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