topics on integrability and n=4 sym shota...

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

I.

Intro to Integrability .

1.

What is integrability ?

Classical ( Quantum ) Meehan is

~ # of conserved charges

= # of d. of .(

tiouthitteegiab.

. liqEf ' H -

. Inset t I wreak'

→ Hk = glut + E WE are

ef double pendulum \,

•

-

• QFT ?→ as

manyd. af

.

→ as manyconserved charges

?

- - → abstract. . .

? physical consequence

@ Physical consequence.

Pb Pa,3-73 scattering .

in non - relativistic RFT

p? §.tp

,

Momentum : Ii,

Pi = FayPj

Energy :

s÷pf=E⇒B?

SupposeI

In An : integer)

In 1Pa ,- - -

, Pm > =

,

Pin Ipa ,-

, pm )

1.

LP, . Pa . B) = hPa. Ps

.Ps }

.

#

• Consider a wave packet

1137 → f.

!dp e-" P - Pm

'

# .

.

-

-- - -

,,

⇒ tix ,-4=1!dp e-a P - Phi

,

i

--

,, ,

-

'

x e- Iiit P 't -1 ipa

#f- Cpi

peak of the wave packet. :

dp tips / = o

p = Pk

⇒ x - Ihh I = o .

yp

"

W

↳ ur. Now

. act einen : ipj.IT"

&"

f-cp ,→ fcp, t i Lupu

⇒ peek :

o = x - Ithkttnxnph"

h= I : Momentum → Shift in redirection

n=2 : Energy.

→ shift in I

n > 2 : Shift depends on Pk

"

÷¥÷÷÷÷÷:* :"

" '

--

- --Isf e iz Arian

S3 → 3 = ( s a)3

④%aIEaxEg.

k¥0 E.

*- -

"

I-

2.

01N ) nonlinear on - model Child )-

Sigma model w/. Targetspace

SM

:÷÷÷÷÷÷÷÷es÷÷÷÷

Z -

- fairere.

:{ AZL

coupling

L -

-

dI.JBtrCn.n=#Input 0 : classically N - I mass les

hjn7iect7o7ticcew@esosw.Yn.n)

→ wrong

Input I : Large N .

Z F¥÷ Sfo e- Setter ]

Seff Co ]= - folk t N together]"

If Tr log

N → as. K . N i fixed

K → o.

"

a 't Hooft coupling

Saddteput

I = I I ¥* ) =L:¥ipe÷Isp .

value .

= ¥ log # ft]⇒

of = A × e-¥ ur cutoff

=3 non perturbative

Tx to

ifI : massive

rich .nu - , )⇒ N massive particles

Input I : At large N N massive'

particles.

{ ( Assume that this is true

( @ finite N )Input 2=7-1

,foaming

.:user red charges.

↳ yay - Baxter eg .

3. Integrable Bootstrap for Our )

Ten.

kinematics. i. j.k.lt .

- - -

.N

l k

Sills) =

5- cpitpsl? Pim '

Xp.

XP.

t - Cp .- ps)

?

i j set = 4in

^

11 I Ls Lo

.jo//---......Et4ssiIfawd.*.-"%gate'¥Tn÷

physical4M

' yt - chanel

.

O→ - O

Pa, a

= m Sinh On .se

Erie mash One.

⇒ s=4m⇒h%Sca - Oz )

• Axioms

* Unit arity .

f.a:÷÷÷÷:.

Sif to ) = Sike lie - O )

* Yay - Baxter eq .

k

Sitio ) =

peona sie sie

+ate.kza

sites !

I, -

N Sg SheI+ be. I

Y.

B. I .

acoz -037-

.

A,

re.

-

i

-

i

=

.

i.

i'

? IBP, µ

-

"

Pz p,

MYAloe -037

Pz

Ix Alfa -03 ) = Alo , -037 ACO . -03)× Ala

-012) + Aco . -02 ) ACO . -03 )

⇒ a÷¥, =a÷¥ta¥E '.

!;

-I

'

ace- on

A 3I

z ALO , -02 )

Aco ,- Oz ) x A ( Oc -03 )

→ at ,

= CEunfixed coeff.

Yang - Baxter 2.

-

E.Ai

an.

f *iki# . iti I =I - fii.in f I -

-

'

,

-

'

/ ,

""

3 ia 3

s3

Iz

b I Oi - 03 ) x ACO ,- Oz ) = b cos -03 ) Aloe - Oz)

C-1%-04⇒ jog:

-

,ibsoIoIbIo¥DTEO ,)- 6¥03)

t

To ,

= - C O t of

#^ / i

Sifton

most! t To .

?f→ r -

+ atIt - I

• Another Y .B

.

( Exercise )The D= - NIµ¥÷Iqf

syn . Sitio)--Tie Lia - o )

-d.TO = e¥o ,

→ * KII

• Unitarily$101 St -01=1

#~0=¥⇒o=Is

.

P x A PT lol =

,

'

r

Similar Fane. Eq .

→ Lecture TV .

T