Yangian symmetry in deformed WZNW models on squashed...
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Yangian symmetry in deformedWZNW models on squashed spheres
Kentaroh Yoshida (Kyoto Univ.)
I. Kawaguchi, D. Orlando and K.Y., arXiv: 1104.0738.
I. Kawaguchi and K.Y., JHEP 1011 (2010) 032 [arXiv:1008.0776]
seminar@IPMU, 2011/05/24
Based on
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Introductory part
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Introduction
No rigorous proof but enormous amount of evidence support this conjecture.
One of the most studied subjects in string theory.
AdS/CFT correspondence
= duality between string (gravity) on AdS space and CFT
1. Integrability in AdS/CFT
2. Applications of AdS/CFT to condensed matter physics
Two topics in AdS/CFT (related to my talk)
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1. Integrability in AdS/CFT
IIB string on AdS5 x S5 4D N=4 SYM theory
Integrable spin chain
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Integrable structure of spin chain Spacetime structure of AdS5 x S5
symmetric coset
[Frolov-Tseytlin, Bena-Polchinski-Roiban, …+ [Minahan-Zarembo, BMSZ, …+
[Beisert-Staudacher, …+
The current understanding
IIB string on AdS5 x S5 4D N=4 SYM theory [Maldacena,1997]
AdS/CFT correspondence
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What is a symmetric coset?
Consider a coset .
: Lie algebra of Gdecompose
is called symmetric coset, when and satisfy the following relations:
Then
, ,
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Lie algebra of HOrtho-complement space
: symmetric coset
AdS background
G = SO(2,4) x SO(6) (global isometry), H = SO(1,4) x SO(5) (local Lorentz)
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2D NLSM is classically integrable, if its target space is given by symmetric coset.
FACT
EX. O(3)-inv. NLSM, SU(2) principal chiral model [Lüscher-Pohlmeyer, 1978]
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Classification of symmetric cosets applicable to AdS/CFT [K. Zarembo, 1003.0465]
(based on the grading-property, vanishing beta-function, )
Symmetric coset classical integrability
= an infinite number of conserved charges
Yangian symmetry (infinite dim. )
In particular
Integrability plays an important role in AdS/CFT
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2. Applications of AdS/CFT to condensed matter physics
Let us consider the gravitational description of condensed matter systems
by assuming the AdS/CFT correspondence.
“Holographic condensed matter physics” (often called AdS/CMP)
e.g. Entanglement entropy, superconductor, (non) Fermi liquid, etc.
Motivation to consider non-relativistic field theories in AdS/CMP
Non-relativistic AdS/CFT
Most of condensed matter systems are non-relativistic.
EX. Schrödinger systems, Lifshitz field theories
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EX. Schrödinger spacetime, Lifshitz spacetime
[S.Schafer-Nameki, M. Yamazaki, K.Y., 0903.4245]
There is a motive to consider non-symmetric cosets in AdS/CMP.
The gravitational background is modified from AdS
NOTE: these backgrounds are described by non-symmetric cosets
Sch.
Lif. z: const.
No one knows whether the NLSM on them are integrable or not, at least so far.
(Probably, it should be a difficult task)
[Son, Balasubramanian-McGreevy] [Kachru-Liu-Mulligan]
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Our aim Find out some examples of non-symmetric and integrable backgrounds
squashed spheres, warped AdS spaces (3 dim.)
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Summary of my talk
1. Yangian symmetry is realized for the squashed S3 .
2. This is the case even after the Wess-Zumino term has been added.
3. RG flow of the squashed WZNW model
IR fixed point is the same as the SU(2) WZW model
We discuss an infinite-dimensional symmetry of NLSMs on squashed spheres.
Motivated by these topics,
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Technical part
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Plan of my talk
2. Yangian symmetry in squashed sigma model
1. BIZZ construction
4. Summary and Discussions
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3. Squashed WZNW model
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1. BIZZ construction
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If a 2D NLSM has a conserved current satisfying the flatness condition,
then an infinite number of conserved non-local charges can be constructed
and the NLSM is classically integrable.
[Brezin,Itzykson,Zinn-Justin,Zuber,1979]
There are some methods, one of which is the BIZZ construction.
FACT2: There exists always a flat conserved current,
if the target space is G itself or M is a symmetric coset.
FACT
KEY INGREDIENT: flat conserved current
(= a conserved current satisfying the flatness condition)
Let us consider a 2D NLSM on G or M=G/H (not necessarily symmetric)
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BIZZ construction
Let’s introduce the covariant derivative:
[Brezin, Itzykson, Zinn-Justin, Zuber, 1979]
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satisfies:
With the covariant derivative, one can construct an infinite number
of non-local charges recursively.
Assume that we have a flat conserved current
NOTE If there is a flat conserved current, then M is not needed to be symmetric.
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Let’s take the Noether current as the 0th current :
Conserved by definition.
Then the next current is defined as
Repeat the same step Infinite number of non-local charges
:conserved
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Non-local charges:
Non-local
Explicit expressions of the charges:
0-th
1-st
where ’s are the generators of G :
What is the algebra that the charges satisfy ?
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Current algebra
Yangian algebra *Drinfel’d,1988+
+ Serre relations
fixed by the classical action
NOTE: Yangian is generated by and .
[a typical form, e.g. O(3)-inv. NLSM]
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Serre relations
S1.
S2.
where
only for A,B,C
:symmetrized product
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2. Yangian symmetry
in squashed sigma model
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I. Kawaguchi and K.Y., JHEP 1011 (2010) 032, [arXiv:1008.0776]
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Isometry:
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a deformation of the round S3
squashing parameter
-fibration
Isometry:
Round S3 with the radius L
Squashed S3
3 angles
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Warped AdS3 a double Wick rotation of squashed S3
2) time-like warped AdS3 :
1) space-like warped AdS3 :
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,
Hereafter we will consider the case of squashed spheres only.
The difference between warped AdS3 and squashed S3 is just signature
at least at classical level.
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Group element representation of squashed sphere
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Let us introduce the SU(2) group element:
Here are the angles of S3 and TA’s are the SU(2) generators:
Then the left-invariant 1-form is expanded as
Finally the metric of squashed S3 is rewritten as
,
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Sigma model action on squashed S3
Global Symmetry:
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SU(2)L Noether current:
Conserved equation of motion
: 2D Minkowski spacetime
NOTE We will not consider the Virasoro conditions
coupling const.
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If the conserved current satisfies the flatness condition,
then an infinite number of conserved non-local charges can be constructed.
BIZZ construction
Check the flatness for the SU(2)L current :
Non-vanishing, because . But total derivative!
Remember
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Ambiguity to add a topological term
Improvement term
New flatness condition:
Current improvement
If we take ,
then the improved current satisfies the flatness condition.
An infinite number of non-local charges (straightforward)
const.
(assume )
BIZZ
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Current algebra
The current algebra is deformed due to the improvement.
Is Yangian algebra still realized?
(non-trivial question)
with (The symbol “imp” is omitted below)
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SU(2)L Yangian algebra
Non-trivial modification only for this part.
Serre relations are also satisfied, although the current algebra is modified.
In summary,
Yangian algebra is realized even after S3 has been squashed.
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Classical r-matrix?
Yangian rational type of r-matrix (naïve expectation)
Lax pair:
Problem: How can we read off the classical r-matrix?
But it’s possible in 3D Schrodinger case (Io’s talk in lunch seminar)
: spectral parameter
Poisson bracket
due to e.o.m. and flat condition
because of
(up to non-ultra local term)
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3. squashed WZNW model
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I. Kawaguchi, D. Orlando, K.Y., arXiv: 1104.0738.
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A simple generalization is to add the Wess-Zumino term.
The classical action
The coefficient is discretized from the consistency to the path integral.
Squashed Wess-Zumino-Novikov-Witten model (SqWZNW model)
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SU(2)L current:
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Check the flatness condition:
Improved current:
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In particular, when
the current improvement is not needed!
Flat current condition
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Current algebra with
The current algebra is fairly modified!
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Yangian algebra
Yangian algebra is realized even after adding the WZ term
+ Serre relations are also satisfied
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Renormalization Group (RG) flow
Renormalized coupling and squashing parameter at 1-loop level
1-loop beta functions:
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Critical surfaceIR fixed point of SU(2) WZW
n=20
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A=0
A=0
A2 > 0
A2 > 0
A2 < 0
A2 < 0
SU(2)L WZW IR fixed pt.
The relation between RG flow and current improvement
On the red line, a flat conserved current is obtained without improvement.
SU(2)L Yangian
SU(2)L Yangian
SU(2)L x SU(2)R Yangian
enhanced
SU(2)L Yangian?
SU(2)L Yangian?
A2<0 is formally possible SU(2)L Yangian?
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4. Summary & Discussions
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Summary
Yangian symmetry is realized for the squashed S3 .
This is the case even after the Wess-Zumino term has been added.
RG flow of the squashed WZNW model
IR fixed point is the same as the SU(2) WZW model
Discussions
• Quantum non-local charges?
• String theory embedding?
An exact marginal deformation of heterotic string background
Check whether anomaly is forbidden from the coset structure.
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[Israel, Kounnas, Orlando, Petropoulos, hep-th/0405213]
[Goldschmidt-Witten]
• How about the SU(2)R symmetry? [Orlando-Reffert-Uruchurtu, 1011.1771]
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Thank you!