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

Dmytro Uvarov

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Dmytro Uvarov. Aspects of Classical Dynamics and Symmetries of the AdS 4 x CP 3 Superstring. 2023. hal-04325013

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# Aspects of Classical Dynamics and Symmetries
of the $AdS_4 \times CP^3$ Superstring

D.V. Uvarov

NSC Kharkiv Institute of Physics and Technology,
61108 Kharkiv, Ukraine

## Abstract

This article, based on author's works, reviews various aspects of classical dynamics and symmetries of the Type IIA superstring in the $AdS_4 \times CP^3$ superspace with the emphasis on isomorphic realization of the orthosymplectic $OSp(4|6)$ isometry of this superspace as the $D = 3 \ N = 6$ superconformal symmetry.

In the sector of superstring dynamics described by the two-dimensional $OSp(4|6)/(SO(1,3) \times U(3))$ $\sigma$-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 $OSp(4|6)/(SO(1,3) \times U(3))$ representative parametrized by coordinates associated with the generators of superconformal algebra. Also found are variations of the Cartan forms and $OSp(4|6)/(SO(1,3) \times U(3))$ coordinates under the global $D = 3 \ N = 6$ superconformal symmetry and derived are the Noether current densities associated with the $\sigma$-model action invariance under this symmetry.

It is also derived the superstring Lagrangian in the $\kappa$-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 $AdS_4 \times CP^3$ superstring. In the partial $\kappa$-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 of the $OSp(4|6)/(SO(1,3) \times U(3))$ $\sigma$-model.

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 $\kappa$-symmetry.

Then it is proved that equations of the models of massless superparticle in the $AdS_4 \times CP^3$ superspace and its $OSp(4|6)/(SO(1,3) \times U(3))$ 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 $\sigma$-model in the $OSp(4|6)/(SO(1,3) \times U(3))$ supercoset space. As a generalization of these results it is proved integrability of the D0-brane equations in the $AdS_4 \times CP^3$ superbackground.
# Contents

1 Introduction 3

2 Superstring in $AdS_4 \times CP^3$ superspace and $N = 6$ superconformal symmetry in (1+2) dimensions 4

2.1 Lagrangian formulation of the $OSp(4|6)/(SO(1,3) \times U(3))$ $\sigma$-model in terms of Cartan forms in conformal basis 6

2.1.1 $osp(4|6)$ superalgebra generators, Cartan forms and Maurer-Cartan equations 7

2.1.2 $OSp(4|6)/(SO(1,3) \times U(3))$ $\sigma$-model Lagrangian 12

2.2 $D=3$ $N=6$ superconformal symmetry of the $\sigma$-model action in the $OSp(4|6)/(SO(1,3) \times U(3))$ superspace 20

2.2.1 Transformation properties of the Cartan forms and coordinates of the $OSp(4|6)/(SO(1,3) \times U(3))$ supercoset 21

2.2.2 Densities of the Noether currents 25

2.3 Lagrangian of the $AdS_4 \times CP^3$ superstring in the $\kappa$-symmetry light-cone gauge 26

2.3.1 $AdS_4 \times S^7$ supermembrane and its reduction to the $AdS_4 \times CP^3$ superstring 31

2.3.2 $AdS_4 \times CP^3$ superstring in the $\kappa$-symmetry light-cone gauge 37

2.4 Light-cone gauge Hamiltonian of the $AdS_4 \times CP^3$ superstring 41

2.4.1 Formulation of the $AdS_4 \times CP^3$ superstring Lagrangian in the $\kappa$-symmetry light-cone gauge in terms of the phase-space variables 41

2.4.2 Light-cone gauge for reparametrization symmetry and the Hamiltonian of the $AdS_4 \times CP^3$ superstring 43

3 Classical integrability of equations of the massless superparticle and D0-brane in the $AdS_4 \times CP^3$ superspace. Minimal extension of the integrable $OSp(4|6)/(SO(1,3) \times U(3))$ $\sigma$-model 47

3.1 Kaluza-Klein gauge for $\kappa$-symmetry and minimal extension of the integrable $OSp(4|6)/(SO(1,3) \times U(3))$ $\sigma$-model 48

3.1.1 $AdS_4 \times S^7$ and $AdS_4 \times CP^3$ supervielbeins in the Kaluza-Klein gauge 49

3.1.2 Minimal extension of the integrable $OSp(4|6)/(SO(1,3) \times U(3))$ $\sigma$-model 52

3.2 Connection between fermionic equations of the $AdS_4 \times CP^3$ superstring and the $OSp(4|6)/(SO(1,3) \times U(3))$ $\sigma$-model 57

3.3 Classical integrability of equations of the massless superparticle and D0-brane in the $AdS_4 \times CP^3$ superbackground 59

3.3.1 Classical integrability of equations of the massless superparticle in the $OSp(4|6)/(SO(1,3) \times U(3))$ superspace 59

3.3.2 Classical integrability of equations of the massless superparticle in the $AdS_4 \times CP^3$ superspace 61
<table>
  <tbody>
    <tr>
      <td>3.3.3</td>
      <td>Classical integrability of equations of the D0-brane in the AdS<sub>4</sub> × CP<sup>3</sup> superbackground</td>
      <td>67</td>
    </tr>
    <tr>
      <td>4</td>
      <td>Conclusion</td>
      <td>70</td>
    </tr>
    <tr>
      <td></td>
      <td>Acknowledgements</td>
      <td>71</td>
    </tr>
    <tr>
      <td></td>
      <td>References</td>
      <td>72</td>
    </tr>
  </tbody>
</table>

# 1 Introduction

The AdS/CFT correspondence [1], [2], [3] 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 [4].

To date the best explored instance of the AdS/CFT correspondence provides dual description of the $D = 10$ Type IIB string theory on the $AdS_5 \times S^5$ superbackground with $N$ units of the Ramond-Ramond five-form flux as the $D = 4 \mathcal{N} = 4$ supersymmetric Yang-Mills theory with $U(N)$ gauge symmetry [1].

Later in [5] another explicit example of the AdS/CFT duality was suggested stating that the $D = 3 \mathcal{N} = 6$ superconformal Chern-Simons-matter theory named ABJM theory gives dual description of the $D = 10$ Type IIA string theory on the $AdS_4 \times CP^3$ superbackground. This IIA supergravity background is known since the mid of 80-s [6], [7], [8] and preserves 24 of 32 supersymmetries. Together with the $SO(2,3) \times SU(4)$ symmetry of the $AdS_4 \times CP^3$ space-time they form the $OSp(4|6)$ isometry supergroup of the $AdS_4 \times CP^3$ superspace. The fact that this superspace is non-maximally supersymmetric, as opposed to the $AdS_5 \times S^5$ one, makes the ABJM correspondence more difficult to investigate.

On the gravity side of this correspondence applicability of the two-dimensional $\sigma$-model [9], [10] description of the $AdS_4 \times CP^3$ superstring dynamics is limited to the $OSp(4|6)/(SO(1,3) \times U(3))$ subspace of $AdS_4 \times CP^3$ superspace in distinction to the $PSU(2,2|4)/(SO(1,4) \times SO(5))$ $\sigma$-model description of the $AdS_5 \times S^5$ superstring [11], [12]. The reason is that the $AdS_4 \times CP^3$ superspace is not isomorphic to a supercoset manifold unlike the $AdS_5 \times S^5$ one. Moreover, although the $OSp(4|6)/(SO(1,3) \times U(3))$ $\sigma$-model is classically integrable, it correctly describes $AdS_4 \times CP^3$ superstring dynamics only in those sectors, where vanishing condition of eight coordinates of the sector of broken supersymmetries represents partial $\kappa$-symmetry gauge condition. This is not the case, e.g. when the superstring moves only within the $AdS_4$ subspace of the $AdS_4 \times CP^3$ space-time [9].

To correctly describe superstring dynamics in this sector one needs to use complete action of the $AdS_4 \times CP^3$ superstring. Since the $AdS_4 \times CP^3$ superspace is not a supercoset manifold the superstring action cannot be constructed in the same way as for the $AdS_5 \times S^5$ superstring. However, to maximally exploit available symmetry in [13] it was used the double-dimensional reduction [14], [15] of the $AdS_4 \times S^7$ supermembrane [16] to obtain the superstring action.
The point is that the $AdS_4 \times S^7$ superspace can be realized as the $OSp(4|8)/(SO(1,3) \times SO(7))$ supercoset manifold so that the coconstruction of the supermembrane action generalizes that of the $AdS_5 \times S^5$ superstring. Further it is known the relation between the $AdS_4 \times S^7$ background of $D=11$ supergravity and the $AdS_4 \times CP^3$ background of the IIA supergravity that relies on the Hopf fibration realization of the 7-sphere $S^7 = CP^3 \times S^1$ [7], [8]. In [13] with relation was lifted to superspace. However, resulting $AdS_4 \times CP^3$ superstring 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 $\kappa$-symmetry and world-sheet reparametrizations. Also one can try to extend the proof of integrability of equations of the $OSp(4|6)/(SO(1,3) \times U(3))$ $\sigma$-model to those of the $AdS_4 \times CP^3$ superstring. This problem was studied in [17], [18] but it still remains unsolved.

In the present review we pursue both ways to explore the structure of the $AdS_4 \times CP^3$ superstring. In the first part considered are dynamics and symmetries of the $OSp(4|6)/(SO(1,3) \times U(3))$ $\sigma$-model that describes the $AdS_4 \times CP^3$ superstring in the partial $\kappa$-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 $AdS_4 \times CP^3$ superstring is examined in various sectors and dynamical regimes.

## 2 Superstring in $AdS_4 \times CP^3$ 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 $OSp(4|6)/(SO(1,3) \times U(3))$ $\sigma$-model [19] in terms of the Cartan forms associated with the generators of the $D=3$ $N=6$ superconformal algebra. Action of the $OSp(4|6)/(SO(1,3) \times U(3))$ $\sigma$-model follows from that of the Type IIA superstring in the $AdS_4 \times CP^3$ superbackground [13] upon imposition of vanishing condition on the coordinates related to eight supersymmetries broken by the background. As was discussed in the Introduction the $\sigma$-model describes superstring dynamics only in those sectors, in which this condition fixes the $\kappa$-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 $ads(1,3)$ isometry algebra of the four-dimensional anti-de Sitter space $AdS_4 = SO(2,3)/SO(1,3)$ and the $conf(1,2)$ conformal algebra of its $D=1+2$ $^2$ Minkowski boundary space. $so(6)$ al-

<footnote>$^1$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. $su(2, 2N)$. 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.</footnote>

<footnote>$^2$Here and below notation $D=T+S$ emphasizes that the space has $T$ time and $S$ space dimensions.</footnote>
gebra is isomorphic to the $su(4)$ algebra that describes isometry of the complex projective
space $\mathbb{CP}^3 = SU(4)/U(3)$ 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 $D = 3 \mathcal{N} = 6$
superconformal algebra. This realization will be exploited in this and subsequent sections.

The $OSp(4|6)/(SO(1,3) \times U(3))$ supercoset manifold is the subspace of the $AdS_4 \times \mathbb{CP}^3$
superspace and is parametrized by 10 bosonic coordinates of the $AdS_4$ and $\mathbb{CP}^3$ spaces and
also 24 Grassmann-odd coordinates. As is known the (super)vielbein and connection one-
forms 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 $OSp(4|6)/(SO(1,3) \times U(3))$ supermanifold equal the $so(2,3)/so(1,3)$ and $su(4)/u(3)$
Cartan forms, and fermionic components equal the Grassmann-odd Cartan forms. Connec-
tion one-form is determined by the Cartan forms for the $so(1,3) \oplus u(3)$ stability subalgebra
generators.

Also in the present subsection fermionic equations of the $\sigma$-model will be presented in
the form similar to that of the fermionic equations of the Green-Schwarz superstrings [20]
and it will be proved their linear dependence. This implies the $\sigma$-model action invariance
under the gauge $\kappa$-symmetry that is the remnant of the $\kappa$-symmetry of the $AdS_4 \times \mathbb{CP}^3$
superstring. To study the non-linear structure of the $\sigma$-model Lagrangian explicit expressions
will be obtained for the constituent Cartan forms. Their form is determined by that of the
$OSp(4|6)/(SO(1,3) \times U(3))$ 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 $D=3 \mathcal{N}=6$ superconformal algebra.

Invariance of the $OSp(4|6)/(SO(1,3) \times U(3))$ $\sigma$-model under global $D=3 \mathcal{N}=6$ su-
perconformal symmetry is examined in subsection 2.2 on the basis of our work [21]. First
examined will be transformation properties of the $OSp(4|6)/(SO(1,3) \times U(3))$ 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 $D=3 \mathcal{N}=6$ supercon-
formal symmetry. Then given will be expressions for the Noether current densities in terms
of coordinates that parametrize the $OSp(4|6)/(SO(1,3) \times U(3))$ element considered in the
previous subsection.

Subsection 2.3 is devoted to derivation of the Lagrangian of the Type IIA superstring in
the $AdS_4 \times \mathbb{CP}^3$ superbackground in the $\kappa$-symmetry light-cone gauge that was obtained in
our work [22]. Considered gauge condition corresponds to the light cone on the conformal
boundary of the anti-de Sitter space-time parametrized the Poincaré coordinates that dis-
tinguishes it from other light-cone gauge conditions previously introduced in [23], [24], [25],
[26], [27], [28], [29].

In the final subsection 2.4 it is derived the Hamiltonian of the $AdS_4 \times \mathbb{CP}^3$ superstring in
the light-cone gauge following our work [30]. To this end obtained in the previous subsection
superstring Lagrangian in the $\kappa$-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 $OSp(4|6)/(SO(1,3) \times U(3))$ $\sigma$-model in terms of Cartan forms in conformal basis

As was mentioned in the Introduction, concrete formulation of the $AdS_4/CFT_3$ duality conjecture was given in Ref. [5], in which constructed was the $D = 3 \mathcal{N} = 6$ 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 $AdS_4 \times CP^3$ 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 $AdS_4 \times CP^3$ superstring. First found was the Lagrangian of the $\sigma$-model in the $OSp(4|6)/(SO(1,3) \times U(3))$ supercoset space[9], [10]$^3$ that is the subspace of the $AdS_4 \times CP^3$ superspace and then constructed was the complete Lagrangian of the superstring [13].

In Refs. [9], [10] the $OSp(4|6)/(SO(1,3) \times U(3))$ $\sigma$-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 $\mathcal{N} = 6$ superconformal symmetry in $D = 1+2$ dimensions [32], it is important for checking the $AdS_4/CFT_3$ duality conjecture to obtain a formulation of the $\sigma$-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 $\sigma$-model Lagrangian it is also important to obtain explicit expressions for relevant Cartan forms that are specified by the $OSp(4|6)/(SO(1,3) \times U(3))$ 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 $D = 3 \mathcal{N} = 6$ 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 $OSp(4|6)/(SO(1,3) \times U(3))$ $\sigma$-model in terms of the Cartan forms for the generators of the $D = 3 \mathcal{N} = 6$ superconformal algebra. Fermionic equations of the $\sigma$-model will be presented in the form similar to that for the Green-Schwarz superstrings and it will be proved

<footnote>$^3$See also [31], where considered was the formulation of the $AdS_4 \times CP^3$ superstring using pure spinors.</footnote>
that in the general case eight of 24 equations are not independent. This implies eight-parameter $\kappa$-invariance of the action. Explicit form of the $\kappa$-symmetry transformations will also be given that allows their comparison with the $\kappa$-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 $OSp(4|6)/(SO(1,3) \times U(3))$ $\sigma$-model is constructed out of the world-sheet mappings of the Cartan one-forms, whose form is determined by the $OSp(4|6)/(SO(1,3) \times U(3))$ supercoset element $\mathcal{G}$. Left-covariant osp(4|6) Cartan one-forms are defined by the expression

$$ \mathcal{G}^{-1} d\mathcal{G} = G_{\text{sp}(4)}(d) + G_{\text{so}(6)}(d) + G_{24\text{susy}}(d) \in \text{osp}(4|6), \quad (2.1) $$

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 $sp(4) = so(2,3) = \text{ads}(1,3) = \text{conf}(1,2)$ and $so(6) = \text{su}(4)$ subalgebras of the $osp(4|6)$ superalgebra $\mathfrak{g}$. 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

$$ \begin{aligned} G_{\text{sp}(4)}(d) &= G^{\alpha\beta}(d)O_{\alpha\beta} = G^{mn}(d)M_{mn} \\ &= 2G^{0'm'}(d)M_{0'm'} + G^{m'n'}(d)M_{m'n'} \end{aligned} \quad (2.2) \\ = G^{mn}(d)M_{mn} + \omega^m(d)P_m + c^m(d)K_m + \Delta(d)D \in \text{sp}(4). $$

In the first of them $O_{\alpha\beta} = O_{\beta\alpha}$ ($\alpha, \beta = 1, \dots, 4$) are generators of the $sp(4)$ algebra and $G^{\alpha\beta}(d)$ are the Cartan one-forms.

The second equality corresponds to the isomorphic realization of $sp(4)$ as the $so(2,3)$ algebra. There $M_{mn} = \frac{i}{4}\gamma_{mn}^{\alpha\beta}O_{\alpha\beta}$ ($\underline{m}, \underline{n} = 0', 0, 1, 2, 3$) are the $so(2,3)$ generators and $\gamma_{mn}^{\alpha\beta}$ denotes antisymmetrized product of $D = 2 + 3$ $\gamma$-matrices (details of the spinor algebra are given in Appendix A of our work [19]). $G^{mn}(d) = -\frac{i}{2}\gamma^{mn}G^{\alpha\beta}(d)$ 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 $OSp(4|6)/(SO(1,3) \times U(3))$ 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

$$
\begin{align}
[M_{0'm'}, M_{0'n'}] &= M_{m'n'}, \quad [M_{0'k'}, M_{m'n'}] = \eta_{k'm'} M_{0'n'} - \eta_{k'n'} M_{0'm'}, \tag{2.3} \\
[M_{k'l'}, M_{m'n'}] &= \eta_{k'n'} M_{l'm'} - \eta_{k'm'} M_{l'n'} - \eta_{l'n'} M_{k'm'} + \eta_{l'm'} M_{k'n'} \nonumber
\end{align}
$$

are invariant under two-element automorphism group $\mathbb{Z}_2$. Under its action generators $M_{0'm'}$
change the sign and $M_{m'n'}$ do not change. This automorphism is extended to the $\mathbb{Z}_4$ auto-
morphism 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

$$
D = -2M_{0'3}, \quad P_m = M_{0'm} - M_{3m}, \quad K_m = M_{0'm} + M_{3m}, \quad m = 0, 1, 2
$$

and respective Cartan forms

$$
\Delta(d) = -G^{0'3}(d), \quad \omega^m(d) = G^{0'm}(d) - G^{3m}(d), \quad c^m(d) = G^{0'm}(d) + G^{3m}(d). \tag{2.4}
$$

For the second summand in (2.1) we give three representations

$$
G_{\mathrm{so}(6)}(d) = \Omega^{IJ}(d)V^{IJ} = \Omega_A^B(d)V_B^A = \Omega_{\tilde{a}}^{\hat{b}}(d)V_{\hat{b}}^{\hat{a}}. \quad (2.5)
$$

In the first representation $V^{IJ} = -V^{JI}$ are the so(6) generators and $\Omega^{IJ}(d)$ 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

$$
\Omega_A^B(d) = \frac{i}{2} \Omega^{IJ}(d) \rho_{AJ}{}^B = \begin{pmatrix} \Omega_a^b(d) & \Omega_a^4(d) \\ \Omega_4^b(d) & \Omega_4^4(d) \end{pmatrix}, \quad \Omega_4^4(d) = -\Omega_a^a(d) \quad (2.6)
$$

and

$$
V_A^B = \frac{i}{4} \rho^{IJ}{}_A{}^B V^{IJ} = \begin{pmatrix} V_a^b & V_a^4 \\ V_4^b & V_4^4 \end{pmatrix}, \quad V_4^4 = -V_a^a, \qquad (2.7)
$$

where $\rho^{IJ}{}_A{}^B = \frac{1}{2}(\rho_I^{AC}\tilde{\rho}^{JCB} - \rho_J^C\tilde{\rho}^{ICB})$ are the so(6) generators in the spinor representation realized by the $D=6$ chiral $\gamma$-matrices $\rho_{AB}^I$ and $\tilde{\rho}^{IAB}$ 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. $\Omega_a^4(d)$ and $\Omega_4^a(d)$ are identified with the complex vielbein components of the $\mathbb{CP}^3$ manifold and $\Omega_a^b(d)$ – with the $u(3)$ connection one-form. Accordingly, $V_a^b$ are the generators of $u(3) \subset su(4)$ 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

$$
\begin{align*}
[V_a^4, V_4^b] &= i(V_a^b + \delta_a^b V_c^c), \quad [V_a^b, V_c^d] = i(\delta_c^b V_a^d - \delta_a^d V_c^b), \\
[V_a^4, V_b^c] &= -i\delta_a^c V_b^4, \quad [V_4^a, V_b^c] = i\delta_b^a V_4^c
\end{align*}
$$

become manifestly invariant under the sign change of the generators of $su(4)/u(3)$ quotient
algebra in analogy with the above mentioned $\mathbb{Z}_2$ automorphism of the ads(1,3) algebra (2.3).
The third representation in (2.5) is given in the $3 \oplus \bar{3}$ basis that corresponds to the decomposition of the $D=6$ vector on the $su(3)$ representations: $6 = 3 \oplus \bar{3}$. Connection between this and the usual bases is given by the relations $O^I = M^{I\hat{a}}O_{\hat{a}}$ and $O_{\hat{a}} = M_{\hat{a}I}^{-1}O^I$, where $O_{\hat{a}}$ are components of a vector in the $3 \oplus \bar{3}$ basis. The form of transformation matrices

$$M^{I\hat{a}} = \frac{1}{2}(\tilde{\rho}^{Ia4}, \rho_{a4}^I), \quad M_{\hat{a}I}^{-1} = \begin{pmatrix} \rho_{4a}^I \\ \tilde{\rho}^{I4a} \end{pmatrix} : \quad MM^{-1} = I$$

are determined by the $\rho_{AB}^I$ and $\tilde{\rho}^{IAB}$ matrices decomposed on the $su(3)$-covariant blocks. Transformation matrices satisfy the relations $\delta^{IJ} = -2M^{I\hat{a}}H_{\hat{a}\hat{b}}M^{T\hat{b}J}$ and $\delta_{IJ} = -\frac{1}{2}M_{I\hat{a}}^{-1T}H^{\hat{a}\hat{b}}M_{\hat{b}J}^{-1}$, where symmetric matrices

$$H_{\hat{a}\hat{b}} = \begin{pmatrix} 0 & \delta_a^b \\ \delta_b^a & 0 \end{pmatrix}, \quad H^{\hat{a}\hat{b}} = \begin{pmatrix} 0 & \delta_b^a \\ \delta_a^b & 0 \end{pmatrix}$$

are metric tensors in the $3 \oplus \bar{3}$ 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

$$\Omega_{\hat{a}}^{\hat{b}}(d) = 2i M_{\hat{a}I}^{-1} \Omega^I_J(d) M^{J\hat{b}} = \begin{pmatrix} \Omega_a^b(d) - \delta_a^b \Omega_c^c(d) & \varepsilon_{acb} \Omega_4^c(d) \\ -\varepsilon^{acb} \Omega_c^4(d) & -\Omega_b^a(d) + \delta_b^a \Omega_c^c(d) \end{pmatrix} \quad (2.8)$$

and

$$V_{\hat{a}}^{\hat{b}} = \frac{i}{2} M_{\hat{a}I}^{-1} V^I_J M^{J\hat{b}} = \frac{1}{2} \begin{pmatrix} V_a^b - \delta_a^b V_c^c & \varepsilon_{acb} V_4^c(d) \\ -\varepsilon^{acb} V_c^4 & -V_b^a + \delta_b^a V_c^c \end{pmatrix}. \quad (2.9)$$

The third summand in (2.1) includes products of the Grassmann-odd Cartan forms with 24 supersymmetry generators from the $osp(4|6)$ superalgebra

$$G_{24\text{susy}}(d) = F_I^\alpha(d)O_\alpha^I = F^{\alpha\hat{a}}(d)O_{\alpha\hat{a}} = \bar{F}^{\alpha\alpha}(d)\bar{O}_{\alpha\alpha} + F_a^\alpha(d)O_\alpha^a. \quad (2.10)$$

Both $F_I^\alpha(d)$ and $O_\alpha^I$ are four-component Majorana spinor $(2,3)$ spinors and $D=6$ vectors. In the second equality in (2.10) they are presented in the $3 \oplus \bar{3}$ basis: $F^{\alpha\hat{a}}(d) = M^{T\hat{a}I}F_I^\alpha$, $O_{\alpha\hat{a}} = M_{\hat{a}I}^{-1}O_\alpha^I$. Supersymmetry generators can be decomposed on the Poincaré and special conformal supersymmetry generators from the $D=3 \mathcal{N}=6$ superconformal algebra

$$O_{\alpha\hat{a}} = \begin{pmatrix} Q_{\mu\hat{a}} \\ S_{\hat{a}}^\mu \end{pmatrix} : \quad Q_{\mu\hat{a}} = \begin{pmatrix} \bar{Q}_{\mu a} \\ Q_\mu^a \end{pmatrix}, \quad S_{\hat{a}}^\mu = \begin{pmatrix} \bar{S}_a^\mu \\ S^{\mu a} \end{pmatrix}.$$

They are two-component $sl(2, \mathbb{R}) = so(1, 2)$ spinors in the (anti)fundamental representation of $su(3)$ algebra and are Hermitian-conjugate to one another: $\bar{Q}_{\mu a} = (Q_\mu^a)^\dagger$, $\bar{S}_a^\mu = (S^{\mu a})^\dagger$. For Cartan forms there hold analogous decompositions

$$F^{\alpha\hat{a}}(d) = \begin{pmatrix} \omega^{\mu\hat{a}}(d) \\ \chi_\mu^{\hat{a}}(d) \end{pmatrix} : \quad \omega^{\mu\hat{a}}(d) = \begin{pmatrix} \bar{\omega}^{\mu a}(d) \\ \omega_a^\mu(d) \end{pmatrix}, \quad \chi_\mu^{\hat{a}}(d) = \begin{pmatrix} \bar{\chi}_\mu^a(d) \\ \chi_{\mu a}(d) \end{pmatrix}$$

with $\bar{\omega}^{\mu a}(d) = (\omega_a^\mu(d))^\dagger$, $\bar{\chi}_\mu^a(d) = (\chi_{\mu a}(d))^\dagger$. Therefore $G_{24\text{susy}}(d)$ can be presented in the following form

$$G_{24\text{susy}}(d) = \omega_a^\mu(d)Q_\mu^a + \bar{\omega}^{\mu a}(d)\bar{Q}_{\mu a} + \chi_{\mu a}(d)S^{\mu a} + \bar{\chi}_\mu^a(d)\bar{S}_a^\mu. \quad (2.11)$$
So we defined isomorphic realization of the osp(4|6) superalgebra as the $D = 3 \mathcal{N} = 6$ superconformal algebra. This realization will be used in what follows.

Taking that there is a basis of the osp(4|6) superalgebra $\mathfrak{g}$, in which generators have definite eigenvalues under the $\mathbb{Z}_4$ automorphism $\Upsilon$: $\Upsilon(\mathfrak{g}_j) = e^{i\pi j/2}\mathfrak{g}_j$, its relations can be written in the concise form

$$ \mathfrak{g} = \bigoplus_{j=0}^{3} \mathfrak{g}_j : \{\mathfrak{g}_i, \mathfrak{g}_j\} = g_{i+j(\text{mod } 4)}. $$

This automorphism is extension 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 $c^i(d)$, then (2.1) acquires the form

$$ \mathcal{G}^{-1} d \mathcal{G} = \sum_{j \in \{0,1,2,3\}} G_j(d) = \sum_{j \in \{0,1,2,3\}} c^j(d) \mathfrak{g}_j. \qquad (2.12) $$

Explicit form of each summand is the following

$$
\begin{align*}
G_o(d) &= G^{mn}(d)M_{mn} + 2G^{3m}(d)M_{3m} + \Omega_a^b(d)V_b^a + \Omega_4^4(d)V_4^4 \\
&= G^{mn}(d)M_{mn} + \frac{1}{2}(c^m(d) - \omega^m(d))(K_m - P_m) + \Omega_a^b(d)V_b^a + \Omega_4^4(d)V_4^4, \\
G_2(d) &= 2G^{0'm}(d)M_{0'm} + 2G^{0'3}(d)M_{0'3} + \Omega_a^4(d)V_4^a + \Omega_4^a(d)V_a^4 \\
&= \frac{1}{2}(c^m(d) + \omega^m(d))(K_m + P_m) + \Delta(d)D + \Omega_a^4(d)V_4^a + \Omega_4^a(d)V_a^4, \\
G_1(d) &= \omega_{(1)a}^{\mu}(d)Q_{(1)\mu}^{a} + \bar{\omega}_{(1)}^{\mu a}(d)\bar{Q}_{(1)\mu a}, \tag{2.13} \\
G_3(d) &= \omega_{(3)a}^{\mu}(d)Q_{(3)\mu}^{a} + \bar{\omega}_{(3)}^{\mu a}(d)\bar{Q}_{(3)\mu a}. \tag{2.14}
\end{align*}
$$

In (2.13) and (2.14) generators from the $\mathfrak{g}_1$ and $\mathfrak{g}_3$ eigenspaces are defined by the linear combinations of the Poincaré and special conformal supersymmetry generators

$$ Q_{(1,3)\mu}^{a} = Q_{\mu}^{a} \pm iS_{\mu}^{a}, \quad \bar{Q}_{(1,3)\mu a} = \bar{Q}_{\mu a} \mp i\bar{S}_{\mu a}. \qquad (2.15) $$

Associated Cartan forms equal

$$ \omega_{(1,3)a}^{\mu}(d) = \frac{1}{2}(\omega_a^{\mu}(d) \pm i\chi_a^{\mu}(d)), \quad \bar{\omega}_{(1,3)}^{\mu a}(d) = \frac{1}{2}(\bar{\omega}^{\mu a}(d) \mp i\bar{\chi}^{\mu a}(d)). \qquad (2.16) $$

These generators can also be written in terms of the four-component spinors

$$ P_{+\alpha}{}^{\beta} \mathcal{O}_{\beta}^{a} = \frac{1}{2} \begin{pmatrix} Q_{(1)\mu}^{a} \\ -iQ_{(1)}^{\mu a} \end{pmatrix}, \quad P_{-\alpha}{}^{\beta} \bar{\mathcal{O}}_{\beta a} = \frac{1}{2} \begin{pmatrix} \bar{Q}_{(1)\mu a} \\ i\bar{Q}_{(1)a}^{\mu} \end{pmatrix} $$

and

$$ P_{-\alpha}{}^{\beta} \mathcal{O}_{\beta}^{a} = \frac{1}{2} \begin{pmatrix} Q_{(3)\mu}^{a} \\ iQ_{(3)}^{\mu a} \end{pmatrix}, \quad P_{+\alpha}{}^{\beta} \bar{\mathcal{O}}_{\beta a} = \frac{1}{2} \begin{pmatrix} \bar{Q}_{(3)\mu a} \\ -i\bar{Q}_{(3)a}^{\mu} \end{pmatrix} $$

with the aid of the $D = 1 + 3$ chiral projectors

$$ P_{\pm \alpha}{}^{\beta} = \frac{1}{2}(\delta_{\alpha}^{\beta} \pm i\Gamma_{\alpha}^{5\beta}), \quad \Gamma_{\alpha}^{5\beta} = (\Gamma^0\Gamma^1\Gamma^2\Gamma^3)_{\alpha}{}^{\beta} = \begin{pmatrix} 0 & \epsilon_{\mu\nu} \\ -\epsilon^{\mu\nu} & 0 \end{pmatrix}. $$
They satisfy the defining relations $P_+ + P_- = I$, $P_\pm P_\pm = P_\pm$, $P_+ P_- = P_- P_+$, $P_+ P_- = P_- P_+$, $P_+ P_+ = P_- P_-$, and also $(P_\pm)^\beta = P_\pm^\beta$, $(P_\pm)^\alpha = P_\pm^\alpha$. Further relations that projectors obey as well as the details of the definition of $D = 1+3$ $\gamma$-matrices and their connection with the $D = 2+3$ and $D = 1+2$ $\gamma$-matrices are given in Ref. [19].

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

$$ 
\begin{aligned}
F_{+a}^{\alpha}(d) &= F_{a}^{\beta}(d) P_{+\beta}^{\alpha} = \left( \omega_{(1)a}^{\mu}(d) - i\omega_{(1)\mu a}(d) \right), \\
\bar{F}_{-a}^{\alpha a}(d) &= \bar{F}_{a}^{\beta a}(d) P_{-\beta}^{\alpha} = \left( \bar{\omega}_{(1)}^{\mu a}(d) - i\bar{\omega}_{(1)\mu a}^{a}(d) \right)
\end{aligned}
\qquad (2.17) $$

and

$$ 
\begin{aligned}
F_{-a}^{\alpha}(d) &= F_{a}^{\beta}(d) P_{-\beta}^{\alpha} = \left( \omega_{(3)a}^{\mu}(d) - i\omega_{(3)\mu a}(d) \right), \\
\bar{F}_{+a}^{\alpha a}(d) &= \bar{F}_{a}^{\beta a}(d) P_{+\beta}^{\alpha} = \left( \bar{\omega}_{(3)}^{\mu a}(d) - i\bar{\omega}_{(3)\mu a}^{a}(d) \right).
\end{aligned}
\qquad (2.18) $$

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 $\sigma$-model in various forms.

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

$$ 
\begin{aligned}
dG^{0'm'}(d) - 2G^{m'}_{\quad n'}(d) \wedge G^{0'n'}(d) - iF_a^\alpha(d) \wedge \gamma_{\alpha\beta}^{0'm'} \bar{F}^{\beta a}(d) &= 0, \\
d\Omega_a^4(d) + i\Omega_{+a}^b(d) \wedge \Omega_b^4(d) - \varepsilon_{abc}\bar{F}^{ab}(d) \wedge C_{\alpha\beta}\bar{F}^{\beta c}(d) &= 0, \\
d\Omega_4^a(d) + i\Omega_4^b(d) \wedge \Omega_{+b}^a(d) + \varepsilon^{abc}F_b^\alpha(d) \wedge C_{\alpha\beta}F_c^\beta(d) &= 0, \\
dF_a^\alpha(d) + \frac{1}{2}F_a^\beta(d) \wedge (2G_{0'm'}(d)\gamma^{0'm'}_{\quad \alpha} + G_{m'n'}(d)\gamma^{m'n'}_{\quad \alpha}) + i\Omega_{-a}^b(d) \wedge F_b^\alpha(d) + i\varepsilon_{acb}\Omega_4^c(d) \wedge \bar{F}^{ab}(d) &= 0, \\
d\bar{F}^{\alpha a}(d) + \frac{1}{2}\bar{F}^{\beta a}(d) \wedge (2G_{0'm'}(d)\gamma^{0'm'}_{\quad \alpha} + G_{m'n'}(d)\gamma^{m'n'}_{\quad \alpha}) + i\bar{F}^{\alpha b}(d) \wedge \Omega_{-b}^a(d) - i\varepsilon^{abc}\Omega_c^4(d) \wedge F_b^\alpha(d) &= 0,
\end{aligned}
\qquad (2.19) $$

where $\Omega_{\pm a}^b(d) = \Omega_a^b(d) \pm \delta_a^b \Omega_c^c(d)$ and $\wedge$ 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 $D=3 \mathcal{N}=6$ superconformal algebra

$$ 
\begin{aligned}
d\omega^m(d) - 2\Delta(d) \wedge \omega^m(d) - 2G^m_n(d) \wedge \omega^n(d) + 2i\omega_a^\mu(d) \wedge \sigma_{\mu\nu}^m \bar{\omega}^{\nu a}(d) &= 0, \\
dc^m(d) + 2\Delta(d) \wedge c^m(d) - 2G^m_n(d) \wedge c^n(d) + 2i\chi_{\mu a}(d) \wedge \tilde{\sigma}^{m\mu\nu} \bar{\chi}_\nu^a(d) &= 0, \\
d\Delta(d) - \omega^m(d) \wedge c_m(d) - i(\bar{\omega}^{\mu a}(d) \wedge \chi_{\mu a}(d) + \omega_a^\mu(d) \wedge \bar{\chi}_\mu^a(d)) &= 0, \\
d\Omega_a^4(d) + i\Omega_{+a}^b(d) \wedge \Omega_b^4(d) - 2\varepsilon_{abc}\bar{\omega}^{\mu b}(d) \wedge \bar{\chi}_\mu^c(d) &= 0, \\
d\Omega_4^a(d) + i\Omega_4^b(d) \wedge \Omega_{+b}^a(d) + 2\varepsilon^{abc}\omega_b^\mu(d) \wedge \chi_{\mu c}(d) &= 0
\end{aligned}
\qquad (2.20) $$
and

$$
\begin{equation}
\begin{aligned}
d\omega_{\hat{a}}^{\mu}(d) - \Delta(d) \wedge \omega_{\hat{a}}^{\mu}(d) &+ \frac{1}{2}\omega_{\hat{a}}^{\nu}(d) \wedge G_{mn}(d)\sigma^{mn}{}^{\mu} \\
&+ \omega^{m}(d) \wedge \tilde{\sigma}_{m}^{\mu\nu}\chi_{\nu\hat{a}}(d) + i\Omega_{\hat{a}}{}^{\hat{b}}(d) \wedge \omega_{\hat{b}}^{\mu}(d) = 0, \\
d\chi_{\mu\hat{a}}(d) + \Delta(d) \wedge \chi_{\mu\hat{a}}(d) &+ \frac{1}{2}G_{mn}(d) \wedge \sigma^{mn}{}^{\mu}\chi_{\nu\hat{a}}(d) \\
&- c_{m}(d) \wedge \sigma_{\mu\nu}^{m}\omega_{\hat{a}}^{\nu}(d) + i\Omega_{\hat{a}}{}^{\hat{b}}(d) \wedge \chi_{\mu\hat{b}}(d) = 0,
\end{aligned}
\tag{2.21}
\end{equation}
$$

where matrices

$$
\sigma_{\mu\nu}^{m} = (I, \sigma^{1}, -\sigma^{3}), \quad \tilde{\sigma}^{m\mu\nu} = \epsilon^{\mu\lambda\nu\rho}\sigma_{\lambda\rho}^{m} = (I, -\sigma^{1}, \sigma^{3})
$$

are $D = 1 + 2$ analogs of relativistic Pauli matrices and $\sigma^{mn}{}^{\mu} = \frac{1}{2}(\sigma_{\mu\lambda}^{m}\tilde{\sigma}^{n\lambda\nu} - \sigma_{\mu\lambda}^{n}\tilde{\sigma}^{m\lambda\nu})$,
$\tilde{\sigma}^{mn\mu}{}^{\nu} = -\sigma^{mn}{}^{\nu\mu}$.

2.1.2 OSp(4|6)/(SO(1,3) × U(3)) σ-model Lagrangian

As in the case of the Green-Schwarz superstrings, action of the $OSp(4|6)/(SO(1,3) \times U(3))$
$\sigma$-model is given by the sum of kinetic and Wess-Zumino terms

$$
S_{\sigma\text{-model, kin}} = S_{\sigma\text{-model, kin}} + S_{\sigma\text{-model, WZ}} : \\
S_{\sigma\text{-model, kin}} = \int_{\Sigma} d^2\xi \mathcal{L}_{\sigma\text{-model, kin}}, \quad S_{\sigma\text{-model, WZ}} = \int_{\Sigma} d^2\xi \mathcal{L}_{\sigma\text{-model, WZ}} \tag{2.22}
$$

where $\xi^i$ are local coordinates of the world-sheet $\Sigma$. The $\sigma$-model Lagrangian is constructed
out of the mappings to the world-sheet of the Cartan forms that are identified with the
$OSp(4|6)/(SO(1,3) \times U(3))$ supervielbein components using the same method as was elab-
orated for construction of the $AdS_5 \times S^5$ superstring Lagrangian [11], [12], [33].

The Polyakov kinetic term has the same form as for the Green-Schwarz superstrings on
curved background and, in particular, on the $AdS_5 \times S^5$ one. It is specified by the mapping
to the world-sheet of the space-time metric

$$
\mathcal{L}_{\sigma\text{-model, kin}} = -\frac{1}{2}\sqrt{-\gamma}\gamma^{ij}(g_{ij}^{AdS_4} + g_{ij}^{CP^3}), \qquad (2.23)
$$

where $g_{ij}^{AdS_4} = G_i{}^{0'm'}\eta_{m'n'}G_j{}^{0'n'}$ and $g_{ij}^{CP^3} = \frac{1}{2}(\Omega_{ia}{}^4\Omega_{j4}{}^a + \Omega_{ja}{}^4\Omega_{i4}{}^a)$ are contributions to induced world-sheet metric determined by the metric tensors of the $AdS_4$ and $\mathbb{CP}^3$ spaces.<fnref n="4" /> They are constructed out of the $so(2,3)/so(1,3)$ and $su(4)/u(3)$ Cartan forms that determine vielbeins of these spaces. $\gamma$ is determinant of the auxiliary world-sheet metric $\gamma_{ij}$, its inverse is $\gamma^{ij}$. Lagrangian (2.23) is invariant under the $\mathbb{Z}_4$ automorphism acting on these Cartan forms by the sign change. $g_{ij}^{AdS_4}$ can also be presented in terms of the Cartan forms associated

<footnote n="4">⁴In 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 $\sigma$-model Lagrangians there enters as the overall factor the dimensionless tension $\tilde{T}_{\text{string}} = \frac{R^2}{2\pi\alpha'}$ that is set to unity. Note that in the M-theory the Regge slope parameter $\alpha'$ is expressed via the $D = 11$ Planck length $l_P$ and string coupling constant $g_s$: $\alpha' = g_s^{-2/3}l_P^2$. In the considered case of the $AdS_4/CFT_3$ correspondence the string coupling equals $g_s = (\frac{R}{kl_P})^{3/2} \ll 1$.</footnote>
with the conf(1,2) algebra generators. Utilization of (2.4) yields the following expressions for the AdS₄ vielbein components

$$G^{0'm}(d) = \frac{1}{2}(\omega^m(d) + c^m(d)), \quad G^{0'3}(d) = -\Delta(d). \qquad (2.24)$$

Then the Lagrangian (2.23) acquires the form

$$\mathcal{L}_{\sigma\text{-model, kin}} = -\frac{1}{2}\sqrt{-\gamma\gamma^{ij}}\left(\frac{1}{4}(\omega_i^m + c_i^m)(\omega_{jm} + c_{jm}) + \Delta_i\Delta_j + \frac{1}{2}(\Omega_{ia}{}^4\Omega_{j4}{}^a + \Omega_{ja}{}^4\Omega_{i4}{}^a)\right). \quad (2.25)$$

The OSp(4|6)/(SO(1,3) × U(3)) σ-model belongs to the family of σ-models on the super-coset spaces G/H with the isometry superalgebras g invariant under the Z₄ automorphism [34]. Its characteristic feature is that the invariant subspace is given by the stability algebra g₀ = h. The Wess-Zumino Lagrangians for the σ-models from this family have the following general form <fnref n="5" />

$$\mathcal{L}_{\sigma\text{-model, WZ}} = \text{sTr} G_1(d) \wedge G_3(d),$$

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 [20], it is given by mapping to the world-sheet of the two-form 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 [35].

In our works [19] and [21] 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 g₁ and g₃ eigenspaces

$$\begin{aligned} \mathcal{L}_{\sigma\text{-model, WZ}} &= -\varepsilon_{\mu\nu}(\omega_{(1)a}{}^{\mu}(d) \wedge \bar{\omega}_{(3)}^{\nu a}(d) + \omega_{(3)a}{}^{\mu}(d) \wedge \bar{\omega}_{(1)}^{\nu a}(d)) \\ &= -\frac{1}{2}(\omega_a^\mu(d) \wedge \varepsilon_{\mu\nu}\bar{\omega}^{\nu a}(d) + \chi_{\mu a}(d) \wedge \varepsilon^{\mu\nu}\bar{\chi}_\nu^a(d)). \end{aligned} \qquad (2.26)$$

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 ⊕ 3̄ basis for D = 6 vectors

$$\begin{aligned} \mathcal{L}_{\sigma\text{-model, WZ}} &= \frac{i}{4}\varepsilon_{\mu\nu}\mathfrak{J}_{\hat{a}}{}^{\hat{b}}\left(\omega_{(1)\hat{b}}{}^{\mu}(d) \wedge \omega_{(3)}^{\nu\hat{a}}(d) + \omega_{(3)\hat{b}}{}^{\mu}(d) \wedge \omega_{(1)}^{\nu\hat{a}}(d)\right) \\ &= \frac{i}{8}\mathfrak{J}_{\hat{a}}{}^{\hat{b}}\left(\omega_{\hat{b}}^{\mu}(d) \wedge \varepsilon_{\mu\nu}\omega^{\nu\hat{a}}(d) + \chi_{\mu\hat{b}}(d) \wedge \varepsilon^{\mu\nu}\chi_{\nu}^{\hat{a}}(d)\right), \end{aligned} \qquad (2.27)$$

where antisymmetric matrix

$$\mathfrak{J}_{\hat{a}\hat{b}} = \mathfrak{J}_{\hat{a}}{}^{\hat{c}} H_{\hat{c}\hat{b}} = 2i \begin{pmatrix} 0 & \delta_a^b \\ -\delta_b^a & 0 \end{pmatrix} \qquad (2.28)$$

<footnote n="5">⁵As 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.</footnote>
defines components of the Kähler two-form of the $\mathbb{CP}^3$ space.<fnref n="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(1,3) spinors

$$ \mathcal{L}_{\sigma-\text{model, WZ}} = \frac{i}{2} \mathcal{F}_a^{\alpha}(d) \wedge C'_{\alpha\beta} \bar{\mathcal{F}}^{\beta a}(d) = \frac{1}{8} \mathcal{J}_{\hat{a}}{}^{\hat{b}} \mathcal{F}_{\hat{b}}^{\alpha}(d) \wedge C'_{\alpha\beta} \mathcal{F}^{\beta\hat{a}}(d), $$

where $C'_{\alpha\beta}$ 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

$$ \mathcal{L}_{\sigma-\text{model, WZ}} = \frac{i}{2} (\mathcal{F}_{+a}^{\alpha}(d) \wedge C'_{\alpha\beta} \bar{\mathcal{F}}_{+}^{\beta a}(d) + \mathcal{F}_{-a}^{\alpha}(d) \wedge C'_{\alpha\beta} \bar{\mathcal{F}}_{-}^{\beta a}(d)) = \frac{1}{4} \mathcal{J}_{\hat{a}}{}^{\hat{b}} \mathcal{F}_{+\hat{b}}^{\alpha}(d) \wedge C'_{\alpha\beta} \mathcal{F}_{-}^{\beta \hat{a}}(d), $$

where one-forms

$$ \mathcal{F}_{+\hat{a}}^{\text{T}\alpha}(d) = (\mathcal{F}_{+a}^{\alpha}(d) \bar{\mathcal{F}}_{-}^{\alpha a}(d)), \quad \mathcal{F}_{-}^{\alpha \hat{a}}(d) = \begin{pmatrix} \bar{\mathcal{F}}_{+}^{\alpha a}(d) \\ \mathcal{F}_{-a}^{\alpha}(d) \end{pmatrix}, $$

that are spin(1,3) spinors and $D=6$ vectors in the $\mathbf{3} \oplus \bar{\mathbf{3}}$ basis and so have 24 real components, will be used below to write down fermionic equations and the $\kappa$-symmetry transformations of the $\sigma$-model in the form similar to fermionic equations and the $\kappa$-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

$$ \delta F(d) = d(i_{\delta} F(d)) + i_{\delta}(dF(d)), \qquad (2.29) $$

where operator $i_{\delta}$ 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

$$ \delta_f S_{\sigma-\text{model}} = \int_{\Sigma} d^2\xi \left( \mathcal{F}_{+i}^{\text{T}\alpha\hat{a}} V_{-}^{ij} M_{j\alpha\hat{a}}^{\beta\hat{b}} C_{\beta\gamma} \mathcal{F}_{+\hat{b}}^{\gamma}(\delta) + \mathcal{F}_{-i}^{\text{T}\alpha\hat{a}} V_{+}^{ij} M_{j\alpha\hat{a}}^{\beta\hat{b}} C_{\beta\gamma} \mathcal{F}_{-\hat{b}}^{\gamma}(\delta) \right). \quad (2.30) $$

Variation (2.30) involves the matrix

$$ M_{i\alpha\hat{a}}^{\beta\hat{b}} = \begin{pmatrix} -\delta_a^b G_i^{0'm'} \Gamma^{m'\beta}_{\alpha} & \delta_{\alpha}^{\beta} \varepsilon_{acb} \Omega_{i4}^c \\ -\delta_{\alpha}^{\beta} \varepsilon^{acb} \Omega_{ic}^4 & -\delta_b^a G_i^{0'm'} \Gamma^{m'\beta}_{\alpha} \end{pmatrix} $$

which enter the world-sheet projections of the $AdS_4$ and $\mathbb{CP}^3$ vielbeins $G_i^{0'm'}$, $\Omega_{i4}^c$ and $\Omega_{ic}^4$ as well as the world-sheet projectors

$$ V_{\pm}^{ij} = \frac{1}{2}(\sqrt{-\gamma} \gamma^{ij} \pm \varepsilon^{ij}). \qquad (2.31) $$

$$ J_A^B = J^{IJ} \rho_{A}^{IJ} = \begin{pmatrix} -2i\delta_a^b & 0 \\ 0 & 6i \end{pmatrix}. $$

The matrix $J_A^B$ can be shown to satisfy the equation $J_A^C J_C^B - 4i J_A^B + 12\delta_A^B = 0$.

<footnote n="6">⁶In conventional basis for $D=6$ vectors respective expression for the components of the Kähler two-form is $J^{IJ} = \frac{i}{2}(\rho_{4a}^I \tilde{\rho}^{J4a} - \rho_{4a}^J \tilde{\rho}^{I4a})$. Upon contraction with the so(6) generators in the spinor representation they form the $4 \times 4$ diagonal matrix</footnote>
These are the same projectors that enter fermionic equations and $\kappa$-symmetry transformations of the Green-Schwarz superstrings. They obey the relations

$$V_{+}^{ij} + V_{-}^{ij} = \sqrt{-\gamma} \gamma^{ij}, \quad V_{\pm}^{ik} \gamma_{kl} V_{\pm}^{jl} = 0, \quad V_{\pm}^{ik} \gamma_{kl} V_{\pm}^{lj} = \sqrt{-\gamma} V_{\pm}^{ij}, \quad V_{\pm}^{ij} V_{\pm}^{kl} = V_{\pm}^{kj} V_{\pm}^{il}$$

and allow to decompose a $D = 1 + 1$ vector $v_i = (v_\tau, v_\sigma)$ over the basis of light-like vectors (see, e.g. [36])

$$V_{\pm}^{ij} v_j = V_{\pm}^{i} v_{\mp \tau}, \quad V_{\pm}^{i} = \frac{1}{2} \left( \frac{1}{\frac{\sqrt{-\gamma \gamma^{\tau \sigma} \mp 1}}{\sqrt{-\gamma \gamma^{\tau \tau}}}} \right), \quad v_{\mp \tau} = \sqrt{-\gamma \gamma^{\tau \tau}} v_{\tau} + (\sqrt{-\gamma \gamma^{\tau \sigma}} \pm 1) v_{\sigma}, \quad (2.32)$$

where $V_{\pm}^{i}$ are basis vectors. In conformal gauge for the metric $\gamma_{ij} = e^{\lambda} \eta_{ij}$ $v_{\pm \tau}$ equal light-cone components of the vector. Considering components of $\mathcal{F}_{\pm \hat{a}}^{\alpha}(\delta)$ as independent variation parameters yields fermionic equations of the $OSp(4|6)/(SO(1,3) \times U(3))$ $\sigma$-model

$$V_{\mp}^{ij} M_{j}^{\mathrm{T}\alpha\hat{a}} \beta\hat{b} \mathcal{F}_{\pm i}^{\beta\hat{b}} = 0, \quad (2.33)$$

which enter the $24 \times 24$ matrices

$$V_{\pm}^{i} M_{\mp}^{\mathrm{T}\alpha\hat{a}} \beta\hat{b} = V_{\pm}^{ij} M_{j}^{\mathrm{T}\alpha\hat{a}} \beta\hat{b}. \quad (2.34)$$

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 $OSp(4|6)/(SO(1,3) \times U(3))$ $\sigma$-model compared to the IIA superstring. As is known the $32 \times 32$ 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 $\gamma^{ij}$. Therefore half of them are linearly dependent that in accordance with the second Noether theorem implies the superstring invariance under the local $16$-parameter $\kappa$-symmetry, whereas independent equations determine dynamics of 16 physical fermionic fields of the superstrings. To show that the $\sigma$-model also describes dynamics of 16 physical fermionic fields it is necessary to calculate the rank of the matrices (2.34).

This can be done using the rank-preserving transformation that brings them to the triangular form

$$\left( \begin{array}{cc} \delta_b^a G_{\pm \tau}^{0' m' \Gamma^{m' \alpha} \beta} & \delta_{\beta}^{\alpha} \epsilon^{acb} \Omega_{\pm \tau c}^4 \\ 0 & -\frac{\Omega_{\pm \tau a}^4 \Omega_{\pm \tau 4}^b}{G_{\pm \tau}^{0' m'} G_{\pm \tau}^{0' m'}} G_{\pm \tau}^{0' m' \Gamma^{m' \alpha} \beta} \end{array} \right) = A_{\pm} B_{\pm}, \quad (2.35)$$

where $G_{\pm \tau}^{0' m'}$, $\Omega_{\pm \tau a}^4$ and $\Omega_{\pm \tau a}^4$ are the components of $G_{i}^{0' m'}$, $\Omega_{i4}^a$ and $\Omega_{i4}^a$ 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 $\gamma_{ij}$ have the form

$$G_{\pm \tau}^{0' m'} G_{\pm \tau}^{0' m'} + \Omega_{\pm \tau a}^4 \Omega_{\pm \tau 4}^a = 0. \quad (2.36)$$

In (2.35) matrices

$$A_{\pm}^{\alpha\hat{a}} \beta\hat{b} = \left( \begin{array}{cc} \delta_b^a G_{\pm\tau}^{0'm'\Gamma^{m'\alpha}\beta} & 0 \\ 0 & -\frac{\Omega_{\pm\tau a}^4 \Omega_{\pm\tau 4}^b}{G_{\pm\tau}^{0'm'} G_{\pm\tau}^{0'm'}} G_{\pm\tau}^{0'm'\Gamma^{m'\alpha}\beta} \end{array} \right)$$
are block-diagonal and

$$B_{\pm \beta \bar{\beta}}^{\alpha \hat{\alpha}} = \begin{pmatrix} \delta_b^\alpha \delta_\beta^\alpha & -\frac{\epsilon^{acb} \Omega_{\pm \tau c}^4}{G_{\pm \tau}^{0'm'} G_{\pm \tau}^{0'm'}} G_{\pm \tau}^{0'm'} \Gamma^{m'\alpha} \beta \\ 0 & \delta_a^b \delta_\beta^\alpha \end{pmatrix}$$

are non-degenerate. Therefore the rank of the matrices (2.35) equals that of matrices $A_\pm$. The 4 × 4 matrices $G_{\pm \tau}^{0'm'} \Gamma^{m'\alpha} \beta$ are non-singular if $G_{\pm \tau}^{0'm'} G_{\pm \tau}^{0'm'} \neq 0$. Then from the Virasoro constraints (2.36) it follows that $\Omega_{\pm \tau a}^4 \neq 0$, $\Omega_{\pm \tau 4}^a \neq 0$. These conditions are met when the superstring moves in the $AdS_4$ and $\mathbb{CP}^3$ spaces. In such case the rank of the 3 × 3 matrices $\Omega_{\pm \tau a}^4 \Omega_{\pm \tau 4}^b$ that enter the lower-diagonal blocks of the matrices $A_\pm$, 7 is unity since it is given by the product of the three-component $\mathbb{CP}^3$ vielbein and its c.c. As a result the rank of $A_\pm$ and $M_\pm^T$ 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 $\sigma$-model [9]. The consequence of degeneracy of the matrix $M_\pm^T$ is that eight fermionic equations of the $\sigma$-model are linearly dependent and are expressed via other 16 equations and its action (2.22) is invariant under the eight-parameter $\kappa$-symmetry. This symmetry is the remnant of the 16-parameter $\kappa$-symmetry of the full action of the Type IIA superstring in the $AdS_4 \times \mathbb{CP}^3$ superbackground [13].

As was noted, when superstring moves only in the $AdS_4$ space, i.e. $\Omega_{\pm \tau a}^4 = \Omega_{\pm \tau 4}^a = 0$, vanishing conditions of the coordinates for broken supersymmetries not only gauge fix the $\kappa$-symmetry but also contain initial data constraints that exclude four fermionic physical degrees of freedom. On the level of the fermionic equations of the $\sigma$-model this manifests itself in that the rank of $M_\pm^T$ reduces to 12 compared to the above considered general case. As a result half of fermionic equations of the $\sigma$-model are independent that implies its action invariance under the 12-parameter $\kappa$-symmetry transformations as was noted in [9]. 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 $\sigma$-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 $AdS_4 \times \mathbb{CP}^3$ superspace.

Above presented proof of the linear dependence of fermionic equations of the $OSp(4|6)/(SO(1,3) \times U(3))$ $\sigma$-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 $\sigma$-model in accordance with the second Noether theorem follows invariance of its action under the $\kappa$-symmetry with local fermionic parameters. Therefore alternative way to find under what conditions the $\sigma$-model correctly describes the superstring dynamics is to determine the number of independent parameters in the $\kappa$-symmetry transformations. To this end it is necessary to calculate the rank of the matrices that enter these transformations. As

<footnote n="7">7 Let us note that in the representation (2.35) lower-diagonal blocks are the Schur complements of the upper-diagonal ones.</footnote>
in the case of the Green-Schwarz superstrings it is convenient to present the $\kappa$-symmetry transformations of the $\sigma$-model in terms of the one-forms in which differentials of the superspace coordinates are replaced by their $\kappa$-variations. For fermionic one-forms respective expressions have the form

$$ \mathcal{F}_{+\hat{a}}^{\alpha}(\delta_{\kappa}) = V_{+}^{ij} V_{+}^{kl} K_{jl\hat{a}\beta}^{\alpha\hat{b}} \tilde{\mathcal{H}}_{+\hat{b}ik}^{\beta}, \quad \mathcal{F}_{-\hat{a}}^{\alpha}(\delta_{\kappa}) = V_{-}^{ij} V_{-}^{kl} K_{jl\hat{a}\beta}^{\alpha\hat{b}} \tilde{\mathcal{H}}_{-\hat{b}ik}^{\beta}, \quad (2.37) $$

where

$$ K_{ij\hat{a}\beta}^{\alpha\hat{b}} = \begin{pmatrix} \delta_{\beta}^{\alpha} (G_{i}^{0'm'} G_{j}^{0'm'} \delta_{a}^{b} + \Omega_{ia}{}^{4} \Omega_{j4}{}^{b}) & G_{i}^{0'm'} \Gamma^{m'\alpha}{}_{\beta} \varepsilon_{acb} \Omega_{j4}{}^{c} \\ -G_{i}^{0'm'} \Gamma^{m'\alpha}{}_{\beta} \varepsilon^{acb} \Omega_{jc}{}^{4} & \delta_{\beta}^{\alpha} (G_{i}^{0'm'} G_{j}^{0'm'} \delta_{b}^{a} + \Omega_{i4}{}^{a} \Omega_{jb}{}^{4}) \end{pmatrix}. $$

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

$$ G^{0'm'}(\delta_{\kappa}) = 0, \quad \Omega_{4}{}^{a}(\delta_{\kappa}) = 0, \quad \Omega_{a}{}^{4}(\delta_{\kappa}) = 0, $$

so the $\kappa$-variation of the $\sigma$-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

$$ \delta_{\kappa}(\sqrt{-\gamma}\gamma^{ij}) = 2i(\mathcal{F}_{+p}^{\alpha\hat{a}} V_{+}^{pr} G_{r}^{0'm'} \Gamma^{m'\alpha}{}_{\gamma} \Gamma^{5}_{\gamma\beta} V_{+}^{ki} V_{+}^{lj} \tilde{\mathcal{H}}_{+\hat{a}kl}^{\beta} + \mathcal{F}_{-p}^{\alpha\hat{a}} V_{-}^{pr} G_{r}^{0'm'} \Gamma^{m'\alpha}{}_{\gamma} \Gamma^{5}_{\gamma\beta} V_{-}^{ki} V_{-}^{lj} \tilde{\mathcal{H}}_{-\hat{a}kl}^{\beta}). $$

Local parameters $\tilde{\mathcal{H}}_{+\hat{a}ij}(\xi)$ and $\tilde{\mathcal{H}}_{-\hat{a}ij}(\xi)$ 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-)self-duality conditions in each of these indices

$$ \frac{1}{\sqrt{-\gamma}} \gamma_{ik} V_{+}^{kl} \tilde{\mathcal{H}}_{-\hat{a}lj}^{\alpha} = \frac{1}{\sqrt{-\gamma}} \gamma_{jk} V_{+}^{kl} \tilde{\mathcal{H}}_{-\hat{a}il}^{\alpha} = \tilde{\mathcal{H}}_{-\hat{a}ij}^{\alpha} $$

and

$$ \frac{1}{\sqrt{-\gamma}} \gamma_{ik} V_{-}^{kl} \tilde{\mathcal{H}}_{+\hat{a}lj}^{\alpha} = \frac{1}{\sqrt{-\gamma}} \gamma_{jk} V_{-}^{kl} \tilde{\mathcal{H}}_{+\hat{a}il}^{\alpha} = \tilde{\mathcal{H}}_{+\hat{a}ij}^{\alpha}. $$

On shell of the Virasoro constraints (2.36) the rank of the matrices $V_{\pm}^{ij} V_{\pm}^{kl} K_{jl\hat{a}\beta}^{\alpha\hat{b}}$ in (2.37) in the general case, when superstring moves both in the $AdS_4$ and in the $\mathbb{CP}^3$ spaces, equals eight. So that only $\frac{1}{3}$ of 24 parameters actually contribute to the $\kappa$-symmetry transformations. To prove this statement first note that their world-sheet vector indices are contracted with indices of the projectors $V_{\pm}^{ij}$ so they have only two independent components $K_{\pm\tau\pm\tau\hat{a}\beta}^{\alpha\hat{b}}$. The rank of these matrices can be found by solving the eigenvalue problem that amounts to computing the determinants of the matrices $K_{\pm\tau\pm\tau} - \lambda_{\pm\tau\pm\tau} I$ using their block structure

$$ \det(K_{\pm\tau\pm\tau} - \lambda_{\pm\tau\pm\tau} I) = \det \begin{pmatrix} A_{\pm\tau\pm\tau a\beta}^{\alpha b} & B_{\pm\tau\pm\tau a\beta b}^{\alpha} \\ C_{\pm\tau\pm\tau}^{\alpha ab} & D_{\pm\tau\pm\tau}^{\alpha a} \end{pmatrix} \quad (2.38) \\ = \det A_{\pm\tau\pm\tau} \det(D_{\pm\tau\pm\tau} - C_{\pm\tau\pm\tau} A_{\pm\tau\pm\tau}^{-1} B_{\pm\tau\pm\tau}), $$

where

$$ \begin{align*} A_{\pm\tau\pm\tau a\beta}^{\alpha b} &= \delta_{\beta}^{\alpha} A_{\pm\tau\pm\tau a}{}^{b}, \\ A_{\pm\tau\pm\tau a}{}^{b} &= (G_{\pm\tau}^{0'm'} G_{\pm\tau}^{0'm'} - \lambda_{\pm\tau\pm\tau}) \delta_{a}^{b} + \Omega_{\pm\tau a}{}^{4} \Omega_{\pm\tau 4}{}^{b}, \\ D_{\pm\tau\pm\tau}{}^{\alpha a}{}_{\beta b} &= \delta_{\beta}^{\alpha} ((G_{\pm\tau}^{0'm'} G_{\pm\tau}^{0'm'} - \lambda_{\pm\tau\pm\tau}) \delta_{b}^{a} + \Omega_{\pm\tau 4}{}^{a} \Omega_{\pm\tau b}{}^{4}), \\ B_{\pm\tau\pm\tau a\beta b}^{\alpha} &= -G_{\pm\tau}^{0'm'} \Gamma^{m'\alpha}{}_{\beta} \varepsilon_{acb} \Omega_{\pm\tau 4}{}^{c}, \\ C_{\pm\tau\pm\tau}{}^{\alpha ab}{}_{\beta} &= G_{\pm\tau}^{0'm'} \Gamma^{m'\alpha}{}_{\beta} \varepsilon^{acb} \Omega_{\pm\tau c}{}^{4}. \end{align*} $$
Due to addition of $\lambda_{\pm\tau\pm\tau}I$ matrices $A_{\pm\tau\pm\tau a}{}^b$ are non-singular: $\det A_{\pm\tau\pm\tau} = -(G_{\pm\tau}{}^{0'm'}G_{\pm\tau}{}^{0'm'} - \lambda_{\pm\tau\pm\tau})^2\lambda_{\pm\tau\pm\tau}$ so exist their inverse matrices

$$A^{-1}_{\mp\tau\mp\tau b}{}^a = \frac{1}{\lambda_{\pm\tau\pm\tau}(G_{\pm\tau}{}^{0'm'}G_{\pm\tau}{}^{0'm'} - \lambda_{\pm\tau\pm\tau})}(\lambda_{\pm\tau\pm\tau}\delta_b^a + \Omega_{\pm\tau b}{}^4\Omega_{\pm\tau 4}{}^a).$$

Then the determinants (2.38) equal

$$\det(K_{\pm\tau\pm\tau} - \lambda_{\pm\tau\pm\tau}I) = \lambda_{\pm\tau\pm\tau}^{16}(\lambda_{\pm\tau\pm\tau} - 2G_{\pm\tau}{}^{0'm'}G_{\pm\tau}{}^{0'm'})^8 = 0.$$

So the rank of the matrix $K_{\pm\tau\pm\tau}$ equals eight in this sector. Matrices $M_{\pm}^T$ and $K_{\pm\tau\pm\tau}$ 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 $OSp(4|6)/(SO(1,3) \times U(3))$ superspace 16 represent gauge-invariant physical degrees of freedom of the $\sigma$-model and eight are pure gauge.

For checking the $AdS_4/CFT_3$ duality conjecture [5] it is important to have explicit expression for the $OSp(4|6)/(SO(1,3) \times U(3))$ $\sigma$-model Lagrangian in terms of the superspace coordinate fields. For this it is necessary to choose the $OSp(4|6)/(SO(1,3) \times U(3))$ supercoset representative in (2.1) and find expressions for the Cartan forms in terms of the superspace coordinates and their differentials. In our work [19] obtained were expressions for the Cartan forms built out of the supercoset element

$$\mathcal{G} = e^{x_m P^m + \theta_a^\mu Q_\mu^a + \bar{\theta}^{\mu a} \bar{Q}_{\mu a}} e^{\eta_{\mu a} S^{\mu a} + \bar{\eta}_\mu^a \bar{S}_a^\mu} e^{z^a V_a^4 + \bar{z}_a V_4^a} e^{\varphi D} \quad (2.39)$$

that includes generators of the $D=3$ $\mathcal{N}=6$ 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 $AdS \times S$-type backgrounds [37], [38], [39], [40], [41], [42]. In (2.39) $x^m$ and $\varphi$ are the Poincaré coordinates of $AdS_4$ space-time and three complex coordinates $z^a$ and $\bar{z}_a$ parametrize $\mathbb{CP}^3$ space. Anticommuting coordinates $\theta_a^\mu$, $\bar{\theta}^{\mu a}$ and $\eta_{\mu a}$, $\bar{\eta}_\mu^a$ 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

$$\begin{aligned} \omega^m(d) &= e^{-2\varphi}\zeta^m(d), \quad \zeta^m(d) = dx^m - id\theta_a^\mu\sigma_{\mu\nu}^m\bar{\theta}^{\nu a} + i\theta_a^\mu\sigma_{\mu\nu}^m d\bar{\theta}^{\nu a}, \\ c^m(d) &= e^{2\varphi}\psi^m(d), \quad \psi^m(d) = -id\eta_{\mu a}\tilde{\sigma}^{m\mu\nu}\bar{\eta}_\nu^a + i\eta_{\mu a}\tilde{\sigma}^{m\mu\nu}d\bar{\eta}_\nu^a \\ &\quad +2(\bar{\eta}_\rho^b\eta_b^\rho) [\eta_{\mu a}\tilde{\sigma}^{m\mu\nu}(d\bar{\theta}_\nu^a + \frac{1}{4}\bar{\zeta}_\nu^a(d)) - (d\theta_{\mu a} + \frac{1}{4}\zeta_{\mu a}(d))\tilde{\sigma}^{m\mu\nu}\bar{\eta}_\nu^a], \\ \Delta(d) &= d\varphi + id\theta_a^\mu\bar{\eta}_\mu^a + id\bar{\theta}^{\mu a}\eta_{\mu a}, \end{aligned} \quad (2.40)$$

where

$$\zeta_a^\mu(d) = -\tilde{\sigma}^{m\mu\nu}\zeta_m(d)\eta_{\nu a}, \quad \bar{\zeta}^{\mu a}(d) = -\tilde{\sigma}^{m\mu\nu}\zeta_m(d)\bar{\eta}_\nu^a.$$

In the bosonic limit these Cartan forms specify the square of infinitesimal length element in the $AdS_4$ space-time in the Poincaré coordinates

$$ds_{AdS_4}^2 = \frac{e^{-4\varphi}}{4} dx^m dx_m + d\varphi d\varphi. \quad (2.41)$$
For given supercoset element (2.39) Cartan forms corresponding to generators of the so(6) algebra are represented by the sum of the purely bosonic summand and that depending on the Grassmann-odd coordinates

$$ \Omega_{\hat{a}}^{\hat{b}}(d) = \Omega_{\text{bos } \hat{a}}^{\hat{b}}(d) + \Omega_{\text{ferm } \hat{a}}^{\hat{b}}(d). \qquad (2.42) $$

Bosonic Cartan forms

$$ \Omega_{\text{bos } \hat{a}}^{\hat{b}}(d) = i T_{\hat{a}}^{\hat{c}} d \bar{T}_{\hat{c}}^{\hat{b}} = \begin{pmatrix} \Omega_{\text{bos } a}^{\text{b}}(d) - \delta_a^b \Omega_{\text{bos } c}^{\text{c}}(d) & \varepsilon_{acb} \Omega_{\text{bos } 4}^{\text{c}}(d) \\ -\varepsilon^{acb} \Omega_{\text{bos } c}^{\text{4}}(d) & -\Omega_{\text{bos } b}^{\text{a}}(d) + \delta_b^a \Omega_{\text{bos } c}^{\text{c}}(d) \end{pmatrix}, $$

are specified by the unitary matrix that in the $\mathbf{3} \oplus \bar{\mathbf{3}}$ basis (2.8) has the form

$$ T_{\hat{a}}^{\hat{b}} = \begin{pmatrix} \cos|z|\delta_a^b + \frac{(1-\cos|z|)}{|z|^2} \bar{z}_a z^b & i \frac{\sin|z|}{|z|} \varepsilon_{acb} z^c \\ -i \frac{\sin|z|}{|z|} \varepsilon^{acb} \bar{z}_c & \cos|z|\delta_b^a + \frac{(1-\cos|z|)}{|z|^2} z^a \bar{z}_b \end{pmatrix}, \qquad (2.43) $$

where $|z|^2 = z^a \bar{z}_a$. So entries of the matrix $\Omega_{\text{bos } \hat{a}}^{\hat{b}}(d)$ equal

$$ \begin{align} \Omega_{\text{bos } a}^{\text{b}}(d) &= i \frac{(1-\cos|z|)}{|z|^2} (\bar{z}_a dz^b - d\bar{z}_a z^b) - i \bar{z}_a z^b \frac{(1-\cos|z|)^2}{2|z|^4} (dz^c \bar{z}_c - z^c d\bar{z}_c), \notag \\ \Omega_{\text{bos } a}^{\text{4}}(d) &= d\bar{z}_a \frac{\sin|z|}{|z|} + \bar{z}_a \frac{\sin|z|(1-\cos|z|)}{2|z|^3} (dz^c \bar{z}_c - z^c d\bar{z}_c) + \bar{z}_a \left(\frac{1}{|z|} - \frac{\sin|z|}{|z|^2}\right) d|z|, \tag{2.44} \\ \Omega_{\text{bos } 4}^{\text{a}}(d) &= dz^a \frac{\sin|z|}{|z|} + z^a \frac{\sin|z|(1-\cos|z|)}{2|z|^3} (z^c d\bar{z}_c - z^c d\bar{z}_c) + z^a \left(\frac{1}{|z|} - \frac{\sin|z|}{|z|^2}\right) d|z|. \notag \end{align} $$

The second term in (2.42) can be presented as

$$ \begin{gathered} \Omega_{\text{ferm } \hat{a}}^{\hat{b}}(d) = (T\Psi(d)\bar{T})_{\hat{a}}^{\hat{b}}, \\ \Psi_{\hat{a}}^{\hat{b}}(d) = \begin{pmatrix} \Psi_a^{\text{b}}(d) - \delta_a^{\text{b}}\Psi_c^{\text{c}}(d) & \varepsilon_{acb}\Psi_4^{\text{c}}(d) \\ -\varepsilon^{acb}\Psi_c^{\text{4}}(d) & -\Psi_b^{\text{a}}(d) + \delta_b^{\text{a}}\Psi_c^{\text{c}}(d) \end{pmatrix}, \end{gathered} \qquad (2.45) $$

where the matrix $\Psi_{\hat{a}}^{\hat{b}}(d)$ has the following elements

$$ \begin{align} \Psi_a^{\text{b}}(d) &= 2(d\theta_a^\mu + \frac{1}{2}\zeta_a^\mu(d))\bar{\eta}_\mu^b - 2(d\bar{\theta}^{\mu b} + \frac{1}{2}\bar{\zeta}^{\mu b}(d))\eta_{\mu a} - \delta_a^b((d\theta_c^\mu + \frac{1}{2}\zeta_c^\mu(d))\bar{\eta}_\mu^c - (d\bar{\theta}^{\mu c} + \frac{1}{2}\bar{\zeta}^{\mu c}(d))\eta_{\mu c}), \notag \\ \Psi_a^{\text{4}}(d) &= 2\varepsilon_{abc}(d\bar{\theta}^{\mu b} + \frac{1}{2}\bar{\zeta}^{\mu b}(d))\bar{\eta}_\mu^c, \quad \Psi_4^{\text{a}}(d) = -2\varepsilon^{abc}(d\theta_b^\mu + \frac{1}{2}\zeta_b^\mu(d))\eta_{\mu c}. \end{align} $$

The form of fermionic Cartan forms is the following

$$ \omega_{\hat{a}}^{\mu}(d) = e^{-\varphi} T_{\hat{a}}^{\hat{b}} \begin{pmatrix} \zeta_{\hat{b}}^{\mu}(d) \\ \bar{\zeta}^{\mu b}(d) \end{pmatrix}, \qquad (2.46) $$

and

$$ \chi_{\mu \hat{a}}(d) = e^{\varphi} T_{\hat{a}}^{\hat{b}} \begin{pmatrix} v_{\mu b}(d) \\ \bar{v}_{\mu}^{b}(d) \end{pmatrix} \qquad (2.47) $$

where

$$ \begin{align*} \zeta_b^\mu(d) &= d\theta_b^\mu + \zeta_b^\mu(d), \quad \bar{\zeta}^{\mu b}(d) = d\bar{\theta}^{\mu b} + \bar{\zeta}^{\mu b}(d), \\ v_{\mu b}(d) &= d\eta_{\mu b} + 2i\bar{\eta}_\mu^c d\theta_c^\nu \eta_{\nu b} + 2i\eta_{\mu c} d\bar{\theta}^{\nu c} \eta_{\nu b} + i(\eta_\mu^\nu \bar{\eta}_\nu^c)\zeta_{\mu b}(d), \\ \bar{v}_\mu^b(d) &= d\bar{\eta}_\mu^b + 2i\eta_{\mu c} d\bar{\theta}^{\nu c} \bar{\eta}_\nu^b + 2i\bar{\eta}_\mu^c d\theta_c^\nu \bar{\eta}_\nu^b + i(\eta_\mu^\nu \bar{\eta}_\nu^c)\zeta_\mu^b(d). \end{align*} $$
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 $\sigma$-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 $AdS_5 \times S^5$ superstring anticommuting coordinates $\theta_a^\mu$ and $\bar{\theta}^{\mu a}$ related to the Poincaré supersymmetry generators enter expressions for the Cartan forms utmost quadratically and the non-linear fermionic contributions are given by the $\eta_{\mu a}$ and $\bar{\eta}_\mu^a$ coordinates that are parameters at the special conformal supersymmetry generators.

For the $AdS_5 \times S^5$ superstring it is possible to fix the $\kappa$-symmetry gauge by the conditions that entirely remove the coordinates $\eta$ so that the Lagrangian becomes quadratic [43] or quartic in the anticommuting coordinates $\theta$ [44], [45], [40], [41]. In the present case it is impossible to gauge away all 12 $\eta$ coordinates by the eight-parameter $\kappa$-symmetry. Instead one can consider, for instance, the $SL(2, \mathbb{R})$ covariant gauge that removes eight $\eta$ coordinates

$$ \eta_{\mu a} = \begin{pmatrix} \eta_{\mu A} \\ \eta_{\mu 3} \end{pmatrix}, \quad \eta_{\mu A} = 0, $$

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) $\Psi_1^2(d) = \Psi_2^1(d) = \Psi_4^3(d) = \Psi_3^4(d) = 0$ but kinetic term of the $\sigma$-model Lagrangian (2.25), however, still includes contributions up to the eighth order in the fermions. The gauge condition

$$ \theta_a^\mu = \begin{pmatrix} \theta_A^\mu \\ \theta_3^\mu \end{pmatrix}, \quad \theta_3^\mu = 0, \quad \eta_{\mu 3} = 0 $$

removes an equal number of $\theta$ and $\eta$ coordinates. In this gauge vanish the following components of the Cartan forms $\Psi_{1,2}^4(d) = \Psi_4^{1,2}(d) = \Psi_{1,2}^3(d) = \Psi_3^{1,2}(d) = 0$ and $\omega_3^\mu(d) = 0$, $\chi_{\mu 3}(d) = 0$. But more substantial simplification can be attained by considering the $SL(2, \mathbb{R})$ non-covariant conditions, e.g.

$$ \eta_{1a} = 0 $$

that partially fixes the $\kappa$-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 $\chi_{1a}(d) = \bar{\chi}_1^a(d) = 0$. 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 $D=3 \mathcal{N}=6$ superconformal symmetry of the $\sigma$-model action in the $OSp(4|6)/(SO(1,3) \times U(3))$ superspace

The $OSp(4|6)/(SO(1,3) \times U(3))$ $\sigma$-model Lagrangian is by construction invariant under global $OSp(4|6)$ symmetry. It can be presented as the $D=3 \mathcal{N}=6$ superconformal symmetry acting on the coordinates of the $OSp(4|6)/(SO(1,3) \times U(3))$ supercoset space. For this
realization the parametrization of the $OSp(4|6)/(SO(1,3) \times U(3))$ supercoset space by the co-ordinates associated with the superconformal algebra generators as in (2.39) is natural. The $OSp(4|6)/(SO(1,3) \times U(3))$ 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<fnref n="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].

In the present subsection there will be carried detailed study of the realization of $D=3 N=6$ superconformal symmetry in the $OSp(4|6)/(SO(1,3) \times U(3))$ $\sigma$-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 $\sigma$-model Lagrangian in terms of the $OSp(4|6)/(SO(1,3) \times U(3))$ 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 $AdS_4 \times CP^3$ superstring in the Grassmann coordinates for the broken supersymmetries. In Ref. [17] it was shown that such current densities can be used as structural blocks for construction of the Lax connection for the $AdS_4 \times CP^3$ superstring.

### 2.2.1 Transformation properties of the Cartan forms and coordinates of the $OSp(4|6)/(SO(1,3) \times U(3))$ supercoset

Consider the $OSp(4|6)/(SO(1,3) \times U(3))$ supercoset representative $\mathcal{G}$. Under global $OSp(4|6)$ symmetry transformation with the parameter $G$ it changes as

$$G\mathcal{G} = \mathcal{G}'H,$$

where $H$ belongs to the $SO(1,3) \times U(3)$ stability group and depends on the coordinates of this coset-space. Under infinitesimal transformation variation of $\mathcal{G}$ equals

$$\delta\mathcal{G} = g\mathcal{G} - \mathcal{G}h, \quad g \in osp(4|6), \quad h \in so(1,3) \oplus u(3).$$

<footnote n="8">⁸Let us note certain ambiguity when using the term 'D = 3 N = 6 superconformal symmetry' both for the transformations of the coordinates of $OSp(4|6)/(SO(1,3) \times U(3))$ 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. [46]).</footnote>
It can also be presented in the following form

$$ \mathcal{G}^{-1} \delta \mathcal{G} = \mathcal{G}^{-1} g \mathcal{G} - h. \qquad (2.48) $$

Consider the $D = 3$ $\mathcal{N} = 6$ superconformal algebra valued infinitesimal transformation parameter

$$ g = a^m P_m + b_m K^m + f D + \frac{1}{2} l^{mn} M_{mn} + y^a V_a^4 + \bar{y}_a V_4^a + w_a^b V_b^a + \varepsilon_a^\mu Q_\mu^a + \bar{\varepsilon}^{\mu a} \bar{Q}_{\mu a} + \xi_{\mu a} S^{\mu a} + \bar{\xi}_\mu^a \bar{S}_a^\mu. \qquad (2.49) $$

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 $D = 3$ $\mathcal{N} = 6$ Poincaré supersymmetry ($\varepsilon_a^\mu, \bar{\varepsilon}^{\mu a}$) and special conformal supersymmetry ($\xi_{\mu a}, \bar{\xi}_\mu^a$). To (2.49) enter also the parameters of the infinitesimal $su(4)$ transformation ($w_a^b, y^a, \bar{y}_a$). 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 $\hat{b}^m, \hat{l}^{mn}$ and $\hat{w}_a^b$

$$ \begin{aligned} \mathcal{G}^{-1} g \mathcal{G} = & (\omega^m(\delta) - \hat{b}^m) P_m + (c^m(\delta) + \hat{b}^m) K_m + \Delta(\delta) D \\ & + (G^{mn}(\delta) + \frac{1}{2} \hat{l}^{mn}) M_{mn} + \Omega_4^a(\delta) V_a^4 + \Omega_a^4(\delta) V_4^a + (\Omega_a^b(\delta) + \hat{w}_a^b) V_b^a \\ & + \omega_a^\mu(\delta) Q_\mu^a + \bar{\omega}^{\mu a}(\delta) \bar{Q}_{\mu a} + \chi_{\mu a}(\delta) S^{\mu a} + \bar{\chi}_\mu^a(\delta) \bar{S}_a^\mu. \end{aligned} $$

From (2.48) it follows that Cartan forms $c(d) = \mathcal{G}^{-1} d\mathcal{G}$ are invariant under global $OSp(4|6)$ symmetry up to compensating transformations

$$ \delta c(d) = -[c, h] - dh. $$

The Cartan forms associated with the quotient algebra generators transform according to their $so(1, 3) \oplus u(3)$ 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

$$ \delta\omega^m(d) + \delta c^m(d) = \hat{l}^{mn}(\omega_n(d) + c_n(d)) + 4\hat{b}^m\Delta(d), \quad \delta\Delta(d) = -\hat{b}_m(\omega^m(d) + c^m(d)), \quad (2.50) $$

where $\hat{l}^{mn}$ and $\hat{b}_m$ are parameters of the infinitesimal local $SO(1, 3)$ transformation. Variation of the $so(1, 2)$ Cartan forms in the spinor representation $G^{\mu\nu}(d) = \varepsilon^{\mu\lambda}\sigma_{mn\lambda}{}^\nu G^{mn}(d)$ has the form

$$ \delta G^{\mu\nu}(d) = \frac{1}{4}(G^{\mu\lambda}(d)\hat{l}_{\lambda}{}^{\nu} + G^{\nu\lambda}(d)\hat{l}_{\lambda}{}^{\mu}) + \hat{b}_{\lambda}{}^{\mu}(\tilde{\omega}(d) - \tilde{c}(d))^{\lambda\nu} + \hat{b}_{\nu}{}^{\lambda}(\tilde{\omega}(d) - \tilde{c}(d))^{\mu\lambda} - \frac{1}{2}d\hat{l}^{\mu\nu}. \quad (2.51) $$

Variation of the $su(4)$ Cartan forms in the $\mathbf{3} \oplus \bar{\mathbf{3}}$ basis is given by the expression

$$ \delta\Omega_{\hat{a}}^{\hat{b}}(d) = i(\Omega_{\hat{a}}^{\hat{c}}(d)\widehat{W}_{\hat{c}}^{\hat{b}} - \widehat{W}_{\hat{a}}^{\hat{c}}\Omega_{\hat{c}}^{\hat{b}}(d)) - d\widehat{W}_{\hat{a}}^{\hat{b}}, \qquad (2.52) $$

where

$$ \widehat{W}_{\hat{a}}^{\hat{b}} = \begin{pmatrix} \hat{w}_a^b - \delta_a^b \hat{w}_c^c & 0 \\ 0 & -\hat{w}_b^a + \delta_b^a \hat{w}_c^c \end{pmatrix} \qquad (2.53) $$
is the matrix of infinitesimal local $U(3)$ transformation. From (2.52) one can find variations of the supervielbein bosonic components tangent to the $\mathbb{CP}^3$ manifold

$$ \delta\Omega_a{}^4(d) = i(\hat{w}_b{}^b\Omega_a{}^4(d) - \hat{w}_a{}^b\Omega_b{}^4(d)), \quad \delta\Omega_4{}^a(d) = -i(\hat{w}_b{}^b\Omega_4{}^a(d) - \Omega_4{}^b(d)\hat{w}_b{}^a) \quad (2.54) $$

and the $u(3)$ connection one-form

$$ \delta\Omega_a{}^b(d) = i(\Omega_a{}^c(d)\hat{w}_c{}^b - \hat{w}_a{}^c\Omega_c{}^b(d)) - d\hat{w}_a{}^b. $$

Variations of the Grassmann-odd Cartan forms that are identified with the supervielbein fermionic components reduce to infinitesimal transformations from the stability group

$$ \delta\omega_{\hat{a}}^{\nu}(d) = \frac{1}{4}\omega_{\hat{a}}^{\lambda}(d)\hat{l}_{\lambda}^{\nu} + \hat{b}^{\nu\lambda}\chi_{\lambda\hat{a}}(d) - i\widehat{W}_{\hat{a}}^{\hat{b}}\omega_{\hat{b}}^{\nu}(d), \quad (2.55) $$

$$ \delta\chi_{\nu\hat{a}}(d) = -\frac{1}{4}\hat{l}_{\nu}^{\lambda}\chi_{\lambda\hat{a}}(d) + \hat{b}_{\nu\lambda}\omega_{\hat{a}}^{\lambda}(d) - i\widehat{W}_{\hat{a}}^{\hat{b}}\chi_{\nu\hat{b}}(d). \quad (2.56) $$

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 of parameters of the transformations from the global symmetry group and the supercoset coordinates. In the case of infinitesimal $D = 3, \mathcal{N} = 6$ 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

$$ \hat{b}^m = e^{2\varphi} A^{-1} b^m(\theta), \quad A = 1 - e^{4\varphi}(\bar{\eta}\eta)^2, \\ \hat{l}^{mn} = l^{mn}(\theta) + ie^{2\varphi} \left[ (\eta_a \hat{b} \sigma^{mn} \bar{\eta}^a) + (\bar{\eta}^a \hat{b} \sigma^{mn} \eta_a) \right], $$

where

$$ b^m(\theta) = b^m - i \left[ (\xi_a(\theta)\tilde{\sigma}^m \bar{\eta}^a) + (\bar{\xi}^a(\theta)\tilde{\sigma}^m \eta_a) \right], \quad \xi_{\mu a}(\theta) = \xi_{\mu a} + b_{\mu\nu}\theta_a^{\nu}, \\ l^{mn}(\theta) = l^{mn} + 2(b^m x^n - b^n x^m) + 2i \left[ (\theta_a \sigma^{mn} \bar{\xi}^a) + (\bar{\theta}^a \sigma^{mn} \xi_a) \right] + i \left[ (\theta_a \sigma^{mn} b \bar{\theta}^a) + (\bar{\theta}^a \sigma^{mn} b \theta_a) \right]. $$

Parameters of infinitesimal transformation from the $U(3)$ stability group that enter (2.53) is convenient to present in the form

$$ \hat{w}_a{}^b = \tilde{w}_a{}^b + \frac{i(1-\cos|z|)}{|z|\sin|z|}(\bar{z}_a \tilde{y}^b - \tilde{y}_a z^b) + \frac{i(1-\cos|z|)^2}{|z|^3\sin2|z|}((\tilde{y}\bar{z}) - (z\tilde{y}))\bar{z}_a z^b, \\ \tilde{w}_a{}^b = w_a{}^b(\theta) - e^{2\varphi}\left[2(\eta_a \hat{b}\bar{\eta}^b) - \delta_a^b(\eta_c \hat{b}\bar{\eta}^c)\right], \quad (2.57) \\ \tilde{y}^a = y^a(\theta) + e^{2\varphi}\varepsilon^{abc}(\eta_b \hat{b}\eta_c), \quad \tilde{y}_a = \bar{y}_a(\theta) - e^{2\varphi}\varepsilon_{abc}(\bar{\eta}^b \hat{b}\bar{\eta}^c) $$

where

and

$$ w_a{}^b(\theta) = w_a{}^b - 2(\xi_{\mu a}\bar{\theta}^{\mu b} + \theta_a^\mu \bar{\xi}_b^c) + \delta_a^b(\xi_{\mu c}\bar{\theta}^{\mu c} + \theta_c^\mu \bar{\xi}_\mu^c) - 2(\theta_a b \bar{\theta}^b) + \delta_a^b(\theta_c b \bar{\theta}^c), \\ y^a(\theta) = y^a + \varepsilon^{abc}(2\xi_{\mu b}\theta_c^\mu + (\theta_b b \theta_c)), \quad \bar{y}_a(\theta) = \bar{y}_a - \varepsilon_{abc}(2\bar{\xi}_\mu^b \bar{\theta}^{\mu c} + (\bar{\theta}^b b \bar{\theta}^c)). $$
Parameters (2.57) are elements of the matrix

$$ \tilde{W}_{\hat{a}}^{\hat{b}} = \begin{pmatrix} \tilde{w}_{a}^{\hat{b}} - \delta_{a}^{\hat{b}} \tilde{w}_{c}^{\hat{c}} & \varepsilon_{acb} \tilde{y}^{c} \\ -\varepsilon^{acb} \tilde{y}_{c} & -\tilde{w}_{b}^{\hat{a}} + \delta_{b}^{\hat{a}} \tilde{w}_{c}^{\hat{c}} \end{pmatrix} \qquad (2.58) $$

that will be used below to write down variation (2.59) of the SU(4)/U(3) coset element under the infinitesimal $D = 3 \mathcal{N} = 6$ superconformal transformations.

Let us also present variations of the coordinates parametrizing the $OSp(4|6)/(SO(1,3) \times U(3))$ supercoset element (2.39). Variations of the coordinates associated with the generators of the $D = 3 \mathcal{N} = 6$ Poincaré superalgebra are given by the expressions

$$ \begin{aligned} \delta x^m &= a^m + l_m x^n + 2f x^m + b^m(x^2 + (\bar{\theta}\theta)^2) - 2x^m b_n x^n \\ &\quad -i [(\varepsilon_a \sigma^m \bar{\theta}^a) + (\bar{\varepsilon}^a \sigma^m \theta_a)] - i [(\xi_a \hat{x} \sigma^m \bar{\theta}^a) + (\bar{\xi}^a \hat{x} \sigma^m \theta_a)] \\ &\quad + e^{2\varphi} \{ \hat{b}^m + i [(\eta_a \hat{b} \sigma^m \bar{\theta}^a) + (\bar{\eta}^a \hat{b} \sigma^m \theta_a)] \}, \\ \delta \theta_a^\mu &= \varepsilon_a^\mu + \frac{1}{4} l^{mn} \theta_a^\nu \sigma_{mn\nu}^\mu + f \theta_a^\mu + i w_b^b \theta_a^\mu - i w_a^b \theta_b^\mu - i \varepsilon_{abc} y^b \bar{\theta}^{\mu c} \\ &\quad + \hat{x}^{\mu\nu} b_{\nu\lambda} \theta_a^\lambda + \hat{x}^{\mu\nu} \xi_{\nu a} - 2i(\theta_b^\mu \bar{\xi}_\nu^b + \bar{\theta}^{\mu b} \xi_{\nu b}) \theta_a^\nu + e^{2\varphi} \hat{b}^{\mu\nu} \eta_{\nu a} \end{aligned} $$

and c.c., where $\hat{x}^{\mu\nu} = \tilde{x}^{\mu\nu} - i\varepsilon^{\mu\nu}(\bar{\theta}\theta)$. Note that terms proportional to the parameters $\hat{b}^m$ vanish in the limit $\varphi \to -\infty$ corresponding to the conformal boundary of $AdS_4$. So in the boundary limit coordinates $(x^m, \theta_a^\mu, \bar{\theta}^{\mu a})$ form the closed set under the $D = 3 \mathcal{N} = 6$ superconformal symmetry transformations and can be chosen for parametrization of the boundary superspace. Variation of the coordinate $\varphi$ parametrizing orthogonal to the boundary dimension of the $AdS_4$ space in the Poincaré coordinates equals

$$ \delta\varphi = f(\theta) = f - b_m x^m + i(\xi_{\mu a} \bar{\theta}^{\mu a} + \bar{\xi}_\mu^a \theta_a^\mu). $$

Variations of the complex coordinates of the $\mathbb{CP}^3$ manifold

$$ \begin{aligned} \delta z^a = iz^b \tilde{w}_b^a + i \tilde{w}_b^b z^a + |z| \cot |z| \tilde{y}^a + \frac{1}{2|z|^2} \left(1 - \frac{2|z|}{\sin 2|z|}\right) (\tilde{y} \tilde{z}) z^a \\ + \frac{1}{2|z|^2} [1 + |z|(\tan |z| - \cot |z|)] (z \tilde{y}) z^a \end{aligned} $$

determine variation of the SU(4)/U(3) coset representative (2.43)

$$ \delta T_{\hat{a}}^{\hat{b}} = i(T_{\hat{a}}^{\hat{c}} \tilde{W}_{\hat{c}}^{\hat{b}} - \tilde{W}_{\hat{a}}^{\hat{c}} T_{\hat{c}}^{\hat{b}}), \quad \delta \bar{T}_{\hat{a}}^{\hat{b}} = -i(\tilde{W}_{\hat{a}}^{\hat{c}} \bar{T}_{\hat{c}}^{\hat{b}} - \bar{T}_{\hat{a}}^{\hat{c}} \tilde{W}_{\hat{c}}^{\hat{b}}), \qquad (2.59) $$

where matrices $\tilde{W}_{\hat{a}}^{\hat{b}}$ and $\tilde{W}_{\hat{a}}^{\hat{c}}$ 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

$$ \begin{aligned} \delta \eta_{\mu a} = \xi_{\mu a}(\theta) - \frac{1}{4} l^{mn}(\theta) \sigma_{mn\mu}^{\nu} \eta_{\nu a} - f(\theta) \eta_{\mu a} + i w_b^b(\theta) \eta_{\mu a} - i w_a^b(\theta) \eta_{\mu b} \\ -i \varepsilon_{abc} y^b(\theta) \bar{\eta}_\mu^c + 2ie^{2\varphi}(\bar{\eta}\eta) \varepsilon_{\mu\lambda} \hat{b}^{\lambda\nu} \eta_{\nu a} \end{aligned} $$

and c.c.
### 2.2.2 Densities of the Noether currents

To obtain expressions for the densities of the Noether currents associated with the invariance of the $\sigma$-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

$$ \begin{aligned} \delta_X \omega^m(d) + \delta_X c^m(d) &= \frac{\partial}{\partial X}(\omega^m(\delta_X) + c^m(\delta_X))dX + \dots = j^m_X dX + \dots, \tag{2.60} \\ \delta_X \Delta(d) &= \frac{\partial}{\partial X} \Delta(\delta_X)dX + \dots = j_X dX + \dots \end{aligned} $$

$$ \begin{aligned} \delta_X \Omega_a^4(d) &= \frac{\partial}{\partial X} \Omega_a^4(\delta_X) dX + \dots = - * \bar{J}_{aX} dX + \dots, \tag{2.61} \\ \delta_X \Omega_4^a(d) &= \frac{\partial}{\partial X} \Omega_4^a(\delta_X) dX + \dots = - * J^a_X dX + \dots \end{aligned} $$

and<fnref n="9" />

$$ \begin{aligned} \delta_X \omega_{\hat{a}}^{\mu}(d) &= \frac{\partial}{\partial X} \omega_{\hat{a}}^{\mu}(\delta_X) dX + \dots = \mathbf{j}_{\hat{a}X}^{\mu} dX + \dots, \tag{2.62} \\ \delta_X \chi_{\mu \hat{a}}(d) &= \frac{\partial}{\partial X} \chi_{\mu \hat{a}}(\delta_X) dX + \dots = \mathbf{J}_{\mu \hat{a}X} dX + \dots \end{aligned} $$

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

$$ \delta_X \Omega_{\hat{a}}^{\hat{b}}(d) = J_{\hat{a}X}^{\hat{b}} dX + \dots, $$

where dots stand for the terms on the r.h.s. of (2.52) and

$$ J_{\hat{a}X}^{\hat{b}} = \frac{\partial}{\partial X} \Omega_{\hat{a}}^{\hat{b}}(\delta_X) = \begin{pmatrix} J_a^b X & J_{abX} \\ -\bar{J}^{ab}_X & -J_b^a X \end{pmatrix}. \tag{2.63} $$

*$J^a_X$ and $*\bar{J}_{aX}$ in (2.61) are related to $J_{abX}$ and $\bar{J}^{abX}$ in (2.63) by dualization

$$ *J^a_X = \frac{1}{2} \varepsilon^{abc} J_{bcX}, \quad *\bar{J}_{aX} = \frac{1}{2} \varepsilon_{abc} \bar{J}^{bc}_X. $$

So densities of the Noether currents have the form of the sum of three summands

$$ \mathcal{J}_X^i(\tau, \sigma) = \mathcal{J}_{AdS_4}^i + \mathcal{J}_{CP^3}^i + \mathcal{J}_{WZ}^i. $$

The first two of them describe contribution of the kinetic term of the $\sigma$-model Lagrangian (2.25). They are determined by contributions to the induced world-sheet metric from metrics of the $AdS_4$

$$ \mathcal{J}_{AdS_4}^i = -\sqrt{-\gamma} \gamma^{ij} \left( \frac{1}{4}(\omega_{jm} + c_{jm})j_X^m + \Delta_j j_X \right) \tag{2.64} $$

<footnote n="9">⁹It is assumed that derivatives with respect to the Grassmann-odd variables act from the right.</footnote>
<table><caption>Table 2.1: Tensors contributing to the Noether current density associated with the $D = 3$ translational invariance</caption>
  <tr>
    <td>δ<sub>a</sub>ω<sup>m</sup>(d) + δ<sub>a</sub>c<sup>m</sup>(d) = j<sup>m</sup><sub>n</sub>da<sup>n</sup> : j<sup>m</sup><sub>n</sub> = <sup>∂(ω<sup>m</sup>(δ<sub>a</sub>) + c<sup>m</sup>(δ<sub>a</sub>))</sup>⁄<sub>∂a<sup>n</sup></sub> = e<sup>-2φ</sup>Aδ<sub>n</sub><sup>m</sup></td>
  </tr>
  <tr>
    <td>δ<sub>a</sub>Δ(d) = 0 : j<sub>m</sub> = 0</td>
  </tr>
  <tr>
    <td>δ<sub>a</sub>Ω<sub>ā</sub><sup>b</sup>(d) = J<sub>ā</sub><sup>b</sup>da<sup>m</sup> : J<sub>ā</sub><sup>b</sup> = <sup>∂</sup>⁄<sub>∂a<sup>m</sup></sub>Ω<sub>ā</sub><sup>b</sup>(δ<sub>a</sub>) = -2(η<sub>ā</sub>σ<sub>m</sub>η<sub>b</sub><sup>̄</sup>)</td>
  </tr>
  <tr>
    <td>δ<sub>a</sub>ω<sub>ā</sub><sup>μ</sup>(d) = j<sub>ām</sub><sup>μ</sup>da<sup>m</sup> : j<sub>ām</sub><sup>μ</sup> = <sup>∂ω<sub>ā</sub><sup>μ</sup>(δ<sub>a</sub>)</sup>⁄<sub>∂a<sup>m</sup></sub> = -e<sup>-φ</sup>τ̃<sub>m</sub><sup>μν</sup>η<sub>νā</sub></td>
  </tr>
  <tr>
    <td>δ<sub>a</sub>χ<sub>μā</sub>(d) = J<sub>μām</sub>da<sup>m</sup> : J<sub>μām</sub> = <sup>∂χ<sub>μā</sub>(δ<sub>a</sub>)</sup>⁄<sub>∂a<sup>m</sup></sub> = -ie<sup>2φ</sup>(η̄η)ε<sub>μν</sub>j<sub>ām</sub><sup>ν</sup></td>
  </tr>
</table>

and $\mathbb{CP}^3$ spaces

$$ \mathcal{J}_{CP^3}^i_X = \frac{1}{2} \sqrt{-\gamma} \gamma^{ij} (\Omega_{ja} {}^4 * J^a_X + \Omega_{j4} {}^a * \bar{J}_{aX}). \qquad (2.65) $$

The last term is given by the contribution of the Wess-Zumino Lagrangian (2.27)

$$ \mathcal{J}_{WZ}^i_X = \frac{i}{4} \epsilon^{ij} \mathcal{J}_a^b \left( \omega_j^{\mu \hat{a}} \epsilon_{\mu \nu} \mathbf{j}_{bX}^{\nu} + \chi_{j\mu}^{\hat{a}} \epsilon^{\mu \nu} \mathbf{J}_{\nu bX} \right). \qquad (2.66) $$

Thus to construct densities of the Noether currents associated with individual symmetries it suffices to substitute in (2.64), (2.65) and (2.66) expressions for the factors at $dX$ from (2.60), (2.61) and (2.62) respectively.

For the $OSp(4|6)/(SO(1,3) \times U(3))$ coset element (2.39) explicit expressions for these factors corresponding to each of the transformations from the $D = 3 \mathcal{N} = 6$ 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 $\mathbf{3} \oplus \bar{\mathbf{3}}$ basis

$$ \theta_{\hat{a}}^{\mu} = \begin{pmatrix} \theta_{a}^{\mu} \\ \bar{\theta}^{\mu a} \end{pmatrix}, \quad \eta_{\mu \hat{a}} = \begin{pmatrix} \eta_{\mu a} \\ \bar{\eta}_{\mu}^{a} \end{pmatrix} $$

and $\theta^{\mu \hat{a}} = H^{\hat{a}b} \theta_{\hat{b}}^{\mu}$, $\eta_{\mu}^{\hat{a}} = H^{\hat{a}b} \eta_{\mu \hat{b}}$. There were also introduced $T$-transformed coordinates endowed with hats

$$ \hat{\theta}_{\hat{a}}^{\mu} = T_{\hat{a}}^{\hat{b}} \theta_{\hat{b}}^{\mu}, \quad \hat{\eta}^{\mu \hat{a}} = H^{\hat{a} \hat{b}} \hat{\eta}_{\hat{b}}^{\mu}. \qquad (2.67) $$

## 2.3 Lagrangian of the $AdS_4 \times \mathbb{CP}^3$ superstring in the $\kappa$-symmetry light-cone gauge

Two previous subsections were devoted to study of the properties of the $\sigma$-model in the $OSp(4|6)/(SO(1,3) \times U(3))$ supercoset space [9]. It describes dynamics of the Type IIA superstring in the $AdS_4 \times \mathbb{CP}^3$ superbackground [13], when the vanishing condition of the eight coordinates for broken supersymmetries represents partial $\kappa$-symmetry gauge-fixing condition as in the case of superstring moving in the anti-de Sitter space and in the $\mathbb{CP}^3$ 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 $\kappa$-symmetry
<table><caption>Table 2.2: Tensors contributing to the Noether current density associated with the invariance under conformal boosts</caption>
  <tbody>
    <tr>
      <td>δ<sub>b</sub>ω<sup>m</sup>(d) + δ<sub>b</sub>c<sup>m</sup>(d) = j<sup>mn</sup>db<sub>n</sub> + ... :</td>
    </tr>
    <tr>
      <td>j<sup>mn</sup> = <div style="display: inline-block; vertical-align: middle; text-align: center; line-height: 1.2em;"><div style="border-bottom: 1px solid black; padding: 0 0.2em;">∂(ω<sup>m</sup>(δ<sub>b</sub>) + c<sup>m</sup>(δ<sub>b</sub>))</div><div style="padding: 0 0.2em;">∂b<sub>n</sub></div></div></td>
    </tr>
    <tr>
      <td>= e<sup>-2φ</sup>A [(x<sup>2</sup> + (θθ)<sup>2</sup>)η<sup>mn</sup> - 2x<sup>m</sup>x<sup>n</sup> + i(θ<sub>ā</sub>σ<sup>m</sup> ̂xσ<sup>n</sup>θ̂<sup>ā</sup>)]</td>
    </tr>
    <tr>
      <td>+ A<sup>∂ ̂b<sup>m</sup> / ∂b<sub>n</sub></sup> + ie<sup>2φ</sup> [(η<sub>ā</sub> ̂xσ<sup>n</sup> ̂σ<sup>m</sup>η̂<sup>ā</sup>) + (η<sub>ā</sub> ̂σ<sup>m</sup>Λ<sub>-</sub>σ<sup>n</sup>θ̂<sup>ā</sup>)]</td>
    </tr>
    <tr>
      <td>+ 2e<sup>2φ</sup>(η̃η)(η<sub>ā</sub>σ<sup>m</sup> ̂xσ<sup>n</sup>θ̂<sup>ā</sup>), Λ<sub>±μ</sub><sup>ν</sup> = δ<sub>μ</sub><sup>ν</sup> ± 2iη<sub>μ</sub><sup>̂</sup>θ<sub>ā</sub><sup>ν</sup></td>
    </tr>
    <tr>
      <td>δ<sub>b</sub>Δ(d) = j<sup>m</sup>db<sub>m</sub> + ... : j<sup>m</sup> = <div style="display: inline-block; vertical-align: middle; text-align: center; line-height: 1.2em;"><div style="border-bottom: 1px solid black; padding: 0 0.2em;">∂Δ(δ<sub>b</sub>)</div><div style="padding: 0 0.2em;">∂b<sub>m</sub></div></div> = -x<sup>m</sup> - i(η<sub>ā</sub> ̂xσ<sup>m</sup>θ̂<sup>ā</sup>)</td>
    </tr>
    <tr>
      <td>δ<sub>b</sub>Ω<sub>ā</sub><sup>̂b</sup>(d) = J<sub>ā</sub><sup>̂bm</sup>db<sub>m</sub> + ... :</td>
    </tr>
    <tr>
      <td>J<sub>ā</sub><sup>̂bm</sup> = <div style="display: inline-block; vertical-align: middle; text-align: center; line-height: 1.2em;"><div style="border-bottom: 1px solid black; padding: 0 0.2em;">∂</div><div style="padding: 0 0.2em;">∂b<sub>m</sub></div></div>Ω<sub>ā</sub><sup>̂b</sup>(δ<sub>b</sub>)</td>
    </tr>
    <tr>
      <td>= -2[(θ̂<sub>ā</sub>σ<sup>m</sup>θ̂<sup>ā</sup>) + (x<sup>2</sup> + (θθ)<sup>2</sup>)(η<sub>ā</sub> ̂σ<sup>m</sup>η̂<sup>ā</sup>)</td>
    </tr>
    <tr>
      <td>- 2x<sup>m</sup>(η̂<sub>ā</sub> ̂x ̂η<sup>ā</sup>) - (η̂<sub>ā</sub> ̂xσ<sup>m</sup> ̂Z<sup>ā</sup>) + (η<sup>̂b</sup> ̂xσ<sup>m</sup> ̂Z<sub>ā</sub>)]</td>
    </tr>
    <tr>
      <td>Z<sub>ā</sub><sup>μ</sup> = θ<sub>ā</sub><sup>μ</sup> - i(θθ)η<sub>ā</sub><sup>μ</sup></td>
    </tr>
    <tr>
      <td>δ<sub>b</sub>ω<sub>ā</sub><sup>μ</sup>(d) = j<sub>ā</sub><sup>μm</sup>db<sub>m</sub> + ... :</td>
    </tr>
    <tr>
      <td>j<sub>ā</sub><sup>μm</sup> = <div style="display: inline-block; vertical-align: middle; text-align: center; line-height: 1.2em;"><div style="border-bottom: 1px solid black; padding: 0 0.2em;">∂ω<sub>ā</sub><sup>μ</sup>(δ<sub>b</sub>)</div><div style="padding: 0 0.2em;">∂b<sub>m</sub></div></div></td>
    </tr>
    <tr>
      <td>= e<sup>-φ</sup>[ ̂x<sup>μν</sup>σ<sub>νλ</sub> ̂Z<sub>ā</sub><sup>λ</sup> - (x<sup>2</sup> + (θθ)<sup>2</sup>) ̂σ<sup>mμν</sup>η<sub>νā</sub></td>
    </tr>
    <tr>
      <td>+ 2x<sup>m</sup> ̂x<sup>μν</sup>η<sub>νā</sub> + i(θθ)ε<sup>μν</sup>η<sub>ρā</sub> ̂x<sup>ρλ</sup>σ<sub>λν</sub><sup>m</sup>]</td>
    </tr>
    <tr>
      <td>δ<sub>b</sub>χ<sub>μā</sub>(d) = J<sub>μā</sub><sup>m</sup>db<sub>m</sub> + ... :</td>
    </tr>
    <tr>
      <td>J<sub>μā</sub><sup>m</sup> = <div style="display: inline-block; vertical-align: middle; text-align: center; line-height: 1.2em;"><div style="border-bottom: 1px solid black; padding: 0 0.2em;">∂χ<sub>μā</sub>(δ<sub>b</sub>)</div><div style="padding: 0 0.2em;">∂b<sub>m</sub></div></div></td>
    </tr>
    <tr>
      <td>= e<sup>φ</sup>(Λ<sub>-μ</sub><sup>ν</sup>σ<sub>νλ</sub><sup>m</sup>θ<sub>ā</sub><sup>λ</sup> - η<sub>νā</sub> ̂x<sup>νλ</sup>σ<sub>λρ</sub><sup>m</sup>Λ<sub>-μ</sub><sup>ρ</sup> - ie<sup>φ</sup>(η̃η)ε<sub>μν</sub>j<sub>ā</sub><sup>νm</sup>)</td>
    </tr>
  </tbody>
</table>

<table><caption>Table 2.3: Tensors contributing to the Noether current density associated with the dilatational invariance</caption>
  <tbody>
    <tr>
      <td>δ<sub>f</sub> ω<sup>m</sup>(d) + δ<sub>f</sub>c<sup>m</sup>(d) = j<sup>m</sup>df :</td>
    </tr>
    <tr>
      <td>j<sup>m</sup> = <div style="display: inline-block; vertical-align: middle; text-align: center; line-height: 1.2em;"><div style="border-bottom: 1px solid black; padding: 0 0.2em;">∂(ω<sup>m</sup>(δ<sub>f</sub>) + c<sup>m</sup>(δ<sub>f</sub>))</div><div style="padding: 0 0.2em;">∂f</div></div> = 2e<sup>-2φ</sup>Ax<sup>m</sup> + 2e<sup>2φ</sup>(η̃η)(η<sub>ā</sub>σ<sup>m</sup>θ̂<sup>ā</sup>)</td>
    </tr>
    <tr>
      <td>δ<sub>f</sub>Δ(d) = jdf : j = <div style="display: inline-block; vertical-align: middle; text-align: center; line-height: 1.2em;"><div style="border-bottom: 1px solid black; padding: 0 0.2em;">∂Δ(δ<sub>f</sub>)</div><div style="padding: 0 0.2em;">∂f</div></div> = 1 + iθ<sub>ā</sub><sup>μ</sup>η<sub>μ</sub><sup>̂</sup></td>
    </tr>
    <tr>
      <td>δ<sub>f</sub>Ω<sub>ā</sub><sup>̂b</sup> = J<sub>ā</sub><sup>̂b</sup>df :</td>
    </tr>
    <tr>
      <td>J<sub>ā</sub><sup>̂b</sup> = <div style="display: inline-block; vertical-align: middle; text-align: center; line-height: 1.2em;"><div style="border-bottom: 1px solid black; padding: 0 0.2em;">∂</div><div style="padding: 0 0.2em;">∂f</div></div>Ω<sub>ā</sub><sup>̂b</sup>(δ<sub>f</sub>) = 2 (<div style="display: inline-block; vertical-align: middle; text-align: center; line-height: 1.2em;"><div style="border-bottom: 1px solid black; padding: 0 0.2em;">Θ<sub>ā</sub><sup>μ</sup>η<sub>μ</sub><sup>̂</sup></div><div style="padding: 0 0.2em;">∂<sup>2</sup>η<sub>μ</sub><sup>̂</sup></div></div> - <div style="display: inline-block; vertical-align: middle; text-align: center; line-height: 1.2em;"><div style="border-bottom: 1px solid black; padding: 0 0.2em;">Θ</div><div style="padding: 0 0.2em;">∂<sup>2</sup>η<sub>μ</sub><sup>̂</sup></div></div><div style="display: inline-block; vertical-align: middle; text-align: center; line-height: 1.2em;"><div style="border-bottom: 1px solid black; padding: 0 0.2em;">η<sub>μ</sub><sup>̂</sup>η<sub>μ</sub><sup>̂</sup></div><div style="padding: 0 0.2em;">∂<sup>2</sup>η<sub>μ</sub><sup>̂</sup></div></div>)</td>
    </tr>
    <tr>
      <td>Θ<sub>ā</sub><sup>μ</sup> = θ<sub>ā</sub><sup>μ</sup> - η<sub>νā</sub> ̂x<sup>νμ</sup></td>
    </tr>
    <tr>
      <td>δ<sub>f</sub>ω<sub>ā</sub><sup>μ</sup>(d) = j<sub>ā</sub><sup>μ</sup>df : j<sub>ā</sub><sup>μ</sup> = <div style="display: inline-block; vertical-align: middle; text-align: center; line-height: 1.2em;"><div style="border-bottom: 1px solid black; padding: 0 0.2em;">∂ω<sub>ā</sub><sup>μ</sup>(δ<sub>f</sub>)</div><div style="padding: 0 0.2em;">∂f</div></div> = e<sup>-φ</sup>(Θ<sub>ā</sub><sup>μ</sup> - 2 ̂x<sup>μν</sup>η<sub>νā</sub>)</td>
    </tr>
    <tr>
      <td>δ<sub>f</sub>χ<sub>μā</sub>(d) = J<sub>μā</sub>df :</td>
    </tr>
    <tr>
      <td>J<sub>μā</sub> = <div style="display: inline-block; vertical-align: middle; text-align: center; line-height: 1.2em;"><div style="border-bottom: 1px solid black; padding: 0 0.2em;">∂χ<sub>μā</sub>(δ<sub>f</sub>)</div><div style="padding: 0 0.2em;">∂f</div></div> = -e<sup>φ</sup>(Λ<sub>-μ</sub><sup>ν</sup>η<sub>νā</sub> + ie<sup>φ</sup>(η̃η)ε<sub>μν</sub>j<sub>ā</sub><sup>ν</sup>)</td>
    </tr>
  </tbody>
</table>
$$
\begin{align*}
\delta_l \omega^m(d) + \delta_l c^m(d) &= j^m_{kn} dl^{kn} + \dots: j^m_{kn} = \frac{\partial(\omega^m(\delta_l) + c^m(\delta_l))}{\partial l^{kn}} = \frac{1}{2} e^{-2\varphi} A (\delta_k^m x_n - \delta_n^m x_k + i(\bar{\theta}\theta) \epsilon^{mn}_k) \\
&\quad + \frac{1}{2} e^{2\varphi} (\bar{\eta}\eta) \{ \delta_k^m (\eta_\hat{a} \sigma_n \bar{\theta}^\hat{a}) - (k \leftrightarrow n) + \epsilon^{mn}_k [i + (\bar{\eta}\theta) + (\bar{\theta}\eta)] \} \\
\delta_l \Delta(d) &= j_{mn} dl^{mn} : j_{mn} = \frac{\partial \Delta(\delta_l)}{\partial l^{mn}} = \frac{i}{4} (\theta_{\hat{a}} \sigma_{mn} \eta^{\hat{a}}) \\
\delta_l \Omega_{\hat{a}}^{\hat{b}}(d) &= J_{\hat{a}}^{\hat{b}}_{mn} dl^{mn} : \\
J_{\hat{a}}^{\hat{b}}_{mn} &= \frac{\partial}{\partial l^{mn}} \Omega_{\hat{a}}^{\hat{b}}(\delta_l) = \frac{1}{2} \left[ (\hat{\Theta}_{\hat{a}} \sigma_{mn} \hat{\eta}^{\hat{b}}) - (\hat{\Theta}^{\hat{b}} \sigma_{mn} \hat{\eta}_{\hat{a}}) \right] \\
\delta_l \omega_{\hat{a}}^{\mu}(d) &= \mathbf{j}_{\hat{a}mn}^{\mu} dl^{mn} + \dots, : \\
\mathbf{j}_{\hat{a}mn}^{\mu} &= \frac{\partial \omega_{\hat{a}}^{\mu}(\delta_l)}{\partial l^{mn}} = \frac{1}{4} e^{-\varphi} \left( \hat{\Theta}_{\hat{a}}^{\nu} \sigma_{mn\nu}^{\mu} - \hat{x}^{\mu\lambda} \sigma_{mn\lambda}^{\nu} \hat{\eta}_{\nu\hat{a}} \right) \\
\delta_l \chi_{\mu\hat{a}}(d) &= \mathbf{J}_{\mu\hat{a}mn} dl^{mn} + \dots: \\
\mathbf{J}_{\mu\hat{a}mn} &= \frac{\partial \chi_{\mu\hat{a}}(\delta_l)}{\partial l^{mn}} = -e^{\varphi} \left( \frac{1}{4} \Lambda_{-\mu}^{\nu} \sigma_{mn\nu}^{\lambda} \hat{\eta}_{\lambda\hat{a}} + ie^{\varphi} (\bar{\eta}\eta) \epsilon_{\mu\nu} \mathbf{j}_{\hat{a}mn}^{\nu} \right)
\end{align*}
$$

<caption>Table 2.4: Tensors contributing to the Noether current density associated with the $D=3$ Lorentz invariance</caption>

$$
\begin{align*}
\delta_w \omega^m(d) + \delta_w c^m(d) &= j^m_a{}^b dw_b^a : \\
&\quad j^m_a{}^b = \frac{\partial(\omega^m(\delta_w) + c^m(\delta_w))}{\partial w_b^a} = 2e^{-2\varphi} A [\delta_a^b (\theta_c \sigma^m \bar{\theta}^c) - (\theta_a \sigma^m \bar{\theta}^b)] \\
&\quad + 2e^{2\varphi} \delta_a^b \{ (\eta_c \tilde{\sigma}^m \bar{\eta}^c) - i(\bar{\eta}\eta) [(\theta_c \sigma^m \bar{\eta}^c) + (\eta_c \sigma^m \bar{\theta}^c)] \} \\
&\quad - 2e^{2\varphi} \{ (\eta_a \tilde{\sigma}^m \bar{\eta}^b) - i(\bar{\eta}\eta) [(\theta_a \sigma^m \bar{\eta}^b) + (\eta_a \sigma^m \bar{\theta}^b)] \} \\
\delta_w \Delta &= j_a{}^b dw_b^a : \\
&\quad j_a{}^b = \frac{\partial \Delta(\delta_w)}{\partial w_b^a} = \delta_a^b (\bar{\theta}^{\mu c} \eta_{\mu c} + \bar{\eta}^c \theta_c^\mu) + \theta_a^\mu \bar{\eta}_\mu^b + \eta_{\mu a} \bar{\theta}^{\mu b} \\
\delta_w \Omega_{\hat{c}}^{\hat{d}}(d) &= J_{\hat{c}}^{\hat{d}}{}^b dw_b^a + \dots : \\
&\quad J_{\hat{c}}^{\hat{d}}{}^b = \frac{\partial}{\partial w_b^a} \Omega_{\hat{c}}^{\hat{d}}(\delta_w) = \mathcal{T}_{\hat{c}}^b \mathcal{T}^{\hat{d}}_a - \mathcal{T}_{\hat{c}a} \mathcal{T}^{\hat{d}b} + \frac{i}{2} \delta_a^b (\mathcal{T}\mathfrak{J}\mathcal{T})_{\hat{c}}^{\hat{d}}, \\
&\quad \mathcal{T}_{\hat{a}}^{\hat{b}} = T_{\hat{a}}^{\hat{b}} + 2i\hat{\eta}_{\nu\hat{a}}\theta^{\nu\hat{b}} \\
\delta_w \omega_{\hat{c}}^{\mu}(d) &= \mathbf{j}_{\hat{c}a}^{\mu}{}^b dw_b^a + \dots : \\
&\quad \mathbf{j}_{\hat{c}a}^{\mu}{}^b = \frac{\partial \omega_{\hat{c}}^{\mu}(\delta_w)}{\partial w_b^a} = e^{-\varphi} \left[ \frac{1}{2} \delta_a^b (\mathcal{T}\mathfrak{J}\theta^\mu)_{\hat{c}} - i\theta_a^\mu \mathcal{T}_{\hat{c}}^b + i\bar{\theta}^{\mu b} \mathcal{T}_{\hat{c}a} \right] \\
\delta_w \chi_{\mu\hat{c}}(d) &= \mathbf{J}_{\mu\hat{c}a}{}^b dw_b^a + \dots : \\
\mathbf{J}_{\mu\hat{c}a}{}^b &= \frac{\partial \chi_{\mu\hat{c}}(\delta_w)}{\partial w_b^a} = e^{\varphi} \left[ \frac{1}{2} \delta_a^b (\mathcal{T}\mathfrak{J}\eta_\mu)_{\hat{c}} - i\eta_{\mu a} \mathcal{T}_{\hat{c}}^b + i\bar{\eta}_\mu^b \mathcal{T}_{\hat{c}a} + ie^{\varphi}(\bar{\eta}\eta)\epsilon_{\mu\nu} \mathbf{j}_{\hat{c}a}^{\nu}{}^b \right]
\end{align*}
$$

<caption>Table 2.5: Tensors contributing to the Noether current density associated with the $u(3)$ invariance</caption>
<table><caption>Table 2.6: Tensors contributing to the Noether current density associated with the $su(4)/u(3)$ invariance</caption>
  <tr>
    <td>
      &delta;<sub>y</sub>&omega;<sup>m</sup>(d) + &delta;<sub>y</sub>c<sup>m</sup>(d) = j<sup>m</sup><sub>a</sub>dy<sup>a</sup> + j<sup>ma</sup>d&ytilde;<sub>a</sub> :<br/>
      j<sup>m</sup><sub>a</sub> = <div style="display: inline-block; vertical-align: middle; text-align: center; line-height: 1.2em;"><span style="display: block; border-bottom: 1px solid black; padding: 0 0.2em;">&part;&omega;<sup>m</sup>(&delta;<sub>y</sub>) + c<sup>m</sup>(&delta;<sub>y</sub>)</span><span style="display: block; padding: 0 0.2em;">&part;y<sup>a</sup></span></div> = -&epsilon;<sub>abc</sub> {e<sup>-2&varphi;</sup>A(&theta;&#773;<sup>b</sup>&sigma;<sup>m</sup>&theta;&#773;<sup>c</sup>) + e<sup>2&varphi;</sup>[(&eta;&#773;<sup>b</sup>&sigma;&#773;<sup>m</sup>&eta;&#773;<sup>c</sup>) - 2i(&eta;&#773;&eta;)&#773;(&theta;&#773;<sup>b</sup>&sigma;&#773;<sup>m</sup>&eta;&#773;<sup>c</sup>)]}
    </td>
  </tr>
  <tr>
    <td>
      &delta;<sub>y</sub>&Delta;(d) = j<sub>a</sub>dy<sup>a</sup> + j&#773;<sup>a</sup>d&ytilde;<sub>a</sub> : j<sub>a</sub> = <div style="display: inline-block; vertical-align: middle; text-align: center; line-height: 1.2em;"><span style="display: block; border-bottom: 1px solid black; padding: 0 0.2em;">&part;&Delta;(&delta;<sub>y</sub>)</span><span style="display: block; padding: 0 0.2em;">&part;y<sup>a</sup></span></div> = &epsilon;<sub>abc</sub>&theta;&#773;<sup>mb</sup>&eta;&#773;<sup>c</sup>:
    </td>
  </tr>
  <tr>
    <td>
      &delta;<sub>y</sub>&Omega;&#773;<sub>b</sub><sup>&#770;</sup>(d) = J<sup>&#770;</sup><sub>b</sub><sup>a</sup>dy<sup>a</sup> + J&#773;<sup>&#770;</sup><sub>b</sub><sup>ca</sup>d&ytilde;<sub>a</sub> + ... :<br/>
      J<sup>&#770;</sup><sub>b</sub><sup>a</sup> = <div style="display: inline-block; vertical-align: middle; text-align: center; line-height: 1.2em;"><span style="display: block; border-bottom: 1px solid black; padding: 0 0.2em;">&part;</sub><sub>y<sup>a</sup></sub>&Omega;&#773;<sup>&#770;</sup><sub>b</sub><sup>&#770;</sup>(&delta;<sub>y</sub>)</span><span style="display: block; padding: 0 0.2em;">&part;</sub><sub>y<sup>a</sup></sub>&Omega;&#773;<sup>ce</sup></span></div>, J&#773;<sup>&#770;</sup><sub>b</sub><sup>ca</sup> = <div style="display: inline-block; vertical-align: middle; text-align: center; line-height: 1.2em;"><span style="display: block; border-bottom: 1px solid black; padding: 0 0.2em;">&part;</sub><sub>y<sub>a</sub></sub>&Omega;&#773;<sup>&#770;</sup><sub>b</sub><sup>&#770;</sup>(&delta;<sub>y</sub>)</span><span style="display: block; padding: 0 0.2em;">&part;</sub><sub>y<sup>a</sup></sub>&Omega;&#773;<sup>ce</sup></span></div> = &epsilon;<sup>ade</sup>T&#773;<sub>bd</sub>T<sup>&#770;</sup><sub>e</sub>
    </td>
  </tr>
  <tr>
    <td>
      &delta;<sub>y</sub>&omega;&#773;<sup>&mu;</sup><sub>b</sub>(d) = j<sup>&mu;</sup><sub>ba</sub>dy<sup>a</sup> + j&#773;<sup>&mu;</sup><sub>b</sub><sup>aa</sup>d&ytilde;<sub>a</sub> + ... :<br/>
      j<sup>&mu;</sup><sub>ba</sub> = <div style="display: inline-block; vertical-align: middle; text-align: center; line-height: 1.2em;"><span style="display: block; border-bottom: 1px solid black; padding: 0 0.2em;">&part;&omega;&#773;<sup>&mu;</sup><sub>b</sub>(&delta;<sub>y</sub>)</span><span style="display: block; padding: 0 0.2em;">&part;y<sup>a</sup></span></div> = ie<sup>-&varphi;</sup>&epsilon;<sub>acd</sub>T&#773;<sup>c</sup><sub>b</sub>&theta;&#773;<sup>md</sup>:
    </td>
  </tr>
  <tr>
    <td>
      &delta;<sub>y</sub>&chi;<sub>&mu;&#773;b</sub>(d) = J<sup>&mu;ba</sup>dy<sup>a</sup> + J&#773;<sup>&mu;</sup><sub>&mu;ba</sub><sup>a</sup>d&ytilde;<sub>a</sub> + ... :<br/>
      J<sup>&mu;ba</sup> = <div style="display: inline-block; vertical-align: middle; text-align: center; line-height: 1.2em;"><span style="display: block; border-bottom: 1px solid black; padding: 0 0.2em;">&part;&chi;<sub>&mu;&#773;b</sub>(&delta;<sub>y</sub>)</span><span style="display: block; padding: 0 0.2em;">&part;y<sup>a</sup></span></div> = ie<sup>&varphi;</sup>(&epsilon;<sub>acd</sub>T&#773;<sup>c</sup><sub>b</sub>&eta;&#773;<sup>d</sup><sub>&mu;</sub> + e<sup>&varphi;</sup>(&eta;&#773;&eta;)&epsilon;<sub>&mu;&nu;</sub>j<sup>&nu;</sup><sub>ba</sub>) and c.c.
    </td>
  </tr>
</table>

<table><caption>Table 2.7: Tensors contributing to the Noether current density associated with invariance under the Poincaré supersymmetry</caption>
  <tr>
    <td>
      &delta;<sub>&epsilon;</sub>&omega;<sup>m</sup>(d) + &delta;<sub>&epsilon;</sub>c<sup>m</sup>(d) = j<sup>ma</sup><sub>&mu;</sub>d&epsilon;<sup>&mu;</sup> - j&#773;<sup>ma</sup><sub>&mu;a</sub>d&xi;&#773;<sup>ma</sup>:
    </td>
  </tr>
  <tr>
    <td>
      j<sup>ma</sup><sub>&mu;</sub> = <div style="display: inline-block; vertical-align: middle; text-align: center; line-height: 1.2em;"><span style="display: block; border-bottom: 1px solid black; padding: 0 0.2em;">&part;&omega;<sup>m</sup>(&delta;<sub>&epsilon;</sub>) + c<sup>m</sup>(&delta;<sub>&epsilon;</sub>)</span><span style="display: block; padding: 0 0.2em;">&part;&epsilon;<sup>&mu;</sup><sub>&alpha;</sub></span></div> = 2&sigma;<sup>m</sup><sub>&mu;&nu;</sub> [ie<sup>-2&varphi;</sup>A&theta;&#773;<sup>&nu;a</sup> + e<sup>2&varphi;</sup>(&eta;&#773;&eta;)&eta;&#773;<sup>&nu;a</sup>]
    </td>
  </tr>
  <tr>
    <td>
      &delta;<sub>&epsilon;</sub>&Delta;(d) = j<sup>a</sup><sub>&mu;</sub>d&epsilon;<sup>&mu;</sup> - j&#773;<sub>&mu;a</sub>d&xi;&#773;<sup>ma</sup>: j<sup>a</sup><sub>&mu;</sub> = <div style="display: inline-block; vertical-align: middle; text-align: center; line-height: 1.2em;"><span style="display: block; border-bottom: 1px solid black; padding: 0 0.2em;">&part;&Delta;(&delta;<sub>&epsilon;</sub>)</span><span style="display: block; padding: 0 0.2em;">&part;&epsilon;<sup>&mu;</sup><sub>&alpha;</sub></span></div> = -i&eta;&#773;<sup>a</sup>
    </td>
  </tr>
  <tr>
    <td>
      &delta;<sub>&epsilon;</sub>&Omega;&#773;<sup>&#770;</sup><sub>b</sub><sup>&#770;</sup>(d) = J<sup>&#770;</sup><sub>b</sub><sup>ca</sup><sub>&mu;</sub>d&epsilon;<sup>&mu;</sup><sub>&alpha;</sub> - J&#773;<sup>&#770;</sup><sub>b</sub><sup>ca</sup><sub>&mu;a</sub>d&xi;&#773;<sup>ma</sup>:
    </td>
  </tr>
  <tr>
    <td>
      J<sup>&#770;</sup><sub>b</sub><sup>ca</sup><sub>&mu;</sub> = <div style="display: inline-block; vertical-align: middle; text-align: center; line-height: 1.2em;"><span style="display: block; border-bottom: 1px solid black; padding: 0 0.2em;">&part;</sub><sub>&epsilon;<sup>&mu;</sup><sub>&alpha;</sub></sub>&Omega;&#773;<sup>&#770;</sup><sub>b</sub><sup>&#770;</sup>(&delta;<sub>&epsilon;</sub>)</span><span style="display: block; padding: 0 0.2em;">&part;</sub><sub>&epsilon;<sup>&mu;</sup><sub>&alpha;</sub></span></div> = 2(&eta;&#773;<sub>&mu;&#773;b</sub>T&#773;<sup>ca</sup> - T&#773;<sup>a</sup><sub>b</sub>&eta;&#773;<sup>&#770;</sup><sub>&mu;</sub>)
    </td>
  </tr>
  <tr>
    <td>
      J&#773;<sup>&#770;</sup><sub>b</sub><sup>ca</sup><sub>&mu;a</sub> = -<div style="display: inline-block; vertical-align: middle; text-align: center; line-height: 1.2em;"><span style="display: block; border-bottom: 1px solid black; padding: 0 0.2em;">&part;</sub><sub>&xi;&#773;<sup>&mu;&alpha;</sup><sub>a</sub></sub>&Omega;&#773;<sup>&#770;</sup><sub>b</sub><sup>&#770;</sup>(&delta;<sub>&epsilon;</sub>)</span><span style="display: block; padding: 0 0.2em;">&part;</sub><sub>&xi;&#773;<sup>&mu;&alpha;</sup><sub>a</sub></span></div> = 2(T&#773;<sub>b&alpha;</sub>&eta;&#773;<sup>&#770;</sup><sub>&mu;</sub> - &eta;&#773;<sub>&mu;&#773;b</sub>T&#773;<sup>ca</sup>)
    </td>
  </tr>
  <tr>
    <td>
      &delta;<sub>&epsilon;</sub>&omega;<sup>&nu;</sup><sub>b</sub>(d) = j<sup>&nu;</sup><sub>b&mu;</sub><sup>a</sup>d&epsilon;<sup>&mu;</sup><sub>&alpha;</sub> + j&#773;<sup>&nu;</sup><sub>b</sub><sup>&nu;&mu;</sup><sub>&mu;a</sub>d&xi;&#773;<sup>ma</sup>:
    </td>
  </tr>
  <tr>
    <td>
      j<sup>&nu;</sup><sub>b</sub><sup>&nu;&mu;</sup><sub>&mu;</sub> = <div style="display: inline-block; vertical-align: middle; text-align: center; line-height: 1.2em;"><span style="display: block; border-bottom: 1px solid black; padding: 0 0.2em;">&part;&omega;<sup>&nu;</sup><sub>b</sub>(&delta;<sub>&epsilon;</sub>)</span><span style="display: block; padding: 0 0.2em;">&part;&epsilon;<sup>&mu;</sup><sub>&alpha;</sub></span></div> = e<sup>-&varphi;</sup>(&delta;<sup>&nu;</sup><sub>&mu;</sub>T&#773;<sup>a</sup><sub>b</sub> + 2i&eta;&#773;<sub>&mu;&#773;b</sub>&theta;&#773;<sup>&nu;a</sup>)
    </td>
  </tr>
  <tr>
    <td>
      &delta;<sub>&epsilon;</sub>&chi;<sub>&nu;&#773;b</sub>(d) = J<sup>&nu;&mu;ba</sup><sup>a</sup>d&epsilon;<sup>&mu;</sup><sub>&alpha;</sub> + J&#773;<sup>&nu;</sup><sub>&nu;&mu;ba</sub><sup>a</sup>d&xi;&#773;<sup>ma</sup>:
    </td>
  </tr>
  <tr>
    <td>
      J<sup>&nu;&mu;ba</sup><sup>a</sup> = <div style="display: inline-block; vertical-align: middle; text-align: center; line-height: 1.2em;"><span style="display: block; border-bottom: 1px solid black; padding: 0 0.2em;">&part;&chi;<sub>&nu;&#773;b</sub>(&delta;<sub>&epsilon;</sub>)</span><span style="display: block; padding: 0 0.2em;">&part;&epsilon;<sup>&mu;</sup><sub>&alpha;</sub></span></div> = ie<sup>&varphi;</sup>(2&eta;&#773;<sub>&mu;&#773;b</sub>&eta;&#773;<sup>a</sup><sub>&nu;</sub> - e<sup>&varphi;</sup>(&eta;&#773;&eta;)&epsilon;<sub>&nu;&lambda;</sub>j<sup>&lambda;&mu;</sup><sub>b</sub><sup>a</sup>) and c.c.
    </td>
  </tr>
</table>
<table><caption>Table 2.8: Tensors contributing to the Noether current density associated with invariance under special conformal supersymmetry</caption>
  <tr>
    <td>δ<sub>ξ</sub>ω<sup>m</sup>(d) + δ<sub>ξ</sub>c<sup>m</sup>(d) = j<sup>mμa</sup>dξ<sub>μa</sub> - j&#773;<sup>mμa</sup>dξ&#773;<sub>μ</sub> + ... :</td>
  </tr>
  <tr>
    <td>j<sup>mμa</sup> = <div style="display: inline-block; vertical-align: middle; text-align: center; line-height: 1.2em;"><div style="border-bottom: 1px solid black; padding: 0 0.2em;">∂(ω<sup>m</sup>(δ<sub>ξ</sub>) + c<sup>m</sup>(δ<sub>ξ</sub>))</div><div style="padding: 0 0.2em;">∂ξ<sub>μa</sub></div></div> = 2[ie<sup>-2φ</sup>Aθ<sup>λa</sup>σ<sup>m</sup><sub>λν</sub> ̂x<sup>νμ</sup> + ie<sup>2φ</sup>η̄<sup>a</sup>σ<sup>mλν</sup>Λ<sub>-νμ</sub> + e<sup>2φ</sup>(η̄η)(η̄<sup>λa</sup>σ<sup>m</sup><sub>λν</sub> ̂x<sup>νμ</sup> - θ̄<sup>a</sup>σ<sup>mλν</sup>Λ<sub>+νμ</sub>)]</td>
  </tr>
  <tr>
    <td>δ<sub>ξ</sub>Δ(d) = j<sup>μa</sup>dξ<sub>μa</sub> - j&#773;<sup>μa</sup>dξ&#773;<sub>μ</sub> + ... : j<sup>μa</sup> = <div style="display: inline-block; vertical-align: middle; text-align: center; line-height: 1.2em;"><div style="border-bottom: 1px solid black; padding: 0 0.2em;">∂Δ(δ<sub>ξ</sub>)</div><div style="padding: 0 0.2em;">∂ξ<sub>μa</sub></div></div> = -i (θ<sup>νa</sup>Λ<sub>-νμ</sub> + η̄<sup>a</sup> ̂x<sup>νμ</sup>)</td>
  </tr>
  <tr>
    <td>δ<sub>ξ</sub>Ω<sub>̂b</sub> ̂c(d) = Ĵ<sub>̂b</sub> ̂c<sup>μa</sup>dξ<sub>μa</sub> - Ĵ<sub>̂b</sub> ̂c<sup>μa</sup>dξ&#773;<sub>μ</sub>:</td>
  </tr>
  <tr>
    <td>Ĵ<sub>̂b</sub> ̂c<sup>μa</sup> = <div style="display: inline-block; vertical-align: middle; text-align: center; line-height: 1.2em;"><div style="border-bottom: 1px solid black; padding: 0 0.2em;">∂</div><div style="padding: 0 0.2em;">∂ξ<sub>μa</sub></div></div>Ω<sub>̂b</sub> ̂c(d<sub>ξ</sub>) = 2(T<sub>̂b</sub><sup>a</sup>Θ<sup>μ̂c</sup> - Θ<sup>μ</sup><sub>̂b</sub>T<sup>ca</sup>),</td>
  </tr>
  <tr>
    <td>Ĵ<sub>̂b</sub> ̂c<sup>μa</sup> = -<div style="display: inline-block; vertical-align: middle; text-align: center; line-height: 1.2em;"><div style="border-bottom: 1px solid black; padding: 0 0.2em;">∂</div><div style="padding: 0 0.2em;">∂ξ&#773;<sub>μ</sub><sup>a</sup></div></div>Ω<sub>̂b</sub> ̂c(d<sub>ξ</sub>) = 2(Θ<sup>μ</sup><sub>̂b</sub>T<sup>ca</sup> - T<sub>ba</sub>Θ<sup>μ̂c</sup>)</td>
  </tr>
  <tr>
    <td>δ<sub>ξ</sub>ω<sup>ν</sup><sub>̂b</sub>(d) = ĵ<sub>̂b</sub><sup>νμa</sup>dξ<sub>μa</sub> + ĵ<sub>ba</sub><sup>νμ</sup>dξ&#773;<sub>μ</sub> + ... :</td>
  </tr>
  <tr>
    <td>ĵ<sub>̂b</sub><sup>νμa</sup> = <div style="display: inline-block; vertical-align: middle; text-align: center; line-height: 1.2em;"><div style="border-bottom: 1px solid black; padding: 0 0.2em;">∂ω<sup>ν</sup><sub>̂b</sub>(δ<sub>ξ</sub>)</div><div style="padding: 0 0.2em;">∂ξ<sub>μa</sub></div></div> = e<sup>-φ</sup>(T<sub>̂b</sub><sup>a</sup> ̂x<sup>νμ</sup> + 2iθ<sup>νa</sup>Θ<sup>μ</sup><sub>̂b</sub>)</td>
  </tr>
  <tr>
    <td>δ<sub>ξ</sub>χ<sub>ν̂b</sub>(d) = J<sub>ν̂b</sub><sup>μa</sup>dξ<sub>μa</sub> + J<sub>νba</sub><sup>μ</sup>dξ&#773;<sub>μ</sub> + ... :</td>
  </tr>
  <tr>
    <td>J<sub>ν̂b</sub><sup>μa</sup> = <div style="display: inline-block; vertical-align: middle; text-align: center; line-height: 1.2em;"><div style="border-bottom: 1px solid black; padding: 0 0.2em;">∂χ<sub>ν̂b</sub>(δ<sub>ξ</sub>)</div><div style="padding: 0 0.2em;">∂ξ<sub>μa</sub></div></div> = e<sup>φ</sup>(Λ<sub>-νμ</sub>T<sub>̂b</sub><sup>a</sup> + 2iη̄<sup>a</sup>Θ<sup>μ</sup><sub>̂b</sub> - ie<sup>φ</sup>(η̄η)ε<sub>νλ</sub>j<sub>̂b</sub><sup>λμa</sup>) and c.c.</td>
  </tr>
</table>

gauges in which physical degrees of freedom are present also in the sector of broken super-symmetries 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 [22].<fnref n="10" />

As was mentioned in the Introduction the complete action of the AdS$_4 \times CP^3$ superstring was derived in Ref. [13] via the double-dimensional reduction [14], [15] of the $D=11$ supermembrane action in the AdS$_4 \times S^7$ superbackground [16]. This approach is based on the possibility of isomorphic realization of the AdS$_4 \times S^7$ superspace as the (11|32)-dimensional $OSp(4|8)/(SO(1,3) \times SO(7))$ supercoset manifold, geometric constituents of which are identified with the $osp(4|8)$ Cartan forms. The AdS$_4 \times S^7$ 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$_5 \times S^5$ superstring [11], [12]. Also to perform the double-dimensional reduction the seven-sphere was realized as the Hopf fibration $CP^3 \times S^1$ [7], [8].

We considered the $OSp(4|8)/(SO(1,3) \times SO(7))$ supercoset element

$$ \mathcal{G} = \mathcal{G} e^{yH} e^{\theta_4^\mu Q_\mu^4 + \bar{\theta}^{\mu 4} \bar{Q}_{\mu 4}} e^{\eta_{\mu 4} S^{\mu 4} + \bar{\eta}_\mu^4 \bar{S}_4^\mu} \in \mathrm{OSp}(4|8)/(\mathrm{SO}(1,3) \times \mathrm{SO}(7)). \quad (2.68) $$

In (2.68) the first factor on the left is the $OSp(4|6)/(SO(1,3) \times U(3))$ supercoset element

$$ \mathcal{G} = e^{x_m P^m + \theta_a^\mu Q_\mu^a + \bar{\theta}^{\mu a} \bar{Q}_{\mu a}} e^{\eta_{\mu a} S^{\mu a} + \bar{\eta}_\mu^a \bar{S}_a^\mu} e^{z^a T_a + \bar{z}_a T^a} e^{\varphi D} \in \mathrm{OSp}(4|6)/(\mathrm{SO}(1,3) \times U(3)) $$

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

<footnote n="10">¹⁰For the AdS$_5 \times S^5$ superstring analogous gauge condition was studied in [40], [41].</footnote>
of the element (2.68) corresponds to the realization of the osp(4|8) isometry superalgebra of the AdS$_4 \times S^7$ superbackground as the $D = 3 \mathcal{N} = 8$ superconformal algebra and results in the Poincaré metric for the anti-de Sitter space (2.41). 32 supersymmetry generators of the $D = 3 \mathcal{N} = 8$ superconformal algebra

$$Q_{\mu}^{A} = \begin{pmatrix} Q_{\mu}^{a} \\ Q_{\mu}^{4} \end{pmatrix}, \quad \bar{Q}_{\mu A} = \begin{pmatrix} \bar{Q}_{\mu a} \\ \bar{Q}_{\mu 4} \end{pmatrix}$$

and

$$S^{\mu A} = \begin{pmatrix} S^{\mu a} \\ S^{\mu 4} \end{pmatrix}, \quad \bar{S}_{A}^{\mu} = \begin{pmatrix} \bar{S}_{a}^{\mu} \\ \bar{S}_{4}^{\mu} \end{pmatrix}$$

are composed of 24 supersymmetry generators $Q_{\mu}^{a}$, $\bar{Q}_{\mu a}$ and $S^{\mu a}$, $\bar{S}_{a}^{\mu}$ of the $D = 3 \mathcal{N} = 6$ superconformal algebra together with additional generators of the Poincaré supersymmetry $Q_{\mu}^{4}$, $\bar{Q}_{\mu 4}$ and special conformal supersymmetry $S^{\mu 4}$, $\bar{S}_{4}^{\mu}$ that are broken by the AdS$_4 \times \mathbb{CP}^3$ 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 $\kappa$-symmetry light-cone gauge is fixed by setting to zero 16 Grassmann-odd coordinates

$$\theta_A^2 = \bar{\theta}^{2A} = \eta_{1A} = \bar{\eta}_1^A = 0 \qquad (2.69)$$

transforming in the $\mathfrak{K}$ and $\bar{\mathfrak{K}}$ representations of the $su(4)$ algebra. These coordinates are associated with the generators $Q_2^A$, $\bar{Q}_{2A}$, $S^{1A}$ and $\bar{S}_A^1$ that have negative weight under the $SO(1, 1) \subset SO(1, 2)$ symmetry generated by $M^{+-} = 2M^{02}$.

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$_4 \times S^7$ superspace coordinates and their differentials. Then the double-dimensional reduction of its action yields the AdS$_4 \times \mathbb{CP}^3$ superstring action. This procedure will be considered in detail in the next paragraph.

### 2.3.1 AdS$_4 \times S^7$ supermembrane and its reduction to the AdS$_4 \times \mathbb{CP}^3$ superstring

The $D = 11$ supermembrane action in the AdS$_4 \times S^7$ superbackground [16] has the form

$$S_{\text{smembrane}}^{\text{AdS}_4 \times S^7} = - \int_V d^3 \xi \sqrt{-g^{(3)}} + S_{\text{WZ}}^{\text{AdS}_4 \times S^7}. \qquad (2.70)$$

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

$$g_{ij}^{(3)} = E_{i}^{\hat{m}} E_{j}^{\hat{m}},$$

where $\hat{\xi}^i = (\tau, \sigma, \rho)$ are the world-volume local coordinates and $E_{i}^{\hat{m}}$ ($\hat{m}, \hat{n} = 0, 1, \dots, 9, 11$) is the mapping to the world-volume of the $D = 11$ supervielbein bosonic components. The
Wess-Zumino term

$$S_{\text{WZ}}^{\text{AdS}_4 \times S^7} = s \int_{\mathcal{M}_4} H_{(4)}$$

is given by the integral of the closed four-form

$$H_{(4)}(d) = \frac{i}{8} F^{\hat{\alpha}}(d) \wedge \mathfrak{g}^{\hat{m}\hat{n}}{}_{\hat{\alpha}}{}^{\hat{\beta}} F_{\hat{\beta}}(d) \wedge E_{\hat{m}}(d) \wedge E_{\hat{n}}(d) + \frac{1}{4} \epsilon_{m'n'k'l'} E^{m'}(d) \wedge E^{n'}(d) \wedge E^{k'}(d) \wedge E^{l'}(d)$$

over the four-dimensional auxiliary hypersurface $\mathcal{M}_4$, whose boundary is the supermembrane's world volume $V$. Kinetic term has the same structure as in the flat superspace [47], while the second summand in $H_{(4)}(d)$ includes contribution of the nonzero four-form of the $AdS_4 \times S^7$ background. The value of the parameter $s = \pm 1$ is fixed by the requirement of the $\kappa$-invariance of the supermembrane action (2.70).

The $D = 11$ supervielbein bosonic components $E^{\hat{m}}(d) = (E^{m'}(d), E^{I'}(d))$ include components $E^{m'}(d)$ in the tangent space to $AdS_4 = SO(2, 3)/SO(1, 3)$ and components $E^{I'}(d)$ ($I' = 1, \dots, 7$) in the tangent space to $S^7 = SO(8)/SO(7)$. They equal the Cartan forms $\mathcal{G}^{0'm'}(d)$ and $\Omega^{8I'}(d)$ 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 $OSp(4|6)/(SO(1, 3) \times U(3))$ superspace and their differentials they may depend also on other coordinates of the $OSp(4|8)/(SO(1, 3) \times SO(7))$ 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 $F^{\hat{\alpha}}(d) = F^{\alpha A'}(d)$ ($\hat{\alpha} = 1, \dots, 32$, $\alpha = 1, \dots, 4$, $A' = 1, \dots, 8$) associated with the supersymmetry generators $O_{\hat{\alpha}} = O_{\alpha A'}$ 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). $\mathfrak{g}^{\hat{m}\hat{n}}{}_{\hat{\alpha}}{}^{\hat{\beta}} = \frac{1}{2}(\mathfrak{g}^{\hat{m}}{}_{\hat{\alpha}}{}^{\hat{\gamma}}\mathfrak{g}^{\hat{n}}{}_{\hat{\gamma}}{}^{\hat{\beta}} - \mathfrak{g}^{\hat{n}}{}_{\hat{\alpha}}{}^{\hat{\gamma}}\mathfrak{g}^{\hat{m}}{}_{\hat{\gamma}}{}^{\hat{\beta}})$ are the spin(1, 10) algebra generators constructed out of the $D = 1 + 10$ $\gamma$-matrices $\mathfrak{g}^{\hat{m}}{}_{\hat{\alpha}}{}^{\hat{\beta}}$.

To perform the double-dimensional reduction [14], [15] of the supermembrane the $D = 11$ supervielbein components need not depend on the coordinate $y \in [0, 2\pi)$ of the spacetime compact dimension that is subject to reduction. However, when the supervielbein bosonic components include the term $dyE_y^{\hat{m}'}$ ($\hat{m}' = 0, 1, \dots, 9$), as is the case for the $OSp(4|8)/(SO(1, 3) \times SO(7))$ supercoset elements considered in [13], [48] and in our works, the local Lorentz rotation in the tangent space has to be performed

$$ \begin{aligned} E^{\hat{m}}(d) &\rightarrow L^{\hat{m}}{}_{\hat{n}} E^{\hat{n}}(d), & L^{\hat{m}}{}_{\hat{n}} &\in SO(1, 10), \\ F^{\hat{\alpha}}(d) &\rightarrow L^{\hat{\alpha}}{}_{\hat{\beta}} F^{\hat{\beta}}(d), & L^{\hat{\alpha}}{}_{\hat{\beta}} &\in Spin(1, 10) \end{aligned} $$

to bring them to the Kaluza-Klein ansatz form

$$(LE)^{\hat{m}'}(d) = \mathcal{E}^{\hat{m}'}(d), \quad (LE)^{11}(d) = \mathcal{E}^{11}(d) = \Phi_L(dy + A_L(d)). \qquad (2.71)$$

Components of the transformed vielbein $\mathcal{E}^{\hat{m}'}(d)$ do not depend on $y$ and $dy$ and are identified with the $D = 10$ supervielbein bosonic components, $\Phi_L = e^{2\phi/3}$ determines the $D = 10$ IIA
dilaton superfield $\phi$ and $A_L(d)$ is the one-form gauge superfield that together with the 3-form 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

$$ (LF)^{\hat{\alpha}}(d) = \mathcal{E}^{\hat{\alpha}}(d) + e^{-2\phi/3} \chi^{\hat{\alpha}} \mathcal{E}^{11}(d) \qquad (2.72) $$

have the form of the sum of the $D = 10$ supervielbein fermionic components

$$ \mathcal{E}^{\hat{\alpha}}(d) = (Lf)^{\hat{\alpha}}(d) - (LF_y)^{\hat{\alpha}}A_L(d) \qquad (2.73) $$

and the term proportional to the dilatino superfield

$$ \chi^{\hat{\alpha}} = (LF_y)^{\hat{\alpha}}. \qquad (2.74) $$

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

$$ F^{\hat{\alpha}}(d) = f^{\hat{\alpha}}(d) + dyF_y^{\hat{\alpha}}, \qquad (2.75) $$

where $f^{\hat{\alpha}}(d)$ and $F_y^{\hat{\alpha}}$ do not depend on $y$ and $f^{\hat{\alpha}}(d)$ does not contain terms proportional to $dy$.

Further the space-time compact dimension is identified with the world-volume compact dimension $y = \rho$ 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

$$ \int_V d^3 \hat{\xi} \sqrt{-g^{(3)}} \rightarrow \int_\Sigma d^2 \xi \Phi_L \sqrt{-g^{(2)}}, $$

where $\xi^i = (\tau, \sigma)$ are world-sheet local coordinates and $g^{(2)} = \det g_{ij}^{(2)}$ is the determinant of the induced world-sheet metric

$$ g_{ij}^{(2)} = E_i^{\hat{m}'} E_{j\hat{m}'} $$

The Wess-Zumino term of the supermembrane reduces to that of the superstring

$$ \int_{M_4} H_{(4)} \rightarrow \int_{M_3} H_{(3)} \qquad (2.76) $$

presented as the integral of the three-form $H_{(3)}(d) = dB_{(2)}(d)$ over auxiliary three-dimensional hypersurface $M_3$ whose boundary is the superstring world-sheet $\Sigma$. The two-form $B_{(2)}(d)$ 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 super-membrane 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 $OSp(4|8)/(SO(1,3) \times SO(7))$ supercoset element $\hat{\mathcal{G}}$ of the general form covariant Cartan forms are defined by the relation

$$ \hat{\mathcal{G}}^{-1} d\hat{\mathcal{G}} = \underline{G}_{\text{sp}(4)}(d) + G_{\text{so}(8)}(d) + G_{32\text{susy}}(d) \in \text{osp}(4|8). \qquad (2.77) $$

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

$$ \begin{aligned} \underline{G}_{\text{sp}(4)}(d) &= \underline{G}^{m'n'}(d)M_{m'n'} + 2\underline{G}^{0'm'}(d)M_{0'm'} \\ &= \underline{G}^{mn}(d)M_{mn} + \underline{\Delta}(d)D + \underline{\omega}^m(d)P_m + \underline{c}^m(d)K_m \in \text{sp}(4). \end{aligned} $$

The Cartan forms $\underline{G}^{0'm'}(d)$ 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)

$$ \underline{G}^{0'm}(d) = \frac{1}{2}(\underline{\omega}^m + \underline{c}^m)(d), \quad \underline{G}^{0'3}(d) = -\underline{\Delta}(d). \qquad (2.78) $$

The second term contains the $so(8)$ generators $V^{IJ}$ together with the Cartan forms $\underline{\Omega}^{IJ}(d)$

$$ G_{\text{so}(8)}(d) = \underline{\Omega}^{IJ}(d)V^{IJ} \in \text{so}(8). \qquad (2.79) $$

It can be presented in the form with manifest $so(7)$ or $so(6)$ covariance

$$ \begin{aligned} G_{\text{so}(8)}(d) &= \underline{\Omega}^{I'J'}(d)V^{I'J'} + \underline{\Omega}^{8I'}(d)V^{8I'} \\ &= \underline{\Omega}^{IJ}(d)V^{IJ} + \underline{\Omega}^{87}(d)V^{87} + \underline{\Omega}^{8I}(d)V^{8I} + 2\underline{\Omega}^{7I}(d)V^{7I}. \end{aligned} $$

The last equality can also be written in the $\mathbf{3} \oplus \bar{\mathbf{3}}$ basis

$$ G_{\text{so}(8)}(d) = \underline{\Omega}_{\hat{a}}^{\hat{b}}(d)V_{\hat{b}}^{\hat{a}} + \underline{\Omega}^{87}(d)V^{87} - \frac{1}{2}\underline{\Omega}^{8\hat{a}}(d)V^{8\hat{a}} - \underline{\Omega}^{7\hat{a}}(d)V^{7\hat{a}}. \qquad (2.80) $$

Definitions of the $so(6)$ Cartan forms and generators in this basis

$$ \underline{\Omega}_{\hat{a}}^{\hat{b}}(d) = \begin{pmatrix} \underline{\Omega}_{\hat{a}}^{b}(d) - \delta_{\hat{a}}^{b}\underline{\Omega}_{\hat{c}}^{c}(d) & \varepsilon_{acb}\underline{\Omega}_{4}^{c}(d) \\ -\varepsilon^{acb}\underline{\Omega}_{4}^{c}(d) & -\underline{\Omega}_{\hat{b}}^{a}(d) + \delta_{\hat{b}}^{a}\underline{\Omega}_{\hat{c}}^{c}(d) \end{pmatrix} $$

and

$$ V_{\hat{a}}^{\hat{b}} = \frac{1}{2} \begin{pmatrix} V_{\hat{a}}^{b} - \delta_{\hat{a}}^{b} V_{\hat{c}}^{c} & \varepsilon_{acb} V_{4}^{c}(d) \\ -\varepsilon^{acb} V_{\hat{c}}^{4} & -V_{\hat{b}}^{a} + \delta_{\hat{b}}^{a} V_{\hat{c}}^{c} \end{pmatrix} $$

coincide with those in (2.8) and (2.9). The $so(8)/(so(2) \times so(6))$ Cartan forms and generators in the $\mathbf{3} \oplus \bar{\mathbf{3}}$ basis are defined as

$$ \underline{\Omega}^{7(8)}_{\hat{a}}(d) = \begin{pmatrix} \underline{\Omega}^{7(8)}_{a}(d) \\ \underline{\Omega}^{7(8)}_{a}(d) \end{pmatrix} = M_{\hat{a}I}^{-1} \underline{\Omega}^{7(8)I}(d) $$
and

$$V^{7(8)}_{\hat{a}} = \begin{pmatrix} V^{7(8)}_{a} \\ V^{7(8)}_{a} \end{pmatrix} = M^{-1}_{\hat{a}I} V^{7(8)I}.$$

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 $su(4) \oplus u(1)$ isometry algebra of $\mathbb{CP}^3 \times S^1$ into the $so(8)$ algebra compared to the standard one that follows from presenting its commutation relations

$$[V^{IJ}, V^{KL}] = \delta^{IL}V^{JK} - \delta^{IK}V^{JL} + \delta^{JK}V^{IL} - \delta^{JL}V^{IK}$$

in the form with manifest $so(6)$ covariance

$$ \begin{align*} [V^{87}, V^{7I}] &= V^{8I}, \quad [V^{87}, V^{8I}] = -V^{7I}, \\ [V^{7I}, V^{7J}] &= [V^{8I}, V^{8J}] = -V^{1J}, \quad [V^{7I}, V^{8J}] = -\delta^{IJ}V^{78}, \\ [V^{IJ}, V^{7(8)K}] &= \delta^{JK}V^{7(8)I} - \delta^{IK}V^{7(8)J}, \\ [V^{IJ}, V^{KL}] &= \delta^{IL}V^{JK} - \delta^{IK}V^{JL} + \delta^{JK}V^{IL} - \delta^{JL}V^{IK}. \end{align*} $$

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 [13]). Taking into account chosen realization of the Kähler two-form components (2.28) new $u(1) \subset so(8)/su(4)$ and $u(3) \subset su(4)$ generators are defined by the relations

$$H = 2V_a^a - V^{78}, \qquad (2.81)$$

$$\tilde{V}_a^b = V_a^b - \frac{1}{2}\delta_a^b V_c^c - \frac{1}{4}\delta_a^b V^{78} = V_a^b - \frac{1}{4}\delta_a^b H \qquad (2.82)$$

and new $su(4)/u(3)$ quotient algebra generators are given by the expressions

$$T_a = \frac{1}{2}(V^7_a - iV^8_a), \quad T^a = -\frac{1}{2}(V^{7a} + iV^{8a}). \qquad (2.83)$$

These generators can be presented in the form that exhibits $so(6)$ covariance in the $\mathbf{3} \oplus \mathbf{3}$ basis

$$T_{\hat{a}} = \begin{pmatrix} T_a \\ -T^a \end{pmatrix} = \frac{1}{2} (V^7_{\hat{a}} - \frac{1}{2} \mathfrak{J}_{\hat{a}} \hat{b}^c V^8_{\hat{b}}).$$

New generators (2.82)-(2.83) indeed satisfy the commutation relations of the $su(4)$ algebra

$$[T_a, T^b] = i(\tilde{V}_a^b + \delta_a^b \tilde{V}_c^c), \quad [T_a, \tilde{V}_b^c] = -i\delta_a^b T_b, \quad [T^a, \tilde{V}_b^c] = i\delta_a^b T^c, \quad [\tilde{V}_a^b, \tilde{V}_c^d] = i(\delta_a^b \tilde{V}_a^d - \delta_a^d \tilde{V}_a^b)$$

and commute with the generator $H$ (2.81). Remaining 12 generators of the $so(8)$ algebra that span the $so(8)/su(4) \times u(1)$ quotient algebra equal

$$\tilde{T}_{\hat{a}} = \begin{pmatrix} -\tilde{T}_a \\ \tilde{T}^a \end{pmatrix} = \frac{1}{2} \left( V^7_{\hat{a}} + \frac{1}{2} \mathfrak{J}_{\hat{a}} \hat{b}^c V^8_{\hat{b}} \right), \quad V_a^4, \quad V_4^a. \qquad (2.84)$$
All commutation relations of the so(8) algebra in this basis are given in the Appendix B of our work [22].

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

$$h(d) = \frac{1}{4}(\Omega^{78}(d) + 2\Omega_a^a(d)) \qquad (2.85)$$

and

$$\tilde{\Omega}_a^b(d) = \Omega_a^b(d) - \delta_a^b \Omega_c^c(d) + \delta_a^b h(d). \qquad (2.86)$$

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

$$\Omega_a(d) = \Omega^7_a(d) - \frac{i}{2}\Omega^8_a(d), \quad \Omega^a(d) = -\Omega^7_a(d) - \frac{i}{2}\Omega^8_a(d) \qquad (2.87)$$

that can be presented in the so(6) covariant way

$$\Omega_{\hat{a}}(d) = \begin{pmatrix} \Omega_a(d) \\ -\Omega^a(d) \end{pmatrix} = \Omega^7_{\hat{a}}(d) - \frac{1}{4} \mathfrak{J}_{\hat{a}} \hat{b} \Omega^8_{\hat{b}}(d).$$

Finally Cartan forms associated with the generators $\tilde{T}_a$ and $\tilde{T}^a$ introduced in (2.84)

$$\tilde{\Omega}_a(d) = -\Omega^7_a(d) - \frac{i}{2}\Omega^8_a(d), \quad \tilde{\Omega}^a(d) = \Omega^7_a(d) - \frac{i}{2}\Omega^8_a(d) \qquad (2.88)$$

can also be presented in the form exhibiting the so(6) covariance in the $\mathbf{3} \oplus \mathbf{3}$ basis

$$\tilde{\Omega}_{\hat{a}}(d) = \begin{pmatrix} -\tilde{\Omega}_a(d) \\ \tilde{\Omega}^a(d) \end{pmatrix} = \Omega^7_{\hat{a}}(d) + \frac{1}{4} \mathfrak{J}_{\hat{a}} \hat{b} \Omega^8_{\hat{b}}(d).$$

As a result decomposition (2.79) is presented in equivalent form corresponding to the embedding of the $su(4) \oplus u(1)$ algebra into so(8) compatible with the Hopf fibration realization of $S^7$

$$G_{\text{so}(8)}(d) = \tilde{\Omega}_a^b(d) \tilde{V}_b^a + \tilde{\Omega}_b^b(d) \tilde{V}_a^a + \Omega_a(d) T^a + \Omega^a(d) T_a + h(d) H \\ + \tilde{\Omega}_a(d) \tilde{T}^a + \tilde{\Omega}^a(d) \tilde{T}_a + \Omega_a^4(d) V_4^a + \Omega_4^a(d) V_a^4.$$

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

$$E_a(d) = \Omega^8_a(d) = i(\Omega_a(d) + \tilde{\Omega}_a(d)), \quad E^a(d) = \Omega^{8a}(d) = i(\Omega^a(d) + \tilde{\Omega}^a(d)), \\ E^7(d) = E^{11}(d) = \Omega^{87}(d) = h(d) + \tilde{\Omega}_a^a(d).$$

$G_{32\text{susy}}(d)$ in (2.77) can be presented in the form similar to that for $G_{24\text{susy}}(d)$ from the paragraph 2.1.1

$$F^{\alpha A'}(d) O_{\alpha A'} = F^{\mu A'}(d) O_{\mu A'} + F_{\mu}^{A'}(d) O_{A'}^{\mu} = \omega^{\mu A'}(d) Q_{\mu A'} + \chi_{\mu}^{A'}(d) S_{A'}^{\mu} \\ = \bar{\omega}^{\mu A}(d) \bar{Q}_{\mu A} + \omega_{A}^{\mu}(d) Q_{\mu}^{A} + \bar{\chi}_{\mu}^{A}(d) \bar{S}_{A}^{\mu} + \chi_{\mu A}(d) S^{\mu A}.$$

$D = 11$ supervielbein fermionic components have the following expressions

$$\frac{1}{\sqrt{2}} F^{\mu A'}(d) = \omega^{\mu A'}(d) = (\bar{\omega}^{\mu A}(d), \omega_{A}^{\mu}(d)), \quad \frac{1}{\sqrt{2}} F_{\mu}^{A'}(d) = \chi_{\mu}^{A'}(d) = (\bar{\chi}_{\mu}^{A}(d), \chi_{\mu A}(d)) \qquad (2.89)$$
in terms of the Cartan forms associated with the generators of Poincaré and special conformal supersymmetries

$$O_{\mu A'} = \frac{1}{\sqrt{2}} Q_{\mu A'} = \frac{1}{\sqrt{2}} \begin{pmatrix} \bar{Q}_{\mu A} \\ Q_{\mu}^{A} \end{pmatrix}, \quad O_{A'}^{\mu} = \frac{1}{\sqrt{2}} S_{A'}^{\mu} = \frac{1}{\sqrt{2}} \begin{pmatrix} \bar{S}_{A}^{\mu} \\ S^{\mu A} \end{pmatrix}.$$

### 2.3.2 AdS$_4 \times \mathbb{CP}^3$ superstring in the $\kappa$-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

$$[M^{+-}, Q_{2A'}] = -Q_{2A'}, \quad [M^{+-}, S_{A'}^1] = -S_{A'}^1.$$

$so(1, 1)$ 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

$$[M^{+-}, Q_{1A'}] = Q_{1A'}, \quad [M^{+-}, S_{A'}^2] = S_{A'}^2.$$

For the coordinates associated with these generators let us introduce the following notation

$$\theta^{1A'} = \begin{pmatrix} \bar{\theta}^{-A} \\ \theta_A^{-} \end{pmatrix} = \begin{pmatrix} \bar{\theta}^A \\ \theta_A \end{pmatrix}, \quad \eta_2^{A'} = \begin{pmatrix} \bar{\eta}^{-A} \\ \eta_A^{-} \end{pmatrix} = \begin{pmatrix} \bar{\eta}^A \\ \eta_A \end{pmatrix}. \qquad (2.90)$$

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 $OSp(4|8)/(SO(1, 3) \times SO(7))$ superspace. In the $\kappa$-symmetry light-cone gauge supervielbein components in the tangent space to AdS$_4$ (2.78) reduce to

$$\begin{align}
E^{+}(d) &= \underline{G}^{0^{+}}(d) = \frac{1}{2}e^{-2\varphi}dx^{+}, \nonumber \\
E^{-}(d) &= \underline{G}^{0^{-}}(d) = \frac{1}{2}e^{-2\varphi}dx^{-} + \varpi(d) - 2e^{-2\varphi}\Theta^2 dx^{+} - 4\Theta dy \tag{2.91} \\
E^{1}(d) &= \underline{G}^{0'1}(d) = \frac{1}{2}e^{-2\varphi}dx^{1}, \quad E^{3}(d) = \underline{G}^{0'3}(d) = -d\varphi, \nonumber
\end{align}$$

where $x^{\pm} = x^2 \pm x^0$, $\Theta = \theta_4 \bar{\theta}^4 + \eta_4 \bar{\eta}^4$ and

$$\varpi(d) = ie^{-2\varphi}(d\theta_a \bar{\theta}^a - \theta_a d\bar{\theta}^a) + i(d\theta_4 \bar{\theta}^4 - \theta_4 d\bar{\theta}^4) + ie^{2\varphi}(d\eta_a \bar{\eta}^a - \eta_a d\bar{\eta}^a) + i(d\eta_4 \bar{\eta}^4 - \eta_4 d\bar{\eta}^4). \quad (2.92)$$

The $S^7$ vielbein components acquire the form

$$\begin{align}
\mathbf{E}_a(d) &= \Omega^8_a(d) = i(\Omega_a(d) + \varepsilon_{abc}\hat{\eta}^b\hat{\eta}^c dx^+ - 2e^{-\varphi}\hat{\eta}_a\eta_4 dx^+), \nonumber \\
\mathbf{E}^a(d) &= \Omega^8_a(d) = i(\Omega^a(d) - \varepsilon^{abc}\hat{\eta}_b\hat{\eta}_c dx^+ + 2e^{-\varphi}\hat{\eta}^a\bar{\eta}^4 dx^+), \tag{2.93}
\end{align}$$

$$\mathbf{E}^7 = h(d) + \tilde{\Omega}_a^a(d) = dy + A(d), \quad A(d) = \Omega_{\text{bos}}^a(d) - e^{-2\varphi}\eta_4\bar{\eta}^4 dx^+, \qquad (2.94)$$
where T-transformed coordinates endowed with hats were introduced in (2.67) and Cartan forms $\Omega_a(d)$, $\Omega^a(d)$ and $\tilde{\Omega}_a^a(d)$ in the light-cone gauge equal respective purely bosonic Cartan forms $\Omega_{\text{bos } a}^4(d)$, $\Omega_{\text{bos } 4}^a(d)$ and $\Omega_{\text{bos } a}^a(d)$ presented in Eq. (2.44).

In accordance with the general expression (2.75) let us single out terms proportional to the differential of the coordinate $y$ of the circle $S^1$ in the Cartan forms (2.89) associated with the generators of Poincaré and special conformal supersymmetries

$$ 
\begin{aligned}
\omega_A^\mu(d) &= \tilde{\omega}_A^\mu(d) + dy\omega_{yA}^\mu, & \bar{\omega}^{\mu A}(d) &= \bar{\tilde{\omega}}^{\mu A}(d) + dy\bar{\omega}_y^{\mu A}, \\
\chi_{\mu A}(d) &= \tilde{\chi}_{\mu A}(d) + dy\chi_{y\mu A}, & \bar{\chi}_\mu^A(d) &= \bar{\tilde{\chi}}_\mu^A(d) + dy\bar{\chi}_{y\mu}^A.
\end{aligned}
$$

In the proposed light-cone gauge the $dy$-independent terms in these one-forms equal

$$ \tilde{\omega}_a^\mu(d) = e^{-\varphi} \begin{pmatrix} \hat{d}\theta_a + dx^1 \hat{\eta}_a \\ dx^+ \hat{\eta}_a \end{pmatrix}, \quad \bar{\tilde{\omega}}^{\mu a}(d) = e^{-\varphi} \begin{pmatrix} \hat{d}\bar{\theta}^a + dx^1 \hat{\eta}^a \\ dx^+ \hat{\eta}^a \end{pmatrix}, \quad (2.95) $$

$$ \tilde{\chi}_{\mu a}(d) = \begin{pmatrix} 0 \\ e^{\varphi} \hat{d}\eta_a \end{pmatrix}, \quad \bar{\tilde{\chi}}_{\mu}^{a}(d) = \begin{pmatrix} 0 \\ e^{\varphi} \hat{d}\bar{\eta}^{a} \end{pmatrix}, $$

where $\hat{d}\theta_{\hat{a}} = T_{\hat{a}}^{\hat{b}} d\theta_{\hat{b}}$ and $\hat{d}\eta_{\hat{a}} = T_{\hat{a}}^{\hat{b}} d\eta^{\hat{b}}$, and

$$ \tilde{\omega}_4^\mu(d) = \begin{pmatrix} d\theta_4 + d\varphi\theta_4 + e^{-2\varphi}dx^1\eta_4 \\ e^{-2\varphi}dx^+\eta_4 \end{pmatrix}, \quad \bar{\tilde{\omega}}^{\mu 4}(d) = \begin{pmatrix} d\bar{\theta}^4 + d\varphi\bar{\theta}^4 + e^{-2\varphi}dx^1\bar{\eta}^4 \\ e^{-2\varphi}dx^+\bar{\eta}^4 \end{pmatrix}, \quad (2.96) $$

$$ \tilde{\chi}_{\mu 4}(d) = \begin{pmatrix} 0 \\ d\eta_4 - d\varphi\eta_4 \end{pmatrix}, \quad \bar{\tilde{\chi}}_{\mu}^4(d) = \begin{pmatrix} 0 \\ d\bar{\eta}^4 - d\varphi\bar{\eta}^4 \end{pmatrix} $$

respectively. At the same time spinors proportional to $dy$ have the following components

$$ \omega_{y4}^{\mu} = \begin{pmatrix} 2i\theta_4 \\ 0 \end{pmatrix}, \quad \bar{\omega}_{y}^{\mu 4} = \begin{pmatrix} -2i\bar{\theta}^4 \\ 0 \end{pmatrix}, \quad (2.97) $$

$$ \chi_{y\mu 4} = \begin{pmatrix} 0 \\ 2i\eta_4 \end{pmatrix}, \quad \bar{\chi}_{y\mu}^4 = \begin{pmatrix} 0 \\ -2i\bar{\eta}^4 \end{pmatrix}. \quad (2.98) $$

From expressions (2.91) it follows that for the supercoset element (2.68) $D = 11$ super-vielbein 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 (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

$$ \begin{pmatrix} \mathcal{E}^{m'}(d) \\ \mathcal{E}^{11}(d) \end{pmatrix} = L \begin{pmatrix} \underline{G}^{0'm'}(d) \\ \Omega^{87}(d) \end{pmatrix}. $$

The entries of the matrix $L$

$$ L = \begin{pmatrix} L_{n'}^{m'} & L_{11}^{m'} \\ L_{m'}^{11} & L_{11}^{11} \end{pmatrix} \in SO(1, 4) \quad (2.99) $$
in the light-cone gauge (2.69) equal

$$L^{m'}_{n'} = \delta^{m'}_{n'} - \frac{1}{2} G^{0'm'}_y G^{0'm'}_{yn'}, \quad L^{11}_{11} = -G^{0'm'}_y, \quad L^{11}_{m'} = G^{0'm'}_{ym'}, \quad L^{11}_{11} = 1, \qquad (2.100)$$

where

$$G^{0'm'}_y = \left( \frac{1}{2}(\omega^m_y + c^m_y), 0 \right) = 2\Theta(1, 0, -1, 0).$$

Note that in this gauge the superfield $\Phi_L = \sqrt{1 + G^{0'm'}_y G^{0'm'}_{ym'}}$ equals unity for the chosen normalization.<fnref n="11" /> As a result bosonic components of the transformed $D = 11$ supervielbein equal

$$\begin{align}
\mathcal{E}^+(d) &= \frac{1}{2} e^{-2\varphi} dx^+, \nonumber \\
\mathcal{E}^-(d) &= \frac{1}{2} e^{-2\varphi} dx^- + \varpi(d) - 2e^{-2\varphi} \Theta^2 dx^+ + 4\Theta(\Omega_{\text{bos } a}^a(d) - e^{-2\varphi} \eta_4 \bar{\eta}^4 dx^+), \tag{2.101} \\
\mathcal{E}^1(d) &= \frac{1}{2} e^{-2\varphi} dx^1, \quad \mathcal{E}^3(d) = -d\varphi, \nonumber \\
\mathcal{E}^{11}(d) &= dy + A_L(d), \quad A_L(d) = A(d) - e^{-2\varphi} \Theta dx^+. \nonumber
\end{align}$$

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

$$L^{\hat{\alpha}}_{\hat{\beta}} = \delta^{\hat{\alpha}}_{\hat{\beta}} - \frac{1}{2} G^{0'm'}_y \mathfrak{g}_{m'}^{\hat{\alpha}} \mathfrak{g}^{11\hat{\gamma}}_{\hat{\beta}}.$$

So $(Lf)^{\hat{\alpha}}(d)$ from (2.73) has the following components

$$(L\tilde{\chi})_{\mu a}(d) = \begin{pmatrix} 0 \\ e^{\varphi} \hat{d}\eta_a + 2ie^{-\varphi}\Theta\hat{\eta}_a dx^+ \end{pmatrix}, \quad (L\bar{\tilde{\chi}})^a_{\mu}(d) = \begin{pmatrix} 0 \\ e^{\varphi} \hat{d}\bar{\eta}^a - 2ie^{-\varphi}\Theta\hat{\eta}^a dx^+ \end{pmatrix}$$

and

$$(L\tilde{\chi})_{\mu 4}(d) = \begin{pmatrix} 0 \\ d\eta_4 - d\varphi\eta_4 + 2ie^{-2\varphi}\Theta\eta_4 dx^+ \end{pmatrix}, \quad (L\bar{\tilde{\chi}})^4_{\mu}(d) = \begin{pmatrix} 0 \\ d\bar{\eta}^4 - d\varphi\bar{\eta}^4 - 2ie^{-2\varphi}\Theta\bar{\eta}^4 dx^+ \end{pmatrix}.$$

One-forms given in (2.95), (2.96), as well as components of $F^\hat{\alpha}_y$ (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 world-sheet metric in the gauge (2.69)

$$g^{(2)}_{l.c.ij} = g^{AdS_4}_{l.c.ij} + g^{CP^3}_{l.c.ij}.$$

The form of two summands given by the mappings of the metric tensors of $AdS_4$ and $CP^3$ spaces is

$$\begin{align}
g^{AdS_4}_{l.c.ij} &= \frac{1}{2}(\mathcal{E}_i^+ \mathcal{E}_j^- + \mathcal{E}_j^+ \mathcal{E}_i^-) + \mathcal{E}_i^1 \mathcal{E}_j^1 + \mathcal{E}_i^3 \mathcal{E}_j^3 \nonumber \\
&= \frac{1}{8}e^{-4\varphi}(\partial_i x^+ \partial_j x^- + \partial_j x^+ \partial_i x^-) + \frac{1}{4}e^{-4\varphi}\partial_i x^1 \partial_j x^1 + \partial_i \varphi \partial_j \varphi \tag{2.102} \\
&\quad + \frac{1}{4}e^{-2\varphi}[\partial_i x^+ (\varpi_j + 4\Theta\Omega_{\text{bos } ja}^a) + \partial_j x^+ (\varpi_i + 4\Theta\Omega_{\text{bos } ia}^a)] - 4e^{-4\varphi}\theta_4 \bar{\theta}^4 \eta_4 \bar{\eta}^4 \partial_i x^+ \partial_j x^+ \nonumber
\end{align}$$

<footnote n="11">¹¹To obtain correct value of the dilaton field $\phi$ it is necessary to restore the dependence of $\Phi_L$ on the radius of the seven-sphere $R$, the rank of the orbifold projection $k$ and the Planck length in dimension $D = 11$ $l_P$: $\Phi_L = \frac{R}{kl_P} \sqrt{1 + G^{0'm'}_y G^{0'm'}_{ym'}}$. So in the proposed gauge the dilaton field equals background value $\phi_0 = \frac{3}{2} \ln \frac{R}{kl_P}$ determined by the vacuum solution of the IIA supergravity equations with the $AdS_4 \times CP^3$ geometry.</footnote>
and

$$
\begin{equation}
\begin{aligned}
g_{\text{l.c. ij}}^{\text{CP}^3} = & -\frac{1}{2}(E_{ia}E_j^a + E_{ja}E_i^a) = \frac{1}{2}(\Omega_{\text{bos }ia}{}^4\Omega_{\text{bos }j4}{}^a + \Omega_{\text{bos }ja}{}^4\Omega_{\text{bos }i}{}^a) \\
& + e^{-\varphi}[\partial_i x^+(\Omega_{\text{bos }ja}{}^4\hat{\eta}^a\bar{\eta}^4 - \Omega_{\text{bos }j4}{}^a\hat{\eta}_a\eta_4) + \partial_j x^+(\Omega_{\text{bos }ia}{}^4\hat{\eta}^a\bar{\eta}^4 - \Omega_{\text{bos }i4}{}^a\hat{\eta}_a\eta_4)] \\
& + \frac{1}{2}[\partial_i x^+(\varepsilon_{abc}\Omega_{\text{bos }j4}{}^a\hat{\eta}^b\bar{\eta}^c - \varepsilon^{abc}\Omega_{\text{bos }ja}{}^4\hat{\eta}_b\hat{\eta}_c) + \partial_j x^+(\varepsilon_{abc}\Omega_{\text{bos }i4}{}^a\hat{\eta}^b\bar{\eta}^c - \varepsilon^{abc}\Omega_{\text{bos }ia}{}^4\hat{\eta}_b\hat{\eta}_c)] \\
& + 2[(\hat{\eta}_a\hat{\eta}^a)^2 + e^{-\varphi}(\varepsilon_{abc}\hat{\eta}^a\hat{\eta}^b\bar{\eta}^c\bar{\eta}^4 + \varepsilon^{abc}\hat{\eta}_a\hat{\eta}_b\hat{\eta}_c\eta_4) + 2e^{-2\varphi}\hat{\eta}_a\hat{\eta}^a\eta_4\bar{\eta}^4]\partial_i x^+ \partial_j x^+.
\end{aligned}
\tag{2.103}
\end{equation}
$$

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 $\kappa$-symmetry light-cone gauge equals

$$
\begin{equation}
\begin{split}
H_{(3)\text{l.c.}}(d) ={}& \frac{i}{4} (\mathcal{E}^{\hat{a}}(d) \mathfrak{g}^{\hat{m}'\hat{n}'}_{\quad \hat{a}\hat{\beta}} \chi_{\hat{\beta}} \wedge \mathcal{E}_{\hat{m}'}(d) \wedge \mathcal{E}_{\hat{n}'}(d) + \mathcal{E}^{\hat{a}}(d) \mathfrak{g}^{\hat{m}'11}_{\quad \hat{a}\hat{\beta}} \wedge \mathcal{E}_{\hat{\beta}}(d) \wedge \mathcal{E}_{\hat{m}'}(d)) \\
& - \varepsilon_{m'n'k'l'} \mathcal{E}^{m'}(d) \wedge \mathcal{E}^{n'}(d) \wedge \mathcal{E}^{k'}(d) L^{l'}_{11}.
\end{split}
\tag{2.104}
\end{equation}
$$

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

$$
\begin{equation}
\begin{aligned}
B_{(2)\text{l.c.}}(d) ={}& \frac{1}{2}e^{-4\varphi}(\theta_4\bar{\theta}^4 + \eta_4\bar{\eta}^4)dx^1 \wedge dx^+ + e^{-2\varphi}\hat{\eta}_a\hat{\eta}^a dx^1 \wedge dx^+ \\
& + \frac{1}{4}e^{-2\varphi}(d\theta_4\bar{\eta}^4 - d\eta_4\bar{\theta}^4 + \eta_4 d\bar{\theta}^4 - \theta_4 d\bar{\eta}^4) \wedge dx^+ \\
& + \frac{1}{2}e^{-2\varphi}(\hat{\eta}_a \hat{d}\bar{\theta}^a + \hat{d}\theta_a \hat{\eta}^a) \wedge dx^+ + ie^{-2\varphi}(\theta_4\bar{\eta}^4 - \eta_4\bar{\theta}^4)dx^+ \wedge \Omega_{\text{bos }a}{}^a(d) \\
& + ie^{-\varphi}\hat{\eta}_a\theta_4 dx^+ \wedge \Omega_{\text{bos }4}{}^a(d) + ie^{-\varphi}\hat{\eta}^a\bar{\theta}^4 dx^+ \wedge \Omega_{\text{bos }a}{}^4(d).
\end{aligned}
\tag{2.105}
\end{equation}
$$

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 $AdS_4 \times \mathbb{CP}^3$ superstring in the gauge (2.69)

$$S_{\text{sstring, l.c.}}^{\text{AdS}_4 \times \text{CP}^3} = \int_{\Sigma} d^2 \xi \mathcal{L}_{\text{sstring, l.c.}}^{\text{AdS}_4 \times \text{CP}^3}$$

with the Lagrangian

$$
\mathcal{L}_{\text{sstring, l.c.}}^{\text{AdS}_4 \times \text{CP}^3} = -\frac{1}{2} \sqrt{-\gamma \gamma^{ij}} g_{\text{l.c.ij}}^{(2)} + s \varepsilon^{ij} B_{(2)\text{l.c.ij}}. \quad (2.106)
$$

From (2.102), (2.103) and (2.105) it follows that kinetic term of the resulting Lagrangian includes contributions up to the fourth order in the superspace Grassmann coordinates (2.90) and the Wess-Zumino term – up to the second order similarly to the $AdS_5 \times S^5$ superstring in the analogous light-cone gauge [40], [41].

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. [40], [41], [49]). Respective gauge conditions will be considered in the next subsection.
## 2.4 Light-cone gauge Hamiltonian of the $AdS_4 \times CP^3$ superstring

In this subsection the $AdS_4 \times CP^3$ superstring in the $\kappa$-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 respective gauge conditions in the superstring models in flat and $AdS_5 \times S^5$ superbackgrounds [50], [40], [41], [51]. 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 [52]. At the same time the Hamiltonian of the $AdS_4 \times CP^3$ superstring in another light-cone gauge obtained in [24], [25], [26] in this limit transfers to the Hamiltonian of the Type IIA superstring in the background that is the $pp$-wave limit [53] of the $AdS_4 \times CP^3$ superbackground and that was described in [54], [23]. 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. [55]) for the $AdS \times S$ space-times, and also for the $AdS_4 \times CP^3$ 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 $\kappa$-symmetry become quadratic and these theories can be quantized [52], [56], [54]. 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 [57].

### 2.4.1 Formulation of the $AdS_4 \times CP^3$ superstring Lagrangian in the $\kappa$-symmetry light-cone gauge in terms of the phase-space variables

To introduce momentum densities conjugate to bosonic coordinates we present the $AdS_4 \times CP^3$ superstring Lagrangian in the $\kappa$-symmetry light-cone gauge (2.69) in the form<fnref n="12" />

$$ \begin{aligned} \mathcal{L}_{\text{sstring, l.c.}}^{AdS_4 \times CP^3} = & -\frac{T}{2}\gamma^{ij} \left\{ \frac{e^{-4\varphi}}{4} \left[ \frac{1}{2}(\partial_i x^+ \partial_j x^- + \partial_j x^+ \partial_i x^-) + \partial_i x^1 \partial_j x^1 \right] \right. \\ & \left. + \partial_i \varphi \partial_j \varphi + g_{MN} \partial_i z^M \partial_j z^N + B \partial_i x^+ \partial_j x^+ \right. \\ & \left. + \frac{e^{-2\varphi}}{4}(\partial_i x^+ \varpi_j + \partial_j x^+ \varpi_i) + (\partial_i x^+ \partial_j z^M + \partial_j x^+ \partial_i z^M) q_M \right\} \\ & + T \varepsilon^{ij} (\tilde{\omega}_i \partial_j x^+ + C \partial_i x^1 \partial_j x^+ + \partial_i x^+ \partial_j z^M \tilde{q}_M), \end{aligned} \quad (2.107) $$

<footnote n="12">¹²Given expression corresponds to the value $s = 1$ of the sign factor in (2.106).</footnote>
in which singled out are the derivatives of these coordinates. Here $z^M$ ($M = 1, ..., 6$) are real coordinates of the $\mathbb{CP}^3$ manifold and form of the metric $g_{MN}(z)$ will be specified later. To simplify further consideration in (2.107) there were introduced the following quantities

$$ \begin{aligned} \tilde{\omega}_i &= \frac{e^{-2\varphi}}{2}(\hat{\eta}_a \hat{\partial}_i \bar{\theta}^a + \hat{\partial}_i \theta_a \hat{\eta}^a) + \frac{e^{-2\varphi}}{4}(\partial_i \theta_4 \bar{\eta}^4 - \partial_i \eta_4 \bar{\theta}^4 + \eta_4 \partial_i \bar{\theta}^4 - \theta_4 \partial_i \bar{\eta}^4), \\ B &= 2[(\hat{\eta}_a \hat{\eta}^a)^2 + e^{-\varphi}(\varepsilon_{abc}\hat{\eta}^a \hat{\bar{\eta}}^b \hat{\bar{\eta}}^c \bar{\eta}^4 + \varepsilon^{abc}\hat{\eta}_a \hat{\eta}_b \hat{\eta}_c \eta_4) + 2e^{-2\varphi}\eta_4 \bar{\eta}^4(\hat{\eta}_a \hat{\eta}^a - e^{-2\varphi}\theta_4 \bar{\theta}^4)], \\ C &= e^{-2\varphi}(\hat{\eta}_a \hat{\bar{\eta}}^a + \frac{e^{-2\varphi}}{2}\Theta), \quad \Theta = \theta_4 \bar{\theta}^4 + \eta_4 \bar{\eta}^4, \\ q_M &= \frac{1}{2}(\Omega_M^a \varepsilon_{abc}\hat{\eta}^b \hat{\bar{\eta}}^c - \Omega_{aM} \varepsilon^{abc}\hat{\eta}_b \hat{\eta}_c) + e^{-\varphi}(\Omega_{aM} \hat{\eta}^a \bar{\eta}^4 - \Omega_M^a \hat{\eta}_a \eta_4) + e^{-2\varphi}\Theta \tilde{\Omega}_a^a M, \\ \tilde{q}_M &= ie^{-\varphi} \left[ \Omega_{aM} \hat{\eta}^a \bar{\theta}^4 + \Omega_M^a \hat{\eta}_a \theta_4 + e^{-\varphi}(\theta_4 \bar{\eta}^4 - \eta_4 \bar{\theta}^4) \tilde{\Omega}_a^a M \right] \end{aligned} $$

and one-form $\varpi(d)$ 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

$$ \begin{aligned} p_{-}(\tau, \sigma) &= -\frac{T e^{-4\varphi}}{8} \gamma^{\tau i} \partial_i x^+, \\ p_{+}(\tau, \sigma) &= -T \gamma^{\tau i} \left( \frac{e^{-4\varphi}}{8} \partial_i x^- + B \partial_i x^+ + q_M \partial_i z^M + \frac{e^{-2\varphi}}{4} \varpi_i \right) - T(C \partial_\sigma x^1 - \tilde{q}_M \partial_\sigma z^M + \tilde{\omega}_\sigma), \quad (2.108) \\ p_1(\tau, \sigma) &= -\frac{T e^{-4\varphi}}{4} \gamma^{\tau i} \partial_i x^+ + T C \partial_\sigma x^+, \\ p_\varphi(\tau, \sigma) &= -T \gamma^{\tau i} \partial_i \varphi. \end{aligned} $$

At the same time for momentum density components conjugate to the $\mathbb{CP}^3$ coordinates we obtain the following expression

$$ p_M(\tau, \sigma) = -T \gamma^{\tau i} (g_{MN} \partial_i z^N + q_M \partial_i x^+) - T \tilde{q}_M \partial_\sigma x^+. \quad (2.109) $$

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

$$ \begin{aligned} \partial_{\tau} x^{+} &= -\frac{8e^{4\varphi}}{T\gamma^{\tau\tau}} p_{-} - \frac{\gamma^{\tau\sigma}}{\gamma^{\tau\tau}} \partial_{\sigma} x^{+}, \\ \partial_{\tau} x^{-} &= -2e^{2\varphi} \varpi_{\tau} + \frac{1}{\gamma^{\tau\tau}} A^{-} - \frac{\gamma^{\tau\sigma}}{\gamma^{\tau\tau}} (\partial_{\sigma} x^{-} + 2e^{2\varphi} \varpi_{\sigma}), \\ \partial_{\tau} x^{1} &= -\frac{4e^{4\varphi}}{T\gamma^{\tau\tau}} (p_{1} - T C \partial_{\sigma} x^{+}) - \frac{\gamma^{\tau\sigma}}{\gamma^{\tau\tau}} \partial_{\sigma} x^{1}, \quad (2.110) \\ \partial_{\tau} \varphi &= -\frac{1}{T\gamma^{\tau\tau}} p_{\varphi} - \frac{\gamma^{\tau\sigma}}{\gamma^{\tau\tau}} \partial_{\sigma} \varphi, \\ \partial_{\tau} z^{M} &= -\frac{g^{MN}}{T\gamma^{\tau\tau}} (p_{N} - 8e^{4\varphi} p_{-} q_{N} + T \tilde{q}_{N} \partial_{\sigma} x^{+}) - \frac{\gamma^{\tau\sigma}}{\gamma^{\tau\tau}} \partial_{\sigma} z^{M}, \end{aligned} $$

where

$$ A^{-} = \frac{8e^{4\varphi}}{T} (p_{+} - 8e^{4\varphi} p_{-} (B - (q \cdot q)) - (p \cdot q)) + 8e^{4\varphi} ((q \cdot \tilde{q}) \partial_{\sigma} x^{+} - C \partial_{\sigma} x^{1} + \tilde{q}_{M} \partial_{\sigma} z^{M} - \tilde{\omega}_{\sigma}). \quad (2.111) $$
In (2.111) and in what follows used is the short-hand notation for the scalar product of $D = 6$ vectors with inverse $\mathbb{CP}^3$ metric $g^{MN}(z)$: $q_M g^{MN} q_N = (q \cdot q)$ etc. From definitions of the momentum densities (2.108) and (2.109) there follow the Virasoro constraints

$$ 
\begin{aligned}
T_1 ={}& \frac{8e^{4\varphi}}{T} p_+ p_- + \frac{T e^{-4\varphi}}{8} \partial_\sigma x^+ \partial_\sigma x^- + \frac{2e^{4\varphi}}{T} p_1^2 + \frac{T e^{-4\varphi}}{8} \partial_\sigma x^1 \partial_\sigma x^1 \\
& + \frac{1}{2T} p_\varphi^2 + \frac{T}{2} \partial_\sigma \varphi \partial_\sigma \varphi + \frac{1}{2T} (p \cdot p) + \frac{T}{2} \partial_\sigma z^M g_{MN} \partial_\sigma z^N \\
& + \frac{32e^{8\varphi}}{T} p_-^2 ((q \cdot q) - B) + 8e^{4\varphi} p_- (\tilde{\omega}_\sigma - \frac{1}{T}(p \cdot q) - (q \cdot \tilde{q}) \partial_\sigma x^+ \\
& + C \partial_\sigma x^1 - \tilde{q}_M \partial_\sigma z^M) - (\frac{T e^{-2\varphi}}{2} \tilde{\omega}_\sigma - T q_M \partial_\sigma z^M - (p \cdot \tilde{q}) \\
& + 4e^{4\varphi} p_1 C) \partial_\sigma x^+ + \frac{T}{2} ((\tilde{q} \cdot \tilde{q}) + B + 4e^{4\varphi} C^2) \partial_\sigma x^+ \partial_\sigma x^+ \approx 0,
\end{aligned}
$$

$$ T_2 = p_+ \partial_\sigma x^+ + p_- \partial_\sigma x^- + p_1 \partial_\sigma x^1 + p_\varphi \partial_\sigma \varphi + p_M \partial_\sigma z^M + 2e^{2\varphi} p_- \tilde{\omega}_\sigma + T \tilde{\omega}_\sigma \partial_\sigma x^+ \approx 0. $$

Substituting expressions for the velocities (2.110) back into (2.107) we obtain the representation for the Lagrangian of the $AdS_4 \times \mathbb{CP}^3$ superstring in the $\kappa$-symmetry light-cone gauge in terms of the phase-space variables

$$ 
\begin{aligned}
\mathcal{L}_{\text{sstring, l.c.}}^{AdS_4 \times \mathbb{CP}^3} ={}& p_+ \partial_\tau x^+ + p_- \partial_\tau x^- + p_1 \partial_\tau x^1 + p_\varphi \partial_\tau \varphi + p_M \partial_\tau z^M \\
& + 2e^{2\varphi} p_- \tilde{\omega}_\tau + T \tilde{\omega}_\tau \partial_\sigma x^+ + \frac{1}{\gamma^{\tau\tau}} T_1 + \frac{\gamma^{\tau\sigma}}{\gamma^{\tau\tau}} T_2.
\end{aligned}
\qquad (2.112) $$

Not that in (2.112) components of the inverse auxiliary metric $\frac{1}{\gamma^{\tau\tau}}$ and $\frac{\gamma^{\tau\sigma}}{\gamma^{\tau\tau}}$ play the role of the Lagrange multipliers for the Virasoro constraints.

## 2.4.2 Light-cone gauge for reparametrization symmetry and the Hamiltonian of the $AdS_4 \times \mathbb{CP}^3$ superstring

Let us fix the light-cone gauge for the reparametrization symmetry by the conditions analogous to those considered in Refs. [50], [40], [41]

$$ x^+(\tau, \sigma) = \tau, \quad p_-(\tau, \sigma) = \frac{1}{2} \tilde{P}_- = \text{const.} $$

Using the equation for $p_+$ it is possible to express $\gamma^{\tau\tau}$ through $\tilde{P}_-$

$$ \gamma^{\tau\tau} = -\frac{4e^{4\varphi}}{T} \tilde{P}_-. $$

Besides in order to bring in (2.112) the kinetic term for fermions $2e^{2\varphi} p_- \tilde{\omega}_\tau$ to the simplest form let us perform the dilatation of the Grassmann coordinates

$$ 
\begin{aligned}
\theta_a(\bar{\theta}^a) \rightarrow \frac{1}{\sqrt{\tilde{P}_-}} \theta_a(\bar{\theta}^a), \quad \eta_a(\bar{\eta}^a) \rightarrow \frac{\exp\left(-\frac{2\varphi}{\sqrt{\tilde{P}_-}}\right)}{\sqrt{\tilde{P}_-}} \eta_a(\bar{\eta}^a), \\
\theta_4(\bar{\theta}^4) \rightarrow \frac{\exp\left(-\frac{\varphi}{\sqrt{\tilde{P}_-}}\right)}{\sqrt{\tilde{P}_-}} \theta_4(\bar{\theta}^4), \quad \eta_4(\bar{\eta}^4) \rightarrow \frac{\exp\left(-\frac{\varphi}{\sqrt{\tilde{P}_-}}\right)}{\sqrt{\tilde{P}_-}} \eta_4(\bar{\eta}^4)
\end{aligned}
\qquad (2.113) $$

together with canonical transformation of the transverse bosonic phase-space variables

$$ 
\begin{aligned}
x^1 \rightarrow \frac{1}{\sqrt{\tilde{P}_-}} x^1, \quad \varphi \rightarrow \frac{1}{\sqrt{\tilde{P}_-}} \varphi, \quad z^M \rightarrow \frac{1}{\sqrt{\tilde{P}_-}} z^M, \\
p_1 \rightarrow \sqrt{\tilde{P}_-} p_1, \quad p_\varphi \rightarrow \sqrt{\tilde{P}_-} p_\varphi, \quad p_M \rightarrow \sqrt{\tilde{P}_-} p_M.
\end{aligned}
\qquad (2.114) $$
Such redefinition, as will be demonstrated below, allows to perform the power-series expansion of the Hamiltonian in the large $\tilde{P}_-$ limit. Then the Lagrangian (2.112) takes the form

$$ \mathcal{L}_{\text{sstring, l.c.}}^{\text{AdS}_4 \times \text{CP}^3} = p_1 \partial_{\tau} x^1 + p_{\varphi} \partial_{\tau} \varphi + p_M \partial_{\tau} z^M + i (\partial_{\tau} \theta_A \bar{\theta}^A - \theta_A \partial_{\tau} \bar{\theta}^A + \partial_{\tau} \eta_A \bar{\eta}^A - \eta_A \partial_{\tau} \bar{\eta}^A) - \mathcal{H}_{\text{l.c.}}, $$

where the density of the light-cone gauge Hamiltonian contains terms up to the fourth power in the Grassmann coordinates

$$ \begin{aligned} \mathcal{H}_{\text{sstring, l.c.}}^{\text{AdS}_4 \times \text{CP}^3}(\tau, \sigma) ={}& \frac{\tilde{T}}{2} \exp\left(-\frac{4\varphi}{\sqrt{\tilde{P}_-}}\right) \left[\hat{\eta}_a \hat{\partial}_\sigma \bar{\theta}^a + \hat{\partial}_\sigma \theta_a \hat{\eta}^a + \frac{1}{2}(\partial_\sigma \theta_4 \bar{\eta}^4 \right. \\ & \left. - \theta_4 \partial_\sigma \bar{\eta}^4 - \partial_\sigma \eta_4 \bar{\theta}^4 + \eta_4 \partial_\sigma \bar{\theta}^4)\right] + \frac{\tilde{T}}{4} \exp\left(-\frac{4\varphi}{\sqrt{\tilde{P}_-}}\right) \left[\frac{2}{\tilde{T}} \exp\left(\frac{4\varphi}{\sqrt{\tilde{P}_-}}\right) p_1^2 \right. \\ & \left. + \frac{\tilde{T}}{8} \exp\left(-\frac{4\varphi}{\sqrt{\tilde{P}_-}}\right) \partial_\sigma x^1 \partial_\sigma x^1 + \frac{1}{2\tilde{T}} p_\varphi^2 + \frac{\tilde{T}}{2} \partial_\sigma \varphi \partial_\sigma \varphi + \frac{1}{2\tilde{T}} (p \cdot p) \right. \\ & \left. + \frac{\tilde{T}}{2} \partial_\sigma z^M g_{MN} \partial_\sigma z^N + \frac{8\tilde{P}_-}{\tilde{T}} \exp\left(\frac{8\varphi}{\sqrt{\tilde{P}_-}}\right) ((q \cdot q) - B) \right. \\ & \left. + 4\sqrt{\tilde{P}_-} \exp\left(\frac{4\varphi}{\sqrt{\tilde{P}_-}}\right) \left(C \partial_\sigma x^1 - \tilde{q}_M \partial_\sigma z^M - \frac{1}{\tilde{T}}(p \cdot q)\right) \right] \end{aligned} \quad (2.115) $$

and $\tilde{T} = \frac{T}{\tilde{P}_-}$. As usual in the (super)string theory in the light-cone gauge the constraint $T_2 \approx 0$ is considered to be explicitly solved with respect to derivative $\partial_\sigma x^-$ 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

$$ \int_{0}^{2\pi} d\sigma [p_1 \partial_{\sigma} x^1 + p_{\varphi} \partial_{\sigma} \varphi + p_M \partial_{\sigma} z^M + i(\partial_{\sigma} \theta_A \bar{\theta}^A - \theta_A \partial_{\sigma} \bar{\theta}^A + \partial_{\sigma} \eta_A \bar{\eta}^A - \eta_A \partial_{\sigma} \bar{\eta}^A)] = 0. $$

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. [49] 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 $\tilde{P}_-$ and $T$ limit with $\tilde{T}$ 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 $\mathbb{CP}^3$ 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 $\bar{z}_a$. So the complex vielbein and the $u(3)$ connection of the $\mathbb{CP}^3$ space are given by the Cartan one-forms (2.44) and the square of the infinitesimal length element equals

$$ ds_{\mathbb{CP}^3}^2 = g_{ab} dz^a dz^b + g^{ab} d\bar{z}_a d\bar{z}_b + 2g_a^b dz^a d\bar{z}_b $$

where the metric components have the form

$$ \begin{aligned} g_{ab} ={}& \frac{1}{4|z|^4} (|z|^2 - \sin^2|z| + \sin^4|z|) \bar{z}_a \bar{z}_b, \quad g^{ab} = \frac{1}{4|z|^4} (|z|^2 - \sin^2|z| + \sin^4|z|) z^a z^b, \\ g_a^b ={}& \frac{\sin^2|z|}{2|z|^2} \delta_a^b + \frac{1}{4|z|^4} (|z|^2 - \sin^2|z| - \sin^4|z|) \bar{z}_a z^b. \end{aligned} \quad (2.116) $$
Components of the inverse metric are defined by the expressions

$$ 
\begin{aligned}
g_{ab}^{-1} &= \frac{|z|^2 - \sin^2|z| + \sin^4|z|}{|z|^2(\sin^4|z| - \sin^2|z|)} \bar{z}_a \bar{z}_b, & g^{-1ab} &= \frac{|z|^2 - \sin^2|z| + \sin^4|z|}{|z|^2(\sin^4|z| - \sin^2|z|)} z^a z^b, \\
g^{-1 a}{}^b &= \frac{2|z|^2}{\sin^2|z|} \delta_a^b + \frac{\sin^2|z|(1+2|z|^2) - \sin^4|z|-|z|^2}{|z|^2(\sin^2|z| - \sin^4|z|)} \bar{z}_a z^b.
\end{aligned} 
\quad (2.117) 
$$

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

$$ 
\begin{aligned}
g_{ab} &= \left(\frac{1}{3} - \frac{8}{45}|z|^2\right) \bar{z}_a \bar{z}_b + O(z^6), & g^{ab} &= \left(\frac{1}{3} - \frac{8}{45}|z|^2\right) z^a z^b + O(z^6), \\
g_a{}^b &= \left(\frac{1}{2} - \frac{1}{6}|z|^2 + \frac{1}{45}|z|^4\right) \delta_a^b - \bar{z}_a z^b \left(\frac{1}{6} - \frac{7}{45}|z|^2\right) + O(z^6)
\end{aligned} 
$$

and

$$ 
\begin{aligned}
g^{-1ab} &= -\left(\frac{4}{3} + \frac{48}{45}|z|^2\right) z^a z^b + O(z^6), & g^{-1}_{ab} &= -\left(\frac{4}{3} + \frac{48}{45}|z|^2\right) \bar{z}_a \bar{z}_b + O(z^6), \\
g^{-1 a}{}^b &= 2\left(1 + \frac{1}{3}|z|^2 + \frac{1}{15}|z|^4\right) \delta_a^b + 2\bar{z}_a z^b \left(\frac{1}{3} + \frac{21}{45}|z|^2\right) + O(z^6).
\end{aligned} 
$$

Analogously one can obtain expansion of the components of the SU(4)/U(3) coset element
(2.43)

$$ 
\begin{aligned}
T_a{}^b &= T_a{}^b = \left(1 - \frac{1}{2}|z|^2 + \frac{1}{24}|z|^4\right) \delta_a^b + \frac{1}{2}\bar{z}_a z^b \left(1 - \frac{1}{12}|z|^2\right) + O(z^6), \\
T_{ab} &= i\epsilon_{acb} z^c \left(1 - \frac{1}{6}|z|^2\right) + O(z^5), \quad T^{ab} = -i\epsilon^{acb} \bar{z}_c \left(1 - \frac{1}{6}|z|^2\right) + O(z^5)
\end{aligned} 
$$

and the CP³ complex vielbein in (2.93) and bosonic part of the Ramond-Ramond one-form
in (2.94)

$$ 
\begin{aligned}
\Omega_a(d) &= \Omega_{a,b} dz^b + \Omega_{a}{}^b d\bar{z}_b : & \Omega_{a,b} &= \bar{z}_a \bar{z}_b \left(\frac{1}{4} + \frac{1}{6}|z| - \frac{1}{16}|z|^2\right) + O(z^6), \\
\Omega_{a}{}^b &= \left(1 - \frac{1}{6}|z|^2 + \frac{1}{120}|z|^4\right) \delta_a^b - \bar{z}_a z^b \left(\frac{1}{4} - \frac{1}{6}|z| - \frac{1}{16}|z|^2\right) + O(z^6); \\
\Omega^a(d) &= \Omega^a{}_{,b} dz^b + \Omega^a{}_{,b} d\bar{z}_b : & \Omega^a{}_{,b} &= \Omega^a{}_{,b}, \\
\Omega^{a,b} &= z^a z^b \left(\frac{1}{4} + \frac{1}{6}|z| - \frac{1}{16}|z|^2\right) + O(z^6); \\
\tilde{\Omega}_a{}^a(d) &= \tilde{\Omega}_a{}^a{}_{,b} dz^b + \tilde{\Omega}_a{}^a{}_{,b} d\bar{z}_b : \\
\tilde{\Omega}_a{}^a{}_{,b} &= \frac{i}{2}\bar{z}_b \left(1 - \frac{1}{3}|z|^2\right) + O(z^5), \quad \tilde{\Omega}_a{}^a{}_{,b} &= -\frac{i}{2}z^b \left(1 - \frac{1}{3}|z|^2\right) + O(z^5).
\end{aligned} 
$$

Taking into account above expansions quadratic part of the superstring Hamiltonian in
the light-cone gauge (2.115) is found to be

$$ \mathcal{H}_{1.c.}^{(2)} = \mathcal{H}_{b}^{(2)} + \mathcal{H}_{f}^{(2)}, $$

where contributions of the bosonic and fermionic fields equal

$$ 
\begin{aligned}
\mathcal{H}_{b}^{(2)} &= \frac{1}{2}(p_1^2 + \frac{\tilde{T}^2}{16}\partial_{\sigma}x^1\partial_{\sigma}x^1) + \frac{1}{8}(p_{\varphi}^2 + \tilde{T}^2\partial_{\sigma}\varphi\partial_{\sigma}\varphi) + \frac{1}{2}(p_a\bar{p}^a + \frac{\tilde{T}^2}{4}\partial_{\sigma}z^a\partial_{\sigma}\bar{z}_a), \\
\mathcal{H}_{f}^{(2)} &= \frac{\tilde{T}}{2}(\eta_{a}\partial_{\sigma}\bar{\theta}^a + \partial_{\sigma}\theta_{a}\bar{\eta}^a) + \frac{\tilde{T}}{4}(\partial_{\sigma}\theta_{4}\bar{\eta}^4 - \partial_{\sigma}\eta_{4}\bar{\theta}^4 + \eta_{4}\partial_{\sigma}\bar{\theta}^4 - \theta_{4}\partial_{\sigma}\bar{\eta}^4).
\end{aligned} 
$$

Respective quadratic Lagrangian of the superstring in the light-cone gauge has the form

$$ 
\begin{aligned}
\mathcal{L}_{\text{l.c.}}^{(2)} &= \frac{1}{2}(\partial_{\tau}x^1\partial_{\tau}x^1 - \frac{\tilde{T}^2}{16}\partial_{\sigma}x^1\partial_{\sigma}x^1) \\
&\quad +2(\partial_{\tau}\varphi\partial_{\tau}\varphi - \frac{\tilde{T}^2}{16}\partial_{\sigma}\varphi\partial_{\sigma}\varphi) + 2(\partial_{\tau}z^a\partial_{\tau}\bar{z}_a - \frac{\tilde{T}^2}{16}\partial_{\sigma}z^a\partial_{\sigma}\bar{z}_a) \\
&\quad -i(\partial_{\tau}\theta_A\bar{\theta}^A - \theta_A\partial_{\tau}\bar{\theta}^A + \partial_{\tau}\eta_A\bar{\eta}^A - \eta_A\partial_{\tau}\bar{\eta}^A) - \frac{\tilde{T}}{2}(\eta_A\partial_{\sigma}\bar{\theta}^A + \partial_{\sigma}\theta_A\bar{\eta}^A).
\end{aligned} 
\quad (2.118) 
$$
Terms including bosonic fields coincide with those in the (super)string Lagrangian in Minkowski (super)space in the light-cone gauge [52]. 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

$$ \Psi_{A'}(\tau, \sigma) = -C_{A'B'}\theta^{B'} + \gamma_{A'B'}^7\eta^{B'}, \quad \bar{\Psi}_{A'} = (\Psi_{A'})^{\dagger}, $$

$$ \Phi_{A'}(\tau, \sigma) = -C_{A'B'}\theta^{B'} - \gamma_{A'B'}^7\eta^{B'}, \quad \bar{\Phi}_{A'} = (\Phi_{A'})^{\dagger}, $$

where $\theta^{B'}$ and $\eta^{B'}$ were introduced in (2.90) and $D = 7$ charge conjugation and $\gamma$-matrices are defined in the Appendix A of our work [22]. Introduced spinor fields satisfy the conditions $\bar{\Psi}^{A'} = -C^{A'B'}\Psi_{B'}$ and $\bar{\Phi}^{A'} = -C^{A'B'}\Phi_{B'}$ 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

$$ -\frac{i}{2}\bar{\Psi}^{A'}\partial_{-}\Psi_{A'} - \frac{i}{2}\bar{\Phi}^{A'}\partial_{+}\Phi_{A'}, $$

where $\partial_{\pm} \equiv \partial_{\tau} \pm \frac{\tilde{T}}{4}\partial_{\sigma}$. Thus we have established that quadratic Lagrangian

$$ \mathcal{L}_{\text{l.c.}}^{(2)} = \frac{1}{2}(\partial_{\tau}x^1\partial_{\tau}x^1 - \frac{\tilde{T}^2}{16}\partial_{\sigma}x^1\partial_{\sigma}x^1) + 2(\partial_{\tau}\varphi\partial_{\tau}\varphi - \frac{\tilde{T}^2}{16}\partial_{\sigma}\varphi\partial_{\sigma}\varphi) \\ +2(\partial_{\tau}z^a\partial_{\tau}\bar{z}_a - \frac{\tilde{T}^2}{16}\partial_{\sigma}z^a\partial_{\sigma}\bar{z}_a) + \frac{i}{2}\Psi^{A'}\partial_{-}\Psi_{A'} + \frac{i}{2}\Phi^{A'}\partial_{+}\Phi_{A'} $$

coincides with the superstring Lagrangian in flat superspace in the light-cone gauge [52].

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

$$ \sqrt{\tilde{P}}\mathcal{H}_{\text{l.c.}}^{(3)} = \mathcal{H}_{\text{b}}^{(3)} + \mathcal{H}_{\text{bf}}^{(3)}, $$

where

$$ \mathcal{H}_{\text{b}}^{(3)} = -\frac{\tilde{T}^2}{4}\varphi\partial_{\sigma}x^1\partial_{\sigma}x^1 - \frac{1}{2}\varphi(p_{\varphi}^2 + \tilde{T}^2\partial_{\sigma}\varphi\partial_{\sigma}\varphi) - 2\varphi(p_a\bar{p}^a + \frac{\tilde{T}^2}{4}\partial_{\sigma}z^a\partial_{\sigma}\bar{z}_a), $$

$$ \mathcal{H}_{\text{bf}}^{(3)} = -2\tilde{T}\varphi(\eta_a\partial_{\sigma}\bar{\theta}^a + \partial_{\sigma}\theta_a\bar{\eta}^a) - i\tilde{T}(\varepsilon^{abc}\eta_a\bar{z}_b\partial_{\sigma}\theta_c + \varepsilon_{abc}\bar{\eta}^a z^b\partial_{\sigma}\bar{\theta}^c) $$

$$ -\tilde{T}\varphi(\partial_{\sigma}\theta_4\bar{\eta}^4 - \partial_{\sigma}\eta_4\bar{\theta}^4 + \eta_4\partial_{\sigma}\bar{\theta}^4 - \theta_4\partial_{\sigma}\bar{\eta}^4) - 2(p_a\bar{\eta}^a\bar{\eta}^4 - \bar{p}^a\eta_a\eta_4) $$

$$ +\varepsilon^{abc}p_a\eta_b\eta_c - \varepsilon_{abc}\bar{p}^a\bar{\eta}^b\bar{\eta}^c + 2\tilde{T}(\eta_a\bar{\eta}^a + \frac{1}{2}\Theta)\partial_{\sigma}x^1 - 2i\tilde{T}(\partial_{\sigma}z^a\eta_a\theta_4 + \partial_{\sigma}\bar{z}_a\bar{\eta}^a\bar{\theta}^4). $$

Quartic-order terms can be presented as the sum of three contributions

$$ \tilde{P}_{-}\mathcal{H}_{\text{l.c.}}^{(4)} = \mathcal{H}_{\text{b}}^{(4)} + \mathcal{H}_{\text{bf}}^{(4)} + \mathcal{H}_{\text{f}}^{(4)}, $$

where

$$ \mathcal{H}_{\text{b}}^{(4)} = \tilde{T}^2\varphi^2\partial_{\sigma}x^1\partial_{\sigma}x^1 + \varphi^2(p_{\varphi}^2 + \tilde{T}^2\partial_{\sigma}\varphi\partial_{\sigma}\varphi) + 4\varphi^2(p_a\bar{p}^a + \frac{\tilde{T}^2}{4}\partial_{\sigma}z^a\partial_{\sigma}\bar{z}_a) $$

$$ -\frac{1}{6}((z^a p_a)^2 + (\bar{p}^a \bar{z}_a)^2 - |z|^2(\bar{p}^a p_a) - (z^a p_a)(\bar{p}^b \bar{z}_b)) $$

$$ +\frac{\tilde{T}^2}{24}((\partial_{\sigma}z^a\bar{z}_a)^2 + (z^a\partial_{\sigma}\bar{z}_a)^2 - |z|^2(\partial_{\sigma}z^a\partial_{\sigma}\bar{z}_a) - (\partial_{\sigma}z^a\bar{z}_a)(z^b\partial_{\sigma}\bar{z}_b)), $$
$$
\begin{aligned}
\mathcal{H}_{\text{bf}}^{(4)} ={}& 4\tilde{T}\varphi^2(\eta_a\partial_\sigma\bar{\theta}^a + \partial_\sigma\theta_a\bar{\eta}^a) + 4i\tilde{T}\varphi(\varepsilon^{abc}\eta_a\bar{z}_b\partial_\sigma\theta_c + \varepsilon_{abc}\bar{\eta}^a z^b\partial_\sigma\bar{\theta}^c) \\
& -\tilde{T}|z|^2(\eta_a\partial_\sigma\bar{\theta}^a + \partial_\sigma\theta_a\bar{\eta}^a) + \tilde{T}((z^a\eta_a)(\partial_\sigma\bar{\theta}^b\bar{z}_b) + (z^a\partial_\sigma\theta_a)(\bar{\eta}^b\bar{z}_b)) \\
& +2\tilde{T}\varphi^2(\partial_\sigma\theta_4\bar{\eta}^4 - \partial_\sigma\eta_4\bar{\theta}^4 + \eta_4\partial_\sigma\bar{\theta}^4 - \theta_4\partial_\sigma\bar{\eta}^4) + 8\varphi(p_a\bar{\eta}^a\bar{\eta}^4 - \bar{p}^a\eta_a\eta_4) \\
& -4\varphi(\varepsilon^{abc}p_a\eta_b\eta_c - \varepsilon_{abc}\bar{p}^a\bar{\eta}^b\bar{\eta}^c) + 2i(\varepsilon^{abc}p_a\bar{z}_b\eta_c\bar{\eta}^4 + \varepsilon_{abc}\bar{p}^a z^b\bar{\eta}^c\eta_4) \\
& +i((z^a p_a) - (\bar{p}^a\bar{z}_a))\Theta - 12\tilde{T}\varphi(\eta_a\bar{\eta}^a + \frac{1}{2}\Theta)\partial_\sigma x^1 \\
& -2i\tilde{T}(\varepsilon^{abc}\eta_a\bar{z}_b\eta_c + \varepsilon_{abc}\bar{\eta}^a z^b\bar{\eta}^c)\partial_\sigma x^1 + 8i\tilde{T}\varphi(\partial_\sigma z^a\eta_a\theta_4 + \partial_\sigma\bar{z}_a\bar{\eta}^a\bar{\theta}^4) \\
& +2\tilde{T}(\varepsilon_{abc}\partial_\sigma z^a z^b\bar{\eta}^c\theta_4 - \varepsilon^{abc}\partial_\sigma\bar{z}_a\bar{z}_b\eta_c\bar{\theta}^4) + \tilde{T}((\partial_\sigma z^a\bar{z}_a) - (z^a\partial_\sigma\bar{z}_a))(\theta_4\bar{\eta}^4 - \eta_4\bar{\theta}^4),
\end{aligned}
$$
$$
\mathcal{H}_{\text{f}}^{(4)} = 8\theta_4\bar{\theta}^4\eta_4\bar{\eta}^4.
$$

### 3 Classical integrability of equations of the mass-less superparticle and D0-brane in the AdS$_4 \times \mathbb{CP}^3$ superspace. Minimal extension of the integrable $OSp(4|6)/(SO(1,3) \times U(3))$ $\sigma$-model

In subsection 3.1 there will be introduced the Kaluza-Klein condition and examined its consequences for the double-dimensional reduction of the $AdS_4 \times S^7$ supermembrane to the $AdS_4 \times \mathbb{CP}^3$ superstring. This condition is imposed on the $D = 11$ supervielbein bosonic components in the tangent space to the $AdS_4 \times \mathbb{CP}^3$ 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 $S^7 = \mathbb{CP}^3 \times S^1$. Its introduction is motivated by the fact that the $OSp(4|8)/(SO(1,3) \times SO(7))$ supercoset representatives considered by other authors [13], [48] and in our work [22] yield expressions for the $D = 11$ supervielbein components that do not depend 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 $AdS_4 \times \mathbb{CP}^3$ superstring Lagrangian.

There will be obtained explicit form of the Kaluza-Klein condition for the $OSp(4|8)/(SO(1,3) \times SO(7))$ supercoset representative parametrized by the coordinates associated with the generators of the $D = 3, N = 8$ superconformal algebra [22]. 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 $AdS_4 \times \mathbb{CP}^3$ superbackground. There will be analyzed $SL(2, \mathbb{R})$-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 $AdS_4 \times S^7$ and $AdS_4 \times CP^3$ superspaces.

For further simplification of the form of the $AdS_4 \times S^7$ supervielbein and $AdS_4 \times CP^3$ superstring Lagrangian considered will be partial $\kappa$-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 $OSp(4|6)/(SO(1,3) \times U(3))$ $\sigma$-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 [58].

In subsection 3.2 it will be proved [59] that, when coordinates of the sector of broken supersymmetries vanish, fermionic equations of the $AdS_4 \times CP^3$ superstring, corresponding to their variation, are the consequences of other 24 fermionic equations. The latter turn into fermionic equations of the $OSp(4|6)/(SO(1,3) \times U(3))$ $\sigma$-model. This confirms that the $\sigma$-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 $OSp(4|6)/(SO(1,3) \times U(3))$ supercoset space [60]. It will also be proved that, analogously to the case of the $AdS_4 \times CP^3$ superstring, fermionic equations of the $AdS_4 \times CP^3$ superparticle corresponding to variation of the coordinates of the sector of broken supersymmetries are the consequences of the fermionic equations of the massless $OSp(4|6)/(SO(1,3) \times U(3))$ superparticle, when these coordinates turn to zero. Besides it will be established the relation between the $\sigma$-model Lax connection and components of the Lax pair of the superparticle in the $OSp(4|6)/(SO(1,3) \times U(3))$ supercoset space. As the generalization of the obtained results it will be proved classical integrability of equations of the massless superparticle [61] and the D0-brane [62] in the $AdS_4 \times CP^3$ superbackground.

## 3.1 Kaluza-Klein gauge for $\kappa$-symmetry and minimal extension of the integrable $OSp(4|6)/(SO(1,3) \times U(3))$ $\sigma$-model

As was noted in Section 2 for considered in [13], [48], [22] elements of the $OSp(4|8)/(SO(1,3) \times SO(7))$ supercoset the $so(2,3)/so(1,3)$ Cartan forms, that determine components of the $AdS_4 \times S^7$ 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)

$$E^{m'}(d) = G^{m'}(d) + dyG_y^{m'}. \qquad (3.1)$$
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 $G_y^{m'}$. This complicates the form of the $D = 11$ supervielbein and the $AdS_4 \times CP^3$ superstring Lagrangian. In our work [58] proposed was the Kaluza-Klein condition

$$G_y^{m'} = 0, \qquad (3.2)$$

imposition of which simplifies derivation of the superstring Lagrangian and its form. For mentioned $OSp(4|8)/(SO(1, 3) \times SO(7))$ supercoset elements $G_y^{m'}$ depends only on eight coordinates related to the supersymmetries broken by the $AdS_4 \times CP^3$ superbackground. Thus for them the Kaluza-Klein condition (3.2) constrains coordinates of the sector of broken supersymmetries.

The study of the $\kappa$-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 $AdS_4 \times CP^3$ superstring equations considered in Refs. [17], [18]. Parametrization of the sector of broken supersymmetries by four (anti-)Majorana $SL(2, R)$ spinor coordinates proposed in our work [58] allows to consider partial $\kappa$-symmetry gauges in which three of them vanish. For each of them it is possible to find the zero-curvature representation for superstring equations that includes the Lax connection of the $OSp(4|6)/(SO(1, 3) \times U(3))$ $\sigma$-model extended by the terms depending on remaining coordinates of the sector of broken supersymmetries. By weakening imposed gauge the $\sigma$-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 $AdS_4 \times CP^3$ 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 $AdS_4 \times CP^3$ 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 [58]. In the proposed gauge there does not turn to zero only spinor coordinate for broken Poincaré supersymmetries that satisfies the Majorana condition $\theta^\mu = \bar{\theta}^\mu$. 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 $OSp(4|6)/(SO(1, 3) \times U(3))$ $\sigma$-model the linear and quadratic terms in $\theta^\mu$ and its differential. These terms constitute the subset of the linear and quadratic terms found in [18].

### 3.1.1 $AdS_4 \times S^7$ and $AdS_4 \times CP^3$ supervielbeins in the Kaluza-Klein gauge

To analyze consequences of the Kaluza-Klein condition (3.2) for the $AdS_4 \times S^7$ supermembrane and the $AdS_4 \times CP^3$ superstring it is necessary to obtain explicit expression for $G_y^{m'}$ in terms of coordinates that requires specifying the $OSp(4|8)/(SO(1, 3) \times SO(7))$ supercoset
representative. Take the representative

$$ \hat{\mathcal{G}} = \mathcal{G} e^{yH} e^{\theta^{\mu} Q_{\mu}^{4} + \bar{\theta}^{\mu} \bar{Q}_{\mu}^{4}} e^{\eta_{\mu} S^{\mu 4} + \bar{\eta}_{\mu} \bar{S}^{\mu 4}} \in \mathrm{OSp}(4|8)/(\mathrm{SO}(1,3) \times \mathrm{SO}(7)) \quad (3.3) $$

that is compatible with the Hopf fibration realization of the seven-sphere. It generalizes the supercoset representative (2.68) considered in Section 2 by including arbitrary $\mathrm{OSp}(4|6)/(\mathrm{SO}(1,3) \times U(3))$ supercoset element $\mathcal{G}$. In (3.3) on the right of it placed are three factors. The first of them contains the generator $H$ of the $U(1)$ isometry of the $S^1$ circle. Other two factors include in the exponent generators of the supersymmetries that are broken by the $AdS_4 \times \mathbb{CP}^3$ superbackground. Such form of the $\mathrm{OSp}(4|8)/(\mathrm{SO}(1,3) \times \mathrm{SO}(7))$ supercoset representative ensures that $AdS_4 \times S^7$ 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 $G_y^{m'}$, that enters (3.1), are expressed through the contributions to the $so(2,3)/so(1,3)$ Cartan forms proportional to $dy$ in the following way

$$ G_y^{m'} = \begin{pmatrix} G_y^m \\ G_y^3 \end{pmatrix} = \begin{pmatrix} \frac{1}{2}(\omega_y^m + c_y^m) \\ -\Delta_y \end{pmatrix}. $$

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

$$ \begin{gathered} \omega_y^m + c_y^m = 4[1 - (\eta\bar{\eta})^2](\theta\sigma^m\bar{\theta}) + 4\{1 - i[(\theta\bar{\eta}) + (\bar{\theta}\eta)]\}(\eta\sigma^m\bar{\eta}), \\ \Delta_y = 2[(\bar{\theta}\eta) - (\theta\bar{\eta})]. \end{gathered} \quad (3.4) $$

In (3.4) there was introduced the following shorthand notations for expressions involving contractions of the $SL(2, \mathbb{R})$ spinors: $(\eta\bar{\eta}) = \eta^\mu\bar{\eta}_\mu$, $(\theta\sigma^m\bar{\theta}) = \theta^\mu\sigma_{\mu\nu}^m\bar{\theta}^\nu$ etc. Expressions (3.4) can be turned to zero in the $SL(2, \mathbb{R})$-covariant way by imposing the (anti-)Majorana condition on the spinor coordinates

$$ \bar{\theta}^{\mu} = s\theta^{\mu}, \quad \bar{\eta}^{\mu} = s\eta^{\mu}, \quad s = \pm 1. \quad (3.5) $$

Conditions (3.5) include common sign factor for coordinates $\theta$ and $\eta$ to turn to zero $\Delta_y$. 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 $\theta^\mu$ and $\eta^\mu$ for coordinates that satisfy the Majorana condition. Then the $AdS_4 \times S^7$ supervielbein bosonic components in the tangent space to $AdS_4$ will have the form

$$ \begin{align} E^m(d) &= G^m(d) = G^{0'm}(d) - \frac{i}{2}\epsilon^{mkl}G_{kl}(d)[(\theta\theta) + (\eta\eta)] \nonumber \\ &\quad -2ic^n(d)(\theta\sigma_n\tilde{\sigma}^m\eta) - c^m(d)(\theta\theta)(\eta\eta) - i(d\theta\sigma^m\theta + d\eta\sigma^m\eta), \tag{3.6} \\ -E^3(d) &= -G^3(d) = [1 + 2i(\theta\eta)]\Delta(d) + iG^{mn}(d)(\theta\sigma_{mn}\eta) + 2i(d\theta\eta), \nonumber \end{align} $$

where $G^{0'm}(d) = \frac{1}{2}(\omega^m + c^m)(d)$ and $-\Delta(d)$ are tangent to the $AdS_4$ components of supervielbein of the $\mathrm{OSp}(4|6)/(\mathrm{SO}(1,3) \times U(3))$ supercoset space (see Eq. (2.24) of Section 2).
$G^{mn}(d)$ is the so(1, 2) Cartan form that is the constituent of the so(1, 3) connection. Tangent to the $\mathbb{CP}^3$ manifold components of the $AdS_4 \times S^7$ supervielbein equal

$$ 
\begin{aligned}
E_a(d) &= i(\Omega_a + \tilde{\Omega}_a)(d) = i[\Omega_a(d) + 2\chi_{\mu a}(d)\theta^\mu - 2\omega_a^\mu(d)\eta_\mu - 2i\chi_a^\mu(d)\eta_\mu(\theta\theta)], \\
E^a(d) &= i(\Omega^a + \tilde{\Omega}^a)(d) = i[\Omega^a(d) - 2\bar{\chi}_\mu^a(d)\theta^\mu + 2\bar{\omega}^{\mu a}(d)\eta_\mu + 2i\bar{\chi}^{\mu a}(d)\eta_\mu(\theta\theta)].
\end{aligned}
\quad (3.7) $$

In (3.7) the $su(4)/u(3)$ Cartan forms $\Omega_a(d)$ and $\Omega^a(d)$ are respective components of the $OSp(4|6)/(SO(1, 3) \times U(3))$ supervielbein. Grassmann-odd Cartan forms $\omega_a^\mu(d)$ and $\chi_{\mu a}(d)$ and c.c. associated with the $D = 3$ $\mathcal{N} = 6$ super-Poincaré and special conformal supersymmetry generators are the $OSp(4|6)/(SO(1, 3) \times U(3))$ 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 $AdS_4 \times \mathbb{CP}^3$ superspace. The 11-th component of the $AdS_4 \times S^7$ supervielbein tangent to the $S^1$ circle has the form

$$ E^{11}(d) = h(d) + \tilde{\Omega}_a^a(d) = \Phi(dy + A(d)) : \quad \Phi = 1 - 4i(\theta\eta), \quad A(d) = \Phi^{-1}\tilde{\Omega}_a^a(d), \quad (3.8) $$

where $h(d)$ is the Cartan form for the generator $H$ of the $U(1)$ isometry group of $S^1$ fiber and $\tilde{\Omega}_a^a(d)$ is the $u(1)$ part of the $u(3)$ connection of the $\mathbb{CP}^3$ manifold $\tilde{\Omega}_a^b(d)$. $\Phi = e^{2\phi/3}$ determines the $D = 10$ dilaton superfield $\phi$. Among fermionic components of the $AdS_4 \times S^7$ supervielbein

$$ \frac{1}{\sqrt{2}} F^{\hat{\alpha}}(d) = \frac{1}{\sqrt{2}} (f^{\hat{\alpha}}(d) + dy F_y^{\hat{\alpha}}) = \begin{pmatrix} \bar{\omega}^{\mu A}(d) \\ \omega_A^{\mu}(d) \\ \bar{\chi}_{\mu}^A(d) \\ \chi_{\mu A}(d) \end{pmatrix} = \begin{pmatrix} \bar{\tilde{\omega}}^{\mu A}(d) \\ \tilde{\omega}_A^{\mu}(d) \\ \bar{\tilde{\chi}}_{\mu}^A(d) \\ \tilde{\chi}_{\mu A}(d) \end{pmatrix} + dy \begin{pmatrix} \bar{\omega}_y^{\mu A} \\ \omega_{yA}^{\mu} \\ \bar{\chi}_{y\mu}^A \\ \chi_{y\mu A} \end{pmatrix} \quad (3.9) $$

the following do not vanish

$$ 
\begin{aligned}
& \tilde{\omega}_a^\mu(d) = \omega_a^\mu(d) + i\chi_a^\mu(d)(\theta\theta), \\
& \tilde{\omega}_4^\mu(d) = d\theta^\mu - 2id\theta^\mu(\theta\eta) + 2i\theta^\mu(d\theta\eta) + \frac{1}{2}G^{mn}(d)\theta^\nu\sigma_{mn\nu}^\mu \\
& \qquad + \Delta(d)\theta^\mu - \omega^m(d)\tilde{\sigma}_m^{\mu\nu}\eta_\nu + iG^{mn}(d)\eta^\nu\sigma_{mn\nu}^\mu(\theta\theta), \\
& \omega_{y4}^\mu = 2i\theta^\mu, \\
& \tilde{\chi}_{\mu a}(d) = \chi_{\mu a}(d) + 4i\chi_{\nu a}(d)\theta^\nu\eta_\mu + i\omega_{\mu a}(d)(\eta\eta) - \chi_{\mu a}(d)(\theta\theta)(\eta\eta), \\
& \tilde{\chi}_{\mu 4}(d) = d\eta_\mu + 2i\eta_\mu(d\theta\eta) + c_m(d)\sigma_{\mu\nu}^m\theta^\nu - \frac{1}{2}G^{mn}(d)\sigma_{mn\nu}^\mu\eta_\nu \\
& \qquad - \Delta(d)\eta_\mu + ic_m(d)\sigma_{\mu\nu}^m\eta^\nu(\theta\theta) + \frac{i}{2}G^{mn}(d)\sigma_{mn\mu}^\nu\theta_\nu(\eta\eta) + 2i\Delta(d)\eta_\mu(\theta\theta), \\
& \chi_{y\mu 4} = 2i\eta_\mu[1 - 2i(\theta\eta)]
\end{aligned}
\quad (3.10) $$

and c.c. They determine fermionic components of the $AdS_4 \times \mathbb{CP}^3$ supervielbein and the dilatino superfield

$$ E^{\hat{\alpha}}(d) = f^{\hat{\alpha}}(d) - F_y^{\hat{\alpha}}A(d), \quad \chi^{\hat{\alpha}} = F_y^{\hat{\alpha}}. $$
### 3.1.2 Minimal extension of the integrable $OSp(4|6)/(SO(1,3) \times U(3))$ $\sigma$-model

In the previous paragraph obtained were expressions for the supervielbein components of the $AdS_4 \times CP^3$ 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 $\kappa$-symmetry gauge, in which remains only Majorana spinor coordinate $\theta^\mu$ for broken Poincaré supersymmetries. This leads to further simplification of the $AdS_4 \times S^7$ supervielbein (3.6)-(3.10). In particular, its bosonic components acquire the form

$$ 
\begin{aligned}
E^m(d) &= G^{0'm}(d) - \frac{i}{2}\epsilon^{mkl}G_{kl}(d)(\theta\theta) - id\theta\sigma^m\theta, & E^3(d) &= -\Delta(d), \\
E_a(d) &= i(\Omega_a(d) + 2\chi_{\mu a}(d)\theta^\mu), & E^a(d) &= i(\Omega^a(d) - 2\bar{\chi}^a(d)\theta^\mu)
\end{aligned}
\quad (3.11) $$

and

$$ E^{11}(d) = dy + \tilde{\Omega}_a^a(d). \quad (3.12) $$

From (3.12) it is seen that $\Phi = 1$ 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 to

$$ 
\begin{aligned}
\tilde{\omega}_a^\mu(d) &= \omega_a^\mu(d) + i\chi_a^\mu(d)(\theta\theta), \\
\tilde{\omega}_4^\mu(d) &= d\theta^\mu + \frac{1}{2}G^{mn}(d)\theta^\nu\sigma_{mn\nu}^\mu + \Delta(d)\theta^\mu, \\
\omega_{y_4}^\mu &= 2i\theta^\mu \\
\tilde{\chi}_{\mu a}(d) &= \chi_{\mu a}(d), \\
\tilde{\chi}_{\mu 4}(d) &= c_m(d)\sigma_{\mu\nu}^m\theta^\nu.
\end{aligned}
\quad (3.13) $$

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 $AdS_4 \times CP^3$ superstring action

$$ S_{\text{sstring, min}}^{AdS_4 \times CP^3} = \int_{\Sigma} d^2\xi \left( \mathcal{L}_{\text{sstring, min, kin}}^{AdS_4 \times CP^3} + \mathcal{L}_{\text{sstring, min, WZ}}^{AdS_4 \times CP^3} \right). $$

can be viewed as the minimal extension of the $OSp(4|6)/(SO(1,3) \times U(3))$ $\sigma$-model. Kinetic and Wess-Zumino terms of the superstring Lagrangian have the form

$$ 
\begin{aligned}
\mathcal{L}_{\text{sstring, min, kin}}^{AdS_4 \times CP^3} &= -\frac{1}{2}\sqrt{-\gamma}\gamma^{ij}(E_i^m E_{jm} + \Delta_i \Delta_j - E_{ia}E_j^a), \\
\mathcal{L}_{\text{sstring, min, WZ}}^{AdS_4 \times CP^3} &= -\frac{1}{2}[(\omega_a^\mu(d) + 2i\Omega_a(d)\theta^\mu) \wedge (\bar{\omega}_\mu^a(d) - 2i\Omega^a(d)\theta_\mu) \\
&\qquad + \chi_{\mu a}(d) \wedge \bar{\chi}^{\mu a}(d)] + (\Omega_a(d) \wedge \Omega^a(d) + 2i\tilde{\Omega}_a^a(d) \wedge \Delta(d))(\theta\theta).
\end{aligned}
$$

To derive superstring equations in the proposed gauge let us choose as independent variation parameters, similarly to the case of the $OSp(4|6)/(SO(1,3) \times U(3))$ $\sigma$-model considered in Section 2, the Cartan forms, associated with generators of the $osp(4|6)/(so(1,3) \times u(3))$ quotient algebra that have definite eigenvalues of the $Z_4$ automorphism, with differentials