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text:
D V Uvarov

Aspects of Classical Dynamics and Symmetries of the AdS 4 × CP 3 Superstring

This article, based on author's works, reviews various aspects of classical dynamics and symmetries of the Type IIA superstring in the AdS 4 × CP 3 superspace with the emphasis on isomorphic realization of the orthosymplectric OSp(4|6) isometry of this superspace as the D = 3 N = 6 superconformal symmetry.

In the sector of superstring dynamics described by the two-dimensional

model its Lagrangian is expressed in terms of the Cartan one-forms associated with the generators of the D = 3 N = 6 superconformal algebra. Explicit expressions for the osp(4|6) Cartan forms in the conformal basis are obtained for the OSp(4|6)/(SO(1, 3) × U (3)) representative parametrized by coordinates associated with the generators of superconformal algebra. Also found are variations of the Cartan forms and OSp(4|6)/(SO(1, 3) × U (3)) coordinates under the global D = 3 N = 6 superconformal symmetry and derived are the Noether current densities associated with the σ-model action invariance under this symmetry. It is also derived the superstring Lagrangian in the κ-symmetry light-cone gauge correponding to the light cone on the conformal boundary of the anti-de Sitter space in the Poincaré coordinates. Then after fixing the light-cone gauge for the reparametrization symmetry it is derived the superstring Hamiltonian that depends only on physical world-sheet fields.

The second part is devoted to study of the integrability of equations of the AdS 4 × CP 3 superstring. In the partial κ-symmetry gauge, in which in the sector of broken supersymmetries there remain non-zero two coordinates associated with broken Poincaré supersymmetries, the superstring equations are presented in the form of the zero-curvature condition for the world-sheet one-form that extends the Lax connection

It is proved that in the general case of the superstring motion both in the AdS 4 and CP 3 spaces the equations, following from the action variation on the coordinates of the sector of broken supersymmetries, become the consequences of other fermionic equations, when these coordinates are gauged away by the κ-symmetry.

Then it is proved that equations of the models of massless superparticle in the AdS 4 × CP 3 superspace and its OSp(4|6)/(SO(1, 3) × U (3)) supercoset subspace are classically integrable. Also it is established the connection between the Lax pair components of the superparticle and the Lax connection of the σ-model in the OSp(4|6)/(SO(1, 3) × U (3)) supercoset space. As a generalization of these results it is proved integrability of the D0-brane equations in the AdS 4 × CP 3 superbackground.

3.3.3 Classical integrability of equations of the D0-brane in the AdS 4 × CP 3 superbackground . . . . . . .

Introduction

The AdS/CF T correspondence [START_REF] Maldacena | The large N limit of superconformal field theories and supergravity[END_REF], [START_REF] Gubser | Gauge theory correlators from noncritical string theory[END_REF], [START_REF] Witten | Anti-de Sitter space and holography[END_REF] being the most symmetric instance of the gauge fields/strings duality is presently considered as the most elaborated way to define quantum gravity in the anti-de Sitter spaces in the framework of superstring theory [START_REF] Cicoli | String cosmology: from the early Universe to today[END_REF].

To date the best explored instance of the AdS/CF T correspondence provides dual description of the D = 10 Type IIB string theory on the AdS 5 × S 5 superbackground with N units of the Ramond-Ramond five-form flux as the D = 4 N = 4 supersymmetric Yang-Mills theory with U (N ) gauge symmetry [START_REF] Maldacena | The large N limit of superconformal field theories and supergravity[END_REF].

Later in [START_REF] Aharony | N = 6 superconformal Chern-Simons-matter theories, M2-branes and their gravity duals[END_REF] another explicit example of the AdS/CF T duality was suggested stating that the D = 3 N = 6 superconformal Chern-Simons-matter theory named ABJM theory gives dual description of the D = 10 Type IIA string theory on the AdS 4 × CP 3 superbackground. This IIA supergravity background is known since the mid of 80-s [START_REF] Watamura | Spontaneous compactification and CP N : SU (3)×SU (2)×U (1), sin 2 θ W , g 3 /g 2 and SU (3)-triplet chiral fermions in four dimensions[END_REF], [START_REF] Nilsson | Hopf fibration of eleven dimensional supergravity[END_REF], [START_REF] Sorokin | Kaluza-Klein theories and spontaneous compactification mechanisms of extra space dimensions[END_REF] and preserves 24 of 32 supersymmetries. Together with the SO(2, 3) × SU (4) symmetry of the AdS 4 × CP 3 space-time they form the OSp(4|6) isometry supergroup of the AdS 4 × CP 3 superspace. The fact that this superspace is non-maximally supersymmetric, as opposed to the AdS 5 × S 5 one, makes the ABJM correspondence more difficult to investigate.

On the gravity side of this correspondence applicability of the two-dimensional σmodel [START_REF] Arutyunov | Superstrings on AdS 4 × CP 3 as a coset sigma-model[END_REF], [START_REF] Stefański | Green-Schwarz action for Type IIA strings on AdS 4 × CP 3[END_REF] description of the AdS 4 × CP 3 superstring dynamics is limited to the OSp(4|6)/(SO(1, 3) × U (3)) subspace of AdS 4 × CP 3 superspace in distinction to the P SU (2, 2|4)/(SO(1, 4) × SO( 5)) σ-model description of the AdS 5 × S 5 superstring [START_REF] Metsaev | Type IIB superstring action in AdS 5 × S 5 background[END_REF], [START_REF] Kallosh | Near horizon superspace[END_REF].

The reason is that the AdS 4 × CP 3 superspace is not isomorphic to a supercoset manifold unlike the AdS 5 × S 5 one. Moreover, although the OSp(4|6)/(SO(1, 3) × U (3)) σ-model is classically integrable, it correctly describes AdS 4 × CP 3 superstring dynamics only in those sectors, where vanishing condition of eight coordinates of the sector of broken sypersymmetries represents partial κ-symmetry gauge condition. This is not the case, e.g. when the superstring moves only within the AdS 4 subspace of the AdS 4 × CP 3 space-time [START_REF] Arutyunov | Superstrings on AdS 4 × CP 3 as a coset sigma-model[END_REF].

To correctly describe superstring dynamics in this sector one needs to use complete action of the AdS 4 ×CP 3 superstring. Since the AdS 4 ×CP 3 superspace is not a supercoset manifold the superstring action cannot be constructed in the same way as for the AdS 5 ×S 5 superstring. However, to maximally exploit available symmetry in [START_REF] Gomis | The complete AdS 4 × CP 3 superspace for type IIA superstring and D-branes[END_REF] it was used the double-dimensional reduction [START_REF] Duff | Superstrings from supermembranes in D = 11[END_REF], [START_REF] Howe | The supermembrane revisited, Classical and Quantum Gravity[END_REF] of the AdS 4 × S 7 supermembrane [START_REF] De Wit | The M-theory two-brane in AdS 4 × S 7 and AdS 7 × S 4[END_REF] to obtain the superstring action.

The point is that the AdS 4 × S 7 superspace can be realized as the OSp(4|8)/(SO(1, 3) × SO [START_REF] Nilsson | Hopf fibration of eleven dimensional supergravity[END_REF]) supercoset manifold so that the costruction of the supermembrane action generalizes that of the AdS 5 × S 5 superstring. Further it is known the relation between the AdS 4 × S 7 background of D = 11 supergravity and the AdS 4 × CP 3 background of the IIA supergravity that relies on the the Hopf fibration realization of the 7-sphere S 7 = CP 3 × S 1 [START_REF] Nilsson | Hopf fibration of eleven dimensional supergravity[END_REF], [START_REF] Sorokin | Kaluza-Klein theories and spontaneous compactification mechanisms of extra space dimensions[END_REF]. In [START_REF] Gomis | The complete AdS 4 × CP 3 superspace for type IIA superstring and D-branes[END_REF] with relation was lifted to superspace. However, resulting AdS 4 × CP 3 supersting action as the functional of the superspace coordinates has complicated and highly non-linear structure.

To simplify its form one can use gauge freedom related to the κ-symmetry and worldsheet reparametrizations. Also one can try to extend the proof of integrability of equations of the OSp(4|6)/(SO(1, 3) × U (3)) σ-model to those of the AdS 4 × CP 3 superstring. This problem was studied in [START_REF] Sorokin | Evidence for the classical integrability of the complete AdS 4 ×CP 3 superstring[END_REF], [START_REF] Cagnazzo | More on integrable structures of superstrings in AdS 4 × CP 3 and AdS 2 × S 2 × T 6 superbackgrounds[END_REF] but it still remains unsolved.

In the present review we pursue both ways to explore the structure of the AdS 4 × CP 3 superstring. In the first part considered are dynamics and symmetries of the OSp(4|6)/(SO(1, 3) × U (3)) σ-model that describes the AdS 4 × CP 3 superstring in the partial κ-symmetry gauge as well as the light-cone gauge corresponding to the light cone on the conformal boundary of the anti-de Sitter space. In the second part integrable structure underlying the AdS 4 ×CP 3 superstring is examined in various sectors and dynamical regimes.

2 Superstring in AdS 4 × CP 3 superspace and N = 6 superconformal symmetry in (1+2) dimensions Subsection 2.1 is devoted to derivation of the novel formulation of the Lagrangian of the OSp(4|6)/(SO(1, 3) × U (3)) σ-model [START_REF] Uvarov | AdS 4 × CP 3 superstring and D = 3 N = 6 superconformal symmetry[END_REF] in terms of the Cartan forms associated with the generators of the D = 3 N = 6 superconformal algebra. Action of the OSp(4|6)/(SO(1, 3) × U (3)) σ-model follows from that of the Type IIA superstring in the AdS 4 × CP 3 superbackground [START_REF] Gomis | The complete AdS 4 × CP 3 superspace for type IIA superstring and D-branes[END_REF] upon imposition of vanishing condition on the coordinates related to eight supersymmetries broken by the background. As was discussed in the Introduction the σ-model describes superstring dynamics only in those sectors, in which this condition fixes the κ-symmetry gauge. The generators of 32-8=24 supersymmetries preserved by this superbackground enter the osp(4|6) orthosymplectic superalgebra of its isometry. This superalgebra contains sp(4) 1 symplectic and so(6) orthogonal subalgebras. As is known sp(4) algebra is isomorphic to the so(2, 3) one. It can be presented as the ads [START_REF] Maldacena | The large N limit of superconformal field theories and supergravity[END_REF][START_REF] Witten | Anti-de Sitter space and holography[END_REF] isometry algebra of the four-dimensional anti-de Sitter space AdS 4 = SO(2, 3)/SO [START_REF] Maldacena | The large N limit of superconformal field theories and supergravity[END_REF][START_REF] Witten | Anti-de Sitter space and holography[END_REF] and the conf (1, 2) conformal algebra of its D = 1 + 2 2 Minkowski boundary space. so(6) al-1 sp(4) denotes the split real form of the complex Lie algebra sp(4, C). For this algebra also used are notations sp(4, R) and sp(2, R). Besides in the designation of real orthosymplectic superalgebras such as osp(4|6) and osp(4|8) we place to the left of the vertical line the even number characterizing the non-compact symplectic subalgebra. Similar notation is used for unitary superalgebras, e.g. su(2, 2|N ). In the literature widely spread is another convention according to which these orthosymplectic superalgebras are denoted as osp(6|4) and osp(8|4) respectively. 2 Here and below notation D = T + S emphasizes that the space has T time and S space dimensions.

gebra is isomorphic to the su(4) algebra that describes isometry of the complex projective space CP 3 = SU (4)/U (3) of real dimension six. Due to these isomorphisms, that will be detailed in paragraph 2.1.1, osp(4|6) superalgebra admits realization as the D = 3 N = 6 superconformal algebra. This realization will be exploited in this and subsequent sections.

The OSp(4|6)/(SO(1, 3) × U (3)) supercoset manifold is the subspace of the AdS 4 × CP 3 superspace and is parametrized by 10 bosonic coordinates of the AdS 4 and CP 3 spaces and also [START_REF] Astolfi | Finite-size corrections in the SU (2) × SU (2) sector of type IIA string theory on AdS 4 × CP 3[END_REF] Grassmann-odd coordinates. As is known the (super)vielbein and connection oneforms of the (super)coset manifolds are built from the Cartan forms corresponding to the generators of their (super)isometry algebras. So bosonic components of the supervielbein of the OSp(4|6)/(SO(1, 3) × U (3)) supermanifold equal the so(2, 3)/so [START_REF] Maldacena | The large N limit of superconformal field theories and supergravity[END_REF][START_REF] Witten | Anti-de Sitter space and holography[END_REF] and su(4)/u(3)

Cartan forms, and fermionic components equal the Grassmann-odd Cartan forms. Connection one-form is determined by the Cartan forms for the so(1, 3) ⊕ u(3) stability subalgebra generators.

Also in the present subsection fermionic equations of the σ-model will be presented in the form similar to that of the fermionic equations of the Green-Schwarz superstrings [START_REF] Green | Covariant description of superstrings[END_REF] and it will be proved their linear dependence. This implies the σ-model action invariance under the gauge κ-symmetry that is the remnant of the κ-symmetry of the AdS 4 × CP 3 superstring. To study the non-linear structure of the σ-model Lagrangian explicit expressions will be obtained for the constituent Cartan forms. Their form is determined by that of the OSp(4|6)/(SO(1, 3)×U (3)) element. Considered will be its parameterization by the Poincaré coordinates for the AdS 4 space that are related to the generators of translations P m and dilatations D of the conformal algebra conf (1, 2) as well as by 24 anticommuting coordinates that are parameters for generators of the Poincaré and special conformal supersymmetries of the D = 3 N = 6 superconformal algebra.

Invariance of the OSp(4|6)/(SO(1, 3) × U (3)) σ-model under global D = 3 N = 6 superconformal symmetry is examined in subsection 2.2 on the basis of our work [START_REF] Uvarov | D = 3 N = 6 superconformal symmetry of the AdS 4 × CP 3 superstring[END_REF]. First examined will be transformation properties of the OSp(4|6)/(SO(1, 3) × U (3)) supercoset element and derived general expressions for the densities of the Noether currents associated with the action invariance under each of the transformations of the D = 3 N = 6 superconformal symmetry. Then given will be expressions for the Noether current densities in terms of coordinates that parametrize the OSp(4|6)/(SO(1, 3) × U (3)) element considered in the previous subsection.

Subsection 2.3 is devoted to derivation of the Lagrangian of the Type IIA superstring in the AdS 4 × CP 3 superbackground in the κ-symmetry light-cone gauge that was obtained in our work [START_REF] Uvarov | AdS 4 ×CP 3 superstring in the light-cone gauge[END_REF]. Considered gauge condition corresponds to the light cone on the conformal boundary of the anti-de Sitter space-time parametrized the Poincaré coordinates that distinguishes it from other light-cone gauge conditions previously introduced in [START_REF] Nishioka | On Type IIA Penrose Limit and N = 6 Chern-Simons Theories[END_REF], [START_REF] Astolfi | Finite-size corrections in the SU (2) × SU (2) sector of type IIA string theory on AdS 4 × CP 3[END_REF], [START_REF] Sundin | The AdS(4) × CP 3 string and its Bethe equations in the near plane wave limit[END_REF], [START_REF] Zarembo | Worldsheet spectrum in AdS 4 /CF T 3 correspondence[END_REF], [START_REF] Bykov | Symmetry algebra of the AdS 4 × CP 3 superstring[END_REF], [START_REF] Kalousios | Factorized tree-level scattering in AdS 4 × CP 3[END_REF], [START_REF] Dukalski | On fermionic reductions of the AdS 4 ×CP 3 superstring[END_REF].

In the final subsection 2.4 it is derived the Hamiltonian of the AdS 4 × CP 3 superstring in the light-cone gauge following our work [START_REF] Uvarov | Light-cone gauge Hamiltonian for AdS 4 × CP 3 superstring[END_REF]. To this end obtained in the previous subsection superstring Lagrangian in the κ-symmetry light-cone gauge will be presented in terms of the phase-space variables. The reparametrization symmetry gauge will be fixed by conditions on one of the space-time light-cone coordinates and on the momentum density conjugate to another light-cone coordinate. This allows to express the Hamiltonian in terms of the gauge-invariant transverse coordinates and momentum density components of the superstring and to show that in the free-field approximation it coincides with the light-cone gauge

Hamiltonian of the IIA superstring in flat superspace.

2.1 Lagrangian formulation of the OSp(4|6)/(SO(1, 3) × U (3)) σ-

model in terms of Cartan forms in conformal basis

As was mentioned in the Introduction, concrete formulation of the AdS 4 /CF T 3 duality conjecture was given in Ref. [START_REF] Aharony | N = 6 superconformal Chern-Simons-matter theories, M2-branes and their gravity duals[END_REF], in which constructed was the D = 3 N = 6 superconformal Chern-Simons-matter theory and found was the domain in its space of parameters, where dual theory can be described in terms of weakly interacting IIA superstrings in the AdS 4 × CP3 superbackground. Verification of the advanced hypothesis requires elaboration of quantum theory of superstrings in the given superbackground. The prerequisite for this is construction of the Lagrangian of classical AdS 4 × CP 3 superstring. First found was the Lagrangian of the σ-model in the OSp(4|6)/(SO(1, 3) × U (3)) supercoset space [START_REF] Arutyunov | Superstrings on AdS 4 × CP 3 as a coset sigma-model[END_REF], [START_REF] Stefański | Green-Schwarz action for Type IIA strings on AdS 4 × CP 3[END_REF] 3

that is the subspace of the AdS 4 × CP 3 superspace and then constructed was the complete Lagrangian of the superstring [START_REF] Gomis | The complete AdS 4 × CP 3 superspace for type IIA superstring and D-branes[END_REF].

In Refs. [START_REF] Arutyunov | Superstrings on AdS 4 × CP 3 as a coset sigma-model[END_REF], [START_REF] Stefański | Green-Schwarz action for Type IIA strings on AdS 4 × CP 3[END_REF] the OSp(4|6)/(SO(1, 3) × U (3)) σ-model Lagrangian was presented in terms of the Cartan one-forms associated with the generators of the ads(1, 3) algebra.

Since in the Chern-Simons-matter theory the OSp(4|6) symmetry is realized as the N = 6 superconformal symmetry in D = 1 + 2 dimensions [START_REF] Bandres | Studies of the ABJM Theory in a Formulation with Manifest SU (4) R-Symmetry[END_REF], it is important for checking the

G -1 dG = G sp(4) (d) + G so(6) (d) + G 24susy (d) ∈ osp(4|6), (2.1)
the r.h.s. of which can be conveniently presented as the sum of three summands. To the first two there enter the products of the Cartan forms and the generators of the sp(4) = so(2, 3) = ads(1, 3) = conf (1, 2) and so(6) = su(4) subalgebras of the osp(4|6) superalgebra g. The last summand includes the products of the Grassmann-odd Cartan forms with 24 supersymmetry generators. Each of three summands can be presented in a variety of ways that will be considered below.

For the first summand in accordance with the isomorphic realizations of the sp(4) algebra we give four representations The second equality corresponds to the isomorphic realization of sp(4) as the so(2, 3)

G sp(4) (d) = G αβ (d)O αβ = G mn (d)M mn = 2G 0 ′ m ′ (d)M 0 ′ m ′ + G m ′ n ′ (d)M m ′ n ′ = G mn (d)M mn + ω m (d)P m + c m (d)K m + ∆(d)D ∈ sp(4).
algebra. There M mn = i 4 γ mn αβ O αβ (m, n = 0 ′ , 0, 1, 2, 3) are the so(2, 3) generators and γ mn αβ denotes antisymmetrized product of D = 2 + 3 γ-matrices (details of the spinor algebra are given in Appendix A of our work [START_REF] Uvarov | AdS 4 × CP 3 superstring and D = 3 N = 6 superconformal symmetry[END_REF]). G mn (d) = -i 2 γ mn αβ G αβ (d) are the so(2, 3) Cartan forms.

In the second line generators of the so(2, 3) algebra were decomposed on M 0 ′ m ′ and M m ′ n ′ (m ′ , n ′ = 0, 1, 2, 3) that corresponds to the realization of the so(2, 3) algebra as the isometry algebra of the four-dimensional anti-de Sitter space AdS 4 . Cartan forms G 0 ′ m ′ (d) will be identified with the bosonic components of the OSp(4|6)/(SO(1, 3) × U (3)) supervielbein in the tangent space to AdS 4 or the AdS 4 vielbein one-form components in the bosonic theory.

G m ′ n ′ (d) is the so(1, 3) connection one-form. Relations of the ads(1, 3) algebra [M 0 ′ m ′ , M 0 ′ n ′ ] = M m ′ n ′ , [M 0 ′ k ′ , M m ′ n ′ ] = η k ′ m ′ M 0 ′ n ′ -η k ′ n ′ M 0 ′ m ′ , [M k ′ l ′ , M m ′ n ′ ] = η k ′ n ′ M l ′ m ′ -η k ′ m ′ M l ′ n ′ -η l ′ n ′ M k ′ m ′ + η l ′ m ′ M k ′ n ′ (2.3)
are invariant under two-element automorphism group Z 2 . Under its action generators M 0 ′ m ′ change the sign and M m ′ n ′ do not change. This automorphism is extended to the Z 4 automorphism of the osp(4|6) superalgebra.

In the last line in (2.2) given is yet another isomorphic realization of the so(2, 3) algebra as the conf (1, 2) conformal algebra of the D = 1 + 2 Minkowski space. We adhere to the following definitions of the generators of dilations, translations, conformal boosts

D = -2M 0 ′ 3 , P m = M 0 ′ m -M 3m , K m = M 0 ′ m + M 3m , m = 0, 1, 2
and respective Cartan forms

∆(d) = -G 0 ′ 3 (d), ω m (d) = G 0 ′ m (d) -G 3m (d), c m (d) = G 0 ′ m (d) + G 3m (d). (2.4)
For the second summand in (2.1) we give three representations

G so(6) (d) = Ω IJ (d)V IJ = Ω A B (d)V B A = Ω âb (d)V bâ . (2.5)
In the first representation V IJ = -V JI are the so(6) generators and Ω IJ (d) are respective Cartan forms. The second representation corresponds to isomorphic realization of the so( 6) algebra as the su(4) one. In this representation Cartan forms and generators are traceless and related to the so(6) generators and Cartan forms in the following way

Ω A B (d) = i 2 Ω IJ (d)ρ IJ A B = Ω a b (d) Ω a 4 (d) Ω 4 b (d) Ω 4 4 (d) , Ω 4 4 (d) = -Ω a a (d) (2.6)
and

V A B = i 4 ρ IJ A B V IJ = V a b V a 4 V 4 b V 4 4 , V 4 4 = -V a a , (2.7
V a 4 , V 4 b = i(V a b + δ b a V c c ), V a b , V c d = i(δ b c V a d -δ d a V c b ), [V a 4 , V b c ] = -iδ c a V b 4 , [V 4 a , V b c ] = iδ a b V
M Iâ = 1 2 (ρ Ia4 , ρ I a4 ), M -1 âI = ρ I 4a ρI4a : M M -1 = I
are determined by the ρ I AB and ρIAB matrices decomposed on the su(3)-covariant blocks. Transformation matrices satisfy the relations δ IJ = -2M Iâ H âb M T bJ and δ IJ =

-1 2 M -1T Iâ H âb M -1
bJ , where symmetric matrices

H âb = 0 δ b a δ a b 0 , H âb = 0 δ a b δ b a 0
are metric tensors in the 3⊕ 3 basis. Cartan forms and generators of the so(6) algebra in this basis are connected with the Cartan forms and generators in the usual one by the relations

Ω âb (d) = 2iM -1 âI Ω I J (d)M J b = Ω a b (d) -δ b a Ω c c (d) ε acb Ω 4 c (d) -ε acb Ω c 4 (d) -Ω b a (d) + δ a b Ω c c (d) (2.8)
and

V âb = i 2 M -1 âI V I J M J b = 1 2 V a b -δ b a V c c ε acb V 4 c (d) -ε acb V c 4 -V b a + δ a b V c c . (2.9)
The third summand in (2.1) includes products of the Grassmann-odd Cartan forms with 24 supersymmetry generators from the osp(4|6) superalgebra

G 24susy (d) = F α I (d)O I α = F αâ (d)O αâ = F αa (d) Ōαa + F α a (d)O a α . ( 2
O αâ = Q µâ S µ â : Q µâ = Qµa Q a µ , S µ â = Sµ a S µa .
They are two-component sl(2, R) = so(1, 2) spinors in the (anti)fundamental representation of su(3) algebra and are Hermitian-conjugate to one another: Qµa = (Q a µ ) † , Sµ a = (S µa ) † . For Cartan forms there hold analogous decompositions

F αâ (d) = ω µâ (d) χ â µ (d) : ω µâ (d) = ωµa (d) ω µ a (d) , χ â µ (d) = χa µ (d) χ µa (d) with ωµa (d) = (ω µ a (d)) † , χa µ (d) = (χ µa (d)) † . Therefore G 24susy (d) can be presented in the following form G 24susy (d) = ω µ a (d)Q a µ + ωµa (d) Qµa + χ µa (d)S µa + χa µ (d) Sµ a .
(2.11)

So we defined isomorphic realization of the osp(4|6) superalgebra as the D = 3 N = 6 superconformal algebra. This realization will be used in what follows.

Taking that there is a basis of the osp(4|6) superalgebra g, in which generators have definite eigenvalues under the Z 4 automorphism Υ: Υ(g j ) = e iπj/2 g j , its relations can be written in the concise form

g = 3 j=0 g j : {g i , g j ] = g i+j(mod 4) .
This automorphism is extention of the discussed above automorphisms of the so(2, 3) and su(4) algebras. With the generators in this basis it is possible to associate Cartan forms c j (d), then (2.1) acquires the form

G -1 dG = j ∈{0,1,2,3} G j (d) = j ∈{0,1,2,3} c j (d)g j .
(2.12)

Explicit form of each summand is the following

G 0 (d) = G mn (d)M mn + 2G 3m (d)M 3m + Ω a b (d)V b a + Ω 4 4 (d)V 4 4 = G mn (d)M mn + 1 2 (c m (d) -ω m (d))(K m -P m ) + Ω a b (d)V b a + Ω 4 4 (d)V 4 4 , G 2 (d) = 2G 0 ′ m (d)M 0 ′ m + 2G 0 ′ 3 (d)M 0 ′ 3 + Ω a 4 (d)V 4 a + Ω 4 a (d)V a 4 = 1 2 (c m (d) + ω m (d))(K m + P m ) + ∆(d)D + Ω a 4 (d)V 4 a + Ω 4 a (d)V a 4 , G 1 (d) = ω µ (1)a (d)Q a (1)µ + ωµa (1) (d) Q(1)µa , (2.13)
G 3 (d) = ω µ (3)a (d)Q a (3)µ + ωµa (3) (d) Q(3)µa . (2.14)
In (2.13) and (2.14) generators from the g 1 and g 3 eigenspaces are defined by the linear combinations of the Poincaré and special conformal supersymmetry generators

Q a (1,3)µ = Q a µ ± iS a µ , Q(1,3)µa = Qµa ∓ i Sµa . (2.15) Associated Cartan forms equal ω µ (1,3)a (d) = 1 2 (ω µ a (d) ± iχ µ a (d)), ωµa (1,3) (d) = 1 2 (ω µa (d) ∓ i χµa (d)). (2.16)
These generators can also be written in terms of the four-component spinors

P +α β O a β = 1 2 Q a (1)µ -iQ µa (1)
, P -α

β Ōβa = 1 2 Q(1)µa i Q µ (1)a and P -α β O a β = 1 2 Q a (3)µ iQ µa (3)
, P +α

β Ōβa = 1 2 Q(3)µa -i Q µ (3)a
with the aid of the D = 1 + 3 chiral projectors

P ± α β = 1 2 (δ β α ± iΓ 5 α β ), Γ 5 α β = (Γ 0 Γ 1 Γ 2 Γ 3 ) α β = 0 ε µν -ε µν 0 .
They satisfy the defining relations P + + P -= I, P ± P ± = P ± , P + P -= P -P + = 0 and also

(P ± α β ) T = P ± β α , (P ± α β ) † = P ± β α .
Further relations that projectors obey as well as the details of the definition of D = 1 + 3 γ-matrices and their connection with the D = 2 + 3 and D = 1 + 2 γ-matrices are given in Ref. [START_REF] Uvarov | AdS 4 × CP 3 superstring and D = 3 N = 6 superconformal symmetry[END_REF].

In the similar way one can write the Cartan forms from (2.16)

F α +a (d) = F β a (d)P +β α = ω µ (1)a (d) -iω (1)µa (d) , Fαa -(d) = Fβa (d)P -β α = ωµa (1) (d) iω a (1)µ (d) (2.17) and F α -a (d) = F β a (d)P -β α = ω µ (3)a (d) iω (3)µa (d) , Fαa + (d) = Fβa (d)P +β α = ωµa (3) (d) -iω a (3)µ (d) .
(2.18)

Both representations for Cartan forms (2.16) and (2.17), (2.18) will be used in the next paragraph to present the Wess-Zumino Lagrangian of the σ-model in various forms.

For future reference we also give the Maurer-Cartan equations for the Cartan forms that enter the OSp(4|6)/(SO(1, 3) × U (3)) σ-model Lagrangian. In the basis corresponding to the so(2, 3) algebra realization as ads(1, 3) they take the form

dG 0 ′ m ′ (d) -2G m ′ n ′ (d) ∧ G 0 ′ n ′ (d) -iF α a (d) ∧ γ 0 ′ m ′ αβ Fβa (d) = 0, dΩ a 4 (d) + iΩ +a b (d) ∧ Ω b 4 (d) -ε abc Fαb (d) ∧ C αβ Fβc (d) = 0, dΩ 4 a (d) + iΩ 4 b (d) ∧ Ω +b a (d) + ε abc F α b (d) ∧ C αβ F β c (d) = 0, dF α a (d) + 1 2 F β a (d) ∧ (2G 0 ′ m ′ (d)γ 0 ′ m ′ α β + G m ′ n ′ (d)γ m ′ n ′ α β ) (2.19) +iΩ -a b (d) ∧ F α b (d) + iε acb Ω 4 c (d) ∧ Fαb (d) = 0, d Fαa (d) + 1 2 Fβa (d) ∧ (2G 0 ′ m ′ (d)γ 0 ′ m ′ α β + G m ′ n ′ (d)γ m ′ n ′ α β ) +i Fαb (d) ∧ Ω -b a (d) -iε acb Ω c 4 (d) ∧ F α b (d) = 0,
where

Ω ±a b (d) = Ω a b (d) ± δ b a Ω c c (
dω m (d)-2∆(d)∧ω m (d)-2G m n (d)∧ω n (d)+2iω µ a (d)∧σ m µν ωνa (d) = 0, dc m (d)+2∆(d)∧c m (d)-2G m n (d)∧c n (d)+2iχ µa (d)∧ σmµν χa ν (d) = 0, d∆(d)-ω m (d) ∧ c m (d)-i(ω µa (d) ∧ χ µa (d) + ω µ a (d) ∧ χa µ (d)) = 0, dΩ a 4 (d) + iΩ +a b (d) ∧ Ω b 4 (d) -2ε abc ωµb (d) ∧ χc µ (d) = 0, dΩ 4 a (d) + iΩ 4 b (d) ∧ Ω +b a (d) + 2ε abc ω µ b (d) ∧ χ µc (d) = 0 (2.20)
and

dω µ â (d) -∆(d) ∧ ω µ â (d) + 1 2 ω ν â (d) ∧ G mn (d)σ mn ν µ +ω m (d) ∧ σµν m χ νâ (d) + iΩ âb (d) ∧ ω µ b (d) = 0, dχ µâ (d) + ∆(d) ∧ χ µâ (d) + 1 2 G mn (d) ∧ σ mn µ ν χ νâ (d) -c m (d) ∧ σ m µν ω ν â (d) + iΩ âb (d) ∧ χ µ b(d) = 0, (2.21)
where matrices σ-model is given by the sum of kinetic and Wess-Zumino terms

S σ-model = S σ-model, kin + S σ-model, WZ : S σ-model, kin = Σ d 2 ξL σ-model, kin , S σ-model, WZ = Σ d 2 ξL σ-model, WZ (2.22)
where ξ i are local coordinates of the world-sheet Σ. The σ-model Lagrangian is constructed out of the mappings to the world-sheet of the Cartan forms that are identified with the OSp(4|6)/(SO(1, 3) × U (3)) supervielbein components using the same method as was elaborated for construction of the AdS 5 × S 5 superstring Lagrangian [START_REF] Metsaev | Type IIB superstring action in AdS 5 × S 5 background[END_REF], [START_REF] Kallosh | Near horizon superspace[END_REF], [START_REF] Roiban | Superstring on AdS 5 × S 5 supertwistor space[END_REF].

The Polyakov kinetic term has the same form as for the Green-Schwarz superstrings on curved background and, in particular, on the AdS 5 × S 5 one. It is specified by the mapping to the world-sheet of the space-time metric

L σ-model, kin = -1 2 √ -γγ ij (g AdS 4 ij + g CP 3 ij ), (2.23)
where

g AdS 4 ij = G i 0 ′ m ′ η m ′ n ′ G j 0 ′ n ′ and g CP 3 ij = 1 2 (Ω i a 4 Ω j 4 a + Ω j a 4 Ω i 4 a
) are contributions to induced world-sheet metric determined by the metric tensors of the AdS 4 and CP 3 spaces. 4 They are constructed out of the so(2, 3)/so [START_REF] Maldacena | The large N limit of superconformal field theories and supergravity[END_REF][START_REF] Witten | Anti-de Sitter space and holography[END_REF] can also be presented in terms of the Cartan forms associated 4 In this and the next sections it is assumed that superspace coordinates are dimensionless implying that the square of the space-time interval includes overall factor R 2 , where R is the radius of the seven-sphere S 7 .

Then the superstring and σ-model Lagrangians there enters as the overall factor the dimensionless tension

T string = R 2 2πα
′ that is set to unity. Note that in the M-theory the Regge slope parameter α ′ is expressed via the D = 11 Planck length l P and string coupling constant g s : α ′ = g -2/3 s l 2 P . In the considered case of the AdS 4 /CF T 3 correspondence the string coupling equals g s = ( R klP ) 3/2 ≪ 1.

with the conf (1, 2) algebra generators. Utilization of (2.4) yields the following expressions for the AdS 4 vielbein components

G 0 ′ m (d) = 1 2 (ω m (d) + c m (d)), G 0 ′ 3 (d) = -∆(d).
(2.24)

Then the Lagrangian (2.23) acquires the form

L σ-model, kin = - 1 2 √ -γγ ij 1 4 (ω m i +c m i )(ω jm +c jm )+∆ i ∆ j + 1 2 (Ω ia 4 Ω j4 a +Ω ja 4 Ω i4 a ) . (2.25)
The OSp(4|6)/(SO(1, 3)×U ( 3)) σ-model belongs to the family of σ-models on the supercoset spaces G /H with the isometry superalgebras g invariant under the Z 4 automorphism [START_REF] Berkovits | Superstring theory on AdS 2 × S 2 as a coset manifold[END_REF]. Its characteristic feature is that the invariant subspace is given by the stability algebra g 0 = h. The Wess-Zumino Lagrangians for the σ-models from this family have the following

general form 5 L σ-model, WZ = sTr G 1 (d) ∧ G 3 (d),
where sTr stands for supertrace of the product of supermatrices in the supermatrix realization of the generators of g. In distinction to the Wess-Zumino Lagrangians of the Green-Schwarz superstrings on flat background [START_REF] Green | Covariant description of superstrings[END_REF], it is given by mapping to the world-sheet of the twoform constructed out of the wedge product of the Grassmann-odd Cartan forms. Detailed discussion of the properties of the strings' Wess-Zumino term in the AdS superbackgrounds can be found, e.g. in [START_REF] Hatsuda | Wess-Zumino term for AdS superstring[END_REF].

In our works [START_REF] Uvarov | AdS 4 × CP 3 superstring and D = 3 N = 6 superconformal symmetry[END_REF] and [START_REF] Uvarov | D = 3 N = 6 superconformal symmetry of the AdS 4 × CP 3 superstring[END_REF] there were obtained a number of novel representations for the Wess-Zumino Lagrangian of the OSp(4|6)/(SO(1, 3) × U (3)) σ-model. In particular, it can written in terms of the sl(2, R) spinor Cartan forms (2.16) associated with the generators (2.15) from the g 1 and g 3 eigenspaces

L σ-model, WZ = -ε µν (ω µ (1)a (d) ∧ ωνa (3) (d) + ω µ (3)a (d) ∧ ωνa (1) (d)) = -1 2 (ω µ a (d) ∧ ε µν ωνa (d) + χ µa (d) ∧ ε µν χa ν (d)) .
(2.26)

In the second line the Wess-Zumino Lagrangian is expressed via the Cartan forms for the generators of Poincaré and special conformal supersymmetries. Lagrangian (2.26) can also be presented in the so(6)-invariant form in the 3 ⊕ 3 basis for D = 6 vectors

L σ-model, WZ = i 4 ε µν J âb ω µ (1) b (d) ∧ ω νâ (3) (d) + ω µ (3) b (d) ∧ ω νâ (1) (d) = i 8 J âb ω µ b (d) ∧ ε µν ω νâ (d) + χ µ b(d) ∧ ε µν χ â ν (d) , (2.27)
where antisymmetric matrix

J âb = J âĉ H ĉb = 2i 0 δ b a -δ a b 0 (2.

28)

5 As for the Green-Schwarz superstrings the Wess-Zumino Lagrangians of the σ-models are defined up to the sign. This ambiguity is compatible with the κ-invariance of the action.

defines components of the Kähler two-form of the CP 3 space. 6 The Wess-Zumino Lagrangian takes more compact form when is expressed in terms of the Grassmann-odd Cartan forms that are four-component spin [START_REF] Maldacena | The large N limit of superconformal field theories and supergravity[END_REF][START_REF] Witten | Anti-de Sitter space and holography[END_REF] spinors

L σ-model, WZ = i 2 F α a (d) ∧ C ′ αβ Fβa (d) = 1 8 J âb F α b (d) ∧ C ′ αβ F βâ (d),
where C ′ αβ is antisymmetric charge conjugation matrix in D = 1 + 3 dimensions. In terms of Cartan forms (2.17) and (2.18) the Wess-Zumino Lagrangian can be written as

L σ-model, WZ = i 2 (F α +a (d) ∧ C ′ αβ Fβa + (d) + F α -a (d) ∧ C ′ αβ Fβa -(d)) = 1 4 J âb F α + b (d) ∧ C ′ αβ F βâ -(d),
where one-forms

F Tα +â (d) = F α +a (d) Fαa -(d) , F αâ -(d) = Fαa + (d) F α -a (d)
, that are spin(1, 3) spinors and D = 6 vectors in the 3 ⊕ 3 basis and so have 24 real components, will be used below to write down fermionic equations and the κ-symmetry transformations of the σ-model in the form similar to fermionic equations and the κ-symmetry transformations of the Green-Schwarz superstrings.

To this end consider contribution of fermionic Cartan forms to variation of the action (2.22). We will make use of the Maurer-Cartan equations (2.19) and known general expression for variation of a differential form

δF (d) = d(i δ F (d)) + i δ (dF (d)), (2.29)
where operator i δ replaces differential with variation and acts on the product of forms in accordance with the Leibniz rule as does the operator of external differentiation. Then one obtains the following expression for required variation

δ f S σ-model = Σ d 2 ξ F Tαâ + i V ij -M jαâ β bC βγ F γ + b (δ) + F Tαâ -i V ij + M jαâ β bC βγ F γ - b (δ) . (2.30)
Variation (2.30) involves the matrix

M iαâ β b = -δ b a G i 0 ′ m ′ Γ m ′ α β δ β α ε acb Ω i4 c -δ β α ε acb Ω ic 4 -δ a b G i 0 ′ m ′ Γ m ′ α β
which enter the world-sheet projections of the AdS 4 and

CP 3 vielbeins G i 0 ′ m ′ , Ω i4 c and Ω ic 4
as well as the world-sheet projectors

V ij ± = 1 2 ( √ -γγ ij ± ε ij ).
(2.31) 6 In conventional basis for D = 6 vectors respective expression for the components of the Kähler two-form is

J IJ = i 2 (ρ I 4a ρJ4a -ρ J 4a ρI4a
). Upon contraction with the so(6) generators in the spinor representation they form the 4 × 4 diagonal matrix

J A B = J IJ ρ IJ A B = -2iδ b a 0 0 6i .
The matrix J A B can be shown to satisfy the equation

J A C J C B -4iJ A B + 12δ B A = 0.
These are the same projectors that enter fermionic equations and κ-symmetry transformations of the Green-Schwarz superstrings. They obey the relations

V ij + + V ij -= √ -γγ ij , V ik ± γ kl V jl ± = 0, V ik ± γ kl V lj ± = √ -γV ij ± , V ij ± V kl ± = V kj ± V il ±
and allow to decompose a D = 1 + 1 vector v i = (v τ , v σ ) over the basis of light-like vectors (see, e.g. [START_REF] Brink | Principles of string theory[END_REF])

V ij ± v j = V i ± v ∓τ , V i ± = 1 2 1 √ -γγ τ σ ∓1 √ -γγ τ τ , v ∓τ = √ -γγ τ τ v τ + ( √ -γγ τ σ ± 1)v σ , (2.32)
where V i ± are basis vectors. In conformal gauge for the metric γ ij = e λ η ij v ±τ equal lightcone components of the vector. Considering components of F α ±â (δ) as independent variation parameters yields fermionic equations of the OSp(4|6)/(SO(1, 3)

× U (3)) σ-model V ij ∓ M T j αâ β bF β b ± i = 0, (2.33)
which enter the 24 × 24 matrices This can be done using the rank-preserving transformation that brings them to the triangular form

V i ± M Tαâ ∓ β b = V ij ± M j Tαâ β b.
  δ a b G ±τ 0 ′ m ′ Γ m ′ α β δ α β ε acb Ω ±τ c 4 0 - Ω ±τ a 4 Ω ±τ 4 b G ±τ 0 ′ m ′ G ±τ 0 ′ m ′ G ±τ 0 ′ m ′ Γ m ′ α β   = A ± B ± , (2.35)
where G ±τ

0 ′ m ′ , Ω ±τ 4 a and Ω ±τ a 4 are the components of G i 0 ′ m ′ , Ω i4 a
and Ω ia 4 obtained using the projectors (2.31). Note that in the basis of the light-like vectors the Virasoro constraints that are equations for the auxiliary metric γ ij have the form

G ±τ 0 ′ m ′ G ±τ 0 ′ m ′ + Ω ±τ a 4 Ω ±τ 4 a = 0. (2.36)
In (2.35) matrices

A αâ ± β b =   δ a b G ±τ 0 ′ m ′ Γ m ′ α β 0 0 - Ω ±τ a 4 Ω ±τ 4 b G ±τ 0 ′ m ′ G ±τ 0 ′ m ′ G ±τ 0 ′ m ′ Γ m ′ α β   are block-diagonal and B αâ ± β b = δ a b δ α β -ε acb Ω ±τ c 4 G ±τ 0 ′ m ′ G ±τ 0 ′ m ′ G ±τ 0 ′ m ′ Γ m ′ α β 0 δ b a δ α β
are non-degenerate. Therefore the rank of the matrices (2.35) equals that of matrices A ± .

The 4 × 4 matrices G ±τ

0 ′ m ′ Γ m ′ α β are non-singular if G ±τ 0 ′ m ′ G ±τ 0 ′ m ′ ̸ = 0.
Then from the Virasoro constraints (2.36) it follows that Ω ±τ a 4 ̸ = 0, Ω ±τ 4 a ̸ = 0. These conditions are met when the superstring moves in the AdS 4 and CP 3 spaces. In such case the rank of the 3 × 3 matrices Ω ±τ a 4 Ω ±τ 4 b that enter the lower-diagonal blocks of the matrices A ± ,7 is unity since it is given by the product of the three-component CP 3 vielbein and its c.c. As a result the rank of A ± and M T ± equals 4 × 3 + 4 × 1 = 16. So this matrix is degenerate and has 16 non-zero eigenvalues. It implies that in this sector the Type IIA superstring dynamics can be described by the σ-model [START_REF] Arutyunov | Superstrings on AdS 4 × CP 3 as a coset sigma-model[END_REF]. The consequence of degeneracy of the matrix M T ± is that eight fermionic equations of the σ-model are linearly dependent and are expressed via other 16 equations and its action (2.22) is invariant under the eight-parameter κ-symmetry. This symmetry is the remnant of the 16-parameter κ-symmetry of the full action of the Type IIA superstring in the AdS 4 × CP 3 superbackground [START_REF] Gomis | The complete AdS 4 × CP 3 superspace for type IIA superstring and D-branes[END_REF].

As was noted, when superstring moves only in the AdS 4 space, i.e. Ω ±τ a 4 = Ω ±τ 4 a = 0, vanishing conditions of the coordinates for broken supersymmetries not only gauge fix the κ-symmetry but also contain initial data constraints that exclude four fermionic physical degrees of freedom. On the level of the fermionic equations of the σ-model this manifests itself in that the rank of M T ± reduces to 12 compared to the above considered general case. As a result half of fermionic equations of the σ-model are independent that implies its action invariance under the 12-parameter κ-symmetry transformations as was noted in [START_REF] Arutyunov | Superstrings on AdS 4 × CP 3 as a coset sigma-model[END_REF]. This agrees with the fact that among eight vanishing conditions of the coordinates for broken supersymmetries only four are the gauge-fixing ones. So the σ-model represents the truncation of the superstring in this sector and in it the superstring dynamics has to be described by the complete action in the AdS 4 × CP 3 superspace.

Above presented proof of the linear dependence of fermionic equations of the OSp(4|6)/(SO(1, 3) × U (3)) σ-model is the novel independent confirmation of the fact that in the general case it describes dynamics of the same number of physical fermionic fields as the Green-Schwarz superstring and belongs to the results submitted to defence.

As was indicated earlier, from the linear dependence of the fermionic equations of the σmodel in accordance with the second Noether theorem follows invariance of its action under the κ-symmetry with local fermionic parameters. Therefore alternative way to find under what conditions the σ-model correctly describes the superstring dynamics is to determine the number of independent parameters in the κ-symmetry transformations. To this end it is necessary to calculate the rank of the matrices that enter these transformations. As in the case of the Green-Schwarz superstrings it is convenient to present the κ-symmetry transformations of the σ-model in terms of the one-forms in which differentials of the superspace coordinates are replaced by their κ-variations. For fermionic one-forms respective expressions have the form

F α +â (δ κ ) = V ij + V kl + K α b jlâβ κ β + bik , F α -â (δ κ ) = V ij -V kl -K α b jlâβ κ β -bik , (2.37)
where

K α b ijâβ = δ α β (G 0 ′ m ′ i G j 0 ′ m ′ δ b a + Ω ia 4 Ω j4 b ) G i 0 ′ m ′ Γ m ′ α β ε acb Ω j4 c -G i 0 ′ m ′ Γ m ′ α β ε acb Ω jc 4 δ α β (G 0 ′ m ′ i G j 0 ′ m ′ δ a b + Ω i4 a Ω jb 4
) .

Replacing differentials of the superspace coordinates by their κ-variations in the supervielbein bosonic components is known to give zero

G 0 ′ m ′ (δ κ ) = 0, Ω 4 a (δ κ ) = 0, Ω a 4 (δ κ ) = 0,
so the κ-variation of the σ-model action (2.22) is given by the expression (2.30) in which (2.37) is substituted. It is compensated by variation of the auxiliary world-sheet metric

δ κ ( √ -γγ ij ) = 2i(F αâ +p V pr + G r 0 ′ m ′ Γ m ′ α γ Γ 5 γβ V ki + V lj + κ β +âkl +F αâ -p V pr -G r 0 ′ m ′ Γ m ′ α γ Γ 5 γβ V ki -V lj -κ β -âkl ).
Local parameters κ β -âij (ξ) and κ β +âij (ξ) carry the pair of the world-sheet vector indices instead of one as in the case of the Green-Schwarz superstrings and satisfy the (anti-)selfduality conditions in each of these indices

1 √ -γ γ ik V kl + κ α -âlj = 1 √ -γ γ jk V kl + κ α -âil = κ α -âij and 1 √ -γ γ ik V kl -κ α +âlj = 1 √ -γ γ jk V kl -κ α +âil = κ α +âij .
On shell of the Virasoro constraints (2.36) the rank of the matrices

V ij ± V kl ± K jl α â b β in (2.37)
in the general case, when superstring moves both in the AdS 4 and in the CP 3 spaces, equals eight. So that only 1 3 of 24 parameters actually contribute to the κ-symmetry transformations. To prove this statement first note that their world-sheet vector indices are contracted with indices of the projectors V ij ± so they have only two independent components K α b ±τ ±τ âβ . The rank of these matrices can be found by solving the eigenvalue problem that amounts to computing the determinants of the matrices K ±τ ±τ -λ ±τ ±τ I using their block structure

det(K ±τ ±τ -λ ±τ ±τ I) = det A αb ±τ ±τ aβ B α ±τ ±τ aβb C αab ±τ ±τ β D αa ±τ ±τ βb = det A ±τ ±τ det(D ±τ ±τ -C ±τ ±τ A -1 ±τ ±τ B ±τ ±τ ), (2.38)
where

A αb ±τ ±τ aβ = δ α β A ±τ ±τ a b , A ±τ ±τ a b = (G ±τ 0 ′ m ′ G ±τ 0 ′ m ′ -λ ±τ ±τ )δ b a + Ω ±τ a 4 Ω ±τ 4 b , D αa ±τ ±τ βb = δ α β ((G ±τ 0 ′ m ′ G ±τ 0 ′ m ′ -λ ±τ ±τ )δ a b + Ω ±τ 4 a Ω ±τ b 4 ), B α ±τ ±τ aβb = -G ±τ 0 ′ m ′ Γ m ′ α β ε acb Ω ±τ 4 c , C αab ±τ ±τ β = G ±τ 0 ′ m ′ Γ m ′ α β ε acb Ω ±τ c 4 .
Due to addition of λ ±τ ±τ I matrices A ±τ ±τ a b are non-singular: det

A ±τ ±τ = -(G ±τ 0 ′ m ′ G ±τ 0 ′ m ′ - λ ±τ ±τ ) 2 λ ±τ ±τ so exist their inverse matrices A -1 ∓τ ∓τ b a = 1 λ ±τ ±τ (G ±τ 0 ′ m ′ G ±τ 0 ′ m ′ -λ ±τ ±τ ) (λ ±τ ±τ δ a b + Ω ±τ b 4 Ω ±τ 4 a ).
Then the determinants (2.38) equal

det(K ±τ ±τ -λ ±τ ±τ I) = λ 16 ±τ ±τ (λ ±τ ±τ -2G ±τ 0 ′ m ′ G ±τ 0 ′ m ′ ) 8 = 0.
So the rank of the matrix K ±τ ±τ equals eight in this sector. Matrices M T ± and K ±τ ±τ are mutually complementary in the sense that the sum of their ranks equals 24. This expresses the fact that out of 24 Grassmann coordinates of the OSp(4|6)/(SO(1, 3) × U (3)) superspace 16 represent gauge-invariant physical degrees of freedom of the σ-model and eight are pure gauge.

For checking the AdS 4 /CF T 3 duality conjecture [START_REF] Aharony | N = 6 superconformal Chern-Simons-matter theories, M2-branes and their gravity duals[END_REF] it is important to have explicit expression for the OSp(4|6)/(SO(1, 3) × U (3)) σ-model Lagrangian in terms of the superspace coordinate fields. For this it is necessary to choose the OSp(4|6)/(SO(1, 3) × U (3)) supercoset representative in (2.1) and find expressions for the Cartan forms in terms of the superspace coordinates and their differentials. In our work [START_REF] Uvarov | AdS 4 × CP 3 superstring and D = 3 N = 6 superconformal symmetry[END_REF] obtained were expressions for the Cartan forms built out of the supercoset element

G = e xmP m +θ µ a Q a µ + θµa Qµa e ηµaS µa +η a µ Sµ a e z a Va 4 +zaV 4 a e φD (2.39)
that includes generators of the D = 3 N = 6 superconformal algebra. Note that supercoset elements of the similar form were used earlier to get expressions for the Lagrangians of strings and branes on the AdS × S-type backgrounds [START_REF] Dall'agata | The OSp(8|4) singleton action from the supermembrane[END_REF], [START_REF] Kallosh | Superconformal actions in Killing gauge[END_REF], [START_REF] Pasti | On gauge-fixed superbrane actions in AdS superbackgrounds[END_REF], [START_REF] Metsaev | Superstring action in AdS 5 ×S 5 : κ-symmetry light cone gauge[END_REF], [START_REF] Metsaev | Light cone superstring in AdS space-time[END_REF], [START_REF] Metsaev | On manifest SU (4) invariant superstring action in AdS 5 × S 5 , Classical and Quantum Gravity[END_REF]. In (2.39) x m and φ are the Poincaré coordinates of AdS 4 space-time and three complex coordinates z a and za parametrize CP 3 space. Anticommuting coordinates θ µ a , θµa and η µa , ηa µ are parameters at the generators of the Poincaré and special conformal supersymmetries. So substitution of (2.39) into (2.1) produces the following expressions for the Cartan forms associated with the generators of translations, conformal boosts and dilatations from the conf (1, 2) algebra

ω m (d) = e -2φ ς m (d), ς m (d) = dx m -idθ µ a σ m µν θνa + iθ µ a σ m µν d θνa , c m (d) = e 2φ υ m (d), υ m (d) = -idη µa σmµν ηa ν + iη µa σmµν dη a ν +2(η b ρ η ρ b ) η µa σmµν d θa ν + 1 4 ζa ν (d) -dθ µa + 1 4 ζ µa (d) σmµν ηa ν , ∆(d) = dφ + idθ µ a ηa µ + id θµa η µa , (2.40)
where

ζ µ a (d) = -σ mµν ς m (d)η νa , ζµa (d) = -σ mµν ς m (d)η a ν .
In the bosonic limit these Cartan forms specify the square of infinitesimal length element in the AdS 4 space-time in the Poincaré coordinates

ds 2 AdS 4 = e -4φ 4 dx m dx m + dφdφ. (2.41)
For given supercoset element (2.39) Cartan forms corresponding to generators of the so [START_REF] Watamura | Spontaneous compactification and CP N : SU (3)×SU (2)×U (1), sin 2 θ W , g 3 /g 2 and SU (3)-triplet chiral fermions in four dimensions[END_REF] algebra are represented by the sum of the purely bosonic summand and that depending on the Grassmann-odd coordinates

Ω âb (d) = Ω bos âb (d) + Ω ferm âb (d). (2.42) Bosonic Cartan forms Ω bos âb (d) = iT âĉ d Tĉ b = Ω bos a b (d) -δ b a Ω bos c c (d) ε acb Ω bos 4 c (d) -ε acb Ω bos c 4 (d) -Ω bos b a (d) + δ a b Ω bos c c (d) ,
are specified by the unitary matrix that in the 3 ⊕ 3 basis (2.8) has the form

T âb =   cos |z|δ b a + (1-cos |z|) |z| 2 za z b i sin |z| |z| ε acb z c -i sin |z| |z| ε acb zc cos |z|δ a b + (1-cos |z|) |z| 2 z a zb   , (2.43)
where |z| 2 = z a za . So entries of the matrix Ω bos âb (d) equal

Ω bos a b (d) = i (1-cos |z|) |z| 2 (z a dz b -dz a z b )-iz a z b (1-cos |z|) 2 2|z| 4 (dz c zc -z c dz c ), Ω bos a 4 (d) = dz a sin |z| |z| + za sin |z|(1-cos |z|) 2|z| 3 (dz c zc -z c dz c )+ za 1 |z| -sin |z| |z| 2 d|z|, Ω bos 4 a (d) = dz a sin |z| |z| +z a sin |z|(1-cos |z|) 2|z| 3 (z c dz c -dz c zc )+z a 1 |z| -sin |z| |z| 2 d|z|.
(

The second term in (2.42) can be presented as

Ω ferm âb (d) = (T Ψ(d) T ) âb , Ψ âb (d) = Ψ a b (d) -δ b a Ψ c c (d) ε acb Ψ 4 c (d) -ε acb Ψ c 4 (d) -Ψ b a (d) + δ a b Ψ c c (d) , (2.45)
where the matrix Ψ âb (d) has the following elements

Ψ a b (d) = 2(dθ µ a + 1 2 ζ µ a (d))η b µ -2(d θµb + 1 2 ζµb (d))η µa -δ b a ((dθ µ c + 1 2 ζ µ c (d))η c µ -(d θµc + 1 2 ζµc (d))η µc ), Ψ a 4 (d) = 2ε abc (d θµb + 1 2 ζµb (d))η c µ , Ψ 4 a (d) = -2ε abc (dθ µ b + 1 2 ζ µ b (d))η µc .
The form of fermionic Cartan forms is the following

ω µ â (d) = e -φ T âb ς µ b (d) ςµb (d) , (2.46)
and

χ µâ (d) = e φ T âb υ µb (d) ῡb µ (d) (2.47)
where

ς µ b (d) = dθ µ b + ζ µ b (d), ςµb (d) = d θµb + ζµb (d), υ µb (d) = dη µb + 2iη c µ dθ ν c η νb + 2iη µc d θνc η νb + i(η ν c ηc ν )ς µb (d), ῡb µ (d) = dη b µ + 2iη µc d θνc ηb ν + 2iη c µ dθ ν c ηb ν + i(η ν c ηc ν )ς b µ (d).
Above presented Cartan forms identically satisfy Maurer-Cartan equations (2.20) and (2.21), and belong to the results submitted to defence.

As is seen from expressions for the Cartan forms the σ-model Lagrangian has rather complicated non-linear structure with the kinetic term containing contributions up to the eighth power in the fermions and the Wess-Zumino term -up to the sixth power. Observe that similarly to the AdS 5 × S 5 superstring anticommuting coordinates θ µ a and θµa related to the Poincaré supersymmetry generators enter expressions for the Cartan forms utmost quadratically and the non-linear fermionic contributions are given by the η µa and ηa µ coordinates that are parameters at the special conformal supersymmetry generators.

For the AdS 5 ×S 5 superstring it is possible to fix the κ-symmetry gauge by the conditions that entirely remove the coordinates η so that the Lagrangian becomes quadratic [START_REF] Kallosh | Simplifying superstring action on AdS 5 × S 5[END_REF] or quartic in the anticommuting coordinates θ [START_REF] Pesando | A κ gauge-fixed Type IIB superstring action[END_REF], [START_REF] Kallosh | The GS action on AdS 5 × S 5[END_REF], [START_REF] Metsaev | Superstring action in AdS 5 ×S 5 : κ-symmetry light cone gauge[END_REF], [START_REF] Metsaev | Light cone superstring in AdS space-time[END_REF]. In the present case it is impossible to gauge away all 12 η coordinates by the eight-parameter κ-symmetry. Instead one can consider, for instance, the SL(2, R) covariant gauge that removes eight η coordinates

η µa = η µA η µ3 , η µA = 0,
where A is the fundamental representation index of SU (2). In this gauge there turn to zero the following entries of the matrix (2.45) In the present subsection there will be carried detailed study of the realization of

Ψ 1 2 (d) = Ψ 2 1 (d) = Ψ 4 3 (d) = Ψ 3 4 (d) = 0 but
D = 3 N = 6 superconformal symmetry in the OSp(4|6)/(SO(1, 3) × U (3)) σ-model.
There will be found general expressions for the Noether current densities for each of the transformations from this symmetry. In the previous subsection there were obtained explicit expressions for the Cartan forms that enter the σ-model Lagrangian in terms of the OSp(4|6)/(SO(1, 3) × U (3)) supercoset coordinates and their differentials (see Eqs. (2.40), (2.42), (2.46) and (2.47)). In this subsection there will be found transformation properties of these coordinates under the D = 3 N = 6 superconformal symmetry and also derived expressions for the Noether current densities in terms of them. These Noether current densities can be viewed as the zeroth-order contributions to the expansion of the current densities related to invariance of the complete action of the AdS 4 × CP 3 superstring in the Grassmann coordinates for the broken supersymmetries. In Ref. [START_REF] Sorokin | Evidence for the classical integrability of the complete AdS 4 ×CP 3 superstring[END_REF] it was shown that such current densities can be used as structural blocks for construction of the Lax connection for the AdS 4 × CP 3 superstring.

Transformation properties of the Cartan forms and coordinates of the

OSp(4|6)/(SO(1, 3) × U (3)) supercoset Consider the OSp(4|6)/(SO(1, 3) × U (3)) supercoset representative G . Under global

OSp(4|6) symmetry transformation with the parameter G it changes as

GG = G ′ H,
where H belongs to the SO(1, 3) × U (3) stability group and depends on the coordinates of this coset-space. Under infinitesimal transformation variation of G equals

δG = gG -G h, g ∈ osp(4|6), h ∈ so(1, 3) ⊕ u(3).
8 Let us note certain ambiguity when using the term 'D = 3 N = 6 superconformal symmetry' both for the transformations of the coordinates of OSp(4|6)/(SO(1, 3) × U (3)) supermanifold from its isometry supergroup and for their limiting case corresponding to superconformal transformations of the boundary superspace coordinates. In general the change of the supercoset coordinates is required in order for its boundary subsuperspace coordinates to have correct transformation properties under this symmetry (see, e.g. discussion in Ref. [START_REF] Claus | Isometries in anti-de Sitter and conformal superspaces[END_REF]).

It can also be presented in the following form

G -1 δG = G -1 gG -h.
(2.48)

Consider the D = 3 N = 6 superconformal algebra valued infinitesimal transformation parameter

g = a m P m + b m K m + f D + 1 2 l mn M mn + y a V a 4 + ȳa V 4 a + w a b V b a +ε µ a Q a µ + εµa Qµa + ξ µa S µa + ξa µ Sµ a .
(

G -1 gG = (ω m (δ) -bm )P m + (c m (δ) + bm )K m + ∆(δ)D +(G mn (δ) + 1 2 lmn )M mn + Ω 4 a (δ)V a 4 + Ω a 4 (δ)V 4 a + (Ω a b (δ) + ŵa b )V b a +ω µ a (δ)Q a µ + ωµa (δ) Qµa + χ µa (δ)S µa + χa µ (δ) Sµ a .
From (2.48) it follows that Cartan forms c(d) = G -1 dG are invariant under global OSp(4|6) symmetry up to compensating transformations

δc(d) = -[c, h] -dh.
The Cartan forms associated with the quotient algebra generators transform according to their so(1, 3) ⊕ u(3) representations, while those associated with the stability algebra generators transform as connections. In particular, variations of the supervielbein bosonic components tangent to the AdS 4 space equal

δω m (d) + δc m (d) = lmn (ω n (d) + c n (d)) + 4 bm ∆(d), δ∆(d) = -bm (ω m (d) + c m (d)), (2.50)
where lmn and bm are parameters of the infinitesimal local SO(1, 3) transformation. Variation of the so(1, 2) Cartan forms in the spinor representation

G µν (d) = ε µλ σ mnλ ν G mn (d) has the form δG µν (d) = 1 4 (G µλ (d) lλ ν +G νλ (d) lλ µ )+ bµ λ (ω(d)-c(d)) λν + bν λ (ω(d)-c(d)) λµ -1 2 d lµν . (2.51)
Variation of the su(4) Cartan forms in the 3 ⊕ 3 basis is given by the expression

δΩ âb (d) = i(Ω âĉ (d) W ĉb -W âĉ Ω ĉb (d)) -d W âb , (2.52)
where

W âb =   ŵa b -δ b a ŵc c 0 0 -ŵb a + δ a b ŵc c   (2.53)
is the matrix of infinitesimal local U (3) transformation. From (2.52) one can find variations of the supervielbein bosonic components tangent to the CP 3 manifold

δΩ a 4 (d) = i ŵb b Ω a 4 (d) -ŵa b Ω b 4 (d) , δΩ 4 a (d) = -i ŵb b Ω 4 a (d) -Ω 4 b (d) ŵb a (2.54)
and the u(3) connection one-form

δΩ a b (d) = i(Ω a c (d) ŵc b -ŵa c Ω c b (d)) -d ŵa b .
Variations of the Grassmann-odd Cartan forms that are identified with the supervielbein fermionic components reduce to infinitesimal transformations from the stability group

δω ν â (d) = 1 4 ω λ â (d) lλ ν + bνλ χ λâ (d) -i W âb ω ν b (d), (2.55
)

δχ νâ (d) = - 1 4 lν λ χ λâ (d) + bνλ ω λ â (d) -i W âb χ ν b(d).
(2.56)

To obtain from the above complete variations of the Cartan forms those corresponding to each of the parameters in (2.49) it is necessary to extract their contributions to the transformation parameters from the stability group.

When the supercoset element is explicitly specified as in (2.39), it becomes possible to find expressions for parameters of the transformations from the stability group as functions

(θ) = b m -i (ξ a (θ)σ m ηa ) + ( ξa (θ)σ m η a ) , ξ µa (θ) = ξ µa + b µν θ ν a , l mn (θ) = l mn +2(b m x n -b n x m )+2i (θ a σ mn ξa )+( θa σ mn ξ a ) +i (θ a σ mn b θa )+( θa σ mn bθ a ) .
Parameters of infinitesimal transformation from the U (3) stability group that enter (2.53) is convenient to present in the form

ŵa b = w a b + i(1-cos |z|) |z| sin |z| (z a y b -ȳa z b ) + i(1-cos |z|) 2 |z| 3 sin 2|z| (( yz) -(z ȳ))z a z b ,
where

w a b = w a b (θ) -e 2φ 2(η a bη b ) -δ b a (η c bη c ) , y a = y a (θ) + e 2φ ε abc (η b bη c ), ȳa = ȳa (θ) -e 2φ ε abc (η b bη c ) (2.57)
and

w a b (θ) = w a b -2(ξ µa θµb + θ µ a ξb µ ) + δ b a (ξ µc θµc + θ µ c ξc µ ) -2(θ a b θb ) + δ b a (θ c b θc ), y a (θ) = y a + ε abc (2ξ µb θ µ c + (θ b bθ c )) , ȳa (θ) = ȳa -ε abc 2 ξb µ θµc + ( θb b θc ) .
Parameters (2.57) are elements of the matrix

W âb =   w a b -δ b a w c c ε acb y c -ε acb ȳc -w b a + δ a b w c c   (2.
δx m = a m + l m n x n + 2f x m + b m (x 2 + ( θθ) 2 ) -2x m b n x n -i (ε a σ m θa ) + (ε a σ m θ a ) -i (ξ a xσ m θa ) + ( ξa xσ m θ a ) +e 2φ bm + i (η a bσ m θa ) + (η a bσ m θ a ) , δθ µ a = ε µ a + 1 4 l mn θ ν a σ mnν µ + f θ µ a + iw b b θ µ a -iw a b θ µ b -iε abc y b θµc + xµν b νλ θ λ a + xµν ξ νa -2i(θ µ b ξb ν + θµb ξ νb )θ ν a + e 2φ bµν
δφ = f (θ) = f -b m x m + i(ξ µa θµa + ξa µ θ µ a ).
Variations of the complex coordinates of the CP 3 manifold

δz a = iz b w b a + i w b b z a + |z| cot |z| y a + 1 2|z| 2 1 -2|z| sin 2|z| ( yz)z a + 1 2|z| 2 [1 + |z|(tan |z| -cot |z|)] (z ȳ)z a
determine variation of the SU (4)/U (3) coset representative (2.43)

δT âb = i(T âĉ W ĉb -W âĉ T ĉb ), δ Tâ b = -i( W âĉ Tĉ b -Tâ ĉ W ĉb ), (2.59)
where matrices W âb and W âb have been introduced in (2.58) and (2.53). Note that variation (2.59) has general form independent of the parametrization of the SU (4)/U (3) coset element.

Finally variations of the Grassmann coordinates associated with the special conformal supersymmetry generators are given by the following expression

δη µa = ξ µa (θ) -1 4 l mn (θ)σ mnµ ν η νa -f (θ)η µa + iw b b (θ)η µa -iw a b (θ)η µb -iε abc y b (θ)η c µ + 2ie
δ X ω m (d)+δ X c m (d) = ∂ ∂X (ω m (δ X )+c m (δ X ))dX +. . . = j m X dX +. . . , δ X ∆(d) = ∂ ∂X ∆(δ X )dX + . . . = j X dX + . . . (2.60) δ X Ω a 4 (d) = ∂ ∂X Ω a 4 (δ X )dX + . . . = - * JaX dX + . . . , δ X Ω 4 a (d) = ∂ ∂X Ω 4 a (δ X )dX + . . . = - * J a X dX + . . . (2.61) and 9 δ X ω µ â (d) = ∂ ∂X ω µ â (δ X )dX + . . . = j µ âX dX + . . . , δ X χ µâ (d) = ∂ ∂X χ µâ (δ X )dX + . . . = J µâX dX + . . . . (2.62)
where X stands for any of the parameters of the D = 3 N = 6 superconformal symmetry and dots stand for the terms on the r.h.s. of (2.50), (2.54), (2.55) and (2.56). Presented in (2.61) variations of the Cartan forms for the su(4)/u(3) quotient-algebra generators can be obtained from the general variation of the su(4) Cartan forms

δ X Ω âb (d) = J âb X dX + . . . ,
where dots stand for the terms on the r.h.s. of (2.52) and

J âb X = ∂ ∂X Ω âb (δ X ) =   J a b X J abX -Jab X -J b a X   . ( 2
J a X = 1 2 ε abc J bcX , * JaX = 1 2 ε abc Jbc X .
So densities of the Noether currents have the form of the sum of three summands

J i X (τ, σ) = J AdS 4 i X + J CP 3 i X + J WZ i X .
The first two of them describe contribution of the kinetic term of the σ-model Lagrangian (2.25). They are determined by contributions to the induced world-sheet metric from metrics of the AdS 4

J AdS 4 i X = - √ -γγ ij 1 4 (ω jm + c jm )j m X + ∆ j j X (2.64) δ a ω m (d) + δ a c m (d) = j m n da n : j m n = ∂(ω m (δa)+c m (δa)) ∂a n = e -2φ Aδ m n δ a ∆(d) = 0 : j m = 0 δ a Ω âb (d) = J âb m da m : J âb m = ∂ ∂a m Ω âb (δ a ) = -2(η âσ m ηb ) δ a ω µ â (d) = j µ âm da m : j µ âm = ∂ω µ â (δa) ∂a m = -e -φ σµν m ηνâ δ a χ µâ (d) = J µâm da m : J µâm = ∂χ µâ (δa) ∂a m = -ie 2φ (ηη)ε µν j ν âm Table 2
J CP 3 i X = 1 2 √ -γγ ij Ω ja 4 * J a X + Ω j4 a *
JaX .

(2.65)

The last term is given by the contribution of the Wess-Zumino Lagrangian (2.27) space. In such gauge physical degrees of freedom of superstring are among 24 coordinates for unbroken supersymmetries. In the present subsection considered is one of the κ-symmetry

J WZ i X = i 4 ε ij J âb ω µâ j ε µν j ν bX + χ â jµ ε µν J ν bX . ( 2
δ b ω m (d) + δ b c m (d) = j mn db n + . . . : j mn = ∂(ω m (δ b )+c m (δ b )) ∂bn = e -2φ A (x 2 + ( θθ) 2 )η mn -2x m x n + i(θ âσ m xσ n θ â) +A ∂ bm ∂bn +ie 2φ (η â xσ n σm η â)+(η â σm Λ -σ n θ â) +2e 2φ (ηη)(η âσ m xσ n θ â), Λ ±µ ν = δ ν µ ± 2iη â µ θ ν â δ b ∆(d) = j m db m + . . . : j m = ∂∆(δ b ) ∂bm = -x m -i(η â xσ m θ â) δ b Ω âb (d) = J âb m db m + . . . : J âb m = ∂ ∂bm Ω âb (δ b ) = -2[( θâ σ m θb )+(x 2 +( θθ) 2 )(η â σm ηb ) -2x m (η â xη b)-( ηâ xσ m Ẑb )+(η b xσ m Ẑâ )], Z µ â = θ µ â -i( θθ)η µ â δ b ω µ â (d) = j µm â db m + . . . : j µm â = ∂ω µ â (δ b ) ∂bm = e -φ [ xµν σ m νλ Ẑλ â -(x 2 + ( θθ) 2 )σ mµν ηνâ +2x m xµν ηνâ +i( θθ)ε µν ηρâ xρλ σ m λν ] δ b χ µâ (d) = J µâ m db m + . . . : J µâ m = ∂χ µâ (δ b ) ∂bm = e φ (Λ -µ ν σ m νλ θλ â -ηνâ xνλ σ m λρ Λ -µ ρ -ie φ (ηη)ε µν j ν âm )
Table 2.2: Tensors contributing to the Noether current density associated with the invariance under conformal boosts

δ f ω m (d) + δ f c m (d) = j m df : j m = ∂(ω m (δ f )+c m (δ f )) ∂f = 2e -2φ Ax m + 2e 2φ (ηη)(η âσ m θ â) δ f ∆(d) = j df : j = ∂∆(δ f ) ∂f = 1 + iθ µ â η â µ δ f Ω âb = J âb df : J âb = ∂ ∂f Ω âb (δ f ) = 2 Θµ â ηb µ -Θµ b ηµâ , Θ µ â = θ µ â -η νâ xνµ δ f ω µ â (d) = j µ â df : j µ â = ∂ω µ â (δf ) ∂f = e -φ ( Θµ â -2x µν ηνâ ) δ f χ µâ (d) = J µâ df : J µâ = ∂χ µâ (δ f ) ∂f = -e φ (Λ -µ ν ηνâ +ie φ (ηη)ε µν j ν â)
Table 2.3: Tensors contributing to the Noether current density associated with the dilatational invariance

δ l ω m (d) + δ l c m (d) = j m kn dl kn + . . . : j m kn = ∂(ω m (δ l )+c m (δ l )) ∂l kn = 1 2 e -2φ A δ m k x n -δ m n x k + i( θθ)ε m kn + 1 2 e 2φ (ηη) δ m k (η âσ n θâ ) -(k ↔ n) + ε m kn i + (ηθ) + ( θη)
δ l ∆(d) = j mn dl mn : j mn = ∂∆(δ l ) ∂l mn = i 4 (θ âσ mn η â) δ l Ω âb (d) = J âb mn dl mn :

J âb mn = ∂ ∂l mn Ω âb (δ l ) = 1 2 ( Θâ σ mn ηb )-( Θb σ mn ηâ )
δ l ω µ â (d) = j µ âmn dl mn + . . . , :

j µ âmn = ∂ω µ â (δ l ) ∂l mn = 1 4 e -φ Θν âσ mnν µ -xµλ σ mnλ ν ηνâ
δ l χ µâ (d) = J µâmn dl mn + . . . :

J µâmn = ∂χ µâ (δ l ) ∂l mn = -e φ 1 4 Λ -µ ν σ mnν λ ηλâ + ie φ (ηη)ε µν j ν âmn Table 2.
δ w ∆ = j a b dw b a : j a b = ∂∆(δw) ∂w b a = δ b a ( θµc η µc + ηc µ θ µ c ) + θ µ a ηb µ + η µa θµb δ w Ω ĉ d(d) = J ĉ da b dw b a + • • • : J ĉ da b = ∂ ∂w b a Ω ĉ d(δ w ) = T ĉb T da -T ĉa T db + i 2 δ b a (T J T ) ĉ d, T âb = T âb + 2iη νâ θ ν b δ w ω µ ĉ (d) = j µ ĉ a b dw b a + . . . : j µ ĉ a b = ∂ω µ ĉ (δw) ∂w b a = e -φ 1 2 δ b a (T J θ µ ) ĉ -iθ µ a T ĉb + i θµb T ĉa δ w χ µĉ (d) = J µĉa b dw b a + . . . : J µĉa b = ∂χ µĉ (δw) ∂w b a = e φ 1 2 δ b a (T Jη µ ) ĉ -iη µa T ĉb +iη b µ T ĉa +ie φ (ηη)ε µν j ν ĉ a b Table 2
invariance δ ε ω m (d) + δ ε c m (d) = j ma µ dε µ a -jm µa dε µa : j ma µ = ∂(ω m (δε)+c m (δε)) ∂ε µ a = 2σ m µν ie -2φ A θνa + e 2φ (ηη)η νa δ ε ∆(d) = j a µ dε µ a -jµa dε µa : j a µ = ∂∆(δε) ∂ε µ a = -iη a µ δ ε Ω b ĉ(d) = J b ĉa µ dε µ a -Jb ĉµa dε µa : J b ĉa µ = ∂ ∂ε µ a Ω b ĉ(δ ε ) = 2(η µ bT ĉa -T ba ηĉ µ ) Jb ĉµa = -∂ ∂ εµa Ω b ĉ(δ ε ) = 2(T ba ηĉ µ -ηµ bT ĉa ) δ ε ω ν b (d) = j ν b a µ dε µ a + jν b µa dε µa : j ν b a µ = ∂ω ν b (δε) ∂ε µ a = e -φ δ ν µ T ba + 2iη µ b θνa δ ε χ ν b(d) = J ν ba µ dε µ a + Jν bµa dε µa : J ν ba µ = ∂χ ν b (δε) ∂ε µ a = ie φ 2η µ b ηa ν -e φ (ηη)ε νλ j λ b a µ
and c.c. gauges in which physical degrees of freedom are present also in the sector of broken supersymmetries and so it can be imposed only in the complete Lagrangian of the superstring. This is the light-cone gauge, in which the light cone is formed by null geodesics on the conformal boundary of the anti-de Sitter space in the Poincaré coordinates [START_REF] Uvarov | AdS 4 ×CP 3 superstring in the light-cone gauge[END_REF]. 10As was mentioned in the Introduction the complete action of the AdS 4 × CP 3 superstring was derived in Ref. [START_REF] Gomis | The complete AdS 4 × CP 3 superspace for type IIA superstring and D-branes[END_REF] via the double-dimensional reduction [START_REF] Duff | Superstrings from supermembranes in D = 11[END_REF], [START_REF] Howe | The supermembrane revisited, Classical and Quantum Gravity[END_REF] of the D = 11 supermembrane action in the AdS 4 × S 7 superbackground [START_REF] De Wit | The M-theory two-brane in AdS 4 × S 7 and AdS 7 × S 4[END_REF]. This approach is based on the possibility of isomorphic realization of the AdS 4 × S 7 superspace as the (11|32)dimensional OSp(4|8)/(SO(1, 3) × SO( 7)) supercoset manifold, geometric constituents of which are identified with the osp(4|8) Cartan forms. The AdS 4 × S 7 supermembrane Lagrangian is constructed out of the Cartan forms for the generators of respective quotient algebra in the similar way as was constructed the Lagrangian of the AdS 5 × S 5 superstring [START_REF] Metsaev | Type IIB superstring action in AdS 5 × S 5 background[END_REF], [START_REF] Kallosh | Near horizon superspace[END_REF]. Also to perform the double-dimensional reduction the seven-sphere was realized as the Hopf fibration CP 3 × S 1 [START_REF] Nilsson | Hopf fibration of eleven dimensional supergravity[END_REF], [START_REF] Sorokin | Kaluza-Klein theories and spontaneous compactification mechanisms of extra space dimensions[END_REF].

j mµa = ∂(ω m (δ ξ )+c m (δ ξ )) ∂ξµa = 2[ie -2φ A θλa σ m λν xνµ + ie 2φ ηa λ σmλν Λ -ν µ +e 2φ (ηη)(η λa σ m λν xνµ -θa λ σmλν Λ +ν µ )] δ ξ ∆(d) = j µa dξ µa -jµ a d ξa µ + . . . : j µa = ∂∆(δ ξ ) ∂ξµa = -i θνa Λ -ν µ + ηa ν xνµ δ ξ Ω b ĉ(d) = Ĵb ĉµa dξ µa -Ĵb ĉµ a d ξa µ : Ĵb ĉµa = ∂ ∂ξµa Ω b ĉ(δ ξ ) = 2(T ba Θµĉ -Θµ b T ĉa ), Ĵb ĉµ a = -∂ ∂ ξa µ Ω b ĉ(δ ξ ) = 2( Θµ b T ĉa -T ba Θµĉ ) δ ξ ω ν b (d) = j ν b µa dξ µa + jνµ ba d ξa µ + . . . : j ν b µa = ∂ω ν b (δ ξ ) ∂ξµa = e -φ (T ba xνµ + 2i θνa Θµ b ) δ ξ χ ν b(d) = J ν bµa dξ µa + Jν bµ a d ξa µ + . . . : J ν bµa = ∂χ ν b (δ ξ ) ∂ξµa = e φ (Λ -ν µ T ba + 2iη a ν Θµ b -ie φ (ηη)ε νλ j λ b µa ) and c.c.
We considered the OSp(4|8)/(SO(1, 3) × SO( 7 The Cartan forms associated with generators of the osp(4|8)/(so(1, 3) × so(7)) quotient algebra, that are constructed out of the element (2.68), determine the Lagrangian of the supermembrane in terms of the AdS 4 × S 7 superspace coordinates and their differentials.

Then the double-dimensional reduction of its action yields the AdS 4 ×CP 3 superstring action. This procedure will be considered in detail in the next paragraph. (2.70)

The kinetic term represents the integral over the world-volume V of the determinant of the induced metric

g (3) ij = E m i E j m,
where ξi = (τ, σ, ρ) are the world-volume local coordinates and E m i ( m, n = 0, 1, ...9, 11) is the mapping to the world-volume of the D = 11 supervielbein bosonic components. The Wess-Zumino term

S AdS 4 ×S 7 WZ = s M 4 H (4)
is given by the integral of the closed four-form

H (4) (d) = i 8 F α(d) ∧ g mn α β F β (d) ∧ E m(d) ∧ E n(d) + 1 4 ε m ′ n ′ k ′ l ′ E m ′ (d) ∧ E n ′ (d) ∧ E k ′ (d) ∧ E l ′ (d)
over the four-dimensional auxiliary hypersurface M 4 , whose boundary is the supermembrane's world volume V . Kinetic term has the same structure as in the flat superspace [START_REF] Bergshoeff | Supermembranes and eleven-dimensional supergravity[END_REF],

while the second summand in H To perform the double-dimensional reduction [START_REF] Duff | Superstrings from supermembranes in D = 11[END_REF], [START_REF] Howe | The supermembrane revisited, Classical and Quantum Gravity[END_REF] of the supermembrane the D = 11 supervielbein components need not depend on the coordinate y ∈ [0, 2π) of the spacetime compact dimension that is subject to reduction. However, when the supervielbein bosonic components include the term dyE m′ y

( m′ = 0, 1, ..., 9), as is the case for the OSp(4|8)/(SO(1, 3) × SO( 7)) supercoset elements considered in [START_REF] Gomis | The complete AdS 4 × CP 3 superspace for type IIA superstring and D-branes[END_REF], [START_REF] Grassi | Simplifying superstring and D-brane actions in AdS 4 × CP 3 superbackground[END_REF] and in our works, the local Lorentz rotation in the tangent space has to be performed

E m(d) → L m nE n(d), L m n ∈ SO(1, 10), F α(d) → L α β F β (d), L α β ∈ Spin(1,
E α(d) = (Lf ) α(d) -(LF y ) αA L (d) (2.73)
and the term proportional to the dilatino superfield

χ α = (LF y ) α.
(2.74)

In the expressions (2.73) and (2.74) there has been used the decomposition of the D = 11 supervielbein fermionic components

F α(d) = f α(d) + dyF α y , (2.75)
where f α(d) and F α y do not depend on y and f α(d) does not contain terms proportional to dy.

Further the space-time compact dimension is identified with the world-volume compact dimension y = ρ and is integrated out. Then the kinetic term of the supermembrane action (2.70) reduces to the kinetic term of the superstring action in the Kaluza-Klein frame (2) , where ξ i = (τ, σ) are world-sheet local coordinates and g (2) = det g

V d 3 ξ -g (3) → Σ d 2 ξΦ L -g
(2)
ij is the determinant of the induced world-sheet metric

g (2) ij = E m′ i E j m′ .
The Wess-Zumino term of the supermembrane reduces to that of the superstring (2.77)

M 4 H (4) → M 3 H (3) (2
The first term takes value in the sp(4) = so(2, 3) algebra and can be expressed in terms of the conf (1, 2) algebra generators and associated Cartan forms

G sp(4) (d) = G m ′ n ′ (d)M m ′ n ′ + 2G 0 ′ m ′ (d)M 0 ′ m ′ = G mn (d)M mn + ∆(d)D + ω m (d)P m + c m (d)K m ∈ sp(4).
The Cartan forms G 0 ′ m ′ (d) are identified with the D = 11 supervielbein bosonic components tangent to AdS 4 and connected to the Cartan forms in conformal basis by the expressions analogous to those in (2.24)

G 0 ′ m (d) = 1 2 (ω m + c m )(d), G 0 ′ 3 (d) = -∆(d).
(2.78)

The second term contains the so(8) generators V IJ together with the Cartan forms Ω IJ (d)

G so(8) (d) = Ω IJ (d)V IJ ∈ so(8). (2.79)
It can be presented in the form with manifest so [START_REF] Nilsson | Hopf fibration of eleven dimensional supergravity[END_REF] or so [START_REF] Watamura | Spontaneous compactification and CP N : SU (3)×SU (2)×U (1), sin 2 θ W , g 3 /g 2 and SU (3)-triplet chiral fermions in four dimensions[END_REF] covariance

G so(8) (d) = Ω I ′ J ′ (d)V I ′ J ′ + Ω 8I ′ (d)V 8I ′ = Ω IJ (d)V IJ + Ω 87 (d)V 87 + Ω 8I (d)V 8I + 2Ω 7I (d)V 7I .
The last equality can also be written in the 3 ⊕ 3 basis

G so(8) (d) = Ω âb (d)V bâ + Ω 87 (d)V 87 - 1 2 Ω 8â (d)V 8 â -Ω 7â (d)V 7 â.
(2.80) Definitions of the so(6) Cartan forms and generators in this basis

Ω âb (d) = Ω a b (d) -δ b a Ω c c (d) ε acb Ω 4 c (d) -ε acb Ω c 4 (d) -Ω b a (d) + δ a b Ω c c (d)
and

V âb = 1 2 V a b -δ b a V c c ε acb V 4 c (d) -ε acb V c 4 -V b a + δ a b V c c
coincide with those in (2.8) and (2.9). The so(8)/(so(2)×so( 6)) Cartan forms and generators in the 3 ⊕ 3 basis are defined as

Ω 7(8) â(d) = Ω 7(8) a (d) Ω 7(8)a (d) = M -1 âI Ω 7(8)I (d) and V 7(8) â = V 7(8) a V 7(8)a = M -1 âI V 7(8)I .
In order to obtain the vielbein of the seven-sphere S 7 in the form that exhibits its Hopf fibration realization it is necessary to consider another embedding of the su(4)⊕u(1) isometry algebra of CP 3 × S 1 into the so(8) algebra compared to the standard one that follows from presenting its commutation relations

[V IJ , V KL ] = δ IL V JK -δ IK V JL + δ JK V IL -δ JL V IK
in the form with manifest so( 6) covariance

[V 87 , V 7I ] = V 8I , [V 87 , V 8I ] = -V 7I , V 7I , V 7J = [V 8I , V 8J ] = -V IJ , [V 7I , V 8J ] = -δ IJ V 78 , V IJ , V 7(8)K = δ JK V 7(8)I -δ IK V 7(8)J , V IJ , V KL = δ IL V JK -δ IK V JL + δ JK V IL -δ JL V IK .
The reason is that generators V 8I and V 87 with which associated are the supervielbein bosonic components in the tangent space to S 7 cannot be identified with the generators of the su(4)/u(3) factor-algebra and u(1) subalgebra of the so(8) algebra since commutation relations of V 8I close on the so(6) = su(4) generators V IJ rather than u(3) ones. Also the generator V 87 does not commute with V 8I (see discussion in [START_REF] Gomis | The complete AdS 4 × CP 3 superspace for type IIA superstring and D-branes[END_REF]). Taking into account chosen realization of the Kähler two-form components (2.28) new u(1) ⊂ so(8)/su(4) and u(3) ⊂ su(4) generators are defined by the relations

H = 2V a a -V 78 , (2.81) Ṽa b = V a b - 1 2 δ b a V c c - 1 4 δ b a V 78 = V a b - 1 4 δ b a H (2.82)
and new su(4)/u(3) quotient algebra generators are given by the expressions

T a = 1 2 (V 7 a -iV 8 a ), T a = - 1 2 (V 7a + iV 8a ). ( 2

.83)

These generators can be presented in the form that exhibits so(6) covariance in the 3 ⊕ 3 basis

T â = T a -T a = 1 2 (V 7 â - 1 2 J âb V 8 b).
New generators (2.82)-(2.83) indeed satisfy the commutation relations of the su(4) algebra

[T a , T b ] = i( Ṽa b + δ b a Ṽc c ), [T a , Ṽb c ] = -iδ c a T b , [T a , Ṽb c ] = iδ a b T c , [ Ṽa b , Ṽc d ] = i(δ b c Ṽa d -δ d a Ṽc b )
and commute with the generator H (2.81). Remaining 12 generators of the so(8) algebra that span the so(8)/su(4) × u(1) quotient algebra equal

Tâ = -Ta T a = 1 2 V 7 â + 1 2 J âb V 8 b , V a 4 , V 4 a .
(2.84)

All commutation relations of the so(8) algebra in this basis are given in the Appendix B of our work [START_REF] Uvarov | AdS 4 ×CP 3 superstring in the light-cone gauge[END_REF].

The Cartan one-forms associated with new u(1) and u(3) generators are expressed in terms of the Cartan forms introduced in (2.80) as follows

h(d) = 1 4 (Ω 78 (d) + 2Ω a a (d)) (2.85) and Ωa b (d) = Ω a b (d) -δ b a Ω c c (d) + δ b a h(d).
(2.86)

The Cartan forms corresponding to the su(4)/u(3) quotient algebra generators are defined by the relations

Ω a (d) = Ω 7 a (d) - i 2 Ω 8 a (d), Ω a (d) = -Ω 7a (d) - i 2 Ω 8a (d) (2.87)
that can be presented in the so(6) covariant way

Ω â(d) = Ω a (d) -Ω a (d) = Ω 7 â(d) - 1 4 J âb Ω 8 b(d).
Finally Cartan forms associated with the generators Ta and T a introduced in (2.84)

Ωa (d) = -Ω 7 a (d) - i 2 Ω 8 a (d), Ωa (d) = Ω 7a (d) - i 2 Ω 8a (d) (2.88)
can also be presented in the form exhibiting the so(6) covariance in the 3 ⊕ 3 basis

Ωâ (d) = -Ωa (d) Ωa (d) = Ω 7 â(d) + 1 4 J âb Ω 8 b(d).
As a result decomposition (2.79) is presented in equivalent form corresponding to the embedding of the su(4) ⊕ u(1) algebra into so(8) compatible with the Hopf fibration realization of S 7 G so(8

) (d) = Ωa b (d) Ṽb a + Ωb b (d) Ṽa a + Ω a (d)T a + Ω a (d)T a + h(d)H + Ωa (d) T a + Ωa (d) Ta + Ω a 4 (d)V 4 a + Ω 4 a (d)V a 4 .
In terms of the Cartan forms (2.85), (2.86), (2.87) and (2.88) components of the S 7 vielbein

E I ′ (d) = (E a (d), E a (d), E 7 (d))
are given by the expressions

E a (d) = Ω 8 a (d) = i(Ω a (d) + Ωa (d)), E a (d) = Ω 8a (d) = i(Ω a (d) + Ωa (d)), E 7 (d) = E 11 (d) = Ω 87 (d) = h(d) + Ωa a (d).
G 32susy (d) in (2.77) can be presented in the form similar to that for G 24susy (d) from the paragraph 2.1.1

F αA ′ (d)O αA ′ = F µA ′ (d)O µA ′ + F A ′ µ (d)O µ A ′ = ω µA ′ (d)Q µA ′ + χ A ′ µ (d)S µ A ′ = ωµA (d) QµA + ω µ A (d)Q A µ + χA µ (d) Sµ A + χ µA (d)S µA .
D = 11 supervielbein fermionic components have the following expressions

1 √ 2 F µA ′ (d) = ω µA ′ (d) = ωµA (d), ω µ A (d) , 1 √ 2 F A ′ µ (d) = χ A ′ µ (d) = χA µ (d), χ µA (d) (2.89)
in terms of the Cartan forms associated with the generators of Poincaré and special conformal supersymmetries

O µA ′ = 1 √ 2 Q µA ′ = 1 √ 2 QµA Q A µ , O µ A ′ = 1 √ 2 S µ A ′ = 1 √ 2 Sµ A S µA .
2.3.2 AdS 4 × CP 3 superstring in the κ-symmetry light-cone gauge As was mentioned above the fermionic light-cone gauge condition (2.69) corresponds to setting to zero Grassmann-odd coordinates of the superspace associated with the generators having negative so(1, 1) weight

[M +-, Q 2A ′ ] = -Q 2A ′ , [M +-, S 1 A ′ ] = -S 1 A ′ .
so(1, 1) generator M +-= 2M 02 is contained in the algebra of the SO(1, 2) Lorentz group acting on the conformal boundary of the AdS 4 space. Other supersymmetry generators have positive so(1, 1) weight

[M +-, Q 1A ′ ] = Q 1A ′ , [M +-, S 2 A ′ ] = S 2 A ′ .
For the coordinates associated with these generators let us introduce the following notation

θ 1A ′ = θ-A θ - A = θA θ A , η A ′ 2 = η-A η - A = ηA η A . (2.90)
They are dynamical fields of the superstring in the gauge (2.69).

Next let us give expressions for the Cartan forms associated with the supercoset element (2.68) that determine the supervielbein of the OSp(4|8)/(SO(1, 3) × SO(7)) superspace. In the κ-symmetry light-cone gauge supervielbein components in the tangent space to AdS 4 (2.78) reduce to

E + (d) = G 0 ′ + (d) = 1 2 e -2φ dx + , E -(d) = G 0 ′ -(d) = 1 2 e -2φ dx -+ ϖ(d) -2e -2φ Θ 2 dx + -4Θdy E 1 (d) = G 0 ′ 1 (d) = 1 2 e -2φ dx 1 , E 3 (d) = G 0 ′ 3 (d) = -dφ, (2.91)
where

x ± = x 2 ± x 0 , Θ = θ 4 θ4 + η 4 η4 and
ϖ(d) = ie -2φ (dθ a θa -θ a d θa )+i(dθ 4 θ4 -θ 4 d θ4 )+ie 2φ (dη a ηa -η a dη a )+i(dη 4 η4 -η 4 dη 4 ). (2.92)
The S 7 vielbein components acquire the form

E a (d) = Ω 8 a (d) = i(Ω a (d) + ε abc ηb ηc dx + -2e -φ ηa η 4 dx + ), E a (d) = Ω 8a (d) = i(Ω a (d) -ε abc ηb ηc dx + + 2e -φ ηa η4 dx + ),
(2.93) In the proposed light-cone gauge the dy-independent terms in these one-forms equal (2.94) match the Kaluza-Klein ansatz, necessary Lorentz rotation L is performed only in the tangent space to AdS 4 and the dimension to be reduced

E 7 = h(d) + Ωa a (d) = dy + A(d), A(d) = Ω bos a a (d) -e -
ωµ a (d) = e -φ dθ a + dx 1 ηa dx + ηa , ωµa (d) = e -φ dθ a + dx 1 ηa dx + ηa , ( 2
E m ′ (d) E 11 (d) = L G 0 ′ m ′ (d) Ω 87 (d) .
The entries of the matrix

L L = L m ′ n ′ L m ′ 11 L 11 m ′ L 11 11 ∈ SO(1, 4) (2.99)
in the light-cone gauge (2.69) equal

L m ′ n ′ = δ m ′ n ′ - 1 2 G 0 ′ m ′ y G 0 ′ yn ′ , L m ′ 11 = -G 0 ′ m ′ y , L 11 m ′ = G 0 ′ ym ′ , L 11 11 = 1, (2.100)
where

G 0 ′ m ′ y = 1 2 (ω m y + c m y ), 0 = 2Θ(1, 0, -1, 0).
Note that in this gauge the superfield Φ L = 1 + G 0 ′ m ′ y G 0 ′ ym ′ equals unity for the chosen normalization. 11 As a result bosonic components of the transformed D = 11 supervielbein equal

E + (d) = 1 2 e -2φ dx + , E -(d) = 1 2 e -2φ dx -+ ϖ(d) -2e -2φ Θ 2 dx + + 4Θ(Ω bos a a (d) -e -2φ η 4 η4 dx + ), E 1 (d) = 1 2 e -2φ dx 1 , E 3 (d) = -dφ, E 11 (d) = dy + A L (d), A L (d) = A(d) -e -2φ Θdx + .
(2.101)

The matrix of respective Lorentz transformation acting on the supervielbein fermionic components has the form

L α β = δ α β - 1 2 G 0 ′ m ′ y g m ′ α γ g 11γ β .
So (Lf ) α(d) from (2.73) has the following components One-forms given in (2.95), (2.96), as well as components of F α y (2.97) and (2.98) do not change under this transformation. D = 10 supervielbein fermionic components and dilatino superfield are expressed in terms of these one-forms in accordance with the general formula (2.72). D = 10 supervielbein bosonic components (2.101) and (2.93) determine induced worldsheet metric in the gauge (2.69)

(L χ) µa (d) = 0 e φ dη a + 2ie -φ Θη a dx + , (L χ) a µ (d) = 0 e φ dη a -2ie -φ Θ ηa dx +
g (2) l.c. ij = g AdS 4 l.c. ij + g CP 3 l.c. ij .
The form of two summands given by the mappings of the metric tensors of AdS 4 and CP 3 spaces is

g AdS 4 l.c. ij = 1 2 (E + i E - j + E + j E - i ) + E 1 i E 1 j + E 3 i E 3 j = 1 8 e -4φ (∂ i x + ∂ j x -+ ∂ j x + ∂ i x -) + 1 4 e -4φ ∂ i x 1 ∂ j x 1 + ∂ i φ∂ j φ + 1 4 e -2φ [∂ i x + (ϖ j +4ΘΩ bos ja a )+∂ j x + (ϖ i +4ΘΩ bos ia a )]-4e -4φ θ 4 θ4 η 4 η4 ∂ i x + ∂ j x +
(2.102) 11 To obtain correct value of the dilaton field ϕ it is necessary to restore the dependence of Φ L on the radius of the seven-sphere R, the rank of the orbifold projection k and the Planck length in dimension D = 11 l P : (2.103)

Φ L = R klP 1 + G 0 ′ m ′ y G 0 ′ ym ′ .
The Wess-Zumino term is specified by the integral of the mapping to the superstring world sheet of the Neveu-Schwarz-Neveu-Schwarz three-form (2.76) that in the proposed κ-symmetry light-cone gauge equals

H (3)l.c. (d) = i 4 (E α(d)g m′ n′ α β χ β ∧E m′ (d)∧E n′ (d)+E α(d)g m′ 11 α β ∧E β (d)∧E m′ (d)) -ε m ′ n ′ k ′ l ′ E m ′ (d) ∧ E n ′ (d) ∧ E k ′ (d)L l ′ 11 .
(2.104)

In the representation (2.104) there were retained vielbein components that turn to zero in the given gauge to have the compact expression for this three-form. So in the first two terms preserved is the manifest SO(1, 9)-invariance and in the third term -the SO(1, 3)-invariance.

Three-form (2.104) can be presented as exterior differential of the two-form potential that in terms of the superspace coordinates and their differentials equals with the Lagrangian and the Wess-Zumino term -up to the second order similarly to the AdS 5 × S 5 superstring in the analogous light-cone gauge [START_REF] Metsaev | Superstring action in AdS 5 ×S 5 : κ-symmetry light cone gauge[END_REF], [START_REF] Metsaev | Light cone superstring in AdS space-time[END_REF].

B (2)l.c. (d) = 1 2 e -4φ (θ 4 θ4 + η 4 η4 )dx 1 ∧ dx + + e -2φ ηa ηa dx 1 ∧ dx + + 1 4 e -2φ (dθ 4 η4 -dη 4 θ4 + η 4 d θ4 -θ 4 dη 4 ) ∧ dx + + 1 2 e -
L AdS 4 ×CP 3 sstring, l.c. = - 1 2 √ -γγ ij g (2) l.c.ij + sε ij B (2)l.c.ij . ( 2
The superstring Lagrangian (2.106) is invariant under world-sheet reparametrizations.

Gauge freedom related to this symmetry can be fixed in a variety of ways, in particular, within the Hamiltonian approach (see, e.g. [START_REF] Metsaev | Superstring action in AdS 5 ×S 5 : κ-symmetry light cone gauge[END_REF], [START_REF] Metsaev | Light cone superstring in AdS space-time[END_REF], [START_REF] Arutyunov | Foundations of the AdS 5 × S 5 superstring. Part I[END_REF]). Respective gauge conditions will be considered in the next subsection. In this subsection the AdS 4 ×CP 3 superstring in the κ-symmetry light-cone gauge (2.69) will be considered as a constrained Hamiltonian system. In order to fix the light-cone gauge for the reparametrization symmetry one of the space-time coordinates in the light-cone basis will be equated to the world-sheet time coordinate and one of the momentum density components will be set to constant similarly to to respective gauge conditions in the superstring models in flat and AdS 5 × S 5 superbackgrounds [START_REF] Goddard | Quantum dynamics of a massless relativistic string[END_REF], [START_REF] Metsaev | Superstring action in AdS 5 ×S 5 : κ-symmetry light cone gauge[END_REF], [START_REF] Metsaev | Light cone superstring in AdS space-time[END_REF], [START_REF] Arutyunov | Uniform light-cone gauge for strings in AdS 5 × S 5 : Solving su(1|1) sector[END_REF]. The main result that will be obtained in this subsection is the expression for the superstring Hamiltonian as a function of the 8+8 gauge-invariant fields of superspace coordinates.

It will be shown that in the quadratic approximation, that is the leading one in the limit of large tension (or mentioned above momentum component), this Hamiltonian reduces to that of the IIA superstring in flat background [START_REF] Green | Superstring theory. V.I. Introduction[END_REF]. At the same time the Hamiltonian of the AdS 4 × CP 3 superstring in another light-cone gauge obtained in [START_REF] Astolfi | Finite-size corrections in the SU (2) × SU (2) sector of type IIA string theory on AdS 4 × CP 3[END_REF], [START_REF] Sundin | The AdS(4) × CP 3 string and its Bethe equations in the near plane wave limit[END_REF], [START_REF] Zarembo | Worldsheet spectrum in AdS 4 /CF T 3 correspondence[END_REF] in this limit transfers to the Hamiltonian of the Type IIA superstring in the background that is the pp-wave limit [START_REF] Penrose | Any space-time has a plane wave as a limit[END_REF] of the AdS 4 × CP 3 superbackground and that was described in [START_REF] Sugiyama | Type IIA string and matrix string on pp-wave[END_REF], [START_REF] Nishioka | On Type IIA Penrose Limit and N = 6 Chern-Simons Theories[END_REF].

This is explained by the fact that in the gauge proposed by us the light cone is formed by null geodesics on the boundary of the AdS 4 space that is D = 3 Minkowski space. As is known (see, e.g. [START_REF] Blau | Penrose limits and maximal supersymmetry[END_REF]) for the AdS × S space-times, and also for the AdS 4 × CP 3 one, there exist two distinct kinds of null geodesics. Whenever null geodesic lies entirely on the boundary of the anti-de Sitter space, the space, arising in the pp-wave limit taken around it, is necessarily flat. Curved pp-wave space-times arise if tangent vector to the null geodesic has a non-zero component in one of the dimensions of the tangent space to the compact manifold. In both cases the superstring Lagrangians on the pp-wave backgrounds upon fixing the light-cone gauge for the κ-symmetry become quadratic and these theories can be quantized [START_REF] Green | Superstring theory. V.I. Introduction[END_REF], [START_REF] Metsaev | Type IIB Green-Schwarz superstring in plane wave Ramond-Ramond background[END_REF], [START_REF] Sugiyama | Type IIA string and matrix string on pp-wave[END_REF]. Geometric interpretation of the higher-order terms in the expansion in powers of the coordinates and momenta of the superstring Lagrangian or Hamiltonian in the light-cone gauge can be found in [START_REF] Blau | Fermi coordinates and Penrose limits[END_REF].

Formulation of the

L AdS 4 ×CP 3 sstring, l.c. = -T 2 γ ij e -4φ 4 1 2 (∂ i x + ∂ j x -+ ∂ j x + ∂ i x -) + ∂ i x 1 ∂ j x 1 +∂ i φ∂ j φ + g M N ∂ i z M ∂ j z N + B∂ i x + ∂ j x + + e -2φ 4 (∂ i x + ϖ j + ∂ j x + ϖ i ) + (∂ i x + ∂ j z M + ∂ j x + ∂ i z M )q M +T ε ij ωi ∂ j x + + C∂ i x 1 ∂ j x + + ∂ i x + ∂ j z M qM , (2.107)
in which singled out are the derivatives of these coordinates. Here z M (M = 1, ..., 6) are real coordinates of the CP 3 manifold and form of the metric g M N (z) will be specified later. To simplify further consideration in (2.107) there were introduced the following quantities ωi = e -2φ 2 (η a ∂i θa + ∂i

θ a ηa ) + e -2φ 4 (∂ i θ 4 η4 -∂ i η 4 θ4 + η 4 ∂ i θ4 -θ 4 ∂ i η4 ), B = 2[(η a ηa ) 2 + e -φ (ε abc ηa ηb ηc η4 + ε abc ηa ηb ηc η 4 ) + 2e -2φ η 4 η4 (η a ηa -e -2φ θ 4 θ4 )], C = e -2φ (η a ηa + e -2φ 2 Θ), Θ = θ 4 θ4 + η 4 η4 , q M = 1 2 (Ω a M ε abc ηb ηc -Ω aM ε abc ηb ηc ) + e -φ (Ω aM ηa η4 -Ω a M ηa η 4 ) + e -2φ Θ Ωa a M , qM = ie -φ Ω aM ηa θ4 + Ω a M ηa θ 4 + e -φ (θ 4 η4 -η 4 θ4 ) Ωa a M
and one-form ϖ(d) is defined in (2.92). There was also restored the dependence on the string tension T .

The momentum density components conjugate to the Poincaré coordinates of the AdS 4 space equal

p -(τ, σ) = -T e -4φ 8 γ τ i ∂ i x + , p + (τ, σ) = -T γ τ i e -4φ 8 ∂ i x -+B∂ i x + +q M ∂ i z M + e -2φ 4 ϖ i -T (C∂ σ x 1 -qM ∂ σ z M + ωσ ), (2.108) p 1 (τ, σ) = -T e -4φ 4 γ τ i ∂ i x 1 + T C∂ σ x + , p φ (τ, σ) = -T γ τ i ∂ i φ.
At the same time for momentum density components conjugate to the CP 3 coordinates we obtain the following expression

p M (τ, σ) = -T γ τ i (g M N ∂ i z N + q M ∂ i x + ) -T qM ∂ σ x + .
(2.109)

With the aid of these relations one can express world-sheet time derivatives of the coordinate fields in terms of the momentum density components

∂ τ x + = -8e 4φ T γ τ τ p --γ τ σ γ τ τ ∂ σ x + , ∂ τ x -= -2e 2φ ϖ τ + 1 γ τ τ A --γ τ σ γ τ τ (∂ σ x -+ 2e 2φ ϖ σ ), ∂ τ x 1 = -4e 4φ T γ τ τ (p 1 -T C∂ σ x + ) -γ τ σ γ τ τ ∂ σ x 1 , (2.110) ∂ τ φ = -1 T γ τ τ p φ -γ τ σ γ τ τ ∂ σ φ, ∂ τ z M = -g M N T γ τ τ (p N -8e 4φ p -q N + T qN ∂ σ x + ) -γ τ σ γ τ τ ∂ σ z M , where A -= -8e 4φ T p + -8e 4φ p -(B -(q•q))-(p•q))+8e 4φ ((q• q)∂ σ x + -C∂ σ x 1 + qM ∂ σ z M -ωσ . (2.111)
In (2.111) and in what follows used is the short-hand notation for the scalar product of D = 6 vectors with inverse CP 3 metric g M N (z): q M g M N q N = (q • q) etc. From definitions of the momentum densities (2.108) and (2.109) there follow the Virasoro constraints

T 1 = 8e 4φ T p + p -+ T e -4φ 8 ∂ σ x + ∂ σ x -+ 2e 4φ T p 2 1 + T e -4φ 8 ∂ σ x 1 ∂ σ x 1 + 1 2T p 2 φ + T 2 ∂ σ φ∂ σ φ + 1 2T (p • p) + T 2 ∂ σ z M g M N ∂ σ z N + 32e 8φ T p 2 -((q • q) -B) + 8e 4φ p -(ω σ -1 T (p • q) -(q • q)∂ σ x + +C∂ σ x 1 -qM ∂ σ z M ) -( T e -2φ 2 ϖ σ -T q M ∂ σ z M -(p • q) +4e 4φ p 1 C)∂ σ x + + T 2 ((q • q) + B + 4e 4φ C 2 )∂ σ x + ∂ σ x + ≈ 0, T 2 = p + ∂ σ x + + p -∂ σ x -+ p 1 ∂ σ x 1 + p φ ∂ σ φ + p M ∂ σ z M + 2e 2φ p -ϖ σ + T ωσ ∂ σ x + ≈ 0.
Substituting expressions for the velocities (2.110) back into (2.107) we obtain the representation for the Lagrangian of the AdS 4 × CP 3 superstring in the κ-symmetry light-cone gauge in terms of the phase-space variables

L AdS 4 ×CP 3 sstring, l.c. = p + ∂ τ x + + p -∂ τ x -+ p 1 ∂ τ x 1 + p φ ∂ τ φ + p M ∂ τ z M +2e 2φ p -ϖ τ + T ωτ ∂ σ x + + 1 γ τ τ T 1 + γ τ σ γ τ τ T 2 .
(2.112)

Not that in (2.112) components of the inverse auxiliary metric 1 γ τ τ and γ τ σ γ τ τ play the role of the Lagrange multipliers for the Virasoro constraints. Let us fix the light-cone gauge for the reparametrization symmetry by the conditions analogous to those considered in Refs. [START_REF] Goddard | Quantum dynamics of a massless relativistic string[END_REF], [START_REF] Metsaev | Superstring action in AdS 5 ×S 5 : κ-symmetry light cone gauge[END_REF], [START_REF] Metsaev | Light cone superstring in AdS space-time[END_REF] x + (τ, σ) = τ, p -(τ, σ) = 1 2 P-= const.

Using the equation for p + it is possible to express γ τ τ through P-

γ τ τ = -4e 4φ T P-.
Besides in order to bring in (2.112) the kinetic term for fermions 2e 2φ p -ϖ τ to the simplest form let us perform the dilatation of the Grassmann coordinates

θ a ( θa ) → 1 √ P- θ a ( θa ), η a (η a ) → exp -2φ √ P- √ P- η a (η a ), θ 4 ( θ4 ) → exp -φ √ P- √ P- θ 4 ( θ4 ), η 4 (η 4 ) → exp -φ √ P- √ P- η 4 (η 4 ) (2.113)
together with canonical transformation of the transverse bosonic phase-space variables

x 1 → 1 √ P- x 1 , φ → 1 √ P- φ, z M → 1 √ P- z M , p 1 → √ P-p 1 , p φ → √ P-p φ , p M → √ P-p M .
(2.114) Such redefinition, as will be demonstrated below, allows to perform the power-series expansion of the Hamiltonian in the large Plimit. Then the Lagrangian (2.112) takes the form

L AdS 4 ×CP 3 sstring, l.c. = p 1 ∂ τ x 1 + p φ ∂ τ φ + p M ∂ τ z M + i(∂ τ θ A θA -θ A ∂ τ θA + ∂ τ η A ηA -η A ∂ τ ηA ) -H l.c. ,
where the density of the light-cone gauge Hamiltonian contains terms up to the fourth power in the Grassmann coordinates

H AdS 4 ×CP 3 sstring, l.c. (τ, σ) = T 2 exp -4φ √ P- [η a ∂σ θa + ∂σ θ a ηa + 1 2 (∂ σ θ 4 η4 -θ 4 ∂ σ η4 -∂ σ η 4 θ4 +η 4 ∂ σ θ4 )] + T 4 exp -4φ √ P- 2 T exp 4φ √ P- p 2 1 + T 8 exp -4φ √ P- ∂ σ x 1 ∂ σ x 1 + 1 2 T p 2 φ + T 2 ∂ σ φ∂ σ φ + 1 2 T (p • p) (2.115) + T 2 ∂ σ z M g M N ∂ σ z N + 8 P- T exp 8φ √ P- ((q • q) -B) +4 √ P-exp 4φ √ P- C∂ σ x 1 -qM ∂ σ z M -1 T (p • q)
and T = T

P-

. As usual in the (super)string theory in the light-cone gauge the constraint T 2 ≈ 0 is considered to be explicitly solved with respect to derivative ∂ σ x -that does not enter the Hamiltonian. So there remains only the zero-mode of this constraint that expresses the level matching condition via the phase-space variables

2π 0 dσ[p 1 ∂ σ x 1 + p φ ∂ σ φ + p M ∂ σ z M + i(∂ σ θ A θA -θ A ∂ σ θA + ∂ σ η A ηA -η A ∂ σ ηA )] = 0.
In the closed (super)string models in flat (super)space this condition reduces to equality of the eigenvalues of the oscillator number operators for oscillators related to the left-and right-moving transverse coordinate fields for each of the physical states.

Since the superstring Hamiltonian in the light-cone gauge (2.115) is highly non-linear, one can consider a number of limiting cases in which its structure simplifies (see, e.g. [START_REF] Arutyunov | Foundations of the AdS 5 × S 5 superstring. Part I[END_REF] for a review). Dilatations of the phase-space variables defined in (2.113) and (2.114) bring the Hamiltonian to the form suitable for its expansion in the large Pand T limit with T fixed. Corresponding series is the power series in the phase-space variables.

To obtain its explicit form it is necessary to choose parametrization of the CP 3 manifold.

As in subsection 2.1.2 consider the SU (4)/U (3) coset element (2.43) parametrized by three complex coordinates z a and their conjugates za . So the complex vielbein and the u(3) connection of the CP 3 space are given by the Cartan one-forms (2.44) and the square of the infinitesimal length element equals

ds 2 CP 3 = g ab dz a dz b + g ab dz a dz b + 2g a b dz a dz b
where the metric components have the form

g ab = 1 4|z| 4 (|z| 2 -sin 2 |z| + sin 4 |z|)z a zb , g ab = 1 4|z| 4 (|z| 2 -sin 2 |z| + sin 4 |z|)z a z b , g a b = sin 2 |z| 2|z| 2 δ b a + 1 4|z| 4 (|z| 2 -sin 2 |z| -sin 4 |z|)z a z b .
(2.116)

Components of the inverse metric are defined by the expressions

g -1 ab = |z| 2 -sin 2 |z|+sin 4 |z| |z| 2 (sin 4 |z|-sin 2 |z|) za zb , g -1ab = |z| 2 -sin 2 |z|+sin 4 |z| |z| 2 (sin 4 |z|-sin 2 |z|) z a z b , g -1 a b = 2|z| 2 sin 2 |z| δ b a + sin 2 |z|(1+2|z| 2 )-sin 4 |z|-|z| 2 |z| 2 (sin 2 |z|-sin 4 |z|) za z b .
(2.117)

Components of the metric (2.116) and its inverse (2.117) admit power-series expansions

g ab = 1 3 -8 45 |z| 2 za zb + O(z 6 ), g ab = 1 3 -8 45 |z| 2 z a z b + O(z 6 ), g a b = 1 2 -1 6 |z| 2 + 1 45 |z| 4 δ b a -za z b 1 6 -7 45 |z| 2 + O(z 6 ) and g -1ab = -4 3 + 48 45 |z| 2 z a z b + O(z 6 ), g -1 ab = -4 3 + 48 45 |z| 2 za zb + O(z 6 ), g -1 a b = 2 1 + 1 3 |z| 2 + 1 15 |z| 4 δ b a + 2z a z b 1 3 + 21 45 |z| 2 + O(z 6
). Analogously one can obtain expansion of the components of the SU (4)/U (3) coset element

(2.43) T a b = T b a = 1 -1 2 |z| 2 + 1 24 |z| 4 δ b a + 1 2 za z b 1 -1 12 |z| 2 + O(z 6 ), T ab = iε acb z c (1 -1 6 |z| 2 ) + O(z 5 ), T ab = -iε acb zc (1 -1 6 |z| 2 ) + O(z 5
) and the CP 3 complex vielbein in (2.93) and bosonic part of the Ramond-Ramond one-form in (2.94)

Ω a (d) = Ω a,b dz b + Ω a ,b dz b : Ω a,b = za zb 1 4 + 1 6 |z| -1 16 |z| 2 + O(z 6 ), Ω a ,b = 1 -1 6 |z| 2 + 1 120 |z| 4 δ b a -za z b 1 4 -1 6 |z| -1 16 |z| 2 + O(z 6 ); Ω a (d) = Ω a ,b dz b + Ω a,b dz b : Ω a ,b = Ω b ,a , Ω a,b = z a z b ( 1 4 + 1 6 |z| -1 16 |z| 2 ) + O(z 6 ); Ωa a (d) = Ωa a ,b dz b + Ωa a,b dz b : Ωa a ,b = i 2 zb (1 -1 3 |z| 2 ) + O(z 5 ), Ωa a,b = -i 2 z b (1 -1 3 |z| 2 ) + O(z 5
). Taking into account above expansions quadratic part of the superstring Hamiltonian in the light-cone gauge (2.115) is found to be

H (2) l.c. = H (2) b + H (2) f
, where contributions of the bosonic and fermionic fields equal

H (2) b = 1 2 (p 2 1 + T 2 16 ∂ σ x 1 ∂ σ x 1 ) + 1 8 (p 2 φ + T 2 ∂ σ φ∂ σ φ) + 1 2 (p a pa + T 2 4 ∂ σ z a ∂ σ za ), H (2) f = T 2 (η a ∂ σ θa + ∂ σ θ a ηa ) + T 4 (∂ σ θ 4 η4 -∂ σ η 4 θ4 + η 4 ∂ σ θ4 -θ 4 ∂ σ η4 ).
Respective quadratic Lagrangian of the superstring in the light-cone gauge has the form

L (2) l.c. = 1 2 (∂ τ x 1 ∂ τ x 1 - T 2 16 ∂ σ x 1 ∂ σ x 1 ) +2(∂ τ φ∂ τ φ - T 2 16 ∂ σ φ∂ σ φ) + 2(∂ τ z a ∂ τ za - T 2 16 ∂ σ z a ∂ σ za ) -i(∂ τ θ A θA -θ A ∂ τ θA + ∂ τ η A ηA -η A ∂ τ ηA ) -T 2 (η A ∂ σ θA + ∂ σ θ A ηA ).
(2.118)

Terms including bosonic fields coincide with those in the (super)string Lagrangian in Minkowski (super)space in the light-cone gauge [START_REF] Green | Superstring theory. V.I. Introduction[END_REF]. To bring the terms quadratic in the fermions to the form that has the Lagrangian for massless spinor field in the world-sheet field theory introduce eight-component spinor fields

Ψ A ′ (τ, σ) = -C A ′ B ′ θ B ′ + γ 7 A ′ B ′ η B ′ , ΨA ′ = (Ψ A ′ ) † , Φ A ′ (τ, σ) = -C A ′ B ′ θ B ′ -γ 7 A ′ B ′ η B ′ , ΦA ′ = (Φ A ′ ) † ,
where θ B ′ and η B ′ were introduced in (2.90) and D = 7 charge conjugation and γ-matrices are defined in the Appendix A of our work [START_REF] Uvarov | AdS 4 ×CP 3 superstring in the light-cone gauge[END_REF]. Introduced spinor fields satisfy the conditions

ΨA ′ = -C A ′ B ′ Ψ B ′ and ΦA ′ = -C A ′ B ′ Φ B ′
-i 2 ΨA ′ ∂ -Ψ A ′ -i 2 ΦA ′ ∂ + Φ A ′ ,
where

∂ ± ≡ ∂ τ ± T 4 ∂ σ .
Thus we have established that quadratic Lagrangian

L (2) l.c. = 1 2 (∂ τ x 1 ∂ τ x 1 - T 2 16 ∂ σ x 1 ∂ σ x 1 ) + 2(∂ τ φ∂ τ φ - T 2 16 ∂ σ φ∂ σ φ) +2(∂ τ z a ∂ τ za - T 2 16 ∂ σ z a ∂ σ za ) + i 2 Ψ A ′ ∂ -Ψ A ′ + i 2 Φ A ′ ∂ + Φ A ′
coincides with the superstring Lagrangian in flat superspace in the light-cone gauge [START_REF] Green | Superstring theory. V.I. Introduction[END_REF].

Cubic terms in the expansion of the Hamiltonian of superstring are given by the sum of the terms that include only bosonic fields and those including products of two fermionic

fields √ P-H (3)
l.c. = H

+ H

= - T 2 4 φ∂ σ x 1 ∂ σ x 1 -1 2 φ(p 2 φ + T 2 ∂ σ φ∂ σ φ) -2φ(p a pa + T 2 4 ∂ σ z a ∂ σ za ), H (3) bf , where H (3) b
bf = -2 T φ(η a ∂ σ θa + ∂ σ θ a ηa ) -i T (ε abc η a zb ∂ σ θ c + ε abc ηa z b ∂ σ θc ) -T φ(∂ σ θ 4 η4 -∂ σ η 4 θ4 + η 4 ∂ σ θ4 -θ 4 ∂ σ η4 ) -2(p a ηa η4 -pa η a η 4 ) +ε abc p a η b η c -ε abc pa ηb ηc + 2 T (η a ηa + 1 2 Θ)∂ σ x 1 -2i T (∂ σ z a η a θ 4 + ∂ σ za ηa θ4 ). (3)
Quartic-order terms can be presented as the sum of three contributions P-H

l.c. = H

+ H

bf + H

, where 7)) supercoset representatives considered by other authors [START_REF] Gomis | The complete AdS 4 × CP 3 superspace for type IIA superstring and D-branes[END_REF],

H (4) b = T 2 φ 2 ∂ σ x 1 ∂ σ x 1 + φ 2 (p 2 φ + T 2 ∂ σ φ∂ σ φ) + 4φ 2 (p a pa + T 2 4 ∂ σ z a ∂ σ za ) -1 6 ((z a p a ) 2 + (p a za ) 2 -|z| 2 (p a p a ) -(z a p a )(p b zb )) + T 2 24 ((∂ σ z a za ) 2 + (z a ∂ σ za ) 2 -|z| 2 (∂ σ z a ∂ σ za ) -(∂ σ z a za )(z b ∂ σ zb )), H (4) bf = 4 T φ 2 (η a ∂ σ θa + ∂ σ θ a ηa ) + 4i T φ(ε abc η a zb ∂ σ θ c + ε abc ηa z b ∂ σ θc ) -T |z| 2 (η a ∂ σ θa + ∂ σ θ a ηa ) + T ((z a η a )(∂ σ θb zb ) + (z a ∂ σ θ a )(η b zb )) +2 T φ 2 (∂ σ θ 4 η4 -∂ σ η 4 θ4 + η 4 ∂ σ θ4 -θ 4 ∂ σ η4 ) + 8φ(p a ηa η4 -pa η a η 4 ) -4φ(ε abc p a η b η c -ε abc pa ηb ηc ) + 2i(ε abc p a zb η c η4 + ε abc pa z b ηc η 4 ) +i((z a p a ) -(p a za ))Θ -12 T φ(η a ηa + 1 2 Θ)∂ σ x 1 -2i T (ε abc η a zb η c + ε abc ηa z b ηc )∂ σ x 1 + 8i T φ(∂ σ z a η a θ 4 + ∂ σ za ηa θ4 ) +2 T (ε abc ∂ σ z a z b ηc θ 4 -ε abc ∂ σ za zb η c θ4 ) + T ((∂ σ z a za ) -(z a ∂ σ za ))(θ 4 η4 -η 4 θ4 ), H (4)
[48] and in our work [START_REF] Uvarov | AdS 4 ×CP 3 superstring in the light-cone gauge[END_REF] yield expressions for the D = 11 supervielbein components that do not dependent on y but those in the tangent space to AdS 4 include summands proportional to dy. So expressions for them differ from the Kaluza-Klein ansatz that determines the form of the D = 11 supervielbein necessary to perform the double-dimensional reduction of the supermembrane. These summands are excluded by the local SO(1, 4) tangent-space rotation of the supervielbein components that complicates their form. Imposition of the Kaluza-Klein condition makes unnecessary such transformation that simplifies reduction procedure and the AdS 4 × CP 3 superstring Lagrangian.

There will be obtained explicit form of the Kaluza-Klein condition for the OSp(4|8)/(SO(1, 3) × SO( 7)) supercoset representative parametrized by the coordinates associated with the generators of the D = 3 N = 8 superconformal algebra [START_REF] Uvarov | AdS 4 ×CP 3 superstring in the light-cone gauge[END_REF]. It will be shown that proportional to dy contributions to the D = 11 supervielbein components in the tangent space to AdS 4 depend only on the eight coordinates for supersymmetries that are broken by the AdS 4 × CP 3 superbackground. There will be analyzed SL(2, R)covariant conditions that turn these contributions to zero. To this end two-component SL(2, R) spinor coordinates of the sector of broken supersymmetries will be decomposed on the four (anti-)Majorana spinors with real components each of which constitutes minimal SL(2, R)-covariant coordinate set. It will be shown that the Kaluza-Klein condition can be satisfied by putting two of them equal zero. Obtained will be respective expressions for the supervielbein components of the AdS 4 × S 7 and AdS 4 × CP 3 superspaces.

For further simplification of the form of the AdS 4 × S 7 supervielbein and AdS 4 × CP 3

superstring Lagrangian considered will be partial κ-symmetry gauge in which in the sector of broken supersymmetries remains the SL(2, R) Majorana spinor coordinate corresponding to generators of the broken Poincaré supersymmetries. Constructed will be the osp(4|6)-superalgebra valued world-sheet one-form that extends the Lax connection of the OSp(4|6)/(SO(1, 3) × U (3)) σ-model by linear and quadratic terms in this Majorana spinor coordinate and its differential and proved that its curvature turns to zero on the superstring equations in this gauge [START_REF] Uvarov | Kaluza-Klein gauge and minimal integrable extension of OSp(4|6)/(SO(1, 3) × U (3)) sigma-model[END_REF].

In subsection 3.2 it will be proved [START_REF] Uvarov | A note about fermionic equations of AdS 4 × CP 3 superstring[END_REF] that, when coordinates of the sector of broken supersymmetries vanish, fermionic equations of the AdS 4 × CP 3 superstring, corresponding to their variation, are the consequences of other 24 fermionic equations. The latter turn into fermionic equations of the OSp(4|6)/(SO(1, 3) × U (3)) σ-model. This confirms that the σ-model consistently describes dynamics of the superstring moving in the AdS 4 and CP 3 spaces.

In subsection 3.3 it will be proved classical integrability of equations of the massless superparticle in the OSp(4|6)/(SO(1, 3) × U (3)) supercoset space [START_REF] Uvarov | Lagrangian mechanics of massless superparticle on AdS 4 × CP 3 superbackground[END_REF]. It will also be proved that, analogously to the case of the AdS 4 × CP 3 superstring, fermionic equations of the AdS 4 × CP 3 superparticle corresponding to variation of the coordinates of the sector of broken supersymmetries are the consequences of the fermionic equations of the massless OSp(4|6)/(SO(1, 3) × U (3)) superparticle, when these coordinates turn to zero. Besides it will be established the relation between the σ-model Lax connection and components of the Lax pair of the superparticle in the OSp(4|6)/(SO(1, 3) × U (3)) supercoset space. As the generalization of the obtained results it will be proved classical integrability of equations of the massless superparticle [START_REF] Uvarov | On integrability of massless AdS 4 ×CP 3 superparticle equations[END_REF] and the D0-brane [START_REF] Uvarov | On integrability of D0-brane equations on AdS 4 ×CP 3 superbackground[END_REF] in the AdS 4 × CP 3 superbackground.

3.1 Kaluza-Klein gauge for κ-symmetry and minimal extension of

the integrable OSp(4|6)/(SO(1, 3) × U (3)) σ-model
As was noted in Section 2 for considered in [START_REF] Gomis | The complete AdS 4 × CP 3 superspace for type IIA superstring and D-branes[END_REF], [START_REF] Grassi | Simplifying superstring and D-brane actions in AdS 4 × CP 3 superbackground[END_REF], [START_REF] Uvarov | AdS 4 ×CP 3 superstring in the light-cone gauge[END_REF] elements of the OSp(4|8)/(SO(1, 3) × SO( 7)) supercoset the so(2, 3)/so(1, 3) Cartan forms, that determine components of the AdS 4 × S 7 supervielbein in the tangent space to AdS 4 , include proportional to dy summands. So they can be presented in the form similar to the fermionic supervielbein components (2.75)

E m ′ (d) = G m ′ (d) + dyG m ′ y . (3.1)
Expression (3.1) differs from the Kaluza-Klein ansatz by the second summand. To exclude it one needs to perform in the tangent-space the local SO(1, 4) transformation, parameters of which depend on G m ′ y . This complicates the form of the D = 11 supervielbein and the AdS 4 × CP 3 superstring Lagrangian. In our work [START_REF] Uvarov | Kaluza-Klein gauge and minimal integrable extension of OSp(4|6)/(SO(1, 3) × U (3)) sigma-model[END_REF] proposed was the Kaluza-Klein condition

G m ′ y = 0, (3.2)
imposition of which simplifies derivation of the superstring Lagrangian and its form. For mentioned OSp(4|8)/(SO(1, 3) × SO( 7)) supercoset elements G m ′ y depends only on eight coordinates related to the supersymmetries broken by the AdS 4 × CP 3 superbackground.

Thus for them the Kaluza-Klein condition (3.2) constrains coordinates of the sector of broken supersymmetries.

The study of the κ-symmetry gauge conditions that restrict the coordinates of the sector of broken supersymmetries is of interest also in connection with the problem of the proof of integrability of the AdS 4 × CP 3 superstring equations considered in Refs. [START_REF] Sorokin | Evidence for the classical integrability of the complete AdS 4 ×CP 3 superstring[END_REF], [START_REF] Cagnazzo | More on integrable structures of superstrings in AdS 4 × CP 3 and AdS 2 × S 2 × T 6 superbackgrounds[END_REF]. Parametrization of the sector of broken supersymmetries by four (anti-)Majorana SL(2, R) spinor coordinates proposed in our work [START_REF] Uvarov | Kaluza-Klein gauge and minimal integrable extension of OSp(4|6)/(SO(1, 3) × U (3)) sigma-model[END_REF] allows to consider partial κ-symmetry gauges in which three of them vanish. For each of them it is possible to find the zerocurvature representation for superstring equations that includes the Lax connection of the OSp(4|6)/(SO(1, 3) × U (3)) σ-model extended by the terms depending on remaining coordinates of the sector of broken supersymmetries. By weakening imposed gauge the σ-model Lax connection can be extended by contributions of other coordinates from this sector while preserving vanishing condition of the curvature on the superstring equations. In the limiting case, when no gauge conditions are imposed, this procedure should give sought for Lax connection of the AdS 4 × CP 3 superstring, the form of which remains unknown. Realization of this approach in the massless superparticle model, that corresponds to the infinite-tension limit of the AdS 4 × CP 3 superstring, allowed, as will be described in subsection 3.3, to prove integrability of its equations and also those of the D0-brane.

In this subsection there will be considered an example of such partial gauge-fixing condition [START_REF] Uvarov | Kaluza-Klein gauge and minimal integrable extension of OSp(4|6)/(SO(1, 3) × U (3)) sigma-model[END_REF]. In the proposed gauge there does not turn to zero only spinor coordinate for broken Poincaré supersymmetries that satisfies the Majorana condition θ µ = θµ . Derived will be the superstring equations in the given gauge and obtained the world-sheet one-form, curvature of which turns to zero on these equations. It includes besides the Lax connection of the OSp(4|6)/(SO(1, 3) × U (3)) σ-model the linear and quadratic terms in θ µ and its differential. These terms constitute the subset of the linear and quadratic terms found in [START_REF] Cagnazzo | More on integrable structures of superstrings in AdS 4 × CP 3 and AdS 2 × S 2 × T 6 superbackgrounds[END_REF].

AdS

Conditions (3.5) include common sign factor for coordinates θ and η to turn to zero ∆ y .

These conditions leave four coordinates in the sector of broken supersymmetries.

For definiteness we will concentrate in the following on the case s = 1 and will continue to use the same notation θ µ and η µ for coordinates that satisfy the Majorana condition.

Then the AdS 4 × S 7 supervielbein bosonic components in the tangent space to AdS 4 will have the form

E m (d) = G m (d) = G 0 ′ m (d) -i 2 ε mkl G kl (d)[(θθ) + (ηη)] -2ic n (d)(θσ n σm η) -c m (d)(θθ)(ηη) -i(dθσ m θ + dησ m η), -E 3 (d) = -G 3 (d) = [1 + 2i(θη)]∆(d) + iG mn (d)(θσ mn η) + 2i(dθη), (3.6)
where G mn (d) is the so(1, 2) Cartan form that is the constituent of the so(1, 3) connection. Tangent to the CP 3 manifold components of the AdS 4 × S 7 supervielbein equal determines the D = 10 dilaton superfield ϕ. Among fermionic components of the AdS 4 × S 7

G 0 ′ m (d) = 1 2 (ω m + c m )(d)
E a (d) = i(Ω a + Ω a )(d) = i[Ω a (d) + 2χ µa (d)θ µ -2ω µ a (d)η µ -2iχ µ a (d)η µ (θθ)], E a (d) = i(Ω a + Ω a )(d) = i[Ω a (d) -2 χa µ (d)θ µ + 2ω µa (d)η µ + 2i χµa (d)η µ (θθ)]. (3.7)
supervielbein 1 √ 2 F α(d) = 1 √ 2 (f α(d) + dyF α y ) =       ωµA (d) ω µ A (d) χA µ (d) χ µA (d)       =       ωµA (d) ωµ A (d) χA µ (d) χµA (d)       + dy       ωµA y ω µ yA χ A yµ χ yµA       (3.9) the following do not vanish ωµ a (d) = ω µ a (d) + iχ µ a (d)(θθ), ωµ 4 (d) = dθ µ -2idθ µ (θη) + 2iθ µ (dθη) + 1 2 G mn (d)θ ν σ mnν µ +∆(d)θ µ -ω m (d)σ µν m η ν + iG mn (d)η ν σ mnν µ (θθ), ω y µ 4 = 2iθ µ , χµa (d) = χ µa (d) + 4iχ νa (d)θ ν η µ + iω µa (d)(ηη) -χ µa (d)(θθ)(ηη), χµ4 (d) = dη µ + 2iη µ (dθη) + c m (d)σ m µν θ ν -1 2 G mn (d)σ mnµ ν η ν -∆(d)η µ + ic m (d)σ m µν η ν (θθ) + i 2 G mn (d)σ mnµ ν θ ν (ηη) + 2i∆(d)η µ (θη), χ yµ4 = 2iη µ [1 -2i(θη)]
(d) = G 0 ′ m (d) -i 2 ε mkl G kl (d)(θθ) -idθσ m θ, E 3 (d) = -∆(d), E a (d) = i(Ω a (d) + 2χ µa (d)θ µ ), E a (d) = i(Ω a (d) -2 χa µ (d)θ µ ) (3.
L AdS 4 ×CP 3 sstring, min, kin = -1 2 √ -γγ ij (E m i E jm + ∆ i ∆ j -E ia E j a ) , L AdS 4 ×CP 3 sstring, min, WZ = -1 2 [(ω µ a (d) + 2iΩ a (d)θ µ ) ∧ (ω a µ (d) -2iΩ a (d)θ µ ) +χ µa (d) ∧ χµa (d)] + (Ω a (d) ∧ Ω a (d) + 2i Ω a a (d) ∧ ∆(d))(θθ).
To derive superstring equations in the proposed gauge let us choose as independent varia- The reason for such choice is that these Cartan forms are used as variation parameters when one derives the OSp(4|6)/(SO(1, 3) × U (3)) σ-model equations and enter its Lax connection [START_REF] Arutyunov | Superstrings on AdS 4 × CP 3 as a coset sigma-model[END_REF], [START_REF] Stefański | Green-Schwarz action for Type IIA strings on AdS 4 × CP 3[END_REF]. 13 It is assumed that it gives leading contribution to the power-series expansion of the Lax connection of the AdS 4 × CP 3 superstring in coordinates of the sector of broken supersymmetries.

So the superstring equations In our work [START_REF] Uvarov | Kaluza-Klein gauge and minimal integrable extension of OSp(4|6)/(SO(1, 3) × U (3)) sigma-model[END_REF] there was found the zero-curvature representation for these equations

δS AdS 4 ×CP 3 sstring, min δG 0 ′ m(δ) = ∂ i ( √ -γγ ij G j 0 ′ m )+2 √ -γγ ij G i mn G j 0 ′ n +2 √ -γγ ij G i 3m ∆ j +2iω (1) µ a (d) ∧ σ m µν ω(1) νa (d) -2iω (3) µ a (d) ∧ σ m µν ω(3) νa (d) + √ -γγ ij [Ω i a (ω ja σ m θ) -Ω ia (ω a j σ m θ)] -i[Ω a (d) ∧ (χ a (d)σ m θ) +Ω a (d)∧( χa (d)σ m θ)]-i∂ i √ -γγ ij [(∂ j θσ m θ)+ 1 2 ε mnp G j np (θθ)] -2i √ -γγ ij G i mn [(∂ j θσ n θ) + 1 2 ε npr G j pr (θθ)]+ √ -γγ ij [(χ ia σ m ωa j ) -(ω ia σ m χa j )](θθ)-4i Ω a a (d) ∧ G 3m (d)(θθ) = 0, δS AdS 4 ×CP 3 sstring, min δ∆(δ) = ∂ i ( √ -γγ ij ∆ j )-2 √ -γγ ij G i 3m G j 0 ′ m +2ω (1) µ a (d) ∧ ω(1) a µ (d) +2ω (3) µ a (d) ∧ ω(3) a µ (d) + √ -γγ ij (Ω ia χj a µ θ µ -Ω i a χ jµa θ µ ) +i(Ω a (d)∧ω µ a (d)θ µ +Ω a (d)∧ ωµa (d)θ µ )+2i √ -γγ ij G i 3m [(∂ j θσ m θ) + 1 2 ε mnp G j np (θθ)] -2 √ -γγ ij χ i µ a χj a µ (θθ) -4i Ω a a (d) ∧ (dθθ) -2Ω a (d)∧Ω a (d)(θθ)+2i(ω µ a (d)∧ χa µ (d)+χ µ a (d)∧ ωa µ (d))(θθ) = 0, δS AdS 4 ×CP 3 sstring, min δΩa(δ) = 1 2 ∂ i ( √ -γγ ij Ω j a )+ i 2 √ -γγ ij Ω i b ( Ω jb a +δ a b Ω jc c ) -iε abc ω (1) µ b (d)∧ω (1)µc (d)-iε abc ω (3) µ b (d)∧ω (3)µc (d)-∂ i ( √ -γγ ij χj a µ θ µ ) -i √ -γγ ij χi b µ θ µ ( Ω jb a +δ a b Ω jc c )+i √ -γγ ij ε abc Ω ib χ jµc θ µ -id(ω µa (d)θ µ ) +ω µb (d)θ µ ∧( Ω b a (d)+δ a b Ω c c (d))+ε abc Ω b (d)∧ω µ c (d)θ µ +2(dθθ)∧Ω a (d) +2∆(d) ∧ Ω a (d)(θθ) + 3ε abc ω µ b (d) ∧ χ µc (d)(θθ) = 0, - δS AdS 4 ×CP 3 sstring, min δ ω(1) a µ (δ) =-2iV ij + G i 0 ′ m σµν m ω (1)jνa -2V ij + ∆ i ω (1)j µ a +2iV ij + ε abc Ω i b ω(1)j µc -i∂ i ( √ -γγ ij Ω ja θ µ )- √ -γγ ij ( Ω ia b -δ b a Ω ic c )Ω jb θ µ -i √ -γγ ij Ω ia (∆ j θ µ + 1 2 G j mn θ ν σ mnν µ + ic m j σµν m θ ν ) + 4i √ -γγ ij ε abc ω(1)i µb χj c ν θ ν -iΩ a (d)∧[dθ µ +(2i Ω b b (d)-∆(d))θ µ + 1 2 G mn (d)θ ν σ mnν µ -iω m (d)σ µν m θ ν ] +ε abc Ω b (d)∧Ω c (d)θ µ -2iε abc ωµb (d)∧ χc ν (d)θ ν -ε abc ωνb (d)∧ ωc ν (d)θ µ -i∂ i [ √ -γγ ij χ j µ a (θθ)] - √ -γγ ij ( Ω ia b -δ b a Ω ic c )χ j µ b (θθ) (3.14) - √ -γγ ij ε abc Ω i b χj µc (θθ) + 2i √ -γγ ij G i 0 ′ m σµν m ω (3)jνa (θθ) -i √ -γγ ij χ i ν a (∆ j δ µ ν + 1 2 G j mn σ mnν µ -ic m j ε µρ σ mνρ )(θθ) -2 √ -γγ ij (∂ i θσ m θ)σ µν m ω (1)jνa + √ -γγ ij G i mn ε µρ σ mnρ ν ω (1)jνa (θθ) +2ω (3) µ a (d) ∧(dθθ)+2∆(d) ∧ω (3) µ a (d)(θθ) +4i Ω c c (d) ∧ω (1) µ a (d)(θθ) -6iε abc Ω b (d) ∧ ω(1) µc (d)(θθ) = 0, - δS AdS 4 ×CP 3 sstring, min δ ω(3) a µ (δ) =-2iV ij -G i 0 ′ m σµν m ω (3)jνa +2V ij -∆ i ω (3)j µ a -2iV ij -ε abc Ω i b ω(3)j µc +i∂ i ( √ -γγ ij Ω ja θ µ )+ √ -γγ ij ( Ω ia b -δ b a Ω ic c )Ω jb θ µ +i √ -γγ ij Ω ia (∆ j θ µ + 1 2 G j mn θ ν σ mnν µ -ic m j σµν m θ ν ) -4i √ -γγ ij ε abc ω(3)i µb χj c ν θ ν -iΩ a (d)∧[dθ µ +(2i Ω b b (d)-∆(d))θ µ + 1 2 G mn (d)θ ν σ mnν µ +iω m (d)σ µν m θ ν ] +ε abc Ω b (d)∧Ω c (d)θ µ -2iε abc ωµb (d)∧ χc ν (d)θ ν +ε abc ωνb (d)∧ ωc ν (d)θ µ +i∂ i [ √ -γγ ij χ j µ a (θθ)] + √ -γγ ij ( Ω ia b -δ b a Ω ic c )χ j µ b (θθ) + √ -γγ ij ε abc Ω i b χj µc (θθ) -2i √ -γγ ij G i 0 ′ m σµν m ω (1)jνa (θθ) (3.15) +i √ -γγ ij χ i ν a (∆ j δ µ ν + 1 2 G j mn σ mnν µ + ic m j ε µρ σ mνρ )(θθ) -2 √ -γγ ij (∂ j θσ m θ)σ µν m ω (3)jνa + √ -γγ ij G i mn ε µρ σ mnρ ν ω (3)jνa (θθ) -2ω (1)
-i∂ i ( √ -γγ ij G 0 ′ m j )σ µν m θ ν -2i √ -γγ ij G 0 ′ m i (σ µν m ∂ j θ ν + 1 2 ε mkl G kl j θ µ ) +2 √ -γγ ij χ i ν a χj a ν θ µ + 2(Ω a (d) ∧ Ω a (d) + 2i∆(d) ∧ Ω a a (d))θ µ (3.16) +i(ω ν a (d)∧ χa ν (d)+χ ν a (d)∧ ωa ν (d))θ µ - √ -γγ ij G mn i ∂ j θ ν σ mnν µ (θθ) + 3 2 √ -γγ ij ∂ ij θ µ (θθ) -3 √ -γγ ij (∂ i θ∂ j θ)θ µ = 0, δS AdS 4 ×CP 3 sstring, min δϑµ = i √ -gγ ij (Ω ia χj µa +Ω i a χ j µ a )+Ω a (d)∧ω µ a (d)-Ω a (d)∧ ωµa (d) -4i √ -gγ ij Ω ia a G j 0 ′ m σµν m θ ν + 2i √ -gγ ij (χ i µ a χj νa + χ i ν a χj µa )θ ν -2iG 0 ′ m (d)∧G 3n (d)θ ν σ mnν µ -2iG 0 ′ m (d)∧G 3 m (d)θ µ -2i∆(d) ∧(dθ µ + 1 2 G mn (d)θ ν σ mnν µ ) -2(ω µ a (d) ∧ χνa (d) + χ ν a (d) ∧ ωµa (d))θ ν (3.17) +i(Ω a (d)∧χ µ a (d)-Ω a (d)∧ χµa (d))(θθ)+6 √ -γγ ij Ω ia a ∂ j θ µ (θθ) -4G 0 ′ m (d) ∧ σµν m dθ ν (θθ) + c m (d) ∧ σµν m dθ ν (θθ) =
δ ηµ = √ -γγ ij (Ω i a ω j µ a -Ω ia ωj µa )+i(Ω a (d)∧χ µ a (d) +Ω a (d) ∧ χµa (d)) -2i √ -γγ ij G i 0 ′ m (G j 3n θ ν σ mnν µ + G j 3 m θ µ ) -2i √ -γγ ij ∆ i (∂ j θ µ + 1 2 G j mn θ ν σ mnν µ ) + 2i √ -γγ ij Ω ia Ω j a θ µ +2 √ -γγ ij (ω i µ a χj νa -χ i ν a ωj µa )θ ν + 4i Ω a a (d) ∧ G 0 ′ m (d)σ µν m θ ν (3.18) +2iχ ν a (d) ∧ χa ν (d)θ µ + 3i √ -γγ ij (Ω i a χ j µ a -Ω ia χj µa )(θθ) - √ -γγ ij (c m i + 2G i 0 ′ m )σ µν m ∂ j θ ν (θθ) -6 Ω a a (d) ∧ dθ µ (θθ) = 0, δS AdS 4 ×CP 3 sstring, min δηµ - δS AdS 4 ×CP 3 sstring, min δ ηµ = √ -γγ ij (Ω i a ω j µ a +Ω ia ωj µa )+i(Ω a (d)∧χ µ a (d) -Ω a (d) ∧ χµa (d)) + 4 √ -γγ ij ∆ i Ω ja a θ µ + 2 √ -γγ ij (ω i µ a χj νa +χ i ν a ωj µa )θ ν -2G 0 ′ m (d) ∧ σµν m (dθ ν -1 2 G kl (d)σ klν λ θ λ ) +2∆(d) ∧ G 3m (d)σ µν m θ ν -2i(χ µ a (d) ∧ χνa (d) + χ ν a (d) ∧ χµa (d))θ ν (3.19) +i √ -γγ ij (Ω i a χ j µ a + Ω ia χj µa )(θθ) -2idθ µ ∧ (dθθ) + i 2 ε lmn G mn (d) ∧ σµν l dθ ν (θθ) -3i∆(d) ∧ dθ µ (θθ) = 0,
dL(d) -L(d) ∧ L(d) = 0, ( 3
= L dτ e G 0 ′ m ′ τ G 0 ′ τ m ′ + Ω τ a Ω τ a . (3.35)
Taking into account (2.24) it can be written in terms of the Cartan forms for generators of the conformal algebra conf (1, 2)

S OSp(4|6)/(SO(1,3)×U (3)) sparticle = L dτ e 1 4 (ω m τ +c m τ )(ω τ m +c τ m )+∆ τ ∆ τ +Ω τ a Ω τ a .
Variation of the action with respect to the Lagrange multiplier e(τ ) leads to the mass-shell

constraint 1 4 (ω m τ + c m τ )(ω τ m + c τ m ) + ∆ τ ∆ τ + Ω τ a Ω τ a = 0. (3.36)
Since this constraint is not used to prove integrability of the dynamical equations of the superparticle we set e(τ ) = 1 by fixing the reparametrization symmetry of the action. Similarly to derivation of the superstring equations in the paragraph 3.1.2 consider as independent variation parameters the Cartan forms, that correspond to generators from g 1,2,3 eigenspaces, with differentials of the coordinates replaced by their variations. As a result we come to the set of dynamical equations of the superparticle

-1 2 δS OSp(4|6)/(SO(1,3)×U (3)) sparticle δG 0 ′ m(δ) = dGτ 0 ′ m dτ + 2G τ mn G τ 0 ′ n + 2G τ 3m ∆ τ = 0, -1 2 δS OSp(4|6)/(SO(1,3)×U (3)) sparticle δ∆(δ) = d∆τ dτ -2G τ 3m G τ 0 ′ m = 0, - δS OSp(4|6)/(SO(1,3)×U (3)) sparticle δΩa(δ) = dΩτ a dτ + iΩ τ b ( Ω τ b a + δ a b Ω τ c c ) = 0 (3.37)
and with the components of the Lax pair

-1 4 δS OSp(4|6)/(SO(1,3)×U (3)) sparticle δ ω(1) a µ (δ) = iG τ 0 ′ m σµν m ω (1)τ νa +∆ τ ω (1) τ µ a -iε abc Ω τ b ω(1)τ µc = 0, -1 4 δS OSp(4|6)/(SO(1,3)×U (3)) sparticle δ ω(3) a µ (δ) = iG τ 0 ′ m σµν m ω (3)τ νa -∆ τ ω (
L OSp(4|6)/(SO(1,3)×U (3)) = 2G τ 0 ′ m M 0 ′ m + ∆ τ D + Ω τ a T a + Ω τ a T a ∈ c 2 τ g 2 (3.40)
and

M = G -1 d dτ G = j∈0,1,2,3 c j τ g j ∈ osp(4|6) (3.41)
(see Eq. (2.12) from Section 2). Since L OSp(4|6)/(SO(1,3)×U (3)) is determined by the world-line projections of the Cartan forms that enter the superparticle's action (3.35) one can present it as a g 2 -valued differential operator applied to the action functional

L OSp(4|6)/(SO(1,3)×U (3)) = M 0 ′ m ∂ ∂Gτ 0 ′ m + 1 2 D ∂ ∂∆τ + T a ∂ ∂Ωτ a + T a ∂ ∂Ωτa S OSp(4|6)/(SO(1,3)×U (3)) sparticle . ( 3
L i = L i + * L i . (3.43)
The second term on the right-hand side includes components The Lax connection (3.43) satisfies the zero curvature condition Reasoning in the opposite direction we conclude that with the aid of known expression for the Lax pair of the superparticle it is possible to recover the dependence on the superspace coordinates (but not on the spectral parameter z) of the part of L i that does not include superspace as the Lax equation that will be obtained in the next paragraph.

* L i = ε ij η jk L k = η ij ε jk L k of
∂ σ L τ -∂ τ L σ -[L τ , L σ ] = 0. ( 3

Classical integrability of equations of the massless superparticle in the

AdS 4 × CP 3 superspace
As was mentioned at the beginning of the subsection 3.1 the problem of proving integrability of equations of the AdS 4 × CP 3 superstring remains unsolved. However, as will be shown in this paragraph it is possible to prove integrability of superstring equations in the limiting case of infinite tension described by the massless superparticle model. This problem was solved in our works [START_REF] Uvarov | Lagrangian mechanics of massless superparticle on AdS 4 × CP 3 superbackground[END_REF] and [START_REF] Uvarov | On integrability of massless AdS 4 ×CP 3 superparticle equations[END_REF] following the approach suggested in [START_REF] Uvarov | Kaluza-Klein gauge and minimal integrable extension of OSp(4|6)/(SO(1, 3) × U (3)) sigma-model[END_REF] that is based on partial fixing the κ-symmetry gauge in the sector of broken supersymmetries. First in [START_REF] Uvarov | Lagrangian mechanics of massless superparticle on AdS 4 × CP 3 superbackground[END_REF] the Lax pair of the OSp(4|6)/(SO(1, 3) × U (3)) superparticle was extended by the contributions of four coordinates associated with broken Poincaré supersymmetries and then in [START_REF] Uvarov | On integrability of massless AdS 4 ×CP 3 superparticle equations[END_REF] obtained was complete expression for the Lax pair of the AdS 4 × CP 3 superparticle that takes into account the dependence on four coordinates for broken special conformal supersymmetries. Obtained components of the Lax pair enter the Lax equation that is satisfied on the AdS 4 × CP 3 superparticle's equations.

To derive equations of the massless AdS 4 × CP 3 superparticle one needs to construct its Lagrangian. Its form is determined by the AdS 4 × CP 3 supervielbein bosonic components that depend on all 24+8 Grassmann coordinates of the superspace. Expressions for them will be obtained below using, as before, the realizations of the osp(4|8) and osp(4|6) superalgebras as D = 3 N = 8 and N = 6 superconformal algebras.

We will consider the OSp(4|8)/(SO(1, 3) × SO( 7)) supercoset element defined in (3.3).

As was mentioned, its structure ensures independence of respective Cartan forms on the coordinate y of the S 1 circle in the Hopf fibration S 7 = CP 3 ×S 1 . However, the so(2, 3)/so [START_REF] Maldacena | The large N limit of superconformal field theories and supergravity[END_REF][START_REF] Witten | Anti-de Sitter space and holography[END_REF] Cartan forms that determine the supervielbein bosonic components in the tangent space to

L m ′ n ′ G n ′ y + L m ′ 11 Φ = 0
, where for the supervielbein components in the tangent space to AdS 4 and S 1 used was the representation (3.1) and E 11 (d) = Φdy + A(d). In general case elements of this matrix have the following form (see [START_REF] Gomis | The complete AdS 4 × CP 3 superspace for type IIA superstring and D-branes[END_REF], [START_REF] Uvarov | Kaluza-Klein gauge and minimal integrable extension of OSp(4|6)/(SO(1, 3) × U (3)) sigma-model[END_REF])

L m ′ n ′ = δ m ′ n ′ + Φ -Φ L (G y • G y )Φ L G m ′ y G yn ′ , L m ′ 11 = - G m ′ y Φ L , L 11 m ′ = G ym ′ Φ L , L 11 11 = Φ Φ L , where Φ L = exp 2ϕ 3 = Φ 2 + (G y • G y ). So transformed bosonic components of the D = 11
supervielbein equal

E m ′ (d) = (LE) m ′ (d) = G m ′ (d)+ Φ -Φ L (G y •G y )Φ L (G y •G(d))G m ′ y - 1 Φ L A(d)G m ′ y , (3.45)
E 11 (d) = (LE) 11 (d) = Φ L (dy + A L (d)),
where

A L (d) = 1 Φ L ((G y • G(d)) + ΦA(d)) (3.46)
is the Ramond-Ramond one-form potential. As was shown in the paragraph 3.1.1, for considered OSp(4|8)/(SO(1, 3) × SO( 7)) supercoset element G m ′ y and Φ depend only on the coordinates from the sector of broken supersymmetries

G m y = 2 1 + 1 2 η 2 η2 (θσ m θ) + 2{1 -i[(θη) + ( θη)]}(ησ m η), G 3 y = 2[(θη) -( θη)], Φ = 1 -2i[(θη) + ( θη)] + 4[(θη)( θη) -(θη)( θη)].
So the Lagrangian of massless AdS 4 × CP 3 superparticle

L AdS 4 ×CP 3 sparticle = Φ L e E τ m ′ E m ′ τ -E τ a E τ a
is expressed through the mappings to the world-line of the supervielbein bosonic components E m ′ (d) and E a (d), E a (d). After substitution of the expressions (3.45) it can be written in the form

L AdS 4 ×CP 3 sparticle = Φ L e G τ m ′ G τ m ′ -Φ -2 L [(G τ • G y ) 2 -G 2 y A 2 τ +2ΦA τ (G τ • G y )]-E τ a E τ a . ( 3
G m ′ (d) = G 0 ′ n (d)M n m ′ + G 3n (d)N n m ′ + ∆(d)L m ′ + G kl (d)K kl m ′ +q m ′ µ dθ µ + qm ′ µ d θµ + s m ′ µ dη µ + sm ′ µ dη µ , E a (d) = iΩ a (d) + u (1) µ ω (1)µa (d) + u (3) µ ω (3)µa (d), E a (d) = iΩ a (d) + ū(1) µ ω(1) a µ (d) + ū(3) µ ω(3) a µ (d), (3.49)
A(d) = Ω a a (d) + G 0 ′ m (d)m m + G 3m (d)n m + ∆(d)l + G mn (d)k mn +h µ dθ µ + hµ d θµ + p µ dη µ + pµ dη µ . (3.50)
In expansions (3.49) bosonic coefficients equal

M n m = δ m n 1 -(θ θ)(η η) + 1 4 (θ 2 θ2 + η 2 η2 ) + 1 8 θ 2 θ2 η 2 η2 -i(θσ n σm η + θσ n σm η) + 2 1 -i 2 [(θη) + ( θη)] (θσ n θ)(ησ m η), N n m = δ m n -(θ θ)(η η) + 1 4 (θ 2 θ2 -η 2 η2 ) + 1 8 θ 2 θ2 η 2 η2 -i(θσ n σm η + θσ n σm η) + 2 1 -i 2 [(θη) + ( θη)] (θσ n θ)(ησ m η), M n 3 = N n 3 = [(θη) -( θη)](θσ n θ), L m = [( θη) -(θη)](ησ m η), -L 3 = 1 + i[(θη) + ( θη)], K kl m = -i 2 ε kl m 1 + 1 2 η 2 η2 (θ θ) + (η η) + 1 2 [( θσ kl η) -(θσ kl η)](ησ m η), K kl 3 = -i 2 [( θσ kl η) + (θσ kl η)] and u (1,3) µ = ∓2{θ µ ± iη µ [1 ± (θ θ)]}, ū µ (1,3) = ∓2{ θµ ∓ iη µ [1 ∓ (θ θ)]}.
Respectively bosonic coefficients in (3.50) have the form

m m = 1 -i[(θη) + ( θη)] (θσ m θ) + 1 -1 2 θ 2 θ2 (ησ m η), n m = 1 -i[(θη) + ( θη)] (θσ m θ) -1 + 1 2 θ 2 θ2 (ησ m η) l = ( θη) -(θη), k mn = 1 2 [( θσ mn η) -(θσ mn η)] -i(θ θ)(ησ mn η).
Non-zero Grassmann-odd coefficients in (3.49) and (3.50) are given by

q mµ = i 2 1 + 1 2 η 2 η2 σmµν θν + 1 2 η2 σmµν η ν , q 3µ = -iη µ , s mµ = i 2 σmµν ην , h µ = -η µ + i( θη)η µ -i( θη)η µ
and c.c. expressions. As is seen all of the above coefficients are functions of eight fermionic coordinates for broken supersymmetries only.

Bosonic equations following from the superparticle's action (3.48) can be brought to the form

-1 2 δS AdS 4 ×CP 3 sparticle δG 0 ′ m(δ) = Ġ 0 ′ m τ + 2(G m τ n G 0 ′ n τ -G m τ n G 0 ′ τ n + ∆ τ G τ 3m -∆ τ G 3m τ ) -2i ω (1)τ a σ m ω(1)τ a -ω (1)τ a σ m ω(1)τ a + ω (3)τ a σ m ω(3)τ a -ω (3)τ a σ m ω(3)τ a = 0, -1 2 δS AdS 4 ×CP 3 sparticle δ∆(δ) = ∆τ + 2G 0 ′ τ m G 3m τ -2G τ 3m G 0 ′ m τ -2 ω (1) τ µ a ω(1)τ a µ -ω (1)τ µ a ω(1) τ a µ -ω (3) τ µ a ω(3)τ a µ + ω (3)τ µ a ω(3) τ a µ = 0, (3.51)
(δ) = ω µ (3)τ a + 1 2 G τ mn ω ν (3)τ a -G mn τ ω (3) ν τ a σ mnν µ +iσ µν m G τ 3m ω (3)τ νa -G 3m τ ω (3)τ νa + iσ µν m G τ 0 ′ m ω (1)τ νa -G 0 ′ m τ ω (1)τ νa +∆ τ ω µ (1)τ a -∆ τ ω (1) µ τ a -i Ω τ a b -δ b a Ω τ c c ω µ (3)τ b -2iw τ ω (3) µ τ a -iε abc Ω τ b ω µc (1)τ -Ω b τ ω(1) µc τ = 0, -1 4 δS AdS 4 ×CP 3 sparticle δ ω(3) a µ (δ) = ω µ (1)τ a + 1 2 G τ mn ω ν (1)τ a -G mn τ ω (1) ν τ a σ mnν µ -iσ µν m G τ 3m ω (1)τ νa -G 3m τ ω (1)τ νa -iσ µν m G τ 0 ′ m ω (3)τ νa -G 0 ′ m τ ω (3)τ νa +∆ τ ω µ (3)τ a -∆ τ ω (3) µ τ a -i Ω τ a b -δ b a Ω τ c c ω µ (1)τ b -2iw τ ω (1) µ τ a -iε abc Ω τ b ω µ (3)τ -Ω b τ ω(3) µc τ = 0. (3.52)
In (3.51) and (3.52) the following bosonic and fermionic quantities have been introduced

      G 0 ′ τ m ∆ τ -G τ 3m -1 2 G τ kl       = Φ L       M m n ′ L n ′ N m n ′ K kl n ′       G τ n ′ -1 Φ 2 L ((G τ • G y ) + ΦA τ )G yn ′ + 1 Φ L       m m l n m k kl       (G 2 y A τ -Φ(G τ • G y )), (3.53
)

Ω a τ = -iΦ L E τ a , Ωτa = -iΦ L E τ a , w τ = 1 2Φ L (G 2 y A τ -Φ(G τ • G y )) (3.54)
and also ω µ L . Then in the m → 0 limit resulting Lagrangian reduces to that of the massless superparticle. The metric e(τ ) is also the Lagrange multiplier for the mass-shell constraint.

Its definition adopted in (3.65) gives the conventional mass term in this constraint. Since the mass-shell constraint is not used for establishing integrability of the dynamical equations of the D0-brane, to simplify further analysis it is convenient to gauge fix the reparametrization symmetry of the action by the condition e = 1 2 . As in the case of the massless superparticle, considering as independent parameters the Cartan forms, associated with generators of the osp(4|6) superalgebra from the g 1,2,3 eigenspaces under the Z 4 automorphism, with differentials of coordinates replaced by their variations yields the set of 10 bosonic and 24 fermionic equations of the D0-brane. By substituting the expressions (3.45) for E m ′ (d) and (3.46) for A L (d) one can bring the D0-brane equations to a system of the first-order ordinary differential equations with the coefficients given by the mappings of the osp(4|6) Cartan forms to its world line

-1 2 δS AdS 4 ×CP 3 D0-brane δG 0 ′ m(δ) = ˙ G 0 ′ m τ + 2(G m τ n G 0 ′ n τ -G m τ n G 0 ′ τ n + ∆ τ G τ 3m -∆ τ G 3m
     G 0 ′ τ m ∆ τ -G τ 3m -1 2 G τ kl       = 1 Φ 2 L       M m n ′ L n ′ N m n ′ K kl n ′       G τ n ′ -1 Φ 2 L ((G τ •G y )+ΦA τ )+ m 2 G yn ′ + 1 Φ 2 L       m m l n m k kl       1 Φ 2 L (G 2 y A τ -Φ(G τ • G y )) + m 2 Φ , (3.69
)

Ω a τ = - i Φ 2 L E τ a , ¯ Ω τ a = - i Φ 2 L E τ a , 2Φ 2 L w τ = 1 Φ 2 L (G 2 y A τ -Φ(G τ • G y )) + m 2 Φ ( 3

Conclusion

At the end let us summarize the results reviewed in the present paper:

-using the isomorphic realization of the osp(4|6) superalgebra as the D = 1 + 2 N = 6

superconformal algebra there has been obtained the expression for the Lagrangian of the σ- -it was proved that fermionic equations of the AdS 4 × CP 3 superstring corresponding to variation of the coordinates of the sector of broken supersymmetries become the consequences of other fermionic equations, when these coordinates turn to zero. Proposed proof generalizes that of Ref. [START_REF] Sorokin | Evidence for the classical integrability of the complete AdS 4 ×CP 3 superstring[END_REF] for equations linearized in the Grassmann coordinates of the AdS 4 × CP 3

superspace.

-equations of the massless superparticle in the OSp(4|6)/(SO(1, 3) × U (3)) supercoset space were presented in the form of the Lax equation and established was the connection between the components of the Lax pair and the σ-model Lax connection. As in the case of the AdS 4 × CP 3 superstring it was proved that fermionic equations of the massless AdS 4 × CP 3 superparticle, corresponding to variation of the coordinates of the sector of broken supersymmetries, are not independent, when these coordinates equal zero. It was proved integrability of equations of the massless superparticle and the D0-brane models in the AdS 4 × CP 3 superbackground.

  AdS 4 × CP 3 superspace and N = 6 superconformal symmetry in (1+2) dimensions 2.1 Lagrangian formulation of the OSp(4|6)/(SO(1, 3) × U (3)) σ-model in terms of Cartan forms in conformal basis . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 osp(4|6) superalgebra generators, Cartan forms and Maurer-Cartan equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.2 OSp(4|6)/(SO(1, 3) × U (3)) σ-model Lagrangian . . . . . . . . . . . 2.2 D = 3 N = 6 superconformal symmetry of the σ-model action in the OSp(4|6)/(SO(1, 3) × U (3)) superspace . . . . . . . . . . . . . . . . . . . . . 2.2.1 Transformation properties of the Cartan forms and coordinates of the OSp(4|6)/(SO(1, 3) × U (3)) supercoset . . . . . . . . . . . . . . . . 2.2.2 Densities of the Noether currents . . . . . . . . . . . . . . . . . . . . 2.3 Lagrangian of the AdS 4 × CP 3 superstring in the κ-symmetry light-cone gauge 2.3.1 AdS 4 × S 7 supermembrane and its reduction to the AdS 4 × CP 3 superstring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 AdS 4 × CP 3 superstring in the κ-symmetry light-cone gauge . . . . . 2.4 Light-cone gauge Hamiltonian of the AdS 4 × CP 3 superstring . . . . . . . . . 2.4.1 Formulation of the AdS 4 × CP 3 superstring Lagrangian in the κsymmetry light-cone gauge in terms of the phase-space variables . . . 2.4.2 Light-cone gauge for reparametrization symmetry and the Hamiltonian of the AdS 4 × CP 3 superstring . . . . . . . . . . . . . . . . . . . . . . 3 Classical integrability of equations of the massless superparticle and D0brane in the AdS 4 × CP 3 superspace. Minimal extension of the integrable OSp(4|6)/(SO(1, 3) × U (3)) σ-model 3.1 Kaluza-Klein gauge for κ-symmetry and minimal extension of the integrable OSp(4|6)/(SO(1, 3) × U (3)) σ-model . . . . . . . . . . . . . . . . . . . . . . 3.1.1 AdS 4 × S 7 and AdS 4 × CP 3 supervielbeins in the Kaluza-Klein gauge 3.1.2 Minimal extension of the integrable OSp(4|6)/(SO(1, 3)×U (3)) σ-model 3.2 Connection between fermionic equations of the AdS 4 × CP 3 superstring and the OSp(4|6)/(SO(1, 3) × U (3)) σ-model . . . . . . . . . . . . . . . . . . . . 3.3 Classical integrability of equations of the massless superparticle and D0-brane in the AdS 4 × CP 3 superbackground . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Classical integrability of equations of the massless superparticle in the OSp(4|6)/(SO(1, 3) × U (3)) superspace . . . . . . . . . . . . . . . . . 3.3.2 Classical integrability of equations of the massless superparticle in the AdS 4 × CP 3 superspace . . . . . . . . . . . . . . . . . . . . . . . . . .

AdS 4 /

 4 CF T 3 duality conjecture to obtain a formulation of the σ-model Lagrangian in terms of the Cartan one-forms that correspond to the generators of the superconformal algebra. To get insight into non-linear structure of the σ-model Lagrangian it is also important to obtain explicit expressions for relevant Cartan forms that are specified by the OSp(4|6)/(SO(1, 3)× U (3)) supercoset element.In paragraph 2.1.1 considered is a number of bases for the generators of the osp(4|6) superalgebra and the Cartan forms associated with them, in particular, one corresponding to its realization as the D = 3 N = 6 superconformal algebra. Also it is discussed the Z 4 automorphism of the osp(4|6) superalgebra and the basis of its generators with definite eigenvalues.In paragraph 2.1.2 obtained are various representations of the Lagrangian of the OSp(4|6)/(SO(1, 3) × U (3)) σ-model in terms of the Cartan forms for the generators of the D = 3 N = 6 superconformal algebra. Fermionic equations of the σ-model will be presented in the form similar to that for the Green-Schwarz superstrings and it will be proved that in the general case eight of 24 equations are not independent. This implies eightparameter κ-invariance of the action. Explicit form of the κ-symmetry transformations will also be given that allows their comparison with the κ-symmetry transformations in the Green-Schwarz superstring models. Besides there will be obtained expressions for the Cartan forms in terms of the coordinates associated with the generators of the D = 3 N = 6 superconformal algebra. 2.1.1 osp(4|6) superalgebra generators, Cartan forms and Maurer-Cartan equations The Lagrangian of the OSp(4|6)/(SO(1, 3) × U (3)) σ-model is constructed out of the world-sheet mappings of the Cartan one-forms, whose form is determined by the OSp(4|6)/(SO(1, 3)×U (3)) supercoset element G . Left-covariant osp(4|6) Cartan one-forms are defined by the expression

(2. 2 )

 2 In the first of them O αβ = O βα (α, β = 1, . . . 4) are generators of the sp(4) algebra and G αβ (d) are the Cartan one-forms.

  the so(6) generators in the spinor representation realized by the D = 6 chiral γ-matrices ρ I AB and ρIAB antisymmetric in the spinor indices. In (2.6) and (2.7) also given are the decompositions of the Cartan forms and generators on the su(3) irreducible representations. Ω a 4 (d) and Ω 4 a (d) are identified with the complex vielbein components of the CP 3 manifold and Ω a b (d) -with the u(3) connection one-form. Accordingly, V a b are the generators of u(3) ⊂ su(4) subalgebra, whereas V a 4 and V 4 a are generators of the su(4)/u(3) quotient algebra. Upon decomposition of the generators of su(4) algebra on the su(3) irreducible representations, its commutation relations

4 c

 4 become manifestly invariant under the sign change of the generators of su(4)/u(3) quotient algebra in analogy with the above mentioned Z 2 automorphism of the ads(1, 3) algebra (2.3). The third representation in (2.5) is given in the 3 ⊕ 3 basis that corresponds to the decomposition of the D = 6 vector on the su(3) representations: 6 = 3 ⊕ 3. Connection between this and the usual bases is given by the relations O I = M Iâ O â and O â = M -1 âI O I , where O â are components of a vector in the 3 ⊕ 3 basis. The form of transformation matrices

. 10 )

 10 Both F α I (d) and O I α are four-component Majorana spin(2, 3) spinors and D = 6 vectors. In the second equality in (2.10) they are presented in the 3 ⊕ 3 basis: F αâ (d) = M TâI F α I , O αâ = M -1 âI O I α . Supersymmetry generators can be decomposed on the Poincaré and special conformal supersymmetry generators from the D = 3 N = 6 superconformal algebra

  d) and ∧ denotes exterior product of differential forms. Another important representation of the Maurer-Cartan equations is in terms of the Cartan forms for the generators of the D = 3 N = 6 superconformal algebra

σ m µν = (I, σ 1 , 2

 12 -σ 3 ), σmµν = ε µλ ε νρ σ m λρ = (I, -σ 1 , σ 3 ) are D = 1 + 2 analogs of relativistic Pauli matrices and σ OSp(4|6)/(SO(1, 3) × U (3)) σ-model Lagrangian As in the case of the Green-Schwarz superstrings, action of the OSp(4|6)/(SO(1, 3) × U (3))

(2. 34 )

 34 Obtained Eqs.(2.33) have the same structure as fermionic equations of the Green-Schwarz superstrings that gives possibility to find dynamics of what number of physical fermionic degrees of freedom describes the OSp(4|6)/(SO(1, 3) × U (3)) σ-model compared to the IIA superstring. As is known the 32 × 32 matrices that enter fermionic equations of the Type II Green-Schwarz superstrings have half zero eigenvalues, when one takes into account equations for the auxiliary metric γ ij . Therefore half of them are linearly dependent that in accordance with the second Noether theorem implies the superstring invariance under the local 16parameter κ-symmetry, whereas independent equations determine dynamics of 16 physical fermionic fields of the superstrings. To show that the σ-model also describes dynamics of 16 physical fermionic fields it is necessary to calculate the rank of the matrices (2.34).

  kinetic term of the σ-model Lagrangian (2.25), however, still includes contributions up to the eighth order in the fermions. The gauge condition number of θ and η coordinates. In this gauge vanish the following components of the Cartan forms Ψ 1,2 4 (d) = Ψ 4 1,2 (d) = Ψ 1,2 3 (d) = Ψ 3 1,2 (d) = 0 and ω µ 3 (d) = 0, χ µ3 (d) = 0. But more substantial simplification can be attained by considering the SL(2, R) non-covariant conditions, e.g. η 1a = 0 that partially fixes the κ-symmetry gauge freedom. In this case c 1 (d) = 0, while other components of the bosonic Cartan forms become quadratic in fermions. Also there turn to zero the components of the fermionic Cartan forms χ 1a (d) = χa 1 (d) = 0. So that the kinetic term of the Lagrangian (2.25) contains fermionic contributions up to the fourth power and the Wess-Zumino term -up to the second power. Remaining gauge freedom can be used to turn to zero extra components of the Cartan forms.

2. 2 D

 2 = 3 N = 6 superconformal symmetry of the σ-model action in the OSp(4|6)/(SO(1, 3) × U (3)) superspace The OSp(4|6)/(SO(1, 3) × U (3)) σ-model Lagrangian is by construction invariant under global OSp(4|6) symmetry. It can be presented as the D = 3 N = 6 superconformal symmetry acting on the coordinates of the OSp(4|6)/(SO(1, 3) × U (3)) supercoset space. For this realization the parametrization of the OSp(4|6)/(SO(1, 3)×U (3)) supercoset space by the coordinates associated with the superconformal algebra generators as in (2.39) is natural. The OSp(4|6)/(SO(1, 3) × U (3)) representative (2.39) includes the Poincaré coordinates for the AdS 4 space and 24 real fermionic coordinates split into two sets of 12 related to the Poincaré and special conformal supersymmetries. In the boundary limit coordinates corresponding to the generators of translations and Poincaré supersymmetries are identified with those of the D = 3 N = 6 Minkowski superspace 8 and its superconformal symmetry is isomorphic to the global symmetry of the Chern-Simons-matter theory proposed by O. Aharony, O. Bergman, D. Jafferis and J. Maldacena [32].

. 49 )

 49 It includes the parameters of the translations a m in the D = 3 Minkowski space-time, conformal boosts b m , dilatations f and Lorentz rotations l mn , as well as anticommuting parameters of the D = 3 N = 6 Poincaré supersymmetry (ε µ a , εµa ) and special conformal supersymmetry (ξ µa , ξa µ ). To (2.49) enter also the parameters of the infinitesimal su(4) transformation (w a b , y a , ȳa ). All together these parameters are denoted X. Then variation (2.48) can be written in terms of the Cartan forms in conformal basis with differentials replaced by variations and parameters of infinitesimal transformations from the stability group bm , lmn and ŵa b

  of parameters of the transformations from the global symmetry group and the supercoset coordinates. In the case of infinitesimal D = 3 N = 6 superconformal transformations with the parameter (2.49) respective parameters of transformations from the so(1, 3) stability algebra in (2.50), (2.51) and (2.55) equal bm = e 2φ A -1 b m (θ), A = 1 -e 4φ (ηη) 2 , lmn = l mn (θ) + ie 2φ (η a bσ mn ηa ) + (η a bσ mn η a ) , where b m

  [START_REF] Uvarov | Kaluza-Klein gauge and minimal integrable extension of OSp(4|6)/(SO(1, 3) × U (3)) sigma-model[END_REF] that will be used below to write down variation(2.59) of the SU (4)/U (3) coset element under the infinitesimal D = 3 N = 6 superconformal transformations. Let us also present variations of the coordinates parametrizing the OSp(4|6)/(SO(1, 3) × U (3)) supercoset element (2.39). Variations of the coordinates associated with the generators of the D = 3 N = 6 Poincaré superalgebra are given by the expressions

  η νa and c.c., where xµν = xµν -iε µν ( θθ). Note that terms proportional to the parameters bm vanish in the limit φ → -∞ corresponding to the conformal boundary of AdS 4 . So in the boundary limit coordinates (x m , θ µ a , θµa ) form the closed set under the D = 3 N = 6 superconformal symmetry transformations and can be chosen for parametrization of the boundary superspace. Variation of the coordinate φ parametrizing orthogonal to the boundary dimension of the AdS 4 space in the Poincaré coordinates equals

. 1 :

 1 Tensors contributing to the Noether current density associated with the D = 3 translational invariance and CP 3 spaces

For

  the OSp(4|6)/(SO(1, 3) × U (3)) coset element (2.39) explicit expressions for these factors corresponding to each of the transformations from the D = 3 N = 6 superconformal symmetry are collected in the Tables 2.1-2.8. For conciseness in some of the formulae from These tables the Grassmann-odd coordinates were grouped into D = 6 vectors in the 3 ⊕ 3 basis θ µ â = θ µ a θµa , η µâ = η µa ηa µ and θ µâ = H âb θ µ b , η â µ = H âb η µ b. There were also introduced T -transformed coordinates endowed with hats θµ â = T âb θ µ b , ηµâ = H âb ηµ b . (2.67) 2.3 Lagrangian of the AdS 4 × CP 3 superstring in the κ-symmetry light-cone gauge Two previous subsections were devoted to study of the properties of the σ-model in the OSp(4|6)/(SO(1, 3) × U (3)) supercoset space [9]. It describes dynamics of the Type IIA superstring in the AdS 4 × CP 3 superbackground [13], when the vanishing condition of the eight coordinates for broken supersymmetries represents partial κ-symmetry gauge-fixing condition as in the case of superstring moving in the anti-de Sitter space and in the CP 3

4 :

 4 Tensors contributing to the Noether current density associated with the D = 3 Lorentz invariance δ w ω m (d) + δ w c m (d) = j (ω m (δw)+c m (δw)) ∂w b a = 2e -2φ A δ b a (θ c σ m θc ) -(θ a σ m θb ) +2e 2φ δ b a (η c σm ηc ) -i(ηη) (θ c σ m ηc ) + (η c σ m θc ) -2e 2φ (η a σm ηb ) -i(ηη) (θ a σ m ηb ) + (η a σ m θb )

. 5 :

 5 Tensors contributing to the Noether current density associated with the u(3) invariance δ y ω m (d) + δ y c m (d) = j m a dy a + jma dȳ a : j m a = ∂(ω m (δy)+c m (δy)) ∂y a = -ε abc e -2φ A( θb σ m θc ) + e 2φ (η b σm ηc ) -2i(ηη)( θb σm ηc ) δ y ∆(d) = j a dy a + ja dȳ a : j a = ∂∆(δy) ∂y a = ε abc θµb ηc µ δ y Ω b ĉ(d) = J b ĉa dy a + Jb ĉa dȳ a + . . . : J b ĉa = ∂ ∂y a Ω b ĉ(δ y ) = -ε ade T bd T ĉe , Jb ĉa = ∂ ∂ ȳa Ω b ĉ(δ y ) = ε ade T bd T ĉe δ y ω µ b (d) = j µ ba dy a + jµa b dȳ a + . . . : j µ ba = ∂ω µ b (δy) ∂y a = ie -φ ε acd T bc θµd δ y χ µ b(d) = J µ ba dy a + Jµ ba dȳ a + . . . : J µ ba = ∂χ µ b (δy) ∂y a= ie φ (ε acd T bc ηd µ + e φ (ηη)ε µν j ν ba ) and c.c.

)) supercoset element Ĝ = G e yH e θ µ 4 Q 4 µ + θµ4 Qµ4 e η µ4 S µ4 +η 4 µ 4 ∈D = 3 N = 8 4 are 2 A

 44443842 Sµ OSp(4|8)/(SO(1, 3) × SO[START_REF] Nilsson | Hopf fibration of eleven dimensional supergravity[END_REF]).(2.68) In (2.68) the first factor on the left is the OSp(4|6)/(SO(1, 3) × U (3)) supercoset element G = e xmP m +θ µ a Q a µ + θµa Qµa e ηµaS µa +η a µ Sµ a e z a Ta+zaT a e φD ∈ OSp(4|6)/(SO(1, 3) × U (3)) that has the same form as (2.39). The only difference is that the SU (4)/U (3) coset element includes generators T a and T a defined below in (2.83) instead of V a 4 and V 4 a . The structure of the element (2.68) corresponds to the realization of the osp(4|8) isometry superalgebra of the AdS 4 × S 7 superbackground as the D = 3 N = 8 superconformal algebra and results in the Poincaré metric for the anti-de Sitter space (2.41). 32 supersymmetry generators of the composed of 24 supersymmetry generators Q a µ , Qµa and S µa , Sµ a of the D = 3 N = 6 superconformal algebra together with additional generators of the Poincaré supersymmetry Q 4 µ , Qµ4 and special conformal supersymmetry S µ4 , Sµ 4 that are broken by the AdS 4 × CP 3 superbackground. Relative positioning of the factors in (2.68) is governed by the requirement of the absence of the supervielbein dependence on the coordinate y of the S 1 circle that will be reduced. It is satisfied by placing e yH factor to the left of the factors that include generators of the broken supersymmetries, whenever external differentiation acts from the right. Proposed κ-symmetry light-cone gauge is fixed by setting to zero 16 Grassmann-odd coordinates θ = θ2A = η 1A = ηA 1 = 0 (2.69) transforming in the 4 and 4 representations of the su(4) algebra. These coordinates are associated with the generators Q A 2 , Q2A , S 1A and S1 A that have negative weight under the SO(1, 1) ⊂ SO(1, 2) symmetry generated by M +-= 2M 02 .

2. 3 . 1

 31 AdS 4 × S 7 supermembrane and its reduction to the AdS 4 × CP 3 superstring The D = 11 supermembrane action in the AdS 4 × S 7 superbackground [16] has the form S AdS 4 ×S 7 smembrane = -V d 3 ξ -g (3) + S AdS 4 ×S 7 WZ .

( 4 )

 4 (d) includes contribution of the nonzero four-form of the AdS 4 × S 7 background. The value of the parameter s = ±1 is fixed by the requirement of the κ-invariance of the supermembrane action (2.70). The D = 11 supervielbein bosonic components Em(d) = (E m ′ (d), E I ′ (d)) include components E m ′ (d) in the tangent space to AdS 4 = SO(2, 3)/SO[START_REF] Maldacena | The large N limit of superconformal field theories and supergravity[END_REF][START_REF] Witten | Anti-de Sitter space and holography[END_REF] and components E I ′ (d) (I ′ = 1, . . . , 7) in the tangent space to S 7 = SO(8)/SO[START_REF] Nilsson | Hopf fibration of eleven dimensional supergravity[END_REF]. They equal the Cartan formsG 0 ′ m ′ (d)and Ω 8I ′ (d) corresponding to the defined below generators M 0 ′ m ′ and V 8I ′ of the so(2, 3)/so(1, 3) and so(8)/so(7) quotient algebras respectively. Here and below underlining of the letters used to denote the Cartan forms for the osp(4|6) generators indicates that besides the coordinates of the (10|24)-dimensional OSp(4|6)/(SO(1, 3) × U (3) superspace and their differentials they may depend also on other coordinates of the OSp(4|8)/(SO(1, 3) × SO(7)) superspace but reduce to the Cartan forms introduced in the paragraph 2.1.1, when these additional coordinates equal zero. Fermionic supervielbein components are identified with the Grassmann-odd Cartan forms F α(d) = F αA ′ (d) (α = 1, . . . , 32, α = 1, . . . , 4, A ′ = 1, . . . , 8) associated with the supersymmetry generators O α = O αA ′ from the osp(4|8) superalgebra. They carry Majorana spinor representation of the spin(1, 10) algebra that decomposes on the product of the Majorana spinor representations of spin(1, 3) and spin(7). the spin(1, 10) algebra generators constructed out of the D = 1 + 10 γ-matrices g m α β .

10 )

 10 to bring them to the Kaluza-Klein ansatz form (LE) m′ (d) = E m′ (d), (LE) 11 (d) = E 11 (d) = Φ L (dy + A L (d)). (2.71) Components of the transformed vielbein E m′ (d) do not depend on y and dy and are identified with the D = 10 supervielbein bosonic components, Φ L = e 2ϕ/3 determines the D = 10 IIA dilaton superfield ϕ and A L (d) is the one-form gauge superfield that together with the 3form gauge superfield make up the set of the Ramond-Ramond superfields in the Type IIA superstring theory. In the spectrum of free superstring they originate from the product of spinor ground states in the sectors of left-and right-moving world-sheet fields. Fermionic components of the transformed D = 11 supervielbein (LF ) α(d) = E α(d) + e -2ϕ/3 χ αE 11 (d) (2.72) have the form of the sum of the D = 10 supervielbein fermionic components

  .76) presented as the integral of the three-form H (3) (d) = dB (2) (d) over auxiliary threedimensional hypersurface M 3 whose boundary is the superstring world-sheet Σ. The twoform B (2) (d) originates in the spectrum of oriented superstrings from decomposition of the product of bosonic oscillators for left-and right-moving parts of the space-time coordinate fields and is called Neveu-Schwarz-Neveu-Schwarz gauge potential. Based on its transformation properties it can be viewed as a stringy generalization of electromagnetic potential with which charged point particles minimally interact. After description of the generalities of the double-dimensional reduction of the supermembrane let us turn to the definition of the osp(4|8) Cartan forms and the bases of the so(8) generators and Cartan forms compatible with the Hopf fibration realization of the seven-sphere. For the OSp(4|8)/(SO(1, 3) × SO(7)) supercoset element Ĝ of the general form covariant Cartan forms are defined by the relation Ĝ -1 d Ĝ = G sp(4) (d) + G so(8) (d) + G 32susy (d) ∈ osp(4|8).

4 edη 4 -dφη 4 respectively.

 444 .95) χµa (d) = 0 e φ dη a , χa µ (d) = 0 e φ dη a , where dθ â = T âb dθ b and dη â = T âb dη b, and ωµ 4 (d) = dθ 4 + dφθ 4 + e -2φ dx 1 η -2φ dx + η 4 , ωµ4 (d) = d θ4 + dφ θ4 + e -2φ dx 1 η4 e -2φ dx + η4 , At the same time spinors proportional to dy have the following components expressions (2.91) it follows that for the supercoset element (2.68) D = 11 supervielbein bosonic components in the tangent space to AdS 4 include terms proportional to dy. As discussed in the paragraph 2.3.1, the local Lorentz rotation in the tangent space needs to be performed to bring the bosonic components of the supervielbein to the Kaluza-Klein ansatz form (2.71). Since vielbein components tangent to the seven-sphere (2.93) and

dη 4 -dη 4 -

 44 dφη 4 +2ie -2φ Θη 4 dx + , (L χ) 4 µ (d) = 0 dφη 4 -2ie -2φ Θη 4 dx + .

( 2 .

 2 105) So presented expressions for the induced world-sheet metric and the two-form B (2)l.c. (d) determine the Polyakov-type action of the AdS 4 × CP 3 superstring in the gauge (2.69) S AdS 4 ×CP 3 sstring, l.c. = Σ d 2 ξL AdS 4 ×CP 3 sstring, l.c.

2. 4

 4 Light-cone gauge Hamiltonian of the AdS 4 × CP 3 superstring

AdS 4 ×

 4 CP3 superstring Lagrangian in the κ-symmetrylight-cone gauge in terms of the phase-space variablesTo introduce momentum densities conjugate to bosonic coordinates we present the AdS 4 × CP3 superstring Lagrangian in the κ-symmetry light-cone gauge (2.69) in the form12

2. 4 . 2

 42 Light-cone gauge for reparametrization symmetry and the Hamiltonian of the AdS 4 × CP 3 superstring

  and represent D = 8 Majorana-Weyl spinors of the different chiralities. As a result the terms with the fermionic fields in the Lagrangian (2.118) acquire the form

f = 8θ 4 θ4 η 4 η4 . 3

 3 Classical integrability of equations of the massless superparticle and D0-brane in the AdS 4 × CP 3 superspace. Minimal extension of the integrable OSp(4|6)/(SO(1, 3) × U (3)) σ-model In subsection 3.1 there will be introduced the Kaluza-Klein condition and examined its consequences for the double-dimensional reduction of the AdS 4 × S 7 supermembrane to the AdS 4 × CP 3 superstring. This condition is imposed on the D = 11 supervielbein bosonic components in the tangent space to the AdS 4 × CP 3 space-time and requires vanishing of the terms proportional to differential of the coordinate y of the 11th dimension presented by the S 1 circle in the Hopf fibration S 7 = CP 3 × S 1 . Its introduction is motivated by the fact that the OSp(4|8)/(SO(1, 3)×SO(

  4 × S 7 and AdS 4 × CP 3 supervielbeins in the Kaluza-Klein gauge To analyze consequences of the Kaluza-Klein condition (3.2) for the AdS 4 × S 7 supermembrane and the AdS 4 × CP 3 superstring it is necessary to obtain explicit expression for G m ′ y in terms of coordinates that requires specifying the OSp(4|8)/(SO(1, 3) × SO(7)) supercoset θν etc. Expressions (3.4) can be turned to zero in the SL(2, R)-covariant way by imposing the (anti-)Majorana condition on the spinor coordinates θµ = sθ µ , ηµ = sη µ , s = ±1.

  and -∆(d) are tangent to the AdS 4 components of supervielbein of the OSp(4|6)/(SO(1, 3) × U (3)) supercoset space (see Eq. (2.24) of Section 2).

In ( 3 . 7 )

 37 the su(4)/u(3) Cartan forms Ω a (d) and Ω a (d) are respective components of the OSp(4|6)/(SO(1, 3) × U (3)) supervielbein. Grassmann-odd Cartan forms ω µ a (d) and χ µa (d) and c.c. associated with the D = 3 N = 6 super-Poincaré and special conformal supersymmetry generators are the OSp(4|6)/(SO(1, 3) × U (3)) supervielbein fermionic components as discussed in detail in Section 2. Because expressions (3.4) turns to zero, one-forms (3.6) and (3.7) represent bosonic components of the supervielbein of the AdS 4 × CP 3 superspace. The 11-th component of the AdS 4 × S 7 supervielbein tangent to the S 1 circle has the formE 11 (d) = h(d) + Ω a a (d) = Φ(dy + A(d)) : Φ = 1 -4i(θη), A(d) = Φ -1 Ω a a (d),(3.8) where h(d) is the Cartan form for the generator H of the U (1) isometry group of S 1 fiber and Ω a a (d) is the u(1) part of the u(3) connection of the CP 3 manifold Ω a b (d). Φ = e 2ϕ/3

(3. 10 ) 3 . 1 . 2

 10312 and c.c. They determine fermionic components of the AdS 4 × CP3 supervielbein and the dilatino superfieldE α(d) = f α(d) -F α y A(d), χ α = F α y . Minimal extension of the integrable OSp(4|6)/(SO(1, 3) × U (3)) σ-modelIn the previous paragraph obtained were expressions for the supervielbein components of the AdS 4 × CP 3 superspace that enter the superstring Lagrangian. However, superstring equations remain too complicated to find their representation in the form of the zero-curvature condition for a world-sheet one-form. They can be simplified by further excluding coordinates in the sector of broken supersymmetries. Therefore in this paragraph considered will be the SL(2, R)-covariant partial κ-symmetry gauge, in which remains only Majorana spinor coordinate θ µ for broken Poincaré supersymmetries. This leads to further simplification of the AdS 4 × S 7 supervielbein (3.6)-(3.10). In particular, its bosonic components acquire the form E m

  11) and E 11 (d) = dy + Ω a a (d).(3.12)From (3.12) it is seen that Φ = 1 as in the light-cone gauge considered in the paragraph 2.3.2. Non-zero fermionic components of the supervielbein in the proposed gauge reduce toωµ a (d) = ω µ a (d) + iχ µ a (d)(θθ), ωµ 4 (d) = dθ µ + 1 2 G mn (d)θ ν σ mnν µ + ∆(d)θ µ , ω y µ 4 = 2iθ µ χµa (d) = χ µa (d), χµ4(d) = c m (d)σ m µν θ ν . (3.13) The D = 11 supervielbein components (3.11)-(3.13) determine the supermembrane Lagrangian and have requisite form to carry out its double-dimensional reduction. Resulting AdS 4 × CP 3 superstring action S AdS 4 ×CP 3 sstring, min = Σ d 2 ξ L AdS 4 ×CP 3 sstring, min, kin + L AdS 4 ×CP 3 sstring, min, WZ . can be viewed as the minimal extension of the OSp(4|6)/(SO(1, 3) × U (3)) σ-model. Kinetic and Wess-Zumino terms of the superstring Lagrangian have the form

  tion parameters, similarly to the case of the OSp(4|6)/(SO(1, 3) × U (3)) σ-model considered in Section 2, the Cartan forms, associated with generators of the osp(4|6)/(so(1, 3) × u(3)) quotient algebra that have definite eigenvalues of the Z 4 automorphism, with differentials of the coordinates replaced by their variations. Such set includes bosonic Cartan forms from the eigenspace g 2 , namely G 0 ′ m (d), ∆(d) and Ω a (d), Ω a (d), as well as fermionic Cartan forms (2.16) corresponding to the supersymmetry generators (2.15) from eigenspaces g 1,3 .

µa

  (d)∧(dθθ)-2∆(d)∧ω (1) µ a (d)(θθ) -4i Ω c c (d)∧ω (3) µ a (d)(θθ)+6iε abc Ω b (d)∧ ω(3) µc (d)(θθ) = 0 and c.c. have been presented in the form of the series in θ µ and dθ µ . Above equations include world-sheet projectorsV ij ± = √ -γγ ij ± ε ij and components of the so(1, 3) connection G 3m (d) = 1 2 (c m -ω m )(d).These equations are extentions of the OSp(4|6)/(SO(1, 3) × U (3)) σ-model equations by linear and quadratic terms in the spinor coordinate θ µ and its differential. From the superstring action besides these equations there also follow equations corresponding to variation of the Grassmann coordinates from the sector of broken super-(Ω a (d)∧ω µ a (d)+Ω a (d)∧ ωµa (d))

  where the SL(2, R) spinor coordinate ϑ µ satisfies the anti-Majorana condition. When θ µ turns to zero, Eqs. (3.16)-(3.19) do not become identities. As will be shown in the next subsection they become the consequences of other fermionic equations.

. 20 )

 20 which enter the one-form L(d) = L so(2,3) (d) + L su(4) (d) + L 24 susy (d) ∈ osp(4|6)(3.21) taking value in the osp(4|6) isometry superalgebra of the AdS 4 × CP 3 superspace.It can be conveniently presented as the sum of three summands. The first summand takes value in the so(2, 3) algebraL so(2,3) (d) = G mn (z, d)M mn + 2G 3m (d)M 3m + 2G 0 ′ m (z, d)M 0 ′ m + ∆(z, d)D ∈ so(2, 3)3.3 Classical integrability of equations of the massless superparticle and D0-brane in the AdS 4 × CP 3 superbackground In this subsection it will be proved classical integrability of the equations of massless superparticle in the OSp(4|6)/(SO(1, 3) × U (3)) supercoset space and in the AdS 4 × CP 3 superspace. Similarly to connection between the OSp(4|6)/(SO(1, 3) × U (3)) σ-model and the AdS 4 × CP 3 superstring, the Lagrangian of massless OSp(4|6)/(SO(1, 3) × U (3)) superparticle follows from that of the massless superparticle in the AdS 4 × CP 3 superspace upon imposing vanishing condition on eight coordinates of the sector of broken supersymmetries. This condition represents partial κ-symmetry gauge fixing when the superparticle moves both in the AdS 4 and in the CP 3 space. It will be shown that in this case equations that follow from variation of the AdS 4 × CP 3 superparticle action on the coordinates of the sector of broken supersymmetries become the consequences of other 24 fermionic equations. The latter turn into fermionic equations of the OSp(4|6)/(SO(1, 3) × U (3)) superparticle. As is known the massless OSp(4|6)/(SO(1, 3) × U (3)) superparticle corresponds to the tension-to-infinity limit of the OSp(4|6)/(SO(1, 3) × U (3)) σ-model. This implies that their integrable structures are connected. In the paragraph 3.3.1 it will be established the relation between the components of the superparticle's Lax pair and the Lax connection of the σmodel. In the final paragraph 3.3.3 it will be proved classical integrability of the equations that follow from the action of D0-brane in the AdS 4 × CP 3 superbackground [48]. 3.3.1 Classical integrability of equations of the massless superparticle in the OSp(4|6)/(SO(1, 3) × U (3)) superspace Action of the massless superparticle model in the OSp(4|6)/(SO(1, 3) × U (3)) supercoset space has the following form [10] S OSp(4|6)/(SO(1,3)×U (3)) sparticle

  also c.c. ones. Let us present Eqs. (3.37) and (3.38) in the form of the Lax equation dL OSp(4|6)/(SO(1,3)×U (3)) dτ + [M, L OSp(4|6)/(SO(1,3)×U (3)) ] = 0 (3.39)

. 42 )

 42 At the end of this paragraph let us establish connection between the components of the Lax pair (3.40) and (3.41) and the Lax connections of the OSp(4|6)/(SO(1, 3) × U (3)) σmodel and the AdS 4 × CP 3 superstring. The Lax connection of the σ-model should follow from that of the superstring when coordinates of the sector of broken supersymmetries turn to zero. In particular, if in (3.21) one sets θ µ = 0 then this one-form turns into the Lax connection of the σ-model. The general structure of the Lax connection L i is the following

  the one-form * L(d) dual to L(d), where η ij and η ij are world-sheet Minkowski metric and its inverse. Both summands in(3.43) depend on the functions ℓ 1 , ℓ 2 , ℓ 3 and ℓ 4 of the spectral parameter z that satisfy equations(3.25). One of their solutions was presented in(3.26).

. 44 )= lim z→1 1 ℓ 2 *

 442 Transition to the superparticle's Lax pair corresponds to dropping the dependence of the superspace coordinate fields on the world-sheet space-like coordinate σ and taking the limit z → 1. In this case L σ = * L τ = 0. Then the Lax pair components are defined by the limiting values of the non-zero components of the Lax connection L L σ , M = lim z→1 L τ and the zero curvature condition (3.44) transforms into the Lax equation for the superparticle. In particular, components of the Lax connection of the OSp(4|6)/(SO(1, 3) × U (3)) σ-model in this limit transfer to the Lax pair components of the OSp(4|6)/(SO(1, 3) × U (3)) superparticle (3.40) and (3.41), and the zero curvature condition -to the Lax equation (3.39).

ℓ 2

 2 and of the part of L i linear in ℓ 2 . So the form of the Lax pair components for the superparticle gives information on the structure of the superstring Lax connection. This explains the interest to presenting equations of the massless superparticle in the AdS 4 × CP3

AdS 4

 4 include summands proportional to dy (see Eq. (3.1)). To remove such contributions and bring the AdS 4 × S 7 supervielbein to the Kaluza-Klein ansatz it is necessary to perform local SO(1, 4) transformation in the tangent-space to AdS 4 × S 1 . Particular case of such transformation was considered in Section 2 when deriving the AdS 4 × CP 3 superstring Lagrangian in the κ-symmetry light-cone gauge (see Eqs. (2.99) and (2.100)). The entries of the SO(1, 4) Lorentz rotation matrix (2.99) are determined by the requirement of dy independence of the transformed supervielbein components in the tangent space to AdS 4 :

. 47 )Ldτ L AdS 4 ×CP 3 sparticle( 3 . 48 )

 473348 As in the case of massless OSp(4|6)/(SO(1, 3) × U (3)) superparticle, the mass-shell constraint, obtained by varying the actionS AdS4 ×CP 3 sparticle = on the Lagrange multiplier e(τ ), is not used in course of the proof of integrability of the dynamical equations. So we set e(τ ) = 1. Let us also note that by redefinition of e(τ ) it is possible to 'absorb' the overall factor of Φ L . Corresponding expression for the Lax pair component L comes about by putting Φ L = 1 in (3.53)-(3.55). The structure of the OSp(4|8)/(SO(1, 3) × SO(7)) supercoset element (3.3) is such that 1-forms G m ′ (d), E a (d), E a (d) and A(d) in (3.47) can be expanded on the osp(4|6) Cartan forms and differentials of the coordinates of the sector of broken supersymmetries

  a + δ a b Ω τ c c -4iw τ Ω τ a +4iε abc ω (1) τ µ b ω (1)τ µc -ω (3) τ µ b ω (3)τ µc = 0. 24 fermionic equations corresponding to variation parameters ω(1)

4 Φ 1 Φ 2 L 2 yA

 4122 [START_REF] Maldacena | The large N limit of superconformal field theories and supergravity[END_REF]τ a = -1 4 Φ L E τ a ū(3) µ , ω µ (3)τ a = 1 4 Φ L E τ a ū(1) µ , L E τ a u (3) µ , ω µa(3)τ = -1 4 Φ L E τ a u (1) µ (3.55) that enter the Lax pair component L AdS 4 ×CP 3 (3.57). Besides variation of the action (3.48)on the coordinates of the sector of broken supersymmetries yields other eight fermionic equationsd dτ Φ L G τ m ′ -((G τ •G y )+ΦA τ )G ym ′ ∂Gτ m ′ A τ -Φ(G τ •G y )) τ A τ G y • ∂Gy ∂υµ -(G τ •G y ) ∂Φ ∂υµ + Φ L G τ m ′ ∂G m ′ τ τ •G y )+ΦA τ ) ∂(Gτ •Gy) ∂υµ + (G 2 y A τ -Φ(G τ •G y )) ∂Aτ ∂υµ = 0. (3.56) Eqs. (3.56) were presented in concise form due to introduced quantities ∂Gτ m ′ ∂ υµ = (q m ′ µ , qm ′ µ , s m ′ µ , sm ′ µ ) and ∂Aτ ∂ υµ = (h µ , hµ , p µ , pµ ), where υ µ = (θ µ , θµ , η µ , ηµ ). Also taking into account the expansions (3.49) and (3.50) we have ∂Gτ m ′ ∂υµ = G τ 0 ′ n ∂Mn m ′ ∂υµ + G τ 3n ∂Nn m ′ and ∂(Gτ • Gy) ∂υµ are defined analogously. As in the case of the OSp(4|6)/(SO(1, 3) × U (3)) superparticle, equations of the superparticle in the AdS 4 × CP 3 superspace (3.51), (3.52) and (3.56) can be presented in the form of the Lax equation dL AdS 4 ×CP 3 dτ + [M, L AdS 4 ×CP 3 ] = 0, where the Lax pair component M was defined in (3.41) and another component equalsL AdS 4 ×CP 3 = L so(2,3) + L su(4) + L 24susy ∈ osp(4|6) : L so(2,3) = 2G 0 ′ m τ M 0 ′ m + ∆ τ D + 2G 3m τ M 3m + G mn τ M mn , L su(4) = Ω a τ T a + Ωτa T a + 4w τ V a a , L 24susy = ω µ (1)τ a Q (1) (3.39) and (3.40)). Taking into account expansions (3.49) and (3.50) it can be presented in the form of the osp(4|6)-valued differential operator applied to the superparticle's action (3.48) L AdS 4 ×CP 3 = M 0 ′ Let us note the arbitrariness in the definition of e(τ ). So by redefining it one can 'absorb' the factor Φ 2

τ)-( 3

 3 2i ω (1)τ a σ m ¯ ω a (1)τ -ω (1)τ a σ m ω(1)τ a + ω (3)τ a σ m ¯ ω a (3)τ -ω (3)τ a σ m ω(3)τ a = 0, a + δ a b Ω τ c c -4i w τ Ω τ a +4iε abc ω µ (1)τ b ω (1)τ µc -ω µ (3)τ b ω (3)τ µc = 0 3m ω (3)τ νa -G 3m τ ω (3)τ νa +iσ µν m G τ 0 ′ m ω (1)τ νa -G 0 ′ m τ ω (1)τ νa +∆ τ ω µ (1)τ -∆ τ ω (1) 3m ω (1)τ νa -G 3m τ ω (1)τ νa -iσ µν m G τ 0 ′ m ω (3)τ νa -G 0 ′ m τ ω (3)τ νa +∆ τ ω µ (3)τ a -∆ τ ω (3) µ τ a -i Ω τ a b -δ b a Ω τ c c ω µ(1)τ -2i w τ ω (1) µ τ a -iε abc Ω τ b also c.c. equations. In Eqs. (3.66), (3.67) and (3.68) the following quantities have been introduced 

2 LG τ m ′ + m 2 - 1 Φ 2 L 2 L (G 2 y 4 L 1 Φ 2 LG 2 Lm 2 Φ + 1 Φ 2 L (G 2 yFor

 2122241222122 to the massless AdS 4 ×CP 3 superparticle model these quantities take into account contribution of the Wess-Zumino term and the overall factor of Φ -2 L in the kinetic term. Finally there are eight equations corresponding to variation of coordinates of the sector of broken supersymmetries((G τ • G y ) + ΦA τ ) G ym ′ ∂Gτ m ′ A τ -Φ(G τ • G y )) A τ A τ (G y • ∂Gy ∂υµ ) + ( m 2 Φ 2 L -(G τ •G y )) ∂Φ ∂υµ + τ m ′ ∂Gτ m ′ ((G τ •G y ) + ΦA τ ) ∂(Gτ •Gy) ∂υµ + A τ -Φ(G τ •G y ))∂Aτ ∂υµ Eqs. (3.66), (3.67), (3.68) and (3.72) we obtained the representation in the form of the Lax equation dL D0-brane dτ + [M, L D0-brane ] = 0. The Lax pair component M is the same as in (3.41) and the Lax component L D0-brane isgiven by the sumL D0-brane = L so(2,3) + L su(4) + L 24susy ∈ osp(4|6),where each summand is the linear combination of the quantities introduced in (3.69)-(3.71)L so(2,3) = 2 G 0 ′ m τ M 0 ′ m + ∆ τ D + 2 G 3m τ M 3m + G mn τ M mn , L su(4) = Ω a τ T a + ¯ Ω τ a T a + 4 w τ V a a , L 24susy = ω µ (1)τ a Q (1) a µ + ¯ ω µa (1)τ Q(1)µa + ω µ (3)τ a Q (3) a µ + ¯ ω µa (3)τ Q(3)µa .L D0-brane takes the same form as for the massless AdS 4 ×CP 3 superparticle modulo definition of the coefficients (3.69)-(3.71). Analogously to the superparticle case the Lax pair component L D0-brane can be presented in the form of the osp(4|6)-valued differential operator applied to the D0-brane action L D0-brane = M 0 ′ m

S

  AdS 4 ×CP 3 D0-brane .

3 N

 3 model in the OSp(4|6)/(SO(1, 3) × U (3)) supercoset space in terms of the Cartan one-forms for generators of this superconformal algebra; -fermionic equations of the σ-model were presented in the form similar to respective equations of the Green-Schwarz superstrings and proved their linear dependence. According to the second Noether theorem this implies invariance of the σ-model action under the κ-symmetry transformations; -there was obtained explicit expression for the σ-model Lagrangian in terms of coordinates of the OSp(4|6)/(SO(1, 3) × U (3)) supercoset space that are associated with the generators of the D = 3 N = 6 superconformal algebra; -there were derived general expressions in terms of the Cartan forms for the Noether current densities corresponding to invariance of the σ-model action under global D = = 6 superconformal symmetry. For the OSp(4|6)/(SO(1, 3) × U (3)) supercoset element parametrized by the Poincaré coordinates for the AdS 4 space and 24 Grassmann coordinates that are parameters for generators of the D = 3 N = 6 Poincaré and special conformal supersymmetries obtained were explicit expressions for the Noether current densities in terms of these coordinates and their world-sheet derivatives. There were also found variations of these coordinates under inifitesimal transformations of the D = 3 N = 6 superconformal symmetry; -obtained was expression for the Lagrangian of the superstring in the κ-symmetry lightcone gauge in terms of the AdS 4 ×CP 3 superspace coordinates and their derivatives. Considered gauge corresponds to the light cone formed by null-geodesics on the conformal boundary of the AdS 4 space-time in the Poincaré coordinates and differs from other light-cone gauge conditions studied earlier. -there was obtained the Hamiltonian of the AdS 4 × CP 3 superstring in the light-cone gauge for the κ-symmetry and world-sheet reparametrizations. Its eigenvalues determine the energy spectrum of quantized superstring that according to the AdS 4 /CF T 3 correspondence conjecture coincides with that of anomalous dimensions of the gauge-invariant operators in the D = 3 N = 6 superconformal Chern-Simons-matter theory. It was shown that quadratic part of obtained Hamiltonian coincides with the light-cone gauge Hamiltonian of the Type IIA superstring in flat background. -there was proposed the Kaluza-Klein condition, which simplifies the double-dimensional reduction of the AdS 4 × S 7 supermembrane and the resultant AdS 4 × CP 3 superstring Lagrangian. For considered in the literature OSp(4|8)/(SO(1, 3) × SO(7)) supercoset elements the Kaluza-Klein condition restricts coordinates of the sector of the supersymmetries broken by the AdS 4 × CP 3 background. Proposed was the parametrization of this sector by four SL(2, R) spinor coordinates that satisfy (anti-)Majorana conditions. It was shown that in this parametrization the Kaluza-Klein condition can be met in the SL(2, R)-covariant way. Obtained were respective expressions for the supervielbein 1-forms of the AdS 4 × S 7 and AdS 4 × CP 3 superspaces; -obtained were the AdS 4 × CP 3 superstring Lagrangian and equations in the partial κ-symmetry gauge, in which there remains single Majorana spinor coordinate in the sector of broken supersymmetries. It is the parameter for generators of the broken Poincaré supersymmetries. For the superstring equations found was the zero-curvature representation. It includes world-sheet one-form that extends the Lax connection of the OSp(4|6)/(SO(1, 3) × U (3)) σ-model by linear and quadratic terms in this coordinate and its differential;

  Densities of the Noether currentsTo obtain expressions for the densities of the Noether currents associated with the invariance of the σ-model action(2.22) under each of the transformations from the global D = 3 N = 6 superconformal symmetry consider variations of the Cartan forms that enter its Lagrangian under these transformations with local parameters. In this case variations of the Cartan

	forms (2.50), (2.54), (2.55) and (2.56), derived in the previous paragraph, acquire additional
	terms proportional to the differential of the transformation parameters. From the general
	formula for variation of a differential form (2.29) it follows that such contributions arise from
	the first summand. For the supercoset Cartan forms these terms have the form

2φ

(ηη)ε µλ bλν η νa and c.c.

2.2.2

Table 2 . 6

 26

: Tensors contributing to the Noether current density associated with the su(4)/u(3)

Table 2 .

 2 7: Tensors contributing to the Noether current density associated with invariance under the Poincaré supersymmetry δ ξ ω m (d)+δ ξ c m (d) = j mµa dξ µa -jmµ

a d ξa µ + . . . :

Table 2 . 8

 28

: Tensors contributing to the Noether current density associated with invariance under special conformal supersymmetry

  [∂ i x + (Ω bos ja 4 ηa η4 -Ω bos j4 a ηa η 4 ) + ∂ j x + (Ω bos ia 4 ηa η4 -Ω bos i4 a ηa η 4 )] + 1 2 [∂ i x + (ε abc Ω bos j4 a ηb ηc -ε abc Ω bos ja 4 ηb ηc )+∂ j x + (ε abc Ω bos i4 a ηb ηc -ε abc Ω bos ia

	and
	g CP 3 l.c. ij = -1 2 (E ia E a j + E ja E a i ) = 1 2 (Ω bos ia	4 Ω bos j4	a + Ω bos ja	4 Ω bos i	a )
	+e -φ 4 ηb ηc )]
	+2[(η a	ηa ) 2 + e -φ (ε abc	ηa ηb ηc η4 + ε abc ηa ηb ηc η 4 ) + 2e -2φ ηa	ηa η 4	η4 ]∂

So in the proposed gauge the dilaton field equals background value ϕ 0 = 3 2 ln R klP determined by the vacuum solution of the IIA supergravity equations with the AdS 4 × CP 3 geometry. i x + ∂ j x + .

See also[START_REF] Fre | Pure Spinor Formalism for OSp(N |4) backgrounds[END_REF], where considered was the formulation of the AdS

× CP 3 superstring using pure spinors.

Let us note that in the representation (2.35) lower-diagonal blocks are the Schur complements of the upper-diagonal ones.

It is assumed that derivatives with respect to the Grassmann-odd variables act from the right.

For the AdS 5 × S 5 superstring analogous gauge condition was studied in[START_REF] Metsaev | Superstring action in AdS 5 ×S 5 : κ-symmetry light cone gauge[END_REF],[START_REF] Metsaev | Light cone superstring in AdS space-time[END_REF].

Given expression corresponds to the value s = 1 of the sign factor in (2.106).

The Lax connection of the OSp(4|6)/(SO(1, 3) × U (3)) σ-model represents particular case of the Lax connections of integrable σ-models in supercoset spaces, stability algebras of which are invariant under the Z 4 automorphism of the isometry superalgebras[START_REF] Bena | Hidden symmetries of the AdS 5 × S 5 superstring[END_REF],[START_REF] Adam | Integrability of Type II superstrings on Ramond-Ramond backgrounds in various dimensions[END_REF],[START_REF] Babichenko | Integrability and the AdS 3 /CF T 2 correspondence[END_REF],[START_REF] Sorokin | Superstrings in AdS 2 ×S 2 ×T 6[END_REF].

Acknowledgements

The author is grateful to Prof. A.A. Zheltukhin for numerous enlightening discussions and suggestions towards improvement of the text.

supercoset representative ensures that AdS 4 × S 7 supervielbein components do not depend on y but the supervielbein bosonic components in tangent space to AdS 4 acquire contributions proportional to dy. Components of the vector G m ′ y , that enters (3.1), are expressed through the contributions to the so(2, 3)/so [START_REF] Maldacena | The large N limit of superconformal field theories and supergravity[END_REF][START_REF] Witten | Anti-de Sitter space and holography[END_REF] Cartan forms proportional to dy in the following way

For the element (3.3) they are functions only of the Grassmann coordinates from the sector of broken supersymmetries

In (3.4) there was introduced the following shorthand notations for expressions involving contractions of the SL(2, R) spinors: (η η) = η µ ηµ , (θσ m θ) = θ µ σ m µν and includes coefficient one-forms

where * denotes dualization of the world-sheet one-forms:

Coefficient one-forms in (3.22), as indicates their first argument, depend on the spectral parameter z through the functions ℓ 1 (z) and ℓ 2 (z), the form of which is determined by the zero-curvature condition. The second summand in (3.21) takes value in the su(4) isometry algebra of the CP 3 manifold

and the coefficient one-forms equal

(3.23)

The last term in (3.21)

is the linear combination of the supersymmetry generators from eigenspaces g 1,3 , and one-

and c.c. depend on the spectral parameter z through the functions ℓ 2 (z), ℓ 3 (z) and ℓ 4 (z).

From the zero-curvature condition it follows that functions ℓ 1 (z), ℓ 2 (z), ℓ 3 (z) and ℓ 4 (z)

satisfy the same set of algebraic equations as in the case of the Lax connection of the

In the literature frequently used is the following solution of these equations turns to zero, these equations do not turn into identities. In this subsection it will be shown that they become the consequences of the fermionic equations of the OSp(4|6)/(SO(1, 3) × U (3)) σ-model. Note that such proof in the linearized approximation in the Grassmann coordinates was given in Ref. [START_REF] Sorokin | Evidence for the classical integrability of the complete AdS 4 ×CP 3 superstring[END_REF].

First let us present fermionic equations of the OSp(4|6)/(SO(1, 3)×U ( 3)) σ-model (2.33) obtained in Section 2 in another form

and

that involves 12 × 12 matrices

Their diagonal blocks are defined as

Note that, when θ µ = 0, fermionic equations of the AdS 4 × CP 3 superstring in the partial κ-symmetry gauge (3.14) and (3.15), obtained in the previous subsection, turn into Eqs.

(3.27) and (3.28). Since the result of the action of the projectors V ij ± on a D = 1 + 1 vector factorizes (see Eq. (2.32) of Section 2)

where

the right-hand side of which enter the first two summands of the Virasoro constraints

In the concluding part of this subsection we shall show that they are consequences of the fermionic equations of the σ-model (3.29) and (3.30).

Consider in detail Eq. (3.29). Applying the rank-eight projection matrix

that satisfies the condition Π 3 (1) -3Π 2 (1) + 2Π (1) = 0, allows to bring this equation to the block-diagonal form

where 2 × 2 matrices when Ω τ a ̸ = 0, Ω τ a ̸ = 0 that corresponds to its motion both the anti-de Sitter space and the CP 3 space. Therefore from the mass-shell constraint for massless superparticle (3.36) it follows that the right-hand side of (3.63) is non-zero.

Consider in detail the system of equations (3.61). Applying the rank-eight projection matrix Analogously it can be shown that from the system (3.62) there follow equations (3.60).