towards the gravity/cybe...

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2016/02/11 Quantum Spacetime’16@Hyrny, Zakopame, Poland

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,

Yuhma Asano, Daisuke Kawai, Andrzej Borowiec, Jerzy Lukierski

0. Introduction

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• background and motivation

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

The integrable structure provides a non-perturbative method to analyze.

It enables us to check the conjectured relations even at finite coupling.

EX anomalous dimensions, amplitudes etc.

Integrability is so powerful!

Indeed, there are many directions of study on the integrability.

the classical integrability on the string-theory side.

The existence of Lax pair (kinematical integrability)

Here, among them, we are concerned with

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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 [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

Integrable deformations

(as exactly solvable models) Solutions of type IIB SUGRAexpect

deformed AdS5 x S5 geometries

Integrability techniques solution generation techniques

EX Integrable twists EX TsT transformations

correspond?

One may consider various kinds of integrable deformations.

There should be many associated solutions of type IIB SUGRA.(This is our observation)

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1. Yang-Baxter defomrations

Along this line, we should follow a systematic way of integrable deformations.

Yang-Baxter deformations

3. Non-standard q-deformations of the AdS5 x S5 superstring

4. Summary & Discussion

The content of this talk

2. The standard q-deformations of the AdS5 x S5 superstring (review)

1. Yang-Baxter deformations

• A brief review of Yang-Baxter deformations

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• An outline of the gravity/CYBE correspondence

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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 deformations [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

Yang-Baxter sigma modelG-principal chiral model

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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

Example: A Yang-Baxter deformation of S3

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Consider the su(2) algebra :

The resulting action:

The unique non-trivial sol. of mCYBE

The r-matrix of Drinfeld-Jimbo type

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The detail of the computation:

Let us first introduce

Then J can be rewritten as

Thus we obtain

The deformed action can be evaluated as

and expand them like

Introducing a coordinate system

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Let us introduce the SU(2) group element:

Here are the angles of S3 . Then J is expanded as

Finally, the metric of squashed S3 is rewritten as

where

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The list of generalizations of Yang-Baxter deformations

(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]


[Delduc-Magro-Vicedo, 1308.3581]

NOTE bi-Yang-Baxter deformation are also possible [Klimcik, 0802.3518, 1402.2105]

(2 classes)

(only for principal chiral models)

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

Integrable deformations of type IIB superstring on AdS5 x S5

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

Conjecture

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！)

solutions of type IIB SUGRA classical r-matrices

Gravity/CYBE correspondence

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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.

But, further studies are necessary for the RR sector and dilaton.

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

The characteristics & current status of the two deformations


1) Standard q-deformation ( -deformation ) ← mCYBE

2) Non-standard q-deformations ( 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


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[Delduc-Magro-Vicedo, 1309.5850]A string action with Drinfeld-Jimbo classical r-matrix

A lot of classical r-matrices

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

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

[Arutyunov-Frolov-Hoare-Roiban-Tseytlin, 1511.05795]Modified SUGRA conjecture

[Arutyunov-Borsato-Frolov, 1507.04239]Super coset construction

A short review of these works (Sec. 2)

[S.J. van Tongeren, 1506.01023]

A brief summary (Sec. 3)

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2. The standard q-deformation

of the AdS5 x S5 superstring

Green-Schwarz action of the AdS5 x S5 superstring[Metsaev-Tseytlin, hep-th/9805028]

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• a coordinate system is introduced via a parameterization of

• Lax pair is constructed classical integrability

[For a nice review, see Arutyunov-Frolov, 0901.4937]

NOTE

,

Maurer-Cartan 1-form Projection op. on the supercoset

Projection on the world-sheet

The standard q-deformed action [Delduc-Magro-Vicedo, PRL122 (2014) 051601, 1309.5850]

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Difference：

• the construction of Lax pair classical integrability

• kappa-symmetry

R-operator:

NOTE：

,: deformation parameter

[Delduc-Magro-Vicedo, 1406.6286]Symmetry q-deformed PSU(2,2|4)

A q-deformation of AdS5 x S5 [Arutyunov-Borsato-Frolov, 1312.3542]

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Singularity!

NS-NS 2-form is turned on both AdS and S

sometimes called -deformation of AdS5 x S5

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The singularity surface

The geometry of the q-deformed AdS5

The UV part is drastically deformed,

while the IR part is not modified.

The geometry around the origin is nothing but AdS5 .

A UV deformation of N=4 SYM?Physical region

Unphysical region

e.g. an omega-deformation

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NOTE: The other fields of type IIB SUGRA have been determined recently.

[Arutyunov-Borsato-Frolov, 1507.04239]Not satisfy type IIB SUGRA EOM!

However, it satisfies a modified type IIB SUGRA EOM!

The RR-field strength is generalized to contain the 2nd order derivatives.

Furthermore, the string world-sheet theory exhibits scale invariance.

The q-deformation deforms type IIB SUGRA itself, rather than a solution?

[Arutyunov-Frolov-Hoare-Roiban-Tseytlin, 1511.05795]

(but not Weyl invariance)

A naked singular surface exists, and the world-sheet beta function doesn’t vanish.

The modified gravity/modified CYBE correspondence?

Some pathologies:

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3. Non-standard q-deformationsof the AdS5 x S5 superstring



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

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Jordanian deformations of the AdS5xS5 superstring

,

Maurer-Cartan 1-form Projection on the group manifold

Projection on the world-sheet


• Construction of Lax pair

• kappa-invarianceAccording to the replacement: mCYBE → CYBE, Lax pair and kappa-trans. are modified.

NOTE：

RJor satisfies CYBE.

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Examples of classical r-matrices & backgrounds

i) 3-parameter gamma-deformations of S5

A special case: Lunin-Maldacena background

Metric:

Abelian r-matrix:

Identification of parameters:

B-field:


[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:


Metric:

B-field:

Note: This background can also be reproduced as a scaling limit of q-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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iv) A new background [S.J. van Tongeren, 1506.01023]

Non-abelian r-matrix:

A deformed AdS5 geometry

NOTE: A slice at is represented by T-dual of 4D de Sitter space.

It can be described as Yang-Baxter deformation of 4D Minkowski spacetimewith the same classical r-matrix.

c.f., another derivation as a limit of q-deformed AdS5 [Pachol and van Tongeren, 1510.02389]

[Matsumoto-Orlando-Reffert-Sakamoto-KY, 1505.04553]

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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


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


[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) Some solutions can also be derived as TsT transformations of AdS5 x S5.

TsT trans. are well described by Yang-Baxter deformations.

In fact, the first 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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4. Summary & Discussion

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Summary

Towards the gravity/CYBE correspondence

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

1) Application to non-integrable cases:

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

[Crichigno-Matsumoto-KY, 1406.2249]

Yang-Baxter deformations can capture TsT-transformations 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, 1510.03083]Relation to kappa-Poincare algebras[Kyono-Sakamoto-KY, 1510.03083]

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Exactly Solvable

Yang-Baxter deformations of 4D Minkowski spacetime

Integrable

Integrable?

[Matsumoto-Orlando-Reffert-Sakamoto-KY, 1505.04553]

[Borowiec-Kyono-Lukierski-Sakamoto-KY, 1510.03083]

[Kyono-Sakamoto-KY, 1510.03083]

A general geometry unifying T-duals of (A)dS4 and pp-wave

A classical r-matrix for general kappa-deformations of Poincare algebra

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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?

What is the associated deformations of N=4 super Yang-Mills?

As a matter of course, it would be significant to consider the fundamental aspects of Yang-Baxter deformations.

There are many open problems!

Stijn’s talk!

towards the gravity/cybe...

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