AdS₄ x CP³ superstring as OSp(4|6)/(SO(1,3) x U(3)) sigma-model in conformal basis
Basic features of Lagrangian formulation for AdS₄ x CP³ superstring in the framework of OSp(4|6)/(SO(1,3) x U(3)) sigma-model approach are reviewed with the emphasis on realization osp(4|6) background isometry superalgebra as D = 3 N = 6 superconformal algebra. Обсуждается лагранжева динамика ІІА су...
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України
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Uvarov, D.V. 2016-10-10T07:47:15Z 2016-10-10T07:47:15Z 2012 AdS₄ x CP³ superstring as OSp(4|6)/(SO(1,3) x U(3)) sigma-model in conformal basis / D.V. Uvarov // Вопросы атомной науки и техники. — 2012. — № 1. — С. 32-36. — Бібліогр.: 21 назв. — англ. 1562-6016 PACS 2010: 11.15.Kc, 11.30.Pb https://nasplib.isofts.kiev.ua/handle/123456789/106976 Basic features of Lagrangian formulation for AdS₄ x CP³ superstring in the framework of OSp(4|6)/(SO(1,3) x U(3)) sigma-model approach are reviewed with the emphasis on realization osp(4|6) background isometry superalgebra as D = 3 N = 6 superconformal algebra. Обсуждается лагранжева динамика ІІА суперструны в подпространстве AdS₄ x CP³ суперпространства, изоморфном OSp(4|6)/(SO(1,3) x U(3)) фактор-многообразию. Ключевую роль при построении лагранжиана суперструны как классически-интегрируемой OSp(4|6)/(SO(1,3) x U(3)) сигма-модели играет Z4 дискретный автоморфизм osp(4|6) супералгебры изометрии AdS4 x CP3 суперпространства. Основное внимание уделяется представлению лагранжиана, следующих из него уравнений движения, а также связности Лакса через формы Картана для генераторов D = 3 N = 6 суперконформной алгебры. Обговорюється лагранжіва динаміка ІІА суперструни в підпросторі AdS₄ x CP³ суперпростору, ізоморфному OSp(4|6)/(SO(1,3) x U(3)) фактор-багатовиду. Ключову роль при побудові лагранжіана суперструни як класично-інтегровної OSp(4|6)/(SO(1,3) x U(3)) сигма-моделі відіграє Z4 дискретний автоморфізм osp(4|6) супералгебри ізометрії AdS4 x CP3 суперпростору. Основну увагу приділено зображенню лагранжіана, рівнянь руху, що з нього випливають, а також зв'язності Лакса через форми Картана для генераторів D = 3 N = 6 суперконформної алгебри. en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Вопросы атомной науки и техники Section A. Quantum Field Theory AdS₄ x CP³ superstring as OSp(4|6)/(SO(1,3) x U(3)) sigma-model in conformal basis AdS₄ x CP³ суперструна как OSp(4|6)/(SO(1,3) x U(3)) сигма-модель в конформном базисе AdS₄ x CP³ суперструна як OSp(4|6)/(SO(1,3) x U(3)) сигма-модель в конформному базисі Article published earlier |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| title |
AdS₄ x CP³ superstring as OSp(4|6)/(SO(1,3) x U(3)) sigma-model in conformal basis |
| spellingShingle |
AdS₄ x CP³ superstring as OSp(4|6)/(SO(1,3) x U(3)) sigma-model in conformal basis Uvarov, D.V. Section A. Quantum Field Theory |
| title_short |
AdS₄ x CP³ superstring as OSp(4|6)/(SO(1,3) x U(3)) sigma-model in conformal basis |
| title_full |
AdS₄ x CP³ superstring as OSp(4|6)/(SO(1,3) x U(3)) sigma-model in conformal basis |
| title_fullStr |
AdS₄ x CP³ superstring as OSp(4|6)/(SO(1,3) x U(3)) sigma-model in conformal basis |
| title_full_unstemmed |
AdS₄ x CP³ superstring as OSp(4|6)/(SO(1,3) x U(3)) sigma-model in conformal basis |
| title_sort |
ads₄ x cp³ superstring as osp(4|6)/(so(1,3) x u(3)) sigma-model in conformal basis |
| author |
Uvarov, D.V. |
| author_facet |
Uvarov, D.V. |
| topic |
Section A. Quantum Field Theory |
| topic_facet |
Section A. Quantum Field Theory |
| publishDate |
2012 |
| language |
English |
| container_title |
Вопросы атомной науки и техники |
| publisher |
Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| format |
Article |
| title_alt |
AdS₄ x CP³ суперструна как OSp(4|6)/(SO(1,3) x U(3)) сигма-модель в конформном базисе AdS₄ x CP³ суперструна як OSp(4|6)/(SO(1,3) x U(3)) сигма-модель в конформному базисі |
| description |
Basic features of Lagrangian formulation for AdS₄ x CP³ superstring in the framework of OSp(4|6)/(SO(1,3) x U(3)) sigma-model approach are reviewed with the emphasis on realization osp(4|6) background isometry superalgebra as D = 3 N = 6 superconformal algebra.
Обсуждается лагранжева динамика ІІА суперструны в подпространстве AdS₄ x CP³ суперпространства, изоморфном OSp(4|6)/(SO(1,3) x U(3)) фактор-многообразию. Ключевую роль при построении лагранжиана суперструны как классически-интегрируемой OSp(4|6)/(SO(1,3) x U(3)) сигма-модели играет Z4 дискретный автоморфизм osp(4|6) супералгебры изометрии AdS4 x CP3 суперпространства. Основное внимание уделяется представлению лагранжиана, следующих из него уравнений движения, а также связности Лакса через формы Картана для генераторов D = 3 N = 6 суперконформной алгебры.
Обговорюється лагранжіва динаміка ІІА суперструни в підпросторі AdS₄ x CP³ суперпростору, ізоморфному OSp(4|6)/(SO(1,3) x U(3)) фактор-багатовиду. Ключову роль при побудові лагранжіана суперструни як класично-інтегровної OSp(4|6)/(SO(1,3) x U(3)) сигма-моделі відіграє Z4 дискретний автоморфізм osp(4|6) супералгебри ізометрії AdS4 x CP3 суперпростору. Основну увагу приділено зображенню лагранжіана, рівнянь руху, що з нього випливають, а також зв'язності Лакса через форми Картана для генераторів D = 3 N = 6 суперконформної алгебри.
|
| issn |
1562-6016 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/106976 |
| citation_txt |
AdS₄ x CP³ superstring as OSp(4|6)/(SO(1,3) x U(3)) sigma-model in conformal basis / D.V. Uvarov // Вопросы атомной науки и техники. — 2012. — № 1. — С. 32-36. — Бібліогр.: 21 назв. — англ. |
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2025-11-25T22:46:26Z |
| last_indexed |
2025-11-25T22:46:26Z |
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| fulltext |
AdS4 × CP 3 SUPERSTRING AS OSp(4|6)/(SO(1, 3)× U(3))
SIGMA-MODEL IN CONFORMAL BASIS
D.V. Uvarov∗
National Science Center “Kharkov Institute of Physics and Technology”, 61108, Kharkov, Ukraine
(Received October 26, 2011)
Basic features of Lagrangian formulation for AdS4×CP 3 superstring in the framework of OSp(4|6)/(SO(1, 3)×U(3))
sigma-model approach are reviewed with the emphasis on realization of osp(4|6) background isometry superalgebra
as D = 3 N = 6 superconformal algebra.
PACS: 11.25.-w; 11.30.Pb; 11.25.Tq
1. INTRODUCTION
Application of relativistic string models to descrip-
tion of non-Abelian gauge theories has rather long
history going back to the dual resonance models [1]
and ’t Hooft approach [2]. The first explicit pro-
posal of duality between gauge theory and string the-
ory was put forward only in 1997 by Maldacena [3]
and intensely exploits the concept of supersymme-
try [4, 5]. Since then many other examples of gauge
fields/strings dualities were proposed (see, e.g. re-
view [6] and references therein), however none of
them has been proved from the first principles yet.
Even their verification appears a non-trivial task be-
cause of non-linearity of the theories involved. To
date pivotal role in testing gauge fields/strings duali-
ties is played by (super)symmetry considerations. For
that reason to the best explored cases belong highly
(super)symmetric ones [3, 7].
This note is devoted to reviewing some aspects
of classical string theory related to the Aharony-
Bergman-Jafferis-Maldacena (ABJM) duality [7] con-
jecturing that D = 3 N = 6 Chern-Simons-matter
theory admits in the ’t Hooft limit [2] dual descrip-
tion as the Type IIA superstring on AdS4 × CP 3
superbackground. This background is known [8] to
solve equations of IIA supergravity and preserves 24
of 32 space-time supersymmetries, i.e. correspond-
ing Killing spinor equation has 24 independent solu-
tions out of 32 maximally allowed. Background sym-
metry group is isomorphic to OSp(4|6) supergroup
that manifests itself as D = 3 N = 6 superconformal
symmetry of the action of dual Chern-Simons-matter
theory. This distinguishes ABJM duality from the
AdS5/CFT4 one [3] relating D = 4 N = 4 super-
Yang-Mills theory and the Type IIB string theory on
AdS5×S5 superbackground that is maximally super-
symmetric solution of IIB supergravity. PSU(2, 2|4)
symmetry of the AdS5×S5 superbackground matches
D = 4 N = 4 superconformal symmetry of the action
of super-Yang-Mills theory.
Symmetry arguments also governed construc-
tion of the classical action for AdS5 × S5 su-
perstring. It was observed [3] that both parts
of the background represent symmetric spaces
AdS5 = SO(2, 4)/SO(1, 4) and S5 = SO(6)/SO(5)
with isometry groups constituting bosonic subgroup
SO(2, 4)×SU(4) of the PSU(2, 2|4) supergroup and
the number of fermionic generators of PSU(2, 2|4)
equals 32 that is the Grassmann-odd dimension of
superspace. This hinted to identify AdS5 × S5 su-
perspace as the PSU(2, 2|4)/(SO(1, 4) × SO(5)) su-
percoset manifold and the AdS5 ×S5 superstring ac-
tion was constructed as the PSU(2, 2|4)/(SO(1, 4)×
SO(5)) supercoset sigma-model [9, 10]. It was then
found that the superstring Lagrangian can be pre-
sented as quadratic polynomial in Cartan forms as-
sociated with the psu(2, 2|4)/(so(1, 4) × so(5)) su-
percoset generators using their decomposition into
eigenspaces of the discrete Z4 automorphism of
psu(2, 2|4) superalgebra [11, 12]. Resulting action is
invariant under global PSU(2, 2|4) supersymmetry,
as well as gauge SO(1, 4)×SO(5) and κ−symmetries
and describes correct number of physical degrees of
freedom. Moreover, corresponding equations of mo-
tion are classically integrable and can be obtained
from the zero-curvature condition for the associated
Lax connection [13].
Above consideration can be generalized to the
AdS4 × CP 3 superstring case. Namely, AdS4 =
SO(2, 3)/SO(1, 3) and CP 3 = SU(4)/U(3) are sym-
metric spaces and their isometry groups can be
combined into bosonic subgroup SO(2, 3) × SO(6)
of the OSp(4|6) supergroup that also includes 24
fermionic generators equal in number to the su-
persymmetries preserved by AdS4 × CP 3 back-
∗E-mail address: uvarov@kipt.kharkov.ua
32 PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY, 2012, N 1.
Series: Nuclear Physics Investigations (57), p. 32-36.
ground. The OSp(4|6)/(SO(1, 3) × U(3)) super-
coset space has 10 space-time and 24 fermionic di-
rections lacking 8 fermionic directions to span full
AdS4 × CP 3 superspace. Nonetheless it is possible
to construct the OSp(4|6)/(SO(1, 3) × U(3)) sigma-
model-type action following the straight-forward
generalization of the prescription used to obtain
PSU(2, 2|4)/(SO(1, 4)×SO(5)) sigma-model [9–12].
In [14,15] there was presented the general structure of
the OSp(4|6)/(SO(1, 3)×U(3)) sigma-model includ-
ing corresponding equations of motion, κ−symmetry
transformation rules and the Lax connection. By
choosing the OSp(4|6)/(SO(1, 3) × U(3)) supercoset
representative explicit form of the Lagrangian was
found to the second order in the world-sheet fields and
checked against known Penrose limit Lagrangian [16].
In Ref. [17] we have presented the
OSp(4|6)/(SO(1, 3)×U(3)) sigma-model-type action
in conformal basis for Cartan forms relying on the
osp(4|6) superalgebra realization as the D = 3 N = 6
superconformal algebra. The proof was also given in
the SO(1, 2) × SU(3) covariant way that among 24
fermionic equations of motion there are only 16 inde-
pendent for non-singular superstring configurations
implying 8-parameter κ−symmetry of the action.
Besides that we have obtained explicit form of the
sigma-model-type Lagrangian to all orders in the
space-time and anticommuting coordinates for the
OSp(4|6)/(SO(1, 3) × U(3)) representative compat-
ible with conformal structure. Such a choice allows
to formulate stringy side of the duality in terms of
coordinates that contain those parametrizing D = 3
N = 6 boundary superspace, where the ABJM gauge
theory [7] can be formulated aiming at getting new
insights into the relation between both theories.
The OSp(4|6)/(SO(1, 3)×U(3)) supercoset space
is the subspace of the full-fledged AdS4 × CP 3 su-
perspace. Hence the OSp(4|6)/(SO(1, 3) × U(3))
sigma-model-type action can be obtained by par-
tial κ−symmetry gauge fixing of the complete
AdS4 × CP 3 superstring action [18]. It is amenable
to describe all possible classical string configura-
tions, in particular those that cannot be consid-
ered within the OSp(4|6)/(SO(1, 3) × U(3)) sigma-
model [14, 18], however it does not correspond to a
supercoset sigma-model and the issue of its integra-
bility remains open [19].
2. OSp(4|6)/(SO(1, 3) × U(3)) SIGMA-MODEL
IN CONFORMAL BASIS
The osp(4|6) superalgebra g of AdS4 × CP 3 super-
background is isomorphic to D = 3 N = 6 supercon-
formal algebra spanned by
g =
⎧⎨
⎩
Mmn, Pm, Km, D
Va
b, Va, V a
Qa
μ, Q̄μa, Sμa, S̄μ
a
⎫⎬
⎭ . (1)
Generators in the first line are that of D = 3 con-
formal group, the second line contains generators of
the SU(4) ∼ SO(6) isometry group of CP 3 mani-
fold. Remaining fermionic generators have been di-
vided into two sets of 12 associated with Poincare
and conformal supersymmetries. We adhere to the
notations of [17], namely small Latin letters from
the middle of the alphabet k, l, m, n = 1, 2, 3 label
SO(1, 2) vectors, while that from the beginning of the
alphabet a, b, c = 1, 2, 3 label objects transforming in
(anti)fundamental representation of SU(3) ⊂ SU(4).
Small Greek letters μ, ν, λ stand for 2-component
spinor indices of Spin (1, 2). (Anti)commutation re-
lations of D = 3 N = 6 superconformal algebra can
be found in [17].
Crucial role in constructing OSp(4|6)/(SO(1, 3)×
U(3)) sigma-model-type Lagrangian is played by Z4
automorphism Ω of the osp(4|6) superalgebra under
which all the generators divide into 4 eigenspaces
with different eigenvalues
g = g0 ⊕ g1 ⊕ g2 ⊕ g3 : Ω(gk) = ikgk. (2)
Invariant eigenspace g0 is spanned by so(1, 3) ⊕ u(3)
stability algebra generators. g2 Eigenspace contains
so(2, 3)/so(1, 3)⊕ su(4)/u(3) coset generators and g1
and g3 – fermionic ones. In terms of the genera-
tors of D = 3 N = 6 superconformal algebra these
eigenspaces can be realized as
g0 = {Mmn, Pm − Km, Va
b},
g2 = {D, Pm + Km, Va, V a},
g1 = {Qa
μ + iSa
μ, Q̄μa − iS̄μa},
g3 = {Qa
μ − iSa
μ, Q̄μa + iS̄μa}. (3)
Left-invariant Cartan forms C(d) in conformal ba-
sis admit the decoposition:
C(d) = G−1dG = ωm(d)Pm+ cm(d)Km+ Δ(d)D
+ Gmn(d)Mmn + Ωa(d)Va + Ωa(d)V a
+ Ωa
b(d)Vb
a + ωμ
a (d)Qa
μ + ω̄μa(d)Q̄μa
+ χμa(d)Sμa + χ̄a
μ(d)S̄μ
a , (4)
where G ∈ OSp(4|6)/(SO(1, 3) × U(3)) is a super-
coset representative. Above division of the osp(4|6)
generators under Ω automorphism induces corre-
sponding division of Cartan forms
C(d) = C0(d) + C2(d) + C1(d) + C3(d) (5)
with individual summands defined as
C0(d) = 1
2 (ωm(d) − cm(d))(Pm − Km)
+ Gmn(d)Mmn + Ωa
b(d)Vb
a; (6)
C2(d) = 1
2 (ωm(d) + cm(d))(Pm + Km)
+ Δ(d)D + Ωa(d)V a + Ωa(d)Va; (7)
C1(d) = 1
2 (ωμ
a (d) + iχμ
a(d))(Qa
μ + iSa
μ)
+ 1
2 (ω̄μa(d) − iχ̄μa(d))(Q̄μa − iS̄μa); (8)
C3(d) = 1
2 (ωμ
a (d) − iχμ
a(d))(Qa
μ − iSa
μ)
+ 1
2 (ω̄μa(d) + iχ̄μa(d))(Q̄μa + iS̄μa). (9)
33
Global OSp(4|6) transformations act on the super-
coset representative according to the rule
LG = G′H, L ∈ OSp(4|6) (10)
with H ∈ SO(1, 3) × U(3) being the compensating
coordinate-dependent rotation under which Cartan
forms from eigenspaces 1,2 and 3 transform homoge-
neously
C′
1,2,3 = HC1,2,3H
−1, (11)
whereas C0 Cartan forms transform as SO(1, 3) ×
U(3) connection
C′
0 = HC0H
−1 − H−1dH. (12)
The OSp(4|6) and Z4−invariant sigma-model-
type action [14] is constructed out of Cartan forms
C1,2,3 according to the general prescription [11–13].
In conformal basis for Cartan forms it is brought to
the form [17]
S = − 1
2
∫
d2ξ
√−ggij
[
1
4 (ωm
i + cm
i )(ωjm + cjm)
+ ΔiΔj + 1
2 (ΩiaΩa
j + ΩjaΩa
i )
]
+ SWZ (13)
with the Wess-Zumimo action given by
SWZ = − 1
4εij
∫
d2ξ
[
(ωμ
ia + iχμ
ia)εμν(ω̄νa
j + iχ̄νa
j )
+ (ωμ
ia − iχμ
ia)εμν(ω̄νa
j − iχ̄νa
j )
]
= − 1
2εij
∫
d2ξ
(
ωμ
iaεμν ω̄νa
j +χiμaεμν χ̄a
jν
)
. (14)
The construction implies identification of Cartan
forms ωm(d) + cm(d), Δ(d) and Ωa(d), Ωa(d) with
tangent to AdS4 and CP 3 components of the su-
pervielbein of OSp(4|6)/(SO(1, 3)×U(3)) supercoset
manifold. Analogously Cartan forms ωμ
a (d) + iχμ
a(d),
ωμ
a (d)− iχμ
a(d) and c.c. are identified with the fermi-
onic components of supervielbein.
Equations of motion resulting from variation of
the supervielbein bosonic components tangent to
AdS4 and CP 3 parts of the background read
∂i(
√−ggij(ωm
j + cm
j )) + 2
√−ggijGmn
i (ωjn
+cjn) + 2
√−ggij(cm
i − ωm
i )Δj
+iεij(ωμ
ia + iχμ
ia)σm
μν(ω̄νa
j − iχ̄νa
j )
−iεij(ωμ
ia − iχμ
ia)σm
μν(ω̄νa
j + iχ̄νa
j ) = 0, (15)
∂i(
√−ggijΔj) + 1
2
√−ggij(ωm
i − cm
i )(ωjm
+cjm) + 1
2εij(ωμ
ia + iχμ
ia)εμν(ω̄νa
j − iχ̄νa
j )
+ 1
2εij(ωμ
ia − iχμ
ia)εμν(ω̄νa
j + iχ̄νa
j ) = 0 (16)
and
∂i(
√−ggijΩa
j ) + i
√−ggijΩb
i(Ωjb
a + δa
b Ωjc
c)
− i
2εijεabc(ωμ
ib + iχμ
ib)εμν(ων
jc + iχν
jc)
− i
2εijεabc(ωμ
ib − iχμ
ib)εμν(ων
jc − iχν
jc) = 0 (17)
respectively. Similarly fermionic equations of motions
can be cast into the form
V ij
+ [(σmμν(ωm
i + cm
i ) + 2iεμνΔi)(ων
ja + iχν
ja)
+2εabcΩb
i(ω̄
μc
j − iχ̄μc
j )] = 0 (18)
for Cartan forms from the C1 eigenspace (8) and
V ij
− [(σmμν(ωm
i + cm
i ) − 2iεμνΔi)(ων
ja − iχν
ja)
−2εabcΩb
i(ω̄
μc
j + iχ̄μc
j )] = 0 (19)
for those from the C3 eigenspace (9) and c.c. equa-
tions. V ij
± = 1
2 (
√−ggij ± εij) are projectors al-
lowing to split world-sheet vectors and tensors into
(anti)self-dual parts. Common feature of the Green-
Schwarz superstring models is that not all fermi-
onic equations are independent. This implies via
the second Noether theorem invariance of the su-
perstring action under the κ−symmetry. In [17]
we gave the proof that in generic case among 24
equations (18), (19) there are only 16 indepen-
dent leading to the 8-parameter κ−symmetry of
the OSp(4|6)/(SO(1, 3) × U(3)) sigma-model-type
action [14]. Such an 8-parameter κ−symmetry is
the ‘remnant’ of 16-parameter symmetry of the full
AdS4 ×CP 3 superstring [18] remained upon gauging
away 8 fermions related to space-time supersymme-
tries broken by the AdS4×CP 3 superbackground. It
should be noted that the above set of equations has to
be supplemented by Virasoro conditions arising upon
variation of (13) on auxiliary 2d metric gij :
1
4 (ωm
i + cm
i )(ωjm + cjm) + ΔiΔj + 1
2 (ΩiaΩa
j
+ΩjaΩa
i )− 1
2gijg
i′j′[ 1
4 (ωm
i′ +cm
i′ )(ωj′m+cj′m)
+Δi′Δj′ + Ωi′aΩa
j′
]
= 0. (20)
Equations of motion (15)-(19) can be obtained
from the zero curvature condition
dL − L ∧ L = 0 (21)
for the Lax connection 1-form L(d) taking value in
the osp(4|6) isometry algebra of the AdS4 ×CP 3 su-
perbackground [14]. Construction of the connection
follows the same steps used to discover Lax repre-
sentation for the AdS5 × S5 superstring equations of
motion [13]. In conformal basis for Cartan forms it
can be written as
L = Lconf3 + Lsu(4) + LF , (22)
where
Lconf3(d) = GmnMmn+ 1
2 (cm − ωm)(Km − Pm)
+ 1
2 [�1(ωm + cm) + �2
∗(ωm + cm)](Pm
+ Km)+(�1Δ+�2
∗Δ)D∈so(2, 3) (23)
and
Lsu(4)(d) = Ωa
bVb
a+Ωa
aVb
b+(�1Ωa+�2
∗Ωa)Va
+ (�1Ωa + �2
∗Ωa)V a ∈ su(4) (24)
are Lax connections for bosonic string on AdS4
and CP 3 manifolds respectively extended by the
34
contributions of 24 anticommuting coordinates of
OSp(4|6)/(SO(1, 3) × U(3)) supercoset space. 2d
Hodge dual of a 1-form a(d) is defined as ∗ai =√−gεijg
jkak. LF is determined by the fermionic gen-
erators and Cartan forms:
LF (d) = �3
1
2 (ωμ
a + iχμ
a)(Qa
μ + iSa
μ)
+ �4
1
2 (ωμ
a − iχμ
a)(Qa
μ − iSa
μ)+c.c. (25)
Checking (21) requires using the Maurer-Cartan
equations for Cartan forms (4) which explicit form
is given in [17]. Also the zero curvature condition
(21) restricts parameters �1,..., �4 to be functions of
a single spectral parameter. One of their possible
parametrizations is as follows:
�1 = 1
2
(
z2 +
1
z2
)
, �2 = 1
2
(
1
z2
− z2
)
,
�3 = z, �4 =
1
z
, (26)
where z is assumed to be complex-valued non-zero.
The Lax connection is defined modulo OSp(4|6)
gauge transformations L′ = GLG−1 − G−1dG. Spe-
cial role is played by such a transformation with
G = G, where G is same as used to define Cartan
forms in (4). Then the transformed Lax connection
can be power series expanded in w = −2 log z around
zero:
L′ = w ∗J + O(w2), (27)
where the leading contribution is given by the Hodge
dual of osp(4|6) superalgebra-valued Noether current
density J associated with OSp(4|6) global symmetry
of the superstring action (13). Complete explicit form
of the Noether currents corresponding to realization
of the OSp(4|6) global symmetry as D = 3 N = 6
superconformal symmetry was obtained in [20].
To obtain explicit form of OSp(4|6)/(SO(1, 3) ×
U(3)) superstring action we have considered the
OSp(4|6)/(SO(1, 3)× U(3)) supercoset element [17]
G = exp (xmPm + θμ
aQa
μ + θ̄μaQ̄μa) exp (ημaSμa
+ η̄a
μS̄μ
a ) exp (zaVa + z̄aV a) expϕD, (28)
parametrized by D = 3 N = 6 super-Poincaré coordi-
nates (xm, θμ
a , θ̄μa), AdS4 radial direction coordinate
ϕ, 3 complex coordinates (za, z̄a) of the CP 3 mani-
fold, and 12 anticommuting coordinates (ημa, η̄a
μ) as-
sociated with D = 3 N = 6 conformal supersymme-
try. Corresponding expressions for Cartan forms and
superstring Lagrangian were derived in [17]. Similar
choice of the supercoset representative was consid-
ered in [21] when studying the AdS5×S5 superstring
in conformal basis for Cartan forms.
3. CONCLUSIONS
We have outlined Lagrangian formulation of AdS4 ×
CP 3 superstring as the OSp(4|6)/(SO(1, 3) ×
U(3)) sigma-model. The procedure behind the
OSp(4|6)/(SO(1, 3) × U(3)) sigma-model construc-
tion [14, 15] is similar to that used previously
to describe the AdS5 × S5 superstring as the
PSU(2, 2|4)/(SO(1, 4)×SO(5)) sigma-model [9–13].
The main distinction is that the osp(4|6) superal-
gebra has 24 supersymmetry generators in contrast
to 32 generators of the psu(2, 2|4) superalgebra. It
is traced back to the fact that AdS4 × CP 3 su-
perbackground preserves 24 space-time supersymme-
tries, whereas the AdS5 × S5 one preserves all 32
space-time supersymmetries. As a consequence the
OSp(4|6)/(SO(1, 3)×U(3)) sigma-model-type action
describes the subsector of the AdS4 × CP 3 super-
string [18] that does not take into account super-
symmetries broken by the background. Important
common feature of both supercoset sigma-models is
that corresponding equations of motion are integrable
and the Lax representation for them is known explic-
itly [13–15].
Aiming to elaborate a representation for the
OSp(4|6)/(SO(1, 3) × U(3)) sigma-model-type La-
grangian that most of all fits symmetry structure of
the dual gauge theory [7] we presented it in conformal
basis for Cartan forms [17]. To this end we worked
out the osp(4|6) superalgebra realization as the D = 3
N = 6 superconformal algebra. There was also
given general derivation of the rank of matrices enter-
ing fermionic equations of motion and κ−symmetry
transformations. Explicit expressions for the osp(4|6)
Cartan forms and OSp(4|6)/(SO(1, 3)×U(3)) sigma-
model-type Lagrangian were found for the supercoset
representative compatible with conformal structure.
For the D = 3 N = 6 superconformal symme-
try of the Lagrangian complete expressions for the
Noether currents were obtained [20]. This results
hopefully will be of use in addressing such issues of
ABJM correspondence as the spectrum identification,
T−duality invariance, Wilson loops/scattering am-
plitudes duality.
References
1. J.H. Schwarz. Dual resonance theory // Phys.
Rept. 1973, v. 4, p. 269-335; Superstring theory
// Phys. Rept. 1982, v. 89, p. 223-322.
2. G. ’t Hooft. A planar diagram theory for strong
interactions // Nucl. Phys. 1974, v. B72, p. 461-
473.
3. J.M. Maldacena. The large N limit of supercon-
formal field theories and supergravity // Adv.
Theor. Math. Phys. 1998, v. 2, p. 231-252.
4. Yu.A. Gol’fand and Ye.P. Likhtman. Extension
of the algebra of Poincare group generators and
violation of P invariance // JETP Lett. 1971,
v. 13, p. 323-326 [Pisma v ZhETF. 1971, v. 13,
p. 452-455].
5. D.V. Volkov and V.P. Akulov. Possible universal
neutrino interaction // JETP Lett. 1972, v. 16,
35
p. 438-440 [Pisma v ZhETF. 1972, v. 16, p.621-
624]; Is the Neutrino a Goldstone Particle? //
Phys. Lett. 1973, v. B46, p. 109-110.
6. K. Peeters and M. Zamaklar. The string/gauge
theory correspodence in QCD // Eur. Phys. J.
2007, v. 152, p. 113-138.
7. O. Aharony, O. Bergman, D.L. Jafferis and
J. Maldacena. N = 6 superconformal Chern-
Simons-matter theories, M2-branes and their
gravity duals // JHEP. 2008, iss. 10, 091.
8. S. Watamura. Spontaneous compactification and
Cp(N): SU(3)×SU(2)×U(1), sin2 θW , g(3)/g(2)
and SU(3) triplet chiral fermions in 4 dimensions
// Phys. Lett. 1984, v. B136, p. 245-250.
9. R.R. Metsaev and A.A. Tseytlin. Type IIB su-
perstring action in AdS5 × S5 background //
Nucl. Phys. 1998, v. B533, p. 109-126.
10. R. Kallosh, J. Rahmfeld, and A. Rajaraman.
Near horizon superspace // JHEP. 1998, iss. 09,
002.
11. N. Berkovits et al. Superstring theory on AdS2×
S2 as a coset supermanifold // Nucl. Phys. 2000,
v. B567, p. 61-86.
12. R. Roiban and W. Siegel. Superstrings on
AdS5 × S5 supertwistor space // JHEP. 2000,
iss. 11, 024.
13. I. Bena, J. Polchinski, and R. Roiban. Hidden
symmetries of the AdS5×S5 superstring // Phys.
Rev. 2004, v. D69, 046002.
14. G. Arutyunov and S. Frolov. Superstrings on
AdS4 × CP 3 as a Coset Sigma-model // JHEP.
2008, iss. 09, 129.
15. B.J. Stefanski. Green-Schwarz action for Type
IIA strings on AdS4 ×CP 3 // Nucl. Phys. 2009,
v. B808, p. 80-87.
16. T. Nishioka and T. Takayanagi. On Type IIA
Penrose limit and N = 6 Chern-Simons theories
// JHEP. 2008, iss. 08, 001.
17. D.V. Uvarov. AdS4×CP 3 superstring and D = 3
N = 6 superconformal symmetry // Phys. Rev.
2009, v. D79, 106007.
18. J. Gomis, D. Sorokin, and L. Wulff. The com-
plete AdS4×CP 3 superspace for type IIA super-
string and D−branes // JHEP. 2009, iss. 03, 015.
P.A. Grassi, D. Sorokin, and L. Wulff. Simplify-
ing superstring and D−brane actions in AdS4 ×
CP 3 superbackground // JHEP. 2009, iss. 08,
060.
19. D. Sorokin and L. Wulff. Evidence for the clas-
sical integrability of the complete AdS4 × CP 3
superstring // JHEP. 2010, iss. 11, 143.
20. D.V. Uvarov. D = 3 N = 6 superconformal sym-
metry of AdS4×CP 3 superstring // Class. Quan-
tum Grav. 2011, v. 28, 235010.
21. R.R. Metsaev and A.A. Tseytlin. Superstring
action in AdS5 × S5: κ−symmetry light cone
gauge // Phys. Rev. 2001, v. D63, 046002.
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