YITP-iTHEMS molecule-type international workshop

Potential Toolkit to Attack
Nonperturbative Aspects of QFT-Resurgence and related topicsSep 7-25, 2020

Opening remarks &Brief introduction to Resurgence
Tatsuhiro MISUMI
Masaru Hongo, Shigeki Sugimoto, Yuya Tanizaki, Hidetoshi Taya@YITP, Kyoto U. & Online 09/07/20

Goals of this workshop
1. Review the novel techniques for nonperturbative effects
of QFT, focusing on resurgence and the related techniques.
2. Study and summarize the very recent results in thetechniques.
3. Raise and consider questions in the techniques and their
physical results.
4. Propose their applications to physical problems otherthan pure QFT.
5. Discuss the questions and applications, and produce newworks by collaborating with the participants.

Goals of this workshop
1. Review the novel techniques for nonperturbative effects
of QFT, focusing on resurgence and the related techniques.
2. Study and summarize the very recent results in thetechniques.
Three Lectures
3. Raise and consider questions in the techniques and their
physical results.
4. Propose their applications to physical problems otherthan pure QFT.
5. Discuss the questions and applications, and produce newworks by collaborating with the participants.

Goals of this workshop
1. Review the novel techniques for nonperturbative effects
of QFT, focusing on resurgence and the related techniques.
2. Study and summarize the very recent results in thetechniques.
3. Raise and consider questions in the techniques and their
physical results. Six Talks &Poster Session

4. Propose their applications to physical problems otherthan pure QFT.
5. Discuss the questions and applications, and produce newworks by collaborating with the participants.

Goals of this workshop
1. Review the novel techniques for nonperturbative effects
of QFT, focusing on resurgence and the related techniques.
2. Study and summarize the very recent results in thetechniques.
3. Raise and consider questions in the techniques and their
physical results.
4. Propose their applications to physical problems otherthan pure QFT.
5. Discuss the questions and applications, and produce newworks by collaborating with the participants.

Free Discussion TimeIt may be a new manner ofholding academic workshops
in Covid-19 and Post-Covid-19 eras.

How to join the workshop
1. Lectures, Invited talks, and Short talks will be held in
Zoom, whose URL has been emailed to all participants.
2. Poster sessions will be held in Mozilla hubs. Its URL andhow-to-use are shown in the email.
3. Discussion sessions will be also held in Mozilla hubs.
All of the important information are shown in
SLACK, whose URL has been sent to you.
We strongly recommend you to join SLACK ASAP

Since some of participants may not be familiar
to resurgence theory in quantum physics,
we are giving its very brief review andrefer to problems to be considered !
See also Prof. Dunne’s lecture on 8th

1. Perturbative series and Borel resummation

Perturbative v.s. Non-perturbative analyses in QT
H = H0 + g 2 H

g21

Perturbation : quantum fluctuation
is calculated as series of g^2 basedon eigenstates of H0
E=

1
X

n=0

cq g 2q

H = H0 + g 2 H

g2Nonperturbative analysis : it isrequired to diagonalize whole
hamiltonian.
cf.)

E⇡e

A
g2

1

Path integral and Saddle points
Z=Z

D exp( S[ ]) =Trivial (perturbative) saddle
1X
Z0 =aq g 2qq=0

Perturbative seriesXZ

2saddles : stationary points

Nontrivial saddles S = 0

Z / e Ssol ⇠ e

Ag2

Non-perturbative contribution

Relation between Pert. and Non-pert.

X

aq g

2q

q=0

Perturbative series

❌

exp



A
g2

Non-perturbative contribution"They are not connected ?
We just have independent contributions ?"No, it is not correct !

!

dxvergence encodes physical information about the saddles of ordinary integr
Perturbation and Borel resummation
ls of quantum mechanics and quantum field theory, as a consequence of Darb
3]. We recall a few relevant definitions and motivate (known) generalizati!
"
tions by using simple quantum
2 mechanics.
H0 + asymptoticg Hpertseries
ψ(x)
=
Eψ(x)
2 ) denote a perturbativethat satisfies the “Gevrey-1” cond2

P (g ) =⇥q=0

q
Perturbative
seriesisoften
aq g ,
Gevrey 1 : |aq | ⇥ CR
aq /q!q!
divergent factorially2q' series tha
sitive constants C and#R [5, 7].$Known
examplesof
perturbative%&
2 series
・Construct an analytic function from asymptotic
m dx condition [29]. We denote themechanics and QFT satisfy the “Gevrey-1”
S
=
dt
−
V
(x)
f P ( ) by BPBorel(t) anddefine it asresummation：Analytic
function which has original2

dt⇥
perturbative
series as asymptotic seriesaq q
BP (t) :=t .
q!unique for one asymptotic series.
Note that the analytic function is not
q=0

$

'

dx
!
call a few relevant definitions and motivate (known) generalizations

sing simple quantum mechanics.
vergence
encodesasymptotic
physical informationabout
the thesaddlesof ordinary
integraperturbative
seriesthat
satisfies“Gevrey-1”
conditio
Perturbation
and Borel
resummation
nsform. The Borel
transform method
is applicableto the following cls of quantum mechanics and quantum field theory, as a consequence of Darb
called Gevrey-1)⇥
3]. We recall a few relevant definitions and motivate (known) generalizatiqµ∂"Ptions(g 2 )by= usinga!qsimple
g 2q , X
1
:
|a
|
⇥
CR
q!
(6.
1 Gevreyqqquantum
22 q mechanics.
1
2H
+
g
H
ψ(x)
=
Eψ(x)
P
(g
)
=
a
(g
)
,
|a
|
∑
Cq!
,
2 ) denoteq=0
q
q
0
pert
a perturbative asymptotic series that satisfies
the “Gevrey-1” condA
q=0
tants C and R [5,⇥7]. Known examples of perturbative series that ari
2 Borel transform
2q Perturbativeq
seriesisoftenonstants.
The
BP
(t)
is
definedas
P (gsatisfy) = theaq g“Gevrey-1”,
Gevrey1 : |aq[29].
| ⇥ CRaWeq!
s and QFT
condition
the Borq /q!denotedivergent factoriallyq=0
1
BP (t) and define it as
X
aq q
'
BP
(t)7].
=$Knownt examples, & of perturbative
sitive constants C and#R [5,series
tha
%
⇥
q!
2
aq=0
q q
m
dx
mechanics and QFT
satisfythe“Gevrey-1”
condition[29].
We
denotetheBP (t) :=
t .
(6.
Borel
transform
S
=
dt
−
V
(x)
q!
fummation
P ( ) by BP(t)
anddefineitas2
q=0
B(g ) is defined as 2dt⇥
Z 1
nsform determines 2“a germ of
a°t/g
holomorphic
function” at t = 0, wi
dt
2aq qt (t).
.
B(g ) =BP (t) 2:=eBP
Borelresummationq!
vergence. Next, one analyticallycontinues
theobtained
germ BP (g q=0
0
x t-plane, called the Borel plane.
We
also
assumethattheanalyt2$ ) reproduces the original sum' P (g2) cothat the Borel resummation B(g!

dxel transform. The Borel transform method is applicable to the follow
eries(calledGevrey-1)
vergence
encodes
physical information about the saddles of ordinary integr
Perturbation and Borel resummationµ∂ls of quantum mechanics andquantumfield
theory,asaconsequenceof Darb
1
q
X
1
2
2 q
3]. We recall a few
definitions
generalizati
P (grelevant)=aq (g ) , and|aq | motivate∑ Cq! (known)
,
A
!
"
q=0
tions by using simple quantum
2 mechanics.
H0 + asymptoticg Hpertseries
ψ(x)
=
Eψ(x)
2 ) denote a perturbativethat satisfies the “Gevrey-1” condare constants. The Borel transform BP (t) is defined as
2

P (g ) =

⇥

q=0

1
q
Perturbative
seriesisoftenXaq g ,
Gevrey a1q: q |aq | ⇥ CR
aq /q!q!
divergent
BP (t) =factoriallyt ,
2q

q=0

q!

' series tha
sitive constants C and#R [5, 7].$Known
examplesof
perturbative
%
&
2
2
In
specialcases,
Borel
resum
may
give
exact resultsrelresummation
B(g)
isdefinedasmdx
mechanics and QFT satisfy the “Gevrey-1” condition [29]. We denote the
S
=
dt
−
V
(x)
Z
1
f P ( ) by BP (t) and define it asdt dt
2°t/g 2
2
B(g ) =eBP (t).
20 ⇥g aq q
BP (t) :=t .
q!2oscillatorcf.) x^4 unharmonic
sily see that the Borel resummation
B(g) reproduces the original sum P (q=0
ne can exchange the integral and the sum. Otherwise, we need to defin
$

'

!

dx

he divergence encodes physical information about the saddles of ordinary integral
vergence
encodes mechanics
physical informationabout
the saddlesof ordinary
integrtegrals of quantum
andquantumfield
theory,asaconsequenceof
Darbo
Perturbation and Borel resummation
mls [1,
3]. We recall
a few relevant
definitions
andmotivate
generalizationof quantum
mechanics
and quantum
field
theory,as a(known)consequenceof Darb
efinitions
by using
simplequantummechanics.
3]. We recall
a few
relevant
definitions
and motivate (known) generalizati
ttionsP (g 2by) denote
perturbative
asymptotic
seriesthat
satisfies
the“Gevrey-1”
condi
usingasimple
quantummechanics.
2!

"

H +g Hψ(x) = Eψ(x)2 ) denote a perturbative
0 ⇥ asymptotic
pertseries that satisfies the “Gevrey-1” cond

P (g 2 ) =⇥aq g 2q ,q=0

Gevrey1 : |aq | ⇥ CRq q!

(qPerturbative
seriesisoften
P (g ) =aq g ,
Gevrey 1 : |aq | ⇥ CR
aq /q!q!
divergentfactoriallyme positive constants C and R [5,7]. Known
examples of perturbative series that
2

2q

q=0

ntum mechanics and QFT satisfy the “Gevrey-1” condition [29]. We denote the B
sitiveconstantsC(t)andRdefine[5, 7].itKnown
examplesof
perturbative
seriestha
rm of P( )Borel
by BP
and
as
transform can have singularities 2
on positive real axis

#

$

%

&

'

m ⇥ dx condition [29]. We denote themechanics and QFT satisfy the “Gevrey-1”
S
=
dt
−
V
(x)
a
q
q
f P ( ) by BP (t) and defineBP
it as(t) :=t .
(2 q=0 dtq!
⇥

aq q
Z 1e±i✏BP (t)“a:=dtgermgt2 of ta .holomorphic
rmal Borel transform
determines
function”
2 ⌥i✏Singularities
on positiveat
realt = 0,B(g e ) =
e
BP
(t)
q!
2 q=0
g
axis leads
to ambiguity
radius of convergence. Next,
theobtained germ B
0 one analytically continues
whole complex t-plane, called the Borel plane. We also assume that the ana

$

'

!

dx

vergence encodes physical information about the saddles of ordinary integr
Perturbation and Borel resummation
ls of quantum mechanics and quantum field theory, as a consequence of Darb
3]. We recall a few relevant definitions and motivate (known) generalizati!
"
tions by using simple quantum
2 mechanics.
H0 + asymptoticg Hpertseries
ψ(x)
=
Eψ(x)
2 ) denote a perturbativethat satisfies the “Gevrey-1” cond2

P (g ) =⇥q
Perturbative
seriesisoften
aq g ,
Gevrey 1 : |aq | ⇥ CR
aq /q!q!
divergent factorially2qq=0' series tha
sitive constants C and#R [5, 7].$Known
examplesof
perturbative
%
&
2 2
2 ⌥i✏
2
B(gQFTe satisfy) = Re[B(giIm[B(g)] [29]. We denote them )] ±dx
mechanics andthe “Gevrey-1”
condition
S
=
dt
−
V
(x)
f P ( ) by BP (t) and define it asA
2

Im[B(g )] ⇡ e

g2

2

dt

This should be cancelled by that from⇥ non-perturbative contribution!

aq qBP(t) :=
t in. perturbative calculation
Non-perturbative
effectreappearsq!
q=0
through imaginary
ambiguity !

$'Possible questions to be asked• A resummation method is not unique. Is there a better
resummation formula ?
• Resummation method to get nonperturbative results
directly ?
Costin, Dunne (17)(20)• Resummation method for the divergent series beyond
Gevrey-1?
• Higher-order perturbative series in QFT ?
cf.) Stochastic perturbation theory

2. Resurgent structure and Trans-series in
ODE and Integral1
ࣜ (ODE) Ͱ‫ࣜܗ‬త΂͖‫਺ڃ‬ղ͔ΒҰൠͷղΛߏ੒͢Δମ"ϕ (z) + ϕ(z) =‫( ܥ‬ODE ͰͷϦαʔδΣϯεཧ࿦) Λઆ໌͠Α͏3) ɽ؆୯ͳ z

͜͜Ͱൺ

ϦΞϯඍ

!∞
Ecalle
(81)
ྫͱͯ͠ɼ
z
=
∞
ʹෆ֬ఆಛҟ఺Λ࣋ͭৗඍ෼ํఔࣜ
Ϧοδํ
Resurgent
structure in ODE Φ0 =
Λߟ͑Δɽ‫ࣜܗ‬త΂͖‫਺ڃ‬ղ
n! z −n−
n=0

−z
1
∆e
=
1
ม‫׵‬͸
BΦ
[t]
=
Ͱ༩͑ΒΕɼਖ਼ͷ্࣮࣠ʹ"0
ϕ (z) + ϕ(z) = 1−t(37)z8ͭ 1ɽҰൠղ͸ɼύϥϝʔλʔσ ∈ R Λಋೖͨ͠
SΦ0
!
X
∞
−n−1
Λߟ͑Δɽ‫ࣜܗ‬త΂͖‫਺ڃ‬ղͷϘϨϧ−zz
Formal solutions
01 == ϕ(z;n!zσ)nΦ=0 ਺
Φ0&+n=0σen!
Ͱ༩͑ΒΕɼͦͷϘϨϧ±i!
around ม‫׵‬͸
z=∞ BΦ0 [t] = 1 q=0"͕Θ͔ΔͰ༩͑ΒΕɼਖ਼ͷ্࣮࣠ʹಛҟ఺Λ࣋∞e1−tperturbative
nonperturbativeΛආ͚ΔඍখҼࢠ% Λ‫ؚ‬Ήԋࢉ S± Φ0 ≡ 08ͭ ɽҰൠղ͸ɼύϥϝʔλʔ σ ∈ R Λಋೖͨ͠τϥϯε‫ڃ‬Λ࢖͏ͱ਺ ϕ(z; σ) = Φ0 + σe−z Ͱ༩͑ΒΕɼͦͷϘϨϧ࿨͸ಛҟ఺
" ∞e±i! −ztΛආ͚ΔඍখҼࢠ % Λ‫ؚ‬Ήԋࢉ S± Φ0 ≡ 0e BΦ0 (t)ࣗ‫ݾ‬ಉ‫ܕ‬

දΘͯ͠

਺͔Βඇ

8 ͜͜Ͱɼz ͸ྔࢠྗֶͷ 1/g ʹʹରԠ͢Δͱࢥ͑͹ྑ͍·Γɼઁ9 ඇઢ‫ܗ‬8 ͜͜Ͱɼz ͸ྔࢠྗֶͷ 1/g ʹʹରԠ͢Δͱࢥ͑͹ྑ͍ɽ77͸ແ‫ݸݶ‬ͷ͜͜Ͱ͸‫ڥ‬ք৚݅ΛఆΊͯ σ ΛܾΊͨͱ͠Α͏ɽͱ͜Ζ͕ɼ
1 ͜͜Ͱൺ
ࣜ (ODE) Ͱ‫ࣜܗ‬త΂͖‫਺ڃ‬ղ͔ΒҰൠͷղΛߏ੒͢Δମ"ϕ (z) + ϕ(z) =શʹܾఆͰม਺ z = |z|eiθ ͕ετʔΫεઢ θ = 0 Λԣ੾Δࡍʹͦͷ઴ۙ‫( ܥ‬ODE ͰͷϦαʔδΣϯεཧ࿦) Λઆ໌͠Α͏3) ɽ؆୯ͳ z ϦΞϯඍͨ‫׬‬શͳτ‫ʹܹٸ͕ܗ‬มԽ͢ΔετʔΫε‫ݱ‬৅͕‫͖ى‬ΔͨΊɼղΛ༩͑!∞−n−Ecalle
(81)
ྫͱͯ͠ɼ
z
=
∞
ʹෆ֬ఆಛҟ఺Λ࣋ͭৗඍ෼ํఔࣜ
Ϧοδํ
Resurgent
structurein
ODEΛߟ͑Δɽ‫ࣜܗ‬త΂͖‫਺ڃ‬ղ
Φ
=
n!
z
0
n=0
ີղ͔Βಘ
Δ σ ͷ஋͕ετʔΫεఆ਺ s = 2πi ͷ෼͚ͩෆ࿈ଓʹมԽ͢
−z
1
∆e
=
1
ม‫׵‬͸
BΦ
[t]
=
Ͱ༩͑ΒΕɼਖ਼ͷ্࣮࣠ʹ"0
ϕ (z) + ϕ(z)
(37)
ʹɼϦαʔ
ΔɽϘϨϧ࿨ S± ʹ‫·ؚ‬ΕΔ±%=͸z1−tθ ͷඍখมԽͱಉ౳ͳͷ8ͭɽҰൠղ͸ɼύϥϝʔλʔσ ∈ R Λಋೖͨ͠1SΦ0΋͜ͷ໛‫ܕ‬!
θ = 0 ্Ͱ͸
→
S
ϕ(z;
σ
s)
ͰɼετʔΫεઢ X
S+∞ϕ(z; σ)+−−n−1Λߟ͑Δɽ‫ࣜܗ‬త΂͖‫਺ڃ‬ղͷϘϨϧ−zz
Formal solutions
01 == ϕ(z;n!zσ)nΦ=0 ਺
Φ0&+n=0σen!
Ͱ༩͑ΒΕɼͦͷϘϨϧͷΑ͏ͳ௓ͼ͕ੜ͡Δ͜ͱʹͳΔɽͦͷ݁Ռɼղͷ࿈ଓੑ±i!
around ม‫׵‬͸
z=∞ BΦ0 [t] = 1 q=0"͕Θ͔ΔͰ༩͑ΒΕɼਖ਼ͷ্࣮࣠ʹಛҟ఺Λ࣋∞e1−tperturbative
nonperturbativeN −1Λආ͚ΔඍখҼࢠ% Λ‫ؚ‬Ήԋࢉ S± Φ0 t=1
≡ 0
শ CP
8
S
ϕ(z;
σ)
=
S
ϕ(z;
σ
+
s)
͔Βɼ
−
ͭ+ɽҰൠղ͸ɼύϥϝʔλʔσ ∈ R Λಋೖͨ͠τϥϯε‫ڃ‬
Λ࢖͏ͱ
ϛΦϯ਺อ
਺ ϕ(z; σ) = Φ0 + σe−z Ͱ༩͑ΒΕɼͦͷϘϨϧ࿨͸ಛҟ఺ࣗ‫ݾ‬ಉ‫ܕ‬−z±i!
" 2πie
S
Φ
(z)
−
S
Φ
(z)
=
(39)
∞e
+
0
−
0
−zt
2 ͷ৔߹)
Λආ͚ΔඍখҼࢠ % Λ‫ؚ‬Ήԋࢉ S± Φ0 ≡ 0e BΦ0 (t)දΘͯ͠ΦϯղΛ‫ٻ‬਺͔Βඇresurgent
structure!……..
butwhy?
ͱ͍͏ؔ܎͕ಘΒΕΔɽ͜ͷΑ͏ʹৗඍ෼ํఔࣜʹ͓͍ͯ
8 ͜͜Ͱɼz ͸ྔࢠྗֶͷ 1/g ʹʹରԠ͢Δͱࢥ͑͹ྑ͍
͜ͱͰશͯ
·Γɼઁ
−z
͸ɼ“ઁಈత”ͳ΂͖‫ ਺ڃ‬Φ0 ͷϘϨϧ࿨͕ e Λ‫ؚ‬Ή “ඇ
9 ඇઢ‫ܗ‬Εʹରͯ͠ઁಈత”‫د‬༩ͱؔ܎෇͍͍ͯΔɽΤΧʔϧ
(Ecalle) ͸ඇઢ‫ܗ‬͸ແ‫ݸݶ‬ͷ
8 ͜͜Ͱɼz ͸ྔࢠྗֶͷ
1/gʹʹରԠ͢Δͱࢥ͑͹ྑ͍ɽ7ཁ੥Λ՝͢ͳ৔߹Λ‫ؚ‬ΉҰൠͷ ODE ʹ͓͍ͯ΋ʮద੾ͳ਺ͷτϥϯ·ΘΓͷઁ
7ε‫਺ڃ‬ύϥϝʔλʔΛಋೖ͢Δ͜ͱͰɼৗඍ෼ํఔࣜͷղߏ੒͢Δ͕͜͜ͷΑ͏ͳ‫ࣜܗ‬తղͷτϥϯε‫͚ॻͯ͠ͱ਺ڃ‬Δʯͱ͍ͷΑ͏ͳτ͏ॏཁͳࣄ࣮Λࣔͨ͠3) ɽྔࢠ࿦Ͱͷ‫ݱ‬৅ΛΑΓਂ͘ཧղ͢ΔͨΊʹɼৗඍ෼ํఔ1
ࣜ (ODE) Ͱ‫ࣜܗ‬త΂͖‫਺ڃ‬ղ͔ΒҰൠͷղΛߏ੒͢Δମ
"

͜͜Ͱൺ
ϕ
(z)
+
ϕ(z)
=
ࣜ (ODE)Ͱ‫ࣜܗ‬త΂͖‫਺ڃ‬ղ͔ΒҰൠͷղΛߏ੒͢Δମ‫( ܥ‬ODEͰͷϦαʔδΣϯεཧ࿦) Λઆ໌͠Α͏3) ɽ؆୯ͳ z ͜͜ͰൺϦΞϯඍ
3)!∞ ϦΞϯඍ‫( ܥ‬ODEྫͱͯ͠ɼ
ͰͷϦαʔδΣϯεཧ࿦)Λઆ໌͠Α͏
ɽ؆୯ͳ
Ecalle(81)z=∞ʹෆ֬ఆಛҟ఺Λ࣋ͭৗඍ෼ํఔࣜ
Resurgent
structure in ODE Φ0 = n=0 Ϧοδํ
Λߟ͑Δɽ‫ࣜܗ‬త΂͖‫਺ڃ‬ղ
n! z −n−
For review for physicists, −zྫͱͯ͠ɼz = ∞ ʹෆ֬ఆಛҟ఺Λ࣋ͭৗඍ෼ํఔࣜ
Ϧοδํ
1
∆e =
Marino(07),
1
9
ม‫׵‬͸
BΦ
[t]
=
Ͱ༩͑ΒΕɼਖ਼ͷ্࣮࣠ʹ"0Λ༻͍ͯ࣍ͷΑ͏ʹදΘͤΔ
ɽ Aniceto,ϕ (z) + ϕ(z) = 1−t(37)
Schiappa,Vonk(13)͍ͨΊɼO(g) ͷ෦෼Λྵͱ͠
−z
z
∆e
=
Cherman,
Dorigoni,
Unsal(14)
8
1
9
"
ͭ+
ɽҰൠղ͸ɼύϥϝʔλʔσ(37)∈ R Λಋೖͨ͠
දΘͤΔ ɽ
Dorigoni(14),1͔ࣜΒಘΒΕΔ∂! E|!=1 ͷ‫ ݫ‬ϕ (z)
ϕ(z)
=
SΦ0
!
X
−z
∞
−n−1
Aniceto,Basar,
Schiappa(18)
σ)
=
S
Φ
(z)
+
σe
.
n
1S
z
± ϕ(z;±
0Λߟ͑Δɽ‫ࣜܗ‬త΂͖‫਺ڃ‬ղ
Φ
=
n!
z
ͷϘϨϧ
−z
Formal(2)solutions
0
=
n!z
n=0
0 ਺ ϕ(z;σ)=Φ+σe Ͱ༩͑ΒΕɼͦͷϘϨϧ&9࠶‫͢ݱ‬ΔɽΛ༻͍ͯ࣍ͷΑ͏ʹදΘͤΔEͷ෦
0np ͷ O(g)
−z
ɽ
SΦ
ྵͱ͠
!∞
1 q=0
±i!0
aroundz=∞"͕Θ͔Δσ) = S± Φ
σe .BΦ0 [t] = (38)0 (z) +ม‫׵‬͸Ͱ༩͑ΒΕɼਖ਼ͷ্࣮࣠ʹಛҟ఺Λ࣋
∞e
−n−1
1−t
Λߟ͑Δɽ‫ࣜܗ‬త΂͖‫਺ڃ‬ղ
Φ0 = n=0nonperturbativen!% zΛ‫ؚ‬ΉԋࢉͷϘϨϧperturbative͑ͯɼ‫ج‬ఈঢ়ଶΤωϧΪʔ͕Λආ͚ΔඍখҼࢠS± Φ0 ≡ 0 ͱ͜͜͜Ͱ͸‫ڥ‬ք৚݅ΛఆΊͯσΛܾΊͨͱ͠Α͏ɽͷ‫ݫ‬8−zͭ ɽҰൠղ͸ɼύϥϝʔλʔσ∈R.Λಋೖͨ͠τϥϯε‫ڃ‬
Λ࢖͏ͱ
1
S
ϕ(z;
σ)
=
S
Φ
(z)
+
σe
(38)
±
±
0
iθ
BΦ0 [t] = 1−tͰ༩͑ΒΕɼਖ਼ͷ্࣮࣠ʹಛҟ఺Λ࣋ͷੑ࣭Λิ͏ͱ‫׬‬શʹܾఆͰΛఆΊͯσ ΛܾΊͨͱ͠Α͏ɽ
ͱ͜Ζ͕ɼ
͕Θ͔Δ
ม਺
) ͷ෦ ม‫׵‬͸−zz = |z|e ͕ετʔΫεઢ9θ = 0 Λԣ੾Δࡍʹͦͷ਺ ϕ(z; σ) = Φ0 + σeͰ༩͑ΒΕɼͦͷϘϨϧ࿨͸ಛҟ఺Λ༻͍ͯ࣍ͷΑ͏ʹදΘͤΔɽࣗ‫ݾ‬ಉ‫ܕ‬͍ͨΊɼO(g)ͷ෦෼Λྵͱ͠
8±i!
εΛ༻͍ͯಋग़ͨ͠‫׬‬શͳτ
"ͱ͜Ζ͕ɼ
ʔΫεઢ=͜͜Ͱ͸‫ڥ‬ք৚݅ΛఆΊͯ
0 Λԣ੾Δࡍʹͦͷ઴ۙ
1.θSolution
is expressed as ‫ʹܹٸ͕ܗ‬มԽ͢ΔετʔΫε‫ݱ‬৅͕‫͖ى‬ΔͨΊɼղΛ
theirσ linear=∞eTrans-seriesͭɽҰൠղ͸ɼύϥϝʔλʔσ ∈combination
R Λಋೖͨ͠τϥϯε‫ڃ‬Ϊʔ͕Λ࢖͏ͱΛܾΊͨͱ͠Α͏ɽ−ztΛආ͚ΔඍখҼࢠ% Λ‫ؚ‬Ήԋࢉ S± Φ0 ≡ 0e BΦ0 (t)දΘ͔ͯࣜ͠ΒಘΒΕΔ
∂
E|
ͷ‫ݫ‬
!
!=1
−z
−z
iθ + σeετʔΫε‫ݱ‬৅͕‫͖ى‬ΔͨΊɼղΛ༩͑ϯΨʔํఔࣜͷ‫ີݫ‬ղ͔ΒಘܾఆͰ ਺ ม਺Δσͷ஋͕ετʔΫεఆ਺s=
2πiͷ෼͚ͩෆ࿈ଓʹม
ϕ(z;
σ)
=
Φ
Ͱ༩͑ΒΕɼͦͷϘϨϧ࿨͸ಛҟ఺z=|z|e͕ετʔΫεઢθ=0Λԣ੾Δࡍʹͦͷ઴ۙ
S
ϕ(z;
σ)
=
S
Φ
(z)
+
σe
.
ࣗ‫ݾ‬ಉ‫ܕ‬
0
±
± 0
(2)
Borel-resum
਺͔Βඇ
±i!
"
࠶‫͢ݱ‬Δɽ
E
ͷ
O(g)
ͷ෦
np
∞e
εఆ਺
s
=
2πi
ͷ෼͚ͩෆ࿈ଓʹมԽ͢શͳτ Λආ͚ΔඍখҼࢠ−zt‫͢ݱ‬Δɽ͜ͷΑ͏ʹɼϦαʔ
8 ͜͜Ͱɼ
‫ʹܹٸ͕ܗ‬มԽ͢ΔετʔΫε‫ݱ‬৅͕‫͖ى‬ΔͨΊɼղΛ༩͑
ΔɽϘϨϧ࿨
±%ʹʹରԠ͢Δͱࢥ͑͹ྑ͍͸ 0θ(t)ͷඍখมԽͱಉ౳±% Λ‫ؚ‬Ήԋࢉ
S± Φz0S͸ྔࢠྗֶͷ≡ʹ‫·ؚ‬ΕΔ
e
BΦ
1/g
දΘͯ͠
0
·Γɼઁ
͑ͯɼ‫ج‬ఈঢ়ଶΤωϧΪʔ͕2.
Atarg[z]=0,anappropriate
value
of
σ
jumps
=
Stokesphenomenon͜͜Ͱ͸‫ڥ‬ք৚݅ΛఆΊͯσΛܾΊͨͱ͠Α͏ɽͱ͜σ͔Βಘ·ΕΔ ±%Δ͸σθͷ஋͕ετʔΫεఆ਺ͷඍখมԽͱಉ౳ͳͷͳ͘ɼͦͷ༗༻ੑ΋͜ͷ໛‫ܕ‬s = 2πi ͷ෼͚ͩෆ࿈ଓʹมԽ͢θ = 0 ্Ͱ͸ S+ ϕ(z; σ) →਺͔Βඇ
S− ϕ(z;ͰɼετʔΫεઢ
9 ඇઢ‫ܗ‬iθͷੑ࣭Λิ͏ͱ‫׬‬શʹܾఆͰ
ม਺
z
=
|z|e
͕ετʔΫεઢθ=0Λԣ੾ΔࡍʹͦͷϦαʔ
s = 2⇡i：Stokes constant→SS±−ʹ‫·ؚ‬ΕΔ
ϕ(z; σ ͷΑ͏ͳ௓ͼ͕ੜ͡Δ͜ͱʹͳΔɽͦͷ݁Ռɼղͷ࿈=0 ্Ͱ͸ ΔɽϘϨϧ࿨S+ ϕ(z; σ)+ s)±% ͸ θ ͷඍখมԽͱಉ౳ͳͷ
8 ͜͜Ͱɼz ͸ྔࢠྗֶͷ
1/g ʹʹରԠ͢Δͱࢥ͑͹ྑ͍ɽ͸ແ‫ݸݶ‬ͷ7·ΓɼઁεΛ༻͍ͯಋग़ͨ͠‫׬‬શͳτ
arg[z]=0+
arg[z]=0N −1ͷ໛‫ܕ‬‫ʹܹٸ͕ܗ‬มԽ͢ΔετʔΫε‫ݱ‬৅͕‫͖ى‬ΔͨΊɼղΛ
Δ͜ͱʹͳΔɽͦͷ݁Ռɼղͷ࿈ଓੑ→ S σϕ(z;ͰɼετʔΫεઢ+ s)‫ ݩ‬N = (2,0) ௒ରশ CP θ = 0 ্Ͱ͸+ ϕ(z;
S+ ϕ(z;Sσ)= Sσ)+ s)σ͔Βɼ− ϕ(z; −9 ඇઢ‫ܗ‬3.
Continuity
ofsolutionleads
toresurgent
relation
betweentwo
sectors13,Δσͷ஋͕ετʔΫεఆ਺s=
2πiͷ෼͚ͩෆ࿈ଓʹมσϯΨʔํఔࣜͷ‫ີݫ‬ղ͔Βಘ+14,s) 15)
͔ΒɼͷΑ͏ͳ௓ͼ͕ੜ͡Δ͜ͱʹͳΔɽͦͷ݁Ռɼղͷ࿈ଓੑɼϑΣϧϛΦϯ਺อ
8 ͜͜Ͱɼ
7 z ͸ྔࢠྗֶͷ 1/g ʹʹରԠ͢Δͱࢥ͑͹ྑ͍ɽN−1−z͸ແ‫ݸݶ‬ͷ‫͢ݱ‬Δɽ͜ͷΑ͏ʹɼϦαʔ
P
ΔɽϘϨϧ࿨
S+ʹ‫·ؚ‬ΕΔ±%͸=θ ͷඍখมԽͱಉ౳
S
2πie
S+ (N
ϕ(z;=
σ)2=ͷ৔߹
S− ϕ(z;σ + s)͔Βɼ±Φ
0 (z) − S− Φ
0 (z)ྗֶ‫ܥ‬ͷҰྫ
)
−z
z)
−
S
Φ
(z)
=
2πie
(39)
− 0
ϯ਺อ
ͳ͘ɼͦͷ༗༻ੑ΋͜ͷ໛‫ܕ‬
θ = 0 ্Ͱ͸ S+ ϕ(z; σ) → S− ϕ(z; σͰɼετʔΫεઢ໛‫Ͱܕ‬શͯͷόΠΦϯղΛ‫ٻ‬−z
S+ Φ0 (z) −ͱ͍͏ؔ܎͕ಘΒΕΔɽ͜ͷΑ͏ʹৗඍ෼ํఔࣜʹ͓S− Φ0 (z) = 2πie
(39)
৔߹) 7
ͷΑ͏ͳ௓ͼ͕ੜ͡Δ͜ͱʹͳΔɽͦͷ݁Ռɼղͷ࿈+− ͱ‫ݺ‬͹ͷ৔߹͸ɼ൑໌͍ͯ͠Δࣔͨ͠ɽ3) Δମ‫ܥ‬͸ΤΠϦΞϯղੳ (Alien calculus)͏ॏཁͳࣄ࣮Λࣔͨ͠ ɽ# ͱΔମ‫ܥ‬͸ΤΠϦΞϯղੳ(Aliencalculus)−ಉ‫ܕ‬SΛఆٛ͢Δɽ
ϯεͷ݁Ռͱͷ‫׬‬࣍‫ݩ‬৔ͷྔࢠ࿦ʹؔ܎͢Δ໛‫Ͱܕ‬଄͕‫׬‬શʹ໌֬Խ͞Εͨͷ͸ॳΊͯͰ͋ಈ‫ͱ਺ڃ‬ඇઁಈ‫਺ڃ‬ͷؔ܎Λ໌֬Խ͠ɼਅͷղΛಋग़͢S+ = S− ◦ S,
S = exp Φe0ઁಈ‫਺ڃ‬ͷؔ܎Λ໌֬Խ͠ɼਅͷղΛಋग़͢Λߟ͑Δɽ‫ࣜܗ‬త΂͖‫਺ڃ‬ղʔδΣϯεͷ݁Ռͱͷ‫׬‬্ͷྫΛΤΠϦΞϯղੳͷ‫ݴ‬༿Ͱ‫ݟ‬௚ͯ͠ΈΑઁಈ‫ͱ਺ڃ‬ඇઁಈ‫਺ڃ‬ͷؔ܎Λ໌֬Խ͠ɼਅͷղΛಋग़͢͜͜Ͱɼ∆ ͸ਖ਼্࣮࣠
t=1্ͷྫΛΤΠϦΞϯղੳͷ‫ݴ‬༿Ͱ‫ݟ‬௚͠
3, 16, 17)શʹ໌֬Խ͞Εͨͷ͸ॳΊͯͰ͋13, ͱ‫ݺ‬͹ΕΔ
16, 17)εཧ࿦ͷ৔ͷྔࢠ࿦΁ͷԠ༻Մೳੑ͕ߴ
‫ํݟ‬Λ͢Ε͹ɼ্#$‫ܥ‬͸ΤΠϦΞϯղੳ(Aliencalculus)ɽ
3, 16,BΦ
17) 0 [t] =
ม‫׵‬͸
Ͱ༩͑ΒΕ
Ξϯղੳ
(Alien calculus)ͱ‫ݺ‬͹ΕΔɽ−zɼผͷ‫ํݟ‬Λ͢Ε͹ɼ্1−tΔମ‫ܥ‬͸ΤΠϦΞϯղੳ(Aliencalculus)ͱ‫ݺ‬͹ΕΔ
ɽ
Resurgent
structure
ODEτʔΫεઢΛ·͙ͨ2͸ਖ਼্࣮࣠
ͭͷϘϨϧ࿨Λͭͳ͙‫܈‬=ͷ৔ͷྔࢠ࿦΁ͷԠ༻Մೳੑ͕ߴ
S− ◦ S,S = expe ∆ τʔΫεઢΛ·͙ͨ(40) ∆in͜͜Ͱɼt=1ͷಛҟ఺ʹͭ
ϯඍ෼ͱ‫ݺ‬͹Εɼަ‫܎ؔ׵‬[
2ͭͷϘϨϧ࿨Λͭͳ
8ྫΛΤΠϦΞϯղੳͷ‫ݴ‬༿Ͱ‫ݟ‬௚ͯ͠ΈΑ͏ɽ·ͣεઁಈత෺ཧྔΛܾͭ ɽҰൠղ͸ɼύϥϝʔλʔσΞϯղੳͷ‫ݴ‬༿Ͱ‫ݟ‬௚ͯ͠ΈΑ͏ɽ·ͣε−z্ͷྫΛΤΠϦΞϯղੳͷ‫ݴ‬༿Ͱ‫ݟ‬௚ͯ͠ΈΑ͏ɽ·ͣεͳ͍ඇઁಈత෺ཧྔΛܾϯඍ෼ͱ‫ݺ‬͹Εɼަ‫܎ؔ׵‬
[e
∆,
∂
]
=
0
ετʔΫεࣗ‫ݾ‬ಉ‫ܕ‬SΛఆٛ͢Δɽ
σϯε‫਺ڃ‬
ϕ(z;
σ)
΁ͷ࡞༻͕
−z
ετʔΫεࣗ‫ݾ‬ಉ‫ܕ‬SΛఆٛ͢Δɽ
ΫεઢΛ·͙ͨ
2ͭͷϘϨϧ࿨Λͭͳ͙‫ͯ͠ͱ༻࡞܈‬ɼ
਺
ϕ(z;
σ)
=
Φ
+
σe
Ͱ༩͑Β
◆Resurgence
theoryand Alien calculus͙ͨ
2 ͭͷϘϨϧ࿨Λͭͳ͙‫ͯ͠ͱ༻࡞܈‬ɼ0ྗ͞Λ͍ࣔͯ͠Δɽ্࣮࣠t=1ͷಛҟ఺ʹ͍ͭͯͷΤΠϦΞ
τʔΫεઢΛ·͙ͨ2ͭͷϘϨϧ࿨Λͭͳ͙‫ͯ͠ͱ༻࡞܈‬ɼϯε‫ ਺ڃ‬ϕ(z; σ) ΁ͷ࡞༻͕ҎԼΛ༩͑Δ
࿦ͷ‫͞ྗڧ‬Λ͍ࣔͯ͠ΔɽͰͷϦαʔδΣϯεʹֶͿ#$ʔΫεࣗ‫ݾ‬ಉ‫ܕ‬SΛఆٛ͢Δɽ−zΛආ͚ΔඍখҼࢠ%Λ‫ؚ‬Ήԋࢉಉ‫ܕ‬S Λఆٛ͢Δɽ−z e−z ∆#ɼަ‫܎ؔ׵‬[e ∆, ∂σS] =0 Λຬͨ͠ɼτϥϦαʔδΣϯεʹֶͿετʔΫεࣗ‫ݾ‬ಉ‫ܕ‬Λఆٛ͢Δɽʹؔ܎͢Δ໛‫Ͱܕ‬
S
=
S
◦
S,
S
=
exp
e
∆ϕ(z;
σ)
+
− =S
ྔࢠ࿦ʹؔ܎͢Δ໛‫Ͱܕ‬・Group action connecting ± Borel resums：Stokes
automorphism
S
◦
S,
S
=
exp
e
−z
+
−
e
∆ϕ(z;
σ)
=
s
∂
ϕ(z;
σ)
ΛΑΓਂ͘ཧղ͢ΔͨΊʹɼৗඍ෼ํఔ
σ
#
$
#
$
΁ͷ࡞༻͕ҎԼΛ༩͑Δɽ#∆$−zͨͷ͸ॳΊͯͰ͋
ਂ͘ཧղ͢ΔͨΊʹɼৗඍ෼ํఔ
−z
−z
S
=
S
◦
S,
S
=
exp
e
(40)
Խ͞Εͨͷ͸ॳΊͯͰ͋
+ S S
−
=
S
◦
S,
=
exp
e
∆
(40)
−
=
S
◦
S,
S
=
exp
e
∆
(40)
+
−
త΂͖‫਺ڃ‬ղ͔ΒҰൠͷղΛߏ੒͢Δମ
͜͜Ͱൺྫ܎਺ͱͯ͠ετʔ͜͜Ͱɼ∆͜͜Ͱൺྫ܎਺ͱͯ͠ετʔΫεఆ਺͸ਖ਼্࣮࣠
t = 1 ͷಛҟ఺ʹ͍ͭͯ‫਺ڃ‬ղ͔ΒҰൠͷղΛߏ੒͢Δମ
ͷԠ༻Մೳੑ͕ߴ
͜͜Ͱɼ
∆
͸ਖ਼্࣮࣠
t
=
1
ͷಛҟ఺ʹͭs−zࢠ࿦΁ͷԠ༻Մೳੑ͕ߴ
3)eʔδΣϯεཧ࿦∆ϕ(z; σ) = s)∂Λઆ໌͠Α͏σ)(41) ϦΞϯඍ෼ͱҰൠͷඍ෼ԋࢉ
σ ϕ(z;
−z
ɽ؆୯ͳ
3)
Ͱɼ
∆
͸ਖ਼্࣮࣠
t
=
1
ͷಛҟ఺ʹ͍ͭͯͷΤΠϦΞ
・Operator
associated
with
each
singularity：Alien
derivativeϯඍ෼ͱ‫ݺ‬͹Εɼަ‫܎ؔ׵‬
[e
∆,
∂
]
=
0
Λຬ
−z
ϯεཧ࿦
)∆Λઆ໌͠Α͏্࣮࣠t=1͸ਖ਼্࣮࣠
ͷಛҟ఺ʹ͍ͭͯͷΤΠϦΞσ͜͜Ͱɼt ɽ؆୯ͳ
= 1 ͷಛҟ఺ʹ͍ͭͯͷΤΠϦΞ
ϦΞϯඍ෼ͱҰൠͷඍ෼ԋࢉΛؔ܎͚ͮΔ
ϯඍ෼ͱ‫ݺ‬͹Εɼަ‫܎ؔ׵‬[e ∆, ∂ ] =8σ ʹʹ͜͜Ͱɼz ͸ྔࢠྗֶͷ 1/g−z −zʹෆ֬ఆಛҟ఺Λ࣋ͭৗඍ෼ํఔࣜz−zϦοδํఔࣜͱ‫ݺ‬͹Εɼ྆ล෼ͱ‫ݺ‬͹Εɼަ‫܎ؔ׵‬
[e
∆,
∂
]
=ϕ(z;
0
Λຬͨ͠ɼτϥ
֬ఆಛҟ఺Λ࣋ͭৗඍ෼ํఔࣜ
σɼަ‫܎ؔ׵‬
[e
∆,
∂
]
=
0
Λຬͨ͠ɼτϥ
[e
,
@
]
=
0
Ϧοδํఔࣜͱ‫ݺ‬͹Εɼ྆ลΛൺֱ͢Δ͜ϯε‫਺ڃ‬σ)΁ͷ࡞༻͕ҎԼΛ༩͑Δɽ
ϯඍ෼ͱ‫ݺ‬͹Εɼަ‫܎ؔ׵‬
[e
∆,
∂
]
=
0
Λຬͨ͠ɼτϥσzσ
ͱͯ͠ετʔΫεఆ਺ s ͕‫ݱ‬ΕΔɽΤΠϯε‫਺ڃ‬
ϕ(z; σ) ΁ͷ࡞༻͕ҎԼΛ༩͑Δ

−z
−z
‫਺ڃ‬
ϕ(z;
σ)
΁ͷ࡞༻͕ҎԼΛ༩͑Δɽ
∆e=0͕ಘΒΕΔɽ͜ͷ∆e=0͕ಘΒΕΔɽ͜ͷ݁Ռͱ(40) ӈ
΁ͷ࡞༻͕ҎԼΛ༩͑Δɽ
1εʹֶͿϯε‫਺ڃ‬ϕ(z;σ)΁ͷ࡞༻͕ҎԼΛ༩͑Δɽ
1"ൠͷඍ෼ԋࢉΛؔ܎͚ͮΔ͜ͷํఔࣜ͸ϒΣϯεʹֶͿ
・Equation
bridgingalien
andstandard
calculus：Bridge
equation(z) +(37) −z7) +ϕϕ(z)= ϕ(z) =(37)−z σ) = s ∂σ ϕ(z; σ)e ∆ϕ(z;
z
z
e
∆ϕ(z;
σ)
=
s
∂
ϕ(z;
σ
σ
−z
‫ݺ‬͹Εɼ྆ลΛൺֱ͢Δ͜ͱͰ
∆Φ
=
s
ͱ
−z
−z
−z
−z
0
Ίʹɼৗඍ෼ํఔ
SΦ0 = Φ0 (41)+se, Φ0 +
Sϕ(z;
σ)
= ϕ(S
e = ∆ϕ(z;
σ)
=
s
∂
ϕ(z;
σ)
SΦ
=
se
,
∆ϕ(z;
σ)
=
s
∂
ϕ(z;
σ)
(41)
e ∆ϕ(z;!
σ)∞se!
∂σ−n−1
ϕ(z;
σ)
(41)
σ
͢ΔͨΊʹɼৗඍ෼ํఔ
0
σ
∞
−n−1
਺ղ
Φ
=
n!
z
ͷϘϨϧ
0
΂͖‫਺ڃ‬ղ
Φ
=
n!
z
ͷϘϨϧ
n=0
ΒΕΔɽ͜ͷ݁Ռͱ(40)ӈࣜΛ༻͍Δͱ0n=0ͷղΛߏ੒͢Δମ
͜͜Ͱൺྫ܎਺ͱͯ͠ετʔΫεఆ਺s͕‫ݱ‬・Bridgeeq.
reveals
relation
betweeneach
sector！
ΒҰൠͷղΛߏ੒͢Δମ
͜͜Ͱൺྫ܎਺ͱͯ͠ετʔΫεఆ਺s1༩͑ΒΕɼਖ਼ͷ্࣮࣠ʹಛҟ఺Λ͕࣋Θ͔Δɽ࠷ऴతʹɼӈࣜ྆ลʹ
S
Λ࡞−Ͱൺྫ܎਺ͱͯ͠ετʔΫεఆ਺s͕‫ݱ‬ΕΔɽΤΠͰ༩͑ΒΕɼਖ਼ͷ্࣮࣠ʹಛҟ఺Λ࣋͜͜Ͱൺྫ܎਺ͱͯ͠ετʔΫεఆ਺s ͕‫ݱ‬ΕΔɽΤΠͱͯ͠ετʔΫεఆ਺s ͕‫ݱ‬ΕΔɽΤΠ͕Θ͔Δɽ࠷ऴతʹɼӈࣜ྆
3)
−t
3)
໌͠Α͏
ɽ؆୯ͳ
−zσ ∈ RϦΞϯඍ෼ͱҰൠͷඍ෼ԋࢉΛؔ܎͚ͮΔ͜ͷΛઆ໌͠Α͏ɽ؆୯ͳʔλʔΛಋೖͨ͠τϥϯε‫ڃ‬
se
,
Sϕ(z;
σ)
=
ϕ(z;
σ +ϦΞϯඍ෼ͱҰൠͷඍ෼ԋࢉΛؔ܎͚ͮΔs) Λ࢖͏ͱ
(42) S+ ϕ(z; σ) = S− ϕ(z; σ + s) ͕ಘΒϯඍ෼ͱҰൠͷඍ෼ԋࢉΛؔ܎͚ͮΔ͜ͷํఔࣜ͸ϒϦΞϯඍ෼ͱҰൠͷඍ෼ԋࢉΛؔ܎͚ͮΔ͜ͷํఔࣜ͸ϒϥϝʔλʔ σ ∈ R Λಋೖͨ͠τϥϯε‫ڃ‬ൠͷඍ෼ԋࢉΛؔ܎͚ͮΔ͜ͷํఔࣜ͸ϒΛ࢖͏ͱ S+ ϕ(z; σ) = S− ϕ(zৗඍ෼ํఔࣜ
Ͱ༩͑ΒΕɼͦͷϘϨϧ࿨͸ಛҟ఺Ϧοδํఔࣜͱ‫ݺ‬͹Εɼ྆ลΛൺֱ͢Δ͜ͱͰࣗ‫ݾ‬ಉ‫ͱܕ‬ΤΠϦΞϯඍ෼͕ϦαʔδΣϯΛ࣋ͭৗඍ෼ํఔࣜ−zϦοδํఔࣜͱ‫ݺ‬͹Εɼ྆ลΛൺֱ͢Δ͜
Ϧοδํఔࣜͱ‫ݺ‬͹Εɼ྆ลΛൺֱ͢Δ͜ͱͰ
δํఔࣜͱ‫ݺ‬͹Εɼ྆ลΛൺֱ͢Δ͜ͱͰ
=0 s=ͱs ͱ‫ݺ‬͹Εɼ྆ลΛൺֱ͢Δ͜ͱͰ∆Φ0 = s ͱ ∆Φ∆Φ0ࣗ‫ݾ‬ಉ‫ͱܕ‬ΤΠϦΞϯඍ෼͕σe Ͱ༩͑ΒΕɼͦͷϘϨϧ࿨͸ಛҟ఺
" ∞e±i! −ztతʹɼӈࣜ྆ลʹ S Λ࡞༻ͤ͞ (40) ࠨࣜEx.) Airy equation' = Ai(z) ⇡ e

3
2 2
3z

'00
S±

z' = 0 (Irregular singularity on z = ∞)

X

an z

3
2n

+

e

3
2 2
3z

S±

X

bn z

3
2n

Re[Ai(z)]0  arg[z]  2⇡

review for physicists,
Resurgent structure in integral For
Cherman, Dorigoni, Unsal(14)Tanizaki (14)

In integral, original contour decomposes into steepest decent contours(Lefschetz thimbles) associated with complex saddles
Thimbles associated with distinct saddles havenontrivial relation via Stokes phenomena· Airy integral
Ai(g 2 ) =
⇡

Z 1
r

1

d exp

g
exp
4⇡

✓



i

✓

2
3g 2

3

3
◆

+

g2

◆

Resurgent structure in integral
Complex saddle contributions in thimble decomposition(Steepest descent method)· Airy integral

◆

i
+
g

Steepest descent method :original contour is decomposed into
thimbles associated with saddle points.

ig

Ai(g 2 ) =

Z 1

1

d exp



i

complex saddle points

C=

X

n J

✓

3

3

+

g2

i
=±
g

Steepest descent contour= Thimble

arg[g 2 ] = 0+

Resurgent structure in integral
Complex saddle contributions in thimble decomposition(Steepest descent method)· Airy integral
Ai(g 2 ) =

Z 1

d exp

1

・J



i

Im[S] = Im[S0 ]Re[S]  Re[S0 ]・ n = hK , Ci
C=

✓

3

3

+

g2

◆

ThimbleIntersection numberof dual thimble K
and original contour

X

n J

i
+
g

i
g

arg[g 2 ] = 0+JResurgent structure in integral
Complex saddle contributions in thimble decomposition(Steepest descent method)· Airy integralJ+

arg[g 2 ] = 0+

K+n+ = hK+ , Ci = 0

n = hK , Ci = 1
C=

X

n J

C=J

2⇡
valid decomposition till arg[g ] =
3

J

2

K

arg[g 2 ] = 0+

Resurgent structure in integral
Complex saddle contributions in thimble decomposition(Steepest descent method)· Airy integral

2⇡
arg[g ] =+32n+ = hK+ , Ci = 1

n = hK , Ci = 1
C=

X

n J

C = J + J+

Stokes phenomenon : at special arg[g2],thimble decomposition discretely changes

K+

K

J+

J
2⇡
arg[g ] =+3
2

Resurgent structure in integral
Complex saddle contributions in thimble decomposition(Steepest descent method)arg[g 2 ] =· Airy integral
22⇡arg[g ] =32⇡arg[g ] =+3

C=J

C = J + J+

2

Thimble decomposition is discretelychanged at Stokes line.
Airy function is continuous even at the
Stokes line.

J



2⇡
3

= J



2⇡ +
+ J+
3

2⇡
!⇡
3

Resurgent structure in integral
Complex saddle contributions in thimble decomposition(Steepest descent method)arg[g 2 ] =· Airy integral
22⇡arg[g ] =32⇡arg[g ] =+3

C=J

C = J + J+

2

Thimble decomposition is discretelychanged at Stokes line.
Airy function is continuous even at the
Stokes line.
Two thimbles have resurgent relation via
ambiguity due to Stokes phenomena !

2⇡
!⇡
3

!

n −nAzϕ± (z; σ) = S± Φ0 (z) +σ e
S± Φn (z)nResurgent structure in ODE
and integral
S+ Φ0 (z) − S− Φ0 (z) ≈ se
Perturbative imaginary
ambiguity−Az

SΦ1 (z)Non-perturbative
effectSθ = Id − Discθ = exp

"!

e

−ωθ z

∆ω θ

#

Resurgent structure is expected to be in quantum theory,
thus perturbative series could include nonpert. information !
−ωθ zeIn a certain∆ωθ ϕ(z;∝ ∂σ ϕ(z;σ)
class σ)of quantum
theories,

we may be able to derive non-perturbative result
from perturbation

$
∞
!

n+l

%

Possible questions to be asked• Thimble decomposition is possible in QFT path integral ?
• Resurgent functions in resurgence theory for ODE can beextended ?
• How can we figure out the intersection number ?
• Resurgent structure in partial differential equations ?

3. Resurgent structure in quantum theory

Example in Double-well QM

Bender-Wu(73)Bogomolny(77) Zinn-Justin(81)・Perturbation in Double-well QM
p2
x2 (1 gx)2
H=
+
2
2
X
E0,pert =aq g 2q
aq =

3q+1
q!
⇡

q=0

2 ⌥i✏

Im[Bpert (g e)] = Im✓

1)

Singularity on
positive real axis

11BPpert (t) =⇡ t 1/3
"Z

(q

1e
0

±i✏

t
g2

dt e
g2 ⇡

Instanton x(⌧ ) = 1 1 + tanh ⌧ ⌧I
2g2solution

#

1
3g 2

1
e
=⌥ 2
t 1/3
g
◆

1
SI = 26g

twice

e fields of S 2 asC.

1 instanton + 1 anti-instanton

The amplitude of one instanton and one anti-instanton amplitude is composed of two configura-I.

tions [I Ī] and [ĪI], as shown configurationin Fig. 2. In these cases, the interaction
between the two constituents
Instanton-antiinstanton= BionINTRODUCTIONV(x)
I. INTRODUCTION
Zinn-Justin (BZJ) prescription [29, 30]: we first evaluate the integral by taking −g > 0, and thename action density is depicted in two 1
ways, as a function
⌧
⌧
⌧
⌧
I
Ī
we
analyticallycontinuetheresult
from
−g
>
0
back
to
g
> 0 in the complex g plane. ThisxI Ī 1=
tanh
tanh
4
neutral bion configurations for λ procedure= 2g
1/1000,λ2 =2 ambiguity depending on2 the path of the analytic continuation
providesthe1/1000
imaginaryquasi moduli integral is ill-defined. Therefore we introduce the Bogomolnyi–
ndence in the action density, withis attractive,
x2 beinganda thecoordinate
2

2

2

x1
0

0

−4

−4
0

−2

x1

yx2
2

0

−2
2

τ

2



!!!!





V (x)



4

4 [I Ī] [ĪI] 









1 s(x1 , x2 )2SIn configurations for λ1 = 1/1000, λ2 = 1/1000
1 Zb

x
!!!!

Bion

!!!!

2

!!!!

as −g 2 = e∓iπ g 2 .

s(x1 , x2 )

0

2



x

e
2
Eb ⇡ Each horizontal
lim line stands=
+
log
±
i⇡
2
2
vacuum in the sine-Gordon gpotential.
density is depicted in two ways, as a function!1
Z0 for the⇡g
4

FIG. 2: A schematic figure of an example of one-instanton and one anti-instanton amplitude ([I Ī], [ĪI]).

the action density, with x2 being a coordinate

0 ≤ φ < 2πe 2SIcancels the imaginary ambiguity⌥ " 22 in perturbation#" 2 #"! ∞|g|$1
g 2 e−R − "R −→ −g Γ(")[I Ī]ξ −2 =dR exp −left of Fig. 2 is obtained as

S 2 as)τ

The amplitude of one-instanton and one anti-instanton configuration [I Ī] corresponding to the2−g 2 =e∓iπ g 2

−

πx1

πx1

−g

0

−→

"

− γ + log22e∓iπ g 2

#

" #
1
+O
+ O(")"

0 ≤ φ < 2πτComplex bion solution and its contribution
2
Behtash, Dunne, Schafer, Sulejmanpasic, Unsal(15)For other models including CPN-1 QM,
See Fujimori, et.al. (16)(17)

d2 z
@V
=
2
d⌧
@z

x ! z = x + iy

・Complex bion solutions
zcb (⌧ ) = z1

(z1

zT )
2

FIG. 1: Real and complex solutions in the inverted tilted double wellpotential. The inverted potential (on the real axis) is shown in black,
the real bounce and associated critical and turning points are shownin red, and the pair of complex bions and turning and critical points⇤
are blue. The blue points correspond to zcr
1 and zT , zT in (6). Notethat the motion0takes place in the real and imaginary
0 parts of thecomplex potential, as explained in the text.

!⌧
coth
2



!(⌧ + ⌧ )tanh
2FIG. 2: Complex bion solution in supersymmetric quantum mechanics with a double well potential. The black and red lines show thereal and imaginary part of the solution for pg = 1 · 10 6 . The characteristic size of the solution is Re[2t0 ] ' 12 log 16
pg . For larger valuesof pg the two tunneling event merge.

!(⌧

⌧0 )

tanhDouble well potential: Consider W (x)zT= x, /3⌧0x, so2thatC2
V (x) is an asymmetric double well potential with an O(g)3where we have used the Cauchy-Riemann equations ∂xVr =∂yVi , and ∂yVr = ∂xVi . An important aspect of (2) is that it
does not describe an ordinary two-dimensional classical mechanical system: the holomorphic classical mechanics is not
the same as the motion of a particle in the two-dimensional
inverted potential Vr (x, y). Instead of the usual Newton
equations with force ~—Vr (x, y), the force in the x-direction isdue to —xVr (x,✏y) while the force in the y-direction is due to2—yVr (x, y). This has interesting consequences.

“tilt”. The ground state energy of the system is zero to all orders in perturbation theory, but non-perturbatively supersymmetry is spontaneously broken and the ground state energy
is non-zero and positive [24]. Note that the positivity of theground state energy is a consequence of the SUSY algebra,
H = 12 {Q, Q̄}, where Q and Q̄ are the SUSY generators.
In the original formulation (3) this can be understood as thecontribution from approximate instanton-anti-instanton solutions of the bosonic potential 12 (W 0 )2 [9]. In the bosonizedversion we seek classical solutions in the inverted potential
Supersymmetric quantum mechanics: Consider supersymV+ . However, the real equations of motion in the inverted
metric quantum mechanics with the superpotential W (x)potential have no finite action configurations except for theZtrivial perturbative saddle, and an exact (real) bounce soluS = dt 12 ẋ2 + 12 (W 0 )2 + [ȳẏ + pW 00 ȳy] ,(3)tion. But this bounce is related to the false vacuum and is
not directly relevant for ground state properties, which are decorresponding to p = 1. The parameter p will be used totermined by saddles starting at the global maximum of thedeform the theory away from the supersymmetric point [9].
inverted potential. But the real motion of a classical particle
We choose W (x) with more than one critical point, so thatstarting at such aof
globalmaximum
is unbounded, and has inFor precise summation
bion
contributions,
there will be real instantons. By projecting to fermion number
finite action.
Sueishi (19).
eigenstates one obtains a pair of Hamiltonianssee
H± [24]:On the other hand, the holomorphic Newton’s equation (5)does support finite action solutions starting from the global
H± = 12 p̂2 +V± (x) , V± (x) = 12 (W 0 (x))2 ± 2p W 00 (x) . (4)maximum. There are exact finite action complex solutions

・Contribution from complex bion
1
3g 2

e
Ecb =
⇡g 2

✓

g
2

◆ 

i⇡
cos(✏⇡) (✏) ±(1 ✏)The imaginary ambiguity cancels that from perturbative series

Possible questions to be asked• Non-trivial Borel singularities always correspond tocomplex solutions ?
• How can we treat systems with fermions ?
By projection? By integrating out?
• Can we clarify relation of Stokes phenomena in semiclassical calculation and Exact WKB analysis ?
Sueishi-san’s talk on 18th

• The same structure exists in quantum field theory ?

Examples in QFT and Matrix model• 2D sigma model with compactification
Dunne, Unsal(12) TM, Nitta, Sakai(14)(15) Fujimori, et.al.(18) Ishikawa, et.al. (19)• 3D Chern-Simons theory (with matter)
Gukov, Marino, Putrov(16-) Honda(16) Fujimori, Honda, Kamata, TM, Sakai(18)• 4D N=2 Super Yang-Mills on S^4

Schiappa, Marino, Aniceto(13~) Honda(16)• Matrix models & Topological String Theories
Marino(07~) Marino, Schiappa, Weiss(09), Hatsuda, Dorigoni(15)Prof. Dunne’s lecture on 8th
Hatsuda-san’s talk on 18th
Honda-san’s talk on 25th
Yoda-san’s poster on 11th

4. Resurgent structure in gauge theorysectionprovidesa first,non-technicalrenormalon
P running This
throughthe chainof loops (seethe right
panel of Figure introduction1), rendering theto
loop
expansion non-Borel
summable(for andreviewclassicsee [10, calculation11]). Another way
saying this it
is thatbegin witha shortandofinterpretafterwards.
the Borel plane contains poles on the real axis, which generate ambiguities in the calculation,
Considerfunctionsdepending on whether
the thepole correlation
is circumvented
from aboveofortwo
fromvector
below. currents
The class jofµ = q̄γµ q o! are referred to as the renormalon
diagrams su↵ering from this problem
diagrams" and the#‘tHooft(79)4−iqx
corresponding non-Borel summability
problem [12].
(−i) isdthe
x e(in)famous#0|Trenormalon(jµ (x)jν (0))|0$= qµ qν − q 2 gµν Π(QBorel non-summability of the perturbation theory is not in itself surprising and was
3argued by Dyson long
2 time ago2 [14]. This problem also appears in quantum mechanics, butwith Q = −q . We now compute the contribution of the fermion
there the divergence is caused by the factorial proliferation of the number of the Feynman
shown
in finds
Fig.that
1 totheAdler isfunction
diagrams. In
fact, onesuch
divergencecured by instanton–anti-instanton events
Then, we can define the perturbative sum for the non-alternating series
2 problem.
[15, 16], and has a priori nothing to do with the renormalon
22tinuation of B
from negative coupling, g 2
0 (g ) in the g complex plane2

Infrared renormalon in QCD

In asymptotically free QFT, another source of Im ambiguity exists.
◆Adler function and renormalon"#ndΠ(Q )
D(Q ) = 4π
2 .
dQ

2 0. This can2be done in one of the two ways as shown inreal axis, g 2 >the positive real axis clock-wise (from2above) and counter-clock-wise (fro

Z 1
1
2
2 2
X
F
(
k̂
)
k̂
Q
2
2
D(Q
)=
↵s
dk̂
0 ↵s log
I. INTRODUCTION
B0 (|g 2 | ± i ) = Re B0 (|g 2 |) ± i Im B0 (|g 2 |)where Im B0 (|g 2 |) ⇤ e 222µ
k̂ The set of selected diagrams is gauge-invariant,0
n=0
but itofisnon-Borel-summability
not the only
kand isI.a manifestationistheambiguous
part,[c
INTRODUCTION
I. ◆INTRODUCTION✓
Adivergence.
definition of the BorelIt
sumisequivalent
to whathere
we described2
selected
for aboilX2↵s 0 that contributes to renormalon
continuation in the complex g -plane is the directional (sectorial) Borel
⇡ ↵s
n!
+
UV
contr.
systematic investigation is postponed to Section 3. Renormalons
were
2
⌅·e
1
" 2
#
2
t/g
n

!"

i

#n

2

S P (g!
) ⇥ B (g= 02t nBP (t)αes (µ) dt,
αs)(µ)βg 0 =BP (t) = α (µ)−

in bubble
αs (µ)β0 t
αs (µ) diagrams (Gross & Neveu 1974;s Lautrup2 1977; 1’t+ αHoofts (µ)β0 t/2nt) = αs (µ)−= "diagrams still(1)#
n feature so prominent in discussions of renormalons that
! 1 +αα
2
(µ)β
t/2
(µ)β
t
αs (µ)
s0sthe vacuum0
Figure 1. Left:polarizationwithall corrections. Right: Particular contribution
to then BP (t) = α (µ)
−
=
(1)
s
t
are
even
identified
with
them.
vacuum polarization2often referred to1as+the
renormalon
diagram.
α
(µ)β
t/2
s
0
2
n
t=−
The Adler function requires no additional subtractions
beyond thαs (µ)β0
It was recently suggested in [17, 18] that IR renormalon ambiguity cancellation can be C+
therenormalized
QCD
Lagrangian. solutions
Therefore
no regularizationis needunderstood
in
terms
of
semi-classicalinstanton-monopole
appearinginthetheory11Nc 2Nf
2
3
1
4
C
=
−1/gdetailed0Singularity
on
real
axis
(
)
t=−
(2)
on R ⇥ S fermion, but which
do notappear on are
R . renormalized.
This idea
wassubstantiated
by
the
loop
insertions
The
renormalized
fermion
lo
2e
12⇡
2
±
αs (µ)β0analysisoftwo-dimensionalmodels on R ⇥ S1 [19–22], which $have extra non-perturbativet=−(2) g2 %2s (µ)β0saddles comparedαtothe theory
on R2 (these are analogous
the instanton-monopolesin− β0f αsto ln(−k/µ2 ) + C
2

Figure 9.
Lateral, or for
right small
and left,L,
Borel
sums. Dark circles are singulargauge theories). Since these theories
reduce◆to4 quantum
mechanics
a resurgent
✓
could
be
relatedto
QCD scalepoints). Whenever a singularity exists between the right and left Borel sums, t
2⇤QCD
q(τ ) corresponds to a Stokes line in1summable.
Thedenote
singularthe
directionin thet-planes to denote the 4-momenta, and small2
Capital letters are ↵usedletters
spatial3-momenta.

i⇡0withea scheme-dependent
constantCand βof0fthe sectorial= Nphysics
thefermioB(↵2es −1/g
) = ReB±f T /(3π)⇡
andlow-energy
see
Fig.2.
The
differencesums
inpassingfrom⇥=
0 to ⇥ = 0
This suppression is caused by the appearance of logarithms in the one loop vacuum polarization diagrams

±

2

g2

Q 6 In the2
acrossMS
a Stokes
schemeC = −5/3.
the0one-loop
β-function.
→(3)
IRray. renormalon+−1/g(see Section 2 and2e[10,11] for more details.), which we revisit in this work on R3 ⇥ S 1 .
3±is6 true2that the perturbation series is divergent,A itspecial(3) argument
Although it
was casepointedout that Dyson’sof this is the lateral Borel sum. The function B ± (g 2
g [13].otherwise stated, αs denotes
Unlessthe strong coupling(x)renormalizedin th
may not be entirely
validcontours just above and just below theVrayat angle ⇥, and is called righmation. If there are no singular points in the ⇥ direction, then the left aQCD(adj.) on R3 × S1

Argyres-Unsal (12)Conjecture : Neutral bion is identified as IR-renormalon
・Non-trivial Polyakov-loop holonomy for small S1
P = diag[1, e

2 i/N

,e

4 i/N

, · · ·, e2 (N 1)i/N( P = 1)N]

1/N fractional instantons (Q=1/N, S=SI/N)cf.)SU(2)BPS(1, 1/2)

KK(−1, 1/2)

For YM on R3×S1 with ’t Hooft twist
see also Yamazaki,Yonekura(17), Itou(19).
For confinement via magnetic bions,see Unsal(07).

Neutral bion = IR-renormalon

???
Prof.
10th
(2,0) Unsal’s lecture on(−2,
0)
BPS
KK
Prof. Cherman’s lecture on 22th(−1, −1/2)(1, −1/2)
Morikawa-san’s talk on 24th
Yamazaki-san’s talk on 25th
Figure 2: (Left)Magnetically and topologically charged monopoles carry compulsory fermion zeromodes. Consequently, they cannot induce a bosonic potential for the dual photon. (Right) Topologi-

of two fractional instantons is given by

3.2

!
N 1 "
ZN -twistedand (104)
its anomaly
H (z) = λ e e CP
+ λ e e , 1model
.
II
0

1

iφ1 − πz
L

2

iφ2

πz
L

model on R × S

This is a composite of (a) and (d) in Fig.10. The inhomogeneous coordinate of CP 1 now reads

2 to R ⇥ S 1 3 (x1 , x2 )We consider
thecirclecompactification
fromR
Dunne-Unsal (12)
N-1
1
1
H
H
Fujimori,et.al. (16)(17)(18)==λ ee+λ e e 2
, 2
(105)
N
1
1
H
H
d CP
model
enceand
L ofits
theanomaly
circle S , i.e. x ⇠ x + L, is regarded to be small. We i
which cannot satisfy the assumption (96) of the Scherk-Schwarz reduction. This is because inboundary
condition,
1N and2 ),inasthereducedsine-Gordon
model
a configuration
endingFig. IR-renormalon
10]
Conjecture:
Complexbion
identified
ircle compactification from
R2 to starting
R
⇥from
Sis
3
(x1at, xS1[(a)wherethe circumfer2
1 2
cannot be connected to another configuration starting from N and ending at~S(x[(d) in,Fig.
10].
The
z
x
+
L)
=
⌦~
z
(x
, x ),
1
2
2
e S , i.e. xformer⇠canxbe connected+ L, onlyis regarded
to be
imposethe ZN -twisted
to a configuration starting
fromsmall.
S and ending We
at N. Thereforethe・Z fractional-twistedboundary
condition on S1 direction
BPS twoN
instanton solution2⇡i/N
cannot be described by the sine-Gordon quantum mechanics
where (! = e)
on,even in the limit of small L. More generally, all the BPS multi-fractional-instanton solutions are
N 1
1 2
1 2
⌦
=
diag(1,
!,
.
.
.
,
!
).
~z (xwith ,thexScherk-Schwarz+ L) =reduction⌦~z (x, xthe),sine-Gordon quantum mechanics(3.8)
inconsistentand hence

CP
II
1
II
2

II
0,1
II
0,2

πx

πx

− L1 i(φ1 − L2 )

1

2

πx1
L

πx

i(φ2 + L2 )fails to capture them. This is consistent with the fact that configurations containing n-instantons

We call this as Z -twisted CP N 1 sigma model, and we denote its p
1/N
as
Z✓,⌦
. fractional⌦=
diag(1,!, . . . , instantons! N 1 ). (Q=1/N, S=SI/N)(3.9)
For our purpose, it is better to regard this twisting matrix ⌦ as
N 1 sigma
Bion model, and we denote
Bion imaginary
ambiguity
seems-twistedCP
its
partitionfunction
at ✓
N
gauge
field
A
along
compactified
direction.
Indeed,theboundary-cond
s(x
,
x
)
s(x1,y1)consistent with IR-renormalon.
gauge transformation makesBut,
theisboundaryof ~z periodic
it related condition
to renormalon
on R2 ? (uose, it is better metry)
to regard
this
twisting⌦is asa backgroundholonomy ofSU
SU
(N
)holonomyx2 to matrix
and
the
price
be
paid
the
(N
)
There have been arguments.
xx1
g compactified direction.
SU (N(19)) theet.al. (19) andYamazaki,
Yonekura
etc….
is gauged, Indeed,
there is thea Zboundary-condition-changing
symmetry
it induces
ZN
N one-formxIshikawa,tion makes the ⌦boundary
conditionof ~z for
periodic(up toU (1)gauge sym7! !⌦. Whatis special
this choiceof
thenontrivial
holonomy ⌦Fujimori-san’s &rice to be paid
iseuclidean
theaction
backgroundSU(N )symmetryholonomy⌦.byWhen(N )FIG. 11: The
density s(x , xinduces) of neutralbion configurationsfor λ = 1/1000,
λ given=/
1/1000transformation
theMorikawa-san’s
talks SUon 24thand φ = π/4 in the CP model on R × S . The same action density is depicted in two ways, as a function
on 25ths a ZN one-form
symmetry and it 0
induces 1
the ZNYamazaki-san’s
zero-form
symmetry,0
1talk
0
of x , x (left) and x (right). There is no x dependence in the action density, with x beinga coordinate
z1
z2
0 1 0 ···
compactied dimension.
s special forof thethis
choice of the nontrivial
holonomy
⌦
is
that
the
above
Z
B z C
Bz C
B 0N0 1 · · ·
B 2 C
B 3C
B

N)

(n ≥ 2) are always non-BPS in the sine-Gordon
N quantum mechanics.

4

1

2

4

2

1 s(x1 , x2 )

0

−4

0

−2

0

2

2

4 −1

0

−4

4

2

0

−2

1

1

1

1

2

1

1

2

1

2

1

2

2

Adiabatic continuity in compactified QFT
Adiabatic continuity conjecture: Vacuum structure & ZN symmetry
persist during ZN-twisted compactification
If adiabatic continuity exists,
the resurgent structure on compactified spacetime
has implications on decompactified spacetime.
•••

Sulejmanpasic (16) Tanizaki, TM, Sakai (17) Hongo, Tanizaki, TM(18)TM, Tanizaki, Unsal(19) Fujimori, et.al. (20)2D ZN-twisted model
QCD(adj.) + ZN-twisted quarks Cherman, Schafer, Unsal (16), (See also Iritani, Itou,TM(15))
YM with ’t Hooft twist Yamazaki,Yonekura (17), Itou (19)Prof. Unsal’s lecture on 10th
Prof. Cherman’s lecture on 22th
Fujimori-san’s talk on 24th
Chen-san’s, Itou-san’s, Tanizaki-san’s posters on 11th

5. Further applications of resurgence theory

Further applications• Condensed matter physics, High-Tc superconductor• Hydrodynamics, Fluid dynamics• Schwinger mechanism• Quantization conditions via exact-WKB, TBA equations• Cosmology & Astrophysics• Phase transition
Prof. Dunne’s lecture on 8th
Hatsuda-san’s talk on 18th
Honda-san’s talk on 25th
Posters by Arraut, Du, Harris, Imaizumi, Shimazaki,Yoda

Goals of this workshop
1. Review the novel techniques for nonperturbative effects
of QFT, focusing on resurgence and the related techniques.
2. Study and summarize the very recent results in thetechniques.
3. Raise and consider questions in the techniques and their
physical results.
4. Propose their applications to physical problems otherthan pure QFT.
5. Discuss the questions and applications, and produce newworks by collaborating with the participants.