# Classical and Quantum Symmetries of (Super)Twistor Formulations of the Models of Relativistic Strings in Minkowski (Super)spaces

Dmytro Uvarov

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Dmytro Uvarov. Classical and Quantum Symmetries of (Super)Twistor Formulations of the Models of Relativistic Strings in Minkowski (Super)spaces. 2023. hal-04248742

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https://hal.science/hal-04248742v1

Preprint submitted on 18 Oct 2023

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<table>
  <tbody>
    <tr>
      <td>4.2</td>
      <td>Dirac bracket algebra of the first-class constraints</td>
      <td>36</td>
    </tr>
    <tr>
      <td>5</td>
      <td>Infinite-dimensional extensions of superconformal symmetry in twistor-string models</td>
      <td>39</td>
    </tr>
    <tr>
      <td>5.1</td>
      <td>Classical higher-spin superconformal symmetries of twistor strings</td>
      <td>40</td>
    </tr>
    <tr>
      <td>5.2</td>
      <td>Quantum higher-spin superconformal symmetries of twistor strings</td>
      <td>45</td>
    </tr>
    <tr>
      <td>5.2.1</td>
      <td>Analysis of the symmetries on the level of superalgebra</td>
      <td>45</td>
    </tr>
    <tr>
      <td>5.2.2</td>
      <td>Properties of the generators of higher-spin symmetries as operators in the world-sheet conformal field theory.</td>
      <td>47</td>
    </tr>
    <tr>
      <td>6</td>
      <td>Conclusion</td>
      <td>49</td>
    </tr>
    <tr>
      <td colspan="2">References</td>
      <td>51</td>
    </tr>
  </tbody>
</table>

# 1 Introduction

In Section 2 there are considered twistor formulations of the bosonic string and $N = 1, 2$ superstrings in flat $D = 4$ (super)space [1], [2]. They will be derived from the classically equivalent representations [3], [4] of the Polyakov Lagrangian for bosonic string [5], [6], [7] and the Green-Schwarz Lagrangian for superstrings [8] and involve spinor Lorentz-harmonic variables [9], [10], [3]. In the (super)twistor formulation Lagrangians of the (super)strings are expressed in terms of the components of two Penrose-Ferber (super)twistors. Unlike the supertwistor formulations of the massless superparticle [11] and tensionless superstring [12] models, tensile superstring action in the supertwistor formulation is invariant under the $\kappa$-symmetry. This is the evidence that there remain pure gauge degrees of freedom in the fermionic sector of superstrings after transition to the supertwistor formulation.

Also in this Section introduced will be reduced supertwistor models as the generalization to the case of tensile superstrings of the twistor formulations for massless superparticle and tensionless superstring. Lagrangians of the reduced supertwistor models are quadratic in the components of (super)twistors. Lagrangian of the reduced model corresponding to the $N = 1$ superstring can be obtained by substituting world-sheet embedding equations [13], [14] into the Wess-Zumino Lagrangian of the $N = 1$ superstring in the supertwistor formulation. Fixing the $\kappa$-symmetry gauge in the $N = 2$ superstring model gives another reduced supertwistor model that represents extention of the reduced model for the $N = 1$ superstring and the twistor formulation of the bosonic string. So in the final paragraph 2.3 reduced model corresponding to the $N = 2$ superstring will be analyzed as a constrained Hamiltonian system in accordance with the Dirac prescription [15]. Identified will be all the primary and secondary constraints and their division on the first- and second-class ones will be carried out.

In Section 3 there are presented formulations [16] of $N = 1$ superstrings in flat $D = 6$ and $D = 10$ superspaces in terms of the supertwistor variables that generalize Penrose-Ferber
supertwistors. Derived will be the constraints on the components of these supertwistors from the requirement of their correspondence to the coordinates of $D = 6, 10 \mathcal{N} = 1$ Minkowski superspaces. Similarly to the case of $D = 4$ superstrings introduced will be the reduced models with Lagrangians quadratic in the supertwistor components.

In the next Section $4 \ D = 10 \ \mathcal{N} = 1$ superstring model in the supertwistor formulation will be studied in the framework of canonical approach as a dynamical system with constraints [17]. Expressions for all the primary and secondary constraints in terms of the supertwistor components will be obtained and divided into the Lorentz-covariant and irreducible sets of the first- and second-class constraints. Then introduced will be the Dirac brackets that take into account the second-class constraints and will be examined the deformation of the first-class constraint algebra upon transition from the Poisson brackets to the Dirac brackets. The role of the deformation parameter is played by the inverse string tension.

It is of interest to compare supertwistor formulation of superstrings considered in preceding Sections with known Berkovits twistor string model [18] and its symmetries. Therefore in the final Section 5 examined will be the global symmetries of the Lagrangians describing dynamics of the left- and right-moving supertwistor variables in the Berkovits twistor string model and its generalization to the case of unconstrained supertwistors [19]. It will be shown that on the classical level these symmetries are infinite-dimensional generalizations of the $D = 4 \ \mathcal{N} = 4$ superconformal symmetry. It will be revealed that expansions of the products of the components of operator of the energy-momentum tensor, corresponding to invariance of these Lagrangians under world-sheet conformal symmetry, and quantum generators of found higher-spin symmetries include anomalous terms. Therefore on the quantum level these infinite-dimensional symmetries break down to the finite-dimensional $(P)SL(4|4, \mathbb{R})$ symmetry.

## 2 Twistor formulations of the bosonic string and superstring models in $D = 4$ Minkowski (super)space in Lagrangian and Hamiltonian approaches

Since the initial period of development of the twistor theory [20], [21], [22] formulations of the models of relativistic point particles in terms of twistors and supertwistors attract attention due to the interplay between the twistor geometry and realizations of the spin degrees of freedom (see [23], [24], [11], [25], [26], [27], [28], [29], [30], [31], [32], [33], [34], [35]). One of the most appealing features of the twistor formulations of supersymmetric particle models, first revealed in Ref. [11], is independence of the Lagrangian on the part of superspace Grassmann coordinates. It is these coordinates that transform as pure gauge quantities under the $\kappa$-symmetry. Transition to the supertwistor variables allows to exclude them without breaking the Lorentz symmetry in contrast to, e.g. the light-cone gauge.
Twistor formulations of the models of extended objects with zero tension in $D = 4$ (super)spaces were considered in [36], [37], [38], [39], [12]. The first attempts to reformulate in terms of twistors equations and constraints of the tensionful bosonic string were undertaken in [40], [41] and [42].

In our work [1] proposed were Lagrangian formulations in terms of Penrose-Ferber (super)twistors of the tensionful bosonic string and superstring models with $N = 1$ and $N = 2$ supersymmetries in the four-dimensional space. Their construction is based on the classically-equivalent Lorentz-harmonic formulations [3], [4] of the Lagrangians of bosonic string and superstrings and uses the fact that in four-dimensional space spinor Lorentz harmonics coincide with the normalized Newman-Penrose dyad [43]. Let us note that there were considered other twistor formulations of the bosonic string [44] and superstrings [45] with non-zero tension in four-dimensional space.

Because the models of massive (super)particles and tensile (super)strings contain dimensionful parameters, characteristic feature of the (super)twistor formulations of their Lagrangians is the presence of the twistor realization of metric tensors in the spaces of two-component $SL(2, \mathbb{C})$ spinors. They break manifestly space-time (super)conformal symmetry and are called infinity twistors [21], [22].

Another class of supersymmetric string models in twistor space constitute twistor string models. For the first time such models were proposed in Refs. [46], [18], [47]. Twistor strings are *ab initio* formulated in the supertwistor space, do not contain dimensionless parameters and hence are invariant under the superconformal symmetry by definition. Their relation of the twistor strings to the twistor formulations of tensionless superstrings was examined in [12].

In the next subsection considered is the twistor formulation of the bosonic string Lagrangian that in what follows is generalized to the case of the $D = 4$ $N = 1$ superstring. Superstring Lagrangian in the supertwistor formulation inherits non-linearities of the super-space formulation and contains pure gauge Grassmann variables unlike supertwistor formulations of the models of massless superparticles and tensionless superstrings. As their generalization proposed will be reduced supertwistor models, Lagrangians of which are quadratic in the (super)twistor components. Also there will be established the relation between reduced models and superstrings in the supertwistor formulation. It will be shown that reduced supertwistor models corresponding to the $N = 1$ superstring and bosonic string model in the twistor formulation are particular cases of reduced model for the $N = 2$ superstring. In the last subsection this model will be examined as the constrained Hamiltonian system. Following the Dirac method [15] identified are all the constraints of the model and divided into the Lorentz-covariant and irreducible first-class ones, that generate gauge symmetries of the action, and the second-class constraints.
## 2.1 Twistor formulation of bosonic string

In the twistor formulation of the bosonic string action that we proposed, basic dynamical variables are a pair of the Penrose twistors

$$Z^{\alpha} = (\mu^{\alpha}, \bar{u}_{\dot{\alpha}}), \quad W^{\alpha} = (\nu^{\alpha}, \bar{v}_{\dot{\alpha}}), \quad \alpha = 1, \dots, 4; \quad \alpha, \dot{\alpha} = 0, 1. \qquad (2.1)$$

$\mu^\alpha$ and $\nu^\alpha$ represent their primary spinor parts but $\bar{u}_{\dot{\alpha}}$ and $\bar{v}_{\dot{\alpha}}$ are projection parts. Dual twistors have the following constituents

$$\bar{Z}_{\alpha} = (u_{\alpha}, \bar{\mu}^{\dot{\alpha}}), \quad \bar{W}_{\alpha} = (v_{\alpha}, \bar{\nu}^{\dot{\alpha}}).$$

Each twistor and its dual have complex components and by definition realize (anti)fundamental representation of the SU(2, 2) group locally isomorphic to conformal group of the four-dimensional Minkowski space-time. It is assumed that projection parts of twistors form normalized Newman-Penrose dyad

$$u^{\alpha} v_{\alpha} = 1, \quad \bar{u}^{\dot{\alpha}} \bar{v}_{\dot{\alpha}} = 1. \qquad (2.2)$$

Conditions (2.2) imply that 2 × 2 matrices

$$v_{(\alpha)}^{\alpha} = (u^{\alpha}, v^{\alpha}), \quad \bar{v}_{(\dot{\alpha})}^{\dot{\alpha}} = (\bar{u}^{\dot{\alpha}}, \bar{v}^{\dot{\alpha}}) \qquad (2.3)$$

have six real independent components and take value in the SL(2, C) group. So the pair of twistors (2.1) has 14 independent components. In the twistor form normalization conditions (2.2)

$$W^{\alpha} I_{\alpha\beta} Z^{\beta} = 1, \quad \bar{W}_{\alpha} I^{\alpha\beta} \bar{Z}_{\beta} = 1 \qquad (2.4)$$

include infinity twistors

$$I_{\alpha\beta} = \begin{pmatrix} 0 & 0 \\ 0 & \varepsilon^{\dot{\alpha}\dot{\beta}} \end{pmatrix}, \quad I^{\alpha\beta} = \begin{pmatrix} \varepsilon^{\alpha\beta} & 0 \\ 0 & 0 \end{pmatrix}$$

that are degenerate matrices and manifestly break conformal symmetry.

In order to associate with the twistor components coordinates of the points of the $D=4$ Minkowski space-time one should impose on them two real zero-norm conditions

$$\bar{Z}_{\alpha} Z^{\alpha} = u_{\alpha} \mu^{\alpha} + \bar{\mu}^{\dot{\alpha}} \bar{u}_{\dot{\alpha}} = 0, \quad \bar{W}_{\alpha} W^{\alpha} = v_{\alpha} \nu^{\alpha} + \bar{\nu}^{\dot{\alpha}} \bar{v}_{\dot{\alpha}} = 0 \qquad (2.5)$$

and two real orthogonality conditions

$$\bar{W}_{\alpha} Z^{\alpha} = v_{\alpha} \mu^{\alpha} + \bar{\nu}^{\dot{\alpha}} \bar{u}_{\dot{\alpha}} = 0, \quad (\bar{W}_{\alpha} Z^{\alpha})^{\dagger} = \bar{Z}_{\alpha} W^{\alpha} = 0. \qquad (2.6)$$

Conditions (2.5) and (2.6) reduce the number of independent components of twistors (2.1) to 10. The same overall number of independent components have the space-time coordinates $x^{\dot{\alpha}\alpha} = x^m \tilde{\sigma}_m^{\dot{\alpha}\alpha}$ and the normalized dyad (2.3). The solution of conditions (2.5) and (2.6) is
given by expressions for the primary spinor parts of twistors and their duals as the products of the space-time coordinates and the projection parts

$$ \mu^{\alpha} = i \bar{u}_{\dot{\alpha}} x^{\dot{\alpha} \alpha}, \quad \nu^{\alpha} = i \bar{v}_{\dot{\alpha}} x^{\dot{\alpha} \alpha}, \qquad (2.7) $$

$$ \bar{\mu}^{\dot{\alpha}} = -i x^{\dot{\alpha} \alpha} u_{\alpha}, \quad \bar{\nu}^{\dot{\alpha}} = -i x^{\dot{\alpha} \alpha} v_{\alpha}. \qquad (2.8) $$

Expressions (2.7) and (2.8) represent instances of the relations that connect twistor components with the space-time coordinates and in the twistor theory are called incidence relations.

Twistor formulation of the $D = 4$ bosonic string proposed in our work [1] is based on the string action [3], [4] that includes tangent to the world sheet components of the local orthonormal Cartan frame [48]. They represent columns of the matrix of vector Lorentz harmonics [49] and can be realized as the products of the Newman-Penrose dyad components [43]. In Ref. [3] there was introduced the following $D = 4$ string action

$$ S_{\text{string}}^{D=4} = \int_{\Sigma} L(\xi), \qquad (2.9) \\ L_{\text{string}}^{D=4}(\xi) = \frac{1}{2(\alpha')^{1/2}} [e^{[+2]} \wedge (\bar{u}^{-} dxu^{-}) - e^{[-2]} \wedge (\bar{v}^{+} dxv^{+})] + \frac{c}{2} e^{[-2]} \wedge e^{[+2]}. $$

It transfers to the Polyakov action [5], [6], [7] upon exclusion of the dyad components

$$ S_{\text{Pstring}}^{D=4} = -\frac{1}{2c\alpha'} \int_{\Sigma} d^2\xi \sqrt{-g} g^{\mu\nu} \partial_{\mu} x^{\nu} \partial_{\nu} x_{\mu}. \qquad (2.10) $$

Lagrangian (2.9) contains components of the zweibein one-form $e^{\pm 2} = d\xi^{\mu} e_{\mu}^{[\pm 2]}$, where $\xi^{\mu} = (\tau, \sigma)$ are local world-sheet coordinates, and the zweibein determinant $e = \frac{1}{2} e^{[-2]} \wedge e^{[+2]}$. Weights of the zweibein and dyad components under the $SO(1, 1)$ group of local symmetry, acting in the tangent space to the world sheet, are given in the form of their superscripts. Dyad components and local frame components orthogonal to the world sheet transform under the local $SO(2) = U(1)$ group action, however their weights are not shown explicitly. $\alpha'$ is the Regge slope parameter, $c$ is the dimensionless constant so the string tension equals $T = \frac{1}{2c\alpha'}$.

To transfer to the twistor formulation let us express in terms of twistors projections of the differentials of the space-time coordinate fields onto the local frame components tangent to the string world sheet

$$ \begin{aligned} -\bar{u}^{-} dxu^{-} &= \frac{i}{2} \omega_{Z}^{[-2]}(d) = \frac{i}{2} (\bar{Z}^{-} dZ^{-} - d\bar{Z}^{-} Z^{-}), \\ -\bar{v}^{+} dxv^{+} &= \frac{i}{2} \omega_{W}^{[+2]}(d) = \frac{i}{2} (\bar{W}^{+} dW^{+} - d\bar{W}^{+} W^{+}). \end{aligned} \qquad (2.11) $$

Then the string action acquires the form

$$ S_{\text{string, tw}}^{D=4} = \int_{\Sigma} L(\xi), \qquad (2.12) \\ L_{\text{string, tw}}^{D=4}(\xi) = \frac{-i}{4(\alpha')^{1/2}} [e^{[+2]} \wedge \omega_{Z}^{[-2]}(d) - e^{[-2]} \wedge \omega_{W}^{[+2]}(d)] + \frac{c}{2} e^{[-2]} \wedge e^{[+2]}. $$
Observe that it is invariant under global SU(2, 2) symmetry that is, however, broken by the constraints (2.4).

To derive equations of motion that follow from the action (2.12) it is necessary to take into account the constraints for twistors (2.5), (2.6) and (2.4). They can either be added to the action (2.12) with Lagrange multipliers or one can apply the method of admissible variations. Its detailed discussion on example of the Lorentz-harmonic formulations of the bosonic string and superstring models is given in Ref. [4]. Admissible variations of twistors were found in our work [1]

$$
\begin{align}
\delta Z^{\alpha-} &= \omega(\delta) Z^{\alpha-} + \tilde{\omega}^{[-2]}(\delta) W^{\alpha+} + \frac{1}{2} \omega_{\bar{W}Z}(\delta) I^{\alpha\beta} \bar{Z}_{\beta-} - \frac{1}{2} \omega_{Z}^{[-2]}(\delta) I^{\alpha\beta} \bar{W}_{\beta+}, \nonumber \\
\delta W^{\alpha+} &= \Omega^{[+2]}(\delta) Z^{\alpha-} - \omega(\delta) W^{\alpha+} + \frac{1}{2} \omega_{\bar{W}}^{[+2]}(\delta) I^{\alpha\beta} \bar{Z}_{\beta-} - \frac{1}{2} \omega_{\bar{W}Z}(\delta) I^{\alpha\beta} \bar{W}_{\beta+}, \nonumber \\
\delta \bar{Z}_{\alpha-} &= \bar{\omega}(\delta) \bar{Z}_{\alpha-} + \tilde{\omega}^{[-2]}(\delta) \bar{W}_{\alpha+} - \frac{1}{2} \omega_{\bar{W}Z}(\delta) I_{\alpha\beta} Z^{\beta+} + \frac{1}{2} \omega_{Z}^{[-2]}(\delta) I_{\alpha\beta} W^{\beta+}, \nonumber \\
\delta \bar{W}_{\alpha+} &= \bar{\Omega}^{[+2]}(\delta) \bar{Z}_{\alpha-} - \bar{\omega}(\delta) \bar{W}_{\alpha+} - \frac{1}{2} \omega_{\bar{W}}^{[+2]}(\delta) I_{\alpha\beta} Z^{\beta-} + \frac{1}{2} \omega_{\bar{W}Z}(\delta) I_{\alpha\beta} W^{\beta+}. \tag{2.13}
\end{align}
$$

The first two terms in each line contain the spin coefficients [43] in which variations have been substituted for differentials

$$
\omega(\delta) = -\frac{1}{2}(\delta Z^{-} I W^{+} + \delta W^{+} I Z^{-}) = \frac{1}{2}(\delta \bar{u}^{\dot{\alpha}-} \bar{v}_{\dot{\alpha}}^{+} + \delta \bar{v}^{\dot{\alpha}+} \bar{u}_{\dot{\alpha}}^{-}), \quad (2.14)
$$

$$
\tilde{\omega}^{[-2]}(\delta) = \delta Z^{-} I Z^{-} = \bar{u}^{\dot{\alpha}-} \delta \bar{u}_{\dot{\alpha}}^{-}, \quad \Omega^{[+2]}(\delta) = -\delta W^{+} I W^{+} = \delta \bar{v}^{\dot{\alpha}+} \bar{v}_{\dot{\alpha}}^{+}. \quad (2.15)
$$

These terms in (2.13) originate from variation of the dyad components and enter both vari-
ations of the projection parts of twistors and of the primary spinor parts via the incidence
relations (2.7) and (2.8). In the orthonormal frame method [48] spin coefficients are inter-
preted as derivation coefficients of the local frame and equal the Cartan one-forms associated
with the generators of the so(1, 3) algebra and constructed out of the frame vectors and their
differentials [13], [14]. Two last terms in each line in (2.13) originate from variations of the
space-time coordinates in the primary spinor parts of twistors. They contain variation pa-
rameters

$$
\omega_Z^{[-2]}(\delta) = (\bar{Z}^- \delta Z^- - \delta \bar{Z}^- Z^-) = 2i \bar{u}^- \delta x u^-, \quad (2.16)
$$

$$
\omega_W^{[+2]}(\delta) = (\bar{W}^+ \delta W^+ - \delta \bar{W}^+ W^+) = 2i \bar{v}^+ \delta x v^+, \quad (2.17)
$$

$$
\omega_{\bar{W}Z}(\delta) = (\bar{W}^+ \delta Z^- - \delta \bar{W}^+ Z^-) = 2i \bar{u}^- \delta x v^+, \quad \omega_{\bar{Z}W}(\delta) = -(\omega_{\bar{W}Z}(\delta))^\dagger, \quad (2.18)
$$

expressions for which can be obtained using the completeness relations for the normalized
dyad (2.2)

$$
u^{\alpha-} v_{\beta}^{+} - u_{\beta}^{-} v^{\alpha+} = \delta_{\beta}^{\alpha}, \quad \bar{u}^{\dot{\alpha}-} \bar{v}_{\dot{\beta}}^{+} - \bar{u}_{\dot{\beta}}^{-} \bar{v}^{\dot{\alpha}+} = \delta_{\dot{\beta}}^{\dot{\alpha}}. \qquad (2.19)
$$

Variation parameters (2.14)-(2.18) are associated, as was shown in [1], with 10 first-order dif-
ferential operators that obey commutation relations of the D = 4 Poincaré algebra *iso(1, 3)*.
Presence of only 10 independent variation parameters is related to the fact that twistor
constraints (2.4) are invariant only under the Poincaré subgroup of the *SU(2, 2)* symmetry.
The substitution of obtained variations of twistors (2.13) to variation of the action (2.12) yields the set of 10 string equations

$$ \frac{\delta S_{\text{string, tw}}^{D=4}}{\delta \omega} = e^{[+2]} \wedge \omega_Z^{[-2]}(d) + e^{[-2]} \wedge \omega_W^{[+2]}(d) = 0, \qquad (2.20) $$

$$ \frac{\delta S_{\text{string, tw}}^{D=4}}{\delta \tilde{\omega}^{[-2]}} = e^{[+2]} \wedge \omega_{\bar{Z}W}(d) = 0, \qquad (2.21) $$

$$ \frac{\delta S_{\text{string, tw}}^{D=4}}{\delta \Omega^{[+2]}} = e^{[-2]} \wedge \omega_{\bar{W}Z}(d) = 0, \qquad (2.22) $$

$$ \frac{\delta S_{\text{string, tw}}^{D=4}}{\delta \omega_Z^{[-2]}} = de^{[+2]} + e^{[+2]} \wedge (\omega + \bar{\omega})(d) = 0, \qquad (2.23) $$

$$ \frac{\delta S_{\text{string, tw}}^{D=4}}{\delta \omega_W^{[+2]}} = de^{[-2]} - e^{[-2]} \wedge (\omega + \bar{\omega})(d) = 0, \qquad (2.24) $$

$$ \frac{\delta S_{\text{string, tw}}^{D=4}}{\delta \omega_{\bar{Z}W}} = e^{[+2]} \wedge \tilde{\omega}^{[-2]}(d) - e^{[-2]} \wedge \bar{\Omega}^{[+2]}(d) = 0. \qquad (2.25) $$

Note that Eqs. (2.20)-(2.22) and (2.25) are complex and should be complemented by their conjugates. To the above set of equations one has to add those defining induced zweibein one-form

$$ e^{[-2]} = \frac{-i}{2c(\alpha')^{1/2}} \omega_Z^{[-2]}(d), \quad e^{[+2]} = \frac{-i}{2c(\alpha')^{1/2}} \omega_W^{[+2]}(d). \qquad (2.26) $$

Now let us find which of the above equations are independent. From Eqs. (2.26) it follows that Eq. (2.20) turns into identity in agreement with the $SO(1, 1) \times SO(2)$ gauge invariance of the action (2.12) (see also [3], [4]). Eqs. (2.21) and (2.22) and their c.c. yield that

$$ \omega_{\bar{W}Z}(d) = \omega_{\bar{Z}W}(d) = 0. \qquad (2.27) $$

Eqs. (2.26) and (2.27) are twistor form of the equations that in the orthonormal frame method fix its orientation with respect to the string world sheet [13], [14]. Eqs. (2.23) and (2.24) represent zero torsion conditions for the induced zweibein and are satisfied identically provided (2.26) and (2.27) are used. Remaining Eq.(2.25) together with the only nontrivial relation following from the integrability conditions of the expressions for twistor differentials (2.13)

$$ e^{+2} \wedge \tilde{\omega}^{-2}(d) + e^{-2} \wedge \bar{\Omega}^{+2}(d) = 0 \qquad (2.28) $$

yield that

$$ e^{+2} \wedge \tilde{\omega}^{-2}(d) = e^{-2} \wedge \bar{\Omega}^{+2}(d) = 0 \qquad (2.29) $$

and the c.c. equations. These equations are conditions of the minimal embedding of the string world sheet, another equivalent representation for which was obtained in [14].

So we have shown that in the twistor formulation one reproduces equations that determine minimal embedding of the string world sheet in the orthonormal frame method.
## 2.2 Supertwistor formulations of $D = 4$ superstrings

Let us turn to consideration of the twistor formulation for the $D = 4$ $\mathcal{N} = 1$ superstring. Basic dynamical variables are $SU(2, 2|1)$ supertwistors and their duals

$$ \mathcal{Z}^A = (\mu^\alpha, \bar{u}_{\dot{\alpha}}, \bar{\eta}), \quad \bar{\mathcal{Z}}_A = (u_\alpha, \bar{\mu}^{\dot{\alpha}}, \eta); \quad \mathcal{W}^A = (\nu^\alpha, \bar{v}_{\dot{\alpha}}, \bar{\zeta}), \quad \bar{\mathcal{W}}_A = (v_\alpha, \bar{\nu}^{\dot{\alpha}}, \zeta) $$

that obey supersymmetric generalizations of the zero norm and orthogonality conditions

$$ \bar{\mathcal{Z}}_A \mathcal{Z}^A = u_\alpha \mu^\alpha + \bar{\mu}^{\dot{\alpha}} \bar{u}_{\dot{\alpha}} + \eta \bar{\eta} = 0, \quad \bar{\mathcal{W}}_A \mathcal{W}^A = v_\alpha \nu^\alpha + \bar{\nu}^{\dot{\alpha}} \bar{v}_{\dot{\alpha}} + \zeta \bar{\zeta} = 0, \quad (2.30) $$

$$ \bar{\mathcal{W}}_A \mathcal{Z}^A = v_\alpha \mu^\alpha + \bar{\nu}^{\dot{\alpha}} \bar{u}_{\dot{\alpha}} + \zeta \bar{\eta} = 0, \quad \bar{\mathcal{Z}}_A \mathcal{W}^A = u_\alpha \nu^\alpha + \bar{\mu}^{\dot{\alpha}} \bar{v}_{\dot{\alpha}} + \eta \bar{\zeta} = 0. $$

These conditions ensure that the number of independent components two supertwistors equals the sum of the number of $D = 4$ $\mathcal{N} = 1$ Minkowski superspace coordinates $(x_{\alpha\dot{\alpha}}, \theta^\alpha, \bar{\theta}^{\dot{\alpha}})$ and the number of components of the normalized Newman-Penrose dyad (2.3). Indeed, each supertwistor has four complex bosonic components subject to four real bosonic conditions (2.30) and two real dyad normalization conditions (2.2). So the number of independent bosonic components of two supertwistors equals the sum of the number of the space-time coordinates and the number of components of the normalized Newman-Penrose dyad. Also each supertwistor has one complex Grassmann-odd component so that pair of supertwistors has the same number of Grassmann-odd degrees of freedom as there are anticommuting coordinates of the $D = 4$ $\mathcal{N} = 1$ superspace. Conditions (2.30) can be solved explicitly and one can express primary spinor parts and Grassmann-odd components of supertwistors in terms of the superspace coordinates. The solution

$$ \begin{aligned} \mu^\alpha &= i \bar{u}_{\dot{\alpha}} x^{\dot{\alpha} \alpha} + \theta^\alpha \bar{\eta}, \quad \bar{\mu}^{\dot{\alpha}} = -i x^{\dot{\alpha} \alpha} u_\alpha + \eta \bar{\theta}^{\dot{\alpha}}, \quad \bar{\eta} = 2 \bar{u}^{\dot{\alpha}} \bar{\theta}_{\dot{\alpha}}, \quad \eta = 2 u^\alpha \theta_\alpha; \\ \nu^\alpha &= i \bar{v}_{\dot{\alpha}} x^{\dot{\alpha} \alpha} + \theta^\alpha \bar{\zeta}, \quad \bar{\nu}^{\dot{\alpha}} = -i x^{\dot{\alpha} \alpha} v_\alpha + \zeta \bar{\theta}^{\dot{\alpha}}, \quad \bar{\zeta} = 2 \bar{v}^{\dot{\alpha}} \bar{\theta}_{\dot{\alpha}}, \quad \zeta = 2 v^\alpha \theta_\alpha \end{aligned} \quad (2.31) $$

represents supersymmetric generalization of the incidence relations (2.7) and (2.8).

$D = 4$ $\mathcal{N} = 1$ superstring action in the Lorentz-harmonic formulation [3] is given by the sum of kinetic and Wess-Zumino terms

$$ S_{\text{sstring}}^{D=4,\mathcal{N}=1} = S_{\text{kin}}^{D=4,\mathcal{N}=1} + S_{\text{WZ}}^{D=4,\mathcal{N}=1}. \quad (2.32) $$

Kinetic term represents supersymmetric generalization of the bosonic string action in the Lorentz-harmonic formulation (2.9) and includes supersymmetric Cartan one-forms $\omega_{\alpha\dot{\alpha}}(d) = dx_{\alpha\dot{\alpha}} + 2i d\theta_\alpha \bar{\theta}_{\dot{\alpha}} - 2i \theta_\alpha d\bar{\theta}_{\dot{\alpha}}$

$$ S_{\text{kin}}^{D=4,\mathcal{N}=1} = \int_{\Sigma} \left( \frac{1}{2(\alpha')^{1/2}} \left[ e^{[+2]} \wedge (\bar{u}^{-} \omega(d) u^{-}) - e^{-[2]} \wedge (\bar{v}^{+} \omega(d) v^{+}) \right] + \frac{c}{2} e^{[2]} \wedge e^{[+2]} \right). $$

In the Wess-Zumino action

$$ S_{\text{WZ}}^{D=4,\mathcal{N}=1} = \frac{is}{c\alpha'} \int_{\Sigma} \omega^{\dot{\alpha}\alpha}(d) \wedge (d\theta_{\alpha}\bar{\theta}_{\dot{\alpha}} - \theta_{\alpha}d\bar{\theta}_{\dot{\alpha}}) \quad (2.33) $$
there has been introduced the sign factor $s = \pm 1$ that reflects arbitrariness in its definition compatible with the $\kappa$-invariance of the complete action (2.32). Upon exclusion of the Lorentz-harmonic variables action (2.32) turns into the $D = 4$ $\mathcal{N} = 1$ Green-Schwarz superstring action [8]

$$S_{\text{GSstring}}^{D=4,\mathcal{N}=1} = S_{\text{GSkin}}^{D=4,\mathcal{N}=1} + S_{\text{WZ}}^{D=4,\mathcal{N}=1}.$$

Its kinetic term

$$S_{\text{GSkin}}^{D=4,\mathcal{N}=1} = -\frac{1}{2c\alpha'} \int_{\Sigma} d^2\xi \sqrt{-gg^{\mu\nu}\omega_{\mu}^{m}\omega_{\nu m}}$$

is the supersymmetric generalization of the Polyakov action (2.10) and Wess-Zumino term is the same as in (2.33).

To transfer to the supertwistor formulation of the kinetic term used are relations

$$\begin{aligned} -\bar{u}^{-}\omega(d)u^{-} &= \frac{i}{2}\omega_{\bar{Z}}^{[-2]}(d) = \frac{i}{2}(\bar{Z}^{-}d\bar{Z}^{-} - d\bar{Z}^{-}\bar{Z}^{-}), \\ -\bar{v}^{+}\omega(d)v^{+} &= \frac{i}{2}\omega_{\bar{W}}^{[+2]}(d) = \frac{i}{2}(\bar{W}^{+}d\bar{W}^{+} - d\bar{W}^{+}\bar{W}^{+}), \end{aligned} \qquad (2.34)$$

that generalize (2.11). As a result we obtain the following action

$$S_{\text{kin, stw}}^{D=4,\mathcal{N}=1} = \int_{\Sigma} \left( \frac{-i}{4(\alpha')^{1/2}} \left[ e^{[+2]} \wedge \omega_{\bar{Z}}^{[-2]}(d) - e^{[-2]} \wedge \omega_{\bar{W}}^{[+2]}(d) \right] + \frac{c}{2} e^{[-2]} \wedge e^{[+2]} \right). \qquad (2.35)$$

Due to completeness relations for the normalized Newman-Penrose dyad (2.19) the Wess-Zumino can be expressed

$$S_{\text{WZ, stw}}^{D=4, \mathcal{N}=1} = -\frac{s}{8c\alpha'} \int_{\Sigma} \left( \omega_{\bar{Z}}^{[-2]}(d) \wedge \omega_{\bar{\zeta}}^{[+2]}(d) + \omega_{\bar{W}}^{[+2]}(d) \wedge \omega_{\bar{\eta}}^{[-2]}(d) \right. \\ \left. - \omega_{\bar{Z}\bar{W}}(d) \wedge \omega_{\bar{\zeta}\bar{\eta}}(d) - \omega_{\bar{W}\bar{Z}}(d) \wedge \omega_{\bar{\eta}\bar{\zeta}}(d) \right) \qquad (2.36)$$

via the one-forms (2.34) and

$$\omega_{\bar{W}\bar{Z}}(d) = \bar{W}^{+}d\bar{Z}^{-} - d\bar{W}^{+}\bar{Z}^{-}, \quad \omega_{\bar{Z}\bar{W}}(d) = -(\omega_{\bar{W}\bar{Z}})^{\dagger}, \qquad (2.37)$$

$$\omega_{\bar{\eta}}^{[-2]}(d) = \eta^{-} \mathcal{D}\bar{\eta}^{-} - \mathcal{D}\eta^{-}\bar{\eta}^{-}, \quad \omega_{\bar{\zeta}}^{[+2]}(d) = \zeta^{+} \mathcal{D}\bar{\zeta}^{+} - \mathcal{D}\zeta^{+}\bar{\zeta}^{+}, \qquad (2.38)$$

$$\omega_{\bar{\zeta}\bar{\eta}}(d) = \zeta^{+} \mathcal{D}\bar{\eta}^{-} - \mathcal{D}\zeta^{+}\bar{\eta}^{-}, \quad \omega_{\bar{\eta}\bar{\zeta}}(d) = -(\omega_{\bar{\zeta}\bar{\eta}})^{\dagger}. \qquad (2.39)$$

One-forms (2.38) and (2.39) include projections of the differentials of the Grassmann-odd superspace coordinates expressed in terms of the Grassmann-odd supertwistor components and spin coefficients

$$\begin{aligned} \mathcal{D}\bar{\eta}^{-} &= 2\bar{u}^{\dot{\alpha}-}d\bar{\theta}_{\dot{\alpha}} = d\bar{\eta}^{-} - \omega(d)\bar{\eta}^{-} - \tilde{\omega}^{[-2]}(d)\bar{\zeta}^{+}, \quad \mathcal{D}\eta^{-} = (\mathcal{D}\bar{\eta}^{-})^{*}, \\ \mathcal{D}\bar{\zeta}^{+} &= 2\bar{v}^{\dot{\alpha}+}d\bar{\theta}_{\dot{\alpha}} = d\bar{\zeta}^{+} + \omega(d)\bar{\zeta}^{+} - \Omega^{[+2]}(d)\bar{\eta}^{-}, \quad \mathcal{D}\zeta^{+} = (\mathcal{D}\bar{\zeta}^{+})^{*}. \end{aligned} \qquad (2.40)$$

Formulation of the Wess-Zumino term in terms of the supertwistor components exhibits the fact that the superstring action is invariant only under the super-Poincaré subgroup that enters $SU(2, 2|1)$ superconformal symmetry. This is explained by presence of the dimensionful
tension $T = \frac{1}{2c\alpha'}$. So in the supertwistor formulation the $\mathcal{N} = 1$ superstring action has the form

$$S_{\text{sstring, stw}}^{D=4, \mathcal{N}=1} = S_{\text{kin, stw}}^{D=4, \mathcal{N}=1} + S_{\text{WZ, stw}}^{D=4, \mathcal{N}=1}. \qquad (2.41)$$

To derive $D = 4$ $\mathcal{N} = 1$ superstring equations in the supertwistor formulation in our work [1] used was the method of admissible variations similarly to the bosonic string case. It was shown that variations of supertwistors compatible with the constraints (2.30) and (2.4) have the form

$$\begin{aligned} \delta \mathcal{Z}^{A-} &= \omega(\delta) \mathcal{Z}^{A-} + \tilde{\omega}^{[-2]}(\delta) \mathcal{W}^{A+} + \left(\frac{1}{2} \omega_{\bar{W}\bar{Z}}(\delta) - \zeta^+ \mathcal{D}(\delta) \bar{\eta}^-\right) \mathcal{I}^{AB} \bar{\mathcal{Z}}_B^{-} \\ &\quad + \left(-\frac{1}{2} \omega_Z^{[-2]}(\delta) + \eta^- \mathcal{D}(\delta) \bar{\eta}^-\right) \mathcal{I}^{AB} \bar{\mathcal{W}}_B^+ + \mathcal{J}_B^A \delta \mathcal{Z}^{B-}, \\ \delta \mathcal{W}^{A+} &= \Omega^{[+2]}(\delta) \mathcal{Z}^{A-} - \omega(\delta) \mathcal{W}^{A+} + \left(\frac{1}{2} \omega_W^{[+2]}(\delta) - \zeta^+ \mathcal{D}(\delta) \bar{\zeta}^+\right) \mathcal{I}^{AB} \bar{\mathcal{Z}}_B^{-} \\ &\quad + \left(-\frac{1}{2} \omega_{\bar{Z}W}(\delta) + \eta^- \mathcal{D}(\delta) \bar{\zeta}^+\right) \mathcal{I}^{AB} \bar{\mathcal{W}}_B^+ + \mathcal{J}_B^A \delta \mathcal{W}^{B+} \end{aligned} \qquad (2.42)$$

and c.c. The supermatrices

$$\mathcal{I}^{AB} = \begin{pmatrix} I^{\alpha\beta} & 0 \\ 0 & 0 \end{pmatrix}, \quad \mathcal{J}_B^A = \begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 1 \end{pmatrix}$$

are supersymmetric generalizations of infinity twistors. In variations (2.42) there are 10 bosonic and four fermionic independent parameters in agreement with the fact that the constraints (2.4) are invariant only under the super-Poincaré subgroup of $SU(2, 2|1)$ symmetry. The first two terms on the r.h.s. include six parameters (2.14) and (2.15) associated with the spin coefficients. They come from variations of the dyad components taking into account the incidence relations (2.31). The third and fourth terms correspond to variations of the superspace coordinates in the primary spinor parts of supertwistors and include one-forms (2.34), (2.37) and (2.40) in which variations were substituted for differentials. The last terms on the r.h.s. of (2.42) represent variations of the Grassmann-odd supertwistor components singled out by the projection supermatrix $\mathcal{J}_B^A$.

From equations for the zweibein components it follows that they are expressed in terms of the one-forms (2.34)

$$e^{[-2]} = \frac{-i}{2c(\alpha')^{1/2}} \omega_{\bar{Z}}^{[-2]}(d), \quad e^{[+2]} = \frac{-i}{2c(\alpha')^{1/2}} \omega_{\bar{W}}^{[+2]}(d). \qquad (2.43)$$

These equations represent supersymmetric generalization of the Eqs. (2.26) for the bosonic string. Substitution of (2.43) into equation

$$\frac{\delta S_{\text{sstring, stw}}^{D=4, \mathcal{N}=1}}{\delta \omega} = e^{[+2]} \wedge \omega_{\bar{Z}}^{[-2]}(d) + e^{[-2]} \wedge \omega_{\bar{W}}^{[+2]}(d) = 0$$

turns it into identity, as for the bosonic string. This is the consequence of the $SO(1, 1) \times SO(2)$ gauge symmetry of the superstring action (2.41). From the equations

$$\frac{\delta S_{\text{sstring, stw}}^{D=4, \mathcal{N}=1}}{\delta \tilde{\omega}^{-2}} = e^{+2} \wedge \omega_{\bar{Z}W}(d) = 0, \quad \frac{\delta S_{\text{sstring, stw}}^{D=4, \mathcal{N}=1}}{\delta \Omega^{+2}} = e^{-2} \wedge \omega_{\bar{W}\bar{Z}}(d) = 0$$
and c.c. ones it follows vanishing of the one-forms

$$ \omega_{\bar{W}\bar{Z}}(d) = \omega_{\bar{Z}W}(d) = 0. \qquad (2.44) $$

Equations (2.43) and (2.44) represent supertwistor form of the equations fixing orientation of the local frame with respect to the world sheet of the superstring. Remaining bosonic equations have the form

$$ \begin{aligned} \frac{\delta S_{\text{sstring, stw}}^{D=4, \mathcal{N}=1}}{\delta \omega_{\bar{Z}}^{[-2]}} &= [de^{[+2]} + e^{[+2]} \wedge (\omega + \bar{\omega})(d)] + \frac{is}{c(\alpha')^{1/2}} \mathcal{D}\zeta^{+} \wedge \mathcal{D}\bar{\zeta}^{+} = 0, \\ \frac{\delta S_{\text{sstring, stw}}^{D=4, \mathcal{N}=1}}{\delta \omega_{\bar{W}}^{[+2]}} &= [de^{[-2]} - e^{[-2]} \wedge (\omega + \bar{\omega})(d)] - \frac{is}{c(\alpha')^{1/2}} \mathcal{D}\eta^{-} \wedge \mathcal{D}\bar{\eta}^{-} = 0, \\ \frac{\delta S_{\text{sstring, stw}}^{D=4, \mathcal{N}=1}}{\delta \omega_{\bar{Z}W}} &= e^{[-2]} \wedge \bar{\Omega}^{[+2]}(d) - e^{[+2]} \wedge \tilde{\omega}^{[-2]}(d) + \frac{is}{c(\alpha')^{1/2}} \mathcal{D}\zeta^{+} \wedge \mathcal{D}\bar{\eta}^{-} = 0. \end{aligned} $$

The last equation should be complemented by the c.c. The first two of these equations express induced world-sheet torsion $T^{[\pm 2]} \equiv de^{[\pm 2]} \pm e^{[\pm 2]} \wedge (\omega + \bar{\omega})(d)$ through the wedge product of the differentials of the Grassmann-odd components of supertwistors (2.40). The third equation together with the nontrivial relation that arises from the integrability conditions for admissible supertwistor differentials (2.42) produce the following equations

$$ e^{[+2]} \wedge \tilde{\omega}^{[-2]}(d) - \frac{i(s+1)}{2c(\alpha')^{1/2}} \mathcal{D}\bar{\eta}^{-} \wedge \mathcal{D}\zeta^{+} = 0, \quad e^{[-2]} \wedge \Omega^{[+2]}(d) + \frac{i(s-1)}{2c(\alpha')^{1/2}} \mathcal{D}\bar{\eta}^{-} \wedge \mathcal{D}\zeta^{+} = 0. $$

From the action (2.41) also follow fermionic equations of motion

$$ \begin{aligned} \frac{\delta S_{\text{sstring, stw}}^{D=4, \mathcal{N}=1}}{\delta \mathcal{D}\bar{\eta}^{-}} &= e^{[+2]} \wedge \mathcal{D}\eta^{-} - \frac{is}{2c(\alpha')^{1/2}} \left[ \omega_{\bar{W}}^{[-2]}(d) \wedge \mathcal{D}\eta^{-} - \omega_{\bar{Z}W}(d) \wedge \mathcal{D}\zeta^{+} \right] = 0, \\ \frac{\delta S_{\text{sstring, stw}}^{D=4, \mathcal{N}=1}}{\delta \mathcal{D}\zeta^{+}} &= e^{[-2]} \wedge \mathcal{D}\zeta^{+} + \frac{is}{2c(\alpha')^{1/2}} \left[ \omega_{\bar{Z}}^{[-2]}(d) \wedge \mathcal{D}\zeta^{+} - \omega_{\bar{W}\bar{Z}}(d) \wedge \mathcal{D}\eta^{-} \right] = 0 \end{aligned} $$

and c.c. equations. The substitution of (2.43) and (2.44) allows to bring them to the form

$$ (s + 1)e^{[+2]} \wedge \mathcal{D}\eta^{-} = 0, \quad (s - 1)e^{[-2]} \wedge \mathcal{D}\zeta^{+} = 0. \qquad (2.45) $$

For each admissible value of the numerical coefficient $s$ half of the above equations turns into identity. According to the second Noether theorem this is related to the $\kappa$-invariance of the superstring action in the supertwistor formulation (2.41). Thus after transition to the supertwistor formulation in the fermionic sector of the $D = 4 \mathcal{N} = 1$ superstring there remain pure gauge degrees of freedom. This distinguishes tensile superstring model from the massless superparticle [11] and tensionless super p-brane [37], [38], [12] models. In these models pure gauge components of the superspace Grassmann coordinates drop out from the Lagrangian upon transition from the superspace to supertwistor formulation that amounts to implicit $\kappa$-symmetry gauge fixing.

Explicit form of variations of the components of supertwistors and zweibein under the $\kappa$-symmetry depends on the value of $s$. When $s = -1$ variations are given by the following expressions

$$ \begin{aligned} \delta_{\kappa} \mathcal{Z}^{A-} &= \tilde{\omega}^{[-2]}(\delta_{\kappa}) \mathcal{W}^{A+} - \zeta^{+} \kappa^{-} \mathcal{I}^{AB} \bar{Z}_{B}^{-} + \eta^{-} \kappa^{-} \mathcal{I}^{AB} \bar{W}_{B}^{+} + \mathcal{J}_{B}^{A} \mathcal{K}^{B-}, \\ \delta_{\kappa} \mathcal{W}^{A+} &= \Omega^{[+2]}(\delta_{\kappa}) \mathcal{Z}^{A-}, \\ \tilde{\omega}^{[-2]}(\delta_{\kappa}) &= \frac{i}{2c(\alpha')^{1/2}} e^{\mu[-2]} \mathcal{D}_{\mu} \zeta^{+} \kappa^{-}, \quad \bar{\Omega}^{[+2]}(\delta_{\kappa}) = -\frac{i}{2c(\alpha')^{1/2}} e^{\mu[+2]} \mathcal{D}_{\mu} \zeta^{+} \kappa^{-}, \\ \delta_{\kappa} e_{\mu}^{[+2]} &= 0, \quad \delta_{\kappa} e_{\mu}^{[-2]} = -\frac{i}{c(\alpha')^{1/2}} (\kappa^{-} \mathcal{D}_{\mu} \eta^{-} - \mathcal{D}_{\mu} \bar{\eta}^{-} \kappa^{-}) \end{aligned} $$
and c.c. ones. Complex Grassmann-odd parameter $\kappa^-(\xi^\mu)$ can be written in the supertwistor form as $\mathcal{K}^{A-} = (0, 0, \kappa^-)$. Accordingly for $s = 1$ variations equal

$$
\begin{align*}
\delta_{\kappa} \mathcal{Z}^{A-} &= \tilde{\omega}^{[-2]}(\delta_{\kappa}) \mathcal{W}^{A+}, \\
\delta_{\kappa} \mathcal{W}^{A+} &= \Omega^{[+2]}(\delta_{\kappa}) \mathcal{Z}^{A-} - \zeta^{+} \kappa^{+} \mathcal{I}^{AB} \bar{\mathcal{Z}}_{B}^{-} + \eta^{-} \kappa^{+} \mathcal{I}^{AB} \bar{\mathcal{W}}_{B}^{+} + \mathcal{J}_{B}^{A} \mathcal{K}^{B+}, \\
\tilde{\omega}^{[-2]}(\delta_{\kappa}) &= -\frac{i}{2c(\alpha')^{1/2}} e^{\mu[-2]} \mathcal{D}_{\mu} \bar{\eta}^{-} \bar{\kappa}^{+}, \quad \bar{\Omega}^{[+2]}(\delta_{\kappa}) = \frac{i}{2c(\alpha')^{1/2}} e^{\mu[+2]} \mathcal{D}_{\mu} \bar{\eta}^{-} \bar{\kappa}^{+}, \\
\delta_{\kappa} e_{\mu}^{[+2]} &= -\frac{i}{c(\alpha')^{1/2}} (\kappa^{+} \mathcal{D}_{\mu} \zeta^{+} - \mathcal{D}_{\mu} \bar{\zeta}^{+} \bar{\kappa}^{+}), \quad \delta_{\kappa} e_{\mu}^{[-2]} = 0.
\end{align*}
$$

The $\mathcal{N} = 1$ superstring Lagrangian in the supertwistor formulation remains non-linear similarly to the original Green-Schwarz Lagrangian and that in the Lorentz-harmonic formulation. Fixing the $\kappa$-symmetry gauge does not exclude does not remove non-linearities. So define reduced models that generalize to the case of tensile strings the twistor formulations of massless superparticle [11] and tensionless string [12] models invariant under the $D = 4$ $\mathcal{N} = 1$ superconformal symmetry. Lagrangians of the reduced models are quadratic in the components of (super)twistors.

Consider the action

$$
S_{\text{sstring, stw}}^{D=4, \mathcal{N}=1/2} = \int_{\Sigma} \left( \frac{-i}{4(\alpha')^{1/2}} \left[ e^{[+2]} \wedge (\bar{\tilde{Z}}_{\mathcal{A}}^{-} d\tilde{Z}^{A-} - d\bar{\tilde{Z}}_{\mathcal{A}}^{-} \tilde{Z}^{A-}) \right. \right. \\
\left. \left. - e^{[-2]} \wedge (\bar{W}_{\mathcal{A}}^{+} dW^{\alpha+} - d\bar{W}_{\mathcal{A}}^{+} W^{\alpha+}) \right] + \frac{c}{2} e^{[-2]} \wedge e^{[+2]} \right). \qquad (2.46)
$$

It contains only half of the Grassmann-odd variables compared to the superstring action in the supertwistor formulation (2.41) that is reflected by the superscript $\mathcal{N} = 1/2$. This model can be obtained from the action (2.41) by substituting equations (2.43) and (2.44) into the Wess-Zumino Lagrangian (2.36) for $s = +1$ and summing up the result with the Lagrangian for the kinetic term (2.35). Then the components of supertwistor $\tilde{Z}$ and its dual have the following incidence relations with the superspace coordinates

$$
\begin{align}
\tilde{Z}^{A-} &= (\tilde{\mu}^{\alpha-}, \bar{u}_{\dot{\alpha}}^{-}, \tilde{\eta}^{-}) : \quad \tilde{\mu}^{\alpha-} = i\bar{u}_{\dot{\alpha}}^{-}x^{\dot{\alpha}\alpha} + \sqrt{2}\theta^{\alpha}\tilde{\eta}^{-}, \quad \tilde{\eta}^{-} = 2\sqrt{2}\bar{u}^{\dot{\alpha}-}\bar{\theta}_{\dot{\alpha}}, \tag{2.47} \\
\tilde{Z}_{\mathcal{A}}^{-} &= (u_{\alpha}^{-}, \tilde{\mu}^{\dot{\alpha}-}, \tilde{\eta}^{-}) : \quad \tilde{\mu}^{\dot{\alpha}-} = -ix^{\dot{\alpha}\alpha}u_{\alpha}^{-} - \sqrt{2}\bar{\theta}^{\dot{\alpha}}\tilde{\eta}^{-}, \quad \tilde{\eta}^{-} = 2\sqrt{2}u^{\alpha-}\theta_{\alpha} \nonumber
\end{align}
$$

and obey the zero-norm condition

$$
\tilde{\mathcal{Z}}_{\mathcal{A}}^{-} \tilde{\mathcal{Z}}^{A-} = u_{\alpha}^{-} \tilde{\mu}^{\alpha-} + \tilde{\mu}^{\dot{\alpha}-} \bar{u}_{\dot{\alpha}}^{-} + \tilde{\eta}^{-} \tilde{\eta}^{-} = 0.
$$

Incidence relations (2.47) differ from (2.31) therefore the supertwistor itself, its primary spinor and Grassmann-odd components are endowed with tilde. $W^{\alpha+}$ and $\bar{W}_{\alpha}^{+}$ are bosonic twistors, components of which satisfy the incidence relations (2.7) and (2.8). Action of the reduced model that corresponds to the $s = -1$ case is obtained from (2.46) by replacing modified supertwistors $\tilde{Z}^{-}$, $\bar{\tilde{Z}}^{-}$ and twistors $W^{+}$, $\bar{W}^{+}$ by bosonic twistors $Z^{-}$, $\bar{Z}^{-}$ and by supertwistors $\tilde{W}^{+}$, $\bar{\tilde{W}}^{+}$ defined analogously to $\tilde{Z}^{-}$, $\bar{\tilde{Z}}^{-}$

$$
\begin{align}
\tilde{W}^{A+} &= (\tilde{\nu}^{\alpha+}, \bar{v}_{\dot{\alpha}}^{+}, \tilde{\zeta}^{+}) : \quad \tilde{\nu}^{\alpha+} = i\bar{v}_{\dot{\alpha}}^{+}x^{\dot{\alpha}\alpha} + \sqrt{2}\theta^{\alpha}\tilde{\zeta}^{+}, \quad \tilde{\zeta}^{+} = 2\sqrt{2}\bar{v}^{\dot{\alpha}+}\bar{\theta}_{\dot{\alpha}}, \nonumber \\
\tilde{W}_{\mathcal{A}}^{+} &= (v_{\alpha}^{+}, \tilde{\nu}^{\dot{\alpha}+}, \tilde{\zeta}^{+}) : \quad \tilde{\nu}^{\dot{\alpha}+} = -ix^{\dot{\alpha}\alpha}v_{\alpha}^{+} - \sqrt{2}\bar{\theta}^{\dot{\alpha}}\tilde{\zeta}^{+}, \quad \tilde{\zeta}^{+} = 2\sqrt{2}v^{\alpha+}\theta_{\alpha}, \tag{2.48} \\
\tilde{W}_{\mathcal{A}}^{+}\tilde{W}^{A+} &= v_{\alpha}^{+}\tilde{\nu}^{\alpha+} + \tilde{\nu}^{\dot{\alpha}+}\bar{v}_{\dot{\alpha}}^{+} + \tilde{\zeta}^{+}\tilde{\zeta}^{+} = 0. \nonumber
\end{align}
$$
Let us conclude this subsection by considering the $D = 4, N = 2$ superstring model. It is known that its action in the Lorentz-harmonic formulation has the form

$$S_{\text{sstring}}^{D=4,\mathcal{N}=2} = S_{\text{kin}}^{D=4,\mathcal{N}=2} + S_{\text{WZ}}^{D=4,\mathcal{N}=2},$$

where

$$S_{\text{kin}}^{D=4,\mathcal{N}=2} = \int_{\Sigma} \left( \frac{1}{2(\alpha')^{1/2}} \left[ e^{[+2]} \wedge (\bar{u}^{-}\omega(d)u^{-}) - e^{-[2]} \wedge (\bar{v}^{+}\omega(d)v^{+}) \right] \right. \\ \left. + \frac{c}{2} e^{-[2]} \wedge e^{[+2]} \right), \quad (2.49)$$

$\omega_{\alpha\dot{\alpha}}(d) = dx_{\alpha\dot{\alpha}} + 2id\theta_{\alpha}^{I}\bar{\theta}_{\dot{\alpha}}^{I} - 2i\theta_{\alpha}^{I}d\bar{\theta}_{\dot{\alpha}}^{I}, I = 1, 2, \text{ and}$

$$S_{\text{WZ}}^{D=4,\mathcal{N}=2} = \frac{is}{c\alpha'} \int_{\Sigma} \omega^{\dot{\alpha}\alpha}(d) \wedge (d\theta_{\alpha}^{1}\bar{\theta}_{\dot{\alpha}}^{1} - \theta_{\alpha}^{1}d\bar{\theta}_{\dot{\alpha}}^{1} - d\theta_{\alpha}^{2}\bar{\theta}_{\dot{\alpha}}^{2} + \theta_{\alpha}^{2}d\bar{\theta}_{\dot{\alpha}}^{2}) \\ + \frac{2s}{c\alpha'} \int_{\Sigma} (d\theta^{\alpha 1}\bar{\theta}^{\dot{\alpha}1} - \theta^{\alpha 1}d\bar{\theta}^{\dot{\alpha}1}) \wedge (d\theta_{\alpha}^{2}\bar{\theta}_{\dot{\alpha}}^{2} - \theta_{\alpha}^{2}d\bar{\theta}_{\dot{\alpha}}^{2}), \quad s = \pm 1. \quad (2.50)$$

Its Lagrangian can be expressed in terms of the components of supertwistors analogously to the $N = 1$ superstring. Extention of the reduced model (2.46) is obtained by the Lorentz-covariant $\kappa$-symmetry gauge fixing $\theta_{\alpha}^{1} = \theta_{\alpha}^{2}$. Then the Wess-Zumino term turns to zero and kinetic term (2.49) is brought to the form

$$S_{\text{sstring, stw}}^{D=4,\mathcal{N}=1/2+1/2} = \int_{\Sigma} \left( \frac{-i}{4(\alpha')^{1/2}} \left[ e^{[+2]} \wedge (\bar{\tilde{Z}}_{\mathcal{A}}^{-} d\tilde{Z}^{A-} - d\tilde{Z}_{\mathcal{A}}^{-} \tilde{Z}^{A-}) \right. \right. \\ \left. - e^{-[2]} \wedge (\tilde{\mathcal{W}}_{\mathcal{A}}^{+} d\tilde{\mathcal{W}}^{A+} - d\tilde{\mathcal{W}}_{\mathcal{A}}^{+} \tilde{\mathcal{W}}^{A+}) \right] + \frac{c}{2} e^{-[2]} \wedge e^{[+2]} \right). \quad (2.51)$$

Incidence relations for the components of supertwistors are the same as given in (2.47) and (2.48). Supertwistors also obey zero-norm and orthogonality conditions $\tilde{\mathcal{W}}_{\mathcal{A}}^{+} \tilde{\mathcal{Z}}^{A-} = 0$ and $\tilde{\mathcal{Z}}_{\mathcal{A}}^{+} \tilde{\mathcal{W}}^{A+} = 0$. Actions of the reduced model (2.46), its counterpart for the $s = -1$ case and that of the bosonic string in the twistor formulation (2.12) follow from (2.51) by setting to zero half or all of the anticommuting components of supertwistors. In the next subsection using the Dirac method [15] the model (2.51) will be analyzed as a Hamiltonian system with constraints.

## 2.3 Hamiltonian mechanics of reduced supertwistor model of $N = 2$ superstring

For transition to the Hamiltonian description of the most general reduced model (2.51) it is convenient to introduce in the action (2.51) inverse zweibein with the components $e^{\mu[\pm 2]}$. In the light-cone basis for the metric in the tangent space to the world sheet they satisfy the relations $e^{\mu[\pm 2]} e_{\mu}^{[\pm 2]} = 0$, $e^{\mu[\pm 2]} e_{\mu}^{[\mp 2]} = 2$ (see Ref. [4]). After introduction of the inverse zweibein the action (2.51) acquires the form

$$S_{\text{sstring, stw}}^{D=4,\mathcal{N}=1/2+1/2} = \int_{\Sigma} d^2\xi \left( \frac{ie}{4(\alpha')^{1/2}} \left[ e^{\mu[+2]} (\tilde{\bar{Z}}_{\mathcal{A}}^{-} \partial_{\mu} \tilde{Z}^{A-} - \partial_{\mu} \tilde{\bar{Z}}_{\mathcal{A}}^{-} \tilde{Z}^{A-}) \right. \right. \\ \left. + e^{\mu[-2]} (\tilde{\mathcal{W}}_{\mathcal{A}}^{+} \partial_{\mu} \tilde{\mathcal{W}}^{A+} - \partial_{\mu} \tilde{\mathcal{W}}_{\mathcal{A}}^{+} \tilde{\mathcal{W}}^{A+}) \right] + ce \right).$$
Further to simplify the analysis we use the unnormalized Newman-Penrose dyad that allows to exclude the pair of the second-class constraints (2.2). In this case the last term in (2.51) is substituted by

$$ce(\xi) \rightarrow ce(\xi)n(\xi)\bar{n}(\xi),$$

where $n(\xi) = u^{\alpha-}v_{\alpha}^{+}$ is complex scalar field and $\bar{n}(\xi) = \bar{u}^{\dot{\alpha}-}\bar{v}_{\dot{\alpha}}^{+}$ is its conjugate. The number of physical degrees of freedom is retained upon this change because the action becomes invariant under the local rescaling of the zweibein and dyad components. Finally define dimensionless zweibein density with the components $\rho^{\mu^{[+2]}} = c(\alpha')^{1/2}ee^{\mu^{[+2]}}$, $\rho^{\mu^{-2]} = c(\alpha')^{1/2}ee^{\mu^{-2]}$ (cf. Ref. [4]). So the action of the reduced supertwistor model of $\mathcal{N} = 2$ superstring will acquire the form

$$S'_{\text{sstring, stw}}^{D=4, \mathcal{N}=1/2+1/2} = \int_{\Sigma} d^2\xi \mathcal{L}^{D=4, \mathcal{N}=1/2+1/2}_{\text{sstring, stw}}(\xi) \qquad (2.52)$$

with the Lagrangian<fnref n="1" />

$$\mathcal{L}_{\text{sstring, stw}}^{D=4, \mathcal{N}=1/2+1/2}(\xi) = \frac{i}{4c\alpha'} [\rho^{\mu^{[+2]}}(\bar{Z}_{\mathcal{A}}^{-} \partial_{\mu} Z^{\mathcal{A}-} - \partial_{\mu} \bar{Z}_{\mathcal{A}}^{-} Z^{\mathcal{A}-}) \\ + \rho^{\mu^{-2]}(\bar{W}_{\mathcal{A}}^{+} \partial_{\mu} W^{\mathcal{A}+} - \partial_{\mu} \bar{W}_{\mathcal{A}}^{+} W^{\mathcal{A}+})] + \frac{1}{2c\alpha'} \epsilon_{\mu\nu} \rho^{\mu^{[+2]}} \rho^{\nu^{[+2]}} n\bar{n}. \qquad (2.53)$$

To this Lagrangian one should add with Lagrange multipliers four constraints on the super-twistor components

$$\chi_{\mathcal{Z}}^{[2]} = \bar{Z}_{\mathcal{A}}^{-} Z^{\mathcal{A}-} \approx 0, \quad \chi_{\mathcal{W}}^{[2]} = \bar{W}_{\mathcal{A}}^{-} W^{\mathcal{A}-} \approx 0, \qquad (2.54)$$

$$\chi_{\bar{W}\mathcal{Z}} = \bar{W}_{\mathcal{A}}^{+} Z^{\mathcal{A}-} \approx 0, \quad \chi_{\bar{Z}W} = \bar{Z}_{\mathcal{A}}^{-} W^{\mathcal{A}+} \approx 0. \qquad (2.55)$$

Introduce general notation for the coordinate fields

$$\mathcal{Q}^{\mathcal{N}}(\tau, \sigma) = (\mu^{\beta-}, \bar{\mu}^{\dot{\beta}-}, u_{\beta}^{-}, \bar{u}_{\dot{\beta}}^{-}, \nu^{\beta+}, \bar{\nu}^{\dot{\beta}+}, v_{\dot{\beta}}^{+}, \bar{v}_{\dot{\beta}}^{+}, \eta^{-}, \bar{\eta}^{-}, \zeta^{+}, \bar{\zeta}^{+}, \rho^{\nu^{[\pm 2]}})$$

and conjugate momentum densities

$$\mathcal{P}_{\mathcal{M}}(\tau, \sigma) = \frac{\delta \mathcal{L}}{\delta \partial_{\tau} \mathcal{Q}^{\mathcal{M}}} \\ = \left( p_{(\mu)\alpha}^{+}, \bar{p}_{(\mu)\dot{\alpha}}^{+}, p_{(u)}^{\alpha+}, \bar{p}_{(u)}^{\dot{\alpha}+}, p_{(\nu)\alpha}^{-}, \bar{p}_{(\nu)\dot{\alpha}}^{-}, p_{(v)}^{\alpha-}, \bar{p}_{(v)}^{\dot{\alpha}-}, p_{(\eta)}^{+}, \bar{p}_{(\eta)}^{+}, p_{(\zeta)}^{-}, \bar{p}_{(\zeta)}^{-}, P_{\mu}^{[\mp 2]} \right)$$

that satisfy the Poisson bracket relations

$$\{\mathcal{P}_{\mathcal{M}}(\sigma), \mathcal{Q}^{\mathcal{N}}(\sigma')\}_{\text{P.B.}} = \delta_{\mathcal{M}\mathcal{N}}\delta(\sigma - \sigma').$$

From the definitions of the momenta conjugate to the components of supertwistors $\mathcal{Z}^{-}$ and $\mathcal{W}^{+}$ there follow the primary constraints

$$\begin{align}
T_{(\mu)\alpha}^{+} &= p_{(\mu)\alpha}^{+} - \frac{i}{4c\alpha'} \rho^{\tau^{[+2]}} u_{\alpha}^{-} \approx 0, & \bar{T}_{(\mu)\dot{\alpha}}^{+} &= \bar{p}_{(\mu)\dot{\alpha}}^{+} + \frac{i}{4c\alpha'} \rho^{\tau^{[+2]}} \bar{u}_{\dot{\alpha}}^{-} \approx 0, \nonumber \\
T_{(u)}^{\alpha+} &= p_{(u)}^{\alpha+} + \frac{i}{4c\alpha'} \rho^{\tau^{[+2]}} \mu^{\alpha-} \approx 0, & \bar{T}_{(u)\dot{\alpha}+}^{\alpha+} &= \bar{p}_{(u)\dot{\alpha}+}^{\dot{\alpha}+} - \frac{i}{4c\alpha'} \rho^{\tau^{[+2]}} \bar{\mu}^{\dot{\alpha}-} \approx 0, \tag{2.56} \\
T_{(\eta)}^{+} &= p_{(\eta)}^{+} + \frac{i}{4c\alpha'} \rho^{\tau^{[+2]}} \bar{\eta}^{-} \approx 0, & \bar{T}_{(\eta)\dot{\alpha}+}^{+} &= \bar{p}_{(\eta)\dot{\alpha}+}^{+} + \frac{i}{4c\alpha'} \rho^{\tau^{[+2]}} \bar{\eta}^{-} \approx 0 \nonumber
\end{align}$$

<footnote n="1">¹In what follows to simplify the notation tildes over the supertwistors and their components will be omitted.</footnote>
and

$$
\begin{align}
T_{(\nu)\alpha}^{-} &= p_{(\nu)\alpha}^{-} - \frac{i}{4c\alpha'} \rho^{\tau[-2]} v_{\alpha}^{+} \approx 0, & \bar{T}_{(\nu)\dot{\alpha}}^{-} &= \bar{p}_{(\nu)\dot{\alpha}}^{-} + \frac{i}{4c\alpha'} \rho^{\tau[-2]} \bar{v}_{\dot{\alpha}}^{+} \approx 0, \nonumber \\
T_{(v)}^{\alpha-} &= p_{(v)}^{\alpha-} + \frac{i}{4c\alpha'} \rho^{\tau[-2]} \nu^{\alpha+} \approx 0, & \bar{T}_{(v)}^{\dot{\alpha}-} &= \bar{p}_{(v)}^{\dot{\alpha}-} - \frac{i}{4c\alpha'} \rho^{\tau[-2]} \bar{v}^{\dot{\alpha}+} \approx 0, \tag{2.57} \\
T_{(\zeta)}^{-} &= p_{(\zeta)}^{-} + \frac{i}{4c\alpha'} \rho^{\tau[-2]} \bar{\zeta}^{+} \approx 0, & \bar{T}_{(\zeta)}^{-} &= \bar{p}_{(\zeta)}^{-} + \frac{i}{4c\alpha'} \rho^{\tau[-2]} \zeta^{+} \approx 0. \nonumber
\end{align}
$$

There are also present four primary constraints that arise because of the absence in the Lagrangian (2.53) of derivatives of the zweibein densities

$$ P_{\mu}^{[\mp 2]} \approx 0. \qquad (2.58) $$

The total Hamiltonian density $H_T(\tau, \sigma)$ is by definition given by the sum of the canonical Hamiltonian density

$$
H_0(\tau, \sigma) = -\frac{i}{4c\alpha'} \left[ \begin{aligned}
& \rho^{\sigma^{[+2]}} (\bar{Z}_A^- \partial_\sigma Z^{A^-} - \partial_\sigma \bar{Z}_A^- Z^{A^-} + 2in\bar{n}\rho^{\tau^{[-2]}}) \\
& + \rho^{\sigma^{[-2]}} (\bar{W}_A^+ \partial_\sigma W^{A^+} - \partial_\sigma \bar{W}_A^+ W^{A^+} - 2in\bar{n}\rho^{\tau^{[+2]}})
\end{aligned} \right]
$$

and the primary constraints (2.54)-(2.58) with arbitrary Lagrange multipliers.

According to the Dirac approach [15] primary constraints (2.54)-(2.58) should be conserved in time. This condition implies that their Poisson brackets with the total Hamiltonian have to vanish in the weak sense. As a result we find bosonic secondary constraints

$$
\begin{align}
T_{\bar{Z}\sigma}^{[-2]} &= -\frac{i}{4}(\bar{Z}_A^- \partial_\sigma Z^{A^-} - \partial_\sigma \bar{Z}_A^- Z^{A^-} + 2in\bar{n}\rho^{\tau^{[-2]}}) \approx 0, \tag{2.59} \\
T_{W\sigma}^{[+2]} &= -\frac{i}{4}(\bar{W}_A^+ \partial_\sigma W^{A^+} - \partial_\sigma \bar{W}_A^+ W^{A^+} - 2in\bar{n}\rho^{\tau^{[+2]}}) \approx 0 \nonumber
\end{align}
$$

and

$$
\begin{align}
\omega_{\bar{W}\bar{Z}\sigma} &= \bar{W}_A^+ \partial_\sigma Z^{A^-} - \partial_\sigma \bar{W}_A^+ Z^{A^-} \approx 0, \tag{2.60} \\
\omega_{\bar{Z}W\sigma} &= \bar{Z}_A^- \partial_\sigma W^{A^+} - \partial_\sigma \bar{Z}_A^- W^{A^+} \approx 0 \nonumber
\end{align}
$$

and also equations for the Lagrange multipliers. These equations allow to express Lagrange multipliers for the primary constraints in terms of the smaller set of independent Lagrange multipliers that correspond to the first-class constraints. So the total Hamiltonian density is expressed as the sum of the first-class constraints with arbitrary Lagrange multipliers

$$
\begin{align}
H_T(\tau, \sigma) ={}& \frac{\rho^{\sigma^{[+2]}}}{c\alpha'} \tilde{T}_{\bar{Z}\sigma}^{[-2]} + \frac{\rho^{\sigma^{[-2]}}}{c\alpha'} \tilde{T}_{W\sigma}^{[+2]} + s_Z^{[+2]} \tilde{\chi}_{\bar{Z}}^{[-2]} + s_W^{[-2]} \tilde{\chi}_{W}^{[+2]} \nonumber \\
& + \beta_{\bar{Z}} \Delta_{\bar{Z}} + \beta_W \Delta_W + \beta^{[+2]} P_{\sigma}^{[-2]} + \beta^{[-2]} P_{\sigma}^{[+2]} \approx 0. \tag{2.61}
\end{align}
$$

Note that components $\rho^{\sigma^{[+2]}}$ and $\rho^{\sigma^{[-2]}}$ of the introduced above zweibein density play the role of the Lagrange multipliers for the first-class constraints $\tilde{T}_{\bar{Z}\sigma}^{[-2]} \approx 0$ and $\tilde{T}_{\bar{Z}\sigma}^{[+2]} \approx 0$ respectively. Additionally note that vanishing in the weak sense of the total Hamiltonian density is the characteristic feature of the models invariant under coordinate diffeomorphisms (see, e.g. discussion in [50]). The first-class constraints that enter the expression for the total
Hamiltonian (2.61) have the form

$$
\begin{align*}
\tilde{T}_{\mathcal{Z}\sigma}^{[-2]} &= T_{\mathcal{Z}\sigma}^{[-2]} + c\alpha'\partial_{\sigma}P_{\tau}^{[-2]} - \frac{c\alpha'}{\rho^{\tau[+2]}}((\partial_{\sigma}\mu^{\alpha-} - i\rho^{\tau[-2]}\bar{n}v^{\alpha+})T_{(\mu)\alpha}^{+} \\
&\quad + (\partial_{\sigma}\bar{\mu}^{\dot{\alpha}-} + i\rho^{\tau[-2]}n\bar{v}^{\dot{\alpha}+})\bar{T}_{(\mu)\dot{\alpha}}^{+} + \partial_{\sigma}u_{\alpha}^{-}T_{(u)}^{\alpha+} + \partial_{\sigma}\bar{u}_{\dot{\alpha}}^{-}\bar{T}_{(u)}^{\dot{\alpha}+} + \partial_{\sigma}\eta^{-}T_{(\eta)}^{+} \\
&\quad + \partial_{\sigma}\bar{\eta}^{-}\bar{T}_{(\eta)}^{+}) + ic\alpha'(n\bar{u}^{\dot{\alpha}-}\bar{T}_{(\nu)\dot{\alpha}}^{-} - \bar{n}u^{\alpha-}T_{(\nu)\alpha}^{-}) - \frac{F}{\rho^{\tau[+2]}}(\chi_{\bar{W}\mathcal{Z}} \\
&\quad + \frac{2ic\alpha'}{\rho^{\tau[+2]}}(\bar{\nu}^{\dot{\alpha}+}\bar{T}_{(\mu)\dot{\alpha}}^{+} + v_{\alpha}^{+}T_{(u)}^{\alpha+} + \zeta^{+}T_{(\eta)}^{+}) - \frac{2ic\alpha'}{\rho^{\tau[-2]}}(\mu^{\alpha-}T_{(\nu)\alpha}^{-} + \bar{u}_{\dot{\alpha}}^{-}\bar{T}_{(v)}^{\dot{\alpha}-} \\
&\quad + \bar{\eta}^{-}\bar{T}_{(\zeta)}^{-})) - \frac{F}{\rho^{\tau[+2]}}(\chi_{\bar{Z}W} - \frac{2ic\alpha'}{\rho^{\tau[+2]}}(\nu^{\alpha+}T_{(\mu)\alpha}^{+} + \bar{v}_{\dot{\alpha}}^{+}\bar{T}_{(u)}^{\dot{\alpha}+} + \bar{\zeta}^{+}\bar{T}_{(\eta)}^{+}) \\
&\quad + \frac{2ic\alpha'}{\rho^{\tau[-2]}}(\bar{\mu}^{\dot{\alpha}-}\bar{T}_{(\nu)\dot{\alpha}}^{-} + u_{\alpha}^{-}T_{(v)}^{\alpha-} + \eta^{-}T_{(\zeta)}^{-})) \approx 0,
\end{align*}
$$

$$
\begin{align*}
\tilde{T}_{W\sigma}^{[+2]} &= T_{W\sigma}^{[+2]} + c\alpha' \partial_{\sigma} P_{\tau}^{[+2]} - \frac{c\alpha'}{\rho^{\tau[-2]}} ((\partial_{\sigma}\nu^{\alpha+} - i\rho^{\tau[+2]}\bar{n}u^{\alpha-}) T_{(\nu)\alpha}^{-} \\
&\quad + (\partial_{\sigma}\bar{\nu}^{\dot{\alpha}+} + i\rho^{\tau[+2]}n\bar{u}^{\dot{\alpha}-}) \bar{T}_{(\nu)\dot{\alpha}}^{-} + \partial_{\sigma}v_{\alpha}^{+}T_{(v)}^{\alpha-} + \partial_{\sigma}\bar{v}_{\dot{\alpha}}^{+}\bar{T}_{(v)}^{\dot{\alpha}-} + \partial_{\sigma}\zeta^{+}T_{(\zeta)}^{-} \\
&\quad + \partial_{\sigma}\bar{\zeta}^{+}\bar{T}_{(\zeta)}^{-}) + ic\alpha' (n\bar{v}^{\dot{\alpha}+}\bar{T}_{(\mu)\dot{\alpha}}^{+} - \bar{n}v^{\alpha+}T_{(\mu)\alpha}^{+}) + \frac{F}{\rho^{\tau[-2]}} (\chi_{\bar{W}\mathcal{Z}} \\
&\quad + \frac{2ic\alpha'}{\rho^{\tau[+2]}} (\bar{\nu}^{\dot{\alpha}+}\bar{T}_{(\mu)\dot{\alpha}}^{+} + v_{\alpha}^{+}T_{(u)}^{\alpha+} + \zeta^{+}T_{(\eta)}^{+}) - \frac{2ic\alpha'}{\rho^{\tau[-2]}} (\mu^{\alpha-}T_{(\nu)\alpha}^{-} + \bar{u}_{\dot{\alpha}}^{-}\bar{T}_{(v)}^{\dot{\alpha}-} \\
&\quad + \bar{\eta}^{-}\bar{T}_{(\zeta)}^{-})) + \frac{F}{\rho^{\tau[-2]}} (\chi_{\bar{Z}W} - \frac{2ic\alpha'}{\rho^{\tau[+2]}} (\nu^{\alpha+}T_{(\mu)\alpha}^{+} + \bar{v}_{\dot{\alpha}}^{+}\bar{T}_{(u)}^{\dot{\alpha}+} + \bar{\zeta}^{+}\bar{T}_{(\eta)}^{+}) \\
&\quad + \frac{2ic\alpha'}{\rho^{\tau[-2]}} (\bar{\mu}^{\dot{\alpha}-}\bar{T}_{(\nu)\dot{\alpha}}^{-} + u_{\alpha}^{-}T_{(v)}^{\alpha-} + \eta^{-}T_{(\zeta)}^{-})) \approx 0,
\end{align*}
$$

$$
\tilde{\chi}_{\mathcal{Z}}^{[-2]} = \chi_{\mathcal{Z}}^{[-2]} - \frac{2ic\alpha'}{\rho^{\tau[+2]}} (\mu^{\alpha-}T_{(\mu)\alpha}^{+} - \bar{\mu}^{\dot{\alpha}-}\bar{T}_{(\mu)\dot{\alpha}}^{+} - u_{\alpha}^{-}T_{(u)}^{\alpha+} + \bar{u}_{\dot{\alpha}}^{-}\bar{T}_{(u)}^{\dot{\alpha}+} - \eta^{-}T_{(\eta)}^{+} + \bar{\eta}^{-}\bar{T}_{(\eta)}^{+}) \approx 0,
$$

$$
\tilde{\chi}_{W}^{[+2]} = \chi_{W}^{[+2]} - \frac{2ic\alpha'}{\rho^{\tau[-2]}} (\nu^{\alpha+}T_{(\nu)\alpha}^{-} - \bar{\nu}^{\dot{\alpha}+}\bar{T}_{(\nu)\dot{\alpha}}^{-} - v_{\alpha}^{+}T_{(v)}^{\alpha-} + \bar{v}_{\dot{\alpha}}^{+}\bar{T}_{(v)}^{\dot{\alpha}-} - \zeta^{+}T_{(\zeta)}^{-} + \bar{\zeta}^{+}\bar{T}_{(\zeta)}^{-}) \approx 0,
$$

$$
\Delta_{\mathcal{Z}} = \mu^{\alpha-}T_{(\mu)\alpha}^{+} + \bar{\mu}^{\dot{\alpha}-}\bar{T}_{(\mu)\dot{\alpha}}^{+} + u_{\alpha}^{-}T_{(u)}^{\alpha+} + \bar{u}_{\dot{\alpha}}^{-}\bar{T}_{(u)}^{\dot{\alpha}+} + \eta^{-}T_{(\eta)}^{+} + \bar{\eta}^{-}\bar{T}_{(\eta)}^{+} - 2\rho^{\mu[+2]}P_{\mu}^{[-2]} \approx 0,
$$

$$
\Delta_W = \nu^{\alpha+} T_{\nu\alpha}^{-} + \bar{\nu}^{\dot{\alpha}+} \bar{T}_{(\nu)\dot{\alpha}}^{-} + v_{\alpha}^{+} T_{(v)}^{\alpha-} + \bar{v}_{\dot{\alpha}}^{+} \bar{T}_{(v)}^{\dot{\alpha}-} + \zeta^{+} T_{(\zeta)}^{-} + \bar{\zeta}^{+} \bar{T}_{(\zeta)}^{-} - 2\rho^{\mu[-2]} P_{\mu}^{[+2]} \approx 0,
$$

where

$$
F = \frac{1}{4} \left( \frac{1}{n\bar{n}} \partial_{\sigma} \bar{Z}_{A}^{-} \partial_{\sigma} W^{A+} - \frac{i}{n} \rho^{\tau[+2]} \bar{\omega}_{\sigma}^{[-2]} - \frac{i}{n} \rho^{\tau[-2]} \Omega_{\sigma}^{[+2]} \right).
$$

The first-class constraints of the reduced supertwistor model associated with the $\mathcal{N} = 1$ superstring are obtained from the above given expressions by setting to zero either $\zeta^+$, $\bar{\zeta}^+$ or $\eta^-$, $\bar{\eta}^-$ and their conjugate momenta depending on the value of $s$. To get the first-class constraints of the bosonic string model in the twistor formulation it is necessary to put to zero both $\eta^-$, $\bar{\eta}^-$ and $\zeta^+$, $\bar{\zeta}^+$ and the conjugate momenta.

On the Poisson brackets obtained first-class constraints generate gauge symmetries of the action (2.52). They were examined in our work [2]. Constraints $\tilde{T}_{\mathcal{Z}\sigma}^{[-2]} \approx 0$ and $\tilde{T}_{W\sigma}^{[+2]} \approx 0$ are generators of the world-sheet reparametrizations. Constraints $\Delta_{\mathcal{Z}} \approx 0$ and $\Delta_{W} \approx 0$ generate dilatations of the supertwistor components and zweibein density

$$
\delta_{d_Z} \mathcal{Z}^{A-} = d_Z \mathcal{Z}^{A-}, \quad \delta_{d_Z} \bar{\mathcal{Z}}_{\mathcal{A}}^{-} = d_Z \bar{\mathcal{Z}}_{\mathcal{A}}^{-}, \quad \delta_{d_Z} \rho^{\mu[+2]} = -2 d_Z \rho^{\mu[+2]},
$$

$$
\delta_{d_W} W^{A+} = d_W W^{A+}, \quad \delta_{d_W} \bar{W}_A^+ = d_W \bar{W}_A^+, \quad \delta_{d_W} \rho^{\mu[-2]} = -2 d_W \rho^{\mu[-2]}.
$$

These dilatations are independent due to introduction of the unnormalized dyad. They represent combination of the local Weyl and $SO(1, 1)$ transformations in the tangent space
to the world sheet as is seen upon transition to the parameters $d_Z = \zeta - \varrho$, $d_W = \zeta + \varrho$. Constraints $\tilde{\chi}_Z^{[-2]} \approx 0$ and $\tilde{\chi}_W^{[+2]} \approx 0$ generate local $U(1)$ rotations of the supertwistor components

$$ \delta_{\varphi_Z} \mathcal{Z}^{A-} = i \varphi_Z \mathcal{Z}^{A-}, \quad \delta_{\varphi_Z} \bar{\mathcal{Z}}_A^- = -i \varphi_Z \bar{\mathcal{Z}}_A^-, \\ \delta_{\varphi_W} \mathcal{W}^{A+} = i \varphi_W \mathcal{W}^{A+}, \quad \delta_{\varphi_W} \bar{\mathcal{W}}_A^+ = -i \varphi_W \bar{\mathcal{W}}_A^+. $$

These symmetries are also independent for unnormalized dyad. Their parameters $\varphi_Z = v - \phi$ and $\varphi_W = v + \phi$ can be expressed through the novel ones $v$ and $\phi$ that correspond to the $U(1)$ subgroup of the $GL(1, \mathbb{C}) = GL(2, \mathbb{C})/SL(2, \mathbb{C})$ group acting on the components of unnormalized dyad and the $SO(2) = U(1) \subset SL(2, \mathbb{C})$ rotation group of the orthogonal to the string world sheet components of the local frame.

Other constraints (2.55), (2.56), (2.57) and (2.60) are of the second-class. Upon transition to the Dirac brackets they need to be considered as strong equalities i.e. as the usual equalities. On the Dirac brackets that take into account primary constraints (2.56) and (2.57) supertwistor components satisfy the relations

$$ \{\mathcal{Z}^{A-}(\sigma), \bar{\mathcal{Z}}_B^-(\sigma')\}_{D.B.} = \frac{2ic\alpha'}{\rho^{\tau[+2]}} \delta_B^A \delta(\sigma - \sigma'), \\ \{\mathcal{W}^{A+}(\sigma), \bar{\mathcal{W}}_B^+(\sigma')\}_{D.B.} = \frac{2ic\alpha'}{\rho^{\tau[-2]}} \delta_B^A \delta(\sigma - \sigma'). $$

These relations are stringy generalization of the Dirac brackets arising in the superparticle models in the supertwistor formulation [11] (see also [51]). Presence of the $\tau$-components of the zweibein density ($\rho^{\mu[+2]}$, $\rho^{\mu[-2]}$) in the second-class constraints (2.56) and (2.57) leads to the nonzero Dirac brackets for the conjugate momenta $P_{\tau}^{[\pm 2]}$ and supertwistors

$$ \{\mathcal{Z}^{A-}(\sigma), P_{\tau}^{[-2]}(\sigma')\}_{D.B.} = \frac{1}{2\rho^{\tau[+2]}} \mathcal{Z}^{A-} \delta(\sigma - \sigma'), \\ \{\mathcal{W}^{A+}(\sigma), P_{\tau}^{[+2]}(\sigma')\}_{D.B.} = \frac{1}{2\rho^{\tau[-2]}} \mathcal{W}^{A+} \delta(\sigma - \sigma') $$

and c.c. relations. Treating the constraints (2.56) and (2.57) as strong equalities results in simplification of the first-class constraints. They reduce to the primary constraints (2.54), (2.58) and

$$ \tilde{T}_{\bar{Z}\sigma}^{[-2]} = T_{\bar{Z}\sigma}^{[-2]} - \frac{F}{\rho^{\tau[+2]}} \chi_{\bar{W}\bar{Z}} - \frac{\bar{F}}{\rho^{\tau[+2]}} \chi_{\bar{Z}W} \approx 0, \\ \tilde{T}_{W\sigma}^{[+2]} = T_{W\sigma}^{[+2]} + \frac{F}{\rho^{\tau[-2]}} \chi_{\bar{W}\bar{Z}} + \frac{\bar{F}}{\rho^{\tau[-2]}} \chi_{\bar{Z}W} \approx 0. $$

The Dirac bracket relations of these first-class constraints between themselves and with the second-class constraints (2.55) and (2.60) are given in [2].

## 3 Twistor formulations of superstrings in $D = 6$ and $D = 10$ Minkowski superspaces

In the preceding Section Lagrangians of the $D = 4$ bosonic string and $\mathcal{N} = 1, 2$ superstrings in the Lorentz-harmonic formulation [3], [4] were presented in terms of the Penrose-Ferber (super)twistors. As is known $\kappa$-invariant action functionals of the classical Green-Schwarz
superstrings exist also in $D = 6$ and $D = 10$ dimensions. In $D = 10$ dimensions due to cancelation of anomalies the Green-Schwarz superstring theories are consistent quantum theories. So in this Section there will be considered supertwistor formulations of the $D = 6$ and $D = 10$ $N = 1$ superstrings that were proposed in our work [16].

Construction of such formulations necessitates generalization of the Ferber supertwistors to dimensions $D > 4$ that is not unique as previous studies indicate [52], [26], [53], [54], [29], [55], [56]. To define supertwistors in higher dimensions impose the same two conditions that were used to construct supertwistor formulations of $D = 4$ superstrings. Firstly, they should carry fundamental representation of respective higher-dimensional superconformal algebra and, secondly, include the spinor Lorentz harmonics.

It should be noted, however, that superconformal algebras, non-compact bosonic subalgebras of which are isomorphic to conformal algebras $so(2, D)$, exist only in dimensions $D \le 6$ (see, e.g. review [57] and references therein). The point is that generators of such non-compact bosonic subalgebras are given by matrices with spinor indices. In general spinor representation has complex dimension $2^{[D/2]}$ and for sufficiently high values of the spacetime dimension it becomes greater than $D$ even after imposition of the Majorana and/or Weyl conditions. That is why in $D = 10$ dimensions minimal supersymmetric extension of the $so(2, 10)$ conformal algebra includes extra bosonic generators and is isomorphic to the $osp(32|1)$ superalgebra [58]. This generalized superconformal algebra includes generators of the $sp(32)$ bosonic symplectic subalgebra and 32 supersymmetry generators that decompose on 16 generators of the Poincaré supersymmetry and 16 special conformal supersymmetry generators given by the $D = 10$ Majorana-Weyl spinors of different chiralities. 528 generators of the $sp(32)$ algebra include generators of the $D = 10$ conformal algebra together with the generators $A_{\hat{m}[4]}$, $Z_{\hat{m}[5]}$ and $Z'_{\hat{m}[5]}$ that are antisymmetric $so(1, 9)$ tensors [58].<fnref n="2" /> Note that according to the conjecture of Ref. [59] the contraction of the $osp(32|1)$ superalgebra is associated with the symmetry underlying the M-theory.

In this Section considered in detail will be the case of the $D = 10$ superstring with the action invariant under minimal $N = 1$ Poincaré supersymmetry. It represents part of the heterotic string action without taking into account degrees of freedom that realize generators of the gauge symmetry in the Yang-Mills theory.<fnref n="3" /> Associated Lagrangian will be expressed in terms of the $OSp(32|1)$ supertwistors [38]. In order to match the number of the physical degrees of freedom in the supertwistor formulation with those in the original Green-Schwarz and Lorentz-harmonic formulations it is necessary to impose the constraints on supertwistors the form of which was identified in our work [16]. They manifestly break (generalized) superconformal symmetry down to $D = 10$ $N = 1$ Poincaré supersymmetry that is the global symmetry of the Green-Schwarz superstrings. These constraints remove from the incidence relations the contributions of the extra coordinates so that remain only

<footnote n="2">²According to adopted notation a number in square brackets following an index denotes the set of anti-symmetrized indices equal to that number.</footnote>

<footnote n="3">³Recall that the Yang-Mills supermultiplet enters the spectrum of the massless excitations of the heterotic string.</footnote>
the Minkowski space coordinates that represent the parameters for momentum generators.

Subsections 3.1 and 3.2 are devoted to the twistor formulation of the $D = 6$ superstring. After defining respective supertwistors obtained will be the formulation of the superstring Lagrangian, given are the equations of motion and $\kappa$-symmetry transformations in terms of their components. In analogy with the $D = 4$ superstring case also introduced will be reduced supertwistor model for the $D = 6 N = 1$ superstring with quadratic Lagrangian. Generalization of the above formulations to the case of the $D = 10$ superstring will be considered in subsections 3.3 and 3.4.

## 3.1 $D = 6$ supertwistors and Lorentz harmonics

As is known $OSp(8^*|2)$ orthosymplectic supergroup is the minimal superconformal group in $D = 1+5$ dimensions [60]. Note that the structure of its $osp(8^*|2)$ superalgebra differs from that of the $osp(32|1)$ superalgebra. There symplectic subalgebra is non-compact, whereas the $osp(8^*|2)$ superalgebra includes unitary-symplectic subalgebra $usp(2) = su(2)$ that is compact and non-compact orthogonal subalgebra $so(8^*) = so(2,6)$ isomorphic to the conformal algebra and also 16 supersymmetry generators. It belongs to the different family of real forms of complex $osp(M|N, \mathbb{C})$ ($M = 2m$) superalgebras (see the review [57]).

Define that $D = 6$ supertwistor transforms under the $OSp(8^*|2)$ fundamental representation

$$ \mathcal{Z}^{\Lambda a} = (\mu^{\alpha a}, v_{\alpha}^{a}, \eta^{i a}). \qquad (3.1) $$

We will use for the components of $D = 6$ supertwistors the same nomenclature as for the Penrose twistors. So $\mu^{\alpha a}$ is the primary spinor part of supertwistor and $v_{\alpha}^a$ its projection part. They transform as $D = 6 SU(2)$-symplectic Majorana-Weyl spinors of different chiralities

$$ (\mu^{\alpha a})^* = C^{\dot{\alpha} \beta} \varepsilon_{ab} \mu^{\beta b}, \quad (v_{\alpha}^a)^* = C^{-1T \dot{\alpha}} \varepsilon_{ab} v_{\beta}^b. \qquad (3.2) $$

Together these spinors make up $SU(2)$-symplectic Majorana-Weyl spinor of $Spin(2,6)$. Note that $D = 6$ charge conjugation matrix $C^{\dot{\alpha} \beta}$ and its inverse $C^{-1\dot{\beta}}$ satisfy the relation $(C^{\dot{\alpha} \beta})^* = -C^{-1\dot{\alpha} \beta}$. Roman-font indices from the beginning of the Latin alphabet denote fundamental representation indices of the $SU(2)$ group under which unit antisymmetric tensors $\varepsilon_{ab}$ and $\varepsilon^{ab}$: $\varepsilon_{ab}\varepsilon^{bc} = \delta_a^c$ are invariant. Projection part of supertwistor $v_{\alpha}^a$ will be identified with one of the $4 \times 2$ rectangular blocks of the matrix of $D = 6$ spinor Lorentz harmonics. Then mentioned $SU(2)$ group represents one of the factors of the $Spin(4) = SU(2) \times SU(2)$ group. The latter is the spin covering of the $SO(4)$ group of rotations of the local frame vectors in the orthogonal space to the string world-sheet. Anticommuting components of supertwistor $\eta^{ia}$ also carry index $i$ of the fundamental representation of another $SU(2)$ group. It is the $R$-symmetry subgroup of $OSp(8^*|2)$ supergroup under which transform Grassmann coordinates of the $D = 6 N = 1$ superspace. Respective invariant metric tensor $\varepsilon_{ij}$ is used in the definition of the $SU(2)$-symplectic Majorana-Weyl condition (3.8) that these coordinates obey.
$D = 6$ spinor Lorentz harmonics [9], [10] satisfy the reality

$$ (v_{\underline{\alpha}}^{(\underline{\alpha})})^* = -C^{(\underline{\alpha})}_{(\underline{\beta})} v_{\underline{\beta}}^{(\underline{\beta})} C^{-1}_{\underline{\beta} \underline{\alpha}} \qquad (3.3) $$

and unimodularity

$$ \det v_{\underline{\alpha}}^{(\underline{\alpha})} = 1 \qquad (3.4) $$

conditions that reduce the number of their independent components to dimension of the Spin(1, 5) group equal 15. The Lorentz harmonics used for description of (super)strings in the orthonormal frame method take value in the coset $SO(1, D-1)/(SO(1, 1) \times SO(D-2))$. Accordingly 4 × 4 matrix of $D = 6$ spinor Lorentz harmonics decomposes on the 4 × 2 rectangular blocks

$$ v_{\underline{\alpha}}^{(\underline{\alpha})} = (v_{\underline{\alpha}}^{+a}, v_{\underline{\alpha}}^{-\dot{a}}) \in \mathrm{Spin}(1, 5)/(SO(1, 1) \times SU(2) \times SU(2)), \qquad (3.5) $$

where italic Latin letters correspond to the indices of the fundamental representations of two SU(2) factors in denominator. Each of these blocks is considered as projection part of OSp(8*|2) supertwistor (3.1). So the Lagrangian of the $D = 6$ superstrings in the super-twistor formulation will contain two OSp(8*|2) supertwistors

$$ \mathcal{Z}^{\Lambda+a} = (\mu^{\alpha+a}, v_{\underline{\alpha}}^{+a}, \eta^{i+a}), \quad \mathcal{Z}^{\Lambda-\dot{a}} = (\mu^{\alpha-\dot{a}}, v_{\underline{\alpha}}^{-\dot{a}}, \eta^{i-\dot{a}}). \qquad (3.6) $$

On the primary spinor parts of supertwistors impose the incidence relations

$$ \mu^{\alpha a} = v_{\underline{\beta}}^{a}(x^{\underline{\beta}\alpha} - 2i\theta_{i}^{\underline{\beta}}\theta^{\alpha i}), \quad \eta^{ia} = 2v_{\underline{\alpha}}^{a}\theta^{\alpha i}, \qquad (3.7) $$

where $D = 6$ Minkowski-space coordinates are realized as antisymmetric 4 × 4 matrices $x^{\underline{\alpha\beta}} = x^{\underline{m}}\tilde{\gamma}_{\underline{m}}^{\underline{\alpha\beta}}$ and $\theta^{\underline{\alpha i}}$ are the $\mathcal{N} = 1$ superspace Grassmann coordinates. Analogously to bosonic components of supertwistors they obey the SU(2)-symplectic Majorana-Weyl condition

$$ (\theta^{\underline{\alpha i}})^* = -C^{\underline{\alpha}}_{\underline{\beta}} \varepsilon_{ij} \theta^{\underline{\beta j}}. \qquad (3.8) $$

So the spinor $\theta^{\underline{\alpha i}}$ has eight real independent components. As a consequence from (3.8) and (3.2) it follows that Grassmann-odd components of supertwistors obey the condition $(\eta^{ia})^* = -\varepsilon_{ab}\varepsilon_{ij}\eta^{bj}$ and thus have four real independent components. In order for the incidence relations (3.7) to hold on the supertwistors (3.6) there have to be imposed 10 constraints

$$ \mathcal{Z}^{\Lambda+a} G_{\Lambda\Sigma} \mathcal{Z}^{\Sigma+b} = \mathcal{Z}^{\Lambda-\dot{a}} G_{\Lambda\Sigma} \mathcal{Z}^{\Sigma-\dot{b}} = \mathcal{Z}^{\Lambda+a} G_{\Lambda\Sigma} \mathcal{Z}^{\Sigma-\dot{a}} = 0. \qquad (3.9) $$

They eliminate contribution of the antisymmetric tensor coordinates $y^{lmn}$ that in the spinor form are given by symmetric 4 × 4 matrix $y^{\underline{\alpha\beta}} = y^{lmn}\tilde{\gamma}_{lmn}^{\underline{\alpha\beta}}$. In (3.9) and below OSp(8*|2)-invariant scalar product of supertwistors is taken using the graded symmetric metric

$$ G_{\Lambda\Sigma} = \begin{pmatrix} 0 & \delta_{\underline{\alpha}}^{\underline{\beta}} & 0 \\ \delta_{\underline{\beta}}^{\underline{\alpha}} & 0 & 0 \\ 0 & 0 & -i\varepsilon_{ij} \end{pmatrix}. \qquad (3.10) $$
Let us show that after imposition of the constraints (3.9) supertwistors (3.6) have the same number of independent components as there are $D = 6$ $\mathcal{N} = 1$ superspace coordinates. As was shown above each supertwistor has four real Grassmann-odd components so their sum equals number of real Grassmann coordinates of the superspace. Also each supertwistor has 16 real bosonic components so that pair of supertwistors constrained by (3.9) and (3.4) has 21 components. Their number equals the sum of the number of $D = 6$ space-time coordinates and independent components of the Lorentz-harmonics.

## 3.2 Supertwistor formulation of $D = 6$ $\mathcal{N} = 1$ superstring

Consider first the action of the $D = 6$ $\mathcal{N} = 1$ superstring in the Lorentz-harmonic formulation

$$S_{\text{sstring}}^{D=6,\mathcal{N}=1} = S_{\text{kin}}^{D=6,\mathcal{N}=1} + S_{\text{WZ}}^{D=6,\mathcal{N}=1}. \qquad (3.11)$$

Its kinetic term

$$S_{\text{kin}}^{D=6,\mathcal{N}=1} = \frac{1}{2(\alpha')^{1/2}} \int_{\Sigma} (e^{[+2]} n_{\underline{m}}^{[-2]} - e^{[-2]} n_{\underline{m}}^{[+2]}) \wedge \omega^{\underline{m}}(d) + \frac{c}{2} \int_{\Sigma} e^{[-2]} \wedge e^{[+2]}$$

contains two tangent to the world sheet light-like vectors $n_{\underline{m}}^{[\pm 2]}(\xi)$ from the local orthonormal frame $n_{\underline{m}}^{(n)}(\xi) = (n_{\underline{m}}^{[+2]}, n_{\underline{m}}^{[-2]}, n_{\underline{m}}^{(i)})$ and one-form $\omega^{\underline{m}}(d) = dx^{\underline{m}} - i d\theta_{i}^{\underline{\alpha}} \gamma_{\underline{m}\alpha\beta} \theta^{\underline{\beta}i}$ invariant under transformations of the $D = 6$ $\mathcal{N} = 1$ Poincaré supersymmetry. Wess-Zumino term in (3.11) is given by the expression

$$S_{\text{WZ}}^{D=6,\mathcal{N}=1} = \frac{is}{c\alpha'} \int_{\Sigma} \omega^{\underline{m}}(d) \wedge d\theta_{i}^{\underline{\alpha}} \gamma_{\underline{m}\alpha\beta} \theta^{\underline{\beta}i}.$$

Vectors of the orthonormal frame have the following realization in terms of the introduced spinor Lorentz-harmonics (3.5)

$$n_{\underline{m}}^{[+2]} = \frac{1}{2} v_{\underline{\alpha}}^{+a} \tilde{\gamma}_{\underline{m}}^{\alpha\beta} v_{\underline{\beta}}^{+b} \varepsilon_{ab}, \quad n_{\underline{m}}^{[-2]} = -\frac{1}{2} v_{\underline{\alpha}}^{-\dot{a}} \tilde{\gamma}_{\underline{m}}^{\alpha\beta} v_{\underline{\beta}}^{-b} \varepsilon_{\dot{a}\dot{b}}, \quad n_{\underline{m}}^{(i)} = -\frac{1}{2} v_{\underline{\alpha}}^{+a} \tilde{\gamma}_{\underline{m}}^{\alpha\beta} v_{\underline{\beta}}^{-b} \sigma_{\dot{a}\dot{b}}^{(i)},$$

where $n_{\underline{m}}^{(i)}$ ((i) = 1, ..., 4) are frame vectors orthogonal to the world sheet and matrices $\sigma_{\dot{a}\dot{b}}^{(i)}$ are $SO(4)$-invariant.

In the twistor formulation projections of the supersymmetric one-form $\omega^{\underline{m}}(d)$ onto the tangent to the world-sheet frame components $n_{\underline{m}}^{[\pm 2]}$ are expressed in terms of the introduced supertwistors (3.6)

$$\begin{align}
\omega^{[+2]}(d) &= n_{\underline{m}}^{[+2]} \omega^{\underline{m}}(d) = \frac{1}{2} \varepsilon_{ab} d\mathcal{Z}^{\Lambda+a} G_{\Lambda\Sigma} \mathcal{Z}^{\Sigma+b} \nonumber \\
&= \frac{1}{2} \varepsilon_{ab} (d\mu^{\alpha+a} v_{\underline{\alpha}}^{+b} + d\nu_{\underline{\alpha}}^{+a} \mu^{\alpha+b} - i \varepsilon_{ij} d\eta^{i+a} \eta^{j+b}), \tag{3.12} \\
\omega^{[-2]}(d) &= n_{\underline{m}}^{[-2]} \omega^{\underline{m}}(d) = -\frac{1}{2} \varepsilon_{\dot{a}\dot{b}} d\mathcal{Z}^{\Lambda-\dot{a}} G_{\Lambda\Sigma} \mathcal{Z}^{\Sigma-\dot{b}} \nonumber \\
&= -\frac{1}{2} \varepsilon_{\dot{a}\dot{b}} (d\mu^{\alpha-\dot{a}} v_{\underline{\alpha}}^{-b} + d\nu_{\underline{\alpha}}^{-\dot{a}} \mu^{\alpha-\dot{b}} - i \varepsilon_{ij} d\eta^{i-\dot{a}} \eta^{j-\dot{b}}). \nonumber
\end{align}$$

Analogously one can present projections of $\omega^{\underline{m}}(d)$ onto the orthogonal to the world-sheet components of the local frame

$$\omega^{(i)}(d) = n_{\underline{m}}^{(i)} \omega^{\underline{m}}(d) = \frac{1}{4} (\mathcal{Z}^{\Lambda+a} G_{\Lambda\Sigma} d\mathcal{Z}^{\Sigma-\dot{a}} - d\mathcal{Z}^{\Lambda+a} G_{\Lambda\Sigma} \mathcal{Z}^{\Sigma-\dot{a}}) \sigma_{\dot{a}\dot{b}}^{(i)}. \qquad (3.13)$$
So that the $D = 6$ $\mathcal{N} = 1$ superstring action (3.11) in twistor formulation acquires the form

$$ S_{\text{string, stw}}^{D=6, \mathcal{N}=1} = \frac{1}{2(\alpha')^{1/2}} \int_{\Sigma} (e^{[+2]} \wedge \omega^{[-2]}(d) - e^{[-2]} \wedge \omega^{[+2]}(d)) + \frac{c}{2} \int_{\Sigma} e^{[-2]} \wedge e^{[+2]} \\ + \frac{is}{c\alpha'} \int_{\Sigma} \left( \left( \frac{1}{2}\omega^{[-2]}(d) \wedge \varphi^{[+2]}(d) + \frac{1}{2}\omega^{[+2]}(d) \wedge \varphi^{[-2]}(d) - \omega^{(i)}(d) \wedge \varphi^{(i)}(d) \right) \right). \quad (3.14) $$

The Wess-Zumino term also includes one-forms quadratic in the Grassmann-odd supertwistor components

$$ \varphi^{[+2]}(d) = -\frac{1}{2}\varepsilon_{ab}\mathcal{D}\eta_i^{+a}\eta^{i+b}, \quad \varphi^{[-2]}(d) = \frac{1}{2}\varepsilon_{\dot{a}\dot{b}}\mathcal{D}\eta_i^{-\dot{a}}\eta^{i-\dot{b}}, \\ \varphi^{(i)}(d) = \frac{1}{4}(\mathcal{D}\eta_i^{+a}\eta^{i-\dot{a}} - \mathcal{D}\eta_i^{-\dot{a}}\eta^{i+a})\sigma_{\dot{a}\dot{a}}^{(i)}, $$

that break $OSp(8^*|2)$ superconformal symmetry down to the $D = 6$ $\mathcal{N} = 1$ super-Poincaré one. Such breaking is explained by the presence of the dimensionful tension parameter in superstring action. Extended differentials of the Grassmann-odd supertwistor components are defined as

$$ \begin{aligned} \mathcal{D}\eta_i^{+a} &= 2v_{\alpha}^{+a} d\theta_i^{\alpha} \\ &= d\eta_i^{+a} + \frac{1}{4}\Omega^{[+2] [-2]}(d)\eta_i^{+a} - \frac{1}{2}\Omega^{[+2](i)}(d)\tilde{\sigma}^{(i)\dot{a}a}\eta_{i\dot{a}}^{-} - \frac{1}{2}\Omega^{(i)(j)}(d)\sigma^{(i)(j)}{}_{b}{}^{a}\eta_i^{+b}, \end{aligned} \quad (3.15) \\ \mathcal{D}\eta_i^{-\dot{a}} &= 2v_{\alpha}^{-\dot{a}} d\theta_i^{\dot{a}} \\ &= d\eta_i^{-\dot{a}} - \frac{1}{4}\Omega^{[+2] [-2]}(d)\eta_i^{-\dot{a}} - \frac{1}{2}\Omega^{[-2](i)}(d)\tilde{\sigma}^{(i)\dot{a}a}\eta_{i\dot{a}}^{+} + \frac{1}{2}\Omega^{(i)(j)}(d)\tilde{\sigma}^{(i)(j)\dot{a}}{}_{b}\eta_i^{-\dot{b}}, \end{aligned} $$

where $\tilde{\sigma}^{(i)(j)\dot{a}}{}_{b}{}^{\dot{a}} = \frac{1}{4}(\tilde{\sigma}^{(i)\dot{a}a}\sigma_{a\dot{b}}^{(j)} - \tilde{\sigma}^{(j)\dot{a}a}\sigma_{a\dot{b}}^{(i)})$, $\sigma^{(i)(j)}{}_{b}{}^{a} = \frac{1}{4}(\sigma_{b\dot{a}}^{(i)}\tilde{\sigma}^{(j)\dot{a}a} - \sigma_{b\dot{a}}^{(j)}\tilde{\sigma}^{(i)\dot{a}a})$. They include decomposed on the $so(1, 1) \oplus so(4)$-covariant constituents $(\Omega^{[+2] [-2]}(d), \Omega^{[\pm 2](i)}(d), \Omega^{(i)(j)}(d)$ left-invariant Cartan forms that generalize spin coefficients (2.14) and (2.15). In the method of admissible variations these one-form with differentials replaced by variations are independent variation parameters for the Lorentz harmonics. Obtained supertwistor formulation of the $D = 6$ $\mathcal{N} = 1$ superstring (3.14) belongs to results submitted to defence.

Let us derive twistor form of the superstring equations. Due to the $SO(1, 1) \times SO(4)$ gauge symmetry of its action in the Lorentz-harmonic (3.11) and supertwistor formulations (3.14) equations corresponding to variation parameters $\Omega^{[+2] [-2]}(\delta)$ and $\Omega^{(i)(j)}(\delta)$ are satisfied identically. Equations associated with the parameters $\Omega^{[\pm 2](i)}(\delta)$ related to the $SO(1, 5)/(SO(1, 1) \times SO(4))$ Cartan forms and variations of the zweibein components can be brought to the form

$$ \omega^{[\pm 2]}(d) = c(\alpha')^{1/2} e^{[\pm 2]}, \quad \omega^{(i)}(d) = 0, \quad (3.16) $$

where one-forms on the l.h.s. are defined in (3.12) and (3.13). Eqs. (3.16) represent supertwistor form of the equations that fix orientation of the local frame with respect to the superstring world sheet. As other independent bosonic variation parameters let us choose those related to one-forms (3.12) and (3.13). Respective equations read

$$ \begin{aligned} de^{[+2]} + \frac{1}{2}e^{[+2]} \wedge \Omega^{[+2] [-2]}(d) + \frac{is}{2c(\alpha')^{1/2}} \mathcal{D}\eta_i^{+a} \wedge \mathcal{D}\eta_a^{i+} &= 0, \\ de^{[-2]} - \frac{1}{2}e^{[-2]} \wedge \Omega^{[+2] [-2]}(d) + \frac{is}{2c(\alpha')^{1/2}} \mathcal{D}\eta_i^{-\dot{a}} \wedge \mathcal{D}\eta_{\dot{a}}^{i-} &= 0, \\ e^{[+2]} \wedge \Omega^{[-2](i)}(d) - e^{[-2]} \wedge \Omega^{[+2](i)}(d) - \frac{is}{c(\alpha')^{1/2}} \mathcal{D}\eta_i^{+a} \wedge \mathcal{D}\eta^{i-\dot{a}} \sigma_{a\dot{a}}^{(i)} &= 0. \end{aligned} $$
As the consequence of the reparametrization invariance of the superstring action the first two of them are satisfied identically when other equations are taken into account. Fermionic equations of the superstring that correspond to variation parameters $\mathcal{D}(\delta)\eta_i^{+a}$, $\mathcal{D}(\delta)\eta_i^{-\dot{a}}$ equal

$$ (1+s)e^{-[2]} \wedge \mathcal{D}\eta_a^{i+} = (1-s)e^{[2]} \wedge \mathcal{D}\eta_a^{i-} = 0. \qquad (3.17) $$

Similarly to fermionic equations of the $D=4$ $N=1$ superstring (cf. (2.45)) for any value of the parameter $s = \pm 1$ one of the above equations turns into identity that according to the second Noether theorem is the consequence of the $\kappa$-invariance of the action (3.14). So also in the case of the $D=6$ $N=1$ superstring after transition from the superspace to the supertwistor formulation there remain pure gauge fermionic degrees of freedom in the Lagrangian.

Explicit form of the $\kappa$-variations of the components of $OSp(8^*|2)$ supertwistors and zweibein depends on the value of the parameter $s$. When $s=1$, we have

$$ \begin{aligned} \delta_{\kappa} \mathcal{Z}^{\Lambda+a} &= \frac{1}{2} \Omega^{[+2](i)}(\delta_{\kappa}) \tilde{\sigma}^{(i)\dot{a}a} \mathcal{Z}_{\dot{a}}^{\Lambda-}, \\ \delta_{\kappa} \mathcal{Z}^{\Lambda-\dot{a}} &= \frac{1}{2} \Omega^{-[2](i)}(\delta_{\kappa}) \tilde{\sigma}^{(i)\dot{a}a} \mathcal{Z}_{\dot{a}}^{\Lambda+} - (K^{\Sigma-\dot{a}} \mathcal{Z}_{\Sigma}^{-\dot{b}}) V_{\dot{b}}^{\Lambda+} - (K^{\Sigma-\dot{a}} \mathcal{Z}_{\Sigma}^{+\dot{b}}) V_{\dot{b}}^{\Lambda-} + K^{\Lambda-\dot{a}}, \\ \delta_{\kappa} e^{[+2]} &= 0, \quad \delta_{\kappa} e^{-[2]} = \frac{1}{c(\alpha')^{1/2}} K^{\Lambda-\dot{a}} \mathcal{D} \mathcal{Z}_{\Lambda\dot{a}}^{-}, \end{aligned} $$

where

$$ \Omega^{\pm 2i}(\delta_{\kappa}) = \pm \frac{1}{c(\alpha')^{1/2}} K^{\Lambda-\dot{b}} e^{\nu[\pm 2]} \mathcal{D}_{\nu} \mathcal{Z}_{\Lambda}^{+a} \sigma_{ab}^{(i)}, $$

and local parameter $\kappa^{i-\dot{a}}(\xi)$ was presented as the Grassmann-odd component of the super-twistor $K^{\Lambda-\dot{a}} = (0, 0, \kappa^{i-\dot{a}})$. For $s=-1$ $\kappa$-variations equal

$$ \begin{aligned} \delta_{\kappa} \mathcal{Z}^{\Lambda+a} &= \frac{1}{2} \Omega^{[+2](i)}(\delta_{\kappa}) \tilde{\sigma}^{i\dot{a}a} \mathcal{Z}_{\dot{a}}^{\Lambda-} - (K^{\Sigma+a} \mathcal{Z}_{\Sigma}^{+\dot{b}}) V_{\dot{b}}^{\Lambda-} - (K^{\Sigma+a} \mathcal{Z}_{\Sigma}^{-\dot{b}}) V_{\dot{b}}^{\Lambda+} + K^{\Lambda+a}, \\ \delta_{\kappa} \mathcal{Z}^{\Lambda-\dot{a}} &= \frac{1}{2} \Omega^{-[2](i)}(\delta_{\kappa}) \tilde{\sigma}^{(i)\dot{a}a} \mathcal{Z}_{\dot{a}}^{\Lambda+}, \\ \delta_{\kappa} e^{[+2]} &= \frac{1}{c(\alpha')^{1/2}} K_{a}^{\Lambda+} \mathcal{D} \mathcal{Z}_{\Lambda}^{+a}, \quad \delta_{\kappa} e^{-[2]} = 0, \end{aligned} $$

where

$$ \Omega^{[\pm 2](i)}(\delta_{\kappa}) = \pm \frac{1}{c(\alpha')^{1/2}} K^{\Lambda+a} e^{\nu[\pm 2]} \mathcal{D}_{\nu} \mathcal{Z}_{\Lambda}^{-\dot{b}} \sigma_{ab}^{(i)} $$

and $K^{\Lambda+a} = (0, 0, \kappa^{i+a})$.

Like in the case of the $D=4$ $N=1$ superstring one can introduce reduced supertwistor model corresponding to the $D=6$ $N=1$ superstring. Its action can be obtained by substituting Eq. (3.16) into the Wess-Zumino Lagrangian. When $s=1$ action of the reduced model takes the form

$$ S_{\text{sstring, stw}, s=1}^{D=6, N=1/2} = \frac{1}{4(\alpha')^{1/2}} \int_{\Sigma} \left( e^{[+2]} \wedge dz_{\dot{a}}^{\dot{\alpha}-} z_{\dot{a}}^{-\dot{\alpha}+} + e^{-[2]} \wedge d\tilde{Z}_{a}^{\Lambda+} \tilde{Z}_{\Lambda}^{+a} \right) \\ + \frac{c}{2} \int_{\Sigma} e^{-[2]} \wedge e^{[+2]}. \qquad (3.18) $$
Accordingly when $s = -1$ it equals

$$ S_{\text{sstring, stw, s=-1}}^{D=6, \mathcal{N}=1/2} = \frac{1}{4(\alpha')^{1/2}} \int_{\Sigma} \left( e^{[+2]} \wedge d\tilde{Z}_{\hat{a}}^{\Lambda-} \tilde{Z}_{\Lambda}^{-\dot{a}} + e^{[-2]} \wedge dz_{\hat{a}}^{\hat{\alpha}+} z_{\hat{\alpha}}^{+\alpha} \right) \\ + \frac{c}{2} \int_{\Sigma} e^{[-2]} \wedge e^{[+2]}. \qquad (3.19) $$

In the action functionals (3.18) and (3.19) $z^{\hat{\alpha}+\alpha}$ and $z^{\hat{\alpha}-\dot{a}}$ are bosonic $D=6$ twistors that obey $SU(2)$-symplectic Majorana-Weyl condition for the Spin(2, 6) spinors

$$ (z^{\hat{\alpha}+\alpha})^* = B^{\hat{\alpha}}_{\hat{\beta}} \varepsilon_{ab} z^{\hat{\beta}+b}, \quad (z^{\hat{\alpha}-\dot{a}})^* = B^{\hat{\alpha}}_{\hat{\beta}} \varepsilon_{\dot{a}b} z^{\hat{\beta}-b}. \qquad (3.20) $$

$\tilde{Z}_{\Lambda}^{+\alpha}$ and $\tilde{Z}_{\Lambda}^{-\dot{a}}$ are redefined supertwistors, whose incidence relations with the superspace coordinates differ from (3.7) and have the form

$$ \tilde{Z}^{\Lambda+a} = (\tilde{\mu}^{\alpha+a}, v_{\underline{\alpha}}^{+\alpha}, \tilde{\eta}^{i+a}): $$

$$ \tilde{\mu}^{\underline{\alpha}+a} = v_{\underline{\beta}}^{+a} (x^{\underline{\beta\alpha}} - 4i \theta_{\underline{i}}^{\underline{\beta}} \theta^{\underline{\alpha i}}), \quad \tilde{\eta}^{i+a} = 2\sqrt{2} v_{\underline{\alpha}}^{+a} \theta^{\underline{\alpha i}}, \qquad (3.21) $$

$$ \tilde{Z}^{\Lambda-\dot{a}} = (\tilde{\mu}^{\underline{\alpha}-\dot{a}}, v_{\underline{\alpha}}^{-\dot{a}}, \tilde{\eta}^{i-\dot{a}}): $$

$$ \tilde{\mu}^{\underline{\alpha}-\dot{a}} = v_{\underline{\beta}}^{-\dot{a}} (x^{\underline{\beta\alpha}} - 4i \theta_{\underline{i}}^{\underline{\beta}} \theta^{\underline{\alpha i}}), \quad \tilde{\eta}^{i-\dot{a}} = 2\sqrt{2} v_{\underline{\alpha}}^{-\dot{a}} \theta^{\underline{\alpha i}}. $$

One can also define reduced supertwistor model corresponding to the $D=6 \mathcal{N} = (2, 0)$ superstring. Its action

$$ S_{\text{sstring, stw}}^{D=6, \mathcal{N}=(1/2+1/2, 0)} = \frac{1}{4(\alpha')^{1/2}} \int_{\Sigma} \left( e^{[+2]} \wedge d\tilde{Z}_{\hat{a}}^{\Lambda-} \tilde{Z}_{\Lambda}^{-\dot{a}} + e^{[-2]} \wedge d\tilde{Z}_{\hat{a}}^{\Lambda+} \tilde{Z}_{\Lambda}^{+\alpha} \right) \\ + \frac{c}{2} \int_{\Sigma} e^{[-2]} \wedge e^{[+2]} \qquad (3.22) $$

can be obtained by fixing the $\kappa$-symmetry gauge by the conditions $\eta^{1i+a} = \eta^{2i+a}$ and $\eta^{1i-\dot{a}} = \eta^{2i-\dot{a}}$ in supertwistor formulation of the $D=6 \mathcal{N} = (2, 0)$ superstring action. This superstring model is formulated in the $D=6$ superspace with two Grassmann coordinates $\theta^{\alpha i1}$ and $\theta^{\alpha i2}$ that are $SU(2)$-symplectic Majorana-Weyl Spin(1, 5) spinors of the same chirality. Its action in the supertwistor formulation is the generalization of the action (3.14). Action of the reduced model (3.22) includes $\mathcal{N}=1$ supertwistors defined in (3.21). It should be noted that (super)twistors, which enter action functionals (3.18), (3.19) and (3.22), satisfy algebraic constraints analogous to (3.9). They represent mixture of the first-class constraints generating $SU(2) \times SU(2)$ gauge symmetry and the second-class ones.

### 3.3 $D=10$ supertwistors and Lorentz harmonics

Consideration of the supertwistor formulation of $D=6 \mathcal{N}=1$ superstring in the preceding subsection served as a preparatory step to the study of supertwistor formulation of the $D=10 \mathcal{N}=1$ superstring to which this subsection is devoted. As discussed in the introduction to this Section $D=10$ supertwistor

$$ Z^{\Lambda} = (\mu^{\hat{\alpha}}, v_{\hat{\alpha}}, \eta) \qquad (3.23) $$
is required to realize fundamental representation of the $OSp(32|1)$ as it is the minimal supergroup containing conformal group in $D = 10$ dimensions. The supertwistor is composed of the pair of the Majorana-Weyl Spin(1,9) spinors of opposite chiralities and the Grassmann-odd scalar. Its projection part is required to be given by the matrix of spinor Lorentz-harmonics so we are led to consider 16-plet of $OSp(32|1)$ supertwistors

$$Z^{\Lambda}(\hat{\alpha}) = (\mu^{\hat{\alpha}}(\hat{\alpha}), v_{\hat{\alpha}}^{(\hat{\alpha})}, \eta^{(\hat{\alpha})}). \qquad (3.24)$$

As is known [3] adapted to description of superstrings 16 × 16 matrix of the $D = 10$ spinor Lorentz-harmonics $v_{\hat{\alpha}}^{(\hat{\alpha})} = (v_{\hat{\alpha}A}^{+}, v_{\hat{\alpha}\hat{A}}^{-})$ takes value in the Spin(1,9)/(SO(1,1) × SO(8)) coset. It is decomposed on two 16 × 8 blocks in accordance with the decomposition of the Majorana-Weyl spinor representation of the right symmetry group Spin(1,9)$_R$ on the $\mathbf{8}_c$ and $\mathbf{8}_s$ Majorana-Weyl spinor representations of the covering of SO(8) group of the local frame rotations in the orthogonal space to the world sheet. Accordingly introduced 16-plet of supertwistors decomposes on two octets of supertwistors

$$Z_A^{\Lambda+} = (\mu_A^{\hat{\alpha}+}, v_{\hat{\alpha}A}^{+}, \eta_A^{+}), \quad Z_A^{\Lambda-} = (\mu_A^{\hat{\alpha}-}, v_{\hat{\alpha}\hat{A}}^{-}, \eta_A^{-}). \qquad (3.25)$$

Further requirement to primary spinor parts and Grassmann-odd components of super-twistors is the incidence to the $D = 10, \mathcal{N} = 1$ Minkowski superspace coordinates $(x^{\hat{m}}, \theta^{\hat{\alpha}})$

$$\mu^{(\hat{\alpha})} = (x^{\hat{\alpha}\hat{\beta}} - 8i\theta^{\hat{\alpha}}\theta^{\hat{\beta}})v_{\hat{\beta}}^{(\hat{\alpha})}, \quad \eta^{(\hat{\alpha})} = 4v_{\hat{\alpha}}^{(\hat{\alpha})}\theta^{\hat{\alpha}}, \qquad (3.26)$$

where $x^{\hat{\alpha}\hat{\beta}} = x^{\hat{m}}\tilde{\sigma}_{\hat{m}}^{\hat{\alpha}\hat{\beta}}$ ($\hat{m} = 0, 1, ..., 9$) and symmetric 16 × 16 matrices $\tilde{\sigma}_{\hat{m}}^{\hat{\alpha}\hat{\beta}}$, $\sigma_{\hat{m}\hat{\alpha}\hat{\beta}}$ are analogues of the relativistic Pauli matrices in $D = 10$ dimensions. For the incidence relations (3.26) to hold supertwistor components have to be constrained. The $OSp(32|1)$-invariant constraints

$$N_{AB}^{+2} = Z_A^{\Lambda+} G_{\Lambda\Sigma} Z_B^{\Sigma+} = 0, \quad N_{\hat{A}\hat{B}}^{-2} = Z_{\hat{A}}^{\Lambda-} G_{\Lambda\Sigma} Z_{\hat{B}}^{\Sigma-} = 0, \\ N_{A\hat{A}} = Z_A^{\Lambda+} G_{\Lambda\Sigma} Z_{\hat{A}}^{\Sigma-} = 0, \qquad (3.27)$$

where

$$G_{\Lambda\Sigma} = \begin{pmatrix} 0 & \delta_{\hat{\alpha}}^{\hat{\beta}} & 0 \\ -\delta_{\hat{\beta}}^{\hat{\alpha}} & 0 & 0 \\ 0 & 0 & -i \end{pmatrix}$$

is the orthosymplectic metric, exclude contribution of 120 coordinates described by antisymmetric tensor $y^{\hat{m}\hat{n}\hat{k}}$. In the spinor form these coordinates are presented by antisymmetric matrix $y^{\hat{\alpha}\hat{\beta}} = y^{\hat{m}\hat{n}\hat{k}}\tilde{\sigma}_{\hat{m}\hat{n}\hat{k}}^{\hat{\alpha}\hat{\beta}}$. Incidence relations (3.26) in general also include contribution of 126 coordinates described by (anti-)self-dual antisymmetric tensor $z^{\hat{m}[5]}$. These coordinates are associated with tensor generators $Z_{\hat{m}[5]}$ of the $osp(32|1)$ superalgebra. In spinor form these coordinates are described by symmetric $\sigma$-traceless matrix $z^{\hat{\alpha}\hat{\beta}} = z^{\hat{m}[5]}\tilde{\sigma}_{\hat{m}[5]}^{\hat{\alpha}\hat{\beta}}$. To remove their contribution to the incidence relations on the supertwistor components let us impose additional constraints

$$N_{\hat{m}_1 \cdots \hat{m}_5} = (\mu_A^{\hat{\gamma}+} v_A^{\hat{\delta}-} + \mu_A^{\hat{\gamma}-} v_A^{\hat{\delta}+}) \sigma_{\hat{m}_1 \cdots \hat{m}_5} \gamma_{\hat{\delta}} = 0. \qquad (3.28)$$
These constraints break $OSp(32|1)$ symmetry down to $D = 10 \mathcal{N} = 1$ super-Poincaré symmetry that is the symmetry of tensile superstrings in $D = 10$ dimensions and contain $16 \times 8$ blocks of the matrix of inverse spinor Lorentz harmonics

$$v_{(\hat{\alpha})}^{\hat{\alpha}} = (v_A^{\hat{\alpha}-}, v_{\hat{A}}^{\hat{\alpha}+}) : \quad v_{(\hat{\alpha})}^{\hat{\alpha}} v_{\hat{\alpha}}^{(\hat{\beta})} = \delta_{(\hat{\alpha})}^{(\hat{\beta})}. \qquad (3.29)$$

These relations can be considered as defining inverse spinor Lorentz harmonics. In that case they have to be added to the set of harmonicity conditions in order not to introduce extra degrees of freedom (see discussion in [4]).

## 3.4 Supertwistor formulation of $D = 10 \mathcal{N} = 1$ superstring

Consider now the representation for $D = 10 \mathcal{N} = 1$ superstring action in terms of the $OSp(32|1)$ supertwistors (3.25). As in the course of discussion of the supertwistor formulations of lower-dimensional superstrings let us start from recalling Lorentz-harmonic formulation of the $D = 10 \mathcal{N} = 1$ superstring action [3], [4]

$$S_{\text{sstring}}^{D=10, \mathcal{N}=1} = S_{\text{kin}}^{D=10, \mathcal{N}=1} + S_{\text{WZ}}^{D=10, \mathcal{N}=1},$$

where kinetic

$$S_{\text{kin}}^{D=10, \mathcal{N}=1} = \frac{1}{2(\alpha')^{1/2}} \int_{\Sigma} \left( e^{[+2]} n_{\hat{m}}^{[-2]} - e^{-[2]} n_{\hat{m}}^{[+2]} \right) \wedge \omega^{\hat{m}}(d) + \frac{c}{2} \int_{\Sigma} e^{-[2]} \wedge e^{[+2]} \qquad (3.30)$$

and Wess-Zumino

$$S_{\text{WZ}}^{D=10, \mathcal{N}=1} = \frac{is}{c\alpha'} \int_{\Sigma} \omega^{\hat{m}}(d) \wedge d\theta^{\hat{\alpha}} \sigma_{\hat{m}\hat{\alpha}\hat{\beta}} \theta^{\hat{\beta}}$$

terms include one-forms $\omega^{\hat{m}}(d) = dx^{\hat{m}} - id\theta^{\hat{\alpha}}\sigma_{\hat{m}\hat{\beta}}\theta^{\hat{\beta}}$ invariant under $D = 10 \mathcal{N} = 1$ Poincaré supersymmetry. Kinetic term (3.30) includes also two tangent to the world sheet light-like vectors $n_{\hat{m}}^{[\pm 2]}(\xi)$ from the local $D = 10$ orthonormal frame $n_{\hat{m}}^{(\hat{\beta})}(\xi) = (n_{\hat{m}}^{[+2]}, n_{\hat{m}}^{[-2]}, n_{\hat{m}}^{(\mathbf{I})})$. These vectors have the following representation in terms of the spinor harmonics

$$n_{\hat{m}}^{[+2]} = \frac{1}{8} v_{\hat{\alpha}A}^{+} \tilde{\sigma}_{\hat{m}}^{\hat{\alpha}\hat{\beta}} v_{\hat{\beta}A}^{+}, \quad n_{\hat{m}}^{[-2]} = \frac{1}{8} v_{\hat{\alpha}\hat{A}}^{-} \tilde{\sigma}_{\hat{m}}^{\hat{\alpha}\hat{\beta}} v_{\hat{\beta}\hat{A}}^{-}, \quad n_{\hat{m}}^{(\mathbf{I})} = \frac{1}{8} v_{\hat{\alpha}A}^{+} \tilde{\sigma}_{\hat{m}}^{\hat{\alpha}\hat{\beta}} v_{\hat{\beta}\hat{A}}^{-} \gamma_{\mathbf{A}\hat{A}}^{(\mathbf{I})} \qquad (3.31)$$

where $(\mathbf{I}) = 1, ..., 8$ is the $SO(8)$ vector index and $\gamma_{\mathbf{A}\hat{A}}^{(\mathbf{I})}$ are $8 \times 8$ chiral $\gamma$-matrices in $D = 8$ dimensions.

Generalizing the discussion of subsection (3.2) to $D = 10$ dimensions, projections of the one-form $\omega^{\hat{m}}(d)$ on the local frame vectors can be expressed in terms of the $OSp(32|1)$ supertwistors (3.25) using the representation (3.31)

$$\begin{aligned} \omega^{[+2]}(d) &= \omega^{\hat{m}}(d) n_{\hat{m}}^{[+2]} = \frac{1}{8} dZ_A^{\Lambda+} G_{\Lambda\Sigma} Z_A^{\Sigma+}, \\ \omega^{-[2]}(d) &= \omega^{\hat{m}}(d) n_{\hat{m}}^{-[2]} = \frac{1}{8} dZ_A^{\Lambda-} G_{\Lambda\Sigma} Z_A^{\Sigma-} \end{aligned} \qquad (3.32)$$
and

$$ \omega^{(\mathbf{I})}(d) = \omega^{\hat{m}}(d) n_{\hat{m}}^{(\mathbf{I})} = \frac{1}{16} \gamma_{AA}^{(\mathbf{I})} (dZ_{A}^{\mathbf{\Lambda}+} G_{\mathbf{\Lambda}\mathbf{\Sigma}} Z_{\hat{A}}^{\mathbf{\Sigma}-} + dZ_{\hat{A}}^{\mathbf{\Lambda}-} G_{\mathbf{\Lambda}\mathbf{\Sigma}} Z_{A}^{\mathbf{\Sigma}+}). \quad (3.33) $$

As a result we obtain the supertwistor formulation of the $D = 10$ $\mathcal{N} = 1$ superstring action

$$ S_{\text{sstring, stw}}^{D=10, \mathcal{N}=1} = \frac{1}{2(\alpha')^{1/2}} \int_{\Sigma} (e^{[+2]} \wedge \omega^{-[2]}(d) - e^{-[2]} \wedge \omega^{[+2]}(d)) \\ + \frac{c}{2} \int_{\Sigma} e^{-[2]} \wedge e^{[+2]} \\ + \frac{is}{c\alpha'} \int_{\Sigma} \left( \frac{1}{2}\omega^{-2}(d) \wedge \varphi^{+2}(d) + \frac{1}{2}\omega^{+2}(d) \wedge \varphi^{-2}(d) - \omega^{\mathbf{I}}(d) \wedge \varphi^{\mathbf{I}}(d) \right). \quad (3.34) $$

To the last line in (3.34) there contribute the one-forms quadratic in the Grassmann-odd supertwistor components

$$ \varphi^{[+2]}(d) = \frac{1}{8} \mathcal{D}\eta_A^+ \eta_A^+, \quad \varphi^{-[2]}(d) = \frac{1}{8} \mathcal{D}\eta_A^- \eta_A^-, \\ \varphi^{(\mathbf{I})}(d) = \frac{1}{16} \gamma_{AA}^{(\mathbf{I})} (\mathcal{D}\eta_A^+ \eta_A^- + \mathcal{D}\eta_A^- \eta_A^+). \quad (3.35) $$

They are invariant only under the $D = 10$ $\mathcal{N} = 1$ super-Poincaré subgroup of the $\text{OSp}(32|1)$ generalized superconformal symmetry. Extended differentials of odd supertwistor components $\eta_A^+$ and $\eta_A^-$

$$ \begin{aligned} \mathcal{D}\eta_A^+ &= d\eta_A^+ + \frac{1}{4}\Omega^{[+2] [-2]}(d)\eta_A^+ - \frac{1}{2}\Omega^{[+2] (\mathbf{I})}(d)\gamma_{AA}^{\mathbf{I}}\eta_A^- - \frac{1}{4}\Omega^{(\mathbf{I}) (\mathbf{J})}(d)\gamma_{AB}^{(\mathbf{I}) (\mathbf{J})}\eta_B^+, \\ \mathcal{D}\eta_A^- &= d\eta_A^- - \frac{1}{4}\Omega^{[+2] [-2]}(d)\eta_A^- - \frac{1}{2}\Omega^{[-2] (\mathbf{I})}(d)\tilde{\gamma}_{AA}^{(\mathbf{I})}\eta_A^+ - \frac{1}{4}\Omega^{(\mathbf{I}) (\mathbf{J})}(d)\tilde{\gamma}_{AB}^{(\mathbf{I}) (\mathbf{J})}\eta_B^- \end{aligned} \quad (3.36) $$

include antisymmetric matrices $\gamma_{AB}^{(\mathbf{I}) (\mathbf{J})} = \frac{1}{2}(\gamma_{AA}^{(\mathbf{I})} \tilde{\gamma}_{AB}^{(\mathbf{J})} - \gamma_{AA}^{(\mathbf{J})} \tilde{\gamma}_{AB}^{(\mathbf{I})})$ and $\tilde{\gamma}_{AB}^{(\mathbf{I}) (\mathbf{J})} = \frac{1}{2}(\tilde{\gamma}_{AA}^{(\mathbf{I})} \gamma_{AB}^{(\mathbf{J})} - \tilde{\gamma}_{AA}^{(\mathbf{J})} \gamma_{AB}^{(\mathbf{I})})$ are the spin(8) algebra generators in the $\mathbf{s}$ and $\mathbf{c}$ representations and the derivation coefficients of the $D = 10$ orthonormal frame equal to the left-invariant Cartan forms in the $\text{so}(1, 1) \oplus \text{so}(8)$ basis. These Cartan forms as well as one-forms (3.32), (3.33) and (3.36), in which differentials are replaced by variations, will be taken as independent parameters in admissible variations of the supertwistors (3.25). The form of variations of the supertwistors compatible with the constraints (3.27) and (3.28) was found in our work [16]. They are applied to derive superstring equations of motion.

The set of independent bosonic equations of the $D = 10$ $\mathcal{N} = 1$ superstring in the super-twistor formulation includes equations that fix orientation of the local frame with respect to the world sheet

$$ \omega^{[\pm 2]}(d) = c(\alpha')^{1/2} e^{\pm 2],} \quad \omega^{(\mathbf{I})}(d) = 0. \quad (3.37) $$

They are analogs of the equations (2.43) and (3.16) for superstrings in lower dimensions. Other independent bosonic equations can be written in the form

$$ e^{[-2]} \wedge \Omega^{[+2](\mathbf{I})}(d) - e^{[+2]} \wedge \Omega^{[-2](\mathbf{I})}(d) - \frac{is}{4c(\alpha')^{1/2}} \mathcal{D}\eta_A^+ \gamma_{AA}^{(\mathbf{I})} \wedge \mathcal{D}\eta_A^- = 0. \quad (3.38) $$

Note that equations associated with the parameters $\hat{\Omega}^{+2-2}(\delta)$ and $\hat{\Omega}^{\mathbf{I}\mathbf{J}}(\delta)$ are satisfied identically as a consequence of the $\text{SO}(1, 1) \times \text{SO}(8)$ gauge invariance of the action (3.34).
Analogously equations associated with the parameters $\omega^{\pm 2}(\delta)$ turn into identities because of the reparametrization symmetry of the action. Using Eqs. (3.37) and (3.38) fermionic equations of the superstring are brought to the following form

$$ (1 + s)e^{[-2]} \wedge \mathcal{D}\eta_A^+ = 0, \quad (1 - s)e^{[+2]} \wedge \mathcal{D}\eta_A^- = 0. \qquad (3.39) $$

Such form of these equations is analogous to that for the $D = 4$ and $D = 6$ superstrings (cf. Eqs. (2.45) and (3.17)) and reflects action invariance under the irreducible $\kappa$-symmetry transformations.

Explicit form of the $\kappa$-variations of the components of supertwistors and zweibein in the case of $s = 1$ is

$$ \begin{aligned} \delta_{\kappa} Z_{A}^{\mathbf{\Lambda}^{+}} &= \frac{1}{2} \Omega^{[+2](\mathbf{I})} (\delta_{\kappa}) \gamma_{A\dot{A}}^{(\mathbf{I})} Z_{\dot{A}}^{\mathbf{\Lambda}^{-}}, \\ \delta_{\kappa} Z_{\dot{A}}^{\mathbf{\Lambda}^{-}} &= \frac{1}{2} \Omega^{[-2](\mathbf{I})} (\delta_{\kappa}) \tilde{\gamma}_{\dot{A}A}^{(\mathbf{I})} Z_{A}^{\mathbf{\Lambda}^{+}} - (K_{\dot{A}}^{\mathbf{\Sigma}^{-}} Z_{\mathbf{\Sigma}\dot{B}}^{-}) V_{\dot{B}}^{\mathbf{\Lambda}^{+}} - (K_{\dot{A}}^{\mathbf{\Sigma}^{-}} Z_{\mathbf{\Sigma}B}^{+}) V_{B}^{\mathbf{\Lambda}^{-}} + K_{\dot{A}}^{\mathbf{\Lambda}^{-}}, \\ \delta_{\kappa} e^{[+2]} &= 0, \quad \delta_{\kappa} e^{[-2]} = \frac{1}{4c(\alpha')^{1/2}} \mathcal{D} Z_{\dot{A}}^{\mathbf{\Lambda}^{-}} K_{\dot{A}\dot{A}}^{-} \end{aligned} $$

where

$$ \Omega^{[\pm 2](\mathbf{I})}(\delta_{\kappa}) = \pm \frac{1}{8c(\alpha')^{1/2}} e^{\mu[\pm 2]} \mathcal{D}_{\mu} Z_{A}^{\mathbf{\Lambda}^{+}} K_{\mathbf{\Lambda}\dot{A}}^{-} \gamma_{A\dot{A}}^{(\mathbf{I})} $$

and $K_{\dot{A}}^{\mathbf{\Lambda}^{-}} = (0, 0, \kappa_{\dot{A}}^{-})$ is the supertwistor form of the gauge parameter $\kappa_{\dot{A}}^{-}(\xi)$. Accordingly, when $s = -1$, variations take the form

$$ \begin{aligned} \delta_{\kappa} Z_{A}^{\mathbf{\Lambda}^{+}} &= \frac{1}{2} \Omega^{[+2](\mathbf{I})} (\delta_{\kappa}) \gamma_{A\dot{A}}^{(\mathbf{I})} Z_{\dot{A}}^{\mathbf{\Lambda}^{-}} - (K_{A}^{\mathbf{\Sigma}^{+}} Z_{\mathbf{\Sigma}B}^{+}) V_{B}^{\mathbf{\Lambda}^{-}} - (K_{A}^{\mathbf{\Sigma}^{+}} Z_{\mathbf{\Sigma}\dot{B}}^{-}) V_{\dot{B}}^{\mathbf{\Lambda}^{+}} + K_{A}^{\mathbf{\Lambda}^{+}}, \\ \delta_{\kappa} Z_{\dot{A}}^{\mathbf{\Lambda}^{-}} &= \frac{1}{2} \Omega^{[-2](\mathbf{I})} (\delta_{\kappa}) \tilde{\gamma}_{\dot{A}A}^{(\mathbf{I})} Z_{A}^{\mathbf{\Lambda}^{+}}, \\ \delta_{\kappa} e^{[+2]} &= \frac{1}{4c(\alpha')^{1/2}} \mathcal{D} Z_{A}^{\mathbf{\Lambda}^{+}} K_{\mathbf{\Lambda}A}^{+}, \quad \delta_{\kappa} e^{[-2]} = 0, \end{aligned} $$

where

$$ \Omega^{[\pm 2](\mathbf{I})}(\delta_{\kappa}) = \pm \frac{1}{8c(\alpha')^{1/2}} e^{\mu[\pm 2]} K_{A}^{\mathbf{\Lambda}^{+}} \mathcal{D}_{\mu} Z_{\mathbf{\Lambda}\dot{A}}^{-} \gamma_{A\dot{A}}^{(\mathbf{I})} $$

with $K_{A}^{\mathbf{\Lambda}^{+}} = (0, 0, \kappa_{A}^{+})$.

Quite analogously to the cases of superstrings in $D = 4$ and $D = 6$ dimensions let us introduce reduced supertwistor model with the Lagrangian quadratic in supertwistors that corresponds to the $D = 10, \mathcal{N} = 1$ superstring. Let us define its action by the integral

$$ S_{\text{sstring, stw}, s=1}^{D=10, \mathcal{N}=1} = \frac{1}{16(\alpha')^{1/2}} \int_{\Sigma} \left( e^{[+2]} \wedge dz_{\dot{A}}^{\hat{\alpha}^{-}} z_{\dot{\alpha}A}^{-} - e^{[-2]} \wedge d\tilde{Z}_{A}^{\mathbf{\Lambda}^{+}} \tilde{Z}_{\mathbf{\Lambda}A}^{+} \right) \\ + \frac{c}{2} \int_{\Sigma} e^{[-2]} \wedge e^{[+2]} \qquad (3.40) $$

or

$$ S_{\text{sstring, stw}, s=-1}^{D=10, \mathcal{N}=1} = \frac{1}{16(\alpha')^{1/2}} \int_{\Sigma} \left( e^{[+2]} \wedge d\tilde{Z}_{\dot{A}}^{\mathbf{\Lambda}^{-}} \tilde{Z}_{\mathbf{\Lambda}\dot{A}}^{-} - e^{[-2]} \wedge dz_{A}^{\hat{\alpha}^{+}} z_{\dot{\alpha}A}^{+} \right) \\ + \frac{c}{2} \int_{\Sigma} e^{[-2]} \wedge e^{[+2]}. \qquad (3.41) $$
Above action functionals can be obtained by substituting equations (3.37) into the action (3.34). Action (3.40) corresponds to the $s = 1$ value of the sign parameter in (3.34), whereas action (3.41) to the $s = -1$ value. Incidence relations for the components of supertwistors that enter (3.40) and (3.41) have the form

$$
\begin{align}
\tilde{Z}_A^{\Lambda^+} &= (\tilde{\mu}_A^{\hat{\alpha}^+}, v_{\hat{\alpha}A}^{+}, \tilde{\eta}_A^{+}) : \nonumber \\
\tilde{\mu}_A^{\hat{\alpha}^+} &= (x^{\hat{\alpha}\hat{\beta}} - 16i\theta^{\hat{\alpha}}\theta^{\hat{\beta}})v_{\hat{\beta}A}^{+}, \quad \tilde{\eta}_A^{+} = 4\sqrt{2}v_{\hat{\alpha}A}^{+}\theta^{\hat{\alpha}}, \tag{3.42} \\
\tilde{Z}_A^{\Lambda^-} &= (\tilde{\mu}_A^{\hat{\alpha}^-}, v_{\hat{\alpha}\hat{A}}^{-}, \tilde{\eta}_A^{-}) : \nonumber \\
\tilde{\mu}_A^{\hat{\alpha}^-} &= (x^{\hat{\alpha}\hat{\beta}} - 16i\theta^{\hat{\alpha}}\theta^{\hat{\beta}})v_{\hat{\beta}\hat{A}}^{-}, \quad \tilde{\eta}_A^{-} = 4\sqrt{2}v_{\hat{\alpha}\hat{A}}^{-}\theta^{\hat{\alpha}}. \nonumber
\end{align}
$$

Action functionals (3.40) and (3.41) also include 32-component Sp(32) twistors $z_A^{\hat{\alpha}^+}$, $z_A^{\hat{\alpha}^-}$. Note that on these (super)twistors there have to be imposed constraints analogous to (3.27) and (3.28) in order to exclude from the incidence relations the coordinates corresponding to tensor generators of the sp(32) algebra.

Along the lines of the above consideration it is possible to get the supertwistor representation for the Type II superstrings in the Lorentz-harmonic formulation [3], [4] and define respective reduced supertwistor model. For instance, action of the reduced supertwistor model for the Type IIB superstring is obtained from its action by fixing the $\kappa$-symmetry gauge by the conditions $\eta_A^{1+} = \eta_A^{2+}$, $\eta_A^{1-} = \eta_A^{2-}$

$$
S_{\text{sstring, stw, gf}}^{D=10, \text{IIB}} = \frac{1}{16(\alpha)^{1/2}} \int_{\Sigma} \left( e^{[+2]} \wedge d\tilde{Z}_{\tilde{A}}^{\Lambda^{-}} G_{\Lambda\Sigma} \tilde{Z}_{\tilde{A}}^{\Sigma^{-}} - e^{[-2]} \wedge d\tilde{Z}_{\tilde{A}}^{\Lambda^{+}} G_{\Lambda\Sigma} \tilde{Z}_{\tilde{A}}^{\Sigma^{+}} \right) \\
+ \frac{c}{2} \int_{\Sigma} e^{[-2]} \wedge e^{[+2]}. \tag{3.43}
$$

Note that it includes both octets of the $\mathcal{N} = 1$ supertwistors (3.42).

Proposed reduced supertwistor models (3.40), (3.41) and (3.43) are submitted to defence.

## 4 Hamiltonian description of $D = 10$ $\mathcal{N} = 1$ superstring in supertwistor formulation

This Section is devoted to study of the $D = 10$ $\mathcal{N} = 1$ superstring in the supertwistor formulation (3.34) as a constrained Hamiltonian system. In analogy to consideration in subsection 2.3 of the Hamiltonian mechanics of $D = 4$ reduced supertwistor model we will use world-sheet vector density in place of the zweibein components, on which the superstring Lagrangian depends non-linearly. Then we come to the superstring action

$$ S_{\text{sstring, stw}}^{D=10, \mathcal{N}=1} = \int_{\Sigma} d^2 \xi \mathcal{L}_{\text{sstring, stw}}^{D=10, \mathcal{N}=1}(\xi) \qquad (4.1) $$

with the Lagrangian density

$$
\mathcal{L}_{\text{sstring, stw}}^{D=10, \mathcal{N}=1}(\xi) = -\frac{1}{2c\alpha'} \left( \rho^{\mu[+2]} \omega_{\mu}^{[-2]} + \rho^{\mu[-2]} \omega_{\mu}^{[+2]} + \epsilon_{\mu\nu} \rho^{\mu[-2]} \rho^{\nu[+2]} \right) \\
+ \frac{is}{c\alpha'} \epsilon^{\mu\nu} \left( \frac{1}{2} \omega_{\mu}^{[+2]} \varphi_{\nu}^{[-2]} + \frac{1}{2} \omega_{\mu}^{[-2]} \varphi_{\nu}^{[+2]} - \omega_{\mu}^{(\mathbf{I})} \varphi_{\nu}^{(\mathbf{I})} \right). \qquad (4.2)
$$
In subsection 4.1 we identify the constraints that follow from the definition of momenta for the dynamical variables in the Lagrangian (4.2) and classify them on the first- and second-class ones. Then in subsection 4.2 there will be described the basis for the second-class constraints, in which the Dirac matrix acquires the form of a sum of the block-diagonal supermatrix proportional to $(\alpha')^{-1}$ and one linear in the constraints, defined the Dirac brackets and evaluated the deformation of the algebra of the first-class constraints to leading order in $\alpha'$.

## 4.1 Total Hamiltonian and the first-class constraints

Let us define momentum densities

$$P_{\mathfrak{M}}(\tau, \sigma) = \frac{\vec{\delta}S'^{D=10,N=1}_{\text{sstring, stw}}}{\delta\partial_{\tau}Q^{\mathfrak{M}}(\tau, \sigma)} = \left\{ p^{-}_{(\mu)\hat{\alpha}A}, p^{+}_{(\mu)\hat{\alpha}\dot{A}}, p^{\hat{\alpha}-}_{(v)A}, p^{\hat{\alpha}+}_{(v)\dot{A}}, \pi^{-}_{A}, \pi^{+}_{\dot{A}}, p^{+}_{(v)\hat{\alpha}A}, p^{-}_{(v)\hat{\alpha}\dot{A}}, P^{\pm 2}_{\mu} \right\}$$

conjugate to the supertwistor components and world-sheet vector density

$$Q^{\mathfrak{M}}(\tau, \sigma) = \left\{ \mu^{\hat{\alpha}+}_{A}, \mu^{\hat{\alpha}-}_{\dot{A}}, v^{+}_{\hat{\alpha}A}, v^{-}_{\dot{\alpha}\dot{A}}, \eta^{+}_{A}, \eta^{-}_{\dot{A}}, v^{\hat{\alpha}-}_{A}, v^{\hat{\alpha}+}_{\dot{A}}, \rho^{\mu\pm 2} \right\}$$

on the Poisson brackets

$$\{P_{\mathfrak{M}}(\sigma), Q^{\mathfrak{M}}(\sigma')\} = \delta_{\mathfrak{M}}^{\mathfrak{M}}\delta(\sigma - \sigma').$$

Note that the matrix of inverse spinor harmonics (3.29) was added to the set of coordinates and to the set of momenta $p^{+}_{(v)\hat{\alpha}A}$ and $p^{-}_{(v)\dot{\alpha}\dot{A}}$. Then relations defining the matrix of inverse spinor harmonics

$$v_{\hat{\alpha}}^{(\hat{\alpha})} v_{(\hat{\beta})}^{\hat{\beta}} - \delta_{(\hat{\beta})}^{(\hat{\alpha})} \approx 0 \qquad (4.3)$$

should be added to the set of the primary constraints of the superstring.

Definition of the momenta densities leads to the primary constraints. Among them there are the constraints that follow from the definitions of momentum densities conjugate to the main spinor parts of supertwistors (3.25)

$$\begin{aligned} \Phi_{\hat{\alpha}A}^{-}(\sigma) &= p_{(\mu)\hat{\alpha}A}^{-} + \frac{1}{16\alpha'} \left( \rho^{\tau[-2]} - i s \varphi_{\sigma}^{[-2]} \right) v_{\hat{\alpha}A}^{+} + \frac{is}{16\alpha'} \varphi_{\sigma}^{(\mathbf{I})} \gamma_{AA}^{(\mathbf{I})} v_{\hat{\alpha}\dot{A}}^{-} \approx 0, \\ \Phi_{\hat{\alpha}\dot{A}}^{+}(\sigma) &= p_{(\mu)\hat{\alpha}\dot{A}}^{+} + \frac{1}{16\alpha'} \left( \rho^{\tau[+2]} - i s \varphi_{\sigma}^{[+2]} \right) v_{\hat{\alpha}\dot{A}}^{-} + \frac{is}{16\alpha'} \varphi_{\sigma}^{(\mathbf{I})} \tilde{\gamma}_{AA}^{(\mathbf{I})} v_{\hat{\alpha}A}^{+} \approx 0 \end{aligned} \qquad (4.4)$$

and the Grassmann-odd components of supertwistors

$$\begin{aligned} D_A^{-}(\sigma) &= \pi_A^{-} + \frac{1}{16\alpha'} \left( i s \omega_{\sigma}^{[-2]} - s \varphi_{\sigma}^{[-2]} - i \rho^{\tau[-2]} \right) \eta_A^{+} + \frac{s}{16\alpha'} \left( \varphi_{\sigma}^{(\mathbf{I})} - i \omega_{\sigma}^{(\mathbf{I})} \right) \gamma_{AA}^{(\mathbf{I})} \eta_{\dot{A}}^{-} \\ &\quad + \frac{i}{2} \eta_B^{+} \left( v_B^{\hat{\alpha}-} p_{(\mu)\hat{\alpha}A}^{-} - v_A^{\hat{\alpha}-} p_{(\mu)\hat{\alpha}B}^{-} \right) + \frac{i}{2} \eta_B^{-} \left( v_B^{\hat{\alpha}+} p_{(\mu)\hat{\alpha}A}^{-} - v_A^{\hat{\alpha}-} p_{(\mu)\hat{\alpha}B}^{+} \right) \approx 0, \\ D_A^{+}(\sigma) &= \pi_A^{+} + \frac{1}{16\alpha'} \left( i s \omega_{\sigma}^{[+2]} - s \varphi_{\sigma}^{[+2]} - i \rho^{\tau[+2]} \right) \eta_A^{-} + \frac{s}{16\alpha'} \left( \varphi_{\sigma}^{(\mathbf{I})} - i \omega_{\sigma}^{(\mathbf{I})} \right) \tilde{\gamma}_{AA}^{(\mathbf{I})} \eta_A^{+} \\ &\quad + \frac{i}{2} \eta_B^{+} \left( v_B^{\hat{\alpha}-} p_{(\mu)\hat{\alpha}\dot{A}}^{+} - v_A^{\hat{\alpha}+} p_{(\mu)\hat{\alpha}\dot{B}}^{-} \right) + \frac{i}{2} \eta_B^{-} \left( v_B^{\hat{\alpha}+} p_{(\mu)\hat{\alpha}\dot{A}}^{+} - v_A^{\hat{\alpha}+} p_{(\mu)\hat{\alpha}\dot{B}}^{+} \right) \approx 0. \end{aligned} \qquad (4.5)$$
The set of the primary constraints also includes constraints that determine momentum densities for spinor harmonics

$$ T_A^{\hat{\alpha}-}(\sigma) = p_{(v)A}^{\hat{\alpha}-} + \frac{1}{16\alpha'} (is\varphi_{\sigma}^{[-2]} - \rho^{\tau[-2]}) \mu_A^{\hat{\alpha}+} - \frac{is}{16\alpha'} \varphi_{\sigma}^{(\mathbf{I})} \gamma_{AA'}^{(\mathbf{I})} \mu_{\hat{A}}^{\hat{\alpha}-} \\ + \frac{is}{256\alpha'} \left[ \frac{1}{2}\omega_{\sigma}^{[-2]} (\eta^{+}\gamma^{(\mathbf{I})(\mathbf{J})}\eta^{+}) + \frac{1}{2}\omega_{\sigma}^{[+2]} (\eta^{-}\tilde{\gamma}^{(\mathbf{I})(\mathbf{J})}\eta^{-}) - \omega_{\sigma}^{(\mathbf{K})} (\eta^{+}\gamma^{(\mathbf{I})(\mathbf{J})(\mathbf{K})}\eta^{-}) \right] \gamma_{AB}^{(\mathbf{I})(\mathbf{J})} v_{\hat{B}}^{\hat{\alpha}-} + \frac{is}{128\alpha'} \omega_{\sigma}^{(\mathbf{I})} (\eta^{+}\gamma^{(\mathbf{I})}\eta^{-}) v_{A}^{\hat{\alpha}-} \\ + \frac{is}{128\alpha'} \left[ \omega_{\sigma}^{[-2]} (\eta^{+}\gamma^{(\mathbf{I})}\eta^{-}) + \omega_{\sigma}^{(\mathbf{J})} (\eta^{-}\tilde{\gamma}^{(\mathbf{I})(\mathbf{J})}\eta^{-}) \right] \gamma_{AA'}^{(\mathbf{I})} v_{\hat{A}}^{\hat{\alpha}+} \approx 0, \quad (4.6) $$

$$ T_A^{\hat{\alpha}+}(\sigma) = p_{(v)A}^{\hat{\alpha}+} + \frac{1}{16\alpha'} (is\varphi_{\sigma}^{[+2]} - \rho^{\tau[+2]}) \mu_A^{\hat{\alpha}-} - \frac{is}{16\alpha'} \varphi_{\sigma}^{(\mathbf{I})} \tilde{\gamma}_{AA'}^{(\mathbf{I})} \mu_A^{\hat{\alpha}+} \\ + \frac{is}{256\alpha'} \left[ \frac{1}{2}\omega_{\sigma}^{[-2]} (\eta^{+}\gamma^{(\mathbf{I})(\mathbf{J})}\eta^{+}) + \frac{1}{2}\omega_{\sigma}^{[+2]} (\eta^{-}\tilde{\gamma}^{(\mathbf{I})(\mathbf{J})}\eta^{-}) - \omega_{\sigma}^{(\mathbf{K})} (\eta^{+}\gamma^{(\mathbf{I})(\mathbf{J})(\mathbf{K})}\eta^{-}) \right] \tilde{\gamma}_{AB}^{(\mathbf{I})(\mathbf{J})} v_{\hat{B}}^{\hat{\alpha}+} - \frac{is}{128\alpha'} \omega_{\sigma}^{(\mathbf{I})} (\eta^{+}\gamma^{(\mathbf{I})}\eta^{-}) v_{\hat{A}}^{\hat{\alpha}+} \\ - \frac{is}{128\alpha'} \left[ \omega_{\sigma}^{[+2]} (\eta^{+}\gamma^{(\mathbf{I})}\eta^{-}) - \omega_{\sigma}^{(\mathbf{J})} (\eta^{+}\gamma^{(\mathbf{I})(\mathbf{J})}\eta^{+}) \right] \tilde{\gamma}_{AA}^{(\mathbf{I})} v_{\hat{A}}^{\hat{\alpha}-} \approx 0 \quad (4.7) $$

and inverse spinor harmonics

$$ p_{(v)\hat{A}}^{+}(\sigma) \approx 0, \quad p_{(v)\hat{A}}^{-}(\sigma) \approx 0. \quad (4.8) $$

Besides there turn to zero momenta for the components of the world-sheet vector density $\rho^{\mu[\pm 2]}$

$$ P_{\tau}^{[\pm 2]}(\sigma) \approx 0, \quad (4.9) $$

$$ P_{\sigma}^{[\pm 2]}(\sigma) \approx 0. \quad (4.10) $$

The set of the primary constraints (4.4)-(4.10) should be extended by adding the constraints on the Lorentz harmonics. 16 × 16 matrix of the spinor harmonics $v_{\hat{\alpha}}^{(\hat{\alpha})}$ satisfies 211 relations [9], [10], [3] reducing the number of its independent components to the dimension of the Spin(1, 9) group equal 45

$$ \begin{gathered} n_{\hat{m}}^{(\hat{k})} v_{\hat{\alpha}}^{(\hat{\alpha})} \tilde{\sigma} \hat{m} \hat{m}_1 \dots \hat{m}_4 \hat{\alpha} \hat{\beta} v_{\hat{\beta}}^{(\hat{\beta})} \sigma_{(\hat{k})(\hat{\alpha})} (\hat{\beta}) \approx 0, \quad (4.11) \\ n_{\hat{m}}^{[+2]} n_{\hat{m}}^{[-2]} - 2 = \frac{1}{64} (v_{\hat{\alpha}A}^{+} \tilde{\sigma} \hat{m} \hat{\beta} v_{\hat{\beta}A}^{+}) (v_{\hat{\gamma}A}^{-} \tilde{\sigma} \hat{m} \hat{\gamma} \delta v_{\hat{\delta}A}^{-}) - 2 \approx 0. \end{gathered} $$

Defining relations (4.3) of the matrix of inverse spinor harmonics, as discussed above, also have to be treated as the primary constraints. Finally 120 + 126 constraints on the components of supertwistors (3.27) and (3.28) should be included in the set of the primary constraints.

As was shown in [4] canonical analysis of the (super)string models in the Lorentz-harmonic formulation simplifies considerably if one treats as strong equalities the constraints defining inverse spinor harmonics (4.3) and the harmonicity conditions (4.11). To this end there were found projections of the constraints that follow from the definitions of momenta for the Lorentz harmonics and that form conjugate pairs of the second-class constraints with the constraints (4.3) and (4.11). Also introduced were respective Dirac brackets. Appealing feature of such Dirac brackets is that they coincide with the Poisson brackets on the subspace
of the phase-space defined by other constraints of the model. In the supertwistor formulation this subspace includes primary constraints (3.27), (3.28), (4.4), (4.5), (4.9), (4.10) and the generators of so(1, 9)$_R$ algebra in the so(1, 1) ⊕ so(8) basis

$$
\begin{align}
\hat{M}^{+2-2}(\sigma) &= v_{\hat{\alpha}A}^{+} T_{\hat{A}}^{\hat{\alpha}-} - v_{\hat{\alpha}\hat{A}}^{-} T_{\hat{A}}^{\hat{\alpha}+} - v_{\hat{A}}^{\hat{\alpha}-} p_{(\nu)\hat{\alpha}A}^{+} + v_{\hat{A}}^{\hat{\alpha}+} p_{(\nu)\hat{\alpha}\hat{A}}^{-} \nonumber \\
&\quad + \mu_{\hat{A}}^{\hat{\alpha}+} \Phi_{\hat{\alpha}A}^{-} - \mu_{\hat{A}}^{\hat{\alpha}-} \Phi_{\hat{\alpha}\hat{A}}^{+} + \eta_{\hat{A}}^{+} D_{\hat{A}}^{-} - \eta_{\hat{A}}^{-} D_{\hat{A}}^{+} \approx 0, \nonumber \\
\hat{M}^{+2\mathbf{I}}(\sigma) &= -\gamma_{A\hat{A}}^{\mathbf{I}} (v_{\hat{\alpha}A}^{+} T_{\hat{A}}^{\hat{\alpha}+} - v_{\hat{A}}^{\hat{\alpha}+} p_{(\nu)\hat{\alpha}A}^{+} + \mu_{\hat{A}}^{\hat{\alpha}+} \Phi_{\hat{\alpha}\hat{A}}^{+} + \eta_{\hat{A}}^{+} D_{\hat{A}}^{+}) \approx 0, \tag{4.12} \\
\hat{M}^{-2\mathbf{I}}(\sigma) &= -\gamma_{A\hat{A}}^{\mathbf{I}} (v_{\hat{\alpha}\hat{A}}^{-} T_{\hat{A}}^{\hat{\alpha}-} - v_{\hat{A}}^{\hat{\alpha}-} p_{(\nu)\hat{\alpha}\hat{A}}^{-} + \mu_{\hat{A}}^{\hat{\alpha}-} \Phi_{\hat{\alpha}A}^{-} + \eta_{\hat{A}}^{-} D_{\hat{A}}^{-}) \approx 0, \nonumber \\
\hat{M}^{\mathbf{I}\hat{J}}(\sigma) &= -\frac{1}{2} \gamma_{AB}^{\mathbf{I}\hat{J}} (v_{\hat{\alpha}A}^{+} T_{\hat{B}}^{\hat{\alpha}-} + v_{\hat{A}}^{\hat{\alpha}-} p_{(\nu)\hat{\alpha}B}^{+} + \mu_{\hat{A}}^{\hat{\alpha}+} \Phi_{\hat{\alpha}B}^{-} + \eta_{\hat{A}}^{+} D_{\hat{B}}^{-}) \nonumber \\
&\quad -\frac{1}{2} \tilde{\gamma}_{\hat{A}\hat{B}}^{\mathbf{I}\hat{J}} (v_{\hat{\alpha}\hat{A}}^{-} T_{\hat{B}}^{\hat{\alpha}+} + v_{\hat{A}}^{\hat{\alpha}+} p_{(\nu)\hat{\alpha}\hat{B}}^{-} + \mu_{\hat{A}}^{\hat{\alpha}-} \Phi_{\hat{\alpha}\hat{B}}^{+} + \eta_{\hat{A}}^{-} D_{\hat{B}}^{+}) \approx 0. \nonumber
\end{align}
$$

Constraints (4.12) complement those on the momentum components for the Lorentz har-
monics that were used to construct Dirac brackets. Using the constraints (3.27) and (3.28)
they can be brought to the form

$$
\begin{equation}
\begin{split}
M^{[+2] [-2]}(\sigma) ={}& v_{\hat{\alpha}A}^{+} p_{(\nu)A}^{\hat{\alpha}-} - v_{\hat{\alpha}\hat{A}}^{-} p_{(\nu)\hat{A}}^{\hat{\alpha}+} - v_{\hat{A}}^{\hat{\alpha}-} p_{(\nu)\hat{\alpha}A}^{+} + v_{\hat{A}}^{\hat{\alpha}+} p_{(\nu)\hat{\alpha}\hat{A}}^{-} \\
& \qquad + \mu_{\hat{A}}^{\hat{\alpha}+} p_{(\mu)\hat{\alpha}A}^{-} - \mu_{\hat{A}}^{\hat{\alpha}-} p_{(\mu)\hat{\alpha}\hat{A}}^{+} + \eta_{\hat{A}}^{+} \pi_{\hat{A}}^{-} - \eta_{\hat{A}}^{-} \pi_{\hat{A}}^{+} \approx 0,
\end{split}
\tag{4.13}
\end{equation}
$$

$$
M^{[+2](\mathbf{I})}(\sigma) = -\gamma_{A\hat{A}}^{(\mathbf{I})} (v_{\hat{\alpha}A}^{+} p_{(\nu)\hat{A}}^{\hat{\alpha}+} - v_{\hat{A}}^{\hat{\alpha}+} p_{(\nu)\hat{\alpha}A}^{+} + \mu_{\hat{A}}^{\hat{\alpha}+} p_{(\mu)\hat{\alpha}\hat{A}}^{+} + \eta_{\hat{A}}^{+} \pi_{\hat{A}}^{+}) \approx 0, \quad (4.14)
$$

$$
M^{[-2](\mathbf{I})}(\sigma) = -\gamma_{A\hat{A}}^{(\mathbf{I})} (v_{\hat{\alpha}\hat{A}}^{-} p_{(\nu)\hat{A}}^{\hat{\alpha}-} - v_{\hat{A}}^{\hat{\alpha}-} p_{(\nu)\hat{\alpha}\hat{A}}^{-} - \mu_{\hat{A}}^{\hat{\alpha}-} p_{(\mu)\hat{\alpha}A}^{-} + \eta_{\hat{A}}^{-} \pi_{\hat{A}}^{-}) \approx 0, \quad (4.15)
$$

$$
\begin{equation}
\begin{split}
M^{(\mathbf{I})(\mathbf{J})}(\sigma) = -\frac{1}{2} \gamma_{AB}^{(\mathbf{I})(\mathbf{J})} & (v_{\hat{\alpha}A}^{+} p_{(\nu)B}^{\hat{\alpha}-} + v_{\hat{A}}^{\hat{\alpha}-} p_{(\nu)\hat{A}B}^{+} + \mu_{\hat{A}}^{\hat{\alpha}+} p_{(\mu)\hat{A}B}^{-} + \eta_{\hat{A}}^{+} \pi_{\hat{B}}^{-}) \\
& -\frac{1}{2} \tilde{\gamma}_{\hat{A}\hat{B}}^{(\mathbf{I})(\mathbf{J})} (v_{\hat{\alpha}\hat{A}}^{-} p_{(\nu)\hat{B}}^{\hat{\alpha}+} + v_{\hat{A}}^{\hat{\alpha}+} p_{(\nu)\hat{\alpha}\hat{B}}^{-} + \mu_{\hat{A}}^{\hat{\alpha}-} p_{(\mu)\hat{\alpha}\hat{B}}^{+} + \eta_{\hat{A}}^{-} \pi_{\hat{B}}^{+}) \approx 0.
\end{split}
\tag{4.16}
\end{equation}
$$

In [4] analogous constraints were named covariant momentum densities. In the supertwistor formulation covariant momentum densities generate infinitesimal SO(1, 9)$_R$ transformations of the supertwistor components. Note that their SO(1, 1) × SO(8) ⊂ SO(1, 9)$_R$ subgroup is the gauge symmetry of the action (4.1) so that the constraint (4.13) after adding contribu- tions of the world-sheet vector density and conjugate momenta and the constraints (4.16) belong to the first class, whereas (4.14) and (4.15) are the second-class constraints.

Analogously constraints on the components of supertwistors (3.27) and (3.28) can also
be treated as strong equalities after introduction of the Dirac brackets. To this end define
projections of the constraints (4.4)

$$
\begin{align}
\Phi_{\hat{A}\hat{B}}^{+2}(\sigma) &= v_{\hat{A}}^{\hat{\alpha}+} \Phi_{\hat{A}\hat{B}}^{+} - v_{\hat{B}}^{\hat{\alpha}+} \Phi_{\hat{A}\hat{A}}^{+} \approx 0, \nonumber \\
\Phi_{\hat{A}\hat{B}}^{-2}(\sigma) &= v_{\hat{A}}^{\hat{\alpha}-} \Phi_{\hat{A}\hat{B}}^{-} - v_{\hat{B}}^{\hat{\alpha}-} \Phi_{\hat{A}\hat{A}}^{-} \approx 0, \nonumber \\
\Phi_{\hat{A}\hat{B}}(\sigma) &= v_{\hat{A}}^{\hat{\alpha}-} \Phi_{\hat{A}\hat{B}}^{+} - v_{\hat{B}}^{\hat{\alpha}+} \Phi_{\hat{A}\hat{A}}^{-} \approx 0, \nonumber \\
\Phi^{\hat{m}[5]}(\sigma) &= v_{\hat{A}\hat{A}}^{+} \tilde{\sigma}^{\hat{m}[5]\hat{\alpha}\hat{\beta}} \Phi_{\hat{\beta}A}^{-} + v_{\hat{\alpha}\hat{A}}^{-} \tilde{\sigma}^{\hat{m}[5]\hat{\alpha}\hat{\beta}} \Phi_{\hat{\alpha}\hat{A}}^{+} \approx 0 \tag{4.17}
\end{align}
$$

that form conjugate pairs of the second-class constraints with the constraints (3.27) and
(3.28)

$$
\{\Phi_{\hat{A}\hat{B}}^{+2}(\sigma), N_{\hat{C}\hat{D}}^{-2}(\sigma')\} = 2(\delta_{\hat{A}\hat{D}}\delta_{\hat{B}\hat{C}} - \delta_{\hat{A}\hat{C}}\delta_{\hat{B}\hat{D}})\delta(\sigma - \sigma'),
$$
$$
\begin{align*}
\{\Phi_{AB}^{-2}(\sigma), N_{CD}^{+2}(\sigma')\} &= 2(\delta_{AD}\delta_{BC} - \delta_{AC}\delta_{BD})\delta(\sigma - \sigma'), \\
\{\Phi_{A\dot{B}}(\sigma), N_{C\dot{D}}(\sigma')\} &= -2\delta_{AC}\delta_{\dot{B}\dot{D}}\delta(\sigma - \sigma'), \\
\{\Phi^{\dot{m}[5]}(\sigma), N^{\dot{m}[5]}(\sigma')\} &= 2(\tilde{\sigma}^{\dot{m}[5]}\sigma^{\dot{m}[5]})\delta(\sigma - \sigma').
\end{align*}
$$

All other Poisson bracket relations between these constraints vanish in the strong sense. For
the subspace of the phase-space that is determined by remaining constraints (4.4)

$$
\begin{align}
\Phi^{[+2]}(\sigma) &= 2v_{\hat{A}}^{\hat{\alpha}+} p_{(\mu)\hat{\alpha}\hat{A}}^{+} + \frac{1}{\alpha'} \left( \rho^{\tau[+2]} - i s \varphi_{\sigma}^{[+2]} \right) \approx 0, \nonumber \\
\Phi^{-[2]}(\sigma) &= 2v_{\hat{A}}^{\hat{\alpha}-} p_{(\mu)\hat{\alpha}A}^{-} + \frac{1}{\alpha'} \left( \rho^{\tau[-2]} - i s \varphi_{\sigma}^{-[2]} \right) \approx 0, \tag{4.18} \\
\Phi^{(\mathbf{I})}(\sigma) &= -\gamma_{A\hat{A}}^{(\mathbf{I})} (v_{\hat{A}}^{\hat{\alpha}-} p_{(\mu)\hat{\alpha}\hat{A}}^{+} + v_{\hat{A}}^{\hat{\alpha}+} p_{(\mu)\hat{\alpha}A}^{-}) - \frac{i s}{\alpha'} \varphi_{\sigma}^{(\mathbf{I})} \approx 0, \nonumber
\end{align}
$$

together with the so(1,9) generators (4.13)-(4.16) and primary constraints (4.5), (4.9), (4.10)
these Dirac brackets coincide with the Poisson brackets.

As a result the total Hamiltonian density

$$
\begin{align}
H_T(\tau, \sigma) ={}& \frac{1}{2\alpha'} \rho^{\sigma^{[+2]}} (\omega_{\sigma}^{[-2]} + \rho^{\tau^{[-2]}}) + \frac{1}{2\alpha'} \rho^{\sigma^{[-2]}} (\omega_{\sigma}^{[+2]} - \rho^{\tau^{[+2]}}) \nonumber \\
& + a^{[+2]} \Phi^{[-2]} + a^{[-2]} \Phi^{[+2]} + a^{(\mathbf{I})} \Phi^{(\mathbf{I})} + b^{\mu^{[+2]}} P_{\mu}^{[-2]} + b^{\mu^{[-2]}} P_{\mu}^{[+2]} \nonumber \\
& + l^{[+2][+2]} M^{[+2][+2]} + l^{[+2](\mathbf{I})} M^{[-2](\mathbf{I})} + l^{[-2](\mathbf{I})} M^{[+2](\mathbf{I})} + l^{(\mathbf{I})(\mathbf{J})} M^{(\mathbf{I})(\mathbf{J})} \tag{4.19} \\
& + \lambda_A^+ D_A^- + \lambda_A^- D_A^+ \nonumber
\end{align}
$$

is determined by the linear combination of the primary constraints (4.18), (4.9), (4.10),
(4.13)-(4.16) and (4.5) with bosonic $a(\tau, \sigma)$, $b(\tau, \sigma)$, $l(\tau, \sigma)$ and fermionic $\lambda(\tau, \sigma)$ Lagrange
multipliers. On the primary constraints imposed is the consistency requirement according
to which they should have weakly vanishing Poisson bracket relations with the total Hamil-
tonian density. This requirement may lead to the secondary constraints or to the equations
that determine Lagrange multipliers. Calculation of the Poisson brackets of the primary
constraints with the total Hamiltonian yields 10 bosonic secondary constraints

$$
\omega_{\sigma}^{[\pm 2]} \mp \rho^{\tau[\pm 2]} \approx 0, \tag{4.20}
$$

$$
\omega_{\sigma}^{(\mathbf{I})} \approx 0, \tag{4.21}
$$

and determines part of the Lagrange multipliers.

Resulting upon consistency analysis of the primary and secondary constraints expres-
sion for the total Hamiltonian density includes the first-class constraints with the Lagrange
multipliers that remained arbitrary

$$
H_{T, s=1}(\tau, \sigma) = \rho^{\sigma^{[+2]}} \Delta_{(-)}^{[-2]} + \rho^{\sigma^{[-2]}} \tilde{\Delta}_{(+)}^{[+2]} + b^{\sigma^{[+2]}} P_{\sigma}^{[-2]} + b^{\sigma^{[-2]}} P_{\sigma}^{[+2]} \\
+ l^{[+2][+2]} \tilde{M}^{[+2][+2]} + l^{(\mathbf{I})(\mathbf{J})} M^{(\mathbf{I})(\mathbf{J})} + \xi_{\hat{A}}^{-} \tilde{D}_{\hat{A}}^{+} \approx 0.
$$

Total Hamiltonian weakly vanishes that, as remarked above, is distinctive feature of the
reparametrization-invariant models. Above expression for the total Hamiltonian corresponds
to the sign parameter's value $s = 1$ in the Lagrangian (4.2). In this case generators of the world-sheet reparametrizations have the form

$$ \Delta_{(-)}^{[-2]}(\sigma) = \frac{1}{2\alpha'}(\omega_{\sigma}^{[-2]} + \rho^{\tau[-2]}) - \frac{1}{2}\Phi^{[-2]} + \partial_{\sigma}P_{\tau}^{[-2]} \\ -\frac{1}{2}\Omega_{\sigma}^{[+2][-2]}P_{\tau}^{[-2]} + \frac{1}{2\rho^{\tau[+2]}}\Omega_{\sigma}^{[-2](\mathbf{I})}M^{[+2](\mathbf{I})} \approx 0, \qquad (4.22) $$

$$ \tilde{\Delta}_{(+)}^{[+2]}(\sigma) = \Delta_{(+)}^{[+2]} - \frac{1}{\rho^{\tau[-2]}} \mathcal{D}_{\sigma} \eta_{A}^{+} \tilde{D}_{A}^{-} \approx 0, \qquad (4.23) $$

where

$$ \Delta_{(+)}^{[+2]}(\sigma) = \frac{1}{2\alpha'}(\omega_{\sigma}^{[+2]} - \rho^{\tau[+2]}) + \frac{1}{2}\Phi^{[+2]} + \partial_{\sigma}P_{\tau}^{[+2]} \\ +\frac{1}{2}\Omega_{\sigma}^{[+2][-2]}P_{\tau}^{[+2]} + \frac{1}{2\rho^{\tau[-2]}}\Omega_{\sigma}^{[+2](\mathbf{I})}M^{[-2](\mathbf{I})} \approx 0 $$

and

$$ \tilde{D}_{A}^{-}(\sigma) = D_{A}^{-} + \frac{i}{16}\eta_{A}^{+}\Phi^{[-2]} - \frac{i}{16}\gamma_{A\dot{A}}^{(\mathbf{I})}\eta_{\dot{A}}^{-}\Phi^{(\mathbf{I})} - \frac{i}{8\rho^{\tau[-2]}}\gamma_{A\dot{A}}^{(\mathbf{I})}\mathcal{D}_{\sigma}\eta_{\dot{A}}^{-}M^{[-2](\mathbf{I})} \approx 0. \qquad (4.24) $$

Lower indices in brackets of $\Delta_{(+)}^{[+2]} \approx 0$ and $\Delta_{(-)}^{[-2]} \approx 0$ indicate the sign of the contribution of the constraints $\Phi^{[\pm 2]} \approx 0$ and should not be confused with their weights under the $SO(1, 1)$ group given by the upper indices. For future use it is convenient to introduce more general linear combinations of the constraints

$$ \Delta_k^{[+2]}(\sigma) = \frac{1}{2\alpha'}(\omega_{\sigma}^{[+2]} - \rho^{\tau[+2]}) + \frac{k}{2}\Phi^{[+2]} + \partial_{\sigma}P_{\tau}^{[+2]} \\ +\frac{1}{2}\Omega_{\sigma}^{[+2][-2]}P_{\tau}^{[+2]} + \frac{1}{2\rho^{\tau[-2]}}\Omega_{\sigma}^{[+2](\mathbf{I})}M^{[-2](\mathbf{I})} \approx 0, \qquad (4.25) $$

$$ \Delta_k^{[-2]}(\sigma) = \frac{1}{2\alpha'}(\omega_{\sigma}^{[-2]} + \rho^{\tau[-2]}) + \frac{k}{2}\Phi^{[-2]} + \partial_{\sigma}P_{\tau}^{[-2]} \\ -\frac{1}{2}\Omega_{\sigma}^{[+2][-2]}P_{\tau}^{[-2]} + \frac{1}{2\rho^{\tau[+2]}}\Omega_{\sigma}^{[-2](\mathbf{I})}M^{[+2](\mathbf{I})} \approx 0. \qquad (4.26) $$

Constraints

$$ \tilde{M}^{[+2][-2]}(\sigma) = M^{[+2][-2]} + 2\rho^{\tau[+2]}P_{\tau}^{[-2]} - 2\rho^{\tau[-2]}P_{\tau}^{[+2]} \approx 0 \qquad (4.27) $$

and (4.16) generate the $SO(1, 1) \times SO(8)$ gauge symmetry of the superstring action (4.1), while eight constraints

$$ \tilde{D}_{A}^{+}(\sigma) = D_{A}^{+} + \frac{i}{16}\eta_{A}^{-}\Phi^{[+2]} - \frac{i}{16}\tilde{\gamma}_{A\dot{A}}^{(\mathbf{I})}\eta_{A}^{+}\Phi^{(\mathbf{I})} \\ -\frac{i}{8\rho^{\tau[-2]}}\tilde{\gamma}_{A\dot{A}}^{(\mathbf{I})}\mathcal{D}_{\sigma}\eta_{A}^{+}M^{[-2](\mathbf{I})} - \frac{i}{4}\mathcal{D}_{\sigma}\eta_{A}^{-}P_{\tau}^{[+2]} \approx 0 \qquad (4.28) $$

are the $\kappa$-symmetry generators. When $s = -1$ expression for the total Hamiltonian density has the form

$$ H_{T, s=-1}(\tau, \sigma) = \rho^{\sigma[+2]}\tilde{\Delta}_{(-)}^{[-2]} + \rho^{\sigma[-2]}\Delta_{(+)}^{[+2]} + b^{\sigma[+2]}P_{\sigma}^{[-2]} + b^{\sigma[-2]}P_{\sigma}^{[+2]} \\ +l^{[+2][-2]}\tilde{M}^{[+2][-2]} + l^{(\mathbf{I})(\mathbf{J})}M^{(\mathbf{I})(\mathbf{J})} + \xi_{A}^{+}\tilde{D}_{A}^{-} \approx 0, $$

where

$$ \tilde{\Delta}_{(-)}^{[-2]}(\sigma) = \Delta_{(-)}^{[-2]} - \frac{1}{\rho^{\tau[+2]}} \mathcal{D}_{\sigma} \eta_{A}^{-} \tilde{D}_{A}^{\tau} \approx 0 \qquad (4.29) $$

and

$$ \tilde{D}_{A}^{\prime +}(\sigma) = D_{A}^{+} + \frac{i}{16}\eta_{A}^{-}\Phi^{[+2]} - \frac{i}{16}\tilde{\gamma}_{A\dot{A}}^{(\mathbf{I})}\eta_{A}^{+}\Phi^{(\mathbf{I})} + \frac{i}{8\rho^{\tau[+2]}}\tilde{\gamma}_{A\dot{A}}^{(\mathbf{I})}\mathcal{D}_{\sigma}\eta_{A}^{+}M^{[+2](\mathbf{I})} \approx 0, $$

whereas the $\kappa$-symmetry generators equal

$$ \tilde{D}_{A}^{\prime -}(\sigma) = D_{A}^{-} + \frac{i}{16}\eta_{A}^{+}\Phi^{[-2]} - \frac{i}{16}\gamma_{A\dot{A}}^{(\mathbf{I})}\eta_{\dot{A}}^{-}\Phi^{(\mathbf{I})} \\ +\frac{i}{8\rho^{\tau[+2]}}\gamma_{A\dot{A}}^{(\mathbf{I})}\mathcal{D}_{\sigma}\eta_{\dot{A}}^{-}M^{[+2](\mathbf{I})} + \frac{i}{4}\mathcal{D}_{\sigma}\eta_{A}^{+}P_{\tau}^{[-2]} \approx 0. $$
## 4.2 Dirac bracket algebra of the first-class constraints

In the previous subsection there were obtained expressions for 33 bosonic and 8 fermionic first-class constraints of the $D = 10$ $N = 1$ superstring model in the supertwistor formulation. Remaining primary and secondary constraints are of the second class. They can be arranged according to their Grassmann parity and the $SO(8)$ representations. $SO(8)$ vector constraints are presented by (4.14), (4.15) and

$$ \begin{aligned} \Delta_k^{(\mathbf{I})}(\sigma) = \frac{1}{\alpha'} \omega_{\sigma}^{(\mathbf{I})} + k \Phi^{(\mathbf{I})} - \Omega_{\sigma}^{[+2](\mathbf{I})} P_{\tau}^{[-2]} - \Omega_{\sigma}^{[-2](\mathbf{I})} P_{\tau}^{[+2]} \\ - \frac{1}{2} \mathcal{D}_{\sigma}^{(\mathbf{I})(\mathbf{J})} \left( \frac{M^{[+2](\mathbf{J})}}{\rho^{\tau[+2]}} \right) - \frac{1}{2} \mathcal{D}_{\sigma}^{(\mathbf{I})(\mathbf{J})} \left( \frac{M^{[-2](\mathbf{J})}}{\rho^{\tau[-2]}} \right) \approx 0 \end{aligned} \quad (4.30) $$

with $k = \pm 1$. In (4.30) $\mathcal{D}_{\sigma}^{(\mathbf{I})(\mathbf{J})} = \delta^{(\mathbf{I})(\mathbf{J})} \partial_{\sigma} - \Omega_{\sigma}^{(\mathbf{I})(\mathbf{J})}$ is the world-sheet pullback of the $SO(8)$-covariant differential. Four scalar constraints can be taken to be (4.9) and $\Delta_{(+)}^{[\pm 2]} \approx 0$ defined in (4.25) and (4.26). Finally as the basis elements in the space of fermionic second-class constraints choose eight constraints (4.24).

To take into account these second-class constraints define the Dirac brackets

$$ \{f(\sigma), g(\sigma')\}_{\text{D.B.}} = \{f(\sigma), g(\sigma')\} - \{f(\sigma), \chi_m\} \mathbf{C}^{-1mn} \{\chi_n, g(\sigma')\}, \quad (4.31) $$

where $\chi_m$ denotes the set of the second-class constraints and $\mathbf{C}^{-1mn}$ is inverse to the Dirac supermatrix

$$ \mathbf{C}_{mn}(\sigma, \sigma') = \{\chi_m(\sigma), \chi_n(\sigma')\}. $$

For the set of the second-class constraints described in the previous subsection the Dirac supermatrix can be presented as a sum

$$ \mathbf{C}_{mn} = J_{mn} + \Lambda_{mn}, \quad (4.32) $$

where $J_{mn}$ is the block-diagonal graded-antisymmetric supermatrix

$$ \alpha' J = $$

<figure><img src="image_2.png" /></figure>
and $\Lambda_{mn}$ depends linearly on the constraints.<fnref n="4" /> Taking into account (4.32) the inverse Dirac supermatrix can be presented as

$$C^{-1} = (I + J^{-1}\Lambda)^{-1}J^{-1}$$

or in the form of the series expansion

$$C^{-1} = J^{-1} - J^{-1}\Lambda J^{-1} + J^{-1}\Lambda J^{-1}\Lambda J^{-1} - J^{-1}\Lambda J^{-1}\Lambda J^{-1} + \dots \quad (4.33)$$

Since supermatrix $J$ is proportional to $(\alpha')^{-1}$ and its inverse depends on $\alpha'$, then inverse of the Dirac supermatrix can be presented as a series in $\alpha'$.<fnref n="5" /> Substituting expansion (4.33) in the definition of the Dirac brackets (4.31) it is possible to calculate them order by order in $J^{-1}$. The leading in $\alpha'$ contribution is of the form

$$ \begin{aligned} \{f(\sigma), g(\sigma')\}_{\text{D.B.}} ={}& \{f(\sigma), g(\sigma')\} - 4i\alpha' \int \frac{d\sigma''}{\rho^{\tau[-2]}(\sigma'')} \{f(\sigma), \tilde{D}_A^-(\sigma'')\} \\ & \times \{\tilde{D}_A^-(\sigma''), g(\sigma')\} - \frac{\alpha'}{2} \int \frac{d\sigma''}{\rho^{\tau[+2]}(\sigma'')} \{\{f(\sigma), M^{[+2](\mathbf{I})}(\sigma'')\}\{\Delta_{(-)}^{(\mathbf{I})}(\sigma''), g(\sigma')\} \\ & - (M^{[+2](\mathbf{I})} \leftrightarrow \Delta_{(-)}^{(\mathbf{I})}) + \frac{\alpha'}{2} \int \frac{d\sigma''}{\rho^{\tau[-2]}(\sigma'')} \{\{f(\sigma), M^{[-2](\mathbf{I})}(\sigma'')\}\{\Delta_{(+)}^{(\mathbf{I})}(\sigma''), g(\sigma')\} \\ & - (M^{[-2](\mathbf{I})} \leftrightarrow \Delta_{(+)}^{(\mathbf{I})}) + \alpha' \int d\sigma'' \{\{f(\sigma), \Delta_{(-)}^{[+2]}(\sigma'')\}\{P_{\tau}^{[-2]}(\sigma''), g(\sigma')\} \\ & - (\Delta_{(-)}^{[+2]} \leftrightarrow P_{\tau}^{[-2]}) - \alpha' \int d\sigma'' \{\{f(\sigma), \Delta_{(+)}^{[-2]}(\sigma'')\}\{P_{\tau}^{[+2]}(\sigma''), g(\sigma')\} \\ & - (\Delta_{(+)}^{[-2]} \leftrightarrow P_{\tau}^{[+2]}) + O(J^{-2}). \end{aligned} \quad (4.34) $$

With the aid of (4.34) one can calculate the Dirac brackets for the superstring phase-space variables. For instance, for two octets of supertwistors (3.25) the Dirac brackets to leading order in $\alpha'$ equal

$$ \begin{aligned} \{Z_A^{\Lambda+}(\sigma), Z_B^{\Sigma+}(\sigma')\}_{\text{D.B.}} ={}& \frac{4i\alpha'}{\rho^{\tau[-2]}} D_{AC}^{\Lambda} D_{BC}^{\Sigma} \delta(\sigma - \sigma') \\ & + \frac{\alpha'}{2\rho^{\tau[-2]}} \gamma_{AA}^{(\mathbf{I})} \gamma_{BB}^{(\mathbf{I})} (V_A^{\Lambda+} Z_B^{\Sigma-} - Z_A^{\Lambda-} V_B^{\Sigma+}) \delta(\sigma - \sigma') \\ & + \frac{\alpha'}{2\rho^{\tau[-2]}} \gamma_{AA}^{(\mathbf{I})} Z_A^{\Lambda-}(\sigma) \gamma_{BB}^{(\mathbf{J})} \mathcal{D}_{\sigma}^{(\mathbf{I})(\mathbf{J})} \left( \frac{1}{\rho^{\tau[-2]}} Z_B^{\Sigma-}(\sigma) \delta(\sigma - \sigma') \right) + O(\alpha'^2), \\ \{Z_A^{\Lambda-}(\sigma), Z_B^{\Sigma-}(\sigma')\}_{\text{D.B.}} ={}& \frac{4i\alpha'}{\rho^{\tau[-2]}} D_{AC}^{\Lambda-} D_{BC}^{\Sigma-} \delta(\sigma - \sigma') \\ & + \frac{\alpha'}{2\rho^{\tau[+2]}} \tilde{\gamma}_{AA}^{(\mathbf{I})} \tilde{\gamma}_{BB}^{(\mathbf{I})} (V_A^{\Lambda-} Z_B^{\Sigma+} - Z_A^{\Lambda+} V_B^{\Sigma-}) \delta(\sigma - \sigma') \\ & - \frac{\alpha'}{2\rho^{\tau[+2]}} \tilde{\gamma}_{AA}^{(\mathbf{I})} Z_A^{\Lambda+}(\sigma) \tilde{\gamma}_{BB}^{(\mathbf{J})} \mathcal{D}_{\sigma}^{(\mathbf{I})(\mathbf{J})} \left( \frac{1}{\rho^{\tau[+2]}} Z_B^{\Sigma+}(\sigma) \delta(\sigma - \sigma') \right) + O(\alpha'^2) \end{aligned} $$

and

$$ \begin{aligned} \{Z_A^{\Lambda+}(\sigma), Z_B^{\Sigma-}(\sigma')\}_{\text{D.B.}} ={}& \frac{4i\alpha'}{\rho^{\tau[-2]}} D_{AC}^{\Lambda} D_{BC}^{\Sigma-} \delta(\sigma - \sigma') \\ & + \frac{\alpha'}{2} \gamma_{AA}^{(\mathbf{I})} \tilde{\gamma}_{BB}^{(\mathbf{I})} \left( \frac{1}{\rho^{\tau[+2]}} V_A^{\Lambda+} Z_B^{\Sigma+} - \frac{1}{\rho^{\tau[-2]}} Z_A^{\Lambda-} V_B^{\Sigma-} \right) \delta(\sigma - \sigma') + O(\alpha'^2), \end{aligned} $$

<footnote n="4">⁴Expressions for the Poisson brackets of the second-class constraints that determine the form of the supermatrix $\Lambda_{mn}$ are given in Appendix B of [17].</footnote>

<footnote n="5">⁵Note, however, that some entries of $\Lambda$ have implicit dependence on $(\alpha')^{-1}$ through the constraints (4.25), (4.26) and (4.30).</footnote>
where 16 × 8 blocks of the inverse spinor harmonic matrix have been written in the super-twistor form as $V_A^{\Lambda-} = (v_A^{\hat{\alpha}-}, 0, 0)$, $V_A^{\Lambda+} = (v_A^{\hat{\alpha}+}, 0, 0)$ and also the following quantities have been introduced

$$ \{\tilde{D}_A^-(\sigma), Z_B^{\Lambda+}(\sigma')\} = D_{AB}^{\Lambda}\delta(\sigma - \sigma'), \\ D_{AB}^{\Lambda} = -D_{BA}^{\Lambda} = \frac{i}{8\rho^{\tau[-2]}}\gamma_{AA}^{(\mathbf{I})} \mathcal{D}_{\sigma}\eta_{\hat{A}}^{\hat{\alpha}}\gamma_{BB}^{(\mathbf{I})}Z_{\hat{B}}^{\Lambda-} + \frac{i}{2}\left(\delta_{AB}\eta_{\hat{B}}^{\hat{\alpha}} + \frac{1}{8}\gamma_{AA}^{(\mathbf{I})}\eta_{\hat{A}}^{\hat{\alpha}}\gamma_{BB}^{(\mathbf{I})}\right)V_{\hat{B}}^{\Lambda+} \\ +\frac{i}{2}\left(\delta_{AB}\eta_{C}^{\hat{\alpha}} - \delta_{AC}\eta_{B}^{\hat{\alpha}} + \frac{1}{8}\delta_{BC}\eta_{A}^{\hat{\alpha}}\right)V_{C}^{\Lambda-} + \delta_{AB}J^{\Lambda}, \quad J^{\Lambda} = (0, 0, 1), $$

$$ \{\tilde{D}_A^-(\sigma), Z_B^{\Lambda-}(\sigma')\} = D_{AB}^{\Lambda-2}\delta(\sigma - \sigma'), \\ D_{AB}^{\Lambda-2} = -D_{BA}^{\Lambda-2} = -\frac{i}{2}\left(\delta_{AB}\eta_{\hat{B}}^{\hat{\alpha}} - \frac{1}{8}\gamma_{AA}^{(\mathbf{I})}\eta_{\hat{A}}^{\hat{\alpha}}\tilde{\gamma}_{BB}^{(\mathbf{I})}\right)V_B^{\Lambda-}. $$

Taking into account the Dirac bracket relations for the phase-space variables it is possible to calculate Dirac bracket algebra of the first-class constraints. The Dirac brackets of the generators of so(1, 1) and so(8) gauge symmetries (4.27) and (4.16) with other first-class constraints coincide with respective Poisson brackets because Poisson brackets of the so(1, 1) and so(8) generators with the second-class constraints are determined by the SO(1, 1) × SO(8) representations of the latter. The Dirac brackets of other first-class constraints in general differ from their Poisson brackets. So for the κ-symmetry generators (4.28) one obtains

$$ \{\tilde{D}_{\hat{A}}^{+}(\sigma), \tilde{D}_{\hat{B}}^{+}(\sigma')\}_{\text{D.B.}} = \frac{i}{4}\delta_{\hat{A}\hat{B}}\tilde{\Delta}_{(+)}^{[2]}\delta(\sigma - \sigma') \\ -\frac{\alpha'}{16\rho^{\tau[+2]}(\rho^{\tau[-2]})^2}\tilde{\gamma}_{AA}^{(\mathbf{I})}\mathcal{D}_{\sigma}\eta_{\hat{A}}^{\hat{\alpha}}\tilde{\gamma}_{BB}^{(\mathbf{J})}\mathcal{D}_{\sigma}\eta_{\hat{B}}^{\hat{\beta}}M^{(\mathbf{I})(\mathbf{J})}\Delta_{(-)}^{[-2]}\delta(\sigma - \sigma') \\ +\frac{\alpha'}{2}\Gamma_{\hat{A}}^{+(\mathbf{I})}(\sigma)\mathcal{D}_{\sigma}^{(\mathbf{I})(\mathbf{J})}\Gamma_{\hat{B}}^{+(\mathbf{J})}(\sigma)\delta(\sigma - \sigma') + O(J^{-2}), $$

where $\Gamma_{\hat{A}}^{+(\mathbf{I})} = -\frac{1}{4\rho^{\tau[+2]\rho^{\tau[-2]}}} \tilde{\gamma}_{AA}^{(\mathbf{J})} \mathcal{D}_{\sigma}\eta_{\hat{A}}^{\hat{\alpha}} (\frac{1}{2}\delta^{(\mathbf{J})(\mathbf{I})} \tilde{M}^{+2-2} + M^{(\mathbf{J})(\mathbf{I})})$. The first term on the r.h.s. is the contribution of the Poisson brackets, while other terms containing products of the so(1, 1), so(8) generators and the reparametrization symmetry generator (4.22) correspond to the leading order deformation. The Dirac brackets of the κ-symmetry and reparametrization generators have the form

$$ \{\tilde{\Delta}_{(+)}^{[+2]}(\sigma), \tilde{D}_{\hat{A}}^{+}(\sigma')\}_{\text{D.B.}} = \frac{i\alpha'}{4\rho^{\tau[-2]}} \tilde{\gamma}_{AA}^{(\mathbf{I})} \mathcal{D}_{\sigma}\eta_{\hat{A}}^{\hat{\alpha}} \left( A^{[+2](\mathbf{I})} - \frac{1}{2} \mathcal{D}_{\sigma}\eta^{\hat{\alpha}}\gamma^{(\mathbf{I})} \tilde{D}^{+} \right) \\ \times \Delta_{(-)}^{[-2]} \delta(\sigma - \sigma') + \frac{i\alpha'}{2} A^{[+2](\mathbf{I})}(\sigma) \mathcal{D}_{\sigma}^{(\mathbf{I})(\mathbf{J})} \Gamma_{\hat{A}}^{+(\mathbf{J})}(\sigma) \delta(\sigma - \sigma') + O(J^{-2}), \quad (4.35) $$

$$ \{\Delta_{(-)}^{[-2]}(\sigma), \tilde{D}_{\hat{A}}^{+}(\sigma')\}_{\text{D.B.}} = -\frac{i}{8}\tilde{\gamma}_{AA}^{(\mathbf{I})}\mathcal{D}_{\sigma}\eta_{\hat{A}}^{\hat{\alpha}}B^{[-2](\mathbf{I})}\delta(\sigma - \sigma') + O(J^{-2}), \quad (4.36) $$

where

$$ \rho^{\tau[+2]}\rho^{\tau[-2]}A^{[+2](\mathbf{I})} = (\mathcal{D}_{\sigma}\eta^{\hat{\alpha}}\gamma^{(\mathbf{I})}\tilde{D}^{+}) + \left(-\frac{1}{2}\delta^{(\mathbf{I})(\mathbf{J})}\tilde{M}^{[+2] [-2]} + M^{(\mathbf{I})(\mathbf{J})}\right) \\ \times \left(\Omega_{\sigma}^{[+2](\mathbf{J})} + \frac{i}{4\rho^{\tau[-2]}}(\mathcal{D}_{\sigma}\eta^{\hat{\alpha}}\gamma^{(\mathbf{J})}\mathcal{D}_{\sigma}\eta^{-})\right), $$

$$ \rho^{\tau[+2]}\rho^{\tau[-2]}B^{[-2](\mathbf{I})} = \left(\frac{1}{2}\delta^{(\mathbf{I})(\mathbf{J})}\tilde{M}^{[+2] [-2]} + M^{(\mathbf{I})(\mathbf{J})}\right) \Omega_{\sigma}^{[-2](\mathbf{J})}. $$

Observe that in (4.35) contribution of the Poisson brackets vanishes and leading in α' terms are quadratic in the first-class constraints. At the same time the Dirac brackets (4.36) to the lowest order in the supermatrix $J^{-1}$ coincide with the Poisson brackets. Finally all the
Dirac brackets of the reparametrization generators include contributions proportional to the products of the first-class constraints

$$
\begin{align*}
\{\tilde{\Delta}_{(+)}^{[+2]}(\sigma), \tilde{\Delta}_{(+)}^{[+2]}(\sigma')\}_{\text{D.B.}} &= -\frac{\alpha'}{2} A^{[+2]}(\mathbf{I})(\sigma) \mathcal{D}_{\sigma}^{(\mathbf{I})(\mathbf{J})} A^{[+2]}(\mathbf{J})(\sigma) \delta(\sigma - \sigma') + O(J^{-2}), \\
\{\Delta_{(-)}^{[-2]}(\sigma), \Delta_{(-)}^{[-2]}(\sigma')\}_{\text{D.B.}} &= \frac{\alpha'}{2} B^{[-2]}(\mathbf{I})(\sigma) \mathcal{D}_{\sigma}^{(\mathbf{I})(\mathbf{J})} B^{[-2]}(\mathbf{J})(\sigma) \delta(\sigma - \sigma') + O(J^{-2}), \\
\{\tilde{\Delta}_{(+)}^{[+2]}(\sigma), \Delta_{(-)}^{[-2]}(\sigma')\}_{\text{D.B.}} &= \frac{1}{2\rho^{\tau[+2]}\rho^{\tau[-2]}} (\mathcal{D}_{\sigma}\eta^{+}\gamma^{(\mathbf{I})}\tilde{D}^{+}) \Omega_{\sigma}^{[-2]}(\mathbf{I}) \left(1 + \frac{\alpha'}{\rho^{\tau[-2]}} \Delta_{(-)}^{[-2]}\right) \delta(\sigma - \sigma') \\
&\quad -\frac{1}{2} \left[ \Omega_{\sigma}^{[+2]}(\mathbf{I}) \left(1 + \frac{\alpha'}{\rho^{\tau[-2]}} \Delta_{(-)}^{[-2]}\right) + \frac{i}{4\rho^{\tau[-2]}} (\mathcal{D}_{\sigma}\eta^{+}\gamma^{(\mathbf{I})}\mathcal{D}_{\sigma}\eta^{-}) \left(1 + \frac{2\alpha'}{\rho^{\tau[-2]}} \Delta_{(-)}^{[-2]}\right) \right] \\
&\quad \times B^{[-2]}(\mathbf{I}) \delta(\sigma - \sigma') + O(J^{-2}).
\end{align*}
$$

5 Infinite-dimensional extensions of superconformal symmetry in twistor-string models

As is known, appearance of the first twistor-string models [46], [18] stimulated active study of the spinor and twistor representations of the scattering amplitudes in gauge theories and allowed to unveil their rich symmetry structure (see, e.g. monograph [61]). Unlike the Neveu-Schwarz-Ramond and Green-Schwarz superstrings the spectrum of twistor string models contains finitely many states. In particular, in the Berkovits model [18] the states of open-string sector (the only one explored to date) are described by the multiplets of the $D = 4$ $\mathcal{N} = 4$ supersymmetric Yang-Mills theory and conformal supergravity that is non-unitary theory [62]. In Ref. [63] it was proved the equivalence of the correlation functions of operators corresponding to particles of the super-Yang-Mills multiplet and tree-level amplitudes in this theory. However, in the loop diagrams it appears impossible to separate contributions of particles of the conformal supergravity multiplet from those of particles of the super-Yang-Mills multiplet that is the drawback of the Berkovits model. Note that later there were proposed other twistor-string models, physical states of which include multiplets of Einstein supergravities [64], [65].

To gain further insights into the properties of twistor strings it is important to identify their classical and quantum symmetries. Since in the $\mathcal{N} = 4$ super-Yang-Mills theory it is observed extension of the superconformal symmetry to the infinite-dimensional one in the limit of zero coupling [66], [67], one could expect appearance of the infinite-dimensional symmetries also in the twistor-string models. In [68] it was found an extension of the obvious global PSL(4|4, $\mathbb{R}$) symmetry of the Berkovits twistor string to non-local infinite-dimensional Yangian symmetry. This extension is similar to corresponding symmetry of the $\mathcal{N} = 4$ supersymmetric Yang-Mills theory related to its integrability [69], [70].

In this Section we analyze classical global symmetries of the Berkovits twistor string model and its extension to the case of unconstrained supertwistors and show that these symmetries represent infinite-dimensional generalizations of the superconformal symmetry.<fnref n="6" />

<footnote n="6">⁶Note that infinite-dimensional conformal symmetries in the massless $\mathcal{N} = 1$ superparticle models in the supertwistor formulation were revealed in [71].</footnote>
First there will be examined global symmetry of the Lagrangian for left(right)-moving free supertwistor variables. It has more simple structure and its generators are given by the set of all monomials constructed from the products of an arbitrary number $L \ge 0$ of PSL(4|4, $\mathbb{R}$) supertwistors with one dual supertwistor. Its finite-dimensional subalgebra is spanned by generators of the $gl(4|4, \mathbb{R})$ superalgebra and single dual supertwistor. These are $L = 1$ and $L = 0$ monomials respectively. In the Berkovits model the Lagrangians for left(right)-moving supertwistor variables include generators of the $gl(1, \mathbb{R}) \subset gl(4|4, \mathbb{R})$ symmetry with arbitrary multiplier (see Eq. (5.9)). So these symmetries are the gauge ones. Therefore the relations of the global symmetry algebra of each of these Lagrangians are obtained from the relations of the infinite-dimensional algebra of these monomials if one sets to zero the generator of respective $gl(1, \mathbb{R})$ symmetry. Finite-dimensional subalgebra of this global symmetry is spanned by the generators of the psl(4|4, $\mathbb{R}$) superalgebra and the generator of 'twisted' $gl_t(1, \mathbb{R})$ symmetry as well as the dual supertwistor. It will be shown that in the quantum theory these classical infinite-dimensional symmetries break down to the $(P)SL(4|4, \mathbb{R})$ one.

## 5.1 Classical higher-spin superconformal symmetries of twistor strings

Consider the action functional of a model with two pairs of unconstrained supertwistors on the world sheet of Lorentzian signature

$$S = \int_{\Sigma} d\tau d\sigma \mathcal{L} \qquad (5.1)$$

with the Lagrangian

$$\mathcal{L} = \mathcal{L}_{\text{tw}} + \mathcal{L}_{\text{matter}},$$

that is the sum of the Lagrangians for the supertwistor fields

$$\mathcal{L}_{\text{tw}} = \frac{1}{2} \mathcal{Y}_{A\mu} V_{+}^{\mu\nu} \partial_{\nu} \bar{Z}^{A} + \frac{1}{2} \bar{\mathcal{Y}}_{A'\mu} V_{-}^{\mu\nu} \partial_{\nu} \bar{Z}^{A'}, \qquad (5.2)$$

and other matter fields $\mathcal{L}_{\text{matter}}$. To the Lagrangian (5.2) there enter the world-sheet projectors $V_{\pm}^{\mu\nu} = \frac{1}{2}(\sqrt{-\gamma}\gamma^{\mu\nu} \pm \varepsilon^{\mu\nu})$. Using the representation for inverse world-sheet metric and the Levi-Civita tensor density via the inverse zweib ein components in the light-cone basis $e_{f}^{\mu} = (e_{[+2]}^{\mu}, e_{[-2]}^{\mu})$ (see, e.g. [4]) these projectors factorize $V_{\pm}^{\mu\nu} = 2ee_{[\pm 2]}^{\mu} e_{[\mp 2]}^{\nu}$. So the Lagrangian (5.2) can be written in the form

$$\mathcal{L}_{\text{tw}} = e \mathcal{Y}_{A[+2]} e_{[-2]}^{\mu} \partial_{\mu} \bar{Z}^{A} + e \bar{\mathcal{Y}}_{A'[-2]} e_{[+2]}^{\mu} \partial_{\mu} \bar{Z}^{A'},$$

where $\mathcal{Y}_{A[+2]} = \mathcal{Y}_{A\nu} e_{[+2]}^{\nu}$, $\bar{\mathcal{Y}}_{A'[-2]} = \bar{\mathcal{Y}}_{A'\nu} e_{[-2]}^{\nu}$. Similar form have the Lagrangians for the fermionic fields and ghosts in the model of the Ramond-Neveu-Schwarz string. Therefore to quantize twistor strings and calculate correlation functions of operators of their physical states one can apply known methods of the two-dimensional conformal field theory.
In the conformal gauge for the zweibein $e_{\mu}^{f} = e_{\mu}^{\frac{\varphi}{2}} \delta_{\mu}^{f}$ and world-sheet metric $\gamma_{\mu\nu} = e^{\varphi} \eta_{\mu\nu}$ the action (5.1) translates to

$$S_{\text{c.g.}} = \int_{\Sigma} d\tau d\sigma \mathcal{L}_{\text{c.g.}} \qquad (5.3)$$

with the Lagrangian

$$\mathcal{L}_{\text{c.g.}} = \mathcal{L}_{\text{tw, c.g.}} + \mathcal{L}_{\text{matter, c.g.}}$$

that is the sum of Lagrangians for left- and right-moving fields. In particular, for the supertwistor fields it takes the form

$$\mathcal{L}_{\text{tw, c.g.}} = \mathcal{L}_{\text{L-tw, c.g.}} + \mathcal{L}_{\text{R-tw, c.g.}} : \qquad (5.4)$$

$$\mathcal{L}_{\text{L-tw, c.g.}} = -2\mathcal{Y}_{\mathcal{A}}\partial_{-}\mathcal{Z}^{\mathcal{A}}, \qquad (5.5)$$

$$\mathcal{L}_{\text{R-tw, c.g.}} = -2\bar{\mathcal{Y}}_{\mathcal{A}'}\partial_{+}\bar{\mathcal{Z}}^{\mathcal{A}'}, \qquad (5.6)$$

where $\sigma^{\pm} = \tau \pm \sigma$ and $\partial_{\pm} = \frac{1}{2}(\partial_{\tau} \pm \partial_{\sigma})$ are world sheet coordinates and derivatives in the light-cone basis. Supertwistors $\mathcal{Z}^{\mathcal{A}} = (Z^{\alpha}, \eta^{\mathcal{A}})$ and $\bar{\mathcal{Z}}^{\mathcal{A}'} = (\bar{Z}^{\alpha'}, \bar{\eta}^{\mathcal{A}'})$ parameterize the product of two supertwistor spaces $\mathcal{T} \times \mathcal{T}'$ and transform as scalars under world-sheet conformal symmetry. Dual supertwistors $\mathcal{Y}_{\mathcal{B}} = (Y_{\beta}, \xi_{\mathcal{B}})$ and $\bar{\mathcal{Y}}_{\mathcal{B}'} = (\bar{Y}_{\beta'}, \bar{\xi}_{\mathcal{B}'})$ transform as (anti-)self-dual world-sheet vectors and in the world-sheet conformal field theory to them there correspond primary fields of conformal weight unity. For brevity world-sheet indices of dual supertwistors will be omitted. Left- and right-moving non-twistor matter fields with the Lagrangian $\mathcal{L}_{\text{matter, c.g.}}$ should give contribution to anomaly of the world-sheet conformal symmetry equal $c = \bar{c} = 26$ to cancel those of the ghost fields for this symmetry $(b, c)$ and $(\bar{b}, \bar{c})$. Such fields may be constituents of the world-sheet currents that realize generators of some Lie algebra (see, e.g. [62]).

From the reality condition of the action (5.3) it follows that left- and right-moving supertwistors and dual supertwistors have to be considered as independent variables with real components. Such supertwistors are associated with the space-time of signature $(+, +, -, -)$ and respective superspaces.<fnref n="7" /> Of the primary interest is the case of $\mathcal{N} = 4$ supertwistors that are dynamical fields in the twistor string models [46], [18]. Penrose transform for homogeneous functions of one supertwistor gives $D = 4$ $\mathcal{N} = 4$ massless supermultiplets including those of the supersymmetric Yang-Mills theory and Einstein supergravity. Conformal group of $D = 4$ space-time of such signature is $SO(3, 3) = SL(4, \mathbb{R})$ and its minimal $\mathcal{N} = 4$ supersymmetric extension is $PSL(4|4, \mathbb{R})$ with the bosonic subgroup $SL(4, \mathbb{R}) \times SL(4, \mathbb{R})$. This implies that bosonic and fermionic components of left-moving supertwistor belong to the fundamental representations of two $SL(4, \mathbb{R})_{L}$ groups, whereas bosonic and fermionic components of the dual supertwistor belong to their antifundamental representations. Accordingly components of $\bar{\mathcal{Z}}^{\mathcal{A}'}$ and $\bar{\mathcal{Y}}_{\mathcal{A}'}$ supertwistors transform in the (anti)fundamental representations of two $SL(4, \mathbb{R})_{R}$ groups.

<footnote n="7">⁷Detailed discussion of the reality conditions of the Berkovits twistor-string Lagrangian on the Lorentzian and Euclidean world sheets as well as real structures in the complex twistor space associated with $D = 4$ space-times of various signatures can be found, e.g. in [72].</footnote>
Lagrangian (5.4) is invariant under global $GL(4|4, \mathbb{R})_L \times GL(4|4, \mathbb{R})_R$ symmetry. Besides transformations from four $SL(4, \mathbb{R})$ subgroups and supersymmetry transformations it includes independent $GL(1, \mathbb{R})$ rescalings of the left- and right-moving supertwistor fields

$$ \delta_{\Lambda} \mathcal{Z}^{A} = \Lambda \mathcal{Z}^{A}, \quad \delta_{\Lambda} \mathcal{Y}_{\Lambda} = -\Lambda \mathcal{Y}_{\Lambda}, \quad \delta_{\bar{\Lambda}} \bar{\mathcal{Z}}^{A'} = \bar{\Lambda} \bar{\mathcal{Z}}^{A'}, \quad \delta_{\bar{\Lambda}} \bar{\mathcal{Y}}_{A'} = -\bar{\Lambda} \bar{\mathcal{Y}}_{A'}. \qquad (5.7) $$

So supertwistors actually parameterize two projective supertwistor spaces $\mathbb{RP}^{3|4}$. Two other scaling symmetries we name 'twisted'

$$ \begin{aligned} \delta_{\Lambda_t} Z^{\alpha} &= \Lambda_t Z^{\alpha}, \quad \delta_{\Lambda_t} \eta^A = -\Lambda_t \eta^A, \quad \delta_{\Lambda_t} Y_{\alpha} = -\Lambda_t Y_{\alpha}, \quad \delta_{\Lambda_t} \xi_A = \Lambda_t \xi_A, \\ \delta_{\bar{\Lambda}_t} \bar{Z}^{\alpha'} &= \bar{\Lambda}_t \bar{Z}^{\alpha'}, \quad \delta_{\bar{\Lambda}_t} \bar{\eta}^A = -\bar{\Lambda}_t \bar{\eta}^A, \quad \delta_{\bar{\Lambda}_t} \bar{Y}_{\alpha} = -\bar{\Lambda}_t \bar{Y}_{\alpha}, \quad \delta_{\bar{\Lambda}_t} \bar{\xi}_{A'} = \bar{\Lambda}_t \bar{\xi}_{A'} \end{aligned} \qquad (5.8) $$

since bosonic and fermionic components of supertwistors and dual supertwistors transform with opposite signs.

In the Berkovits model $GL(1, \mathbb{R})_L \times GL(1, \mathbb{R})_R$ symmetry is gauged by adding to the Lagrangian (5.4) constraints $T = \mathcal{Y}_A \mathcal{Z}^A \approx 0$ and $\bar{T} = \bar{\mathcal{Y}}_{A'} \bar{\mathcal{Z}}^{A'} \approx 0$ with the Lagrange multipliers

$$ \mathcal{L}_{GL(1, \mathbb{R})_L \times GL(1, \mathbb{R})_R} = \lambda T + \bar{\lambda} \bar{T}. \qquad (5.9) $$

On the quantum level this necessitates adding two units to the central charges of the matter fields to compensate contributions of the $(b, c)$-ghosts and ghosts for this gauge symmetry.

Let us note that in the open string sector one equates left- and right-moving fields on the world-sheet boundary $\partial\Sigma = \{\sigma = 0, \pi\}$: $\mathcal{Z}^A|_{\partial\Sigma} = \bar{\mathcal{Z}}^{A'}|_{\partial\Sigma}, \mathcal{Y}_A|_{\partial\Sigma} = \bar{\mathcal{Y}}_{A'}|_{\partial\Sigma}$. This leads to identification of the left and right $GL(4|4, \mathbb{R})$ supergroups and the constraints in (5.4) and (5.9). As the consequence of the introduction of local $GL(1, \mathbb{R})$ symmetry vertex operators are required to be functions of the supertwistors of homogeneity degree zero. This reduces the set of physical states to the multiplets of $\mathcal{N} = 4$ supersymmetric Yang-Mills theory and conformal supergravity [62]. In the extended model (5.3) there can exist states corresponding to all supermultiplets of $D = 4$ $\mathcal{N} = 4$ Poincaré supersymmetry and superconformal symmetry.

Let us focus on the sector of left-movers with the Lagrangian (5.5). Definition of momenta conjugate to supertwistors yields the second-class constraints. So introducing the Dirac brackets we obtain the following non-zero relations between the components of supertwistor and the dual supertwistor

$$ \{\mathcal{Z}^A(\sigma), \mathcal{Y}_B(\sigma')\}_{D.B.} = \delta_B^A \delta(\sigma - \sigma') \qquad (5.10) $$

that are graded-antisymmetric

$$ \{\mathcal{Y}_B(\sigma), \mathcal{Z}^A(\sigma')\}_{D.B.} = -(-)^a \delta_B^A \delta(\sigma - \sigma'). $$

In the component form they read

$$ \{Z^{\alpha}(\sigma), Y_{\beta}(\sigma')\}_{D.B.} = \delta_{\beta}^{\alpha} \delta(\sigma - \sigma'), \quad \{\eta^A(\sigma), \xi_B(\sigma')\}_{D.B.} = \delta_B^A \delta(\sigma - \sigma'). $$
Analogous relations also hold for the right-movers described by the Lagrangian (5.6).

In our work [19] it was shown that global symmetry of the Lagrangian (5.5) is infinite-dimensional. Variations of the supertwistors follow from their Dirac brackets relations (5.10) with the generating functional

$$G = \int d\sigma \sum_{L \ge 0} G^{(L)}(\sigma), \quad G^{(L)}(\sigma) = \mathcal{Y}_B(\sigma) \Lambda^{\mathcal{B}}_{A_L \dots A_1} \mathcal{Z}^{A_1}(\sigma) \dots \mathcal{Z}^{A_L}(\sigma), \qquad (5.11)$$

where the parameters $\Lambda^{\mathcal{B}}_{A_L \dots A_1}$ are graded-symmetric in the lower indices. Grassmann parity of their components is determined by the sum of parities of the indices assuming that indices of bosonic components of supertwistors have parity zero and indices of the fermionic components have parity unity. So that the global symmetry includes both bosonic higher-spin symmetries and higher-spin supersymmetries. For fixed value of L transformation rules for supertwistors are the following

$$\begin{aligned} \delta_{\Lambda_L} \mathcal{Z}^{\mathcal{A}}(\sigma) &= \Lambda^{\mathcal{A}}_{\mathcal{B}(L)} \mathcal{Z}^{\mathcal{B}(L)}(\sigma), \\ \delta_{\Lambda_L} \mathcal{Y}_{\mathcal{A}}(\sigma) &= -L \mathcal{Y}_{\mathcal{C}}(\sigma) \Lambda^{\mathcal{C}}_{\mathcal{A}\mathcal{B}(L-1)} \mathcal{Z}^{\mathcal{B}(L-1)}(\sigma). \end{aligned} \qquad (5.12)$$

Associated Noether current densities up to a numerical factor are given by the monomials

$$T^{(L)}_{\mathcal{B}}{}^{A(L)}(\sigma) = \mathcal{Y}_{\mathcal{B}} \mathcal{Z}^{A(L)}, \quad L \ge 0. \qquad (5.13)$$

In (5.12) and (5.13) introduced was a concise notation for the product of supertwistors $\mathcal{Z}^{A(L)} = \mathcal{Z}^{A_1} \dots \mathcal{Z}^{A_L}$ that is graded-symmetric. It is in general assumed graded symmetry of the sets of PSL(4|4, $\mathbb{R}$) indices denoted by the same letters.<fnref n="8" /> Similar concise notation is used for the products of bosonic and fermionic components of supertwistors $Z^{\alpha(L)} = Z^{\alpha_1} \dots Z^{\alpha_L}$ and $\eta^{A[M]} = \eta^{A_1} \dots \eta^{A_M}$ ($M \le \mathcal{N} = 4$) that are (anti)symmetric. On the Dirac brackets Noether current densities (5.13) generate infinite-dimensional superalgebra that in Ref. [19] was named the twistor-string superalgebra

$$\begin{aligned} & \{T^{(L)}_{\mathcal{B}}{}^{A(L)}(\sigma), T^{(M)}_{\mathcal{D}}{}^{C(M)}(\sigma')\}_{\mathrm{D.B.}} \\ & = (\delta^{A(1)}_{\mathcal{D}} T^{(L+M-1)}_{\mathcal{B}}{}^{A(L-1)C(M)} - \delta^{C(1)}_{\mathcal{B}} T^{(L+M-1)}_{\mathcal{D}}{}^{A(L)C(M-1)}) \delta(\sigma - \sigma'). \end{aligned} \qquad (5.14)$$

Finite-dimensional subalgebra of the twistor-string algebra is spanned, apart from the order-zero generator $\mathcal{Y}_{\mathcal{A}}(\sigma)$, that generates constant shifts of the supertwistor components, by the quadratic monomial

$$T_{\mathcal{A}}^{\mathcal{B}}(\sigma) = \mathcal{Y}_{\mathcal{A}} \mathcal{Z}^{\mathcal{B}}. \qquad (5.15)$$

Dirac brackets of monomials (5.15) generate the $gl(4|4, \mathbb{R})$ superalgebra relations

$$\{T_{\mathcal{A}}^{\mathcal{B}}(\sigma), T_{\mathcal{C}}^{\mathcal{D}}(\sigma')\}_{\mathrm{D.B.}} = (\delta_{\mathcal{C}}^{\mathcal{B}} T_{\mathcal{A}}^{\mathcal{D}} - (-)^{\varepsilon_a^b \varepsilon_c^d} \delta_{\mathcal{A}}^{\mathcal{D}} T_{\mathcal{C}}^{\mathcal{B}})(\sigma) \delta(\sigma - \sigma'), \varepsilon_a^b = a+b. \qquad (5.16)$$

<footnote n="8">⁸ Subscript L in the notation of the symmetry groups and algebras will be omitted in what follows as the discussion is concentrated on the sector of left-movers only.</footnote>
$gl(4|4, \mathbb{R})$ current density (5.15) decomposes on the following irreducible components

$$
\begin{align}
T_A^B(\sigma) &= \{\tilde{T}_\alpha^\beta, \tilde{T}_A^B; Q_\alpha^B, Q_A^\beta; T, U\} : \nonumber \\
\tilde{T}_\alpha^\beta &= Y_\alpha Z^\beta - \frac{1}{4}\delta_\alpha^\beta(YZ), \quad \tilde{T}_A^B = \xi_A \eta^A - \frac{1}{4}\delta_A^B(\xi\eta); \tag{5.17} \\
Q_\alpha^B &= Y_\alpha \eta^B, \quad Q_A^\beta = \xi_A Z^\beta; \nonumber \\
T &= \mathrm{Tr} T_A^B = Y_\alpha Z^\alpha + \xi_A \eta^A, \quad U = s\mathrm{Tr} T_A^B = Y_\alpha Z^\alpha - \xi_A \eta^A, \nonumber
\end{align}
$$

where tilde over the current densities denotes their tracelessness. The densities of two sl(4, R) currents T̃_αβ(σ) and T̃_A^B(σ) and those of the supersymmetry currents Q_α^B(σ) and Q_A^β(σ) span psl(4|4, R) superalgebra. Together with T(σ) they span sl(4|4, R) superalgebra. On the Dirac brackets current densities T̃_αβ(σ) and T̃_A^B(σ) generate infinitesimal rotations of the supertwistor and dual supertwistor components with parameters Ξ_α^β and Υ_A^B

$$
\begin{align*}
\delta_{\Xi} Z^{\alpha}(\sigma) &= \Xi^{\alpha}_{\beta} Z^{\beta}(\sigma), & \delta_{\Xi} Y_{\alpha}(\sigma) &= -Y_{\beta}(\sigma) \Xi^{\beta}_{\alpha}, & \Xi^{\alpha}_{\alpha} &= 0; \\
\delta_{\Upsilon} \eta^{A}(\sigma) &= \Upsilon^{A}_{B} \eta^{B}(\sigma), & \delta_{\Upsilon} \xi_{A}(\sigma) &= -\xi_{B}(\sigma) \Upsilon^{B}_{A}, & \Upsilon^{A}_{A} &= 0.
\end{align*}
$$

Current densities $Q_{\alpha}^{B}(\sigma)$ and $Q_{A}^{\beta}(\sigma)$ are generators of the supersymmetry transformations realized on the supertwistor components

$$
\begin{align*}
\delta_{\epsilon} Z^{\alpha}(\sigma) &= \epsilon^{\alpha}_{A} \eta^{A}(\sigma), \quad \delta_{\epsilon} \xi_{A}(\sigma) = -Y_{\alpha}(\sigma) \epsilon^{\alpha}_{A}; \\
\delta_{\epsilon} Y_{\alpha}(\sigma) &= -\xi_{A}(\sigma) \epsilon^{A}_{\alpha}, \quad \delta_{\epsilon} \eta^{A}(\sigma) = \epsilon^{A}_{\alpha} Z^{\alpha}(\sigma),
\end{align*}
$$

where $\epsilon^\alpha_A$ and $\epsilon^A_\alpha$ are independent Grassmann-odd parameters with 16 real components each. Current densities $T(\sigma)$ and $U(\sigma)$ generate $GL(1, \mathbb{R})$ and $GL_t(1, \mathbb{R})$ transformations of components of the supertwistors given in (5.7) and (5.8).

$gl(4|4, \mathbb{R})$ superalgebra relations (5.16) can be presented in the component form

$$
\begin{align}
\{\tilde{T}_{\alpha}^{\beta}(\sigma), \tilde{T}_{\gamma}^{\delta}(\sigma')\}_{\text{D.B.}} &= (\delta_{\gamma}^{\beta}\tilde{T}_{\alpha}^{\delta} - \delta_{\alpha}^{\delta}\tilde{T}_{\gamma}^{\beta})(\sigma)\delta(\sigma - \sigma'), \nonumber \\
\{\tilde{T}_{A}^{B}(\sigma), \tilde{T}_{C}^{D}(\sigma')\}_{\text{D.B.}} &= (\delta_{C}^{B}\tilde{T}_{A}^{D} - \delta_{A}^{D}\tilde{T}_{C}^{B})(\sigma)\delta(\sigma - \sigma'), \nonumber \\
\{Q_{\alpha}^{B}(\sigma), Q_{C}^{\delta}(\sigma')\}_{\text{D.B.}} &= (\delta_{C}^{B}\tilde{T}_{\alpha}^{\delta} + \delta_{\alpha}^{\delta}\tilde{T}_{C}^{B} + \frac{1}{4}\delta_{\alpha}^{\delta}\delta_{C}^{B}T)(\sigma)\delta(\sigma - \sigma'), \nonumber \\
\{\tilde{T}_{\alpha}^{\beta}(\sigma), Q_{\gamma}^{D}(\sigma')\}_{\text{D.B.}} &= (\delta_{\gamma}^{\beta}Q_{\alpha}^{D} - \frac{1}{4}\delta_{\alpha}^{\beta}Q_{\gamma}^{D})(\sigma)\delta(\sigma - \sigma'), \tag{5.18} \\
\{\tilde{T}_{\alpha}^{\beta}(\sigma), Q_{C}^{\delta}(\sigma')\}_{\text{D.B.}} &= -(\delta_{\alpha}^{\delta}Q_{C}^{\beta} - \frac{1}{4}\delta_{\alpha}^{\beta}Q_{C}^{\delta})(\sigma)\delta(\sigma - \sigma'), \nonumber \\
\{\tilde{T}_{A}^{B}(\sigma), Q_{\gamma}^{D}(\sigma')\}_{\text{D.B.}} &= -(\delta_{A}^{D}Q_{\gamma}^{B} - \frac{1}{4}\delta_{A}^{B}Q_{\gamma}^{D})(\sigma)\delta(\sigma - \sigma'), \nonumber \\
\{\tilde{T}_{A}^{B}(\sigma), Q_{C}^{\delta}(\sigma')\}_{\text{D.B.}} &= (\delta_{C}^{B}Q_{A}^{\delta} - \frac{1}{4}\delta_{A}^{B}Q_{C}^{\delta})(\sigma)\delta(\sigma - \sigma'), \nonumber \\
\{U(\sigma), Q_{\alpha}^{B}(\sigma')\}_{\text{D.B.}} &= 2Q_{\alpha}^{B}(\sigma)\delta(\sigma - \sigma'), \nonumber \\
\{U(\sigma), Q_{A}^{\beta}(\sigma')\}_{\text{D.B.}} &= -2Q_{A}^{\beta}(\sigma)\delta(\sigma - \sigma'). \nonumber
\end{align}
$$

Note that $T(\sigma)$ commutes on the Dirac brackets with all other $gl(4|4, \mathbb{R})$ current densities thus forming an Abelian ideal. The density of the 'twisted' $gl_t(1, \mathbb{R})$ current $U(\sigma)$ does not enter the r.h.s. of the relations (5.18). So the $gl(4|4, \mathbb{R})$ superalgebra has the structure of the semidirect sum of $sl(4|4, \mathbb{R})$ and $gl_t(1, \mathbb{R})$.
## 5.2 Quantum higher-spin superconformal symmetries of twistor strings

It was shown in [68] that $SL(4|4, \mathbb{R})$ symmetry is preserved in the twistor-string model at the quantum level, whereas the generator of the 'twisted' $GL_t(1, \mathbb{R})$ symmetry $U$ has anomalous operator-product expansion with the world-sheet stress-energy tensor. This implies breaking of the corresponding symmetry. Thus infinite-dimensional symmetry that could exist in the quantum theory has to include $sl(4|4, \mathbb{R})$ as finite-dimensional subalgebra.

### 5.2.1 Analysis of the symmetries on the level of superalgebra

Identification of the possible set of generators for such $sl$-type superalgebra it is convenient to start from consideration of those generators of the twistor-string algebra that are irreducible $sl(4, \mathbb{R})$ tensors composed of the bosonic components of supertwistors. Among the level-one generators (5.17) these are $\tilde{T}_{\alpha}^{\beta}$. Respective level-$L$ ($L > 1$) generators have the form

$$ \tilde{T}_{\alpha}^{\beta(L)} = Y_{\alpha} Z^{\beta(L)} - \frac{1}{L+3} (YZ) \delta_{\alpha}^{\beta(1)} Z^{\beta(L-1)}. \qquad (5.19) $$

From the relations (5.14) one infers that the Dirac brackets of the generators of level $L$ and $M$ of the twistor-string superalgebra close on the generators of the level $L+M-1$. In particular, the Dirac brackets of the level-one generators (5.17) and level-$L$ generators yield level-$L$ generators. These Dirac brackets determine transformation properties of the higher-level generators under two $sl(4, \mathbb{R})$ symmetries and 32 supersymmetries from the $psl(4|4, \mathbb{R})$ superalgebra. This feature of the level-one generators can be used to obtain expressions for irreducible level-$L$ generators by successive calculation of the Dirac brackets of the bosonic generators (5.19) with the supersymmetry generators $Q_{\alpha}^{\beta}$ and $Q_{\alpha}^{B}$.

Application of this procedure to level-two generators $\tilde{T}_{\gamma}^{\delta(2)}(\sigma)$

$$ \{Q_{\alpha}^{\beta}(\sigma), \tilde{T}_{\gamma}^{\delta(2)}(\sigma')\}_{\text{D.B.}} = (\delta_{\gamma}^{\beta} Q_{\alpha}^{\delta(2)} - \frac{1}{5} \delta_{\gamma}^{(\delta_1} Q_{\alpha}^{\delta_2)\beta)}(\sigma)\delta(\sigma - \sigma'), \qquad (5.20) $$

$$ \{Q_{\alpha}^{B}(\sigma), \tilde{T}_{\gamma}^{\delta(2)}(\sigma')\}_{\text{D.B.}} = -\left(\delta_{\alpha}^{(\delta_1} \tilde{Q}_{\gamma}^{\delta_2)B} - \frac{1}{5} \delta_{\gamma}^{(\delta_1} \tilde{Q}_{\alpha}^{\delta_2)B}\right)(\sigma)\delta(\sigma - \sigma') \qquad (5.21) $$

allows to define level-two supersymmetry generators

$$ Q_{\alpha}^{\delta(2)} = \xi_{\alpha} Z^{\delta(2)}, \quad \tilde{Q}_{\gamma}^{\delta A} = \tilde{T}_{\gamma}^{\delta} \eta^{A}. \qquad (5.22) $$

From the expressions for non-zero Dirac brackets of these generators with $Q_{\alpha}^{B}$ and $Q_{\alpha}^{\beta}$ one can determine other bosonic level-two generators

$$ \{Q_{\alpha}^{B}(\sigma), Q_{C}^{\delta(2)}(\sigma')\}_{\text{D.B.}} = \delta_{C}^{B} \tilde{T}_{\alpha}^{\delta(2)}(\sigma)\delta(\sigma - \sigma') + \delta_{\alpha}^{(\delta_1} (\tilde{T}_{C}^{\delta_2)B} + \delta_{C}^{B} (\frac{9}{40}T - \frac{1}{40}U) Z^{\delta_2}) (\sigma)\delta(\sigma - \sigma'), \qquad (5.23) $$

$$ \{Q_{\alpha}^{\beta}(\sigma), \tilde{Q}_{\gamma}^{\delta D}(\sigma')\}_{\text{D.B.}} = (\delta_{\gamma}^{\beta} \tilde{T}_{A}^{\delta D} - \frac{1}{4} \delta_{\gamma}^{\delta} \tilde{T}_{A}^{\beta D})(\sigma)\delta(\sigma - \sigma') + \delta_{A}^{D} \left[ \tilde{T}_{\gamma}^{\beta \delta} + \left(\frac{9}{40}T - \frac{1}{40}U\right) \left(\delta_{\gamma}^{\beta} Z^{\delta} - \frac{1}{4} \delta_{\gamma}^{\delta} Z^{\beta}\right) \right](\sigma)\delta(\sigma - \sigma'), \qquad (5.24) $$
where

$$ \tilde{T}_A^{\beta B} = \tilde{T}_A^B Z^{\beta}. \qquad (5.25) $$

Proceeding further one finds the complete set of irreducible level-two generators

$$ \begin{gathered} \tilde{T}_{\alpha}^{\beta(2)}, \quad \tilde{T}_{A}^{\alpha B}, \quad T_{\alpha}^{B[2]} = Y_{\alpha} \eta^{B[2]}; \\ Q_{A}^{\alpha(2)}, \quad \tilde{Q}_{\alpha}^{\beta B}, \quad \tilde{Q}_{A}^{B[2]} = \xi_{A} \eta^{B[2]} - \frac{1}{3}(\xi \eta) \delta_{A}^{[B_1}] \eta^{B_2} \end{gathered} \qquad (5.26) $$

and

$$ TZ^{\alpha}, \quad UZ^{\alpha}, \quad T\eta^{A}, \quad U\eta^{A}. \qquad (5.27) $$

The operators associated with the generators (5.27), as will be shown below, are not the primary fields in the world-sheet conformal field theory and hence corresponding symmetries are broken on the quantum level. Since these generators appear on the r.h.s. of relations (5.23) and (5.24) this implies breaking of the level-two supersymmetries generated by $Q_A^{\alpha(2)}$ and $\tilde{Q}_\alpha^{\beta B}$. Accordingly from the relations (5.20) and (5.21) it follows that bosonic symmetry generated by $\tilde{T}_\gamma^{\delta(2)}$ also gets broken. So that classical level-two symmetries break in the quantum theory.

Analogous calculation of the Dirac bracket relations of the level-$L$ ($L > 2$) bosonic generators (5.19) and level-one supersymmetry generators

$$ \begin{gathered} \{Q_A^{\beta}(\sigma), \tilde{T}_{\gamma}^{\delta(L)}(\sigma')\}_{\text{D.B.}} \\ = \left(\delta_{\gamma}^{\beta} Q_A^{\delta(L)} - \frac{1}{L+3} \delta_{\gamma}^{(1)} Q_A^{\delta(L-1)\beta}\right)(\sigma)\delta(\sigma - \sigma'), \end{gathered} \qquad (5.28) $$

$$ \begin{gathered} \{Q_{\alpha}^{B}(\sigma), \tilde{T}_{\gamma}^{\delta(L)}(\sigma')\}_{\text{D.B.}} \\ = - \left( \delta_{\alpha}^{\delta(1)} \tilde{Q}_{\gamma}^{\delta(L-1)B} - \frac{1}{L+3} \delta_{\gamma}^{\delta(1)} \tilde{Q}_{\alpha}^{\delta(L-1)B} \right) (\sigma) \delta(\sigma - \sigma') \end{gathered} \qquad (5.29) $$

yields level-$L$ supersymmetry generators

$$ Q_A^{\delta(L)} = \xi_A Z^{\delta(L)}, \quad \tilde{Q}_\gamma^{\delta(L-1)B} = \tilde{T}_\gamma^{\delta(L-1)} \eta^B. \qquad (5.30) $$

In its turn expressions for other bosonic generators are determined by the Dirac bracket relations of the generators (5.30) with the level-one supersymmetry generators

$$ \begin{gathered} \{Q_{\alpha}^{B}(\sigma), Q_{C}^{\delta(L)}(\sigma')\}_{\text{D.B.}} = \delta_{C}^{B} \tilde{T}_{\alpha}^{\delta(L)}(\sigma) \delta(\sigma - \sigma') \\ + \delta_{\alpha}^{\delta(1)} \left[ \tilde{T}_{C}^{\delta(L-1)B} + \delta_{C}^{B} \left( \frac{L+7}{8(L+3)} T - \frac{L-1}{8(L+3)} U \right) Z^{\delta(L-1)} \right] (\sigma) \delta(\sigma - \sigma') \end{gathered} \qquad (5.31) $$

and

$$ \begin{aligned} \{Q_A^{\beta}(\sigma), \tilde{Q}_{\gamma}^{\delta(L-1)D}(\sigma')\}_{\text{D.B.}} = & \left( \delta_{\gamma}^{\beta} \tilde{T}_A^{\delta(L-1)l} - \frac{1}{L+2} \delta_{\gamma}^{\delta(1)} \tilde{T}_A^{\beta \delta(L-2)D} \right) (\sigma) \delta(\sigma - \sigma') \\ & + \delta_A^D \left[ \tilde{T}_\gamma^{\beta \delta(L-1)} + \left( \frac{L+7}{8(L+3)} T - \frac{L-1}{8(L+3)} U \right) \right. \\ & \left. \times \left( \delta_\gamma^\beta Z^{\delta(L-1)} - \frac{1}{L+2} \delta_\gamma^{\delta(1)} Z^\beta Z^{\delta(L-2)} \right) \right] (\sigma) \delta(\sigma - \sigma'), \end{aligned} \qquad (5.32) $$

where

$$ \tilde{T}_C^{\delta(L-1)B} = \tilde{T}_C^B Z^{\delta(L-1)} \qquad (5.33) $$
and so on. Resulting complete set of irreducible level-L generators includes the following bosonic

$$ \begin{aligned} \tilde{T}_{\alpha}^{\beta(p)B[q]} &= \tilde{T}_{\alpha}^{\beta(p)} \eta^{B[q]}, \quad q = 0, 2, 4, \quad p+q = L; \\ \tilde{T}_{A}^{\beta(p)B[q]} &= \tilde{T}_{A}^{B[q]} Z^{\beta(p)}, \quad q = 1, 3, \quad p+q = L \end{aligned} \qquad (5.34) $$

and fermionic generators

$$ \begin{aligned} \tilde{Q}_{\alpha}^{\beta(p)B[q]} &= \tilde{T}_{\alpha}^{\beta(p)} \eta^{B[q]}, \quad q = 1, 3, \quad p+q = L; \\ \tilde{Q}_{A}^{\beta(p)B[q]} &= \tilde{Q}_{A}^{B[q]} Z^{\beta(p)}, \quad q = 0, 2, 4, \quad p+q = L. \end{aligned} \qquad (5.35) $$

Relevant (traceless) products of the bosonic components of supertwistors are defined in (5.19) and the definition of the (traceless) products of fermionic components is given in (5.17), (5.26) and by the expressions

$$ \begin{aligned} \tilde{T}_{A}^{B[3]} &= \xi_{A} \eta^{B[3]} - \frac{1}{2} (\xi \eta) \delta_{A}^{[B_{1}} \eta^{B_{2}} \eta^{B_{3}]}, \\ Q_{A} &= \xi_{A}, \quad \tilde{Q}_{A}^{B[4]} = \xi_{A} \eta^{B[4]} - (\xi \eta) \delta_{A}^{[B_{1}} \eta^{B_{2}} \eta^{B_{3}} \eta^{B_{4}]}. \end{aligned} \qquad (5.36) $$

In this set there are also the generators

$$ TZ^{\alpha(p)} \eta^{A[q]}, \quad UZ^{\alpha(p)} \eta^{A[q]}, \quad 0 \le q \le 4, \quad p+q = L-1 \qquad (5.37) $$

to which, as will be demonstrated in the next paragraph, do not correspond primary operators in the world-sheet conformal field theory. They arise on the r.h.s. of the relations (5.31) and (5.32) and cause breaking of the level-L symmetries in analogy with those of level-two symmetries.

In the Berkovits twistor string model $GL(1, \mathbb{R})$ symmetry is the gauge one so that generators carrying the factor of $T$ need to be set to zero. To exclude other generators, to which do not correspond primary fields, it would be necessary to also consider 'twisted' $GL_t(1, \mathbb{R})$ symmetry as a gauge one. However, to the generator of this symmetry $U$ also does not correspond a primary field and therefore generators having the factor of $U$ cannot be set to zero. So we conclude that on any level $L > 1$ it is not possible to find a set of the generators with closed Dirac bracket relations to which there would correspond primary fields in the world-sheet conformal field theory. As a result on the quantum level consistent global symmetry of the Lagrangian (5.4) for left- and right-moving supertwistors is the $SL(4|4, \mathbb{R}) \times SL(4|4, \mathbb{R})$ supergroup in the sector of closed strings and $SL(4|4, \mathbb{R})$ in the sector of open strings. Accordingly quantum global symmetry of the Berkovits model is $PSL(4|4, \mathbb{R}) \times PSL(4|4, \mathbb{R})$ or $PSL(4|4, \mathbb{R})$.

### 5.2.2 Properties of the generators of higher-spin symmetries as operators in the world-sheet conformal field theory

This paragraph is devoted to study of the operators in the world-sheet conformal field theory with the Lagrangian (5.4) that correspond to the classical generators of twistor-string algebra. It is aimed to prove the statements used in the previous paragraph.
To apply the technique of the two-dimensional conformal field theory Lagrangian (5.4) should be defined on the Euclidian signature world-sheet. To this end Wick rotation needs to be performed

$$ \tau \to i\sigma^2, \sigma \to \sigma^1 \Rightarrow \sigma^+ \to z = \sigma^1 + i\sigma^2, \sigma^- \to -\bar{z} = -(\sigma^1 - i\sigma^2). $$

This results in the following changes of the world-sheet derivatives

$$ \partial_+ \to \partial_z = \frac{1}{2}(\partial_1 - i\partial_2) \equiv \partial, \quad \partial_- \to -\partial_{\bar{z}} = -\frac{1}{2}(\partial_1 + i\partial_2) \equiv -\partial_{\bar{z}}, $$

the world-sheet area element

$$ d\tau d\sigma \to i d\sigma^1 d\sigma^2 = \frac{i}{2} d^2 z $$

and supertwistors

$$ \mathcal{Y}_A(\tau, \sigma) \to \mathcal{Y}_A(z, \bar{z}), \quad \bar{\mathcal{Y}}_A(\tau, \sigma) \to -\bar{\mathcal{Y}}_A(z, \bar{z}). $$

So that the Euclidean action for supertwistor fields takes the form

$$ S_E = \int_{\Sigma_E} d^2 z (\mathcal{Y}_A \bar{\partial} \mathcal{Z}^A + \bar{\mathcal{Y}}_A \partial \bar{\mathcal{Z}}^A). \qquad (5.38) $$

Two-dimensional conformal field theory defined by the action on the Euclidean world-sheet (5.38) consists of the conformal theory for bosonic components of supertwistors that belongs to the family of $\beta\gamma$ conformal field theories and conformal theory for fermionic components of supertwistors from the family of bc conformal field theories (see, e.g. [73]). From the expression for the action (5.38) one can determine the form of the singular terms in the expansion of the operator-product of supertwistors. In the left-moving (holomorphic) sector, on which we focus, they equal

$$ \mathcal{Z}^A(z) \mathcal{Y}_B(w) \sim \frac{\delta_B^A}{z-w}, \quad \mathcal{Y}_B(z) \mathcal{Z}^A(w) \sim -\frac{(-)^a \delta_B^A}{z-w}. $$

For the product of the supertwistor components singular terms have the form

$$ Z^{\alpha}(z) Y_{\beta}(w) \sim \frac{\delta_{\beta}^{\alpha}}{z-w}, \quad \eta^{A}(z) \xi_{B}(w) \sim \frac{\delta_{B}^{A}}{z-w}. $$

By definition (see, e.g. [73]) primary field $\mathcal{O}(z)$ is characterized by the following general form of the operator-product expansion with the world-sheet stress-energy tensor $L(z)$

$$ L(z) \mathcal{O}(w) \sim \frac{h}{(z-w)^2} \mathcal{O}(w) + \frac{1}{z-w} \partial \mathcal{O}(w), \qquad (5.39) $$

where $h$ is its conformal weight.<fnref n="9" /> The holomorphic stress-energy tensor in the theory for two free supertwistors (5.38) has the form

$$ L_{\text{tw}}(z) = -\mathcal{Y}_A \partial \mathcal{Z}^A, $$

<footnote n="9">⁹It is assumed that local composite operators are normal-ordered but normal ordering signs `::` will be omitted.</footnote>
where $\mathcal{Y}_B$ and $\mathcal{Z}^A$ are primary fields of conformal weight unity and zero respectively.

From the perspective of the world-sheet conformal field theory the requirement of consistency on the quantum level for the considered in the previous subparagraphs global symmetries is the correspondence of the primary fields to their generators, i.e. their operator-product expansions with the stress-energy tensor need to be anomaly free (see, e.g. [73]). In other words, on the r.h.s. of the products like (5.39) there should not appear terms with poles of the order higher than two. As will be shown the generators containing the factors of $T$ or $U$ do not satisfy this requirement.

Calculating singular terms in the operator expansion of the product of the stress-energy tensor with the densities of $gl(4, \mathbb{R})$ currents

$$Y_{\gamma} \partial Z^{\gamma}(z) Y_{\beta} Z^{\alpha}(w) \sim \frac{\delta_{\beta}^{\alpha}}{(z-w)^3} - \frac{1}{(z-w)^2} Y_{\beta} Z^{\alpha}(w) - \frac{1}{(z-w)} \partial(Y_{\beta} Z^{\alpha})(w)$$

and

$$\xi_C \partial \eta^C(z) \xi_B \eta^A(w) \sim - \frac{\delta_B^A}{(z-w)^3} - \frac{1}{(z-w)^2} \xi_B \eta^A(w) - \frac{1}{z-w} \partial(\xi_B \eta^A)(w),$$

one infers that quantum $sl(4|4, \mathbb{R})$ generators $\tilde{T}_{\alpha}^{\beta}$, $\tilde{T}_{A}^{B}$, $Q_{\alpha}^{B}$, $Q_{A}^{\beta}$ and $T$ are indeed the primary fields of unit weight, while operator $U$ is not a primary field [68]

$$L_{\text{tw}}(z)U(w) \sim \frac{-8}{(z-w)^3} + \frac{1}{(z-w)^2}U(w) + \frac{1}{z-w}\partial U(w).$$

To the higher-level generators (5.34) and (5.35) there also correspond primary fields of unit weight. At the same time the operator-products of the generators (5.37) with the stress-energy tensor include anomalous contributions

$$L_{\text{tw}}(z)TZ^{\alpha(p)}\eta^{A[q]}(w) \sim -\frac{p+q}{(z-w)^3}Z^{\alpha(p)}\eta^{A[q]}(w) + \mathcal{O}((z-w)^{-2}),$$

$$L_{\text{tw}}(z)UZ^{\alpha(p)}\eta^{A[q]}(w) \sim -\frac{8+p-q}{(z-w)^3}Z^{\alpha(p)}\eta^{A[q]}(w) + \mathcal{O}((z-w)^{-2}).$$

When $p = q = 0$ these operator-product expansions coincide with the operator-products of the $gl(1, \mathbb{R})$ and $gl_t(1, \mathbb{R})$ generators with the stress-energy tensor. For $p \neq 0$, $q \neq 0$ both anomalous terms do not vanish so that associated symmetries are broken. Since generators (5.34) and (5.35) are linked with other level-$L$ generators by level-one supersymmetries (cf. Eqs. (5.28)-(5.32)) higher-spin symmetries are broken for all values of $L > 1$. For $L = 1$ consistent on the quantum level global symmetry is isomorphic to $SL(4|4, \mathbb{R})$.

# 6 Conclusion

Present paper reviewed the following results obtained by the author:

- construction of (super)twistor formulations of the Lagrangians of bosonic string and $N = 1$ superstring in $D = 4$ dimensions. It was shown that Lagrangians of tensile superstrings in the supertwistor formulation are non-linear and include pure gauge fermionic degrees of freedom unlike the supertwistor formulations of massless superparticle and tensionless superstring models;
- construction of reduced supertwistor models corresponding to the $\mathcal{N} = 1$ and $\mathcal{N} = 2$ superstrings that generalize supertwistor formulations of the massless superparticle and tensionless superstring. Their Lagrangians are quadratic in the (super)twistor components. The reduced supertwistor model corresponding to the $\mathcal{N} = 2$ superstring includes as particular cases reduced supertwistor model for the $\mathcal{N} = 1$ superstring and the bosonic string model in the twistor formulation. The relation of the reduced models to the supertwistor formulations of superstrings was also discussed;

- analysis of the reduced supertwistor model for the $\mathcal{N} = 2$ superstring was as constrained Hamiltonian system. There were obtained Lorentz-covariant and irreducible first- and second-class constraints and identified the gauge symmetries generated by the first-class constraints;

- construction of the formulations of $D = 6$ and $D = 10$ $\mathcal{N} = 1$ superstrings in terms of the supertwistors that realize fundamental representations of minimal superconformal groups $OSp(8^*|2)$ and $OSp(32|1)$. There were determined the constraints on the supertwistor components that ensure their correspondence to the coordinates of $D = 6$ and $D = 10$ $\mathcal{N} = 1$ Minkowski superspaces. Obtained superstring Lagrangians in $D = 6$ and $D = 10$ dimensions are non-linear in the supertwistor components similarly to Lagrangians in the superspace formulation and invariant under the $\kappa$-symmetry transformations. In analogy to the superstrings in $D = 4$ dimensions proposed were reduced supertwistor models with Lagrangians quadratic in the supertwistor components;

- analysis of the $D = 10$ $\mathcal{N} = 1$ superstring model in the supertwistor formulation as a Hamiltonian system with constraints. There have been found expressions for the primary and secondary constraints in terms of the supertwistor components and their division on the irreducible and Lorentz-covariant sets of the first- and second-class constraints was carried out. To take into account the second-class constraints there have been introduced the Dirac brackets. It has been shown that the Dirac bracket relations of the first-class constraints to the leading order in the inverse superstring tension include quadratic summands in the first-class constraints. This reveals non-linearity of the algebra of the first-class constraints;

- unveiling of the infinite-dimensional global symmetries of the classical Lagrangians for left- and right-moving supertwistor fields in the Berkovits twistor string model and its extension to unconstrained supertwistors. Their finite-dimensional constituent is given by the $D = 4$ $\mathcal{N} = 4$ superconformal symmetry group;

- study of the expansions of the operator products of generators of the found higher-spin symmetries with the components of the stress-energy tensor corresponding to the world-sheet conformal invariance of the supertwistor Lagrangians. It was shown that they include anomalous terms. Therefore on the quantum level only the finite-dimensional part of the infinite-dimensional symmetry remains unbroken.
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