Chaos and Non-‐Integrability in AdS/CFT

“Chaos and Turbulence in AdS/CFT” @ Osaka University Sep. 8 th, 2014 Chaos and Non-­‐Integrability in AdS/CFT
Kentaroh Yoshida (Dept. of Phys., Kyoto U.)
In collaboraQon with Daisuke Kawai (Dept. of Phys., Kyoto U.)
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Make a map of the space of AdS/CFTs from the viewpoint of integrability
Islands: Integrable Models
Sea: Non-­‐Integrable Models
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0. Classical Integrability
A review of the Liouville integrability in classical mechanics
What is the classical integrability in field theories in 1+1 dimensions?
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The Liouville integrability
(in classical mechanics)
Let us consider a Hamiltonian system with N parQcles. Then moQon of the parQcles is described by If the system possesses N conserved charges , in involuQon, then the soluQon of e. o. m. is obtained by ``quadrature’’.
There is a canonical transformaQon (depend only on )
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The soluQon is trivially given by
Liouville integrability (or complete integrability)
NOTE: the concept of the Liouville integrability is definite when the parQcle number is finite.
But then, it would NOT be definite in 1+1 D classical field theories. It is because the number of d. o. f. is conQnuously infinite, while the number of conserved charges is countably infinite. So the Liouville theorem does not work apparently.
However, everyone says `` classical integrability of string theory on AdS5 x S5 ’’ !!
What is it at all?
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Two concepts of integrability in 1+1 D classical field theories
1) kinemaQcal integrability (used in the textbook by Faddeev & Takhtajan)
The existence of the Lax pair (or the zero curvature representaQon)
Inverse scakering method solitons
2) complete integrability The explicit construcQon of acQon-­‐angle variables
EX sine-­‐Gordon model, NL Schrödinger eq., Heisenberg magnet
In general, it is quite difficult to show the complete integrability.
In the case of AdS/CFT, the classical integrability is used in the sense of kinemaQcal one.
In parQcular, the complete integrability has not been shown in AdS/CFT.
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1.  A brief summary of the recent progress A list of integrable and non-­‐integrable backgrounds
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A recent progress in string theory
Integrability in AdS/CFT EX
[Heng-­‐Yu’s review talk]
Type IIB string on AdS5 x S5 4D planar N=4 super Yang-­‐Mills
Type IIA string on AdS4 x CP3 3D N=6 Chern-­‐Simons maker system
An enormous amount of works have been done.
Why is the integrability so akracQve?
The integrability provides very powerful tools.
It enables us to check its conjectured relaQons without SUSY, even at finite coupling, in spite of severe quantum correcQons.
EX anomalous dimensions, amplitudes etc.
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QuesQon
How prevalent is integrability in various kinds AdS/CFTs?
NOTE: Integrable subsectors are ubiquitous. EX integrable subsectors in large N QCD in 4 D
So we will concentrate on the full integrability below.
There are various kinds of AdS/CFT
Real (or complex) beta-­‐deformaQons [Lunin-­‐Maldacena, Frolov]
Gravity duals for NC gauge theories [Hashimoto-­‐Itzhaki, Maldacena-­‐Russo]
AdS5 x T1,1 [Klebanov-­‐Wiken]
AdS5 x Yp,q [Gauntlek-­‐Martelli-­‐Sparks-­‐Waldram]
Klebanov-­‐Strassler, Maldacena-­‐Nunez
AdS BH [Horowitz-­‐Strominger] AdS solitons [Wiken, Horowitz-­‐Myers]
AdS/NRCFT [Son,Balasubramanian-­‐McGreevy, Kachru-­‐Liu-­‐Mulligan]
q-­‐deformaQon of AdS5xS5 [Delduc-­‐Magro-­‐Vicedo, Arutyunov-­‐Borsato-­‐Frolov]
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How can one show (non-­‐)integrability ?
To show integrability
Explicitly construct the Lax pair kinemaQcal integrability
A more general scheme: Yang-­‐Baxter sigma model approach
[Delduc-­‐Magro-­‐Vicedo, Kawaguchi-­‐Matsumoto-­‐KY]
By idenQfying classical r-­‐matrices corresponding to SUGRA sols, one can show the kinemaQcal integrability (the gravity/CYBE correspondence)
To show non-­‐integrability
A reducQon to a 1D system If a chaos is shown, then it’s non-­‐integrable.
LOGIC
A certain subsector is non-­‐integrable the full system cannot be integrable. There may be some integrable subsectors though
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A classificaQon list (would not be complete)
Integrable backgrounds
Real beta-­‐deformaQons
[Frolov, hep-­‐th/0503201]
Gravity duals for NC gauge theories
[Matsumoto-­‐KY, 1403.2703]
q-­‐deformaQon of AdS5 x S5
[Delduc-­‐Magro-­‐Vicedo, 1309.5850, Arutyunov-­‐Borsato-­‐Frolov, 1312.3542]
[Hubeny-­‐Rangamani-­‐Ross , hep-­‐th/0504034]
[Dhokarh-­‐Haque-­‐Hashimoto , 0801.3812]
[Kawaguchi-­‐Matsumoto-­‐KY, 1401.4855]
TsT transformaQos of AdS5 Non-­‐integrable backgrounds
Complex beta-­‐deformaQons AdS5 x T1,1 [Basu-­‐Pando Zayas, 1103.4107]
[Giataganas-­‐Pando Zayas-­‐Zoubos, 1311.3241]
AdS5 x Yp,q [Basu-­‐Pando Zayas, 1105.2540]
AdS BH [Pando Zayas-­‐Terrero Escalante, 1007.0277] AdS solitons [Basu-­‐Pand Zayas, 1103.4101]
Klebanov-­‐Strassler, Maldacena-­‐Nunez [Basu-­‐Das-­‐Ghosh-­‐Pando Zayas, 1201.5634]
Schrödinger spaceQme with [Giataganas-­‐Sfetsos, 1403.2703]
Lifshitz space (with hyper-­‐scaling violaQon) [Giataganas-­‐Sfetsos, 1403.2703]
[Bai-­‐Chen-­‐Lee-­‐Moon, 1406.5816]
p-­‐brane backgrounds [Stepanchuk-­‐Tseytlin, 1211.3727] [Chervonyi-­‐Lunin, 1311.1521]
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In the following, I will focus upon two examples.
The plan of my talk
2. Chaos in AdS5 x T1,1
(A review)
3. Tests of complete integrability in q-­‐deformed AdS5 x S5
4. Summary & Discussion
As another example,
For chaos in Lifshitz spaceQmes, Bum-­‐Hoon’s talk
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2. Chaos in AdS5 x T1,1
P. Basu and L.A. Pando Zayas, 1103.4107.
A lot of typos are contained in this paper. So, all of the figures, which appear below, have been reproduced by D. Kawai.
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String theory on AdS5 x T1,1
(only the bosonic part)
The metric of AdS5 x T1,1
The Polyakov string acQon (in conformal gauge)
The Virasoro constraints
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An ansatz for wrapped strings EquaQons of moQon
(constants of moQon)
The constraint condiQon
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NOTE The system is reduced to a 1D system like two coupled oscillators.
The concept of the integrability is definite.
SQll, an analyQcal study of the differenQal equaQon system, so we will check the behavior of soluQons numerically.
If the reduced system exhibits a chaoQc behavior, the full system is shown to be non-­‐integrable.
In the following, we will show Poincare secQons of the phase space with and with the condiQon,
by changing the value of the energy .
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KAM tori
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A similarity structure may be found.
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Tend to be ordered again
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The potenQal shape
At low energy, the moQon is bounded by the potenQal wall.
However, it is unbounded at high energy.
The behavior is very similar to the Henon-­‐Heiles system.
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The largest Lyapunov index
: iniQal condiQon
Two different orbits
: deviaQon of iniQal condiQons
This index measures the sensiQvity of the dependence on the iniQal condiQon.
If it is non-­‐zero and posiQve, then the orbits tend to separate away exponenQally.
ChaoQc behavior
The system is shown to be non-­‐integrable
NOTE: If the system is integrable, then the index is zero.
The moQon is represented by
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The largest Lyapunov index
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3.  Tests of complete integrability in q-­‐deformed AdS5 x S5
With Daisuke Kawai (work in progress)
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A q-­‐deformaQon of AdS5 x S5 [Delduc-­‐Magro-­‐Vicedo, 1309.5850]
[Arutyunov-­‐Borsato-­‐Frolov, 1312.3542]
someQmes called -­‐deformaQon of AdS5 x S5
Singularity!
DeformaQon parameter：
NS-­‐NS 2-­‐form is turned on both AdS and S
NOTE: The other fields in type IIB SUGRA have not been determined yet.
The RR fluxes may be complex. [Hoare-­‐Roiban-­‐Tseytlin, 1403.5517]
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The known results
The kinemaQcal integrability has already been shown. The Lax pair is constructed.
[Delduc-­‐Magro-­‐Vicedo, 1309.5850]
The deformed AdS5 part contains a singularity surface.
[Arutyunov-­‐Borsato-­‐Frolov, 1312.3542]
QUESTIONS
Does the deformed AdS5 part lead to complete integrability?
What is the moQon of strings on the deformed AdS5?
It may be interesQng to check whether chaos appears or not.
The strategy is quite similar to the AdS5 x T1,1 case.
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A reducQon ansatz for the deformed AdS5
For the deformed S5, a 1D system
Check the behavior of soluQons numerically.
NOTE The equaQons of moQon and the Virasoro constraints are so complicated, so will not be presented here.
We will show Poincare secQons and the largest Lyapunov index below.
The results support the complete integrability.
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Poincare secQon for
(the undeformed AdS5)
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Poincare secQon for
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Poincare secQon for
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The largest Lyapunov index with
(the undeformed AdS5)
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The largest Lyapunov index with
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The largest Lyapunov index with
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4. Summary & Discussion
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Summary
We have discussed two examples:
1) Chaos and non-­‐integrability in AdS5 xT1,1
2) CompaQbility to complete integrability in q-­‐deformed AdS5 x S5
Discussion
•  Gauge theory interpretaQon of chaoQc strings?
c.f., AdS5 x T1,1
Kolmogorov-­‐Sinai entropy, a fractal-­‐like structure, InformaQon loss due to chaos [Opening talk by Koji]
•  The complete integrability in q-­‐deformed AdS5 ?
So far, the complete integrability has been supported. Is there a curious ansatz to exhibit chaos? 38
Kolmogorov-­‐Sinai entropy
Yakov G. Sinai won Abel prize in 2014!
This entropy measures unpredictability of a dynamical system.
One can predict what happens in a dynamical system in the short term. But, when analyzed in the long term, one cannot predict the moQon.
(due to the chaoQc behavior)
c.f., weather forecast One can rely on it for the next ten minutes, but not for the next ten days.
This unpredictability leads to the entropy producQon.
NOTE
In typical cases, the KS entropy is represented by the sum of all of the posiQve Lyapunov indices.
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Thank you!
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