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ESI Higher spin Holography 2019
Chaos
in 3D HS Gravity
Junggi Yoon
Korea Institute for Advanced Study( K IHS )
Chaos in 3D Higher spin Gravity
Based on 1903 .
08761 Prithvi Narayan , TY
( 1903 . 09086 Viktor Jahnke,Keun - Young Kim ,JY)
↳ rotating BTZ Black hole .
I.
Introduction.
* Butterfly effect.
* .
" "
• Classical.
fuel% ,
next .
• Quantum ( [ VCA ,Wco )? My
,n e
*
• out - of - time - ordered correlate.
⇒ ( Vce ) Wco ) Vce ) WCO ) ) n l - E e't
'
B
* Bound on chaos.
:
XL € 2¥Malda cena ,Shenker , Stanford .
* Black hole :
maximally chaotic .
thyEoe .si#asai:er.ing
of
exchangeofgraviton .
⇒ t.
: saturate !r e
spin - s field exchange?
)f ⇒ e⇒cs - he
D= ⇒ Cst ) ;violate chaos bound
for s > 2.
HS gravity .
i ) HS symmetry is broken.
Ti ) HSsymmetry is not broken
. .infinite cower of Its field
.
@ Too > - It e¥tte tt - - .
= ÷¥ - A- O. .
.
iii ) Finite cower of Hs field.
3D SLU ) CSgravity .
5=2.3 .- - -
- N.
: semi - classical limit of ↳ EAI HS gravity .
( x → - N ) ⇒ truncation
non -
unitary . Causality in Ape function ??
E.Perlmutter : W vacuum block
.
Motivation :
How to calculate spin - s field contributionto Ape function in SLCN.ci ) CS
)spi§gravity ?
Cf ) JT gravity / Schwarzman theory / STK model.
S = Jdt . Sch team -4¥ ) , Ef f : boundary graviton( pseudo Goldstone boson
=fde Sch # es , Eft 2¥ If-
SSBIESB of
reparameeri Eat ion
Coddjoint orbit action.
DiffLcs ) .
o O
Schutt . -4.
-
- fEff÷5
o
D Quadratic level : foes = t t Eee )
& mode expansion : = et E En e-"t
-
S" '
= n-
Cn-
- I ) E- n En
.
'
. vanishes at n -- O , I I
③ Gct , its ) -- LOGGED > ~
so
[email protected]_Gcei.t .it Efface . .es
-feet -- Et Eine htt
.. .
-
i③ / LEN E- n > Senga 't " Se. .Gets . eat
,
-
Questions.
I.
What is W - generalizationof Schwarz ian action ?
Ooguri . Bershadsky ( 1.989) & Wei Li . Theisen 420157
Grum i ler etc . ( 2018 )
D How to get the on - shell action ?
( in CS gravity )
- Boundary term , Boundary condition.
② finite temperature Schwarz ion ?
( smoothgauge
transfers Large gauge transf )
Sch [ feel . - it ⇒ Solitaire " . tff : R → R fee ) :S
'
→ s'
TI.
( boundary - to - boundary ) two point function.
① What kind of ewo point function .
pure CS : no matter
topological : difficult to
couple to matter .
② Ws - transformation of two - point function
③ Why do we calculate
< En E - n > SEUG FENG ?
I.
W - generalized Schwarz ian :
* Quick derivation :
A = bib ( d tacet ) bar) : CS connection
r Lob = e
..
⇒ Az = g¥BzfG ) & Gauss Decomposition g- LDU
=L:{ EtD SLCN.cl ) CS gravity
A. Tt
.
Los = fTr( ADA -15A ' ) -
.
--
. .
I "
SIG n f Tr( ASA ) r .e.Ezra
= for , Tr( Az SAE - A -zfAz )
Az = O .
. good variational principle .
Spin -3 charged black hole : X( chemical potential .
)
A = b-' ( dta # b b -
-emo .
Asympeic Ads condition '
[ Ln .hn?--@-m)Ln-im .
A - Aids ~ OU )Wn .
→ a # = Le - 2 dell - it Wiz ) KI - z
r
Jan de Boer , Juan Jotter. 1302.0816 .
\ Solid torus .
I✓
£ ~ ZTZTL N ETI, ,
In variation,consider St : periodicity can be changed
w ~ Wtt ~ w t2TLi..
dwrdut = #td÷⇐, . .
d5= dzdE=
Upshot :
• keep volume dando =
4TdE_Tenth ) i
④ Add boundary term
⇒ Variational principle .
Az -
- O,
82=0.
⇒ Ion- shell = iL÷ff÷⇒Tr[ Eat - EEE )
=i fE÷a
, @ Ice - Esea )(a. c:¥¥⇒
* No spin - 3 chemical potential .
⇒ No NCE ) in action : future work.
* fo : const . csolution ⇒ BTF..
( Lo > O )→ ABIE = Li - fol - l
AE hI£Ao¥ Iz ) htt ) .
h : smooth residualgauge
transf !↳ does not chang hdonomy ( can be connected
* Holonomy alongEuclidean eine direst identity .
Hol CA ) =b-lh-ie-whbcju-raceiblecyde.ca#nmge )
smoothness condition
W = Caz f- I Az ~ Zai Lo.
Euclidean BH .
⇒ e - if2TCLo a
Change holonomy → change e.
fluctuation→ See =o implies So : fixed ( around fixed
② finite temperature Schwarzman ? background .
* Difficult to parametrize large & smoothgauge
transfseparately .
⇒ Give. up the finite transf
.
he large h- or
do -_Je LEO
•
( smooch fluctuation aroundLIFO
stationary background )* Consider fixed fo → infinitesimal residual gauge→
( stationary background) transf- y
'
. g-uant.ee smooth residualgauge
transf.
→ h= I teh "t 's E
' "D 't * ' I.
3×3K "
.I ' ' : traceless # aerix
8 d. of 8. d. of
Geqs .
2 parameter .
1st order :
gauge fixing condition ⇒ N, §
2nd order e. . → n
'
. 5475 .
begs .
fat,87
2 parameter .
Ion- Ey Io t C- I "
t EI" '
± . -- .
-
toed derivative
-Vanishes at a- 0.tl
,
&vanish at n' -0 , It ,I2
.
graviton spin 3 field.
It.
Vasiliev equation ( SL CN ) ).
DC TAC - CF -
- O.
* For seaeionarybackground
Ccr ,E. Z ) = BID e- AKZZCoe-aem-Eb-l.gg
* For non - constant A.
shr '
e- t.ee- fi :#
⇐ e-si :*
he - so Genii classical )Q
Icr , zrzizsz ' )
= ercot-erflim.me - IIe. e-
SEE )r.s.ae
'
.
Wilson line.
Trc )s
-
.
↳ °o°.
"
highest weight"
* OI CEE . . E. E.) = finesse" "
Ecr . Ez , ⇒ E. )
: boundary - to -
boundary propagator.
-⇒ geodesic distance .
* For Non - constant background ⑦⇒ gravitationally dressed
Wilson line Cmaseer field )
by boundary graviton( soft )
* Soft mode expansionh= I t NCH t . - - -
⇒ §dressed
=
↳ Mn& anode expansion )
⇐ i. ⇒ Boca . . Ht znnfn.
@dressed
, . .
,
I
↳ ( Zi , Zz )
: two - point function
in constant background
( Ezdressedcz . .z . ) &dressed
⇒ S
= GG.4Gcs.ci ) t cnn.n-nyfmEU.nfnfks.ci/
= + e . .
* Soft mode eigenfunction [email protected] ,-
= Er Cir )
=
& #dressed
#
GUY
X -
- lzczitza ) , G- Iz CF - Zz )
quantumfluctuate -
②. @
dressed=
'
+ qfnngfdres.ie .-
n
µ ,@dressed
y ~ fun @dressed - - -
Ward ideneiey
* Ws - transf of two point function ?
At least , a special operator .
Observation : null relation in Way minimal modelin Large C .
( II :O ) ( dual to scalar field )
- . .
La → Iz,
(( z → e-"# for finite temperature .
u
§ 197 = Gain fan ( x , o )
* Conjecture .
for null relations.
HE '0 n CL-it "¢
gthen
,we can fix coefficient
.
→ Sgs , L 7 = Hypergeometric functions.
appears in STK model
as basis in limit q→N , n → 1,
They are solution of
* Recursion relation.
We know
tfrom 04 > transf ← from coyfoz.mg/eransf .
- f. ( X ,= Zz fun ( XO )
1h
* Can we conserve fan ( XD ) out of fssn ?
51=1 , 2,
- - -
,S - C
.
fan I >
e-" " Xgs.ncoystffs.n.fssn.ws :& ,
transl ihy of center of coordinates( ska ) talk ) )
* We found that- 5lgsmCo )Gses , a
= I Gain ( O ) t45-1
Then, we have .
Js , n (G) = O for n = O , II. 12 ,
- -
; Est .
( fan ,fs
, a > ~ Sss ' fan '
=) Solution for recursion relations.
* Four point functions.
¥¥dresed!④
in Gaa )
one spin - s fieldexchange .
es, es ,dressed
dressed
{ 5ns - n > 898 SEE.
I A
AnalyticFaction 070C ~1e¥GtH .
C -
Lyapunov exponent : ⇒ Cst )
: violate -the bound on chaos ( for s > 2)
⇒ Semi - classical limit : non - unitary .
I corrections .
( E correction toLyeayygnon)
Interesting to study⇐ 3. a > SEELE
( 5=2 ) £÷→t Ise . - -. ) t - - -
loop correction.
+ f t + ..
E
Gis - n > Tsm .su#nSnEf.nS.nEl Ee - c- - E .
E E t - - -
t t + . . .
Es - c - Ia Ia - C - I ,
5555788 ESE.
