Paper
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Hidden Yangian symmetry in sigma model on squashed sphere
Authors
Io Kawaguchi,Kentaroh Yoshida
Naomichi Tani,Thomas Flesch,Stephen Neate,Andrzej Z Gorski,Andi Hektor,Zygmunt Lalak,Jerome Paillet,Theodora Ioannidou,Nerea Zabala Unzalu,Alfonso V. Ramallo,Ramaswamy Jagannathan,Min-Chun Ku,Michael Cockram,Jacob VanderPlas,Prahathees Eswara,Петр Мангилев,Jeffrey Harvey,Joachim Dengler,Anna Paszkowska,Martin Goodchild,Trevor Glasbey,Srinivasan Jagannathan,Saša Obradović,Raul Rosenvald,Michele Portolan,Garrett Poe,Josef Kraus,Kasper Peeters,Alain Jaccard,Carolina Abadía Quintero,Marc Torrens,Mikołaj Baranowski,Benjamin Bellenie,Jacek Domagała,Boban Dašić,Andy Van Brocklin,Luis Garcés-Erice,Bernard Gagnon,David Van Wie,Gianluca Calcagni,Vishnu Ram OV,Felipe Lumbreras,N/A ,Marie Connett,Aaron Balog,Silviu Pufu,Lorenzo Tancredi,Christopher Herzog,David Clarke,Anselmo Meregaglia,Andre Lukas,Dan Andersen,Fabio Maltoni,Joaquim Matias,Gaurav Sharma,Jan Holzer,Tadeusz Lulek,kanaka b,Piotr Stopa,Patricia Cleave,Lee Fitzgerald,Santiago Folgueras,Polly Livermore,Yasunori Nomura,Pei-Ming Ho,Piret Lõhmus,Maria Jesus Cocero,Phillip Szepietowski,Dean Coble,Sakura Schafer-Nameki,Michele Cicoli,Irene Valenzuela,Ingo Jordan,Ivonne Zavala,Nilanjan Brahma,Radu Popescu,Noah Goldberg,Shlomo Sergei Razamat,Mathew Bullimore,Gaurav Rattan,B. B. Levchenko,Prabhat Ranjan Pujahari,Michael Baker,Lukasz Kumiega,Antonio Padilla,Marco Drewes,Montserrat Armas Padilla,edouard alphandéry,Stepan Jenicek,Lionel Mason,Clare Burrage,Ulf Gran,Kyohei Mukaida,Raúl Emilio Real Delor,Nico Callewaert,Iosif Bena,Francesco Montanari,Jon Sertucha,Masahito Yamazaki,Carlo Angelantonj,Robert LoGalbo,Marinei Lopes Pedralli,Antonio Marti,David Wands,Harvey Reall,Andrew Zentner,Ashutosh Kumar Pandey,Alejandra Romero,Justyna Tomaszewska,Hao Liu,Fabian Schmidt,Cristian Randieri,Hayato Motohashi,Peter Arnold,Daniele Binosi,Jing Ren,Anibal Medina,Pau Novella,Di-Lun Yang,Nicholas Rodd,Donghui Jeong,Sumanta Chakraborty,Joshua Spitz,Ely Kovetz,Pedro Carrilho,Ivan Sipiran,Pedro Pérez,Benjamin Bustos,Chan Youn Park,Paul Sutcliffe,Olga Dyomina,Iván Barrios,Rosa Garcia,Simon Catterall,Enis Tuncer,Alexandre Barros,Sandra Schneider,Li Li,Monica Rodriguez Garcia-Risco,masataka nishikawa,Gavin Salam,Wilmar Alberto Cardona Castro,Takeshi Kobayashi,Carlos Urzúa,Aleksandr N. Ignatov,Valerie Domcke,Georgios Papathanasiou,Javier Tarrio,pedro candeias,Gerhard Kresbach,Shehu AbdusSalam,Haitang Yang,Michael Katharakis,Laura Andrianopoli,Ryo Saito,Joseph Conlon,Hanna Pawlowska,Antonio Amariti,Alejandro Ibarra,JOSE FRANCISCO DE ASSIS DIAS,Alessandro Minotti,Francisco A. 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arXiv:1008.0776v2 [hep-th] 11 Aug 2010
KUNS-2286
Hidden Yangian symmetry in sigma model
on squashed sphere
Io Kawaguchi∗ and Kentaroh Yoshida†
Department of Physics, Kyoto University
Kyoto 606-8502, Japan
Abstract
We discuss a hidden symmetry of a two-dimensional sigma model on a squashed
S3. The SU (2) current can be improved so that it can be regarded as a flat con-
nection. Then we can obtain an infinite number of conserved non-local charges and
show the Yangian algebra by directly checking the Serre relation. This symmetry is
also deduced from the coset structure of the squashed sphere. The same argument
is applicable to the warped AdS3 spaces via double Wick rotations.
∗E-mail: io@gauge.scphys.kyoto-u.ac.jp
†E-mail: kyoshida@gauge.scphys.kyoto-u.ac.jp
KUNS-2286
Hidden Yangian symmetry in sigma model
on squashed sphere
Io Kawaguchi∗ and Kentaroh Yoshida†
Department of Physics, Kyoto University
Kyoto 606-8502, Japan
Abstract
We discuss a hidden symmetry of a two-dimensional sigma model on a squashed
S3. The SU (2) current can be improved so that it can be regarded as a flat con-
nection. Then we can obtain an infinite number of conserved non-local charges and
show the Yangian algebra by directly checking the Serre relation. This symmetry is
also deduced from the coset structure of the squashed sphere. The same argument
is applicable to the warped AdS3 spaces via double Wick rotations.
∗E-mail: io@gauge.scphys.kyoto-u.ac.jp
†E-mail: kyoshida@gauge.scphys.kyoto-u.ac.jp
1 Introduction
It is well known that two-dimensional sigma models on symmetric spaces are classically
integrable [1]. Simple examples are the O(3) non-linear sigma model and the SU(2) prin-
cipal chiral model. The integrability of the sigma models survives at quantum level and
physical quantities such as S-matrix can be computed exactly [2,3]. Thus the integrability
allows us to study non-perturbative aspects of quantum field theory. There is a hidden
symmetry [4, 5], called the Yangian symmetry [6], behind the integrable structure.
The integrable structure on symmetric spaces may appear as classical integrability of
type IIB superstring on AdS5 × S5 [7] and now it has a new perspective and potential
applications in the AdS/CFT correspondence [8]. The integrable string backgrounds have
recently been classified based on symmetric coset structure [9]. Then an interesting issue
is to consider integrability and hidden symmetry of the sigma models on non-symmetric
spaces and discuss a generalization of the AdS/CFT dictionary. Simple examples of non-
symmetric spaces are squashed spheres and warped AdS spaces.
In this letter, as a simple example, we will discuss a hidden symmetry of a sigma
model on a squashed S3. This squashed geometry is realized as a U(1)-fibration (ψ) over
S2 (θ, φ) and the metric is given by
ds2 = L2
4
[ dθ2 + cos2 θ dφ2 + (1 + C)(dψ + sin θ dφ)2 ] . (1.1)
The constant parameter C measures a deformation of S3 and C = 0 just describes the
round S3 with the radius L . The SU(2)L × SU(2)R symmetry for the round S3 is broken
to SU(2)L ×U(1)R for C 6 = 0 . For C = 0 the Yangian symmetry is well known as a hidden
symmetry. But it remains to be clarified for C 6 = 0 whether or not the SU(2)L × U(1)R
can enhance to an infinite-dimenisonal symmetry.
2 Yangian symmetry
We start from the classical action of the sigma model:
S = −1
2
∫ ∫
dtdx
[
(∂µθ)2 + cos2 θ(∂µφ)2 + (1 + C)(∂µψ + sin θ∂µφ)2
]
. (2.1)
For simplicity, we will not take the Virasoro conditions into account and hence the base
space is assumed to be a two-dimensional Minkowski spacetime with the coordinates
xµ = (t, x) and the metric ηµν = (−1, 1) .
1
It is well known that two-dimensional sigma models on symmetric spaces are classically
integrable [1]. Simple examples are the O(3) non-linear sigma model and the SU(2) prin-
cipal chiral model. The integrability of the sigma models survives at quantum level and
physical quantities such as S-matrix can be computed exactly [2,3]. Thus the integrability
allows us to study non-perturbative aspects of quantum field theory. There is a hidden
symmetry [4, 5], called the Yangian symmetry [6], behind the integrable structure.
The integrable structure on symmetric spaces may appear as classical integrability of
type IIB superstring on AdS5 × S5 [7] and now it has a new perspective and potential
applications in the AdS/CFT correspondence [8]. The integrable string backgrounds have
recently been classified based on symmetric coset structure [9]. Then an interesting issue
is to consider integrability and hidden symmetry of the sigma models on non-symmetric
spaces and discuss a generalization of the AdS/CFT dictionary. Simple examples of non-
symmetric spaces are squashed spheres and warped AdS spaces.
In this letter, as a simple example, we will discuss a hidden symmetry of a sigma
model on a squashed S3. This squashed geometry is realized as a U(1)-fibration (ψ) over
S2 (θ, φ) and the metric is given by
ds2 = L2
4
[ dθ2 + cos2 θ dφ2 + (1 + C)(dψ + sin θ dφ)2 ] . (1.1)
The constant parameter C measures a deformation of S3 and C = 0 just describes the
round S3 with the radius L . The SU(2)L × SU(2)R symmetry for the round S3 is broken
to SU(2)L ×U(1)R for C 6 = 0 . For C = 0 the Yangian symmetry is well known as a hidden
symmetry. But it remains to be clarified for C 6 = 0 whether or not the SU(2)L × U(1)R
can enhance to an infinite-dimenisonal symmetry.
2 Yangian symmetry
We start from the classical action of the sigma model:
S = −1
2
∫ ∫
dtdx
[
(∂µθ)2 + cos2 θ(∂µφ)2 + (1 + C)(∂µψ + sin θ∂µφ)2
]
. (2.1)
For simplicity, we will not take the Virasoro conditions into account and hence the base
space is assumed to be a two-dimensional Minkowski spacetime with the coordinates
xµ = (t, x) and the metric ηµν = (−1, 1) .
1
The isometry of the metric (1.1) is SU(2)L × U(1)R and hence the action (2.1) is
invariant under the following transformations:
δ1 (φ, ψ, θ) = ǫ (−1 , 0 , 0) ,
δ2 (φ, ψ, θ) = ǫ
(
− sin φ tan θ , sin φ
cos θ , − cos φ
)
,
δ3 (φ, ψ, θ) = ǫ
(
cos φ tan θ , − cos φ
cos θ , − sin φ
)
,
δ4 (φ, ψ, θ) = ǫ (0 , − 1 , 0) .
Here ǫ is an infinitesimal constant. Note that the transformations are independent of
the squashing parameter C . The first three are the SU(2)L generators and the last is
the U(1)R one. By following the Noether’s theorem, we can construct the four conserved
currents
j1
µ = ∂µφ + sin θ ∂µψ + C sin θ (∂µψ + sin θ ∂µφ) ,
j2
µ = cos φ ∂µθ − sin φ cos θ ∂µψ
−C sin φ cos θ (∂µ ψ + sin θ ∂µφ) ,
j3
µ = sin φ ∂µθ + cos φ cos θ ∂µψ
+C cos φ cos θ (∂µψ + sin θ ∂µφ) ,
j4
µ = (1 + C)(∂µψ + sin θ ∂µφ) .
After some algebra, for the SU(2)L current jA
µ (A = 1, 2, 3), we obtain that
ǫµν
(
∂µjA
ν − 1
2 ε A
BC jB
µ jC
ν
)
= CnA ǫµν ∂µ(sin θ)∂ν φ
with an anti-symmetric tensor ǫµν normalized as ǫtx = +1 and nA is a unit vector on the
S2
nA ≡ (sin θ, − sin φ cos θ, cos φ cos θ) .
Thus the SU(2)L current does not satisfy the flatness condition. But it can be improved
with the ambiguity of the Noether current so that it satisfies the flatness condition. To
improve the current, the following topological current IA
µ has to be added to jA
µ :
IA
µ ≡ ±√Cǫµν ∂ν nA . (2.2)
This term is obviously conserved: ∂µIA
µ = 0 . We will discuss the geometrical meaning of
(2.2) later.
2
invariant under the following transformations:
δ1 (φ, ψ, θ) = ǫ (−1 , 0 , 0) ,
δ2 (φ, ψ, θ) = ǫ
(
− sin φ tan θ , sin φ
cos θ , − cos φ
)
,
δ3 (φ, ψ, θ) = ǫ
(
cos φ tan θ , − cos φ
cos θ , − sin φ
)
,
δ4 (φ, ψ, θ) = ǫ (0 , − 1 , 0) .
Here ǫ is an infinitesimal constant. Note that the transformations are independent of
the squashing parameter C . The first three are the SU(2)L generators and the last is
the U(1)R one. By following the Noether’s theorem, we can construct the four conserved
currents
j1
µ = ∂µφ + sin θ ∂µψ + C sin θ (∂µψ + sin θ ∂µφ) ,
j2
µ = cos φ ∂µθ − sin φ cos θ ∂µψ
−C sin φ cos θ (∂µ ψ + sin θ ∂µφ) ,
j3
µ = sin φ ∂µθ + cos φ cos θ ∂µψ
+C cos φ cos θ (∂µψ + sin θ ∂µφ) ,
j4
µ = (1 + C)(∂µψ + sin θ ∂µφ) .
After some algebra, for the SU(2)L current jA
µ (A = 1, 2, 3), we obtain that
ǫµν
(
∂µjA
ν − 1
2 ε A
BC jB
µ jC
ν
)
= CnA ǫµν ∂µ(sin θ)∂ν φ
with an anti-symmetric tensor ǫµν normalized as ǫtx = +1 and nA is a unit vector on the
S2
nA ≡ (sin θ, − sin φ cos θ, cos φ cos θ) .
Thus the SU(2)L current does not satisfy the flatness condition. But it can be improved
with the ambiguity of the Noether current so that it satisfies the flatness condition. To
improve the current, the following topological current IA
µ has to be added to jA
µ :
IA
µ ≡ ±√Cǫµν ∂ν nA . (2.2)
This term is obviously conserved: ∂µIA
µ = 0 . We will discuss the geometrical meaning of
(2.2) later.
2
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