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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

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type IIB string on AdS5 x S5 4D SU(N) SYM

The AdS/CFT correspondence

Recent progress: the discovery of integrability behind the AdS/CFT

The integrable structure provides powerful tools.

It enables us to check the conjectured relations in the AdS/CFT without SUSY, even at finite coupling.

the classical integrability on the string-theory side.

EX anomalous dimensions, amplitudes etc.

An enormous amount of works have been done, so far.

The existence of Lax pair (kinematical integrability)

[For a big review, Beisert et al., 1012.3982]

Motivated by this success, our interest here is

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The coset structure of AdS5 x S5 is closely related to the integrability.

Z2-grading 　　　　　　　 classical integrability

: symmetric coset

The classical integrability of the AdS5 x S5 superstring

Including fermions

: super coset

Z4-grading 　　　　　　　 classical integrability elucidated by [Bena-Polchinski-Roiban, 2003]

This fact is the starting point of our later argument.

[Metsaev-Tseytlin, 1998]

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Integrable deformations of the AdS5 x S5 superstring

A classification of SUGRA solutions based on integrable deformations

Motive

The next issue

One may consider various kinds of integrable deformations.

There should be many associated solutions of type IIB SUGRA.

Integrability techniques solution generation techniques

EX Integrable twists EX TsT transformations

Integrable deformations deformed AdS5 x S5 geometries

Solutions of type IIB SUGRA(as exactly solvable models)

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A systematic deformation scheme

• An integrable deformation is specified by a classical r-matrix.

satisfying modified classical Yang-Baxter eq. (mCYBE)

R : a linear op. 　　　　　　　 a classical r-matrix

Yang-Baxter sigma model [Klimcik, 2002, 2008]

Integrable deformation!

: const

• Given a classical r-matrix, a Lax pair follows automatically.

The insertion of is a characteristic of this method.

NOTE

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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

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The list of generalizations of Yang-Baxter sigma models

(i) modified classical Yang-Baxter eq. (trigonometric)

(ii) classical Yang-Baxter eq. (rational)

1) Principal chiral model

2) Symmetric coset sigma model

3) Type IIB superstring on AdS5xS5

1) Principal chiral model

2) Symmetric coset sigma model

3) Type IIB superstring on AdS5xS5

[Delduc-Magro-Vicedo, 1309.5850]

[Kawaguchi-Matsumoto-KY, 1401.4855]

[Klimcik, hep-th/0210095, 0802.3518]

[Matsumoto-KY, 1501.03665]

[Matsumoto-KY, 1501.03665]

[Delduc-Magro-Vicedo, 1308.3581]

NOTE 　 bi-Yang-Baxter sigma models are also constructed. [Klimcik, 0802.3518, 1402.2105]

(2 classes)

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solutions of type IIB SUGRA 　 　　　 　 classical r-matrices

Gravity/CYBE correspondence

[Delduc-Magro-Vicedo, 1309.5850][Kawaguchi-Matsumoto-KY, 1401.4855]

Integrable deformations of the AdS5 x S5 superstring

The moduli space of a certain class of solutions of type IIB SUGRA is described by the classical Yang-Baxter equation.

Claim

c.f., a droplet picture of free fermion in the bubbling geometry [Lin-Lunin-Maldacena, 2005]

One may construct a new bubbling-like scenario.

Droplet classical r-matrixfree fermion CYBE (Just an analogy ！ )

The current status of the two deformations

[Kawaguchi-Matsumoto-KY, 1401.4855]

1) Standard q-deformation ← 　 mCYBE 　　 　

2) Non-standard q-deformation (Jordanian deformations) ← CYBE

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]

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[Delduc-Magro-Vicedo, 1309.5850]Classical r-matrix of Drinfeld-Jimbo type

A lot of classical r-matrices

[Matsumoto-KY, 1404.1838, 1404.3657, 1412.3658, 1502.00740]

Metric and B-field [Arutyunov-Borsaro-Frolov, 1312.3542]

RR fluxes and dilaton for AdS2 x S2 and AdS3 x S3 [Lunin-Roiban-Tseytlin, 1411.1066]

[Kameyama-KY, 1410.5544, 1510.NNNNN]Minimal surfaces, quark-antiquark potential

Talks by Arutyunov, Hoare

c.f., Drukker-Forini, 1105.5144 Poseter by Kameyama

Towards the complete gravity solution [Arutyunov-Borsaro-Frolov, 1507.04239][Hoare-Tseytlin, 1508.01150]

AdSn x Sn [Hoare-Roiban-Tseytlin, 1403.5517]

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Jordanian deformation of the AdS5 x S5 superstring [Kawaguchi-Matsumoto-KY, 1401.4855]

• Lax pair is constructed classical integrability

• Kappa invariance a consistency as string theory

RJor satisfies CYBE.

The undeformed limit: the Metsaev-Tseytlin action[Metsaev-Tseytlin, hep-th/9805028]

The replacement: mCYBE → CYBE modifies Lax pair and kappa-transformation.

This is a generalization of the work of Delduc-Magro-Vicedo for mCYBE.

NOTE

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3 Examples:

i) 3-parameter gamma-deformations of S5

A special case: Lunin-Maldacena background

Metric:

Abelian r-matrix:

Identification of parameters:

B-field:

[Matsumoto-KY, 1404.1838]

[Lunin-Maldacena, Frolov, 2005]

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ii) Gravity duals for SYM on non-commutative space

[Hashimoto-Itzhaki, Maldacena-Russo, 1999]

Abelian Jordanian r-matrix:

where , ,

Identification of parameters:

Note:

[Matsumoto-KY, 1404.3657]

Metric:

B-field:

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]

[Bobev-Kundu, 0904.2873]

Metric:

B-field:

Mixed r-matrix:

[Herzog-Rangamani-Ross, 0807.1099][Maldacena-Martelli-Tachikawa, 0807.1100]

[Adams-Balasubramanian-McGreevy, 0807.1111]

S5 -coordinates:

Classical r-matrix for 3-parameter cases

[Bobev-Kundu-Pilch, 0905.0673]Classical r-matrix for general Melvin twists

NOTE:

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AdS pp-wave background

Other examples: NC deformations of global AdS5

[Hubeby-Rangamani-Ross, hep-th/0504034]

[Dhokarh-Haque-Hashimoto, 0801.3812]

Dipole backgrounds

[Kawaguchi-Matsumoto-KY, 1402.6147]

[Bergman-Dasgupta-Ganor-Karzmarek-Rajesh, hep-th/0103090]

[Matsumoto-KY, 1412.3658]

[McLaughlin-Swanson, hep-th/0605018]

Schrödinger AdS pp-wave

2) 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 .

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, [Kameyama-Kyono-Sakamoto-KY, 1509.00173]

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Summary

Towards the gravity/CYBE correspondence

EX 　　 Lunin-Maldacena-Frolov, Maldacena-Russo, Schrödinger spacetimes

1) Application to non-integrable case:

EX AdS5 x T1,1 : non-integrability & chaos 　　　

[Crichigno-Matsumoto-KY, 1406.2249]

Yang-Baxter sigma models can capture deformations of T1,1 as well.

[Basu-Pando Zayas, 1103.4107]

2) Yang-Baxter deformations of Minkowski spacetime:[Matsumoto-Orlando-Reffert-Sakamoto-KY, 1505.04553]

[Asano-Kawai-Kyono-KY, 1505.07583]

Further generalizations

Need to make efforts for the RR-sector and dilaton

[Borowiec-Kyono-Lukierski-Sakamoto-KY, 1510.NNNNN]Relation to kappa-Poincare algebras

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Thank you!

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

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Back up

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How prevalent is integrability in various kinds AdS/CFTs?

NOTE: Integrable subsectors are ubiquitous. EX integrable subsectors in large N QCD in 4 D

Question

There are various kinds of AdS/CFT

Real (or complex) beta-deformations [Lunin-Maldacena, Frolov]

Gravity duals for NC gauge theories [Hashimoto-Itzhaki, Maldacena-Russo]

AdS5 x T1,1 [Klebanov-Witten]

AdS BH [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]

Klebanov-Strassler, Maldacena-Nunez

AdS5 x Yp,q [Gauntlett-Martelli-Sparks-Waldram]

So we will concentrate on the full integrability below.

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A classification list

Integrable backgrounds

Non-integrable backgrounds

[Bai-Chen-Lee-Moon, 1406.5816]

Real beta-deformations

Complex beta-deformations [Giataganas-Pando Zayas-Zoubos, 1311.3241]

Gravity duals for NC gauge theories

AdS5 x T1,1 [Basu-Pando Zayas, 1103.4107]

AdS solitons [Basu-Das-Ghosh, 1103.4101]AdS BH [Pando Zayas-Terrero Escalante, 1007.0277]

AdS5 x Yp,q [Basu-Pando Zayas, 1105.2540]

[Matsumoto-KY, 1403.2703]

[Frolov, hep-th/0503201]

(would not be complete)

Lifshitz space (with hyper-scaling violation)

[Giataganas-Sfetsos, 1403.2703]

p-brane backgrounds [Stepanchuk-Tseytlin, 1211.3727] [Chervonyi-Lunin, 1311.1521]

q-deformation of AdS5 x S5[Delduc-Magro-Vicedo, 1309.5850, Arutyunov-Borsato-Frolov, 1312.3542]

Schrödinger spacetime with [Giataganas-Sfetsos, 1403.2703]

TsT transformatios of AdS5 [Kawaguchi-Matsumoto-KY, 1401.4855]

[Hubeny-Rangamani-Ross , hep-th/0504034][Dhokarh-Haque-Hashimoto , 0801.3812]

Klebanov-Strassler, Maldacena-Nunez 　　　　　　 [Basu-Das-Ghosh-Pando Zayas, 1201.5634]

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Deformed AdS5xS5

superstring action

substitute

from CYBE

A schematic picture

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.

The metric (string frame)

& NS-NS 2-form

rewrite the action & read off

A gravity solutionof type IIB SUGRA

compare!

correspondence

(to see the gravity/CYBE correspondence)

A classical r-matrix

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Jordanian deformation of the AdS5 x S5 superstring

,

Maurer-Cartan 1-form Projection on the group manifold

Projection on the world-sheet

RJor satisfies CYBE.

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Exactly solvable

Yang-Baxter deformations of 4D Minkowski spacetime

Integrable

Integrable?

The results obtained so far support the integrability of YB-deformed action.