2014/03/04``Progress in the synthesis of integrabilities arising from gauge‐string duality’’

Recent progress on q‐deformations
q deformationsof the AdS5 x S5 superstring

Kentaroh Yoshida (Kyoto Univ.)
In collaboration withIo Kawaguchi (Kyoto Univ.) and Takuya Matsumoto (Utrecht Univ.)
Refs 1401.4855
Refs.
1401 4855 and 1402.6147
1402 6147
1

0.

Introduction

An overview of integrability behind AdS/CFT
A strategy to consider integrable deformations2A recent progress in string theory
The discovery of integrability behind AdS/CFTThe current understanding[Beisert‐Staudacher, …]Integrable
teg ab e spin
sp chain,c a ,Bethe ansatz
IIB string on AdS5 x S54D

[Frolov‐Tseytlin, Bena‐Polchinski‐Roiban, Arutyunov‐Frolov…]

The integrable structure

Next step:pSU(N) SYM[Minahan‐Zarembo, BMSZ, …]

the structure of AdS5 x S5

Integrableg
deformations of AdS/CFT3

NOTE

AdS/CFT belongs to the rational class (like XXX‐model)

There are some kinds of integrable deformationsEXbeta‐deformations, gamma‐deformations

These are based on TsT transformations and closed within the rational class

Our interest is a trigonometric‐type deformation of AdS/CFT(like XXZ‐model)

to understand the wider structure behind AdS/CFTc.f.,

XXX model is nothing but a degenerate limit of XXZ model.
XXZ model has a richer structure than XXX model.

EX phase transition

The study of it may reveal some new aspects of AdS/CFT (Our motive)
4Strategy to consider trigonometric deformationsAq
q‐deformed symmetry
yy is a characteristic of the trigonometricgclass.
EX

q‐deformation of su(2)[Drinfeld, Jimbo,1985]

By taking the limit,the su(2) algebra is reproduced.

Three approaches:Spin chain side1) q‐deformed
d fd SS‐matrixti[Beisert,Koroteev,Galleas,Matsumoto,Hoare‐Hollowood‐Miramontes,A t
Arutyunov‐ded Leeuw‐vanLTTongeren]]Mathematically done, but the geometry is unknown

String theory sidea little understood2) q‐deformed geometry
3D is easy, the very recent progress on 10D[Cherednik, Balog‐Forgacs‐Palla,Kawaguchi‐Matsumoto‐KY,Basso‐Rej, Orlando‐Reffert‐Uruchurtu, Klimcik
Delduc‐Magro‐Vicedo, Arutyunov‐Borsato‐Frolov]

Gauge theory side3) deformed N=4 SYM??
Complex beta deformation?
[Berenstein‐Cherkis, Correa, Lunin‐Maldacena]5In this talk, we will concentrate on the string theory side.
• A summary of the recent progress on q‐deformations of the AdS5 x S5 superstring• AnA iintroductiont d ti tot our recentt works,
k 1401.4855
1401 4855 & 1402
1402.6147.
6147
Io Kawaguchi will talk about them.

The contents of this talk1.

Yang‐Baxter sigma models

2.

q‐deformations of the AdS5 x S5 superstring

3.

Summary and Discussion

61. Yang‐Baxter sigma models• A review of squashed S3 and its higher‐dimensional generalization• An introduction of modified Yang‐Baxter eq. & linear R‐operators
C. Klimcik, hep‐th/0210095, 0802.3518
Kawaguchi‐KY 1107.3662,
Kawaguchi‐KY,
1107 3662 Delduc‐Magro‐Vicedo,
Delduc‐Magro‐Vicedo 1308.3581
1308 3581
7

A simple example of deformed sigma models ‐ squashed S3 sigma models
What is squashed S3 ?
Round S3 with the radius L
3 anglesl‐fibrationfIsometry:

a deformation of round S3

C=0Squashed S3Isometry:squashing parameter8

Sigma model action on squashed S3[Cherednik, 1981](): 2D Minkowski spacetime,Symmetry

hybrid integrability[Kawaguchi‐Matsumoto‐KY, 1203.3400]

SU(2)L symmetry
ttwot SU(2)L Yangians
Y iequivalence[Kawaguchi‐KY, 1008.0776]

U(1)R symmetry

q‐deformed SU(2)Rquantum affine algebra[Kawaguchi‐KY, 1107.3662][Kawaguchi‐Matsumoto‐KY, 1201.3058]

This plays an important role in studying the integrability of Yang‐Baxter sigma models
9Yang‐Baxter sigma models‐ the principal chiral model (PCM) case ‐Let us start from a G‐principal chiral model, whose symmetry is
The deformed PCM action.

[C. Klimcik, hep‐th/0210095]: deformation
parameterp:a solution of modified classical Yang‐Baxter eq. (mCYBE)

A solution of mCYBE

An integrable deformation

Symmetry
NOTE

The construction of Lax pairs[C. Klimcik, 0802.3518]

The enhancement to q
q‐deformed[Kawaguchi‐KY, 1107.3662,
Delduc Magro Vicedo 1308.3581]
Delduc‐Magro‐Vicedo,
1308 3581]

Possibly, quantum affine algebra (Not yet shown)

10R‐operator & modified classical Yang‐Baxter eq.
St t ffrom
Start

Yang‐Baxter eq.

Expanding the R‐matrix asClassical rr‐matrix
matrixClassical Yang‐Baxter eq. (CYBE)

With an r‐matrix, one may introduce
A linear R‐operator

11

Then the CYBE can be rewritten as(CYBE)
if and only if• the bilinear form is non‐degenerate• the classical r‐matrix is skew‐symmetric

Then the modified classical Yang‐Baxter eq. (mCYBE) is given by(We setby rescaling).
12

An example of R‐operator for squashed S3Consider the su(2) algebra :The Drinfeld‐Jimbo type r‐matrix:
satisfies mCYBE (unique solution)

The resulting action:

13

Computation:
We first introduce
and expandpthem as

Then J can be rewritten as

Thus we obtain
Thus the action is computed as

14

The symmetric coset case

[Delduc‐Magro‐Vicedo, 1308.3581]

One may consider the Yang‐Baxter extension for symmetric coset M=G/H.
The decomposition
pof Lie algebra:g((direct pproduct as vector spaces)
p)

0 00,

00 10,

11 11,0The deformed coset action,The symmetry
q‐deformation
q deformation of G

[Delduc Magro Vicedo 1308.3581][Delduc‐Magro‐Vicedo,1308 3581]
152.

q‐deformations ofthe AdS5 x S5 superstring• A review of the AdS5 x S5 superstring (MT)• A q‐deformation of the AdS5 x S5 superstring (DMV)• The explicit metric (ABF)• Jordanian deformations (KMY)16

The AdS5 x S5 backgroundThe symmetric coset exhibits the Z2‐grading.

I l di ffermions
Includingi

The super coset:
The Z4‐grading ensures the classical integrability[Bene‐Polchinski‐Roiban, 2003]

Decomposition
pof su(2,2|4)( , | )

These are used to construct the Green‐Schwarz (GS) string action on AdS5 x S5 .
17[Metsaev‐Tseytlin, hep‐th/9805028][For a nice review, see 0901.4937]

The GS action of the AdS5 x S5 superstring

hwhereNOTE,• By
B taking
ki a parameterizationi i off theh group element,lthe coordinate system can be introduced.
• The Lax pair is constructed.
There are many works on the classical integrability.
18

A q‐deformation of the AdS5 x S5 superstring

where[Delduc‐Magro‐Vicedo,PRL122(2014)051601, 1309.5850],NOTE• R‐operation:• kappa‐invariant• theth LaxL pairi hash beenb
constructedt t d19

The deformed metric

whereNOTE(in the string frame)[Arutyunov‐Borsato‐Frolov, 1312.3542]

is a deformation parameter.
thish metric is singular.
l20

The NS‐NS 2‐form

NS‐NS 2‐form is turned on both the AdS and sphere parts.

NOTE:

the other components of type IIB SUGRA have NOT been fixed yet.

The world‐sheet S‐matrix
The perturbative world‐sheet S‐matrix of the bosonic particles have been computed.
It exactly agrees the q‐deformed S‐matrix(in the large string tension limit)

21Another kind of integrable deformations[Kawaguchi‐Matsumoto‐KY, 1401.4855]

There are two types of q‐deformations

1)

Standard q‐deformation (Drinfeld‐Jimbo type)2)

Non‐standard q‐deformation (Jordanian deformations)

Delduc‐Magro‐Vicedo, Arutyunov‐Borsato‐Frolov

1)standard

Kawaguchi Matsumoto KY
Kawaguchi‐Matsumoto‐KY

2)

Jordanian

Advantages of Jordanian deformations
i)

Partial deformations are possible (for example, only the AdS5 part)ii)

A type IIB SUGRA solution has been found[Kawaguchi‐Matsumoto‐KY, 1402.6147]

In the next talk22

3. Summary and Discussion

23Summary
An overview of the recent progress on
q‐deformationsq
deformations of the AdS5 x S5 superstring
Yang‐Baxter sigma model description

Future problems(for the standard q‐deformation)

SUGRA solutions

find out dilaton and R‐R fluxes
singularity is resolved?

One‐loop beta function?
For Jordanian deformations of the string action & type IIB SUGRA solution,
Io Kawaguchi’s talk24Thank you!

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