of the coordinates replaced by their variations. Such set includes bosonic Cartan forms from the eigenspace $\mathfrak{g}_2$, namely $G^{0'm}(d)$, $\Delta(d)$ and $\Omega_a(d)$, $\Omega^a(d)$, as well as fermionic Cartan forms (2.16) corresponding to the supersymmetry generators (2.15) from eigenspaces $\mathfrak{g}_{1,3}$. The reason for such choice is that these Cartan forms are used as variation parameters when one derives the $OSp(4|6)/(SO(1,3) \times U(3))$ $\sigma$-model equations and enter its Lax connection [9], [10].$^{13}$ It is assumed that it gives leading contribution to the power-series expansion of the Lax connection of the $AdS_4 \times \mathbb{CP}^3$ superstring in coordinates of the sector of broken supersymmetries.

So the superstring equations

$$ \begin{aligned} \frac{\delta S^{AdS_4 \times \mathbb{CP}^3}_{\text{sstring, min}}}{\delta G^{0'm}(\delta)} = & \partial_i(\sqrt{-\gamma\gamma^{ij}G_j}^{0'm}) + 2\sqrt{-\gamma\gamma^{ij}}G_i^{mn}G_j^{0'n} + 2\sqrt{-\gamma\gamma^{ij}}G_i^{3m}\Delta_j \\ & + 2i\omega_{(1)}^\mu_a(d) \wedge \sigma_{\mu\nu}^m \bar{\omega}_{(1)}^{\nu a}(d) - 2i\omega_{(3)}^\mu_a(d) \wedge \sigma_{\mu\nu}^m \bar{\omega}_{(3)}^{\nu a}(d) \\ & + \sqrt{-\gamma\gamma^{ij}}[\Omega_i^a(\omega_{ja}\sigma^m\theta) - \Omega_{ia}(\bar{\omega}_j^a\sigma^m\theta)] - i[\Omega^a(d) \wedge (\chi_a(d)\sigma^m\theta) \\ & + \Omega_a(d)\wedge(\bar{\chi}^a(d)\sigma^m\theta)] - i\partial_i\{\sqrt{-\gamma\gamma^{ij}}[(\partial_j\theta\sigma^m\theta) + \frac{1}{2}\epsilon^{mnp}G_{jnp}(\theta\theta)]\} \\ & - 2i\sqrt{-\gamma\gamma^{ij}}G_i^{mn}[(\partial_j\theta\sigma_n\theta) + \frac{1}{2}\epsilon_{npr}G_j^{pr}(\theta\theta)] + \sqrt{-\gamma\gamma^{ij}}[(\chi_{ia}\sigma^m\bar{\omega}_j^a) \\ & - (\omega_{ia}\sigma^m\bar{\chi}_j^a)](\theta\theta) - 4i\tilde{\Omega}_a^a(d) \wedge G^{3m}(d)(\theta\theta) = 0, \end{aligned} $$

$$ \begin{aligned} \frac{\delta S^{AdS_4 \times \mathbb{CP}^3}_{\text{sstring, min}}}{\delta \Delta(\delta)} = & \partial_i(\sqrt{-\gamma\gamma^{ij}\Delta_j}) - 2\sqrt{-\gamma\gamma^{ij}}G_i^{3m}G_j^{0'm} + 2\omega_{(1)}^\mu_a(d) \wedge \bar{\omega}_{(1)}^a_\mu(d) \\ & + 2\omega_{(3)}^\mu_a(d) \wedge \bar{\omega}_{(3)}^a_\mu(d) + \sqrt{-\gamma\gamma^{ij}}(\Omega_{ia}\bar{\chi}_j^a\theta^\mu - \Omega_i^a\chi_{j\mu a}\theta^\mu) \\ & + i(\Omega^a(d) \wedge \omega_a^\mu(d)\theta_\mu + \Omega_a(d) \wedge \bar{\omega}^{\mu a}(d)\theta_\mu) + 2i\sqrt{-\gamma\gamma^{ij}}G_i^{3m}[(\partial_j\theta\sigma_m\theta) \\ & + \frac{1}{2}\epsilon_{mnp}G_j^{np}(\theta\theta)] - 2\sqrt{-\gamma\gamma^{ij}}\chi_{ia}^\mu\bar{\chi}_j^a\theta(\theta) - 4i\tilde{\Omega}_a^a(d) \wedge (d\theta\theta) \\ & - 2\Omega_a(d) \wedge \Omega^a(d)(\theta\theta) + 2i(\omega_a^\mu(d) \wedge \bar{\chi}_\mu^a(d) + \chi_a^\mu(d) \wedge \bar{\omega}_\mu^a(d))(\theta\theta) = 0, \end{aligned} $$

$$ \begin{aligned} \frac{\delta S^{AdS_4 \times \mathbb{CP}^3}_{\text{sstring, min}}}{\delta \Omega_a(\delta)} = & \frac{1}{2}\partial_i(\sqrt{-\gamma\gamma^{ij}\Omega_j^a}) + \frac{i}{2}\sqrt{-\gamma\gamma^{ij}}\Omega_i^b(\tilde{\Omega}_{jb}^a + \delta_b^a\tilde{\Omega}_{jc}^c) \\ & - i\epsilon^{abc}\omega_{(1)b}^\mu(d)\wedge\omega_{(1)\mu c}(d) - i\epsilon^{abc}\omega_{(3)b}^\mu(d)\wedge\omega_{(3)\mu c}(d) - \partial_i(\sqrt{-\gamma\gamma^{ij}\bar{\chi}_j^a\theta^\mu}) \\ & - i\sqrt{-\gamma\gamma^{ij}\bar{\chi}_i^b\theta^\mu}(\tilde{\Omega}_{jb}^a + \delta_b^a\tilde{\Omega}_{jc}^c) + i\sqrt{-\gamma\gamma^{ij}}\epsilon^{abc}\Omega_{ib}\chi_{j\mu c}\theta^\mu - id(\bar{\omega}^{\mu a}(d)\theta_\mu) \\ & + \bar{\omega}^{\mu b}(d)\theta_\mu\wedge(\tilde{\Omega}_b^a(d) + \delta_b^a\tilde{\Omega}_c^c(d)) + \epsilon^{abc}\Omega_b(d)\wedge\omega_c^\mu(d)\theta_\mu + 2(d\theta\theta)\wedge\Omega^a(d) \\ & + 2\Delta(d)\wedge\Omega^a(d)(\theta\theta) + 3\epsilon^{abc}\omega_b^\mu(d)\wedge\chi_{\mu c}(d)(\theta\theta) = 0, \end{aligned} $$

$$ \begin{aligned} -\frac{\delta S^{AdS_4 \times \mathbb{CP}^3}_{\text{sstring, min}}}{\delta \bar{\omega}_{(1)}^a_\mu(\delta)} = & -2iV_+^{ij}G_i^{0'm}\tilde{\sigma}_m^{\mu\nu}\omega_{(1)j\nu a} - 2V_+^{ij}\Delta_i\omega_{(1)ja}^\mu + 2iV_+^{ij}\epsilon_{abc}\Omega_i^b\bar{\omega}_{(1)j}^{\mu c} \\ & -i\partial_i(\sqrt{-\gamma\gamma^{ij}\Omega_{ja}\theta^\mu}) - \sqrt{-\gamma\gamma^{ij}}(\tilde{\Omega}_{ia}^b - \delta_a^b\tilde{\Omega}_{ic}^c)\Omega_{jb}\theta^\mu - i\sqrt{-\gamma\gamma^{ij}}\Omega_{ia}(\Delta_j\theta^\mu \\ & + \frac{1}{2}G_j^{mn}\theta^\nu\sigma_{mn\nu}^\mu + ic_j^m\tilde{\sigma}_m^{\mu\nu}\theta_\nu) + 4i\sqrt{-\gamma\gamma^{ij}}\epsilon_{abc}\bar{\omega}_{(1)i}^{\mu b}\bar{\chi}_{j\nu}^c\theta^\nu \end{aligned} $$

<footnote>$^{13}$The Lax connection of the $OSp(4|6)/(SO(1,3) \times U(3))$ $\sigma$-model represents particular case of the Lax connections of integrable $\sigma$-models in supercoset spaces, stability algebras of which are invariant under the $\mathbb{Z}_4$ automorphism of the isometry superalgebras [63], [64], [65], [66].</footnote>
$$
\begin{align}
& -i\Omega_a(d) \wedge [d\theta^\mu + (2i\tilde{\Omega}_b^b(d) - \Delta(d))\theta^\mu + \frac{1}{2}G^{mn}(d)\theta^\nu\sigma_{mn\nu}^\mu - i\omega^m(d)\tilde{\sigma}_m^{\mu\nu}\theta_\nu] \nonumber \\
& + \varepsilon_{abc}\Omega^b(d)\wedge\Omega^c(d)\theta^\mu - 2i\varepsilon_{abc}\bar{\omega}^{\mu b}(d)\wedge\bar{\chi}_\nu^c(d)\theta^\nu - \varepsilon_{abc}\bar{\omega}^{\nu b}(d)\wedge\bar{\omega}_\nu^c(d)\theta^\mu \nonumber \\
& \quad -i\partial_i[\sqrt{-\gamma}\gamma^{ij}\chi_{ja}^\mu(\theta\theta)] - \sqrt{-\gamma}\gamma^{ij}(\tilde{\Omega}_{ia}^b - \delta_a^b\tilde{\Omega}_{ic}^c)\chi_{jb}^\mu(\theta\theta) \nonumber \\
& \quad -\sqrt{-\gamma}\gamma^{ij}\varepsilon_{abc}\Omega_i^b\bar{\chi}_j^{\mu c}(\theta\theta) + 2i\sqrt{-\gamma}\gamma^{ij}G_i^{0'm}\tilde{\sigma}_m^{\mu\nu}\omega_{(3)j\nu a}(\theta\theta) \nonumber \\
& \quad -i\sqrt{-\gamma}\gamma^{ij}\chi_{ia}^\nu(\Delta_j\delta_\nu^\mu + \frac{1}{2}G_j^{mn}\sigma_{mn\nu}^\mu - ic_j^m\varepsilon^{\mu\rho}\sigma_{mn\rho})(\theta\theta) \nonumber \\
& -2\sqrt{-\gamma}\gamma^{ij}(\partial_i\theta\sigma^m\theta)\tilde{\sigma}_m^{\mu\nu}\omega_{(1)j\nu a} + \sqrt{-\gamma}\gamma^{ij}G_i^{mn}\varepsilon^{\mu\rho}\sigma_{mn\rho}^\nu\omega_{(1)j\nu a}(\theta\theta) \nonumber \\
& \quad +2\omega_{(3)a}^\mu(d) \wedge (d\theta\theta) + 2\Delta(d) \wedge \omega_{(3)a}^\mu(d)(\theta\theta) \nonumber \\
& +4i\tilde{\Omega}_c^c(d) \wedge \omega_{(1)a}^\mu(d)(\theta\theta) - 6i\varepsilon_{abc}\Omega^b(d) \wedge \bar{\omega}_{(1)}^{\mu c}(d)(\theta\theta) = 0, \tag{3.14}
\end{align}
$$

$$
\begin{align}
-\frac{\delta S_{\text{sstring, min}}^{AdS_4 \times CP^3}}{\delta \bar{\omega}_{(3)\mu}^{a}(\delta)} &= -2iV_{-}^{ij}G_{i}^{0'm}\tilde{\sigma}_{m}^{\mu\nu}\omega_{(3)j\nu a} + 2V_{-}^{ij}\Delta_{i}\omega_{(3)j}^{\mu} - 2iV_{-}^{ij}\varepsilon_{abc}\Omega_{i}^{b}\bar{\omega}_{(3)j}^{\mu c} \nonumber \\
&\quad + i\partial_{i}(\sqrt{-\gamma}\gamma^{ij}\Omega_{ja}\theta^{\mu}) + \sqrt{-\gamma}\gamma^{ij}(\tilde{\Omega}_{ia}^{b} - \delta_{a}^{b}\tilde{\Omega}_{ic}^{c})\Omega_{jb}\theta^{\mu} + i\sqrt{-\gamma}\gamma^{ij}\Omega_{ia}(\Delta_{j}\theta^{\mu} \nonumber \\
&\quad + \frac{1}{2}G_{j}^{mn}\theta^{\nu}\sigma_{mn\nu}^{\mu} - ic_{j}^{m}\tilde{\sigma}_{m}^{\mu\nu}\theta_{\nu}) - 4i\sqrt{-\gamma}\gamma^{ij}\varepsilon_{abc}\bar{\omega}_{(3)i}^{\mu b}\bar{\chi}_{j\nu}^{c}\theta^{\nu} \nonumber \\
&\quad -i\Omega_{a}(d)\wedge[d\theta^{\mu} + (2i\tilde{\Omega}_{b}^{b}(d) - \Delta(d))\theta^{\mu} + \frac{1}{2}G^{mn}(d)\theta^{\nu}\sigma_{mn\nu}^{\mu} + i\omega^{m}(d)\tilde{\sigma}_{m}^{\mu\nu}\theta_{\nu}] \nonumber \\
&\quad + \varepsilon_{abc}\Omega^{b}(d)\wedge\Omega^{c}(d)\theta^{\mu} - 2i\varepsilon_{abc}\bar{\omega}^{\mu b}(d)\wedge\bar{\chi}_{\nu}^{c}(d)\theta^{\nu} + \varepsilon_{abc}\bar{\omega}^{\nu b}(d)\wedge\bar{\omega}_{\nu}^{c}(d)\theta^{\mu} \nonumber \\
&\quad + i\partial_{i}[\sqrt{-\gamma}\gamma^{ij}\chi_{ja}^{\mu}(\theta\theta)] + \sqrt{-\gamma}\gamma^{ij}(\tilde{\Omega}_{ia}^{b} - \delta_{a}^{b}\tilde{\Omega}_{ic}^{c})\chi_{jb}^{\mu}(\theta\theta) \tag{3.15} \\
&\quad + \sqrt{-\gamma}\gamma^{ij}\varepsilon_{abc}\Omega_{i}^{b}\bar{\chi}_{j}^{\mu c}(\theta\theta) - 2i\sqrt{-\gamma}\gamma^{ij}G_{i}^{0'm}\tilde{\sigma}_{m}^{\mu\nu}\omega_{(1)j\nu a}(\theta\theta) \nonumber \\
&\quad + i\sqrt{-\gamma}\gamma^{ij}\chi_{ia}^{\nu}(\Delta_{j}\delta_{\nu}^{\mu} + \frac{1}{2}G_{j}^{mn}\sigma_{mn\nu}^{\mu} + ic_{j}^{m}\varepsilon^{\mu\rho}\sigma_{mn\rho})(\theta\theta) \nonumber \\
&\quad -2\sqrt{-\gamma}\gamma^{ij}(\partial_{j}\theta\sigma^{m}\theta)\tilde{\sigma}_{m}^{\mu\nu}\omega_{(3)j\nu a} + \sqrt{-\gamma}\gamma^{ij}G_{i}^{mn}\varepsilon^{\mu\rho}\sigma_{mn\rho}^{\nu}\omega_{(3)j\nu a}(\theta\theta) \nonumber \\
&\quad -2\omega_{(1)a}^{\mu}(d)\wedge(d\theta\theta) - 2\Delta(d)\wedge\omega_{(1)a}^{\mu}(d)(\theta\theta) \nonumber \\
&\quad -4i\tilde{\Omega}_{c}^{c}(d)\wedge\omega_{(3)a}^{\mu}(d)(\theta\theta) + 6i\varepsilon_{abc}\Omega^{b}(d)\wedge\bar{\omega}_{(3)}^{\mu c}(d)(\theta\theta) = 0 \nonumber
\end{align}
$$

and c.c. have been presented in the form of the series in $\theta^\mu$ and $d\theta^\mu$. Above equations include world-sheet projectors $V_\pm^{ij} = \sqrt{-\gamma}\gamma^{ij} \pm \varepsilon^{ij}$ and components of the so(1,3) connection $G^{3m}(d) = \frac{1}{2}(c^m - \omega^m)(d)$. These equations are extentions of the $OSp(4|6)/(SO(1,3) \times U(3))$ $\sigma$-model equations by linear and quadratic terms in the spinor coordinate $\theta^\mu$ and its differential. From the superstring action besides these equations there also follow equations corresponding to variation of the Grassmann coordinates from the sector of broken supersymmetries

$$
\begin{align}
\frac{\delta S_{\text{sstring, min}}^{AdS_4 \times CP^3}}{\delta \theta_{\mu}} &= \sqrt{-\gamma} \gamma^{ij} (\Omega_{ia} \bar{\chi}_j^{\mu a} - \Omega_i^a \chi_{ja}^{\mu}) + i (\Omega^a(d) \wedge \omega_a^\mu(d) + \Omega_a(d) \wedge \bar{\omega}^{\mu a}(d)) \nonumber \\
&\quad -i \partial_i (\sqrt{-\gamma} \gamma^{ij} G_j^{0'm} \tilde{\sigma}_m^{\mu\nu} \theta_\nu - 2i \sqrt{-\gamma} \gamma^{ij} G_i^{0'm} (\tilde{\sigma}_m^{\mu\nu} \partial_j \theta_\nu + \frac{1}{2} \varepsilon_{mkl} G_j^{kl} \theta^\mu) \nonumber \\
&\quad +2\sqrt{-\gamma} \gamma^{ij} \chi_{ia}^\nu \bar{\chi}_{j\nu}^a \theta^\mu + 2(\Omega_a(d) \wedge \Omega^a(d) + 2i\Delta(d) \wedge \tilde{\Omega}_a^a(d))\theta^\mu \tag{3.16} \\
&\quad +i(\omega_a^\nu(d) \wedge \bar{\chi}_\nu^a(d) + \chi_a^\nu(d) \wedge \bar{\omega}_\nu^a(d))\theta^\mu - \sqrt{-\gamma} \gamma^{ij} G_i^{mn} \partial_j \theta^\nu \sigma_{mn\nu}^\mu(\theta\theta) \nonumber
\end{align}
$$
$$
\begin{align}
& + \frac{3}{2} \sqrt{-\gamma \gamma^{ij}} \partial_{ij} \theta^{\mu} (\theta \theta) - 3 \sqrt{-\gamma \gamma^{ij}} (\partial_i \theta \partial_j \theta) \theta^{\mu} = 0, \nonumber \\
& \frac{\delta S_{\text{sstring, min}}^{\text{AdS}_4 \times \mathbb{CP}^3}}{\delta \vartheta_{\mu}} = i \sqrt{-g} \gamma^{ij} (\Omega_{ia} \bar{\chi}_j^{\mu a} + \Omega_{i^a} \chi_{ja}^{\mu}) + \Omega^a(d) \wedge \omega_a^{\mu}(d) - \Omega_a(d) \wedge \bar{\omega}^{\mu a}(d) \nonumber \\
& \qquad - 4i \sqrt{-g} \gamma^{ij} \tilde{\Omega}_{ia}{}^a G_j{}^{0'm} \tilde{\sigma}_m^{\mu\nu} \theta_\nu + 2i \sqrt{-g} \gamma^{ij} (\chi_{ia}^\mu \bar{\chi}_j^{\nu a} + \chi_{ia}^\nu \bar{\chi}_j^{\mu a}) \theta_\nu \nonumber \\
& \qquad - 2i G^{0'm}(d) \wedge G^{3n}(d) \theta^\nu \sigma_{mn\nu}{}^\mu - 2i G^{0'm}(d) \wedge G^3{}_m(d) \theta^\mu - 2i \Delta(d) \wedge (d\theta^\mu \nonumber \\
& \qquad + \frac{1}{2} G^{mn}(d) \theta^\nu \sigma_{mn\nu}{}^\mu) - 2(\omega_a^\mu(d) \wedge \bar{\chi}^{\nu a}(d) + \chi_a^\nu(d) \wedge \bar{\omega}^{\mu a}(d))\theta_\nu \nonumber \\
& \qquad + i(\Omega^a(d) \wedge \chi_a^\mu(d) - \Omega_a(d) \wedge \bar{\chi}^{\mu a}(d))(\theta \theta) + 6\sqrt{-\gamma \gamma^{ij}} \tilde{\Omega}_{ia}{}^a \partial_j \theta^\mu(\theta \theta) \nonumber \\
& \qquad - 4G^{0'm}(d) \wedge \tilde{\sigma}_m^{\mu\nu} d\theta_\nu(\theta \theta) + c^m(d) \wedge \tilde{\sigma}_m^{\mu\nu} d\theta_\nu(\theta \theta) = 0, \tag{3.17}
\end{align}
$$

$$
\begin{align}
& \frac{\delta S_{\text{sstring, min}}^{\text{AdS}_4 \times \mathbb{CP}^3}}{\delta \bar{\eta}_\mu} + \frac{\delta S_{\text{sstring, min}}^{\text{AdS}_4 \times \mathbb{CP}^3}}{\delta \bar{\eta}_\mu} = \sqrt{-\gamma \gamma^{ij} (\Omega_i^a \omega_{ja}^\mu - \Omega_{ia} \bar{\omega}_j^{\mu a})} + i (\Omega^a(d) \wedge \chi_a^\mu(d) \nonumber \\
& \qquad + \Omega_a(d) \wedge \bar{\chi}^{\mu a}(d)) - 2i \sqrt{-\gamma \gamma^{ij}} G_i{}^{0'm} (G_j{}^{3n} \theta^\nu \sigma_{mn\nu}{}^\mu + G_j{}^3{}_m \theta^\mu) \nonumber \\
& \qquad - 2i \sqrt{-\gamma \gamma^{ij}} \Delta_i (\partial_j \theta^\mu + \frac{1}{2} G_j{}^{mn} \theta^\nu \sigma_{mn\nu}{}^\mu) + 2i \sqrt{-\gamma \gamma^{ij}} \Omega_{ia} \Omega_j{}^a \theta^\mu \nonumber \\
& \qquad + 2 \sqrt{-\gamma \gamma^{ij}} (\omega_{ia}^\mu \bar{\chi}_j^{\nu a} - \chi_{ia}^\nu \bar{\omega}_j^{\mu a}) \theta_\nu + 4i \tilde{\Omega}_a{}^a(d) \wedge G^{0'm}(d) \tilde{\sigma}_m^{\mu\nu} \theta_\nu \nonumber \\
& \qquad + 2i \chi_a^\nu(d) \wedge \bar{\chi}_\nu^a(d) \theta^\mu + 3i \sqrt{-\gamma \gamma^{ij}} (\Omega_i^a \chi_{ja}^\mu - \Omega_{ia} \bar{\chi}_j^{\mu a})(\theta \theta) \nonumber \\
& \qquad - \sqrt{-\gamma \gamma^{ij}} (c_i^m + 2G_i{}^{0'm}) \tilde{\sigma}_m^{\mu\nu} \partial_j \theta_\nu(\theta \theta) - 6 \tilde{\Omega}_a{}^a(d) \wedge d\theta^\mu(\theta \theta) = 0, \tag{3.18}
\end{align}
$$

$$
\begin{align}
& \frac{\delta S_{\text{sstring, min}}^{\text{AdS}_4 \times \mathbb{CP}^3}}{\delta \bar{\eta}_\mu} - \frac{\delta S_{\text{sstring, min}}^{\text{AdS}_4 \times \mathbb{CP}^3}}{\delta \bar{\eta}_\mu} = \sqrt{-\gamma \gamma^{ij} (\Omega_i^a \omega_{ja}^\mu + \Omega_{ia} \bar{\omega}_j^{\mu a})} + i (\Omega^a(d) \wedge \chi_a^\mu(d) \nonumber \\
& \qquad - \Omega_a(d) \wedge \bar{\chi}^{\mu a}(d)) + 4\sqrt{-\gamma \gamma^{ij}} \Delta_i \tilde{\Omega}_{ja}{}^a \theta^\mu + 2\sqrt{-\gamma \gamma^{ij}} (\omega_{ia}^\mu \bar{\chi}_j^{\nu a} \nonumber \\
& \qquad + \chi_{ia}^\nu \bar{\omega}_j^{\mu a}) \theta_\nu - 2G^{0'm}(d) \wedge \tilde{\sigma}_m^{\mu\nu}(d\theta_\nu - \frac{1}{2}G^{kl}(d)\sigma_{kl\nu}\lambda^{\lambda}\theta_\lambda) \nonumber \\
& \qquad + 2\Delta(d) \wedge G^{3m}(d) \tilde{\sigma}_m^{\mu\nu} \theta_\nu - 2i(\chi_a^\mu(d) \wedge \bar{\chi}^{\nu a}(d) + \chi_a^\nu(d) \wedge \bar{\chi}^{\mu a}(d))\theta_\nu \nonumber \\
& \qquad + i\sqrt{-\gamma \gamma^{ij}} (\Omega_i^a \chi_{ja}^\mu + \Omega_{ia} \bar{\chi}_j^{\mu a})(\theta \theta) - 2id\theta^\mu \wedge (d\theta \theta) \nonumber \\
& \qquad + \frac{i}{2}\varepsilon^{lmn} G_{mn}(d) \wedge \tilde{\sigma}_l^{\mu\nu} d\theta_\nu(\theta \theta) - 3i\Delta(d) \wedge d\theta^\mu(\theta \theta) = 0, \tag{3.19}
\end{align}
$$

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

In our work [58] there was found the zero-curvature representation for these equations

$$ d\mathcal{L}(d) - \mathcal{L}(d) \wedge \mathcal{L}(d) = 0, \tag{3.20} $$

which enter the one-form

$$ \mathcal{L}(d) = \mathcal{L}_{\text{so}(2,3)}(d) + \mathcal{L}_{\text{su}(4)}(d) + \mathcal{L}_{24 \text{ susy}}(d) \in \text{osp}(4|6) \tag{3.21} $$

taking value in the osp(4|6) isometry superalgebra of the AdS$_4 \times \mathbb{CP}^3$ superspace.

It can be conveniently presented as the sum of three summands. The first summand takes value in the so(2, 3) algebra

$$ \mathcal{L}_{\text{so}(2,3)}(d) = \mathbf{G}^{mn}(z,d)M_{mn} + 2G^{3m}(d)M_{3m} + 2\mathbf{G}^{0'm}(z,d)M_{0'm} + \mathbf{\Delta}(z,d)D \in \text{so}(2,3) $$
and includes coefficient one-forms

$$
\begin{align}
\mathbf{G}^{mn}(z, d) &= G^{mn}(d) + i\ell_2\epsilon^{mnk}(\ell_2 G^{0'k}(d) + \ell_1 * G^{0'k}(d))(\theta\theta), \nonumber \\
\mathbf{G}^{0'm}(z, d) &= \ell_1 G^{0'm}(d) + \ell_2 * [G^{0'm}(d) - i(d\theta\sigma^m\theta) - \frac{i}{2}\epsilon^{mnk}G_{nk}(d)(\theta\theta)], \tag{3.22} \\
\Delta(z, d) &= \ell_1\Delta(d) + \ell_2 * \Delta(d) + 2i\ell_2\tilde{\Omega}_a^a(d)(\theta\theta), \nonumber
\end{align}
$$

where * denotes dualization of the world-sheet one-forms: $a(d) = d\xi^i a_i \to *a(d) = d\xi^i * a_i$,
*$a_i = \sqrt{-\gamma}\epsilon_{ij}\gamma^{jk}a_k = \frac{1}{\sqrt{-\gamma}}\gamma_{ij}\epsilon^{jk}a_k$. Coefficient one-forms in (3.22), as indicates their first
argument, depend on the spectral parameter $z$ through the functions $\ell_1(z)$ and $\ell_2(z)$, the
form of which is determined by the zero-curvature condition. The second summand in (3.21)
takes value in the $su(4)$ isometry algebra of the $\mathbb{CP}^3$ manifold

$$
\mathcal{L}_{su(4)}(d) = \tilde{\Omega}_a^b(z, d) \tilde{V}_b^a + \tilde{\Omega}_b^b(z, d) \tilde{V}_a^a + \Omega^a(z, d) T_a + \Omega_a(z, d) T^a \in su(4)
$$

and the coefficient one-forms equal

$$
\begin{align}
\tilde{\Omega}_a^b(z, d) &= \tilde{\Omega}_a^b(d) - i\ell_2 \delta_a^b (\ell_1 \Delta(d) + \ell_2 * \Delta(d))(\theta\theta), \tag{3.23} \\
\Omega^a(z, d) &= \ell_1 \Omega^a(d) + \ell_2 * \Omega^a(d) - 2i\ell_2 \bar{\omega}_\mu^a(d) \theta^\mu - 2\ell_2 * \bar{\chi}_\mu^a(d) \theta^\mu + 2\ell_2 \Omega^a(d) (\theta\theta).
\end{align}
$$

The last term in (3.21)

$$
\mathcal{L}_{24 \text{ susy}}(d) = \boldsymbol{\omega}_{(1)a}^{\mu}(z, d) Q_{(1)\mu}^{a} + \bar{\boldsymbol{\omega}}_{(1)}^{\mu a}(z, d) \bar{Q}_{(1)\mu a} + \boldsymbol{\omega}_{(3)a}^{\mu}(z, d) Q_{(3)\mu}^{a} + \bar{\boldsymbol{\omega}}_{(3)}^{\mu a}(z, d) \bar{Q}_{(3)\mu a}
$$

is the linear combination of the supersymmetry generators from eigenspaces $\mathfrak{g}_{1,3}$, and one-
forms

$$
\begin{align}
\boldsymbol{\omega}_{(1)a}^{\mu}(z, d) &= \ell_3 \omega_{(1)a}^{\mu}(d) - \frac{i}{2} \ell_2 \ell_4 \Omega_a(d) \theta^{\mu} - \frac{i}{2} \ell_2 \ell_4 * \Omega_a(d) \theta^{\mu} \nonumber \\
&\quad - \frac{1}{2} \ell_2 \ell_4 \omega_{(1)a}^{\mu}(d)(\theta\theta) - \frac{i}{2} \ell_2 \ell_4 * \chi_a^{\mu}(d)(\theta\theta), \tag{3.24} \\
\boldsymbol{\omega}_{(3)a}^{\mu}(z, d) &= \ell_4 \omega_{(3)a}^{\mu}(d) + \frac{i}{2} \ell_2 \ell_3 \Omega_a(d) \theta^{\mu} - \frac{i}{2} \ell_2 \ell_3 * \Omega_a(d) \theta^{\mu} \nonumber \\
&\quad - \frac{1}{2} \ell_2 \ell_3 \omega_{(3)a}^{\mu}(d)(\theta\theta) - \frac{i}{2} \ell_2 \ell_3 * \chi_a^{\mu}(d)(\theta\theta)
\end{align}
$$

and c.c. depend on the spectral parameter $z$ through the functions $\ell_2(z)$, $\ell_3(z)$ and $\ell_4(z)$.
From the zero-curvature condition it follows that functions $\ell_1(z)$, $\ell_2(z)$, $\ell_3(z)$ and $\ell_4(z)$
satisfy the same set of algebraic equations as in the case of the Lax connection of the
$OSp(4|6)/(SO(1,3) \times U(3))$ $\sigma$-model

$$
\ell_1^2 - \ell_2^2 = \ell_3 \ell_4 = 1, \quad (\ell_1 - \ell_2) \ell_4 = \ell_3, \quad (\ell_1 + \ell_2) \ell_3 = \ell_4. \tag{3.25}
$$

In the literature frequently used is the following solution of these equations

$$
\ell_1(z) = \frac{1}{2} \left( \frac{1}{z^2} + z^2 \right), \quad \ell_2(z) = \frac{1}{2} \left( \frac{1}{z^2} - z^2 \right), \quad \ell_3(z) = z, \quad \ell_4(z) = \frac{1}{z}. \tag{3.26}
$$

At $z=1$ one-forms (3.22)-(3.24) reduce to the osp(4|6) Cartan forms in conformal basis.
Then (3.21), as the Lax connection of the $\sigma$-model, reduces to the defining relation of these
Cartan forms (see Eqs. (2.1), (2.2), (2.5) and (2.11) of Section 2). Also the 1-form (3.21)
similarly to the Lax connection of the $\sigma$-model is invariant under the $\mathbb{Z}_4$ automorphism of the
osp(4|6) superalgebra if transformation of its generators is accompanied by transformation
of the spectral parameter $z \to -iz$. Accordingly from (3.26) it follows that the functions of
the spectral parameter that enter the Lax connection transform as $\ell_{1,2} \to -\ell_{1,2}$, $\ell_3 \to -i\ell_3$
and $\ell_4 \to i\ell_4$.
## 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

In the previous subsection, when proving vanishing of the curvature (3.20) of the one-form (3.21), used were also fermionic equations (3.16)-(3.19) corresponding to variation of the coordinates for broken supersymmetries. It was noted that, when the spinor coordinate $\theta^\mu$ turns to zero, these equations do not turn into identities. In this subsection it will be shown that they become the consequences of the fermionic equations of the $OSp(4|6)/(SO(1,3) \times U(3))$ $\sigma$-model. Note that such proof in the linearized approximation in the Grassmann coordinates was given in Ref. [17].

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

$$V_{+}^{ij} \mathcal{M}_{(1)\hat{a}\hat{\nu}}^{\mu\hat{b}} \begin{pmatrix} \omega_{(1)}{}_{jb}^{\nu} \\ \bar{\omega}_{(1)}{}_{j}^{\nu b} \end{pmatrix} = 0 \qquad (3.27)$$

and

$$V_{-}^{ij} \mathcal{M}_{(3)\hat{a}\hat{\nu}}^{\mu\hat{b}} \begin{pmatrix} \omega_{(3)}{}_{jb}^{\nu} \\ \bar{\omega}_{(3)}{}_{j}^{\nu b} \end{pmatrix} = 0 \qquad (3.28)$$

that involves 12 × 12 matrices

$$\mathcal{M}_{(1)\hat{a}\hat{\nu}}^{\mu\hat{b}} = \begin{pmatrix} \delta_a^b m_i^\mu \nu & -i \delta_\nu^\mu \varepsilon_{acb} \Omega_i^c \\ -i \delta_\nu^\mu \varepsilon^{acb} \Omega_{ic} & -\delta_b^a \bar{m}_i^\mu \nu \end{pmatrix}, \quad \mathcal{M}_{(3)\hat{a}\hat{\nu}}^{\mu\hat{b}} = \begin{pmatrix} -\delta_a^b \bar{m}_i^\mu \nu & i \delta_\nu^\mu \varepsilon_{acb} \Omega_i^c \\ i \delta_\nu^\mu \varepsilon^{acb} \Omega_{ic} & \delta_b^a m_i^\mu \nu \end{pmatrix}.$$

Their diagonal blocks are defined as

$$m_i^\mu \nu = -i G_i^{0'm} \sigma_m^\mu \nu + \Delta_i \delta_\nu^\mu, \quad \bar{m}_i^\mu \nu = i G_i^{0'm} \sigma_m^\mu \nu + \Delta_i \delta_\nu^\mu.$$

Note that, when $\theta^\mu = 0$, fermionic equations of the $AdS_4 \times CP^3$ superstring in the partial $\kappa$-symmetry gauge (3.14) and (3.15), obtained in the previous subsection, turn into Eqs. (3.27) and (3.28). Since the result of the action of the projectors $V_\pm^{ij}$ on a $D = 1+1$ vector factorizes (see Eq. (2.32) of Section 2)

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

one can write Eqs. (3.27) and (3.28) in the form

$$\mathcal{M}_{(1)+\hat{a}\hat{\nu}}^{\mu\hat{b}} \begin{pmatrix} \omega_{(1)}{}_{-b}^{\nu} \\ \bar{\omega}_{(1)}{}_{-}^{\nu b} \end{pmatrix} = 0, \qquad (3.29)$$

$$\mathcal{M}_{(3)-\hat{a}\hat{\nu}}^{\mu\hat{b}} \begin{pmatrix} \omega_{(3)}{}_{+b}^{\nu} \\ \bar{\omega}_{(3)}{}_{+}^{\nu b} \end{pmatrix} = 0, \qquad (3.30)$$
where

$$ \mathcal{M}_{(1)+\hat{a}\nu}^{\mu\hat{b}} = \begin{pmatrix} \delta_a^b m_{+}^{\mu}{}_{\nu} & -i\delta_{\nu}^{\mu} \varepsilon_{acb} \Omega_{+}{}^{c} \\ -i\delta_{\nu}^{\mu} \varepsilon^{acb} \Omega_{+c} & -\delta_b^a \bar{m}_{+}{}^{\mu}{}_{\nu} \end{pmatrix}, \quad \mathcal{M}_{(3)-\hat{a}\nu}^{\mu\hat{b}} = \begin{pmatrix} -\delta_a^b \bar{m}_{-}{}^{\mu}{}_{\nu} & i\delta_{\nu}^{\mu} \varepsilon_{acb} \Omega_{-}{}^{c} \\ i\delta_{\nu}^{\mu} \varepsilon^{acb} \Omega_{-c} & \delta_b^a m_{-}{}^{\mu}{}_{\nu} \end{pmatrix} $$

are 12 × 12 matrices of the rank eight. It can be shown that 2 × 2 matrices $m_{\pm}^{\mu}{}_{\nu}$ and $\bar{m}_{\pm}{}^{\mu}{}_{\nu}$ satisfy the relations

$$ m_{\pm}^{\mu}{}_{\nu} \bar{m}_{\pm}{}^{\nu}{}_{\lambda} = \delta_{\lambda}^{\mu} (G_{\pm}^{\quad 0'}{}_{m} G_{\pm}^{\quad 0'm} + \Delta_{\pm}^{2}), \qquad (3.31) $$

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

$$ G_{\pm}^{\quad 0'}{}_{m} G_{\pm}^{\quad 0'm} + \Delta_{\pm}^{2} + \Omega_{\pm a} \Omega_{\pm}^{a} = 0. $$

Under conditions $\Omega_{\pm a} \neq 0$, $\Omega_{\pm}^{a} \neq 0$ that determine applicability of the $\sigma$-model to description of the $AdS_4 \times CP^3$ superstring dynamics the right-hand side in (3.31) does not turn to zero.

Consider Eqs. (3.16)-(3.19) when $\theta^{\mu} = 0$. They can be written in terms of the mappings to the world-sheet of the osp(4|6)/(so(1, 3) × u(3)) Cartan forms associated with the generators that have definite eigenvalues under the $\mathbb{Z}_4$ automorphism

$$ \Omega_{+}{}^{a} \omega_{-(1)}{}^{\mu}_{a} = \Omega_{+a} \bar{\omega}_{-(1)}{}^{\mu a} = 0, \qquad (3.32) $$

$$ \Omega_{-}{}^{a} \omega_{+(3)}{}^{\mu}_{a} = \Omega_{-a} \bar{\omega}_{+(3)}{}^{\mu a} = 0. \qquad (3.33) $$

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

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

$$ \Pi_{(1)\hat{a}\nu}^{\mu\hat{b}} = \begin{pmatrix} \delta_a^b \delta_{\nu}^{\mu} & -i \frac{m_{+}^{\mu}{}_{\nu}}{G_{+} \cdot G_{+} + \Delta_{+}^2} \varepsilon_{acb} \Omega_{+}{}^{c} \\ i \frac{\bar{m}_{+}{}^{\mu}{}_{\nu}}{G_{+} \cdot G_{+} + \Delta_{+}^2} \varepsilon^{acb} \Omega_{+c} & \delta_b^a \delta_{\nu}^{\mu} \end{pmatrix} $$

that satisfies the condition $\Pi_{(1)}^3 - 3\Pi_{(1)}^2 + 2\Pi_{(1)} = 0$, allows to bring this equation to the block-diagonal form

$$ \begin{pmatrix} m_{+}{}^{\mu}{}_{\nu} \Omega_{+a} \Omega_{+}{}^{b} & 0 \\ 0 & -\bar{m}_{+}{}^{\mu}{}_{\nu} \Omega_{+}{}^{a} \Omega_{+b} \end{pmatrix} \begin{pmatrix} \omega_{(1)-}{}^{\nu} \\ \bar{\omega}_{(1)-}{}^{\nu b} \end{pmatrix} = 0. \qquad (3.34) $$

Using the relation (3.31) the system (3.34) transforms to equivalent system of four equations

$$ \Omega_{+}{}^{a} \omega_{(1)-}{}^{\mu}_{a} = \Omega_{+a} \bar{\omega}_{(1)-}{}^{\mu a} = 0 $$

that coincide with Eq. (3.32). In analogous way it can be verified that Eq. (3.33) follows from (3.30).
### 3.3 Classical integrability of equations of the massless superparticle and D0-brane in the $AdS_4 \times CP^3$ superbackground

In this subsection it will be proved classical integrability of the equations of massless superparticle in the $OSp(4|6)/(SO(1,3) \times U(3))$ supercoset space and in the $AdS_4 \times CP^3$ superspace. Similarly to connection between the $OSp(4|6)/(SO(1,3) \times U(3))$ $\sigma$-model and the $AdS_4 \times CP^3$ superstring, the Lagrangian of massless $OSp(4|6)/(SO(1,3) \times U(3))$ superparticle follows from that of the massless superparticle in the $AdS_4 \times CP^3$ superspace upon imposing vanishing condition on eight coordinates of the sector of broken supersymmetries. This condition represents partial $\kappa$-symmetry gauge fixing when the superparticle moves both in the $AdS_4$ and in the $CP^3$ space. It will be shown that in this case equations that follow from variation of the $AdS_4 \times CP^3$ superparticle action on the coordinates of the sector of broken supersymmetries become the consequences of other 24 fermionic equations. The latter turn into fermionic equations of the $OSp(4|6)/(SO(1,3) \times U(3))$ superparticle.

As is known the massless $OSp(4|6)/(SO(1,3) \times U(3))$ superparticle corresponds to the tension-to-infinity limit of the $OSp(4|6)/(SO(1,3) \times U(3))$ $\sigma$-model. This implies that their integrable structures are connected. In the paragraph 3.3.1 it will be established the relation between the components of the superparticle's Lax pair and the Lax connection of the $\sigma$-model.

In the final paragraph 3.3.3 it will be proved classical integrability of the equations that follow from the action of D0-brane in the $AdS_4 \times CP^3$ superbackground [48].

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

Action of the massless superparticle model in the $OSp(4|6)/(SO(1,3) \times U(3))$ supercoset space has the following form [10]

$$S_{\text{sparticle}}^{\text{OSp}(4|6)/(SO(1,3) \times U(3))} = \int_L \frac{d\tau}{e} \left( G_{\tau}^{0'm'} G_{\tau m'}^{0'} + \Omega_{\tau a} \Omega_{\tau}^{a} \right). \qquad (3.35)$$

Taking into account (2.24) it can be written in terms of the Cartan forms for generators of the conformal algebra $conf(1,2)$

$$S_{\text{sparticle}}^{\text{OSp}(4|6)/(SO(1,3) \times U(3))} = \int_L \frac{d\tau}{e} \left( \frac{1}{4}(\omega_{\tau}^m + c_{\tau}^m)(\omega_{\tau m} + c_{\tau m}) + \Delta_{\tau}\Delta_{\tau} + \Omega_{\tau a}\Omega_{\tau}^{a} \right).$$

Variation of the action with respect to the Lagrange multiplier $e(\tau)$ leads to the mass-shell constraint

$$\frac{1}{4}(\omega_{\tau}^{m} + c_{\tau}^{m})(\omega_{\tau m} + c_{\tau m}) + \Delta_{\tau}\Delta_{\tau} + \Omega_{\tau a}\Omega_{\tau}^{a} = 0. \qquad (3.36)$$

Since this constraint is not used to prove integrability of the dynamical equations of the superparticle we set $e(\tau) = 1$ by fixing the reparametrization symmetry of the action. Similarly to derivation of the superstring equations in the paragraph 3.1.2 consider as independent
variation parameters the Cartan forms, that correspond to generators from $\mathfrak{g}_{1,2,3}$ eigenspaces, with differentials of the coordinates replaced by their variations. As a result we come to the set of dynamical equations of the superparticle

$$ 
\begin{aligned}
-\frac{1}{2} \frac{\delta S_{\text{sparticle}}^{\text{OSp}(4|6)/(SO(1,3) \times U(3))}}{\delta G^{0'm}(\delta)} &= \frac{dG_{\tau}^{0'm}}{d\tau} + 2G_{\tau}^{mn}G_{\tau}^{0'}{}_{n} + 2G_{\tau}^{3m}\Delta_{\tau} = 0, \\
-\frac{1}{2} \frac{\delta S_{\text{sparticle}}^{\text{OSp}(4|6)/(SO(1,3) \times U(3))}}{\delta \Delta(\delta)} &= \frac{d\Delta_{\tau}}{d\tau} - 2G_{\tau}^{3m}G_{\tau}^{0'}{}_{m} = 0, \qquad (3.37) \\
-\frac{\delta S_{\text{sparticle}}^{\text{OSp}(4|6)/(SO(1,3) \times U(3))}}{\delta \Omega_{a}(\delta)} &= \frac{d\Omega_{\tau}^{a}}{d\tau} + i\Omega_{\tau}^{b}(\tilde{\Omega}_{\tau b}^{a} + \delta_{b}^{a}\tilde{\Omega}_{\tau c}^{c}) = 0
\end{aligned}
$$

and

$$ 
\begin{aligned}
-\frac{1}{4} \frac{\delta S_{\text{sparticle}}^{\text{OSp}(4|6)/(SO(1,3) \times U(3))}}{\delta \tilde{\omega}_{(1)}^{a}(\delta)} &= iG_{\tau}^{0'm} \tilde{\sigma}_{m}^{\mu\nu} \omega_{(1)\tau\nu a} + \Delta_{\tau}\omega_{(1)\tau a}^{\mu} - i\epsilon_{abc}\Omega_{\tau}^{b}\bar{\omega}_{(1)\tau}^{\mu c} = 0, \\
-\frac{1}{4} \frac{\delta S_{\text{sparticle}}^{\text{OSp}(4|6)/(SO(1,3) \times U(3))}}{\delta \tilde{\omega}_{(3)}^{a}(\delta)} &= iG_{\tau}^{0'm} \tilde{\sigma}_{m}^{\mu\nu} \omega_{(3)\tau\nu a} - \Delta_{\tau}\omega_{(3)\tau a}^{\mu} + i\epsilon_{abc}\Omega_{\tau}^{b}\bar{\omega}_{(3)\tau}^{\mu c} = 0
\end{aligned}
\qquad (3.38) $$

and also c.c. ones.

Let us present Eqs. (3.37) and (3.38) in the form of the Lax equation

$$ \frac{d\mathcal{L}_{\text{OSp}(4|6)/(SO(1,3) \times U(3))}}{d\tau} + [\mathfrak{M}, \mathcal{L}_{\text{OSp}(4|6)/(SO(1,3) \times U(3))}] = 0 \qquad (3.39) $$

with the components of the Lax pair

$$ \mathcal{L}_{\text{OSp}(4|6)/(SO(1,3) \times U(3))} = 2G_{\tau}^{0'm} M_{0'm} + \Delta_{\tau} D + \Omega_{\tau a} T^{a} + \Omega_{\tau}^{a} T_{a} \in \mathfrak{c}_{\tau}^{2} \mathfrak{g}_{2} \qquad (3.40) $$

and

$$ \mathfrak{M} = \mathcal{G}^{-1} \frac{d}{d\tau} \mathcal{G} = \sum_{j \in 0,1,2,3} \mathfrak{c}_{\tau}^{j} \mathfrak{g}_{j} \in \text{osp}(4|6) \qquad (3.41) $$

(see Eq. (2.12) from Section 2). Since $\mathcal{L}_{\text{OSp}(4|6)/(SO(1,3) \times U(3))}$ is determined by the world-line projections of the Cartan forms that enter the superparticle's action (3.35) one can present it as a $\mathfrak{g}_2$-valued differential operator applied to the action functional

$$ \mathcal{L}_{\text{OSp}(4|6)/(SO(1,3) \times U(3))} = \left( M_{0'm} \frac{\partial}{\partial G_{\tau}^{0'm}} + \frac{1}{2} D \frac{\partial}{\partial \Delta_{\tau}} + T^{a} \frac{\partial}{\partial \Omega_{\tau}^{a}} + T_{a} \frac{\partial}{\partial \Omega_{\tau a}} \right) S_{\text{sparticle}}^{\text{OSp}(4|6)/(SO(1,3) \times U(3))} \qquad (3.42) $$

At the end of this paragraph let us establish connection between the components of the Lax pair (3.40) and (3.41) and the Lax connections of the $\text{OSp}(4|6)/(SO(1,3) \times U(3))$ $\sigma$-model and the $AdS_4 \times \mathbb{CP}^3$ superstring. The Lax connection of the $\sigma$-model should follow from that of the superstring when coordinates of the sector of broken supersymmetries turn to zero. In particular, if in (3.21) one sets $\theta^{\mu} = 0$ then this one-form turns into the Lax connection of the $\sigma$-model. The general structure of the Lax connection $\mathcal{L}_i$ is the following

$$ \mathcal{L}_i = \hat{\mathcal{L}}_i + *\tilde{\mathcal{L}}_i. \qquad (3.43) $$

The second term on the right-hand side includes components $*\tilde{\mathcal{L}}_i = \epsilon_{ij} \eta^{jk} \tilde{\mathcal{L}}_k = \eta_{ij} \epsilon^{jk} \tilde{\mathcal{L}}_k$ of the one-form $*\tilde{\mathcal{L}}(d)$ dual to $\tilde{\mathcal{L}}(d)$, where $\eta_{ij}$ and $\eta^{ij}$ are world-sheet Minkowski metric and its
inverse. Both summands in (3.43) depend on the functions $\ell_1, \ell_2, \ell_3$ and $\ell_4$ of the spectral parameter $z$ that satisfy equations (3.25). One of their solutions was presented in (3.26). The Lax connection (3.43) satisfies the zero curvature condition

$$ \partial_{\sigma} \mathcal{L}_{\tau} - \partial_{\tau} \mathcal{L}_{\sigma} - [\mathcal{L}_{\tau}, \mathcal{L}_{\sigma}] = 0. \qquad (3.44) $$

Transition to the superparticle's Lax pair corresponds to dropping the dependence of the superspace coordinate fields on the world-sheet space-like coordinate $\sigma$ and taking the limit $z \to 1$. In this case $\hat{\mathcal{L}}_{\sigma} = *\hat{\mathcal{L}}_{\tau} = 0$. Then the Lax pair components are defined by the limiting values of the non-zero components of the Lax connection

$$ \mathcal{L} = \lim_{z \to 1} \frac{1}{\ell_2} * \tilde{\mathcal{L}}_{\sigma}, \quad \mathfrak{M} = \lim_{z \to 1} \hat{\mathcal{L}}_{\tau} $$

and the zero curvature condition (3.44) transforms into the Lax equation for the superparticle. In particular, components of the Lax connection of the $OSp(4|6)/(SO(1,3) \times U(3))$ $\sigma$-model in this limit transfer to the Lax pair components of the $OSp(4|6)/(SO(1,3) \times U(3))$ superparticle (3.40) and (3.41), and the zero curvature condition – to the Lax equation (3.39).

Reasoning in the opposite direction we conclude that with the aid of known expression for the Lax pair of the superparticle it is possible to recover the dependence on the superspace coordinates (but not on the spectral parameter $z$) of the part of $\hat{\mathcal{L}}_i$ that does not include $\ell_2$ and of the part of $\tilde{\mathcal{L}}_i$ linear in $\ell_2$. So the form of the Lax pair components for the superparticle gives information on the structure of the superstring Lax connection. This explains the interest to presenting equations of the massless superparticle in the $AdS_4 \times CP^3$ superspace as the Lax equation that will be obtained in the next paragraph.

### 3.3.2 Classical integrability of equations of the massless superparticle in the $AdS_4 \times CP^3$ superspace

As was mentioned at the beginning of the subsection 3.1 the problem of proving integrability of equations of the $AdS_4 \times CP^3$ superstring remains unsolved. However, as will be shown in this paragraph it is possible to prove integrability of superstring equations in the limiting case of infinite tension described by the massless superparticle model. This problem was solved in our works [60] and [61] following the approach suggested in [58] that is based on partial fixing the $\kappa$-symmetry gauge in the sector of broken supersymmetries. First in [60] the Lax pair of the $OSp(4|6)/(SO(1,3) \times U(3))$ superparticle was extended by the contributions of four coordinates associated with broken Poincaré supersymmetries and then in [61] obtained was complete expression for the Lax pair of the $AdS_4 \times CP^3$ superparticle that takes into account the dependence on four coordinates for broken special conformal supersymmetries. Obtained components of the Lax pair enter the Lax equation that is satisfied on the $AdS_4 \times CP^3$ superparticle's equations.

To derive equations of the massless $AdS_4 \times CP^3$ superparticle one needs to construct its Lagrangian. Its form is determined by the $AdS_4 \times CP^3$ supervielbein bosonic components
that depend on all 24+8 Grassmann coordinates of the superspace. Expressions for them will be obtained below using, as before, the realizations of the osp(4|8) and osp(4|6) superalgebras as $D = 3 \mathcal{N} = 8$ and $\mathcal{N} = 6$ superconformal algebras.

We will consider the $OSp(4|8)/(SO(1,3) \times SO(7))$ supercoset element defined in (3.3). As was mentioned, its structure ensures independence of respective Cartan forms on the coordinate $y$ of the $S^1$ circle in the Hopf fibration $S^7 = \mathbb{CP}^3 \times S^1$. However, the $so(2,3)/so(1,3)$ Cartan forms that determine the supervielbein bosonic components in the tangent space to $AdS_4$ include summands proportional to $dy$ (see Eq. (3.1)). To remove such contributions and bring the $AdS_4 \times S^7$ supervielbein to the Kaluza-Klein ansatz it is necessary to perform local $SO(1,4)$ transformation in the tangent-space to $AdS_4 \times S^1$. Particular case of such transformation was considered in Section 2 when deriving the $AdS_4 \times \mathbb{CP}^3$ superstring Lagrangian in the $\kappa$-symmetry light-cone gauge (see Eqs. (2.99) and (2.100)). The entries of the $SO(1,4)$ Lorentz rotation matrix (2.99) are determined by the requirement of $dy$ independence of the transformed supervielbein components in the tangent space to $AdS_4$: $L_{n'}^{m'} G_y^{n'} + L_{11}^{m'} \Phi = 0$, where for the supervielbein components in the tangent space to $AdS_4$ and $S^1$ used was the representation (3.1) and $E^{11}(d) = \Phi dy + A(d)$. In general case elements of this matrix have the following form (see [13], [58])

$$L_{n'}^{m'} = \delta_{n'}^{m'} + \frac{\Phi - \Phi_L}{(G_y \cdot G_y)\Phi_L} G_y^{m'} G_{yn'}, \quad L_{11}^{m'} = -\frac{G_y^{m'}}{\Phi_L}, \quad L_{11}^{11} = \frac{G_{ym'}}{\Phi_L}, \quad L_{11}^{11} = \frac{\Phi}{\Phi_L},$$

where $\Phi_L = \exp(\frac{2\phi}{3}) = \sqrt{\Phi^2 + (G_y \cdot G_y)}$. So transformed bosonic components of the $D=11$ supervielbein equal

$$\mathcal{E}^{m'}(d) = (LE)^{m'}(d) = G^{m'}(d) + \frac{\Phi - \Phi_L}{(G_y \cdot G_y)\Phi_L} (G_y \cdot G(d)) G_y^{m'} - \frac{1}{\Phi_L} A(d) G_y^{m'}, \quad (3.45)$$

$$\mathcal{E}^{11}(d) = (LE)^{11}(d) = \Phi_L(dy + A_L(d)),$$

where

$$A_L(d) = \frac{1}{\Phi_L}((G_y \cdot G(d)) + \Phi A(d)) \quad (3.46)$$

is the Ramond-Ramond one-form potential. As was shown in the paragraph 3.1.1, for considered $OSp(4|8)/(SO(1,3) \times SO(7))$ supercoset element $G_y^{m'}$ and $\Phi$ depend only on the coordinates from the sector of broken supersymmetries

$$G_y^m = 2 \left(1 + \frac{1}{2}\eta^2 \bar{\eta}^2\right) (\theta \sigma^m \bar{\theta}) + 2\{1 - i[(\theta \bar{\eta}) + (\bar{\theta} \eta)]\}(\eta \sigma^m \bar{\eta}), \\ G_y^3 = 2[(\theta \bar{\eta}) - (\bar{\theta} \eta)], \\ \Phi = 1 - 2i[(\theta \bar{\eta}) + (\bar{\theta} \eta)] + 4[(\theta \eta)(\bar{\theta} \bar{\eta}) - (\theta \bar{\eta})(\bar{\theta} \eta)].$$

So the Lagrangian of massless $AdS_4 \times \mathbb{CP}^3$ superparticle

$$\mathcal{L}_{\text{sparticle}}^{AdS_4 \times \mathbb{CP}^3} = \frac{\Phi_L}{e} \left( \mathcal{E}_{\tau m'} \mathcal{E}_{\tau}^{m'} - \mathbf{E}_{\tau a} \mathbf{E}_{\tau}^{a} \right)$$
is expressed through the mappings to the world-line of the supervielbein bosonic components $\mathcal{E}^{m'}(d)$ and $\mathbf{E}_a(d)$, $\mathbf{E}^a(d)$. After substitution of the expressions (3.45) it can be written in the form

$$ \mathcal{L}_{\text{sparticle}}^{\text{AdS}_4 \times \mathbb{CP}^3} = \frac{\Phi_L}{e} (G_{\tau m'} G_{\tau}^{m'} - \Phi_L^{-2} [(G_{\tau} \cdot G_y)^2 - G_y^2 A_{\tau}^2 + 2\Phi A_{\tau} (G_{\tau} \cdot G_y)] - \mathbf{E}_{\tau a} \mathbf{E}_{\tau}^a). \quad (3.47) $$

As in the case of massless $OSp(4|6)/(SO(1,3) \times U(3))$ superparticle, the mass-shell constraint, obtained by varying the action

$$ S_{\text{sparticle}}^{\text{AdS}_4 \times \mathbb{CP}^3} = \int_L d\tau \mathcal{L}_{\text{sparticle}}^{\text{AdS}_4 \times \mathbb{CP}^3} \quad (3.48) $$

on the Lagrange multiplier $e(\tau)$, is not used in course of the proof of integrability of the dynamical equations. So we set $e(\tau) = 1$. Let us also note that by redefinition of $e(\tau)$ it is possible to 'absorb' the overall factor of $\Phi_L$. Corresponding expression for the Lax pair component $\mathcal{L}$ comes about by putting $\Phi_L = 1$ in (3.53)-(3.55).

The structure of the $OSp(4|8)/(SO(1,3) \times SO(7))$ supercoset element (3.3) is such that 1-forms $G^{m'}(d)$, $\mathbf{E}_a(d)$, $\mathbf{E}^a(d)$ and $A(d)$ in (3.47) can be expanded on the $osp(4|6)$ Cartan forms and differentials of the coordinates of the sector of broken supersymmetries

$$ \begin{aligned} G^{m'}(d) = G^{0'n}(d)M_n^{m'} + G^{3n}(d)N_n^{m'} + \Delta(d)L^{m'} + G^{kl}(d)K_{kl}^{m'} & \\ + q^{m'\mu}d\theta_\mu + \bar{q}^{m'\mu}d\bar{\theta}_\mu + s^{m'\mu}d\eta_\mu + \bar{s}^{m'\mu}d\bar{\eta}_\mu, & \end{aligned} \quad (3.49) $$

$$ \mathbf{E}_a(d) = i\Omega_a(d) + u_{(1)}{}^{\mu}\omega_{(1)\mu a}(d) + u_{(3)}{}^{\mu}\omega_{(3)\mu a}(d), $$

$$ \mathbf{E}^a(d) = i\Omega^a(d) + \bar{u}_{(1)}{}^{\mu}\bar{\omega}_{(1)}{}^a_{\mu}(d) + \bar{u}_{(3)}{}^{\mu}\bar{\omega}_{(3)}{}^a_{\mu}(d), $$

$$ \begin{aligned} A(d) = \tilde{\Omega}_a^a(d) + G^{0'm}(d)m_m + G^{3m}(d)n_m + \Delta(d)l + G^{mn}(d)k_{mn} & \\ + h^\mu d\theta_\mu + \bar{h}^\mu d\bar{\theta}_\mu + p^\mu d\eta_\mu + \bar{p}^\mu d\bar{\eta}_\mu. & \end{aligned} \quad (3.50) $$

In expansions (3.49) bosonic coefficients equal

$$ \begin{aligned} M_n^m = \delta_n^m [1 - (\theta\bar{\theta})(\eta\bar{\eta}) + \frac{1}{4}(\theta^2\bar{\theta}^2 + \eta^2\bar{\eta}^2) + \frac{1}{8}\theta^2\bar{\theta}^2\eta^2\bar{\eta}^2] & \\ -i(\theta\sigma_n\tilde{\sigma}^m\bar{\eta} + \bar{\theta}\sigma_n\tilde{\sigma}^m\eta) + 2\{1 - \frac{i}{2}[(\theta\bar{\eta}) + (\bar{\theta}\eta)]\} (\theta\sigma_n\bar{\theta})(\eta\sigma^m\bar{\eta}), & \end{aligned} $$

$$ \begin{aligned} N_n^m = \delta_n^m [-(\theta\bar{\theta})(\eta\bar{\eta}) + \frac{1}{4}(\theta^2\bar{\theta}^2 - \eta^2\bar{\eta}^2) + \frac{1}{8}\theta^2\bar{\theta}^2\eta^2\bar{\eta}^2] & \\ -i(\theta\sigma_n\tilde{\sigma}^m\bar{\eta} + \bar{\theta}\sigma_n\tilde{\sigma}^m\eta) + 2\{1 - \frac{i}{2}[(\theta\bar{\eta}) + (\bar{\theta}\eta)]\} (\theta\sigma_n\bar{\theta})(\eta\sigma^m\bar{\eta}), & \end{aligned} $$

$$ M_n^3 = N_n^3 = [(\theta\bar{\eta}) - (\bar{\theta}\eta)](\theta\sigma_n\bar{\theta}), $$

$$ L^m = [(\bar{\theta}\eta) - (\theta\bar{\eta})](\eta\sigma^m\bar{\eta}), \quad -L^3 = 1 + i[(\theta\bar{\eta}) + (\bar{\theta}\eta)], $$

$$ K_{kl}^m = -\frac{i}{2}\epsilon_{kl}^m \{ (1 + \frac{1}{2}\eta^2\bar{\eta}^2)(\theta\bar{\theta}) + (\eta\bar{\eta}) \} + \frac{1}{2}[(\bar{\theta}\sigma_{kl}\eta) - (\theta\sigma_{kl}\bar{\eta})](\eta\sigma^m\bar{\eta}), $$

$$ K_{kl}^3 = -\frac{i}{2}[(\bar{\theta}\sigma_{kl}\eta) + (\theta\sigma_{kl}\bar{\eta})] $$

and

$$ u_{(1,3)}{}^{\mu} = \mp 2\{\theta^{\mu} \pm i\eta^{\mu}[1 \pm (\theta\bar{\theta})]\}, \quad \bar{u}_{(1,3)}{}^{\mu} = \mp 2\{\bar{\theta}^{\mu} \mp i\bar{\eta}^{\mu}[1 \mp (\theta\bar{\theta})]\}. $$
Respectively bosonic coefficients in (3.50) have the form

$$
\begin{align*}
m_m &= \{1 - i[(\theta\bar{\eta}) + (\bar{\theta}\eta)]\} (\theta\sigma_m\bar{\theta}) + (1 - \frac{1}{2}\theta^2\bar{\theta}^2) (\eta\sigma_m\bar{\eta}), \\
n_m &= \{1 - i[(\theta\bar{\eta}) + (\bar{\theta}\eta)]\} (\theta\sigma_m\bar{\theta}) - (1 + \frac{1}{2}\theta^2\bar{\theta}^2) (\eta\sigma_m\bar{\eta}) \\
l &= (\bar{\theta}\eta) - (\theta\bar{\eta}), \quad k_{mn} = \frac{1}{2}[(\bar{\theta}\sigma_{mn}\eta) - (\theta\sigma_{mn}\bar{\eta})] - i(\theta\bar{\theta})(\eta\sigma_{mn}\bar{\eta}).
\end{align*}
$$

Non-zero Grassmann-odd coefficients in (3.49) and (3.50) are given by

$$
\begin{align*}
q^{m\mu} &= \frac{i}{2} \left(1 + \frac{1}{2}\eta^2\bar{\eta}^2\right) \tilde{\sigma}^{m\mu\nu}\bar{\theta}_\nu + \frac{1}{2}\bar{\eta}^2\tilde{\sigma}^{m\mu\nu}\eta_\nu, \quad q^{3\mu} = -i\bar{\eta}^\mu, \\
s^{m\mu} &= \frac{i}{2}\tilde{\sigma}^{m\mu\nu}\bar{\eta}_\nu, \quad h^\mu = -\bar{\eta}^\mu + i(\bar{\theta}\eta)\bar{\eta}^\mu - i(\bar{\theta}\bar{\eta})\eta^\mu
\end{align*}
$$

and c.c. expressions. As is seen all of the above coefficients are functions of eight fermionic coordinates for broken supersymmetries only.

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

$$
\begin{align}
-\frac{1}{2} \frac{\delta S_{\text{sparticle}}^{\text{AdS}_4 \times \mathbb{CP}^3}}{\delta G_{0'm}^{0'm}} &= \dot{G}_{\tau}^{0'm} + 2(G_{\tau}^{m} G_{\tau}^{0'n} - G_{\tau}^{m} G_{\tau}^{0'n} + \Delta_{\tau} G_{\tau}^{3m} - \Delta_{\tau} G_{\tau}^{3m}) \nonumber \\
&\quad -2i (\omega_{(1)\tau a} \sigma^m \bar{\omega}_{(1)\tau}^a - \omega_{(1)\tau a} \sigma^m \bar{\omega}_{(1)\tau}^{a*} + \omega_{(3)\tau a} \sigma^m \bar{\omega}_{(3)\tau}^a - \omega_{(3)\tau a} \sigma^m \bar{\omega}_{(3)\tau}^{a*}) = 0, \nonumber \\
-\frac{1}{2} \frac{\delta S_{\text{sparticle}}^{\text{AdS}_4 \times \mathbb{CP}^3}}{\delta \Delta(\delta)} &= \dot{\Delta}_{\tau} + 2G_{\tau}^{0'm} G_{\tau}^{3m} - 2G_{\tau 3m} G_{\tau}^{0'm} \tag{3.51} \\
-2 (\omega_{(1)\tau a} \bar{\omega}_{(1)\tau}^a - \omega_{(1)\tau a} \bar{\omega}_{(1)\tau}^{a*} - \omega_{(3)\tau a} \bar{\omega}_{(3)\tau}^a + \omega_{(3)\tau a} \bar{\omega}_{(3)\tau}^{a*}) &= 0, \nonumber \\
-\frac{\delta S_{\text{sparticle}}^{\text{AdS}_4 \times \mathbb{CP}^3}}{\delta \Omega_a(\delta)} &= \dot{\Omega}_{\tau}^a + i\Omega_{\tau}^b (\tilde{\Omega}_{\tau b}^a + \delta_b^a \tilde{\Omega}_{\tau c}^b) - 4i\mathbf{w}_{\tau} \Omega_{\tau}^a \nonumber \\
&\quad +4i\varepsilon^{abc} (\omega_{(1)\tau b} \omega_{(1)\tau mc} - \omega_{(3)\tau b} \omega_{(3)\tau mc}) = 0. \nonumber
\end{align}
$$

24 fermionic equations corresponding to variation parameters $\bar{\omega}_{(1)}_{\mu}^{a}(\delta)$, $\bar{\omega}_{(3)}_{\mu}^{a}(\delta)$ and c.c. are given by the expressions

$$
\begin{align}
\frac{1}{4} \frac{\delta S_{\text{sparticle}}^{\text{AdS}_4 \times \mathbb{CP}^3}}{\delta \bar{\omega}_{(1)}_{\mu}^{a}(\delta)} &= \dot{\omega}_{(3)\tau a}^{\mu} + \frac{1}{2} \left( G_{\tau}^{mn} \omega_{(3)\tau a}^{\nu} - G_{\tau}^{mn} \omega_{(3)\tau a}^{\nu} \right) \sigma_{mn\nu}^{\mu} \nonumber \\
&\quad + i \tilde{\sigma}_{m}^{\mu\nu} \left( G_{\tau}^{3m} \omega_{(3)\tau\nu a} - G_{\tau}^{3m} \omega_{(3)\tau\nu a} \right) + i \tilde{\sigma}_{m}^{\mu\nu} \left( G_{\tau}^{0'm} \omega_{(1)\tau\nu a} - G_{\tau}^{0'm} \omega_{(1)\tau\nu a} \right) \nonumber \\
&\quad + \Delta_{\tau} \omega_{(1)\tau a}^{\mu} - \Delta_{\tau} \omega_{(1)\tau a}^{\mu} - i \left( \tilde{\Omega}_{\tau a}^{b} - \delta_{a}^{b} \tilde{\Omega}_{\tau c}^{b} \right) \omega_{(3)\tau b}^{\mu} \nonumber \\
&\quad - 2i \mathbf{w}_{\tau} \omega_{(3)\tau a}^{\mu} - i \varepsilon_{abc} \left( \Omega_{\tau}^{b} \bar{\omega}_{(1)\tau}^{\mu c} - \Omega_{\tau}^{b} \bar{\omega}_{(1)\tau}^{\mu c} \right) = 0, \tag{3.52} \\
-\frac{1}{4} \frac{\delta S_{\text{sparticle}}^{\text{AdS}_4 \times \mathbb{CP}^3}}{\delta \bar{\omega}_{(3)}_{\mu}^{a}(\delta)} &= \dot{\omega}_{(1)\tau a}^{\mu} + \frac{1}{2} \left( G_{\tau}^{mn} \omega_{(1)\tau a}^{\nu} - G_{\tau}^{mn} \omega_{(1)\tau a}^{\nu} \right) \sigma_{mn\nu}^{\mu} \nonumber \\
&\quad - i \tilde{\sigma}_{m}^{\mu\nu} \left( G_{\tau}^{3m} \omega_{(1)\tau\nu a} - G_{\tau}^{3m} \omega_{(1)\tau\nu a} \right) - i \tilde{\sigma}_{m}^{\mu\nu} \left( G_{\tau}^{0'm} \omega_{(3)\tau\nu a} - G_{\tau}^{0'm} \omega_{(3)\tau\nu a} \right) \nonumber \\
&\quad + \Delta_{\tau} \omega_{(3)\tau a}^{\mu} - \Delta_{\tau} \omega_{(3)\tau a}^{\mu} - i \left( \tilde{\Omega}_{\tau a}^{b} - \delta_{a}^{b} \tilde{\Omega}_{\tau c}^{b} \right) \omega_{(1)\tau b}^{\mu} \nonumber \\
&\quad - 2i \mathbf{w}_{\tau} \omega_{(1)\tau a}^{\mu} - i \varepsilon_{abc} \left( \Omega_{\tau}^{b} \bar{\omega}_{(3)\tau}^{\mu c} - \Omega_{\tau}^{b} \bar{\omega}_{(3)\tau}^{\mu c} \right) = 0. \nonumber
\end{align}
$$
In (3.51) and (3.52) the following bosonic and fermionic quantities have been introduced

$$ \begin{pmatrix} \mathbf{G}_{\tau \ m}^{0'} \\ \mathbf{\Delta}_{\tau} \\ -\mathbf{G}_{\tau 3m} \\ -\frac{1}{2}\mathbf{G}_{\tau kl} \end{pmatrix} = \Phi_L \begin{pmatrix} M_m^{n'} \\ L^{n'} \\ N_m^{n'} \\ K_{kl}^{n'} \end{pmatrix} \left[ G_{\tau n'} - \frac{1}{\Phi_L^2}((G_\tau \cdot G_y) + \Phi A_\tau)G_{yn'} \right] \\ + \frac{1}{\Phi_L} \begin{pmatrix} m_m \\ l \\ n_m \\ k_{kl} \end{pmatrix} (G_y^2 A_\tau - \Phi(G_\tau \cdot G_y)), \qquad (3.53) $$

$$ \Omega_{\tau}^{a} = -i\Phi_{L}\mathbf{E}_{\tau}^{a}, \quad \bar{\Omega}_{\tau a} = -i\Phi_{L}\mathbf{E}_{\tau a}, \quad \mathbf{w}_{\tau} = \frac{1}{2\Phi_{L}}(G_{y}^{2}A_{\tau} - \Phi(G_{\tau} \cdot G_{y})) \qquad (3.54) $$

and also

$$ \begin{align} \omega_{(1)\tau a}^{\mu} &= -\frac{1}{4} \Phi_L \mathbf{E}_{\tau a} \bar{u}_{(3)}^{\mu}, & \omega_{(3)\tau a}^{\mu} &= \frac{1}{4} \Phi_L \mathbf{E}_{\tau a} \bar{u}_{(1)}^{\mu}, \tag{3.55} \\ \bar{\omega}_{(1)\tau}^{\mu a} &= \frac{1}{4} \Phi_L \mathbf{E}_{\tau}^{a} u_{(3)}^{\mu}, & \bar{\omega}_{(3)\tau}^{\mu a} &= -\frac{1}{4} \Phi_L \mathbf{E}_{\tau}^{a} u_{(1)}^{\mu} \nonumber \end{align} $$

that enter the Lax pair component $\mathcal{L}_{AdS_4 \times \mathbb{CP}^3}$ (3.57). Besides variation of the action (3.48) on the coordinates of the sector of broken supersymmetries yields other eight fermionic equations

$$ \begin{aligned} & -\frac{d}{d\tau} \left\{ \Phi_L \left[ G_{\tau m'} - \frac{1}{\Phi_L^2}((G_\tau \cdot G_y) + \Phi A_\tau)G_{ym'} \right] \frac{\partial G_\tau^{m'}}{\partial \dot{v}_\mu} \right. \\ & \left. \qquad + \frac{1}{\Phi_L} (G_y^2 A_\tau - \Phi(G_\tau \cdot G_y)) \frac{\partial A_\tau}{\partial \dot{v}_\mu} \right\} + \frac{1}{2} \frac{\partial \mathcal{L}_{\text{sparticle}}^{AdS_4 \times \mathbb{CP}^3}}{\partial \Phi_L} \frac{\partial \Phi_L}{\partial v_\mu} + \frac{1}{\Phi_L} A_\tau \left( A_\tau \left( G_y \cdot \frac{\partial G_y}{\partial v_\mu} \right) \right. \\ & \qquad \left. - (G_\tau \cdot G_y) \frac{\partial \Phi}{\partial v_\mu} \right) + \Phi_L \left[ G_{\tau m'} \frac{\partial G_\tau^{m'}}{\partial v_\mu} - \frac{1}{2} \left( \mathbf{E}_{\tau a} \frac{\partial \mathbf{E}_\tau^a}{\partial v_\mu} + \mathbf{E}_\tau^a \frac{\partial \mathbf{E}_{\tau a}}{\partial v_\mu} \right) \right] \\ & \qquad \left. - \frac{1}{\Phi_L} \left[ \left( (G_\tau \cdot G_y) + \Phi A_\tau \right) \frac{\partial (G_\tau \cdot G_y)}{\partial v_\mu} + (G_y^2 A_\tau - \Phi(G_\tau \cdot G_y)) \frac{\partial A_\tau}{\partial v_\mu} \right] = 0. \end{aligned} \qquad (3.56) $$

Eqs. (3.56) were presented in concise form due to introduced quantities $\frac{\partial G_\tau^{m'}}{\partial \dot{v}_\mu} = (q^{m'\mu}, \bar{q}^{m'\mu}, s^{m'\mu}, \bar{s}^{m'\mu})$ and $\frac{\partial A_\tau}{\partial \dot{v}_\mu} = (h^\mu, \bar{h}^\mu, p^\mu, \bar{p}^\mu)$, where $v_\mu = (\theta_\mu, \bar{\theta}_\mu, \eta_\mu, \bar{\eta}_\mu)$. Also taking into account the expansions (3.49) and (3.50) we have

$$ \begin{aligned} \frac{\partial G_{\tau}^{m'}}{\partial v_{\mu}} ={}& G_{\tau}^{0'n} \frac{\partial M_{n}^{m'}}{\partial v_{\mu}} + G_{\tau}^{3n} \frac{\partial N_{n}^{m'}}{\partial v_{\mu}} + \Delta_{\tau} \frac{\partial L^{m'}}{\partial v_{\mu}} + G_{\tau}^{kl} \frac{\partial K_{kl}^{m'}}{\partial v_{\mu}} \\ & - \dot{\theta}_{\nu} \frac{\partial q^{m'\nu}}{\partial v_{\mu}} - \dot{\theta}_{\nu} \frac{\partial \bar{q}^{m'\nu}}{\partial v_{\mu}} - \dot{\eta}_{\nu} \frac{\partial s^{m'\nu}}{\partial v_{\mu}} - \dot{\bar{\eta}}_{\nu} \frac{\partial \bar{s}^{m'\nu}}{\partial v_{\mu}}, \\ - \frac{\partial \mathbf{E}_{\tau a}}{\partial v_{\mu}} ={}& \omega_{(1)\tau\nu a} \frac{\partial u_{(1)}^{\nu}}{\partial v_{\mu}} + \omega_{(3)\tau\nu a} \frac{\partial u_{(3)}^{\nu}}{\partial v_{\mu}}, \quad - \frac{\partial \mathbf{E}_{\tau}^{a}}{\partial v_{\mu}} = \bar{\omega}_{(1)\tau\nu}^{a} \frac{\partial \bar{u}_{(1)}^{\nu}}{\partial v_{\mu}} + \bar{\omega}_{(3)\tau\nu}^{a} \frac{\partial \bar{u}_{(3)}^{\nu}}{\partial v_{\mu}}. \end{aligned} $$

$\frac{\partial A_\tau}{\partial v_\mu}$ and $\frac{\partial (G_\tau \cdot G_y)}{\partial v_\mu}$ are defined analogously.

As in the case of the $\mathcal{O}Sp(4|6)/(SO(1,3) \times U(3))$ superparticle, equations of the super-particle in the $AdS_4 \times \mathbb{CP}^3$ superspace (3.51), (3.52) and (3.56) can be presented in the form of the Lax equation

$$ \frac{d\mathcal{L}_{AdS_4 \times \mathbb{CP}^3}}{d\tau} + [\mathfrak{M}, \mathcal{L}_{AdS_4 \times \mathbb{CP}^3}] = 0, $$
where the Lax pair component $\mathfrak{M}$ was defined in (3.41) and another component equals

$$ \begin{aligned} \mathcal{L}_{AdS_4 \times \mathbb{CP}^3} &= \mathcal{L}_{so(2,3)} + \mathcal{L}_{su(4)} + \mathcal{L}_{24\text{susy}} \in \text{osp}(4|6): \\ \mathcal{L}_{so(2,3)} &= 2\mathbf{G}_{\tau}^{0'm} M_{0'm} + \mathbf{\Delta}_{\tau} D + 2\mathbf{G}_{\tau}^{3m} M_{3m} + \mathbf{G}_{\tau}^{mn} M_{mn}, \tag{3.57} \\ \mathcal{L}_{su(4)} &= \mathbf{\Omega}_{\tau}^{a} T_a + \bar{\mathbf{\Omega}}_{\tau a} T^a + 4\mathbf{w}_{\tau} \tilde{V}_a^a, \\ \mathcal{L}_{24\text{susy}} &= \boldsymbol{\omega}_{(1)\tau a}^{\mu} Q_{(1)\mu}^{a} + \bar{\boldsymbol{\omega}}_{(1)\tau}^{\mu a} \bar{Q}_{(1)\mu a} + \boldsymbol{\omega}_{(3)\tau a}^{\mu} Q_{(3)\mu}^{a} + \bar{\boldsymbol{\omega}}_{(3)\tau}^{\mu a} \bar{Q}_{(3)\mu a} \end{aligned} $$

(cf. (3.39) and (3.40)). Taking into account expansions (3.49) and (3.50) it can be presented in the form of the osp(4|6)-valued differential operator applied to the superparticle's action (3.48)

$$ \begin{aligned} \mathcal{L}_{AdS_4 \times \mathbb{CP}^3} = & \left( M_{0'm} \frac{\partial}{\partial G_{\tau}^{0'm}} + \frac{1}{2} D \frac{\partial}{\partial \Delta_{\tau}} - M_{mn} \frac{\partial}{\partial G_{\tau mn}} - M_{3m} \frac{\partial}{\partial G_{\tau 3m}} \right. \\ & \left. + T_a \frac{\partial}{\partial \Omega_{\tau a}} + T^a \frac{\partial}{\partial \Omega_{\tau}^a} + \tilde{V}_a^a \frac{\partial}{\partial \tilde{\Omega}_{\tau b}^a} - \frac{1}{4} Q_{(1)\mu}^a \frac{\partial}{\partial \tilde{\omega}_{(3)\tau}^a} \right. \\ & \left. + \frac{1}{4} \bar{Q}_{(1)\mu a} \frac{\partial}{\partial \omega_{(3)\tau\mu a}} + \frac{1}{4} Q_{(3)\mu}^a \frac{\partial}{\partial \bar{\omega}_{(1)\tau}^a} - \frac{1}{4} \bar{Q}_{(3)\mu a} \frac{\partial}{\partial \omega_{(1)\tau\mu a}} \right) S_{\text{sparticle}}^{AdS_4 \times \mathbb{CP}^3} \end{aligned} \tag{3.58} $$

When coordinates of the sector of broken supersymmetries equal zero, Eqs. (3.57) and (3.58) turn into respective Eqs. (3.40) and (3.42) for the Lax pair components of the $OSp(4|6)/(SO(1,3) \times U(3))$ superparticle.

When $v_{\mu} = 0$, the $AdS_4 \times \mathbb{CP}^3$ superparticle equations (3.51) and (3.52) turn into bosonic and fermionic equations of the $OSp(4|6)/(SO(1,3) \times U(3))$ superparticle (3.37) and (3.38) respectively. Eqs. (3.56) become the consequences of the fermionic equations (3.38). Below we will present the proof of the latter statement. Analogous statement for the $AdS_4 \times \mathbb{CP}^3$ superstring was proved in subsection 3.2.

Conditions $v_{\mu} = 0$ represent partial $\kappa$-symmetry fixing, when $\Omega_{\tau}^{a} \neq 0$, $\Omega_{\tau a} \neq 0$. Then Eqs. (3.56) reduce to

$$ \Omega_{\tau}^{a} \omega_{\tau a}^{\mu} = \Omega_{\tau a} \bar{\omega}_{\tau}^{\mu a} = 0, \quad \Omega_{\tau}^{a} \chi_{\tau a}^{\mu} = \Omega_{\tau a} \bar{\chi}_{\tau}^{\mu a} = 0. $$

They can be presented in equivalent form as equations for mappings to the world-line of one-forms $\omega_{(1,3)\tau a}^{\mu}$ and $\bar{\omega}_{(1,3)\tau}^{\mu a}$

$$ \Omega_{\tau}^{a} \omega_{(1)\tau a}^{\mu} = \Omega_{\tau a} \bar{\omega}_{(1)\tau}^{\mu a} = 0, \tag{3.59} $$

$$ \Omega_{\tau}^{a} \omega_{(3)\tau a}^{\mu} = \Omega_{\tau a} \bar{\omega}_{(3)\tau}^{\mu a} = 0. \tag{3.60} $$

To show that they follow from Eqs. (3.38) let us present the latter in equivalent form

$$ \mathcal{M}_{(1)\tau \hat{a} \nu}^{\mu \hat{b}} \begin{pmatrix} \omega_{(1)\tau b}^{\nu} \\ \bar{\omega}_{(1)\tau}^{\nu b} \end{pmatrix} = 0, \tag{3.61} $$

$$ \mathcal{M}_{(3)\tau \hat{a} \nu}^{\mu \hat{b}} \begin{pmatrix} \omega_{(3)\tau b}^{\nu} \\ \bar{\omega}_{(3)\tau}^{\nu b} \end{pmatrix} = 0. \tag{3.62} $$

Eqs. (3.61) and (3.62) involve the $12 \times 12$ matrices $\mathcal{M}_{(1,3)\tau \hat{a} \nu}^{\mu \hat{b}}$ of rank eight

$$ \mathcal{M}_{(1)\tau \hat{a} \nu}^{\mu \hat{b}} = \begin{pmatrix} \delta_a^b m_{\tau}^{\mu} & -i \delta_{\nu}^{\mu} \varepsilon_{acb} \Omega_{\tau}^c \\ -i \delta_{\nu}^{\mu} \varepsilon^{acb} \Omega_{\tau c} & -\delta_b^a \bar{m}_{\tau}^{\mu} \end{pmatrix}, \quad \mathcal{M}_{(3)\tau \hat{a} \nu}^{\mu \hat{b}} = \begin{pmatrix} -\delta_a^b \bar{m}_{\tau}^{\mu} & i \delta_{\nu}^{\mu} \varepsilon_{acb} \Omega_{\tau}^c \\ i \delta_{\nu}^{\mu} \varepsilon^{acb} \Omega_{\tau c} & \delta_b^a m_{\tau}^{\mu} \end{pmatrix}, $$
where 2 × 2 matrices

$$m_{\tau}^{\mu}{}_{\nu} = -iG_{\tau}{}^{0'm} \sigma_{m}{}^{\mu}{}_{\nu} + \Delta_{\tau} \delta_{\nu}^{\mu}, \quad \bar{m}_{\tau}{}^{\mu}{}_{\nu} = iG_{\tau}{}^{0'm} \sigma_{m}{}^{\mu}{}_{\nu} + \Delta_{\tau} \delta_{\nu}^{\mu}$$

obey the relation

$$m_{\tau}^{\mu}{}_{\nu} \bar{m}_{\tau}{}^{\nu}{}_{\lambda} = \delta_{\lambda}^{\mu} (G_{\tau} \cdot G_{\tau} + \Delta_{\tau}^{2}). \qquad (3.63)$$

Recall that similarly to the case of the σ-model, the model of massless $OSp(4|6)/(SO(1,3) \times U(3))$ superparticle describes dynamics of the superparticle in the $AdS_4 \times CP^3$ superspace, when $\Omega_{\tau}^a \neq 0$, $\Omega_{\tau a} \neq 0$ that corresponds to its motion both the anti-de Sitter space and the $CP^3$ space. Therefore from the mass-shell constraint for massless superparticle (3.36) it follows that the right-hand side of (3.63) is non-zero.

Consider in detail the system of equations (3.61). Applying the rank-eight projection matrix

$$\Pi_{(1)}^{\mu \hat{b}} = \begin{pmatrix} \delta_a^b \delta_\nu^\mu & -i \frac{m_\tau^\mu_\nu}{G_\tau \cdot G_\tau + \Delta_\tau^2} \varepsilon_{acb} \Omega_\tau^c \\ i \frac{\bar{m}_\tau^\mu_\nu}{G_\tau \cdot G_\tau + \Delta_\tau^2} \varepsilon^{acb} \Omega_{\tau c} & \delta_b^a \delta_\nu^\mu \end{pmatrix}$$

that satisfies $\Pi_{(1)}^3 - 3\Pi_{(1)}^2 + 2\Pi_{(1)} = 0$ and recalling that $G_\tau \cdot G_\tau + \Delta_\tau^2 \neq 0$ one can bring Eq. (3.61) to the block-diagonal form

$$\begin{pmatrix} m_{\tau}^{\mu}{}_{\nu} \Omega_{\tau a} \Omega_{\tau}^{b} & 0 \\ 0 & -\bar{m}_{\tau}^{\mu}{}_{\nu} \Omega_{\tau}^{a} \Omega_{\tau b} \end{pmatrix} \begin{pmatrix} \omega_{(1)}{}_{\tau b}^{\nu} \\ \bar{\omega}_{(1)}{}_{\tau}^{\nu b} \end{pmatrix} = 0. \qquad (3.64)$$

Further with the use of the relation (3.63) the system (3.64) reduces to four equations (3.59). Analogously it can be shown that from the system (3.62) there follow equations (3.60).

### 3.3.3 Classical integrability of equations of the D0-brane in the $AdS_4 \times CP^3$ superbackground

As a generalization of the results of the previous paragraph, in this paragraph proved will be integrability of equations of the D0-brane in the $AdS_4 \times CP^3$ superbackground.

The D0-brane action

$$S_{D0-\text{brane}}^{AdS_4 \times CP^3} = \int_L d\tau \mathcal{L}_{D0-\text{brane}}^{AdS_4 \times CP^3}$$

is given by the integral over its world-line L of the Lagrangian

$$\mathcal{L}_{D0-\text{brane}}^{AdS_4 \times CP^3} = -\frac{m}{\Phi_L^2} \sqrt{-(\mathcal{E}_{\tau m'} \mathcal{E}_{\tau}^{m'} - \mathbf{E}_{\tau a} \mathbf{E}_{\tau}^a) + mA_{L\tau}},$$

that represents the sum of kinetic and Wess-Zumino terms. The latter is determined by the mapping to the world line of the Ramond-Ramond one-form potential $A_L(d)$ and m is the charge of D0-brane equal its mass. Introducing auxiliary world-line metric $e(\tau)$ it is possible to exclude the square root from the Lagrangian

$$\mathcal{L}_{D0-\text{brane}}^{AdS_4 \times CP^3} = \frac{1}{2} \left[ \frac{1}{e\Phi_L^2} (\mathcal{E}_{\tau m'} \mathcal{E}_{\tau}^{m'} - \mathbf{E}_{\tau a} \mathbf{E}_{\tau}^a) - em^2 \right] + mA_{L\tau}. \qquad (3.65)$$
Let us note the arbitrariness in the definition of $e(\tau)$. So by redefining it one can 'absorb' the factor $\Phi_L^2$. Then in the $m \to 0$ limit resulting Lagrangian reduces to that of the massless superparticle. The metric $e(\tau)$ is also the Lagrange multiplier for the mass-shell constraint. Its definition adopted in (3.65) gives the conventional mass term in this constraint. Since the mass-shell constraint is not used for establishing integrability of the dynamical equations of the D0-brane, to simplify further analysis it is convenient to gauge fix the reparametrization symmetry of the action by the condition $e = \frac{1}{2}$.

As in the case of the massless superparticle, considering as independent parameters the Cartan forms, associated with generators of the osp(4|6) superalgebra from the $\mathfrak{g}_{1,2,3}$ eigenspaces under the $\mathbb{Z}_4$ automorphism, with differentials of coordinates replaced by their variations yields the set of 10 bosonic and 24 fermionic equations of the D0-brane. By substituting the expressions (3.45) for $\mathcal{E}^{m'}(d)$ and (3.46) for $A_L(d)$ one can bring the D0-brane equations to a system of the first-order ordinary differential equations with the coefficients given by the mappings of the osp(4|6) Cartan forms to its world line

$$ 
\begin{aligned}
-\frac{1}{2} \frac{\delta S_{D0-brane}^{AdS_4 \times CP^3}}{\delta G^{0'm}(\delta)} &= \dot{\tilde{G}}_{\tau}^{0'm} + 2(G_{\tau}^{mn} \tilde{G}_{\tau}^{0'n} - \tilde{G}_{\tau}^{mn} G_{\tau}^{0'n} + \tilde{\Delta}_{\tau} G_{\tau}^{3m} - \Delta_{\tau} \tilde{G}_{\tau}^{3m}) \\
&\quad -2i (\omega_{(1)\tau a} \sigma^m \tilde{\omega}_{(1)\tau}^a - \tilde{\omega}_{(1)\tau a} \sigma^m \bar{\omega}_{(1)\tau}^a + \omega_{(3)\tau a} \sigma^m \tilde{\omega}_{(3)\tau}^a - \tilde{\omega}_{(3)\tau a} \sigma^m \bar{\omega}_{(3)\tau}^a) = 0, \\
&\quad -\frac{1}{2} \frac{\delta S_{D0-brane}^{AdS_4 \times CP^3}}{\delta \Delta(\delta)} = \dot{\tilde{\Delta}}_{\tau} + 2G_{\tau}^{0'm} \tilde{G}_{\tau}^{3m} - 2G_{\tau 3m} \tilde{G}_{\tau}^{0'm} \\
&\quad -2 (\omega_{(1)\tau a} \tilde{\omega}_{(1)\tau}^a - \tilde{\omega}_{(1)\tau a} \bar{\omega}_{(1)\tau}^a - \omega_{(3)\tau a} \tilde{\omega}_{(3)\tau}^a + \tilde{\omega}_{(3)\tau a} \bar{\omega}_{(3)\tau}^a) = 0, \\
&\quad -\frac{\delta S_{D0-brane}^{AdS_4 \times CP^3}}{\delta \Omega_a(\delta)} = \dot{\tilde{\Omega}}_{\tau}^a + i \tilde{\Omega}_{\tau}^b (\tilde{\Omega}_{\tau b}^a + \delta_b^a \tilde{\Omega}_{\tau c}^c) - 4i \tilde{\mathbf{w}}_{\tau} \Omega_{\tau}^a \\
&\quad +4i \varepsilon^{abc} (\omega_{(1)\tau b} \tilde{\omega}_{(1)\tau}^a - \omega_{(3)\tau b} \tilde{\omega}_{(3)\tau}^a) = 0
\end{aligned} 
\quad (3.66) $$

and

$$ 
\begin{aligned}
\frac{1}{4} \frac{\delta S_{D0-brane}^{AdS_4 \times CP^3}}{\delta \tilde{\omega}_{(1)\mu}^a(\delta)} &= \dot{\tilde{\omega}}_{(3)\tau a}^{\mu} + \frac{1}{2} \left( G_{\tau}^{mn} \tilde{\omega}_{(3)\tau a}^{\nu} - \tilde{G}_{\tau}^{mn} \omega_{(3)\tau a}^{\nu} \right) \sigma_{mn\nu}^{\mu} \\
&\quad + i \tilde{\sigma}_{m}^{\mu\nu} \left( G_{\tau}^{3m} \tilde{\omega}_{(3)\tau\nu a} - \tilde{G}_{\tau}^{3m} \omega_{(3)\tau\nu a} \right) + i \tilde{\sigma}_{m}^{\mu\nu} \left( G_{\tau}^{0'm} \tilde{\omega}_{(1)\tau\nu a} - \tilde{G}_{\tau}^{0'm} \omega_{(1)\tau\nu a} \right) \\
&\quad + \Delta_{\tau} \tilde{\omega}_{(1)\tau}^{\mu} - \tilde{\Delta}_{\tau} \omega_{(1)\tau a}^{\mu} - i (\tilde{\Omega}_{\tau a}^{b} - \delta_a^b \tilde{\Omega}_{\tau c}^c) \tilde{\omega}_{(3)\tau b}^{\mu} \\
&\quad -2i \tilde{\mathbf{w}}_{\tau} \omega_{(3)\tau a}^{\mu} - i \varepsilon_{abc} (\Omega_{\tau}^{b} \tilde{\bar{\omega}}_{(1)\tau}^{\mu c} - \tilde{\Omega}_{\tau}^{b} \bar{\omega}_{(1)\tau}^{\mu c}) = 0,
\end{aligned} 
\quad (3.67) $$

$$ 
\begin{aligned}
-\frac{1}{4} \frac{\delta S_{D0-brane}^{AdS_4 \times CP^3}}{\delta \tilde{\omega}_{(3)\mu}^a(\delta)} &= \dot{\tilde{\omega}}_{(1)\tau a}^{\mu} + \frac{1}{2} \left( G_{\tau}^{mn} \tilde{\omega}_{(1)\tau a}^{\nu} - \tilde{G}_{\tau}^{mn} \omega_{(1)\tau a}^{\nu} \right) \sigma_{mn\nu}^{\mu} \\
&\quad -i \tilde{\sigma}_{m}^{\mu\nu} \left( G_{\tau}^{3m} \tilde{\omega}_{(1)\tau\nu a} - \tilde{G}_{\tau}^{3m} \omega_{(1)\tau\nu a} \right) - i \tilde{\sigma}_{m}^{\mu\nu} \left( G_{\tau}^{0'm} \tilde{\omega}_{(3)\tau\nu a} - \tilde{G}_{\tau}^{0'm} \omega_{(3)\tau\nu a} \right) \\
&\quad + \Delta_{\tau} \tilde{\omega}_{(3)\tau a}^{\mu} - \tilde{\Delta}_{\tau} \omega_{(3)\tau a}^{\mu} - i (\tilde{\Omega}_{\tau a}^{b} - \delta_a^b \tilde{\Omega}_{\tau c}^c) \tilde{\omega}_{(1)\tau}^{\mu} \\
&\quad -2i \tilde{\mathbf{w}}_{\tau} \omega_{(1)\tau a}^{\mu} - i \varepsilon_{abc} (\Omega_{\tau}^{b} \tilde{\bar{\omega}}_{(3)\tau}^{\mu c} - \tilde{\Omega}_{\tau}^{b} \bar{\omega}_{(3)\tau}^{\mu c}) = 0,
\end{aligned} 
\quad (3.68) $$

and also c.c. equations. In Eqs. (3.66), (3.67) and (3.68) the following quantities have been
introduced

$$
\begin{pmatrix}
\tilde{\mathbf{G}}_{\tau}^{0'm} \\
\tilde{\mathbf{\Delta}}_{\tau} \\
-\tilde{\mathbf{G}}_{\tau 3m} \\
-\frac{1}{2}\tilde{\mathbf{G}}_{\tau kl}
\end{pmatrix}
=
\frac{1}{\Phi_L^2}
\begin{pmatrix}
M_m^{n'} \\
L^{n'} \\
N_m^{n'} \\
K_{kl}^{n'}
\end{pmatrix}
\left\{
\begin{aligned}
& G_{\tau n'} - \left[ \frac{1}{\Phi_L^2} \left( (G_\tau \cdot G_y) + \Phi A_\tau \right) + \frac{m}{2} \right] G_{yn'} \\
& + \frac{1}{\Phi_L^2} \left[ \frac{1}{\Phi_L^2} (G_y^2 A_\tau - \Phi(G_\tau \cdot G_y)) + \frac{m}{2} \Phi \right]
\end{aligned}
\right\},
\tag{3.69}
$$

$$
\tilde{\Omega}_{\tau}^{a} = -\frac{i}{\Phi_L^2} \mathbf{E}_{\tau}^{a}, \quad \tilde{\Omega}_{\tau a} = -\frac{i}{\Phi_L^2} \mathbf{E}_{\tau a}, \quad 2\Phi_L^2 \tilde{\mathbf{w}}_{\tau} = \frac{1}{\Phi_L^2} (G_y^2 A_{\tau} - \Phi(G_{\tau} \cdot G_y)) + \frac{m}{2}\Phi \quad (3.70)
$$

and

$$
\begin{align}
\tilde{\omega}_{(1)\tau a}^{\mu} &= -\frac{1}{4\Phi_L^2} \mathbf{E}_{\tau a} \bar{u}_{(3)}^{\mu}, \quad \tilde{\omega}_{(3)\tau a}^{\mu} = \frac{1}{4\Phi_L^2} \mathbf{E}_{\tau a} \bar{u}_{(1)}^{\mu}, \tag{3.71} \\
\tilde{\omega}_{(1)\tau}^{\mu a} &= \frac{1}{4\Phi_L^2} \mathbf{E}_{\tau}^{a} u_{(3)}^{\mu}, \quad \tilde{\omega}_{(3)\tau}^{\mu a} = -\frac{1}{4\Phi_L^2} \mathbf{E}_{\tau}^{a} u_{(1)}^{\mu}. \nonumber
\end{align}
$$

Compared to the massless AdS$_4 \times$CP$^3$ superparticle model these quantities take into account
contribution of the Wess-Zumino term and the overall factor of $\Phi_L^{-2}$ in the kinetic term.
Finally there are eight equations corresponding to variation of coordinates of the sector of
broken supersymmetries

$$
\begin{equation}
\begin{split}
& -\frac{d}{d\tau} \left\{ \frac{1}{\Phi_L^2} \left[ G_{\tau m'} + \left[ \frac{m}{2} - \frac{1}{\Phi_L^2} \left( (G_\tau \cdot G_y) + \Phi A_\tau \right) \right] G_{ym'} \right] \frac{\partial G_\tau^{m'}}{\partial \dot{v}_\mu} \right. \\
& \qquad \left. + \frac{1}{\Phi_L^2} \left[ \frac{m}{2} \Phi + \frac{1}{\Phi_L^2} (G_y^2 A_\tau - \Phi(G_\tau \cdot G_y)) \right] \frac{\partial A_\tau}{\partial \dot{v}_\mu} \right\} + \frac{1}{2} \frac{\partial \mathcal{L}_{D0-\text{brane}}^{\text{AdS}_4 \times \text{CP}^3}}{\partial \Phi_L} \frac{\partial \Phi_L}{\partial v_\mu} \\
& + \frac{1}{\Phi_L^4} A_\tau \left[ A_\tau (G_y \cdot \frac{\partial G_y}{\partial v_\mu}) + \left( \frac{m}{2} \Phi_L^2 - (G_\tau \cdot G_y) \right) \frac{\partial \Phi}{\partial v_\mu} \right] + \frac{1}{\Phi_L^2} \left\{ G_{\tau m'} \frac{\partial G_\tau^{m'}}{\partial v_\mu} \right. \\
& \qquad \left. - \frac{1}{2} \left( \mathbf{E}_{\tau a} \frac{\partial \mathbf{E}_\tau^a}{\partial v_\mu} + \mathbf{E}_\tau^a \frac{\partial \mathbf{E}_{\tau a}}{\partial v_\mu} \right) + \left[ \frac{m}{2} - \frac{1}{\Phi_L^2} \left( (G_\tau \cdot G_y) + \Phi A_\tau \right) \right] \frac{\partial (G_\tau \cdot G_y)}{\partial v_\mu} \right. \\
& \qquad \left. + \left[ \frac{m}{2} \Phi + \frac{1}{\Phi_L^2} (G_y^2 A_\tau - \Phi(G_\tau \cdot G_y)) \right] \frac{\partial A_\tau}{\partial v_\mu} \right\} = 0.
\end{split}
\tag{3.72}
\end{equation}
$$

For Eqs. (3.66), (3.67), (3.68) and (3.72) we obtained the representation in the form of
the Lax equation

$$
\frac{d\mathcal{L}_{D0-\text{brane}}}{d\tau} + [\mathfrak{M}, \mathcal{L}_{D0-\text{brane}}] = 0.
$$

The Lax pair component $\mathfrak{M}$ is the same as in (3.41) and the Lax component $\mathcal{L}_{D0-\text{brane}}$ is
given by the sum

$$
\mathcal{L}_{D0-\text{brane}} = \tilde{\mathcal{L}}_{\text{so}(2,3)} + \tilde{\mathcal{L}}_{\text{su}(4)} + \tilde{\mathcal{L}}_{24\text{susy}} \in \text{osp}(4|6),
$$

where each summand is the linear combination of the quantities introduced in (3.69)-(3.71)

$$
\begin{align*}
\tilde{\mathcal{L}}_{\text{so}(2,3)} &= 2\tilde{\mathbf{G}}_{\tau}^{0'm} M_{0'm} + \tilde{\mathbf{\Delta}}_{\tau} D + 2\tilde{\mathbf{G}}_{\tau}^{3m} M_{3m} + \tilde{\mathbf{G}}_{\tau}^{mn} M_{mn}, \\
\tilde{\mathcal{L}}_{\text{su}(4)} &= \tilde{\mathbf{\Omega}}_{\tau}^{a} T_a + \tilde{\mathbf{\Omega}}_{\tau a} T^a + 4 \tilde{\mathbf{w}}_{\tau} \tilde{V}_a^a, \\
\tilde{\mathcal{L}}_{24\text{susy}} &= \tilde{\mathbf{\omega}}_{(1)\tau a}^{\mu} Q_{(1)\mu}^{a} + \tilde{\mathbf{\omega}}_{(1)\tau}^{\mu a} \bar{Q}_{(1)\mu a} + \tilde{\mathbf{\omega}}_{(3)\tau a}^{\mu} Q_{(3)\mu}^{a} + \tilde{\mathbf{\omega}}_{(3)\tau}^{\mu a} \bar{Q}_{(3)\mu a}.
\end{align*}
$$
$\mathcal{L}_{D0-\text{brane}}$ takes the same form as for the massless $AdS_4 \times \mathbb{CP}^3$ superparticle modulo definition of the coefficients (3.69)-(3.71). Analogously to the superparticle case the Lax pair component $\mathcal{L}_{D0-\text{brane}}$ can be presented in the form of the $osp(4|6)$-valued differential operator applied to the $D0$-brane action

$$ \mathcal{L}_{D0-\text{brane}} = \left( M_{0'm} \frac{\partial}{\partial G_{\tau'0'm}} + \frac{1}{2} D \frac{\partial}{\partial \Delta_{\tau}} - M_{mn} \frac{\partial}{\partial G_{\tau mn}} - M_{3m} \frac{\partial}{\partial G_{\tau 3m}} \right. \\ \left. + T_a \frac{\partial}{\partial \Omega_{\tau a}} + T^a \frac{\partial}{\partial \Omega_{\tau a}} + \tilde{V}_a^a \frac{\partial}{\partial \tilde{\Omega}_{\tau b}^a} - \frac{1}{4} Q_{(1)\mu}^a \frac{\partial}{\partial \tilde{\omega}_{(3)\tau \mu}^a} \right. \\ \left. + \frac{1}{4} \tilde{Q}_{(1)\mu a} \frac{\partial}{\partial \omega_{(3)\tau \mu a}} + \frac{1}{4} Q_{(3)\mu}^a \frac{\partial}{\partial \tilde{\omega}_{(1)\tau \mu}^a} - \frac{1}{4} \tilde{Q}_{(3)\mu a} \frac{\partial}{\partial \omega_{(1)\tau \mu a}} \right) S_{D0-\text{brane}}^{AdS_4 \times \mathbb{CP}^3} $$

# 4 Conclusion

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

- using the isomorphic realization of the $osp(4|6)$ superalgebra as the $D = 1 + 2N = 6$ superconformal algebra there has been obtained the expression for the Lagrangian of the $\sigma$-model in the $OSp(4|6)/(SO(1,3) \times U(3))$ supercoset space in terms of the Cartan one-forms for generators of this superconformal algebra;

- fermionic equations of the $\sigma$-model were presented in the form similar to respective equations of the Green-Schwarz superstrings and proved their linear dependence. According to the second Noether theorem this implies invariance of the $\sigma$-model action under the $\kappa$-symmetry transformations;

- there was obtained explicit expression for the $\sigma$-model Lagrangian in terms of coordinates of the $OSp(4|6)/(SO(1,3) \times U(3))$ supercoset space that are associated with the generators of the $D = 3N = 6$ superconformal algebra;

- there were derived general expressions in terms of the Cartan forms for the Noether current densities corresponding to invariance of the $\sigma$-model action under global $D = 3N = 6$ superconformal symmetry. For the $OSp(4|6)/(SO(1,3) \times U(3))$ supercoset element parametrized by the Poincaré coordinates for the $AdS_4$ space and 24 Grassmann coordinates that are parameters for generators of the $D = 3N = 6$ Poincaré and special conformal supersymmetries obtained were explicit expressions for the Noether current densities in terms of these coordinates and their world-sheet derivatives. There were also found variations of these coordinates under infinitesimal transformations of the $D = 3N = 6$ superconformal symmetry;

- obtained was expression for the Lagrangian of the superstring in the $\kappa$-symmetry light-cone gauge in terms of the $AdS_4 \times \mathbb{CP}^3$ superspace coordinates and their derivatives. Considered gauge corresponds to the light cone formed by null-geodesics on the conformal boundary of the $AdS_4$ space-time in the Poincaré coordinates and differs from other light-cone gauge conditions studied earlier.

- there was obtained the Hamiltonian of the $AdS_4 \times \mathbb{CP}^3$ superstring in the light-cone gauge for the $\kappa$-symmetry and world-sheet reparametrizations. Its eigenvalues determine the energy spectrum of quantized superstring that according to the $AdS_4/CFT_3$ correspondence
conjecture coincides with that of anomalous dimensions of the gauge-invariant operators in the $D = 3 \mathcal{N} = 6$ superconformal Chern-Simons-matter theory. It was shown that quadratic part of obtained Hamiltonian coincides with the light-cone gauge Hamiltonian of the Type IIA superstring in flat background.

- there was proposed the Kaluza-Klein condition, which simplifies the double-dimensional reduction of the $AdS_4 \times S^7$ supermembrane and the resultant $AdS_4 \times \mathbb{CP}^3$ superstring Lagrangian. For considered in the literature $OSp(4|8)/(SO(1,3) \times SO(7))$ supercoset elements the Kaluza-Klein condition restricts coordinates of the sector of the supersymmetries broken by the $AdS_4 \times \mathbb{CP}^3$ background. Proposed was the parametrization of this sector by four $SL(2, \mathbb{R})$ spinor coordinates that satisfy (anti-)Majorana conditions. It was shown that in this parametrization the Kaluza-Klein condition can be met in the $SL(2, \mathbb{R})$-covariant way. Obtained were respective expressions for the supervielbein 1-forms of the $AdS_4 \times S^7$ and $AdS_4 \times \mathbb{CP}^3$ superspaces;

- obtained were the $AdS_4 \times \mathbb{CP}^3$ superstring Lagrangian and equations in the partial $\kappa$-symmetry gauge, in which there remains single Majorana spinor coordinate in the sector of broken supersymmetries. It is the parameter for generators of the broken Poincaré supersymmetries. For the superstring equations found was the zero-curvature representation. It includes world-sheet one-form that extends the Lax connection of the $OSp(4|6)/(SO(1,3) \times U(3))$ $\sigma$-model by linear and quadratic terms in this coordinate and its differential;

- it was proved that fermionic equations of the $AdS_4 \times \mathbb{CP}^3$ superstring corresponding to variation of the coordinates of the sector of broken supersymmetries become the consequences of other fermionic equations, when these coordinates turn to zero. Proposed proof generalizes that of Ref. [17] for equations linearized in the Grassmann coordinates of the $AdS_4 \times \mathbb{CP}^3$ superspace.

- equations of the massless superparticle in the $OSp(4|6)/(SO(1,3) \times U(3))$ supercoset space were presented in the form of the Lax equation and established was the connection between the components of the Lax pair and the $\sigma$-model Lax connection. As in the case of the $AdS_4 \times \mathbb{CP}^3$ superstring it was proved that fermionic equations of the massless $AdS_4 \times \mathbb{CP}^3$ superparticle, corresponding to variation of the coordinates of the sector of broken supersymmetries, are not independent, when these coordinates equal zero. It was proved integrability of equations of the massless superparticle and the D0-brane models in the $AdS_4 \times \mathbb{CP}^3$ superbackground.

# Acknowledgements

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

[1] J.M. Maldacena, The large *N* limit of superconformal field theories and supergravity, Advances in Theoretical and Mathematical Physics, 1998, V.2, P.231-252. arXiv:hep-th/9711200.

[2] S.S. Gubser, I.R. Klebanov, A.M. Polyakov, Gauge theory correlators from noncritical string theory, Physics Letters B, 1998, V.428, P.105-114. arXiv:hep-th/9802109.

[3] E. Witten, Anti-de Sitter space and holography, Advances in Theoretical and Mathematical Physics, 1998, V.2, P.253-291. arXiv:hep-th/9802150.

[4] M. Cicoli, J.P. Conlon, A. Maharana, S. Parameswaran, F. Quevedo, Y. Zavala, String cosmology: from the early Universe to today, arXiv:2303.04819 [hep-th].

[5] O. Aharony, O. Bergman, D.L. Jafferis, J. Maldacena, *N* = 6 superconformal Chern-Simons-matter theories, M2-branes and their gravity duals, Journal of High Energy Physics, 2008, V.0810, 091, arXiv:0806.1218 [hep-th].

[6] S. Watamura, Spontaneous compactification and $CP_N$: $SU(3) \times SU(2) \times U(1)$, $\sin^2 \theta_W$, $g_3/g_2$ and $SU(3)$-triplet chiral fermions in four dimensions, Physics Letters B, 1984, V.136, P.245-250.

[7] B.E.W. Nilsson, C. Pope, Hopf fibration of eleven dimensional supergravity, Classical and Quantum Gravity, 1984, V.1, P.499-515.

[8] D.P. Sorokin, V.I. Tkach, D.V. Volkov, Kaluza-Klein theories and spontaneous compactification mechanisms of extra space dimensions, in Quantum Gravity, eds. M.A. Markov, V.A. Berezin, V.P. Frolov. Singapore: World Scientific, 1986. - pp.376-392.

D.P. Sorokin, V.I. Tkach, D.V. Volkov, On the relationship between compactified vacua of $D = 11$ and $D = 10$ supergravities, Physics Letters B, 1985, V.161, P.301-306.

[9] G. Arutyunov, S. Frolov, Superstrings on $AdS_4 \times CP^3$ as a coset sigma-model, Journal of High Energy Physics, 2008, V.0809, 129. arXiv:0806.4940 [hep-th].

[10] B.J. Stefański, Green-Schwarz action for Type IIA strings on $AdS_4 \times CP^3$, Nuclear Physics B, 2009, V.808, P.80-88. arXiv:0806.4948 [hep-th].

[11] R.R. Metsaev, A.A. Tseytlin, Type IIB superstring action in $AdS_5 \times S^5$ background, Nuclear Physics B, 1998, V.533, P.109-126. arXiv:hep-th/9805028.

[12] R. Kallosh, J. Rahmfeld, A. Rajaraman, Near horizon superspace, Journal of High Energy Physics, 1998, V.9809, 002. arXiv:hep-th/9805217.
[13] J. Gomis, D. Sorokin, L. Wulff, The complete $AdS_4 \times CP^3$ superspace for type IIA superstring and D-branes, Journal of High Energy Physics, 2009, V.0903, 015. arXiv:0811.1566 [hep-th].

[14] M.J. Duff, T. Inami, P.S. Howe, K.S. Stelle, Superstrings from supermembranes in $D = 11$, Physics Letters B, 1987, V.191, P.70-74.

[15] P.S. Howe, E. Sezgin, The supermembrane revisited, Classical and Quantum Gravity, 2005, V.22, P.2167-2200. arXiv:hep-th/0412245.

[16] B. de Wit, K. Peeters, J. Plefka, A. Sevrin, The M-theory two-brane in $AdS_4 \times S^7$ and $AdS_7 \times S^4$, Physics Letters B, 1998, V.443, P.153-158. arXiv:hep-th/9808052.

[17] D. Sorokin, L. Wulff, Evidence for the classical integrability of the complete $AdS_4 \times CP^3$ superstring, Journal of High Energy Physics, 2010, V.1011, 143. arXiv:1009.3498 [hep-th].

[18] A. Cagnazzo, D. Sorokin, L. Wulff, More on integrable structures of superstrings in $AdS_4 \times CP^3$ and $AdS_2 \times S^2 \times T^6$ superbackgrounds, Journal of High Energy Physics, 2012, V.1201, 004. arXiv:1111.4197 [hep-th].

[19] D.V. Uvarov, $AdS_4 \times CP^3$ superstring and $D = 3, N = 6$ superconformal symmetry, Physical Review D, 2009, V.79, 106007. arXiv:0811.2813 [hep-th].

[20] M.B. Green, J.H. Schwarz, Covariant description of superstrings, Physics Letters B, 1984, V.136, P.367-370.
M.B. Green, J.H. Schwarz, Properties of the covariant formulation of superstring theories, Nuclear Physics B, 1984, V.243, P.285-306.

[21] D.V. Uvarov, $D = 3, N = 6$ superconformal symmetry of the $AdS_4 \times CP^3$ superstring, Classical and Quantum Gravity, 2011, V.28, 235010. arXiv:1011.5457 [hep-th].

[22] D.V. Uvarov, $AdS_4 \times CP^3$ superstring in the light-cone gauge, Nuclear Physics B, 2010, V.826, P.294-312. arXiv:0906.4699 [hep-th].

[23] T. Nishioka, T. Takayanagi, On Type IIA Penrose Limit and $N = 6$ Chern-Simons Theories, Journal of High Energy Physics, 2008, V.0808, 001. arXiv:0806.3391 [hep-th].

[24] D. Astolfi, V.G.M. Puletti, G. Grignani, T. Harmark, M. Orselli, Finite-size corrections in the $SU(2) \times SU(2)$ sector of type IIA string theory on $AdS_4 \times CP^3$, Nuclear Physics, 2009, V.810, P.150-173. arXiv:0807.1527 [hep-th].

[25] P. Sundin, The $AdS(4) \times CP^3$ string and its Bethe equations in the near plane wave limit, Journal of High Energy Physics, 2009, V.0902, 046, arXiv:0811.2775 [hep-th].

[26] K. Zarembo, Worldsheet spectrum in $AdS_4/CFT_3$ correspondence, Journal of High Energy Physics, 2009, V.0904, 135. arXiv:0903.1747 [hep-th].
[27] D. Bykov, Symmetry algebra of the $AdS_4 \times CP^3$ superstring, Theoretical and Mathematical Physics, 2010, V.163, P.496-510. arXiv:0904.0208 [hep-th].

[28] C. Kalousios, C.Vergu, A. Volovich, Factorized tree-level scattering in $AdS_4 \times CP^3$, Journal of High Energy Physics, 2009, V.0909, 049. arXiv:0905.4702 [hep-th].

[29] M. Dukalski, S.J. van Tongeren, On fermionic reductions of the $AdS_4 \times CP^3$ superstring, Physical Review D, 2009, V.80, 046005. arXiv:0906.0706 [hep-th].

[30] D.V. Uvarov, Light-cone gauge Hamiltonian for $AdS_4 \times CP^3$ superstring, Modern Physics Letters A, 2010, V.25, P.1251-1265. arXiv:0912.1044 [hep-th].

[31] P. Fre, P.A. Grassi, Pure Spinor Formalism for $OSp(N|4)$ backgrounds, International Journal of Modern Physics A, 2012, V.27, 1250185. arXiv:0807.0044 [hep-th].
G. Bonelli, P.A. Grassi, H. Safaai, Exploring pure spinor string theory on $AdS_4 \times CP^3$, Journal of High Energy Physics, 2008, V.0810, 085. arXiv:0808.1051 [hep-th].
R. D'Auria, P. Fre, P.A. Grassi, M. Trigiante, Superstrings on $AdS_4 \times CP^3$ from Supergravity, Physical Review D, 2009, V.79, 086001. arXiv:0808.1282 [hep-th].

[32] M. Bandres, A. Lipstein, J.H. Schwarz, Studies of the ABJM Theory in a Formulation with Manifest $SU(4) R$-Symmetry, Journal of High Energy Physics, 2008, V.0809, 027. arXiv:0807.0880 [hep-th].

[33] R. Roiban, W. Siegel, Superstring on $AdS_5 \times S^5$ supertwistor space, Journal of High Energy Physics, 2000, V.0012, 024. arXiv:hep-th/0010104.

[34] N. Berkovits, M. Bershadsky, T. Hauer, S. Zhukov, B. Zweibach, Superstring theory on $AdS_2 \times S^2$ as a coset manifold, Nuclear Physics B, 2000, V.567, P.61-86. arXiv:hep-th/990200.

[35] M. Hatsuda, M. Sakaguchi, Wess-Zumino term for $AdS$ superstring, Physical Review D, V.66, 045020. arXiv:hep-th/0205092.

[36] L. Brink, M. Henneaux, Principles of string theory. New York: Plenum Press, 1988. - 297p.

[37] G. Dall'Agata, D. Fabbri, C. Fraser, P. Fre, P. Termonia, M. Trigiante, The $OSp(8|4)$ singleton action from the supermembrane, Nuclear Physics B, 1999, V.542, P.157-194. arXiv:hep-th/9807115.

[38] R. Kallosh, Superconformal actions in Killing gauge, arXiv:hep-th/9807206.

[39] P. Pasti, D.P. Sorokin, M. Tonin, On gauge-fixed superbrane actions in $AdS$ super backgrounds, Physics Letters B, 1999, V.447, P.251-256. arXiv:hep-th/9809213.

[40] R.R. Metsaev, A.A. Tseytlin, Superstring action in $AdS_5 \times S^5$: $\kappa$-symmetry light cone gauge, Physical Review D, 2001, V.63, 046002. arXiv:hep-th/0007036.
[41] R.R. Metsaev, C.B. Thorn, A.A. Tseytlin, Light cone superstring in AdS space-time, Nuclear Physics B, 2001, V.596, P.151-184. arXiv:hep-th/0009171.

[42] R.R. Metsaev, On manifest $SU(4)$ invariant superstring action in $AdS_5 \times S^5$, Classical and Quantum Gravity, 2001, V.18, P.1245-1260. arXiv:hep-th/0012026.

[43] R. Kallosh, A.A. Tseytlin, Simplifying superstring action on $AdS_5 \times S^5$, Journal of High Energy Physics, 1998, V.9810, 016. arXiv:hep-th/9808088.

[44] I. Pesando, A $\kappa$ gauge-fixed Type IIB superstring action, Journal of High Energy Physics, 1998, V.9811, 002. arXiv:hep-th/9808020.

[45] R. Kallosh, J. Rahmfeld, The GS action on $AdS_5 \times S^5$, Physics Letters B, 1998, V.443, P.143-146. arXiv:hep-th/9808038.

[46] P. Claus, J. Rahmfeld, H. Robins, J. Tannenhauser, Y. Zunger, Isometries in anti-de Sitter and conformal superspaces, Journal of High Energy Physics, 2000, V.0007, 047. arXiv:hep-th/0007099.

[47] E. Bergshoeff, E. Sezgin, P.K. Townsend, Supermembranes and eleven-dimensional supergravity, Physics Letters B, 1987, V.189, P.75-78.

[48] P.A. Grassi, D. Sorokin and L. Wulff, Simplifying superstring and $D$-brane actions in $AdS_4 \times CP^3$ superbackground, Journal of High Energy Physics, 2009, V.0908, 060. arXiv:0903.5407 [hep-th].

[49] G. Arutyunov, S. Frolov, Foundations of the $AdS_5 \times S^5$ superstring. Part I, Journal of Physics A: Mathematical and Theoretical, 2009, V.42, 254003. arXiv:0901.4937 [hep-th].

[50] P. Goddard, J. Goldstone, C. Rebbi, C.B. Thorn, Quantum dynamics of a massless relativistic string, Nuclear Physics B, 1973, V.56, P.109-135.

[51] G. Arutyunov, S. Frolov, Uniform light-cone gauge for strings in $AdS_5 \times S^5$: Solving $su(1|1)$ sector, Journal of High Energy Physics, 2006, V.0601, 055. arXiv:hep-th/0510208.
G. Arutyunov, S. Frolov, M. Zamaklar, Finite-size effects from giant magnons, Nuclear Physics B, 2007, V.778, P.1-35. arXiv:hep-th/0606126.

[52] M.B. Green, J.H. Schwarz, E. Witten, Superstring theory. V.I. Introduction. Cambridge: Cambridge University Press, 1987. - 470p.

[53] R. Penrose, Any space-time has a plane wave as a limit, In Differential Geometry and Relativity. Mathematical Physics and Applied Mathematics, eds. M. Cahen, M. Flato. Dordrecht: Reidel, 1976. - pp.271-275.
[54] K. Sugiyama, K. Yoshida, Type IIA string and matrix string on *pp*-wave, Nuclear Physics B, 2002, V.644, P.128-150. arXiv:hep-th/0208029.

S. Hyun, H. Shin, $N = (4, 4)$ type IIA string theory on *pp*-wave background, Journal of High Energy Physics, 2002, V.0210, 070. arXiv:hep-th/0208074.

P. Sundin, On the worldsheet theory of the type IIA $AdS_4 \times CP^3$ superstring, Journal of High Energy Physics, 2010, V.1004, 014. arXiv:0909.0697 [hep-th].

[55] M. Blau, J.M. Figueroa-O’Farrill, C. Hull, G. Papadopoulos, Penrose limits and maximal supersymmetry, Classical and Quantum Gravity, 2002, V.19, L87-L95. arXiv:hep-th/0201081.

M. Blau, J.M. Figueroa-O’Farrill, G. Papadopoulos, Penrose limits, supergravity and brane dynamics, Classical and Quantum Gravity, 2002, V.19, P.4753-4807. arXiv:hep-th/0202111.

[56] R.R. Metsaev, Type IIB Green-Schwarz superstring in plane wave Ramond-Ramond background, Nuclear Physics B, 2002, V.625, P.70-96. arXiv:hep-th/0112044.

R.R. Metsaev, A.A. Tseytlin, Exactly solvable model of superstring in Ramond-Ramond plane wave background, Physical Review D, 2002, V.65, 126004. arXiv:hep-th/0202109.

[57] M. Blau, D. Frank, S. Weiss, Fermi coordinates and Penrose limits, Classical and Quantum Gravity, 2006, V.23, P.3993-4010. arXiv:hep-th/0603109.

M. Blau, S. Weiss, Penrose Limits vs String Expansions, Classical and Quantum Gravity, 2008, V.25, 125014. arXiv:0710.3480 [hep-th].

[58] D.V. Uvarov, Kaluza-Klein gauge and minimal integrable extension of $OSp(4|6)/(SO(1,3) \times U(3))$ sigma-model, International Journal of Modern Physics A, 2012, V.27, 1250118. arXiv:1203.3041 [hep-th].

[59] D.V. Uvarov, A note about fermionic equations of $AdS_4 \times CP^3$ superstring, Bulletin of TSPU, 2012, N13(128), P.164-169. arXiv:1210.0715 [hep-th].

[60] D.V. Uvarov, Lagrangian mechanics of massless superparticle on $AdS_4 \times CP^3$ superbackground, Nuclear Physics B, 2013, V.B867, P.354-369. arXiv:1205.5388 [hep-th].

[61] D.V. Uvarov, On integrability of massless $AdS_4 \times CP^3$ superparticle equations, Modern Physics Letters A, 2014, V.29, 1350183. arXiv:1212.3157 [hep-th].

[62] D.V. Uvarov, On integrability of D0-brane equations on $AdS_4 \times CP^3$ superbackground, Journal of Physics: Conference Series, 2014, V.482, 012043. arXiv:1306.0732 [hep-th].

[63] I. Bena, J. Polchinski, R. Roiban, Hidden symmetries of the $AdS_5 \times S^5$ superstring, Physical Review D, 2004, V.69, 046002. arXiv:hep-th/0305116.
[64] I. Adam, A. Dekel, L. Mazzucato, Y. Oz, Integrability of Type II superstrings on Ramond-Ramond backgrounds in various dimensions, Journal of High Energy Physics, 2007, V.0706, 085. arXiv:hep-th/0702083.

[65] A. Babichenko, B. Stefanski, K. Zarembo, Integrability and the *AdS*<sub>3</sub>/<i>CFT</i><sub>2</sub> correspondence, Journal of High Energy Physics, 2010, V.1003, 058. arXiv:0912.1723 [hep-th].

[66] D. Sorokin, A. Tseytlin, L. Wulff, K. Zarembo, Superstrings in *AdS*<sub>2</sub> × *S*<sup>2</sup> × *T*<sub>6</sub>, Journal of Physics A: Mathematical and Theoretical, 2011, V.44, 275401. arXiv:1104.1793 [hep-th].