Topics on Integrability and N=4 SYM

Shota Komatsu

TopicsouIntegvak.li#&N--4SYMI

.

Intro to Integrability ( O CN ) model )

I.

From N -- 4 SYM to Spin Chain

III. OD E I IM correspondence

Lecture Tv :

' '

Bootstrapping"

Quantum Mechanics

.

1811.04-812I Ito

,Marino

,Shu]

. 1002.

2459 C Alday etat ]

I. Motivation

.

A harmonic oscillator

¥ifE ;HEH

Bohn.- Sommerfeld

I. C .

Double - well'

⇒→ dimpled

II§

,,iE ,

da --

cnn.r.it/hFiuiteB:T=yenh-y→ degeneracy lifted

.

←

VE"

Hire .

→"

Interactions"

among cycles.

*¥e÷:::÷÷÷÷Tf-

Goal : Have a control over

interactions among cycles

Method : Analytic propertiesin complex x & hi

2.fr#SchridyertofuucTiondeel .

Steph Rewriting Schrodinger eq .

f- It'

die Vix ,- E) Ya ,

= o

I

Realignpolynomial.

[ - y'

d.? t art 't £

break) tea ,

Tbio

new

'

to I = 0

View in )

we father assume

I / via ,

,.

r : odd

2.

All zeros of tix , → real axis

I Technical ) ( realitycondition )

ftp.T.in.no.

Steps.solution @ Incl → A

by → as ⇒ f- 54ft It 't . - - - ] taro

steading

4cal

rexpftffz.se#)2/differemtsol.

2 so I :exponentially I

largesmall

-

ef

¥14- exp FIFE] sired

t : sural'

- : large - - I 'Anti - Stokes

" .

. l-

Stokes sector.

1- i S-

- : e f- focus,

④ Comment.

small sole.

: Well - defined

large Sol .

: Ambiguous

(large )'

n C large ) t ④ Gar all )

④ Comment 2 : Y → CIOS.

⇒Compensated by x → ei% x

Lt - expft.IE )- -

: i%

.

In particular .

0=2I

T

EI.io?os-.eieg-

① More generally , } → Ei"

} rotates

sectors by 1 unit

! Iqs,

Street.

\ r

-

. .

,

- - - - redline.

Stale"

3) xsktil 's)

⇒ sole"k

's ) x Skl 's )-

Define. Spells ) : = Sole

" kg )a-

Quantizatioucoudit.io#

Energy eigenstate on x ER

→ 4 must decayboth at Is

→

ls.islxsr.LI#f(Via , =

Wtae uergy )→ I. C . determines bo

Step Wronski an.

For E 422 : thx , ] 14.2=0

Wronski an i

< 4 . ,4 . > =

4Intellect=EatsI. a Ebt.LY/

. Lt . ,4) (5) ← independent of x .

. L 4 ,47 = 0

Q.

C.

: Go, SHI > Ly) = O .

Step Functional Eq .

Identity:

/EabfcdtEadE#+ Eac Edb = O-

→Gi , Sj7 ( Sk

.Se ) t Csi

.Se > csj.sk )

+ ( Si.See > Cse

, Sj7 to-

← Pliickerrel.

DEI Y- function

Ysj (3) = G-J.SPG-j-i.si#CS-j-i.S-jSCSj.Sj+, >

(3)

Ysj , (3) =GIiSj3Csj⇒g

( Sj -2 ,S - j - D Csi . Sju ) )

[.

peas ,- -

t

¥¥÷¥i:÷÷÷¥¥#¥JRoot Plucker rel

.

Rootlet )no =

0

yo = (S-oi= O

. - - - - - -

Y,

=Gt

Go.

Saks . ,s , >I

E!3)

go

Y.

-

. Gt.? .

"

S-2 53

= O I ,

Ss = IS-2.y

,

't 'Y

,

"= I

.in

- a

⇒ iEsoaET¥⇐⇐=

fsa.AT?Go.-ST.5sEi.-soJfE.-sa7

- a

= I-

Inguinal

o-- ok

.. . .

it.

.. .

. ..

.- .

!

In"

°

-

r

④ Physical Meaning of Y - function

A.

WKB approx.

⇒y.ci .

C Se.

Go,

s, >

→ WKB.

.- snexpf.tl?r.iid.J

Zo Hols > I --na!

. E"

Ust - expfffjrrnd.ci

at.

so > -

exp ft 's I ! Fri , da )i

-

⇒.. e. a.:¥

exp f- fo FEETdoc)

¥t÷ . .s

.Feria .

Y 's → Nou perturbative / Quantum

B.S . period.

-

.¥n.

← r=3

? ttft-

get%)I ' I

lY

, !i-iiyya.

.

- ixIl l

"

'

' *-

t' 54 :

B.

d. C . .

.

.si;

-

Stz So

④ f- I Sz S2,

-

in,

Y,

H=Gibess

'

G-z.S-DCSo.sc ?Sss s!

I. C.

i 5-2 Tso X 5-12

→ yet , - I ⇐ try ,

"-

- o .

-

④ f- 3 Idouble - well)

yz¥¥4y,

I Soos

f S3

( It Y, )"

I t 4314 ]Gt .

SD Go,

- =-

✓ T G- i.So > 02,53 )

12 Pliicker ↳3

= - I

I. C. Iet( HY . )

"Atl )

"= O

[ instantly

et . . - t.AT .

* I derived this this morning .

-

3. Solving .ft system

Ys" '

Ys"

= ( It Ys - , ) It Tsu )- - -

- -- - - - -

-- - -

log Ys"

t log Ys "= log It Ys - i )

+ log Ut Tsu I

Ys - exp t TAI B. s. period 20

. ls Cb ) i = log Ys 13 ) t

-

,-

- -

,

t.si#ds5.?-.logUtYs-i)-lgUtYsti

)

} = e- O O : rapidity .

• Keo ) =1-

2K Cosho .

← kernel

& Consider Convolution

K *

test't

Ks * lls" 't lst '

)

HIit

b.IIE)

= [adat ashot e

y → as

Types-

"

-8¥ ⇒ . - - - iFest, real. axis

Ho

= So a¥siuhcoyT Mst

=

- -

- - g-

• I assumed that Is ly ) doesn't

have poles inside the strip .

Ey. reality condition.

L} → o . you can show it

by WEB.

-

ls = Kx testes )

= K * flog Ht Yost ) t log Ht Tsai ) ]

log Ys =

-Mstf!d÷log thistle lost - - I

t. Xiii

"

TB Aeg!Y

- exp f-FJ

He,

Harmonic Osc.

log I = - Ms Eto- -

E, B. S

. period .

I. C. Y,

"-11=0 .

⇒ after,= 2T Cutt )

← BeS. quantization

F.utuvedireeTiousaSUS4QM.c-N-foldsusY@M.fa.oi3.H

for ,at } -

- fat )

giroud St .

→ N-th excitedState

. large c confound block.

\

r . , a

TfL's blockc- irregular Con