2015/09/22
Selected Topics in Theoretical High Energy physics@Tbilisi State Univ., Georgia
Towards the gravity/CYBE correspondence
Kentaroh Yoshida (Dept. of Phys., Kyoto U.)
Based on collaboration with Takuya Matsumoto (Nagoya U.)
Io Kawaguchi, Takashi Kameyama, Marcos Crichigno,
Domenico Orlando, Susanne Reffert, Jun-ichi Sakamoto,
Hideki Kyono, Andrzej Borowiec, Jerzy Lukierski
1
The AdS/CFT correspondence
type IIB string on AdS5 x S5
Recent progress:
4D
SU(N) SYM
the discovery of integrability behind the AdS/CFT
[For a big review, Beisert et al., 1012.3982]
The integrable structure provides powerful tools.
It enables us to check the conjectured relations in the AdS/CFT
without SUSY, even at finite coupling.
EX anomalous dimensions,
amplitudes etc.
An enormous amount of works have been done, so far.
Motivated by this success, our interest here is
the classical integrability on the string-theory side.
The existence of Lax pair (kinematical integrability)
2
The classical integrability of the AdS5 x S5 superstring
The coset structure of AdS5 x S5 is closely related to the integrability.
: symmetric coset
Z2-grading
classical integrability
Including fermions
: super coset
[Metsaev-Tseytlin, 1998]
Z4-grading
classical integrability
elucidated by
[Bena-Polchinski-Roiban, 2003]
This fact is the starting point of our later argument.
3
The next issue
Integrable deformations of the AdS5 x S5 superstring
Integrable deformations
(as exactly solvable models)
Integrability techniques
EX Integrable twists
deformed AdS5 x S5 geometries
Solutions of type IIB SUGRA
solution generation techniques
EX TsT transformations
One may consider various kinds of integrable deformations.
There should be many associated solutions of type IIB SUGRA.
Motive
A classification of SUGRA solutions based on integrable deformations
4
A systematic deformation scheme
Yang-Baxter sigma model
Integrable deformation!
[Klimcik, 2002, 2008]
: const
R : a linear op.
a classical r-matrix
satisfying modified classical Yang-Baxter eq. (mCYBE)
The insertion of
is a characteristic of this method.
NOTE
• An integrable deformation is specified by a classical r-matrix.
• Given a classical r-matrix, a Lax pair follows automatically.
5
R-operators and classical r-matrices
A linear R-operator
A skew-symmetric classical r-matrix
Two sources of classical r-matrix
1) modified classical Yang-Baxter eq. (mCYBE)
the original work by Klimcik
2) classical Yang-Baxter eq. (CYBE)
a possible generalization
6
The list of generalizations of Yang-Baxter sigma models
(2 classes)
(i) modified classical Yang-Baxter eq. (trigonometric)
1) Principal chiral model
[Klimcik, hep-th/0210095, 0802.3518]
2) Symmetric coset sigma model
[Delduc-Magro-Vicedo, 1308.3581]
3) Type IIB superstring on AdS5xS5
[Delduc-Magro-Vicedo, 1309.5850]
(ii) classical Yang-Baxter eq. (rational)
NOTE
1) Principal chiral model
[Matsumoto-KY, 1501.03665]
2) Symmetric coset sigma model
[Matsumoto-KY, 1501.03665]
3) Type IIB superstring on AdS5xS5
[Kawaguchi-Matsumoto-KY, 1401.4855]
bi-Yang-Baxter sigma models are also constructed.
[Klimcik, 0802.3518, 1402.2105]
7
Integrable deformations of the AdS5 x S5 superstring
[Delduc-Magro-Vicedo, 1309.5850]
[Kawaguchi-Matsumoto-KY, 1401.4855]
Gravity/CYBE correspondence
solutions of type IIB SUGRA
classical r-matrices
Claim
The moduli space of a certain class of solutions of type IIB SUGRA
is described by the classical Yang-Baxter equation.
c.f., a droplet picture of free fermion in the bubbling geometry
Droplet
free fermion
classical r-matrix
CYBE
[Lin-Lunin-Maldacena, 2005]
(Just an analogy！)
One may construct a new bubbling-like scenario.
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The current status of the two deformations
1)
Standard q-deformation
← mCYBE
Talks by Arutyunov, Hoare
Classical r-matrix of Drinfeld-Jimbo type
Metric and B-field
[Delduc-Magro-Vicedo, 1309.5850]
[Arutyunov-Borsaro-Frolov, 1312.3542]
AdSn x Sn
[Hoare-Roiban-Tseytlin, 1403.5517]
RR fluxes and dilaton for AdS2 x S2 and AdS3 x S3
Towards the complete gravity solution
[Arutyunov-Borsaro-Frolov, 1507.04239]
Minimal surfaces, quark-antiquark potential
[Kameyama-KY, 1410.5544, 1510.NNNNN]
c.f., Drukker-Forini, 1105.5144
2)
[Lunin-Roiban-Tseytlin, 1411.1066]
[Hoare-Tseytlin, 1508.01150]
Poseter by Kameyama
Non-standard q-deformation (Jordanian deformations) ← CYBE
A lot of classical r-matrices
[Kawaguchi-Matsumoto-KY, 1401.4855]
i)
Partial deformations are possible (i.e., only AdS5 or only S5 )
ii)
Many r-matrices have been identified with solutions of type IIB SUGRA
[Kawaguchi-Matsumoto-KY, 1402.6147] [Matsumoto-KY, 1404.1838, 1404.3657, 1412.3658, 1502.00740]
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Jordanian deformation of the AdS5 x S5 superstring
[Kawaguchi-Matsumoto-KY,
1401.4855]
RJor satisfies CYBE.
The undeformed limit:
the Metsaev-Tseytlin action
[Metsaev-Tseytlin, hep-th/9805028]
• Lax pair is constructed
classical integrability
• Kappa invariance
a consistency as string theory
NOTE
This is a generalization of the work of Delduc-Magro-Vicedo for mCYBE.
The replacement: mCYBE → CYBE modifies Lax pair and kappa-transformation.
10
3 Examples:
i) 3-parameter gamma-deformations of S5
[Matsumoto-KY, 1404.1838]
Abelian r-matrix:
Metric:
B-field:
[Lunin-Maldacena, Frolov, 2005]
A special case:
Lunin-Maldacena background
Identification of parameters:
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ii) Gravity duals for SYM on non-commutative space
[Matsumoto-KY, 1404.3657]
Abelian Jordanian r-matrix:
where
,
,
[Hashimoto-Itzhaki, Maldacena-Russo, 1999]
Metric:
B-field:
Identification of parameters:
Note:
This background can be reproduced as a special limit of -deformed AdS5
[Arutyunov-Borsaro-Frolov, 1507.04239] [Kameyama-Kyono-Sakamoto-KY, 1509.00173]
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iii) Schrödinger spacetimes
[Matsumoto-KY, 1502.00740]
Mixed r-matrix:
Metric:
[Herzog-Rangamani-Ross, 0807.1099]
[Maldacena-Martelli-Tachikawa, 0807.1100]
B-field:
[Adams-Balasubramanian-McGreevy, 0807.1111]
S5 -coordinates:
NOTE:
Classical r-matrix for 3-parameter cases
[Bobev-Kundu, 0904.2873]
Classical r-matrix for general Melvin twists
[Bobev-Kundu-Pilch, 0905.0673]
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Other examples:
NC deformations of global AdS5
[McLaughlin-Swanson, hep-th/0605018]
[Dhokarh-Haque-Hashimoto, 0801.3812]
AdS pp-wave background
[Hubeby-Rangamani-Ross, hep-th/0504034]
Schrödinger AdS pp-wave
[Kawaguchi-Matsumoto-KY, 1402.6147]
[Matsumoto-KY, 1412.3658]
Dipole backgrounds
[Bergman-Dasgupta-Ganor-Karzmarek-Rajesh,
hep-th/0103090]
Remarks
1)
Solutions can also be derived as TsT transformations of AdS5 x S5.
TsT trans. are well described by Yang-Baxter deformations.
In fact, 3 examples presented here can be interpreted as
the undeformed theory with twisted periodic b.c.
c.f., Frolov, Alday-Arutyunov-Frolov,
2)
[Kameyama-Kyono-Sakamoto-KY, 1509.00173]
RR sector and dilaton have not been confirmed yet.
Schrödinger spacetimes are good candidates.
The dilaton is const. and the RR sector is the same as the usual AdS5 x S5 .
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Summary
Towards the gravity/CYBE correspondence
EX
Lunin-Maldacena-Frolov, Maldacena-Russo, Schrödinger spacetimes
Need to make efforts for the RR-sector and dilaton
Further generalizations
1) Application to non-integrable case:
EX
AdS5 x T1,1 :
non-integrability & chaos
[Basu-Pando Zayas, 1103.4107]
[Asano-Kawai-Kyono-KY, 1505.07583]
Yang-Baxter sigma models can capture deformations of T1,1 as well.
[Crichigno-Matsumoto-KY, 1406.2249]
2) Yang-Baxter deformations of Minkowski spacetime:
[Matsumoto-Orlando-Reffert-Sakamoto-KY, 1505.04553]
Relation to kappa-Poincare algebras
[Borowiec-Kyono-Lukierski-Sakamoto-KY, 1510.NNNNN]
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The limitation of the gravity/CYBE correspondence?
To what extent the correspondence can work?
What is the mathematical origin of Yang-Baxter deformations?
The associated deformations of N=4 super Yang-Mills theory?
It is significant to study the fundamental aspect of
Yang-Baxter deformations
Thank you!
16
Back up
17
Question
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-deformations
Gravity duals for NC gauge theories
AdS5 x T1,1
[Klebanov-Witten]
Klebanov-Strassler,
AdS BH
[Lunin-Maldacena, Frolov]
[Hashimoto-Itzhaki, Maldacena-Russo]
AdS5 x Yp,q
[Gauntlett-Martelli-Sparks-Waldram]
Maldacena-Nunez
[Horowitz-Strominger]
AdS solitons
[Witten, Horowitz-Myers]
AdS/NRCFT [Son,Balasubramanian-McGreevy, Kachru-Liu-Mulligan]
q-deformation of AdS5xS5
[Delduc-Magro-Vicedo, Arutyunov-Borsato-Frolov]
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A classification list
(would not be complete)
Integrable backgrounds
Real beta-deformations
[Frolov, hep-th/0503201]
Gravity duals for NC gauge theories
[Matsumoto-KY, 1403.2703]
q-deformation 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 transformatios of AdS5
Non-integrable backgrounds
Complex beta-deformations
AdS5 x T1,1
AdS BH
[Giataganas-Pando Zayas-Zoubos, 1311.3241]
[Basu-Pando Zayas, 1103.4107]
[Pando Zayas-Terrero Escalante, 1007.0277]
Klebanov-Strassler, Maldacena-Nunez
AdS5 x Yp,q
[Basu-Pando Zayas, 1105.2540]
AdS solitons
[Basu-Das-Ghosh, 1103.4101]
[Basu-Das-Ghosh-Pando Zayas, 1201.5634]
Schrödinger spacetime with
[Giataganas-Sfetsos, 1403.2703]
Lifshitz space (with hyper-scaling violation)
[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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A schematic picture
(to see the gravity/CYBE correspondence)
A classical r-matrix
from CYBE
A gravity solution
of type IIB SUGRA
correspondence
substitute
compare!
rewrite the action
& read off
Deformed AdS5xS5
superstring action
The metric (string frame)
&
NS-NS 2-form
Note: The metric & NS-NS 2-form have successfully been obtained so far.
However, further studies are necessary for the RR sector and dilaton,
though the kappa-symmetry may ensure that they can be reproduced as well.
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Jordanian deformation of the AdS5 x S5 superstring
RJor satisfies CYBE.
Maurer-Cartan 1-form
Projection on the group manifold
,
Projection on the world-sheet
21
Yang-Baxter deformations of 4D Minkowski spacetime
Exactly
solvable
Integrable
Integrable?
The results obtained so far support the integrability of YB-deformed action.
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