

tachyon free# (Strings with SUSY) < # (Strings without SUSY)• Non-SUSY “Heterotic” string theoryに焦点を当てる（後述）
3

Introduction
ØNon-SUSY stringの問題点：宇宙定数が極めて⼤きくなる
Our approach：Interpolating model（例）
10D SUSY
string∞

0𝑅
9D Interpolating
model

SUSYが漸近的に回復する 𝑅 ≈ ∞ の領域での宇宙定数：

10D Non-SUSY
stringItoyama, Taylor (1986): positive const.
, : # of massless B, F

• 𝑛! = 𝑛" ⇒ 指数関数的に抑圧された宇宙定数をもつ• しかしモジュライ不安定性の問題（最⼩値negative等）があった• 構成法：1⽅向に対してのみ 𝒁# Scherk-Schwarz twists

任意の次元数に対して 𝒁. twistsして得られる模型を調べた
4

Outline
1. Introduction
2. Non-SUSY Hetero with general Z2 twists
3. Endpoint limits & Interpolations
4. Cosmological Constant
5. Summary5Outline
1. Introduction
2. Non-SUSY Hetero with general Z2 twists
3. Endpoint limits & Interpolations
4. Cosmological Constant
5. Summary

6

Heterotic Superstring

Gross, Hervey, Martinec, Rohm (1985)ØHybrid(“heterotic”) theory including only closed strings:• Left: bosonic string (26D)• Right: superstring (10D)ØCompactified on 𝑇 2 (with maximal SUSY)
16

10

Compact
space

𝑋!$

"

𝑋!

L
bosonic string
(26D)

'

Targetspacetime

"

"
𝑋# 𝜓#

R
superstring(10D)

Compactified
on 𝑇 $

16

𝑋!$

𝑑

𝑋!%

𝑋#% 𝜓#%

10 − 𝑑

𝑋!

"

"
𝑋# 𝜓#

L
bosonic string
(26D)

"

R
superstring
(10D)

',)

(,)
• 𝑋$,& , 𝑋$,&
/ 𝜓& : bosonic / fermionic coordinates(𝜇 = 0, … 9 − 𝑑 , 𝑖 = 10 − 𝑑, … , 9 , 𝐼 = 1, … , 16)
7

Heterotic SuperstringØThe internal momenta 𝑃 = (ℓ4 , 𝑝4 ; 𝑝5 ) ∈ Γ6782,2• Narain lattice: even self-dual lattice w/ Lorentz. sign. (16 + 𝑑, 𝑑)• Labeled by an integer vector 𝑍 = 𝑞( , 𝑚) , 𝑛) ∈ 𝒁*+ ×𝒁, ×𝒁,
winding numbers

KK momenta• Turn on full moduli: 𝑑 𝑑 + 16 = 𝑑 # + 16𝑑 ⇒ 𝐺)- + 𝐵)- + 𝐴()• Consider a rectangular 𝑑-torus: 𝐺)- = 𝑅)# 𝛿)<latexit sha1_base64="Av6CpM4PMEDkMGq6Ieq2YddjlnE=">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</latexit>`IL = ⇡ Ii

m AIi ,

Narain, Sarmadi, Witten, (1986)

✓
✓
◆◆
1
1
pLi = p⇡ · Ai + ni + mj Gij + Bij
Ai · Aj
2
2Ri
✓
✓
◆◆
1
1
pRi = p⇡ · Ai + ni mj Gij Bij + Ai · Aj
2
2Ri

16

ℓ$!

𝑑

𝑝!%

𝑝#%

𝜋 % ≡ 𝑞 % 𝛼&' ∈ Γ&' ⇐ 𝑆𝑝𝑖𝑛(32)/𝒁# or 𝐸! ×𝐸! lattice8

Construction of Non-SUSY Hetero
Ø 𝑍# freely acting orbifold (stringy Scherk-Schwarz comp.)• Project out SUSY hetero on 𝑇 , by𝑍# generator : (−)+ 𝛼

Dixon, Harvey (1986)
Ginsparg, Vafa (1987)(+ twisted sec. added)

𝐹: spacetime fermion #𝛼: shift of order 2 such as• 𝛿 is called a shift vector : 2𝛿 ∈ Γ-./,,,• labeled by a vector 𝑍! = 𝑞$ ! , 𝑚' " , 𝑛$ " whose components are 0 or 1

Non-SUSY strings depend on 𝑍F• Splitting the Narain lattice Γ#$%&,& into Γ%#$%&,& and Γ(#$%&,& :16+d,d=+<latexit sha1_base64="qNW8hZSa2CpDK+NMzZzrUZ7R2F0=">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</latexit><latexit sha1_base64="WfCOt/a+5O6/WX3cqrbyI9wRs8c=">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</latexit>

16+d,dP2

16+d,d

·P 2Z

= P2

16+d,d· P 2 Z + 1/2

Boson/Fermion lives in Γ(&'($,$ / Γ*&'($,$ respectively

SUSY breaking
9

1-loop Partition Function• Heterotic strings on 𝑇 2 (with maximal SUSY)""

𝑋! , 𝑋#"

𝜓# , 𝜓#%𝛼!

𝑋!$ , 𝑋!% , 𝑋#%𝜏 = 𝛼! /𝛼"
8<latexit sha1_base64="gJSQ6vZlDpKr/GeUieUm0GfGnVI=">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</latexit>

(8 d)

ZB

= ⌧2

d
2

(8 d)

(⌘ ⌘¯)<latexit sha1_base64="L7XWMfYrh4vF+mpZVN+UgLLYdjs=">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</latexit>

orbifoldingby (−)* 𝛼𝛼"Z 16+d,d = ⌘(16+d)

⌘¯

d

X

q

1 2
2 pL

q̄

16

𝑋!%

𝑑

𝑋!$

10 − 𝑑

𝑋!"

1 2
2 pR

p2 16+d,d𝑞 = 𝑒 +,"- , 𝑉., 𝑆., 𝑂., 𝐶.: 𝑆𝑂 8 characters

𝑋#$ 𝜓#$

"
"
𝑋# 𝜓#

L

R

• Non-SUSY Heterotic strings
n
⇠
(8 d)
SU⇠
SY
⇠
Z(Ẑ)
= ZB
V̄8 Z 16+d,d<latexit sha1_base64="V/l/6Ik3KblExq/xIZSdCMq4/lU=">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</latexit>+

vectorS̄8 Z 16+d,d + Ō8 Z 16+d,d +spinorscalar±C̄8 Z 16+d,d +⌥co-spinoro10Outline
1. Introduction
2. Non-SUSY Hetero with general Z2 twists
3. Endpoint limits & Interpolations
4. Cosmological Constant
5. Summary11

Endpoint limits &InterpolationsØConsider 𝑑 = 1,2 cases with 𝐴 = 𝐵 = 0• 1-loop partition function:
n
⇠
(8 d)
SU⇠
SY
⇠
Z(Ẑ)
= ZB
V̄8 Z 16+d,d<latexit sha1_base64="V/l/6Ik3KblExq/xIZSdCMq4/lU=">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</latexit>+

S̄8 Z 16+d,d + Ō8 Z 16+d,d +±C̄8 Z 16+d,d +⌥o• The behavior of 𝑍K2345,5 (8L) in the limits of 𝑅M → 0, ∞±• Take 𝑅) → ∞ ⇒ only 𝑚) = 0 contributes• Take 𝑅) → 0 ⇒ only 𝑛) = 0 contributes(𝑖 = 1,2)

Example in 𝑑 = 1 with (𝑚P & , 𝑛R& ) = (1,0):
R1
R1 !1
1
! p (⌘ ⌘¯) Z 16 , Z 17,1 +
! 0,
SUSY
±
±
⌧2
1
1
R !0
R1 !0
1
1
17,1
Z 17,1 1 !
,Z!
(⌘ ⌘¯) Z 16⇡
ˆ .
p (⌘ ⌘¯) Z 16
p
+
±
±+2
±
±
R 1 ⌧2R 1 ⌧2<latexit sha1_base64="bUvHloC0AbznCsEIjvQW40yQqJQ=">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</latexit>

Z 17,1

R1 !1

Non-SUSY
12Endpoint limits &Interpolations• 9D Non-SUSY heterotic models (𝑑 = 1)•class (1): |𝜋|
3 = 0 (mod 4), (𝑚;7 𝑛)
3 = (0; 0)𝑅0non-SUSY heterotic strings on a circle•(𝑍6 = (:𝑞, 𝑚,
< 𝑛)
: ∈ 𝒁&' ×𝒁×𝒁)
𝜋: = 𝑞𝛼: &' ∈ Γ&'!

10D non-SUSY

Itoyama, Koga, Nkajima (2021)∞

10D non-SUSY

class (2): |𝜋|
3 ! = 0 (mod 4), (𝑚,7 𝑛)
3 = (1; 0)
10D non-SUSY

𝑅
0∞

10D SUSY

Itoyama, Taylor ’86• class (3): |𝜋|
3 ! = 0 (mod 4), (𝑚,7 𝑛)
3 = (0; 1)𝑅

10D non-SUSY
0∞interpolation between SUSY and non-SUSY vacua (Interpolating model)
10D SUSY•

class (4): |𝜋|
3 ! = 2 (mod 4), (𝑚,7 𝑛)
3 = (1; 1)
10D SUSY0SUSY restored at both of the endpoints

𝑅∞

10D SUSY

13

Endpoint limits & Interpolations• 8D Non-SUSY heterotic models (𝑑 = 2)•

class (1) & (2) :|𝜋|
3 ! = 0 (mod 4), (𝑚;7 𝑛)
3 = (1,0; 0,0)class(1)
10D non-SUSY•

class(2)
9D non-SUSY

class (1) & (4) :|𝜋|
3 ! = 2 (mod 4), (𝑚;7 𝑛)
3 = (1,0; 1,0)class(4)10D SUSY

10D SUSY

9D non-SUSY∞ 𝑅(
9D non-SUSY𝑅&

9D SUSY

9D SUSY∞

0𝑅&

9D SUSY

00
10D non-SUSY

10D SUSY

∞ 𝑅(
∞

0

Koga (2022)

9D non-SUSY

10D SUSY

10D SUSY
class(2)

9D non-SUSY

class(4)10D SUSYNo class(1)<latexit sha1_base64="O2b90bdCu0X4mI1rrAesBu+nJgo=">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</latexit>

10D (Non-)SUSY
condition

Limits of R1 , R2
(R1 , R2 ) ! (1, 1)(R1 , R2 ) ! (1, 0)(R1 , R2 ) ! (0, 1)(R1 , R2 ) ! (0, 0)

10D SUSY modelm̂1 + m̂2 > 0m̂1 + n̂2 > 0n̂1 + m̂2 > 0n̂1 + n̂2 > 0

10D Non-SUSY modelm̂1 + m̂2 = 0m̂1 + n̂2 = 0n̂1 + m̂2 = 0n̂1 + n̂2 = 0

14

Outline
1. Introduction
2. Non-SUSY Hetero with general Z2 twists
3. Endpoint limits & Interpolations
4. Cosmological Constant
5. Summary

15

Cosmological Constant• 1-loop cosmological constant (vacuum energy density) :<latexit sha1_base64="nZY/dCREpKOvXrstYEqNrEK4bfU=">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</latexit>⇤(10 d)=p1(2⇡ ↵0 ) (10 d)2Z⇠
d2 ⌧ ⇠
SU⇠
SY
Z
2
(Ẑ)
F ⌧2

Fundamental Region :<latexit sha1_base64="taqNyC3yv30O+odJAjZZyUzUKnk=">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</latexit>F=⇢⌧ = ⌧1 + i⌧2 2 C11
 ⌧1  , |⌧ |2
2

1

• decompose ℱ into ℱ,& = 𝜏 ∈ ℱ 𝜏# ≥ 1 and ℱ-& = 𝜏 ∈ ℱ 𝜏# < 1
𝜏(

1

-1

1

ℱ,& = 𝜏 ∈ ℱ 𝜏# ≥ 1

The integral can be evaluatedℱ-& = 𝜏 ∈ ℱ 𝜏# < 1exp. suppressed𝜏&16(10−𝑑)D non-SUSY Heterotic models• Consider compact coordinates Y i (i = 1, . . . , d) below :<latexit sha1_base64="yVv2VQFd3pZGwV0AUScAKPFs9vs=">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</latexit>

i = a(2) + a(4) + b(3) + b(1)𝐷<latexit sha1_base64="YT9Cd9ddzHdjC2TtPghfYEU1lQI=">AAAChnichVHLSsNAFD3GV62PVt0IbtSiVIQwqfWBIBTduFRrVdBSkjjq0DQJSVrQ0rXgD7hwpSAi4lY/wI0/4MJPEJcV3LjwJg2IinpD5tw5c8+dMzOabQjXY+ypSWpuaW1rj3REO7u6e2Lx3r511yo7Os/plmE5m5rqckOYPOcJz+CbtsPVkmbwDa246K9vVLjjCstc8w5sni+pe6bYFbrqEVWID4t5tVBNpsZrEz6mCTXCyRCV8VohnmAyC2LoZ6KESQJhLFvxS2xjBxZ0lFEChwmPcgMqXPq2oIDBJi6PKnEOZSJY56ghStoyVXGqUIkt0rhHs62QNWnu93QDtU67GPQ7pBzCKHtkV6zOHtg1e2bvv/aqBj18LweEWkPL7ULseCD79q+qROhh/1P1p2cPu5gNvArybgeMfwq9oa8cntSzc6uj1TF2zl7I/xl7Yvd0ArPyql+s8NVTROkBlO/X/TNZT8nKtJxeSScyC+FTRDCIESTpvmeQwRKWkaN9j3CDW9xJEUmWpqSZRqnUFGr68SWkzAfuSpQS</latexit>

class (#) in 9D

𝑌

(

𝑑−𝐷

𝐷!

𝐷 − 𝐷!

𝐷'𝑑 − 𝐷' − 𝐷class (2)class (4)class (3)class (1)• Assignment of (m̂, n̂) :<latexit sha1_base64="HqU6JO9TUSDSSpwflUuN3lZuPlw=">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</latexit>(m̂a(2) , n̂a(2) ) = (1, 0)

for a(2) = 1, . . . , D2(m̂a(4) , n̂a(4) ) = (1, 1)

for a(4) = D2 + 1, . . . , D(m̂b(3) , n̂b(3) ) = (0, 1)

for b(3) = D + 1, . . . , D + D3(m̂b(1) , n̂b(1) ) = (0, 0)

for b(1) = D + D3 + 1, . . . , d<latexit sha1_base64="sFCEbuouT6LsdXjA1QiQakT6YX0=">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</latexit>

SUSY at 𝑅% → ∞Non-SUSY at 𝑅& → ∞17Formula for CC• Consider 𝐷 ≥ 1 : SUSY is restored at all 𝑅M ≈ ∞• Up to exponentially suppressed terms,!

<latexit sha1_base64="8NH5CCaSewf6ce6+hH3Tq+V/fgI=">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</latexit>

d 1

d
Y

(D
X X

) 5

d
X

4! · 2pRi(2na 1)2 Ra2 +(2nb )2 Rb2⇡ 15 d ( ↵0 )10 d i=1
n
a=1
b=D+1
0
"
(D
)#1
d
X
X
X
⇥ 8 @24 +exp 2⇡i(2na 1)(⇡ · Aa ) +nb (⇡ · Ab ) A⇤(10 d) ⇠

⇡2

a=1

g

b=D+1

Δ. : nonzero roots of 𝑆𝑂(32) or 𝐸! ×𝐸!
⇒ CC does not depend on all the other endpoint models

masslesscondition<latexit sha1_base64="K+1rra5WzLvtTWEob76Hc0Uj5So=">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</latexit>

2⇡ · Aa 2 Z⇡ · Ab 2 Z<latexit sha1_base64="K5bZVSeMRtZUcN81piNhcmAWom4=">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</latexit>⇤(10 d) ⇠d 1

4! · 2⇡ 15 d

d
Y

i=1

Ri

!

(
X X
!
n

a

(2na

1)2 Ra2 +X
b(2nb )2 Rb2

) 5

(nF

nB )

<latexit sha1_base64="ppW7IVBMnYNn98ZU02q8Jqf+Mfk=">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</latexit>

18

Solutions of 𝑛! = 𝑛"ØSUSY 𝑆𝑂 32 endpoint models:<latexit sha1_base64="nYj06/GqlumOKY/5bBOcr5d3xkg=">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</latexit>

SO(32) =±, ±, 014ØSimplest configurations:• 𝐴(E (𝑎 = 1, … , 𝐷) are the same configuration• 𝐴(F (𝑏 = 𝐷 + 1, … , 𝑑) is taken to be 0<latexit sha1_base64="6E61CXHP/rG+ecEaWU3Hmexq8VE=">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</latexit>

Aa =

✓

✓ ◆q ◆
10p ,(p + q = 16) , Ab = 016
2• 𝐷 ∈ 2𝒁:
𝑛! − 𝑛" = −504 ≠ 0• 𝐷 ∈ 2𝒁 + 1: 𝑛! − 𝑛" = 4𝑝𝑞 − 2𝑝 𝑝 − 1 + 2𝑞 𝑞 − 1− 24

𝑛Z = 𝑛[ ⇒ 𝑝, 𝑞 = 9,7 , (7,9)
CC is exp. supp. when gauge group is 𝑆𝑂 18 ×𝑆𝑂(14)19Wilson-line Moduli Stability (1)• SUSY 𝑆𝑂 32 endpoint models:<latexit sha1_base64="XuLfo7VH08x3saUGpXd8XBwaAOs=">AAADJXichVFLaxNRFD4zvmp9NOpGcDMYW1Il4UwIVcRFsRtbXPRhmkpuGmYmN8ml83LuTaAOsxb8AwquFETEn+DSjRt3KnTlWlxWcCPiyc1Q0dJ6hplz7ne+79z5OG7sC6kQdwzzyNFjx09MnJw8dfrM2anCufPrMhokHq97kR8lG64juS9CXldC+XwjTrgTuD5vuFsLo35jyBMpovCe2o55K3B6oegKz1EEtQtP2V0id5zNtGRjuTObWTNMisAqMzkI2ilzgzTMsoW9ivm8q0pWtXatNmbYhDywFm8t6WzPZcyLpGY1qywWFlN9rmj8YsYS0eur1gH9pb2+TrPtQhErqMPaX9h5UYQ8lqPCW2DQgQg8GEAAHEJQVPvggKSnCTYgxIS1ICUsoUroPocMJkk7IBYnhkPoFn17dGrmaEjn0Uyp1R7d4tObkNKCafyEr3EX3+Mb/Io/D5yV6hmjf9mm7I61PG5PPb649uO/qoCygv4f1SEKl9iad/8qf9RsHOpOQRduaFeCXMYaGfn1xhOGD5/srt1cnU5n8AV+I6fPcQffkddw+N17ucJXn9Gi7H/Xsr9Yr1bsuUptpVacv52vbAIuwWUo0V6uwzzcgWWo062/jCtG2aiYr8wP5kfz85hqGrnmAvwV5pffp4vM2w==</latexit>⇤(10 d)

X⇠<latexit sha1_base64="LkFvN1QfUZBiqsz++oguSRJpoD4=">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</latexit>

I

✓ =

D
X

0

X

n

Cn @24 + 4

1I<J16(2na

1)AIa +

nb AIb

a=1

d
X

⇥
⇤
⇥
⇤
cos 2⇡✓I cos 2⇡✓J A

sum of WLs

b=D+1<latexit sha1_base64="0mOPdmq1B90+fBC3bTreDZePENE=">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</latexit>

4! · 2d+2p
Cn =⇡ 15 d ( ↵0 )10 d

1

d
Y

i=1

Ri

!( D
X

a=1

(2na

1)2 Ra2 +dX(2nb )2 Rb2

b=D+1

) 5

• Simplest configurations are critical points:<latexit sha1_base64="6E61CXHP/rG+ecEaWU3Hmexq8VE=">AAADKXichVHPSxtBGP3c/rLW1theCr2EBkVbCd+KRBEErZce/dEYJdGwu/2SDG52191JwC6bq9B/oAdPCj0U/4DScy+eCj0ItVfxIB4VerHFL5OVtopxhp158+a9NzvzmZ4tAom436Hdun3n7r3O+10Puh8+6kn0Pl4I3JpvUdZybddfNI2AbOFQVgpp06Lnk1E1bcqZq9PN/Vyd/EC4zhu57tFy1Sg7oiQsQzJVTFSmiqERTRRsKsmBJK6EXjSUbK0KJd+wQj0Kh6OCL8oVObgSrl3ARkvjvVyb0DMxN9RocJr5b5qeuTAUEylMo2rJq0CPQQriNuMm9qAAb8EFC2pQBQIHJGMbDAi450EHBI+5ZQiZ8xkJtU8QQRd7a6wiVhjMrvJY5lU+Zh1eNzMD5bb4FJs/n51J6MM9/IQnuIs7eIRn12aFKqP5L+s8my0vecWe90/nf93oqvIsofLX1cZhslrpll7QRj7XRmkxcngkvk/EN/nCN/iKn/Gw7YtIKMGYegnBbk8xpHLUqfV3H07mx+f6wn7cxmN+nS3c59RddOqn1sdZmtvk4uqXS3kVLAyn9Ux6ZHYkNfkqLnMnPIPnMMC1HIVJeA0zkOVTv8Ep/IY/2o72Xfuh/WxJtY7Y8wT+a9rBOavFydw=</latexit>

Aa =

✓

✓ ◆q ◆
10p ,(p + q = 16) , Ab = 016
2<latexit sha1_base64="MkUOPd4dIjJADprsW4VpFrV0vD0=">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</latexit>@⇤(10 d)⇠0@AIi

(I = 1, . . . , 16, i = 1, . . . , d)20

Wilson-line Moduli Stability (2)ØHessian matrix:• Simplest configurations<latexit sha1_base64="6E61CXHP/rG+ecEaWU3Hmexq8VE=">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</latexit>

Aa =

✓

✓ ◆q ◆
10p ,(p + q = 16) , Ab = 016
2• 𝐷 ∈ 2𝒁:<latexit sha1_base64="ZUGAvLZ5Bhiu3piUxFxjVHCYi/M=">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</latexit>@ 2 ⇤(10 d)⇠ ⇠ IJ ij
IJ@Ai @Aj(I, J = 1, . . . , 16, i, j = 1, . . . , d) ⇠ > 0<latexit sha1_base64="tJ2cIRqca5UCGlzaanBKVLJkOWA=">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</latexit>⇒ Hessian is positive definiteØ A global minimum when the gauge group is 𝑆𝑂(32)and no massless fermions exist (Λ < 0)21

Wilson-line Moduli Stability (3)ØHessian matrix:• Simplest configurations<latexit sha1_base64="6E61CXHP/rG+ecEaWU3Hmexq8VE=">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</latexit>

Aa =

✓

✓ ◆q ◆
10p ,(p + q = 16) , Ab = 016
2• 𝐷 ∈ 2𝒁 + 1:
<latexit sha1_base64="UzH0UrIe7YnqvsK/+JLpOaut+wA=">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</latexit>2(10 d)@ ⇤⇠@AIi @AJj((2p 17) ⇠ 0 IJ ij( 2p + 15) ⇠ 0 IJ ij(I = 1, . . . , p),(I = p + 1, . . . , 16)⇠0 > 0<latexit sha1_base64="ap7X/v24V5PiKrbYn3aNKdxLmOI=">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</latexit>⇒ Hessian is positive/negative definite for 𝑝 = 0,16/ 𝑝 = 8
Ø A global minimum when the gauge group is 𝑆𝑂(32)while a local maximum when the gauge group is 𝑆𝑂 16 × 𝑆𝑂(16)Ø 𝑝 = 7,9 (𝑛! = 𝑛" ) ⇒ saddle points
22Outline
1. Introduction
2. Non-SUSY Hetero with general Z2 twists
3. Endpoint limits & Interpolations
4. Cosmological Constant
5. Summary

23

Summary• 10 − 𝑑 D Non-SUSY models are constructed by orbifolding by (−)! 𝛼( 𝛼 : shift of order 2 in Narain lattice)• Various interpolation are shown in 𝑑 = 2 case• Cosmological constant of (10 − 𝑑)D Non-SUSY models in 𝑅) ≈ ∞ is<latexit sha1_base64="9fSyzceI6wKcHx+MZqpd+hJt9k4=">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</latexit>⇤(10 d) ⇠d 1

4! · 2⇡ 15 d

d
Y

i=1

Ri

!

(
X X
!
n

a

(2na

1)2 Ra2 +X
b(2nb )2 Rb2

) 5

(nF

p

0

nB ) + O(e R/ ↵ )• Find the configurations of WLs which gives exp. supp. CC• Analyze WL-moduli stability: 𝑛! = 𝑛" ↔ saddle points
Out look
Higher-loop correction, (meta)stable vacua, cosmology
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