Towards the quantization of the non-relativistic D2-brane in the pure spinor formalism

Abstract

An attempt is made to apply the pure spinor formalism to the non-relativistic IIA D2-brane. The fermionic constraints corresponding to the rescaled fermionic coordinates are given. Two commuting spinor fields are introduced, each one corresponding to a fermionic constraint. A BRST charge is constructed via the ansatz proposed by Berkovits in (JHEP 04:018, arXiv:hep-th/0001035, 2000, JHEP 09:046, arXiv:hep-th/0006003, 2000, JHEP 09:016, arXiv:hep-th/0105050, 2001, ICTP Lect. Notes Ser. 13:57, arXiv:hep-th/0209059, 2003). The nilpotency of the BRST charge leads to a set of constraints for the two spinor fields including pure spinor constraints. A novel non-trivial solution is given for one of the spinor fields which can be written as a sum of two pure spinors.

1 Introduction

The pure spinor formalism introduced by Berkovits [1,2,3,4] is a successful attempt to solve the longstanding problem of finding a manifestly supersymmetric and covariant superstring formalism. The basic ingredient is the BRST-like operator $Q=\int dz \lambda ^{\alpha }d_{\alpha }$ where $d_{\alpha }$ is the fermionic constraint that appears in the conventional Green-Schwarz formalism and $\lambda _{\alpha }$ is a bosonic chiral spinor that plays the role of the associated “ghost”. For Q to be regarded as a BRST operator must be nilpotent and this leads to the relation $\lambda ^{\alpha }\gamma _{\alpha \beta }^{m}\lambda ^{\beta }=0$. This in 10 dimensions is the condition for $\lambda ^{\alpha }$ to be characterized as a pure spinor.

An important property of Q is that its cohomology correctly reproduces the spectrum of the superstring. The pure spinor formalism has been used as well for the covariant quantization of the superparticle [3] and also to study several aspects of string theory, for example the propagation of strings in curved backgrounds [5,6,7,8].

Another important application of the pure spinor formalism is the calculation of scattering amplitudes within the framework of superstring theory [9,10,11,12,13]. The manifest Lorentz covariance and spacetime supersymmetry make the calculation much easier than in other formalisms. Thus, pure spinors play a crucial role within string perturbation theory. However, within the context of D-branes, there are no non-trivial solutions reported in the literature. Moreover, so far it is not known whether this formalism can be consistently extended or generalized to be applied to non-relativistic systems with kappa symmetry, a fact that could allow us to draw some conclusions for the general case too, especially if there are problems in solving the pure spinor constraints in that case.

In this paper we try to extend the pure spinor formalism to the case of the non-relativistic IIA D2-brane. The non-relativistic limit of string theories [14, 15] give us a deeper understanding of string theories themselves. The non-relativistic limit of Dp branes has been studied in [16, 17]. It is important to note that in this limit the kappa symmetry is maintained and this allows us to treat non-relativistic Dp branes in the framework of the pure spinor formalism.

Here we present a novel non-trivial solution for the non-relativistic D2-brane within the pure spinor formalism. This fact could lead to the quantization of branes with interesting and relevant results.

Our starting point is the action of a IIA D2-brane in a flat 10d background. The fields consist of the 10d superspace coordinates ($x^m,\theta$) and an Abelian gauge field $A_\mu$ [18,19,20]:

$$\begin{aligned} S=-T\int d^{3}\sigma \sqrt{-\det (G_{\mu \nu }+{{{\mathcal {F}}}}_{\mu \nu }})+\int {{{\mathcal {L}}}}_{3}^{WZ}, \end{aligned}$$

(1.1)

where T is the string tension and the Wess-Zumino action reads

(1.2)

$G_{\mu \nu }=\eta _{mn}\Pi ^{m}_{\mu }\Pi ^{n}_{\nu }\$ and here $\ \Pi ^{m}_{\mu }=\frac{\partial X^{m}}{\partial \sigma ^{\mu }}-{\overline{\theta }}\Gamma ^{m}\frac{\partial \theta }{\partial \sigma ^{\mu }}.$

$$\begin{aligned} {{{\mathcal {F}}}}_{\mu \nu }= & {} \frac{\partial A_{\nu }}{\partial \sigma ^{\mu }}-\frac{\partial A_{\mu }}{\partial \sigma ^{\nu }}+\left( {\overline{\theta }}\Gamma _{11}\Gamma _{m}\partial _{\nu }\theta \right) \left( \Pi ^{m}_{\mu }+\frac{1}{2}{\overline{\theta }}\Gamma ^{m}\partial _{\mu }\theta \right) \nonumber \\&- \left( {\overline{\theta }}\Gamma _{11}\Gamma _{m}\partial _{\mu }\theta \right) \left( \Pi ^{m}_{\nu }+\frac{1}{2}{\overline{\theta }}\Gamma ^{m}\partial _{\nu }\theta \right) \end{aligned}$$

(1.3)

(1.4)

with

(1.5)

(1.6)

(1.7)

(1.8)

and $m,n=0,...,9/ \mu ,\nu =0,1,2.$

The action (1.1) has a global supersymmetry and also a local supersymmetry (kappa symmetry).

The action for the non-relativistic D2-brane is obtained from (1.1) by doing the following rescaling [16, 17]:

$$\begin{aligned}&X^{\mu }=\omega x^{\mu } \end{aligned}$$

(1.9)

$$\begin{aligned}&X^{a}=x^{a} \end{aligned}$$

(1.10)

$$\begin{aligned}&T=\frac{1}{\omega }T_{NR} \end{aligned}$$

(1.11)

$$\begin{aligned}&A_{i}=\omega w_{i} \end{aligned}$$

(1.12)

where the subindex $_{NR}$ stands for non-relativistic and $i,\mu =0,1,2/a=3,...,9.$

$$\begin{aligned} \theta =\sqrt{\omega }\theta _{-}+\frac{1}{\sqrt{\omega }}\theta _{+} \end{aligned}$$

(1.13)

where $\theta _{+},\theta _{-}$ are eigenstates of

$$\begin{aligned}&\Gamma _{\star }\equiv \Gamma _{0}\Gamma _{1}\Gamma _{2} \end{aligned}$$

(1.14)

$$\begin{aligned}&\Gamma _{\star }\theta _{\pm }=\pm \theta _{\pm } \end{aligned}$$

(1.15)

The action of the non-relativistic D2-brane is obtained by expanding (1.1) in powers of $\omega$ and keeping the finite part as $\omega \rightarrow \infty$ [17]

$$\begin{aligned} S_{NR}=\int d^{3}\sigma {{{\mathcal {L}}}}^{DBI}_{NR}+\int {{{\mathcal {L}}}}^{WZ}_{NR} \end{aligned}$$

(1.16)

where

$$\begin{aligned} \mathcal{L}^{DBI}_{NR}= & {} T_{NR}(\epsilon _{ijk}R^{0}_{i}R^{1}_{j}R^{2}_{k})\left[ -\left( {\overline{\theta }}_{+}\widehat{\gamma ^{0}}\frac{\partial \theta _{+}}{\partial \sigma ^{0}}+ {\overline{\theta }}_{+}\widehat{\gamma ^{1}}\frac{\partial \theta _{+}}{\partial \sigma ^{1}}\nonumber \right. \right. \\&\left. \left. +{\overline{\theta }}_{+}\widehat{\gamma ^{2}}\frac{\partial \theta _{+}}{\partial \sigma {2}}\right) \right. \nonumber \\&\left. +\frac{1}{2}{\widehat{g}}\,^{il}(\eta _{a{a\acute{}}}\,u^{a}_{i}u^{a\acute{}}_{l})+\frac{1}{4}\widetilde{\mathcal{F}}^{(1)}_{il} \widetilde{\mathcal{F}}^{(1)}_{jk}{\widehat{g}}\,^{ij}{\widehat{g}}\,^{lk}\right] \end{aligned}$$

(1.17)

$$\begin{aligned} {{{\mathcal {L}}}}^{WZ}_{NR}= & {} T_{NR}\left[ \frac{1}{2}({\overline{\theta }}_{+}\Gamma _{\mu \nu }d\theta _{+}) \right. \Big ((dx^{\mu }+{\overline{\theta }}_{-}\Gamma ^{\mu }d\theta _{-}) \Big . \nonumber \\&(dx^{\nu }\Big .+{\overline{\theta }}_{-}\Gamma ^{\nu }d\theta _{-}) -({\overline{\theta }}_{-}\Gamma ^{\mu }d\theta _{-}) (dx^{\nu }+\frac{2}{3}{\overline{\theta }}_{-}\Gamma ^{\nu }d\theta _{-})\Big )\nonumber \\&+ \frac{1}{2}({\overline{\theta }}_{-}\Gamma _{\mu \nu }d\theta _{-}) ({\overline{\theta }}_{+}\Gamma ^{\mu }d\theta _{+})\left( dx^{\nu } +\frac{2}{3}{\overline{\theta }}_{-}\Gamma ^{\nu }d\theta _{-}\right) \nonumber \\&+\frac{1}{2}({\overline{\theta }}_{-}\Gamma _{ab}d\theta _{-})\Big (\left( dx^{a}+{\overline{\theta }}_{+}\Gamma ^{a}d\theta _{-}+{\overline{\theta }}_{-}\Gamma ^{a}d\theta _{+}\right) \nonumber \\&\times (dx^{b}+{\overline{\theta }}_{+}\Gamma ^{b}d\theta _{-}+{\overline{\theta }}_{-}\Gamma ^{b}d\theta _{+}) \Big . \nonumber \\&\left. -({\overline{\theta }}_{+}\Gamma ^{a}d\theta _{-}+ +{\overline{\theta }}_{-}\Gamma ^{a}d\theta _{+})\nonumber \right. \\&\times \left( dx^{b}\right. \left. \left. +\frac{2}{3}({\overline{\theta }}_{-}\Gamma ^{b}d\theta _{+}+{\overline{\theta }}_{+}\Gamma ^{b}d\theta _{-})\right) \right) \nonumber \\&+({\overline{\theta }}_{+}\Gamma _{\mu a}d\theta _{-}+{\overline{\theta }}_{-}\Gamma _{\mu a}d\theta _{+})\left( (dx^{\mu }+{\overline{\theta }}_{-}\Gamma ^{\mu }d\theta _{-})\nonumber \right. \\&\times \left( dx^{a}\right. \left. \left. +\frac{1}{2}({\overline{\theta }}_{+}\Gamma ^{a}d\theta _{-}+{\overline{\theta }}_{-} \Gamma ^{a}d\theta _{+}\right) \right) \Bigg )\nonumber \\&-\frac{1}{2}({\overline{\theta }}_{-}\Gamma ^{\mu }d\theta _{-})\left( dx^{a}+\frac{1}{3}({\overline{\theta }}_{-}\Gamma ^{a}d\theta _{+}+{\overline{\theta }}_{+}\Gamma ^{a}d\theta _{-})\right) \nonumber \\&+\frac{1}{2}({\overline{\theta }}_{+}\Gamma _{\mu }\Gamma _{11}d\theta _{-}+{\overline{\theta }}_{-}\Gamma _{\mu }\Gamma _{11}d\theta _{+})\nonumber \\&\times \left( dx^{\mu }+\frac{2}{3}{\overline{\theta }}_{-}\Gamma ^{\mu }d\theta _{-}\right) ({\overline{\theta }}_{+}\Gamma _{11}d\theta _{-}+{\overline{\theta }}_{-}\Gamma _{11}d\theta _{+}) \nonumber \\&+\frac{1}{2}({\overline{\theta }}_{-}\Gamma _{b}\Gamma _{11}d\theta _{-})\Big (dx^{b}+\frac{2}{3}({\overline{\theta }}_{-}\Gamma ^{b}d\theta _{+} \Big . \nonumber \\&+\Big . {\overline{\theta }}_{+}\Gamma ^{b}d\theta _{-})\Big )({\overline{\theta }}_{+}\Gamma _{11}d\theta _{-}+{\overline{\theta }}_{-}\Gamma _{11}d\theta _{+})\nonumber \\&+ (d{\overline{\theta }}_{-}\Gamma _{11}\theta _{+}+d{\overline{\theta }}_{+}\Gamma _{11}\theta _{-})(f-({\overline{\theta }}_{-}\Gamma _{\mu }\Gamma _{11}d\theta _{+}\nonumber \\&+\left. {\overline{\theta }}_{+}\Gamma _{\mu }\Gamma _{11}d\theta _{-})\left( dx^{\mu }+\frac{1}{2}{\overline{\theta }}_{-}\Gamma ^{\mu }d\theta _{-}\right) \nonumber \right. \\&\left. {-}({\overline{\theta }}_{-} \Gamma _{a}\Gamma _{11}d\theta _{-})\left( dx^{a}{+}\frac{1}{2}({\overline{\theta }}_{-}\Gamma ^{a}d\theta _{+}{+}{\overline{\theta }}_{+}\Gamma ^{a}d\theta _{-})\right) \Bigg )\right] \nonumber \\ \end{aligned}$$

(1.18)

where

$$\begin{aligned} R^{\mu }_{i}= & {} \partial _{i}x^{\mu }-{\overline{\theta }}_{-}\Gamma ^{\mu }\partial _{i}\theta _{-} \end{aligned}$$

(1.19)

$$\begin{aligned} u^{a}_{i}= & {} \partial _{i}x^{a}-{\overline{\theta }}_{-}\Gamma ^{a}\partial _{i}\theta _{+}-{\overline{\theta }}_{+}\Gamma ^{a}\partial _{i}\theta _{-} \end{aligned}$$

(1.20)

$i,\mu =0,1,2/a=3,...,9;$ also

$$\begin{aligned} \widehat{\gamma ^{0}}= & {} \frac{1}{\left( \epsilon _{ijk}R^{0}_{i}R^{1}_{j}R^{2}_{k}\right) }\left( \epsilon _{ijk}\Gamma ^{i}R^{j}_{1}R^{k}_{2}\right) \end{aligned}$$

(1.21)

$$\begin{aligned} \widehat{\gamma ^{1}}= & {} -\frac{1}{\left( \epsilon _{ijk}R^{0}_{i}R^{1}_{j}R^{2}_{k}\right) }\left( \epsilon _{ijk}\Gamma ^{i}R^{j}_{0}R^{k}_{2}\right) \end{aligned}$$

(1.22)

$$\begin{aligned} \widehat{\gamma ^{2}}= & {} \frac{1}{\left( \epsilon _{ijk}R^{0}_{i}R^{1}_{j}R^{2}_{k}\right) }\left( \epsilon _{ijk}\Gamma ^{i}R^{j}_{0}R^{k}_{1}\right) \end{aligned}$$

(1.23)

$$\begin{aligned} {\widehat{g}}^{jk}= & {} \eta _{\mu \nu }R^{\mu }_{j}R^{\nu }_{k} \end{aligned}$$

(1.24)

where we have introduced the following quantities $\epsilon _{012}=1,\$ $\eta _{\mu \nu }=\text {diag}(-1,1,1),\$ $\eta _{a{a\acute{}}}=\text {diag}(1,...,1),\$ $\mu ,\nu =0,1,2;\$ $a,a\acute{}=3,...,9;$

$$\begin{aligned} \widetilde{\mathcal{F}}^{(1)}_{ij}= & {} f_{ij}+\left[ ({\overline{\theta }}_{-}\Gamma _{11}\Gamma _{\mu }\partial _{j}\theta _{+}+{\overline{\theta }}_{+} \Gamma _{11}\Gamma _{\mu }\partial _{j}\theta _{-})\nonumber \right. \\&\left. \times \left( R^{\mu }_{i}+\frac{1}{2}{\overline{\theta }}_{-}\Gamma ^{\mu }\partial _{i}\theta _{-}\right) -(i\leftrightarrow j)\right] \nonumber \\&+\left[ ({\overline{\theta }}_{-}\Gamma _{11}\Gamma _{a}\partial _{j}\theta _{-}) \left( u^{a}_{i}+\frac{1}{2}({\overline{\theta }}_{-}\Gamma ^{a}\partial _{i}\theta _{+}\nonumber \right. \right. \\&\left. \left. + {\overline{\theta }}_{+}\Gamma ^{a}\partial _{i}\theta _{-})\right) -(i\leftrightarrow j )\right] , \end{aligned}$$

(1.25)

where $f_{ij}=\partial _{i}w_{j}-\partial _{j}w_{i},\$ $f=\frac{1}{2}f_{ij}d\sigma ^{i}d\sigma ^{j}, \$ and $\mu ,\nu ,i,j=0,1,2/a=3,...,9.$

We denote by $[{{{\mathcal {L}}}}^{WZ}_{NR}]_{3}$ the 3-form coefficient of ${{{\mathcal {L}}}}^{WZ}_{NR}$ given in (1.18). The Lagrangian density of the non-relativistic D2-brane is given by:

$$\begin{aligned} {{{\mathcal {L}}}}_{NR}={{{\mathcal {L}}}}^{DBI}_{NR}+[{{{\mathcal {L}}}}^{WZ}_{NR}]_{3} \end{aligned}$$

(1.26)

The action (1.16) is invariant under the non-relativistic counterpart of the global and local supersymmetric transformations that leave the relativistic D2-brane action (1.1) invariant [16].

The conjugate momenta of the variables $x^{\mu },x^{a},\theta _{+},\theta _{-},w_{i}$ are given by:

$$\begin{aligned} p_{\mu }= & {} \frac{\partial {{{\mathcal {L}}}}_{NR}}{\partial {{\dot{x}}^{\mu }}}\nonumber \\= & {} \frac{\partial {{{\mathcal {L}}}}_{NR}}{\partial R^{\mu }_{0}}+ \frac{\partial {{{\mathcal {L}}}}_{NR}}{\partial {\widetilde{\mathcal{F}}^{(1)}_{0i}}}({\overline{\theta }}_{-}\Gamma _{11}\Gamma _{\mu }\partial _{i}\theta _{+}+{\overline{\theta }}_{+}\Gamma _{11}\Gamma _{\mu }\partial _{i}\theta _{-})\nonumber \\ \end{aligned}$$

(1.27)

$\mu =0,1,2/i=1,2;$

$$\begin{aligned} p_{a}= & {} \frac{\partial {{{\mathcal {L}}}}_{NR}}{\partial {{\dot{x}}^{a}}}\nonumber \\= & {} \frac{\partial \mathcal{L}_{NR}}{\partial {{\dot{u}}^{a}_{0}}}+\frac{\partial \mathcal{L}_{NR}}{\partial \widetilde{\mathcal{F}}^{(1)}_{0i}}({\overline{\theta }}_{-}\Gamma _{11}\Gamma _{a}\partial _{i}\theta _{-}) \end{aligned}$$

(1.28)

$a=3,...,9;$

$$\begin{aligned} J_{+l}= & {} \frac{\partial ^{r}\mathcal{L}_{NR}}{\partial {{\dot{\theta }}_{+l}}}-\frac{\partial \mathcal{L}_{NR}}{\partial u^{a}_{0}}( {\overline{\theta }}_{-}\Gamma ^{a})_{l}\nonumber \\&-\frac{\partial {{{\mathcal {L}}}}_{NR}}{\partial \widetilde{\mathcal{F}}^{(1)}_{0i}}\left[ \left( R^{\mu }_{i}+\frac{1}{2}{\overline{\theta }}_{-}\Gamma ^{\mu }\partial _{i}\theta _{-}\right) ({\overline{\theta }}_{-}\Gamma _{11}\Gamma _{\mu })_{l}\nonumber \right. \\&\left. + \frac{1}{2}({\overline{\theta }}_{-}\Gamma _{11}\Gamma _{a}\partial _{i}\theta _{-})({\overline{\theta }}_{-}\Gamma ^{a})_{l}\right] \end{aligned}$$

(1.29)

and

$$\begin{aligned} J_{-l}= & {} \frac{\partial ^{r}\mathcal{L}_{NR}}{\partial {{\dot{\theta }}_{-l}}}=\frac{\partial ^{r}[\mathcal{L}^{WZ}_{NR}]_{3}}{\partial {{\dot{\theta }}_{-l}}}-\frac{\partial \mathcal{L}_{NR}}{\partial R^{\mu }_{0}}({\overline{\theta }}_{-}\Gamma ^{\mu })_{l} \nonumber \\&-\frac{\partial {{{\mathcal {L}}}}_{NR}}{\partial u^{a}_{0}}({\overline{\theta }}_{+}\Gamma ^{a})_{l}\nonumber \\&-\frac{\mathcal{L}_{NR}}{\partial \widetilde{{{{\mathcal {F}}}}}^{(1)} _{0i}}\left[ \frac{1}{2}({\overline{\theta }}_{-}\Gamma _{11}\Gamma _{\mu }\partial _{i}\theta _{+}{+}{\overline{\theta }}_{+}\Gamma _{11}\Gamma _{\mu }\partial _{i}\theta _{-}) ({\overline{\theta }}_{-}\Gamma ^{\mu })_{l} \right. \nonumber \\&+\frac{1}{2}({\overline{\theta }}_{-}\Gamma _{11}\Gamma _{a}\partial _{i}\theta _{-})({\overline{\theta }}_{+}\Gamma ^{a})_{l}\nonumber \\&+\left( R^{\mu }_{i} + \frac{1}{2}{\overline{\theta }}_{-}\Gamma ^{\mu }\partial _{i}\theta _{-}\right) ({\overline{\theta }}_{+}\Gamma _{11}\Gamma _{\mu })_{l}\nonumber \\&+\left. \left( u^{a}_{i}{+}\frac{1}{2}({\overline{\theta }}_{-}\Gamma ^{a}\partial _{i}\theta _{+}{+}{\overline{\theta }}_{+}\Gamma ^{a}\partial _{i}\theta _{-})\right) ({\overline{\theta }}_{-}\Gamma _{11}\Gamma _{a})_{l}\right] ,\nonumber \\ \end{aligned}$$

(1.30)

where $l=1,...,32;$ and

$$\begin{aligned} E^{i}=\frac{\partial \mathcal{L}_{NR}}{\partial {{\dot{w}}_{i}}}=\frac{\partial \mathcal{L}_{NR}}{\partial \widetilde{{{{\mathcal {F}}}}}^{(1)}_{0i}}. \end{aligned}$$

(1.31)

The fermionic constraints are given by

$$\begin{aligned} F_{+l}= & {} J_{+l}+p_{a}({\overline{\theta }}_{-}\Gamma ^{a})_{l}\nonumber \\&+E^{i}\left[ \left( R^{\mu }_{i}+\frac{1}{2}{\overline{\theta }}_{-}\Gamma ^{\mu }\partial \theta _{-}\right) ({\overline{\theta }}_{-}\Gamma _{11}\Gamma _{\mu })_{l} \right. \nonumber \\&\left. -\frac{1}{2}({\overline{\theta }}_{-}\Gamma _{11}\Gamma _{a}\partial _{i}\theta _{-})({\overline{\theta }}_{-}\Gamma ^{a})_{l}\right] \nonumber \\&+T_{NR}\epsilon _{ijk} ({\overline{\theta }}_{+}\Gamma ^{i})_{l}R^{j}_{1}R^{k}_{2}-\frac{\partial ^{r}[{{{\mathcal {L}}}}^{WZ}_{NR}]_{3}}{\partial {{\dot{\theta }}_{+l}}} \end{aligned}$$

(1.32)

where the last derivative does not include differentiation with respect to $u^{a}_{0},\widetilde{{{{\mathcal {F}}}}}^{(1)}_{0i}$, and

$$\begin{aligned} F_{-l}= & {} J_{-l}+p_{a}({\overline{\theta }}_{+}\Gamma ^{a})_{l}+p_{\mu }({\overline{\theta }}_{-}\Gamma ^{\mu })_{l}\nonumber \\&+E^{i}\left[ \left( R^{\mu }_{i} +\frac{1}{2}({\overline{\theta }}_{-}\Gamma ^{\mu }\partial _{i}\theta _{-}\right) ({\overline{\theta }}_{+}\Gamma _{11}\Gamma _{\mu })_{l} \right. \nonumber \\&+\left( u^{a}_{i}+\frac{1}{2}({\overline{\theta }}_{-}\Gamma ^{a}\partial _{i}\theta _{+}+{\overline{\theta }}_{+}\Gamma ^{a}\partial _{i}\theta _{-}\right) ({\overline{\theta }}_{-}\Gamma _{11}\Gamma _{a})_{l}\nonumber \\&-\frac{1}{2}({\overline{\theta }}_{-}\Gamma _{11}\Gamma _{\mu }\partial _{i}\theta _{+} +\left. {\overline{\theta }}_{+}\Gamma _{11}\Gamma _{\mu }\partial _{i}\theta _{-})({\overline{\theta }}_{-}\Gamma ^{\mu })_{l}\nonumber \right. \\&\left. -\frac{1}{2}({\overline{\theta }}_{-} \Gamma _{11}\Gamma _{a}\partial _{i}\theta _{-})({\overline{\theta }}_{+}\Gamma ^{a})_{l}\right] -\frac{\partial ^{r}[{{{\mathcal {L}}}}^{WZ}_{NR}]_{3}}{\partial {{\dot{\theta }}_{-l}}},\nonumber \\ \end{aligned}$$

(1.33)

where the last derivative does not include differentiation with respect to $u^{a}_{0}, \,R^{\mu }_{0}, \,\widetilde{\mathcal{F}}^{(1)}_{0i}.$

We introduce now two commuting spinor fields $\lambda _{+},\lambda _{-}$ corresponding to the fermionic constraints $F_{+},F_{-}$ and we write down a BRST charge according to the ansatz proposed in [1,2,3,4]:

$$\begin{aligned} Q= & {} \int d^{2}\sigma \left[ \lambda ^{l}_{+}J_{+l}+\lambda ^{l}_{-}J_{-l}+p_{a}({\overline{\theta }}_{-}\Gamma ^{a}\lambda _{+}+{\overline{\theta }}_{+}\Gamma ^{a}\lambda _{-} )\nonumber \right. \\&+p_{\mu }({\overline{\theta }}_{-}\Gamma ^{\mu }\lambda _{-}) +E^{i}\left[ \left( R^{\mu }_{i}+\frac{1}{2}{\overline{\theta }}_{-}\Gamma ^{\mu }\partial _{i}\theta _{-}\right) \nonumber \right. \\&\times ({\overline{\theta }}_{+}\Gamma _{11}\Gamma _{\mu }\lambda _{-} +{\overline{\theta }}_{-}\Gamma _{11}\Gamma _{\mu }\lambda _{+}) +\Big (u^{a}_{i}+\frac{1}{2}({\overline{\theta }}_{-}\Gamma ^{a}\partial _{i}\theta _{+} \Big . \nonumber \\&+\Big . {\overline{\theta }}_{+}\Gamma ^{a}\partial _{i}\theta _{-})\Big )({\overline{\theta }}_{-}\Gamma _{11}\Gamma _{a}\lambda _{-})\nonumber \\&-\frac{1}{2}({\overline{\theta }}_{-} \Gamma _{11}\Gamma _{\mu }\partial _{i}\theta _{+}+{\overline{\theta }}_{+}\Gamma _{11}\Gamma _{\mu }\partial _{i}\theta _{-})({\overline{\theta }}_{-}\Gamma ^{\mu } \lambda _{-})\nonumber \\&-\left. \frac{1}{2}({\overline{\theta }}_{-}\Gamma _{11}\Gamma _{a}\partial _{i}\theta _{-})({\overline{\theta }}_{+}\Gamma ^{a}\lambda _{-}+{\overline{\theta }}_{-} \Gamma ^{a}\lambda _{+})\right] \nonumber \\&+T_{NR}\epsilon _{ijk}({\overline{\theta }}_{+}\Gamma ^{i}\lambda _{+})R^{j}_{1}R^{k}_{2}-\left( \lambda ^{T}_{+}\frac{\partial ^{r} [{{{\mathcal {L}}}}^{WZ}_{NR}]_{3}}{\partial {{\dot{\theta }}_{+}}}\right) \nonumber \\&\left. -\left( \lambda ^{T}_{-}\frac{\partial ^{r}[{{{\mathcal {L}}}}^{WZ}_{NR}]_{3}}{\partial {{\dot{\theta }}_{-}}}\right) \right] \end{aligned}$$

(1.34)

with $i,j,k,\mu =0,1,2/a=3,...,9.$

A set of BRST transformations is the following:

$$\begin{aligned} s\theta ^{l}_{+}= & {} \lambda ^{l}_{+},\nonumber \\ s\theta ^{l}_{-}= & {} \lambda ^{l}_{-},\nonumber \\ sx^{\mu }= & {} {\overline{\theta }}_{-}\Gamma ^{\mu }\lambda _{-},\nonumber \\ sx^{a}= & {} {\overline{\theta }}_{-}\Gamma ^{a}\lambda _{+}+{\overline{\theta }}_{+}\Gamma ^{a}\lambda _{-},\nonumber \\ sR^{\mu }_{\nu }= & {} 2(\partial _{\nu }{\overline{\theta }}_{-}\Gamma ^{\mu }\lambda _{-}),\nonumber \\ su^{a}_{\mu }= & {} 2(\partial _{\mu }{\overline{\theta }}_{-}\Gamma ^{a}\lambda _{+}+\partial _{\mu }{\overline{\theta }}_{+}\Gamma ^{a}\lambda _{-}),\nonumber \\ s\widetilde{{{{\mathcal {F}}}}}^{(1)}_{ij}= & {} 2R^{\mu }_{j}({\overline{\lambda }}_{-}\Gamma _{11}\Gamma _{\mu }\partial _{i}\theta _{+}+{\overline{\lambda }}+ \Gamma _{11}\Gamma _{\mu }\partial _{i}\theta _{-})\nonumber \\&+2u^{a}_{j}(\partial _{i}{\overline{\theta }}_{-}\Gamma _{11}\Gamma ^{a}\lambda _{-})-(i\leftrightarrow j) \end{aligned}$$

(1.35)

where again $\mu ,\nu =0,1,2/i,j=1,2/a=3,...,9/l=1,... 32;\$ from which we obtain

$$\begin{aligned} s^{2}\theta ^{l}_{+}= & {} s^{2}\theta ^{l}_{-}=0,\nonumber \\ s^{2}x^{\mu }= & {} {\overline{\lambda }}_{-}\Gamma ^{\mu }\lambda _{-},\nonumber \\ s^{2}x^{a}= & {} 2({\overline{\lambda }}_{-}\Gamma ^{a}\lambda _{+}),\nonumber \\ s^{2}R^{\mu }_{i}= & {} \frac{\partial }{\partial \sigma ^{i}}({\overline{\lambda }}_{-}\Gamma ^{\mu }\lambda _{-}),\nonumber \\ s^{2}u^{a}_{i}= & {} 2\frac{\partial }{\partial \sigma ^{i}}({\overline{\lambda }}_{+}\Gamma ^{a}\lambda _{-}) \end{aligned}$$

(1.36)

and

$$\begin{aligned} s^{2}\widetilde{\mathcal{F}}^{(1)}_{ij}= & {} 2R^{\mu }_{j}\frac{\partial }{\partial \sigma ^{i}}({\overline{\lambda }}_{+}\Gamma _{11}\Gamma _{\mu }\lambda _{-})+u^{a}_{j}\frac{\partial }{\partial \sigma ^{i}} ({\overline{\lambda }}_{-}\Gamma _{11}\Gamma _{a}\lambda _{-})\nonumber \\&+2(\partial _{j}{\overline{\theta }}_{-}\Gamma _{11}\Gamma ^{\mu }\partial _{i}\theta _{-})({\overline{\lambda }}_{-}\Gamma _{\mu }\lambda _{-})\nonumber \\&+2(\partial _{j}{\overline{\theta }}_{-}\Gamma ^{a}\partial _{i}\theta _{+})({\overline{\lambda }}_{-}\Gamma _{11}\Gamma _{a}\lambda _{-})\nonumber \\&+2(\partial _{j}{\overline{\theta }}_{-}\Gamma _{11}\Gamma _{a}\partial _{i}\theta _{-})({\overline{\lambda }}_{-}\Gamma ^{a}\lambda _{+})\nonumber \\&+2(\partial _{j}\theta _{-} \Gamma ^{\mu }\partial _{i}\theta _{-})({\overline{\lambda }}_{-}\Gamma _{11}\Gamma _{\mu }\lambda _{+}) \end{aligned}$$

(1.37)

where we used the ansatz

$$\begin{aligned} \lambda _{-}= & {} P_{-}\lambda ,\nonumber \\ P_{-}= & {} \frac{1}{2}(1-\Gamma _{0}\Gamma _{1}\Gamma _{2}) \end{aligned}$$

(1.38)

The full set of constraints required for the nilpotency of the BRST charge is obtained by studying the equal time Poisson bracket $\{Q(\sigma ),Q(\sigma ')\}$ where $Q(\sigma )$ is given in (1.34). A basis for expanding a $32\times 32$ matrix M is given in Appendix A.

Thus, the expression for the equal time Poisson bracket $\{Q(\sigma ),Q(\sigma ')\}$ where $Q(\sigma )$ is the full BRST charge defined in (1.34) is given by:

$$\begin{aligned}&\{Q(\sigma ),Q(\sigma ')\}\nonumber \\&\quad =\int d^{2}\sigma (\partial _{i}E^{i})[({\overline{\theta }}_{-}\Gamma ^{a}\lambda _{+}+{\overline{\theta }}_{+}\Gamma ^{a}\lambda _{-})({\overline{\theta }}_{-}\Gamma _{11}\Gamma _{a} \lambda _{-})\nonumber \\&\qquad +({\overline{\theta }}_{-}\Gamma ^{\mu }\lambda _{-})({\overline{\theta }}_{+}\Gamma _{11}\Gamma _{\mu }\lambda _{-}+{\overline{\theta }}_{-}\Gamma _{11}\Gamma _{\mu }\lambda _{+})]\nonumber \\&\qquad +2T_{NR}\epsilon _{\mu \nu \rho }\int d^{2}\sigma R^{\nu }_{1}R^{\rho }_{2}({\overline{\lambda }}_{+}\Gamma ^{\mu }\lambda _{+})\nonumber \\&\qquad +T_{NR}\epsilon _{0jk}\int d\sigma ^{2}R^{\mu }_{j}R^{\nu }_{k}({\overline{\lambda }}_{+}\Gamma _{\mu \nu }\lambda _{+})\nonumber \\&\qquad +\int d^{2}\sigma [F_{\mu }({\overline{\lambda }}_{-}\Gamma ^{\mu }\lambda _{-})\nonumber \\&\quad +F_{a}({\overline{\lambda }}_{-}\Gamma ^{a}\Gamma _{11}\lambda _{-}) +F_{ab}( {\overline{\lambda }}_{-}\Gamma ^{ab}\lambda _{-})\nonumber \\&\qquad +F^{\nu abdf}({\overline{\lambda }}_{-}\Gamma _{\nu abdf}\lambda _{-}){+}{\overline{F}}({\overline{\lambda }}_{-}\Gamma _{11}\lambda _{+}) {+} {\overline{F}}_{a}({\overline{\lambda }}_{-}\Gamma ^{a}\lambda _{+})\nonumber \\&\qquad +{\overline{F}}_{\nu b}({\overline{\lambda }}_{-}\Gamma ^{\nu b}\lambda _{+})+{\overline{F}}_{\nu }({\overline{\lambda }}_{-}\Gamma ^{\nu }\Gamma _{11}\lambda _{+})+ {\overline{F}}_{\nu abc}\nonumber \\&\qquad ({\overline{\lambda }}_{-}\Gamma ^{\nu abc}\lambda _{+})+{\overline{F}}^{\nu ab}({\overline{\lambda }}_{+}\Gamma _{\nu ab}\Gamma _{11}\lambda _{-})\nonumber \\&\qquad +{\overline{F}}_{fglmn}({\overline{\lambda }}_{+}\Gamma ^{fglmn}\lambda _{-})+{\overline{F}}_{lmn}({\overline{\lambda }}_{-}\Gamma ^{lmn}\lambda _{+})] \end{aligned}$$

(1.39)

where now $\epsilon _{012}=1,i=1,2/\mu ,\nu ,\rho =0,1,2/a,b,c,d,f,g,l,m,n=3,...,9;$ and the ansatz $\lambda _{-}=P_{-}\lambda$ was used.

The expressions for $F_{\mu },F_{a},F_{ab},F_{\nu abdf},{\overline{F}},{\overline{F}}_{a}, {\overline{F}}_{\nu },{\overline{F}}_{\nu b },{\overline{F}}_{\nu abc},{\overline{F}}_{\nu ab},{\overline{F}}_{fglm},{\overline{F}}_{lmn}$ are given in Appendix B. These expressions are different from zero, so the following constraints guarantee the nilpotency of the BRST charge:

$$\begin{aligned}&\partial _{i}E^{i}=0 \qquad (\text {Gauss law}),\nonumber \\&{\overline{\lambda }}_{+}\Gamma _{\mu }\lambda _{+}=0, \qquad {\overline{\lambda }}_{+}\Gamma _{\mu \nu }\lambda _{+}=0,\nonumber \\&{\overline{\lambda }}_{-}\Gamma _{\mu }\lambda _{-}=0,\qquad {\overline{\lambda }}_{-}\Gamma _{a}\Gamma _{11}\lambda _{-}=0,\qquad \nonumber \\&{\overline{\lambda }}_{-}\Gamma _{ab}\lambda _{-}=0, \qquad {\overline{\lambda }}_{-}\Gamma _{\mu abcd}\lambda _{-}=0,\nonumber \\&{\overline{\lambda }}_{-}\Gamma _{11}\lambda _{+}=0,\qquad {\overline{\lambda }}_{-}\Gamma _{a}\lambda _{+}=0,\qquad \nonumber \\&{\overline{\lambda }}_{-}\Gamma _{\mu a}\lambda _{+}=0,\qquad {\overline{\lambda }}_{-}\Gamma _{\mu }\Gamma _{11}\lambda _{+}=0,\nonumber \\&{\overline{\lambda }}_{-}\Gamma _{\mu abc}\lambda _{+}=0, \qquad {\overline{\lambda }}_{+}\Gamma _{\mu ab}\Gamma _{11}\lambda _{-}=0, \qquad \nonumber \\&{\overline{\lambda }}_{+}\Gamma _{fglmn}\lambda _{-}=0, \qquad {\overline{\lambda }}_{-}\Gamma _{lmn}\lambda _{+}=0 \end{aligned}$$

(1.40)

with the ansatz $\lambda _{-}=P_{-}\lambda ,( i=1,2/\mu ,\nu =0,1,2/a,b,c,d,f,g,l,m,n=3,..,9).$

2 Solving the constraints

In order to solve the constraints (1.40) we use the methodology used in [21, 22]. The Dirac matrices $\Gamma ^{m}\, (m=0,1,..,9)$ are combined into five creation operators $a^{i}\, (i=1,..,5)$ and five annihilation operators $a_{i}\, (i=1,..,5)$ as follows:

$$\begin{aligned} a^{1}= & {} \frac{1}{2}(\Gamma ^{1}-i\Gamma ^{2}),\qquad a^{2}=\frac{1}{2}(\Gamma ^{3}-i\Gamma ^{4}),\qquad \nonumber \\ a^{3}= & {} \frac{1}{2}(\Gamma ^{5}-i\Gamma ^{6}),\nonumber \\ a^{4}= & {} \frac{1}{2}(\Gamma ^{7}-i\Gamma ^{8}),\qquad a^{5}=\frac{1}{2}(\Gamma ^{9}+\Gamma ^{0}),\nonumber \\ a_{1}= & {} \frac{1}{2}(\Gamma ^{1}+i\Gamma ^{2}),\qquad a_{2}=\frac{1}{2}(\Gamma ^{3}+i\Gamma ^{4}),\qquad \nonumber \\ a_{3}= & {} \frac{1}{2}(\Gamma ^{5}+i\Gamma ^{6}),\nonumber \\ a_{4}= & {} \frac{1}{2}(\Gamma ^{7}+i\Gamma ^{8}),\qquad a_{5}=\frac{1}{2}(\Gamma ^{9}-\Gamma ^{0}) \end{aligned}$$

(2.1)

The following identities hold:

$$\begin{aligned} \{a_{i},a^{j}\}= & {} \delta _{ij},\{a^{i},a^{j}\}=0,\qquad \{\Gamma _{11},a^{i}\}=0,\qquad \nonumber \\ \{\Gamma _{11},a_{j}\}= & {} 0,\qquad a_{i}=a^{i\dag },\nonumber \\&\text {with now}\quad i,j=1,..,5. \end{aligned}$$

(2.2)

Here we introduce the vacuum state $\mid 0>$ and the state $<0\mid$ with the equations:

$$\begin{aligned} a_{i}\mid 0> =0, \qquad <0\mid a^{i}=0, \qquad i=1,..,5. \end{aligned}$$

(2.3)

We decompose a 32-component Dirac spinor into a sum of a positive chirality 16-component part and a negative chirality 16-component part as follows:

$$\begin{aligned} \mid \lambda>= & {} \left( \lambda _{+}\mid 0>+ \frac{1}{2}\lambda _{ij}a^{j}a^{i}\mid 0>\nonumber \right. \\&\left. +\frac{1}{4!}\lambda ^{i}\epsilon _{ijklm}a^{j}a^{k}a^{l}a^{m} \mid 0>\right) \nonumber \\&+\left( \frac{1}{5!}\lambda '_{+}\epsilon _{ijklm}a^{i}a^{j}a^{k}a^{l}a^{m}\mid 0> \nonumber \right. \\&\left. +\frac{1}{3!}\lambda '^{ij}\epsilon _{ijklm}a^{k}a^{l}a^{m}\mid 0>+ \lambda '_{i}a^{i}\mid 0>\right) \nonumber \\= & {} (1+{\overline{10}}+5)+(1+10+{\overline{5}}), \end{aligned}$$

(2.4)

where $\lambda _{ij}=-\lambda _{ji},\lambda '^{ij}=-\lambda '^{ji}.$

Let us try to solve from (1.40) the constraints ${\overline{\lambda }}_{+}\Gamma _{\mu }\lambda _{+}=0$ and ${\overline{\lambda }}_{+}\Gamma _{\mu \nu }\lambda _{+}=0\,$ (where $\mu ,\nu =0,1,2$).

It is interesting to note that $\lambda _{+}$ can be written as a sum of two pure spinors $\lambda _{1+}=P_{+}\lambda _{+}$ and $\lambda _{2+}=P_{-}\lambda _{+}$ where $P_{\pm }=\frac{1}{2}(1\pm \Gamma _{0}\Gamma _{1}\Gamma _{2})$ and the constraints ${\overline{\lambda }}_{+}\Gamma _{\mu }\lambda _{+}=0$ , ${\overline{\lambda }}_{+}\Gamma _{\mu \nu } \lambda _{+}=0, (\mu ,\nu =0,1,2)$ can be expressed equivalently as two pure spinor constraints: ${\overline{\lambda }}_{1+}\Gamma _{m}\lambda _{1+}=0$, ${\overline{\lambda }}_{2+}\Gamma _{m} \lambda _{2+}=0, (m=0,...,9)$.

The constraints ${\overline{\lambda }}_{+}\Gamma _{\mu }\lambda _{+}=0$ can be written as follows:

$$\begin{aligned}<\lambda _{+}\mid Ca_{1}\mid \lambda _{+}>= & {} 0,\nonumber \\<\lambda _{+}\mid Ca^{1}\mid \lambda _{+}>= & {} 0,\nonumber \\<\lambda _{+}\mid Ca_{5}\mid \lambda _{+}>= & {} <\lambda _{+}\mid Ca^{5}\mid \lambda _{+}>\,, \end{aligned}$$

(2.5)

whereas the constraints ${\overline{\lambda }}_{+}\Gamma _{\mu \nu }\lambda _{+}=0, (\mu ,\nu =0,1,2)$ can be expressed as:

$$\begin{aligned}<\lambda _{+}\mid C(a^{5}-a_{5})(a^{1}+a_{1})\mid \lambda _{+}>= & {} 0,\nonumber \\<\lambda _{+}\mid C(a^{1}+a_{1})(a^{1}-a_{1})\mid \lambda _{+}>= & {} 0,\nonumber \\ <\lambda _{+}\mid C(a^{5}-a_{5})(a^{1}-a_{1})\mid \lambda _{+}>= & {} 0. \end{aligned}$$

(2.6)

For a general $\mid \lambda _{+}>$ we can write

$$\begin{aligned} \mid \lambda _{+}>= & {} \lambda _{++}\mid 0>+\frac{1}{2}\lambda _{+ij}a^{j}a^{i}\mid 0>\nonumber \\&+\frac{1}{4!}\lambda ^{i}_{+}\epsilon _{ijklm}a^{j}a^{k}a^{l}a^{m}\mid 0>\nonumber \\&+\lambda '_{+i}a^{i}\mid 0>+\frac{1}{3!}\lambda '^{ij}_{+}\epsilon _{ijlkm}a^{l}a^{k}a^{m}\mid 0>\nonumber \\&+\frac{1}{5!}\lambda '_{++}\epsilon _{ijklm}a^{i}a^{j}a^{k}a^{l}a^{m}\mid 0>, \end{aligned}$$

(2.7)

where now $\epsilon _{12345}=1.$

Thus we have the following relations:

$$\begin{aligned}&<\lambda _{+}\mid Ca^{1}\mid \lambda _{+}>\nonumber \\&\quad =2\left( \lambda _{++}\lambda ^{1}_{+}+\frac{1}{8}\epsilon _{1ijkl}\lambda _{+ij}\lambda _{+lk}\right) -4\lambda '_{+i}\lambda ^{'i1}_{+}=0, \end{aligned}$$

(2.8)

$$\begin{aligned}&<\lambda _{+}\mid Ca_{1}\mid \lambda _{+}>\nonumber \\&\quad = 2\lambda ^{i}\lambda _{+i1}-2\lambda '_{+1}\lambda '_{++}+\epsilon _{1iji'j'}\lambda ^{'ij}_{+} \lambda ^{'i'j'}_{+}=0, \end{aligned}$$

(2.9)

$$\begin{aligned}&<\lambda _{+}\mid Ca^{5}\mid \lambda _{+}> =<\lambda _{+}\mid Ca_{5}\mid \lambda _{+}> \nonumber \\&\quad \Rightarrow 2(\lambda _{++}\lambda ^{5}_{+}+\frac{1}{8}\epsilon _{5ijkl}\lambda _{+ij}\lambda _{+lk})-4\lambda '_{+i}\lambda ^{'i5}_{+} \nonumber \\&\quad = 2\lambda ^{i}_{+}\lambda _{+i5}-2\lambda '_{+5}\lambda '_{++}+\epsilon _{5iji'j'}\lambda ^{'ij}_{+}\lambda ^{'i'j'}_{+}. \end{aligned}$$

(2.10)

On the other hand the constraints ${\overline{\lambda }}_{+}\Gamma ^{\mu \nu }\lambda _{+}=0$ can be rewritten as follows:

$$\begin{aligned}&\frac{1}{2}<\lambda _{+}\mid C(a_{1}a^{1}-a^{1}a_{1})\mid \lambda _{+}>\nonumber \\&\quad =-\lambda _{++}\lambda '_{++}+4\lambda _{+i1}\lambda ^{'i1}_{+}-\lambda _{+ij}\lambda ^{'ij}\nonumber \\&\qquad +\lambda ^{i}_{+}\lambda '_{+i}-2\lambda ^{1}_{+} \lambda '_{+1}=0,\nonumber \\&\quad<\lambda _{+}\mid C(a^{5}-a_{5})a^{1}\mid \lambda _{+}>\nonumber \\&\quad =4\lambda _{++}\lambda ^{'51}_{+}+4\lambda ^{'i1}_{+}\lambda _{+i5}+2\lambda '_{+5}\lambda ^{1}_{+}\nonumber \\&\qquad -\epsilon _{15iji'}\lambda _{+ij}\lambda '_{+i'}=0,\nonumber \\&\frac{1}{2}<\lambda _{+}\mid C(a^{5}-a_{5})a_{1}\mid \lambda _{+}>\nonumber \\&\quad =-2\lambda _{+i1}\lambda ^{'i5}_{+}+\lambda _{+51}\lambda '_{++}+\lambda ^{5}_{+}\lambda '_{+1}\nonumber \\&\qquad -\epsilon _{ijk15}\lambda ^{'ij}_{+}\lambda ^{k} _{+}=0. \end{aligned}$$

(2.11)

From (2.8)–(2.11) we obtain the non-trivial solution

$$\begin{aligned} \lambda ^{1}_{+}= & {} \frac{1}{\lambda }_{++}\left( 2\lambda '_{+i}\lambda ^{'i1}_{+}+\frac{1}{8}\epsilon _{1ijkl}\lambda _{+ij}\lambda _{+kl}\right) ,\nonumber \\ \lambda '_{+1}= & {} \frac{1}{2\lambda '_{++}}\left( 2\lambda ^{i}_{+}\lambda _{+i1}+\epsilon _{1ijkl}\lambda ^{'ij}_{+}\lambda ^{'kl}_{+}\right) ,\nonumber \\ \lambda ^{5}_{+}= & {} \frac{1}{\lambda _{++}}\left( \lambda ^{i}_{+}\lambda _{+i5}-\lambda '_{+5}\lambda '_{++}+2\lambda '_{+i}\lambda ^{'i5}_{+}\nonumber \right. \\&\left. +\frac{1}{2}\epsilon _{5ijkl} \lambda ^{'ij}_{+}\lambda ^{'kl}_{+}+\frac{1}{8}\epsilon _{5ijkl}\lambda _{+ij}\lambda _{+kl}\right) ,\nonumber \\ \lambda ^{'51}_{+}= & {} \frac{1}{4\lambda _{++}}\left( -4\lambda ^{'i1}_{+}\lambda _{+i5}-2\lambda '_{+5}\lambda ^{1}_{+}+\epsilon _{15ijk}\lambda _{+ij}\lambda '_{+k}\right) ,\nonumber \\ \lambda _{+51}= & {} \frac{1}{\lambda '_{++}}\left( 2\lambda _{+i1}\lambda ^{'i5}_{+}-\lambda ^{5}_{+}\lambda '_{+1}+\epsilon _{15ijk}\lambda ^{'ij}\lambda ^{k}_{+}\right) ,\nonumber \\ \lambda '_{++}= & {} \frac{1}{\lambda _{++}}\left( 4\lambda _{+i1}\lambda ^{'i1}_{+}-\lambda _{+ij}\lambda ^{'ij}_{+}+\lambda ^{i}_{+}\lambda '_{+i}-2\lambda ^{1}_{+}\lambda '_{+1}\right) .\nonumber \\ \end{aligned}$$

(2.12)

Now from (1.40) we deal with the following constraints:

$$\begin{aligned} {\overline{\lambda }}_{-}\Gamma ^{\mu }\lambda _{-}= & {} 0, \qquad {\overline{\lambda }}_{-}\Gamma ^{a}\Gamma _{11}\lambda _{-}=0, \qquad \nonumber \\ {\overline{\lambda }}_{-}\Gamma ^{ab}\lambda _{-}= & {} 0, \qquad {\overline{\lambda }}_{-}\Gamma ^{\mu abcd}\lambda _{-}=0, \end{aligned}$$

(2.13)

where $\mu =0,1,2/a,b,c,d=3,..,9$ with $\lambda _{-}=P_{-}\lambda$ ($P_{-}$ is given in (1.38)). Thus, we can write

$$\begin{aligned} P_{-}=\frac{1}{2}\left[ 1+i(a^{5}-a_{5})(a_{1}a^{1}-a^{1}a_{1})\right] . \end{aligned}$$

(2.14)

We have that $P^{T}_{-}C=CP_{-}$ where C is the charge conjugation matrix. So for example the constraints ${\overline{\lambda }}_{-}\Gamma ^{\mu }\lambda _{-}=0$ can be rewritten as follows:

$$\begin{aligned}&<\lambda \mid CP_{-}a^{1}P_{-}\mid \lambda>=0,\nonumber \\&<\lambda \mid CP_{-}a_{1}P_{-}\mid \lambda>=0,\nonumber \\&<\lambda \mid CP_{-}a^{5}P_{-}\mid \lambda>=<\lambda \mid CP_{-}a_{5}P_{-}\mid \lambda >. \end{aligned}$$

(2.15)

The solution of the constraints in (2.13) is the trivial one. This can be shown as follows. Let us consider the following 16 constraints:

$$\begin{aligned} {\overline{\lambda }}_{-}\Gamma ^{13579}\lambda _{-}= & {} 0, \quad {\overline{\lambda }}_{-}{\Gamma }^{13589}\lambda _{-}=0, \quad {\overline{\lambda }}_{-}\Gamma ^{13679}\lambda _{-}=0, \quad \nonumber \\ {\overline{\lambda }}_{-}\Gamma ^{13689}\lambda _{-}= & {} 0\nonumber \\ {\overline{\lambda }}_{-}\Gamma ^{14579}\lambda _{-}= & {} 0,\quad {\overline{\lambda }}_{-}\Gamma ^{14679}\lambda _{-}=0,\quad {\overline{\lambda }}_{-}\Gamma ^{14589}\lambda _{-}=0,\nonumber \\ {\overline{\lambda }}_{-}\Gamma ^{14689}\lambda _{-}= & {} 0\nonumber \\ {\overline{\lambda }}_{-}\Gamma ^{23579}\lambda _{-}= & {} 0,\quad {\overline{\lambda }}_{-}\Gamma ^{23589}\lambda _{-}=0,\quad {\overline{\lambda }}_{-}\Gamma ^{23679}\lambda _{-}=0,\nonumber \\ {\overline{\lambda }}_{-}\Gamma ^{23689}\lambda _{-}= & {} 0,\nonumber \\ {\overline{\lambda }}_{-}\Gamma ^{24579}\lambda _{-}= & {} 0,\quad {\overline{\lambda }}_{-}\Gamma ^{24589}\lambda _{-}=0,\quad {\overline{\lambda }}_{-}\Gamma ^{24679}\lambda _{-}=0,\nonumber \\ {\overline{\lambda }}_{-}\Gamma ^{24689}\lambda _{-}= & {} 0. \end{aligned}$$

(2.16)

which can be rewritten as:

$$\begin{aligned}&<\lambda \mid Ca^{1}((a^{5}+a_{5})-i(a_{5}a^{5}-a^{5}a_{5}))a^{2}a^{3}a^{4}\mid \lambda >\nonumber \\&\quad = -(\lambda _{+})^{2}+(\lambda '_{5})^{2}+2i(\lambda _{+} \lambda '_{5})=0 \end{aligned}$$

(2.17)

$$\begin{aligned}&<\lambda \mid Ca^{1}((a^{5}+a_{5})-i(a_{5}a^{5}-a^{5}a_{5}))a^{2}a^{3}a_{4}\mid \lambda >\nonumber \\&\quad = -(\lambda _{45})^{2}+(\lambda '_{4})^{2}-2i(\lambda '_{4}\lambda _{45})=0 \end{aligned}$$

(2.18)

$$\begin{aligned}&<\lambda \mid Ca^{1}((a^{5}+a_{5})-i(a_{5}a^{5}-a^{5}a_{5}))a^{2}a_{3}a^{4}\mid \lambda >\nonumber \\&\quad = -(\lambda _{35})^{2}+(\lambda '_{3})^{2}-2i(\lambda '_{3}\lambda _{35})=0 \end{aligned}$$

(2.19)

$$\begin{aligned}&<\lambda \mid Ca^{1}((a^{5}+a_{5})-i(a_{5}a^{5}-a^{5}a_{5}))a_{2}a^{3}a^{4}\mid \lambda >\nonumber \\&\quad = -(\lambda _{52})^{2}+(\lambda '_{2})^{2}-2i(\lambda '_{2}\lambda _{52})=0 \end{aligned}$$

(2.20)

$$\begin{aligned}&<\lambda \mid Ca^{1}((a^{5}+a_{5})-i(a_{5}a^{5}-a^{5}a_{5}))a^{2}a_{3}a_{4}\mid \lambda >\nonumber \\&\quad = -(\lambda _{34})^{2}+4(\lambda ^{'12})^{2}-4i(\lambda _{34}\lambda ^{'12})=0 \end{aligned}$$

(2.21)

$$\begin{aligned}&<\lambda \mid Ca^{1}((a^{5}+a_{5})-i(a_{5}a^{5}-a^{5}a_{5}))a_{2}a^{3}a_{4}\mid \lambda >\nonumber \\&\quad = -(\lambda _{42})^{2}+4(\lambda ^{'13})^{2}-4i(\lambda _{42}\lambda ^{'13})=0 \end{aligned}$$

(2.22)

$$\begin{aligned}&<\lambda \mid Ca^{1}((a^{5}+a_{5})-i(a_{5}a^{5}-a^{5}a_{5}))a_{2}a_{3}a^{4}\mid \lambda >\nonumber \\&\quad = -(\lambda _{23})^{2}+4(\lambda ^{'14})^{2}-4i(\lambda _{23}\lambda ^{'14})=0 \end{aligned}$$

(2.23)

$$\begin{aligned}&<\lambda \mid Ca^{1}((a^{5}+a_{5})-i(a_{5}a^{5}-a^{5}a_{5}))a_{2}a_{3}a_{4}\mid \lambda >\nonumber \\&\quad = -(\lambda ^{1})^{2}+4(\lambda ^{'15})^{2}-4i(\lambda ^{1}\lambda ^{'15})=0 \end{aligned}$$

(2.24)

$$\begin{aligned}&<\lambda \mid Ca_{1}((a^{5}+a_{5})-i(a_{5}a^{5}-a^{5}a_{5}))a^{2}a^{3}a^{4}\mid \lambda >\nonumber \\&\quad = -(\lambda _{15})^{2}+(\lambda '_{1})^{2}+2i(\lambda '_{1}\lambda _{15})=0 \end{aligned}$$

(2.25)

$$\begin{aligned}&<\lambda \mid Ca_{1}((a^{5}+a_{5})-i(a_{5}a^{5}-a^{5}a_{5}))a^{2}a^{3}a_{4}\mid \lambda >\nonumber \\&\quad = -(\lambda _{14})^{2}+4(\lambda ^{'23})^{2}+4i(\lambda _{14}\lambda ^{'23})=0 \end{aligned}$$

(2.26)

$$\begin{aligned}&<\lambda \mid Ca_{1}((a^{5}+a_{5})-i(a_{5}a^{5}-a^{5}a_{5}))a^{2}a_{3}a^{4}\mid \lambda >\nonumber \\&\quad = -(\lambda _{13})^{2}+4(\lambda ^{'24})^{2}-4i(\lambda _{13}\lambda ^{'24})=0 \end{aligned}$$

(2.27)

$$\begin{aligned}&<\lambda \mid Ca_{1}((a^{5}+a_{5})-i(a_{5}a^{5}-a^{5}a_{5}))a_{2}a^{3}a^{4}\mid \lambda >\nonumber \\&\quad = -(\lambda _{12})^{2}+4(\lambda ^{'34})^{2}+4i(\lambda _{12}\lambda ^{'34})=0 \end{aligned}$$

(2.28)

$$\begin{aligned}&<\lambda \mid Ca_{1}((a^{5}+a_{5})-i(a_{5}a^{5}-a^{5}a_{5}))a^{2}a_{3}a_{4}\mid \lambda >\nonumber \\&\quad = -(\lambda ^{2})^{2}+4(\lambda ^{'52})^{2}-4i(\lambda ^{2}\lambda ^{'52})=0 \end{aligned}$$

(2.29)

$$\begin{aligned}&<\lambda \mid Ca_{1}((a^{5}+a_{5})-i(a_{5}a^{5}-a^{5}a_{5}))a_{2}a^{3}a_{4}\mid \lambda >\nonumber \\&\quad = -(\lambda ^{3})^{2}+4(\lambda ^{'53})^{2}+4i(\lambda ^{3}\lambda ^{'53})=0 \end{aligned}$$

(2.30)

$$\begin{aligned}&<\lambda \mid Ca_{1}((a^{5}+a_{5})-i(a_{5}a^{5}-a^{5}a_{5}))a_{2}a_{3}a^{4}\mid \lambda >\nonumber \\&\quad = -(\lambda ^{4})^{2}+4(\lambda ^{'45})^{2}+4i(\lambda ^{4}\lambda ^{'45})=0 \end{aligned}$$

(2.31)

$$\begin{aligned}&<\lambda \mid Ca_{1}((a^{5}+a_{5})-i(a_{5}a^{5}-a^{5}a_{5}))a_{2}a_{3}a_{4}\mid \lambda >\nonumber \\&\quad = -(\lambda ^{5})^{2}+(\lambda '_{+})^{2}-2i(\lambda ^{5}\lambda '_{+})=0 \end{aligned}$$

(2.32)

and the expression for $\mid \lambda>$ is given in (2.4). According to Comment 2 in [21, 22] we choose all the $\lambda$ s and $\lambda '$ s to be real. However the Dirac spinor becomes complex in a general Lorentz frame. Then the relations (2.17)-(2.32) imply that all the components of $\mid \lambda>$ are zero and the solution of the constraints in (2.16) is the trivial one. It is interesting to note that $\lambda _{-}$ corresponds to $\theta _{-}$ which is a gauge degree of freedom as pointed out in [16].

Thus, the novel nontrivial solution (2.12) found for one of the spinor fields actually proves that the pure spinor formalism can be successfully applied to non-relativistic systems with kappa symmetry as is the case of the IIA D2-brane.

3 Conclusions

This paper is the first attempt to extend the application of the pure spinor formalism to non-relativistic systems with kappa symmetry expecting to draw some conclusions for the general case as well. This can very useful especially in cases where it is difficult to solve the pure spinor constraints in the general case. We treated the non-relativistic IIA D2 brane in the framework of this formalism [23] and we derived the fermionic constraints corresponding to the rescaled fermionic coordinates. We introduced two commuting spinor fields each one corresponding to a fermionic coordinate. The nilpotency of the BRST charge leads to a set of constraints for the two spinor fields including pure spinor constraints. Nontrivial solutions are found for the spinor field $\lambda _{+}$ which corresponds to the fermionic coordinate $\theta _{+}$. It is interesting to note that this solution can be written as the sum of two pure spinors $\lambda _{1+}=P_+\lambda _+$ and $\lambda _{2+}=P_-\lambda _+$ where $P_\pm = \frac{1}{2}(1\pm \Gamma _0\Gamma _1\Gamma _2)$. The solution for the spinor field $\lambda _{-}$ corresponding to $\theta _{-}$ which, according to the proof given in [16] constitutes a gauge degree of freedom, is the trivial one. So for example the expression of the BRST charge for $\theta _{-}$= 0 in the Schrödinger representation is given by $Q=\int d^2\sigma [-i(\lambda _{1+}+\lambda _{2+})^\alpha \frac{\partial }{\partial \theta _{+}^\alpha } + 2T_{NR}\epsilon _{ijk}({\bar{\theta }}_{+}\Gamma ^{i}\lambda _{1+})\partial _{1}x^{j}\partial _{2}x^{k}]$. Cohomological issues concerning the BRST charge are currently under investigation [26,27,28,29]. This study can also be performed for more general manifolds and for general dimensions as well. We would like to finally mention that the treatment of the relativistic Dp-brane in the framework of the pure spinor formalism has been reported in [24].

Appendices

Appendix A

For the matrices $\Gamma ^{m}\, (m=0,..,9)$ we use the Majorana representation ($\Gamma ^{0}\,$ is real antisymmetric, $\Gamma ^{i}\, (i=1,...,9)$ are real symmetric)

$$\begin{aligned} \{\Gamma ^{m},\Gamma ^{n}\}=2\eta ^{mn} \end{aligned}$$

(4.1)

where $\eta ^{mn}=diag(-1,1,...,1)$ and $m,n=0,...,9;$ and

$$\begin{aligned} C= & {} \Gamma _{0},\nonumber \\ (\Gamma ^{m})^{T}= & {} -\Gamma _{0}\Gamma ^{m}\Gamma ^{-1}_{0}. \end{aligned}$$

(4.2)

A basis for the $32\times 32$ matrices is given by (see Appendix B in [25] for instance):

$$\begin{aligned} B= & {} \left\{ I,\Gamma ^{A}\Gamma ^{AB},\Gamma ^{ABC},\Gamma ^{ABCD},\Gamma ^{ABCDE},\Gamma ^{ABCD}\Gamma _{11},\right. \nonumber \\&\left. \Gamma ^{AB}\Gamma _{11},\Gamma ^{A}\Gamma _{11},\Gamma _{11}\right\} , \end{aligned}$$

(4.3)

with $A<B<C<D<E.$

For the differential form $d\theta$ we use the convention given in [18]:

$$\begin{aligned} d\theta =d\sigma ^{\mu }\partial _{\mu }\theta =-\partial _{\mu }\theta d\sigma ^{\mu }. \end{aligned}$$

(4.4)

The following identities hold for $\lambda _{-}=P_{-}\lambda$:

$$\begin{aligned} \Gamma _{\mu \nu }\lambda _{-}= & {} -\epsilon _{\mu \nu \rho }\Gamma ^{\rho }\lambda _{-}\nonumber \\ \Gamma _{\mu \nu \rho }\lambda _{-}= & {} -\epsilon _{\mu \nu \rho }\lambda _{-} \end{aligned}$$

(4.5)

$\epsilon _{012}=1.$

Appendix B

$$\begin{aligned} F_{\mu }= & {} 2[p_{\mu }-E^{i}({\overline{\theta }}_{-}\Gamma _{11}\Gamma _{\mu }\partial _{i}\theta _{+}+{\overline{\theta }}_{+} \Gamma _{11}\Gamma _{\mu }\partial _{i}\theta _{-})]\nonumber \\&+2T_{NR}\epsilon _{0jk}[-2({\overline{\theta }}_{+}\Gamma _{\mu \nu }\partial _{j}\theta _{+})R^{\nu }_{k} \nonumber \\&+({\overline{\theta }}_{+}\Gamma _{a\mu }\partial _{j}\theta _{-}+{\overline{\theta }}_{-}\Gamma _{a\mu }\partial _{j}\theta _{+})u^{a}_{k}]\nonumber \\&+\frac{T_{NR}}{96} \epsilon _{0jk}[({\overline{\theta }}_{-}\Gamma ^{\nu }\partial _{k}\theta _{-})(-89({\overline{\theta }}_{+}\Gamma _{\mu \nu }\partial _{j}\theta _{+}) \nonumber \\&+22({\overline{\theta }}_{+}\partial _{j}\theta _{+})\eta _{\mu \nu })+({\overline{\theta }}_{+}\Gamma ^{\nu }\partial _{k}\theta _{+})(83({\overline{\theta }}_{-} \Gamma _{\mu \nu }\partial _{j}\theta _{-})\nonumber \\&+6({\overline{\theta }}_{-}\partial _{j}\theta _{-})\eta _{\mu \nu })-2({\overline{\theta }}_{-}\Gamma ^{a}\partial _{k}\theta _{+}\nonumber \\&+{\overline{\theta }}_{+}\Gamma ^{a}\partial _{k}\theta _{-})(47({\overline{\theta }}_{+}\Gamma _{\mu a }\partial _{j}\theta _{-})+23({\overline{\theta }}_{-}\Gamma _{\mu a }\partial _{j}\theta _{+}))\nonumber \\&-5({\overline{\theta }}_{+}\Gamma _{\mu }\Gamma _{\nu a }\partial _{j}\theta _{-})(\partial _{k}{\overline{\theta }}_{+}\Gamma ^{\nu a }\theta _{-})\nonumber \\&-\frac{1}{3}(\partial _{k}{\overline{\theta }}_{+}\Gamma ^{a}\theta _{-})(37(\partial _{j}{\overline{\theta }}_{-}\Gamma _{\mu a }\theta _{+})\nonumber \\&-66(\partial _{j}{\overline{\theta }}_{+}\Gamma _{\mu a }\theta _{-}))-(\partial _{j}\theta _{-}\Gamma ^{a}\theta _{+}) (21(\partial _{k}{\overline{\theta }}_{+}\Gamma _{\mu a }\theta _{-})\nonumber \\&-2(\partial _{k}\theta _{-}\Gamma _{\mu a}\theta _{+}))-2({\overline{\theta }}_{+}\Gamma _{11}\partial \theta _{-}+{\overline{\theta }}_{-}\Gamma _{11}\partial \theta _{+})\nonumber \\&\times (25({\overline{\theta }}_{+}\Gamma _{11}\Gamma _{\mu }\partial _{j}\theta _{-})+31({\overline{\theta }}_{-}\Gamma _{11}\Gamma _{\mu }\partial _{j}\theta _{+}))\nonumber \\&+2({\overline{\theta }}_{-}\Gamma _{11}\Gamma ^{\nu }\partial _{k}\theta _{+}+{\overline{\theta }}_{+}\Gamma _{11}\Gamma ^{\nu }\partial _{k}\theta _{-})\nonumber \\&\times ({\overline{\theta }}_{+}\Gamma _{11}\Gamma _{\mu }\Gamma _{\nu }\partial _{j}\theta _{-}+2({\overline{\theta }}_{-}\Gamma _{11}\partial _{j}\theta _{+})\eta _{\mu \nu })\nonumber \\&- (\partial _{j}{\overline{\theta }}_{-}\Gamma _{11}\Gamma ^{\nu }\theta _{-})(64(\partial _{k}{\overline{\theta }}_{+}\Gamma _{11}\theta _{-})\nonumber \\&-11(\partial _{k}{\overline{\theta }}_{+}\Gamma _{11}\Gamma _{\mu \nu }\theta _{-})+46(\partial _{k}{\overline{\theta }}_{-}\Gamma _{11}\theta _{+})\eta _{\mu \nu }\nonumber \\&- 2(\partial _{k}{\overline{\theta }}_{-}\Gamma _{11}\Gamma _{\nu }\Gamma _{\mu }\theta _{+}))+({\overline{\theta }}_{-}\Gamma _{11}\Gamma _{\nu }\partial _{k}\theta _{+}) \nonumber \\&\times (9({\overline{\theta }}_{-}\Gamma _{11}\Gamma _{\nu }\Gamma _{\mu }\theta _{+})+34({\overline{\theta }}_{-}\Gamma _{11}\partial _{j}\theta _{+})\eta _{\mu \nu }\nonumber \\&+23({\overline{\theta }}_{+}\Gamma _{11}\partial _{j}\theta _{-})\eta _{\mu \nu })-7(\partial _{j}{\overline{\theta }}_{-}\Gamma _{11}\Gamma _{a}\theta _{-})\nonumber \\&\times (\partial _{k}{\overline{\theta }}_{+}\Gamma _{11}\Gamma _{\mu a}\theta _{+})+5(\partial _{k}{\overline{\theta }}_{+}\Gamma _{11}\Gamma ^{\nu a }\Gamma _{\mu }\theta _{+}) \end{aligned}$$

$$\begin{aligned}&\qquad ({\overline{\theta }}_{-}\Gamma _{\nu a}\Gamma _{11}\partial _{j}\theta _{-})+({\overline{\theta }}_{-}\Gamma ^{\nu a }\Gamma _{11}\Gamma _{\mu }\partial _{j}\theta _{-})\nonumber \\&\qquad \times (\partial _{k}{\overline{\theta }}_{+}\Gamma _{\nu a}\Gamma _{11}\theta _{+})-(\partial _{k}{\overline{\theta }}_{+}\Gamma _{11}\Gamma ^{a}\theta _{+})\nonumber \\&\qquad (\partial _{j}{\overline{\theta }}_{-}\Gamma _{11}\Gamma _{\mu a }\theta _{-})+({\overline{\theta }}_{+}\Gamma _{\nu a}\partial _{j}\theta _{-})({\overline{\theta }}_{-}\Gamma ^{\nu a}\Gamma _{\mu }\partial _{k}\theta _{+})]\nonumber \\ \end{aligned}$$

(5.1)

$$\begin{aligned} F_{a}= & {} -2E_{i}u_{a}^{i}+2T_{NR}\epsilon _{0jk}({\overline{\theta }}_{+}\Gamma _{11}\partial _{j}\theta _{-}+{\overline{\theta }}_{-}\Gamma _{11} \partial _{j}\theta _{+})u_{ak}\nonumber \\&+\frac{T_{NR}}{3}\epsilon _{0jk}[-\frac{1}{16}({\overline{\theta }}_{+}\Gamma ^{b}\partial _{j}\theta _{-})\nonumber \\&\times (10({\overline{\theta }}_{-}\Gamma _{ab}\Gamma _{11}\partial _{k}\theta _{+})-({\overline{\theta }}_{+}\Gamma _{ab}\Gamma _{11}\partial _{k}\theta _{-})\nonumber \\&+3( \partial _{k}\theta _{-}\Gamma _{11}\theta _{+})\eta _{ab}+45(\partial _{k}{\overline{\theta }}_{+}\Gamma _{11}\theta _{-})\eta _{ab})\nonumber \\&+\frac{3}{16}({\overline{\theta }}_{-}\Gamma ^{b}\partial _{j}\theta _{+})(-8(\partial _{k}{\overline{\theta }}_{+}\Gamma _{ab}\Gamma _{11}\theta _{-})\nonumber \\&+12 ({\overline{\theta }}_{-}\Gamma _{11}\partial _{k}\theta _{+})\eta _{ab}-({\overline{\theta }}_{+}\Gamma _{11}\partial _{k}\theta _{-})\eta _{ab})\nonumber \\&+\frac{7}{8}({\overline{\theta }}_{+}\Gamma ^{\mu }\partial _{k}\theta _{+})(\partial _{j}{\overline{\theta }}_{-}\Gamma _{\mu a}\Gamma _{11}\theta _{-})\nonumber \\&+\frac{1}{48}({\overline{\theta }}_{+}\Gamma _{11}\Gamma ^{\mu }\partial _{k}\theta _{-})(5(\partial _{j}{\overline{\theta }}_{+}\Gamma _{\mu a}\theta _{-})-6(\partial _{j}{\overline{\theta }}_{-}\Gamma _{\mu a}\theta _{+}))\nonumber \\&-\frac{1}{16}(\partial _{j}{\overline{\theta }}_{-}\Gamma _{11}\Gamma ^{b}\theta _{-})(13(\partial _{k}{\overline{\theta }}_{+}\Gamma _{ab}\theta _{+})\nonumber \\&+3(\overline{\theta }_{+}\partial _{k}\theta _{+})\eta _{ab})-\frac{1}{32}(\partial _{j}{\overline{\theta }}_{-}\Gamma _{abc}\theta _{+})\nonumber \\&\times (2(\partial _{k}{\overline{\theta }}_{+}\Gamma ^{bc} \Gamma _{11}\theta _{-})+\partial _{k}{\overline{\theta }}_{-}\Gamma ^{bc}\Gamma _{11}\theta _{+})\nonumber \\&+\frac{1}{16}(\partial _{j}{\overline{\theta }}_{-}\Gamma ^{\mu b}\theta _{+})(-2(\partial _{k}{\overline{\theta }}_{-}\Gamma _{\mu ab}\Gamma _{11}\theta _{+})\nonumber \\&-6(\partial _{k}{\overline{\theta }}_{+}\Gamma _{11}\Gamma _{\mu ab}\theta _{-})+9({\overline{\theta }}_{-}\Gamma _{11}\Gamma _{\mu }\partial _{k}\theta _{+})\eta _{ab}\nonumber \\&+({\overline{\theta }}_{+}\Gamma _{11}\Gamma _{\mu }\partial _{k} \theta _{-})\eta _{ab})-\frac{9}{16}({\overline{\theta }}_{-}\Gamma ^{\mu }\partial _{j}\theta _{-})\nonumber \\&\times ({\overline{\theta }}_{+}\Gamma _{\mu a}\Gamma _{11}\partial _{k}\theta _{+})+\frac{3}{8}({\overline{\theta }}_{+}\Gamma ^{\mu b}\Gamma _{11}\partial _{k}\theta _{+})(\partial _{j}{\overline{\theta }}_{-}\Gamma _{\mu ab}\theta _{-})\nonumber \\&+\frac{117}{64}({\overline{\theta }}_{-}\partial _{j}\theta _{-})({\overline{\theta }}_{+}\Gamma _{11}\Gamma _{a}\partial _{k}\theta _{+})\nonumber \\&+\frac{137}{64}({\overline{\theta }}_{-}\Gamma _{ab}\partial _{j}\theta _{-})({\overline{\theta }}_{+}\Gamma _{11}\Gamma ^{b}\partial _{k}\theta _{+})\nonumber \\&+ \frac{1}{16}({\overline{\theta }}_{-}\Gamma ^{\mu b}\partial _{j}\theta _{+})(\partial _{k}{\overline{\theta }}_{-}\Gamma _{11}\Gamma _{\mu ab}\theta _{+})\nonumber \\&- \frac{1}{16}({\overline{\theta }}_{-}\Gamma _{11}\Gamma ^{\mu b}\partial _{j}\theta _{-})\nonumber \\&\times (\partial _{k}{\overline{\theta }}_{+}\Gamma _{\mu ab}\theta _{+})-\frac{1}{16}({\overline{\theta }}_{-}\Gamma ^{bc}\partial _{k}\theta _{-})({\overline{\theta }}_{+}\Gamma _{abc}\Gamma _{11}\partial _{j}\theta _{+})]\nonumber \\ \end{aligned}$$

(5.2)

$$\begin{aligned} F_{ab}= & {} T_{NR}\epsilon _{0jk}u_{aj}u_{bk}+\frac{T_{NR}}{16}\epsilon _{0jk}[({\overline{\theta }}_{+}\Gamma ^{d}\partial _{j}\theta _{-})\nonumber \\&\times (-\frac{14}{3} (\partial _{k}{\overline{\theta }}_{-}\Gamma _{ab}\Gamma _{d}\theta _{+})\nonumber \\&+\frac{14}{3}(\partial _{k}{\overline{\theta }}_{-}\Gamma _{d}\Gamma _{ab}\theta _{+})+\frac{35}{6}(\partial _{k}{\overline{\theta }}_{+}\Gamma _{ab}\Gamma _{d}\theta _{-})\nonumber \\&-\frac{15}{2}( \partial _{k}{\overline{\theta }}_{+}\Gamma _{d}\Gamma _{ab}\theta _{-})+\frac{16}{3}({\overline{\theta }}_{+}\Gamma _{b}\partial _{k}\theta _{-})\eta _{ad}\nonumber \\&+\frac{32}{3}({\overline{\theta }}_{-}\Gamma _{b}\partial _{k}\theta _{+})\eta _{ad})+\frac{1}{6}({\overline{\theta }}_{-}\Gamma ^{d}\partial _{j}\theta _{+})(7(\partial _{k} {\overline{\theta }}_{+}\Gamma _{d}\Gamma _{ab}\theta _{-})\nonumber \\&+35(\partial _{k}{\overline{\theta }}_{+}\Gamma _{ab}\Gamma _{d}\theta _{-})\nonumber \\&+4(\partial _{k}{\overline{\theta }}_{-}\Gamma _{d}\Gamma _{ab}\theta _{+})-33(\partial _{k}{\overline{\theta }}_{-}\Gamma _{ab}\Gamma _{d}\theta _{+})\nonumber \\&-32({\overline{\theta }} _{-}\Gamma _{a}\partial _{k}\theta _{+})\eta _{bd})+\frac{1}{6}({\overline{\theta }}_{-}\Gamma _{11}\Gamma ^{\mu }\partial _{k}\theta _{+})\nonumber \\&\times (-4(\partial _{j}{\overline{\theta }}_{+}\Gamma _{11}\Gamma _{\mu ab}\theta _{-})+37(\partial _{j}{\overline{\theta }}_{-}\Gamma _{11}\Gamma _{\mu ab}\theta _{-}))\nonumber \\ \end{aligned}$$

$$\begin{aligned}&\qquad +\frac{15}{4}({\overline{\theta }}_{+}\Gamma ^{\mu d}\Gamma _{11}\partial _{k}\theta _{+})(\partial _{j}{\overline{\theta }}_{-}\Gamma _{11}\Gamma _{\mu d}\Gamma _{ab}\theta _{-})\nonumber \\&\qquad +\frac{25}{12}(\partial _{j}{\overline{\theta }}_{-}\Gamma ^{\mu d}\theta _{+})(\partial _{k}{\overline{\theta }}_{+}\Gamma _{\mu d}\Gamma _{ab}\theta _{-}) \nonumber \\&\qquad - \frac{77}{12}({\overline{\theta }}_{+}\Gamma ^{d}\Gamma _{11}\partial _{k}\theta _{+})(\partial _{j}{\overline{\theta }}_{-}\Gamma _{11}\Gamma _{d}\Gamma _{ab}\theta _{-})\nonumber \\&\qquad -\frac{35}{4}(\partial _{j}{\overline{\theta }}_{-}\Gamma _{11}\theta _{+})(\partial _{k}{\overline{\theta }}_{+}\Gamma _{11}\Gamma _{ab}\theta _{-})\nonumber \\&\qquad -\frac{45}{4}({\overline{\theta }} _{-}\Gamma _{ab}\partial _{j}\theta _{-})({\overline{\theta }}_{+}\partial _{k}\theta _{+})+\frac{37}{4}({\overline{\theta }}_{+}\Gamma ^{\mu }\partial _{k}\theta _{+})\nonumber \\&\qquad \times (\partial _{j}{\overline{\theta }}_{-}\Gamma _{\mu ab}\theta _{-})+\frac{1}{6}({\overline{\theta }}_{-}\Gamma _{11}\Gamma ^{d}\partial _{j}\theta _{-})(5(\partial _{k}{\overline{\theta }}_{+}\Gamma _{11}\Gamma _{ab}\Gamma _{d} \theta _{+})\nonumber \\&\qquad +39(\partial _{k}{\overline{\theta }}_{+}\Gamma _{d}\Gamma _{ab}\Gamma _{11}\theta _{+}))-3({\overline{\theta }}_{-}\Gamma _{11}\partial _{k}\theta _{+})(\partial _{j}{\overline{\theta }}_{-}\Gamma _{11}\Gamma _{ab}\theta _{+})\nonumber \\&\qquad +({\overline{\theta }}_{-}\Gamma ^{\mu d}\partial _{j}\theta _{+})(\partial _{k}{\overline{\theta }}_{-}\Gamma _{ab}\Gamma _{\mu d}\theta _{+})-({\overline{\theta }}_{-}\Gamma _{df}\partial _{k}\theta _{-})\nonumber \\&\qquad \times ({\overline{\theta }}_{+}\Gamma ^{d}\Gamma _{ab}\Gamma ^{f}\partial _{j}\theta _{+})-\frac{1}{4}({\overline{\theta }}_{+}\Gamma _{11}\Gamma ^{\mu }\partial _{k}\theta _{-})( \partial _{j}{\overline{\theta }}_{+}\Gamma _{11}\Gamma _{\mu ab}\theta _{-})\nonumber \\&\qquad -\frac{1}{2}({\overline{\theta }}_{-}\partial _{j}\theta _{-})(\partial _{k}{\overline{\theta }}_{+}\Gamma _{ab}\theta _{+})\nonumber \\&\qquad -\frac{1}{6}({\overline{\theta }}_{-}\Gamma _{11}\Gamma ^{\mu d}\partial _{j}\theta _{-})(\partial _{k}{\overline{\theta }}_{+}\Gamma _{11}\Gamma _{\mu ab}\Gamma _{d}\theta _{+})\nonumber \\&\qquad +\frac{1}{6}(\partial _{k}{\overline{\theta }}_{+}\Gamma _{df}\Gamma _{11}\theta _{-})(\partial _{j}{\overline{\theta }}_{-}\Gamma _{11}\Gamma ^{d} \Gamma _{ab}\Gamma ^{f}\theta _{+})\nonumber \\&\qquad -4({\overline{\theta }}_{-}\Gamma ^{\mu }\partial _{j}\theta _{-})(\partial _{k}{\overline{\theta }}_{+}\Gamma _{\mu ab}\theta _{+})] \end{aligned}$$

(5.3)

$$\begin{aligned} {\overline{F}}= & {} 2T_{NR}\epsilon _{0jk}\Big [\widetilde{{{{\mathcal {F}}}}}_{jk}^{(1)}-\frac{1}{3}(({\overline{\theta }}_{-}\Gamma ^{\mu }\partial _{k}\theta _{-})({\overline{\theta }}_{+}\Gamma _{\mu }\Gamma _{11}\partial _{j}\theta _{-} \nonumber \\&+ {\overline{\theta }}_{-}\Gamma _{\mu }\Gamma _{11}\partial _{j}\theta _{+})\nonumber \\&+({\overline{\theta }}_{-}\Gamma ^{a}\Gamma _{11}\partial _{j}\theta _{-})({\overline{\theta }}_{-}\Gamma _{a}\partial _{k}\theta _{+}+{\overline{\theta }}_{+}\Gamma _{a} \partial _{k}\theta _{-}))\Big ] \end{aligned}$$

(5.4)

$$\begin{aligned} {\overline{F}}_{\nu }= & {} -4E^{i}R_{\nu i}{+}4T_{NR}\epsilon _{0jk}R_{\nu k}({\overline{\theta }}_{+}\Gamma _{11}\partial _{j}\theta _{-}{+}{\overline{\theta }} {-}\Gamma _{11}\partial _{j}\theta _{+})\nonumber \\&+\frac{1}{48}T_{NR}\epsilon _{0jk}[-({\overline{\theta }}_{-}\Gamma ^{\mu a }\partial _{j}\theta _{+})\nonumber \\&\times (16(\partial _{k}{\overline{\theta }}_{-}\Gamma _{11}\Gamma _{a}\theta _{-})\eta _{\mu \nu }-5(\partial _{k}{\overline{\theta }}_{-}\Gamma _{\mu \nu a }\Gamma _{11}\theta _{-}))\nonumber \\&-12(\partial _{k}{\overline{\theta }}_{-}\Gamma _{11}\Gamma ^{a}\theta _{-})({\overline{\theta }}_{+}\Gamma _{\nu a }\partial _{j}\theta _{-})\nonumber \\&-({\overline{\theta }}_{-}\Gamma ^{\mu }\partial _{j}\theta _{-})(-37(\partial _{k}{\overline{\theta }}_{+}\Gamma _{\mu \nu }\Gamma _{11}\theta _{-}) \nonumber \\&+ 41(\partial _{k}{\overline{\theta }}_{+}\Gamma _{11}\theta _{-})\eta _{\mu \nu }+18(\partial _{k}{\overline{\theta }}_{-}\Gamma _{11}\theta _{+})\eta _{\mu \nu }) \nonumber \\&+(\partial _{j}{\overline{\theta }}_{-}\Gamma _{\nu a}\Gamma _{b}\theta _{-})(-5(\partial _{k}{\overline{\theta }}_{+}\Gamma ^{b}\Gamma ^{a}\Gamma _{11}\theta _{-})\nonumber \\&+2(\partial _{k}{\overline{\theta }}_{-} \Gamma ^{b}\Gamma ^{a}\Gamma _{11}\theta _{+}))-({\overline{\theta }}_{-}\Gamma ^{ab}\partial _{k}\theta _{-})\nonumber \\&\times (7({\overline{\theta }}_{-}\Gamma _{\nu ab}\Gamma _{11}\partial _{j}\theta _{+})+2({\overline{\theta }}_{+}\Gamma _{\nu ab }\Gamma _{11}\partial _{j}\theta _{-}))\nonumber \\&+(\partial _{j}{\overline{\theta }}_{+}\Gamma _{11}\Gamma ^{\mu }\theta _{-})(70({\overline{\theta }}_{-}\partial _{k}\theta _{-})\eta _{\mu \nu }\nonumber \\&+59(\partial _{k}{\overline{\theta }}_{-}\Gamma _{\mu \nu }\theta _{-}))+(\partial _{j}{\overline{\theta }}_{-}\Gamma _{\nu a}\Gamma _{11}\theta _{-})\nonumber \\&\times (49({\overline{\theta }}_{-}\Gamma ^{a}\partial _{k}\theta _{+})+24({\overline{\theta }}_{+}\Gamma ^{a}\partial _{k}\theta _{-})) \nonumber \\&-5(\partial _{k}{\overline{\theta }}_{+}\Gamma ^{\mu a}\Gamma _{\nu }\theta _{-})({\overline{\theta }}_{-}\Gamma _{11}\Gamma _{\mu a }\partial _{j}\theta _{-})] \end{aligned}$$

(5.5)

$$\begin{aligned} {\overline{F}}_{a}= & {} 4(p_{a}-E^{i}({\overline{\theta }}_{-}\Gamma _{11}\Gamma _{a}\partial _{i}\theta _{-}))\nonumber \\&-2T_{NR}\epsilon _{0jk}[({\overline{\theta }} _{-}\Gamma _{\mu a}\partial _{k}\theta _{+}+{\overline{\theta }}_{+}\Gamma _{\mu a}\partial _{k}\theta _{-})R^{\mu }_{j}\nonumber \\&-({\overline{\theta }}_{-}\Gamma _{ab}\partial _{k}\theta _{-})u^{b}_{j}]+\frac{2}{3}T_{NR}\epsilon _{0jk}[\frac{1}{16}({\overline{\theta }}_{-} \Gamma ^{\mu }\partial _{j}\theta _{-})\nonumber \\&\times (25(\partial _{k}{\overline{\theta }}_{+}\Gamma _{\mu a }\theta _{-})+19(\partial _{k}\theta _{-}\Gamma _{\mu a}\theta _{+}))\nonumber \\&+\frac{1}{8}({\overline{\theta }}_{-}\partial _{j}\theta _{-})(21(\partial _{k}{\overline{\theta }}_{+}\Gamma _{a}\theta _{-})+9(\partial _{k} {\overline{\theta }}_{-}\Gamma _{a}\theta _{+})) \end{aligned}$$

$$\begin{aligned}&\quad -\frac{1}{32}(\partial _{j}{\overline{\theta }}_{-}\Gamma _{\mu ab }\theta _{-})(5({\overline{\theta }}_{-}\Gamma ^{\mu b}\partial _{k}\theta _{+})\nonumber \\&\quad -2({\overline{\theta }}_{+}\Gamma ^{\mu b}\partial _{k}\theta _{-}))+\frac{1}{32}({\overline{\theta }}_{-}\Gamma _{ab}\partial _{k}\theta _{-})(7({\overline{\theta }}_{-}\Gamma ^{b}\partial _{j} \theta _{+})\nonumber \\&\quad +12({\overline{\theta }}_{+}\Gamma ^{b}\partial _{j}\theta _{-}))+\frac{1}{16}(\partial _{j}{\overline{\theta }}_{-}\Gamma _{11}\Gamma _{\mu a }\theta _{-})\nonumber \\&\quad \times (3(\partial _{k}{\overline{\theta }}_{-}\Gamma _{11}\Gamma ^{\mu }\theta _{+})+10(\partial _{k}{\overline{\theta }}_{+}\Gamma _{11}\Gamma ^{\mu }\theta _{-})) \nonumber \\&\quad +\frac{1}{16}({\overline{\theta }}_{-}\Gamma _{a}\Gamma _{11}\partial _{j}\theta _{-})(29({\overline{\theta }}_{+}\Gamma _{11}\partial _{k}\theta _{-})\nonumber \\&\quad +20({\overline{\theta }}_{-}\Gamma _{11}\partial _{k}\theta _{+} ))+\frac{1}{32}({\overline{\theta }}_{-}\Gamma _{11}\Gamma ^{b}\partial _{k}\theta _{-})\nonumber \\&\quad \times (2({\overline{\theta }}_{+}\Gamma _{11}\Gamma _{ab}\partial _{j}\theta _{-})-{\overline{\theta }}_{-}\Gamma _{11}\Gamma ^{ab}\partial _{j}\theta _{+})\nonumber \\&\quad +\frac{1}{32}(\partial _{j}{\overline{\theta }}_{-}\Gamma _{11}\Gamma _{abc}\theta _{-})({\overline{\theta }}_{+}\Gamma _{11}\Gamma ^{bc}\partial _{k}\theta _{-}\nonumber \\&\quad - 5({\overline{\theta }}_{-}\Gamma _{11}\Gamma ^{bc}\partial _{k}\theta _{+}))+\frac{1}{32}({\overline{\theta }}_{-}\Gamma ^{bc}\partial _{j}\theta _{-})\nonumber \\&\quad \times (7(\partial _{k}{\overline{\theta }}_{+}\Gamma _{abc}\theta _{-})+2(\partial _{k}{\overline{\theta }}_{-}\Gamma _{abc}\theta _{+}))\nonumber \\&\quad -\frac{5}{32}(\partial _{k} {\overline{\theta }}_{+}\Gamma _{11}\Gamma _{\mu ab}\theta _{-})({\overline{\theta }}_{-}\Gamma _{11}\Gamma ^{\mu b}\partial _{j}\theta _{-})] \end{aligned}$$

(5.6)

$$\begin{aligned} {\overline{F}}_{\nu a}= & {} 4T_{NR}\epsilon _{0jk}R_{\nu j}u_{ak}+\frac{1}{48}T_{NR}\epsilon _{0jk}[({\overline{\theta }}_{-}\Gamma ^{\mu } \partial _{j}\theta _{+})\nonumber \\&\times (37({\overline{\theta }}_{-}\Gamma _{a}\partial _{k}\theta _{+})\eta _{\mu \nu }\nonumber \\&+34({\overline{\theta }}_{+}\Gamma _{a}\partial _{k}\theta _{-}) \eta _{\mu \nu }-5({\overline{\theta }}_{-}\Gamma _{\mu \nu a}\partial _{k}\theta _{+})\nonumber \\&+6({\overline{\theta }}_{+}\Gamma _{\mu \nu a}\partial _{k}\theta _{-}))+ ({\overline{\theta }}_{-}\Gamma _{11}\Gamma ^{b}\partial _{j}\theta _{-})\nonumber \\&\times (5(\partial _{k}{\overline{\theta }}_{+}\Gamma _{11}\Gamma _{\nu }(\eta _{ab}+ \Gamma _{ba})\theta _{-})\nonumber \\&-2(\partial _{k}{\overline{\theta }}_{-}\Gamma _{11}\Gamma _{\nu ba}\theta _{+}))-({\overline{\theta }}_{-}\Gamma ^{b}\partial _{k}\theta _{+})\nonumber \\&\times (-19(\partial _{j}{\overline{\theta }}_{-}\Gamma _{\nu ba}\theta _{-})+9(\partial _{j}{\overline{\theta }}_{-}\Gamma _{\nu }\theta _{-})\eta _{ab}) \nonumber \\&+ ({\overline{\theta }}_{-}\Gamma ^{b\mu }\Gamma _{11}\partial _{j}\theta _{-})(\partial _{k}{\overline{\theta }}_{+}\Gamma _{\mu }\Gamma _{11}\Gamma _{\nu a}\Gamma _{b}\theta _{-})\nonumber \\&+5({\overline{\theta }}_{-}\Gamma ^{\mu }\Gamma _{11}\partial _{k}\theta _{+})(\partial _{j}{\overline{\theta }}_{-}\Gamma _{\mu }\Gamma _{11}\Gamma _{\nu a}\theta _{-})\nonumber \\&-({\overline{\theta }}_{-}\Gamma ^{bc}\Gamma _{11}\partial _{k}\theta _{+})(\partial _{j}{\overline{\theta }}_{-}\Gamma _{11}\Gamma _{\nu abc}\theta _{-})\nonumber \\&+(\partial _{k}{\overline{\theta }}_{+}\Gamma ^{b\mu }\theta _{-})(\partial _{j}{\overline{\theta }}_{-}\Gamma _{\mu }\Gamma _{\nu a}\Gamma _{b}\theta _{-}) \nonumber \\&- 5({\overline{\theta }}_{-}\Gamma _{11}\partial _{k}\theta _{+})({\overline{\theta }}_{-}\Gamma _{\nu a}\Gamma _{11}\partial _{j}\theta _{-})- 22({\overline{\theta }}_{+}\Gamma ^{b}\partial _{j}\theta _{-})\nonumber \\&\times ({\overline{\theta }}_{-}\Gamma _{\nu ba}\partial _{k}\theta _{-})+2({\overline{\theta }}_{+}\Gamma _{11}\Gamma ^{\mu }\partial _{k}\theta _{-})(- \partial _{j}{\overline{\theta }}_{-}\Gamma _{11}\Gamma _{\mu \nu a}\theta _{-}\nonumber \\&+2(\partial _{j}{\overline{\theta }}_{-}\Gamma _{11}\Gamma _{a}\theta _{-})\eta _{\mu \nu })\nonumber \\&+2(\partial _{k}{\overline{\theta }}_{-}\Gamma ^{b\mu }\theta _{+})(2(\partial _j{\overline{\theta }}_{-}\Gamma _{\mu \nu }\theta _{-})\eta _{ab}+(\partial _{j} {\overline{\theta }}_{-}\Gamma _{\mu \nu ba}\theta _{-})\nonumber \\&-2(\partial _{j}{\overline{\theta }}_{-}\Gamma _{ba}\theta _{-})\eta _{\mu \nu }\nonumber \\&-(\partial _{j}{\overline{\theta }}_{-}\theta _{-})\eta _{\mu \nu }\eta _{ab})-2(\partial _{j}{\overline{\theta }}_{-}\Gamma _{11}\theta _{+})(\partial _{k} {\overline{\theta }}_{-}\Gamma _{11}\Gamma _{\nu a}\theta _{-})\nonumber \\&+(\partial _{k}{\overline{\theta }}_{-}\Gamma ^{bc}\theta _{-})\nonumber \\&\times (\partial _{j}{\overline{\theta }} _{-}\Gamma _{\nu abc}\theta _{+}-2(\partial _{j}{\overline{\theta }}_{+}\Gamma _{\nu abc}\theta _{-}))] \end{aligned}$$

(5.7)

$$\begin{aligned} {\overline{F}}^{\nu ab}= & {} \frac{1}{96}T_{NR}\epsilon _{0jk}[(\partial _{k}{\overline{\theta }}_{-}\Gamma ^{\mu }\Gamma _{11}\Gamma ^{\nu ab}\Gamma ^{d}\theta _{-})\nonumber \\&\times (4(\partial _{j}{\overline{\theta }}_{-}\Gamma _{d\mu }\theta _{+})-\partial _{j}{\overline{\theta }}_{+}\Gamma _{d\mu }\theta _{-})\nonumber \\&-6(\partial _{k}{\overline{\theta }}_{-}\Gamma ^{d}\theta _{+})(\partial _{j}{\overline{\theta }}_{-}(\Gamma _{d}\Gamma ^{\nu ab}+3\Gamma ^{\nu ab}\Gamma _{d})\Gamma _{11}\theta _{-})\nonumber \\&+3({\overline{\theta }}_{-}\Gamma ^{d}\partial _{k}\theta _{+})\nonumber \\&\times (\partial _{j}{\overline{\theta }}_{-}(\Gamma _{d}\Gamma ^{\nu ab}+6\Gamma ^{\nu ab}\Gamma _{d})\Gamma _{11}\theta _{-})\nonumber \\&+(\partial _{k}{\overline{\theta }} _{+}\Gamma _{dc}\Gamma _{11}\theta _{-})(\partial _{j}{\overline{\theta }}_{-}\Gamma ^{d}\Gamma ^{\nu ab}\Gamma ^{c}\theta _{-})\\&-3(\partial _{k}{\overline{\theta }}_{+}\Gamma _{11}\theta _{-})(\partial _{j}{\overline{\theta }}_{-}\Gamma ^{\nu ab}\theta _{-}) \end{aligned}$$

$$\begin{aligned}&+ 3(\partial _{j}{\overline{\theta }}_{-}\Gamma ^{\mu }\theta _{-})(\partial _{k}{\overline{\theta }}_{+}\Gamma _{\mu }\Gamma _{11}\Gamma ^{\nu ab}\theta _{-})\nonumber \\&-({\overline{\theta }}_{-}\Gamma _{cd}\partial _{j}\theta _{-})(\partial _{k}{\overline{\theta }}_{+}\Gamma ^{c}\Gamma _{11}\Gamma ^{\nu ab}\Gamma ^{d}\theta _{-})\nonumber \\&+({\overline{\theta }}_{-}\Gamma _{d\mu }\Gamma _{11}\partial _{j}\theta _{-})(\partial _{k}{\overline{\theta }}_{+}\Gamma ^{\mu }\Gamma ^{\nu ab}\Gamma ^{d}\theta _{-})\nonumber \\&+3({\overline{\theta }}_{-}\Gamma ^{\mu }\Gamma _{11}\partial _{k}\theta _{+})(\partial _{j}{\overline{\theta }}_{-}\Gamma _{\mu }\Gamma ^{\nu ab}\theta _{-})\nonumber \\&+ 3({\overline{\theta }}_{-}\partial _{j}\theta _{-})(\partial _{k}{\overline{\theta }}_{+}\Gamma _{11}\Gamma ^{\nu ab}\theta _{-})\nonumber \\&-3({\overline{\theta }}_{-}\Gamma _{11}\Gamma ^{d}\partial _{j}\theta _{-})({\overline{\theta }}_{-}\Gamma ^{\nu ab}\Gamma _{d}\partial _{k}\theta _{-})] \end{aligned}$$

(5.8)

$$\begin{aligned} {\overline{F}}^{lmn}= & {} \frac{1}{48}T_{NR}\epsilon _{0jk}[-\frac{1}{4}({\overline{\theta }}_{+}\Gamma ^{d}\partial _{k}\theta _{-})\nonumber \\&\times ({\overline{\theta }}_{-}\Gamma ^{lmn} \Gamma _{d}\partial _{j}\theta _{-}+{\overline{\theta }}_{-}\Gamma _{d}\Gamma ^{lmn}\partial _{j}\theta _{-})\nonumber \\&-\frac{1}{32}({\overline{\theta }}_{-}\Gamma ^{d}\partial _{k}\theta _{+})(5({\overline{\theta }}_{-}\Gamma ^{lmn}\Gamma _{d}\partial _{j}\theta _{-})\nonumber \\&+6({\overline{\theta }} _{-}\Gamma _{d}\Gamma ^{lmn}\partial _{j}\theta _{-}))+\frac{1}{32}({\overline{\theta }}_{-}\Gamma _{\mu }\partial _{j}\theta _{-})\nonumber \\&\times (\partial _{k}{\overline{\theta }}_{+}\Gamma ^{\mu lmn}\theta _{-})+\frac{1}{32}({\overline{\theta }}_{-}\Gamma _{\mu d}\partial _{j}\theta _{+})\nonumber \\&\times ({\overline{\theta }}_{-}\Gamma ^{d}\Gamma ^{\mu lmn}\partial _{k}\theta _{-})+\frac{1}{32}(\partial _{k}{\overline{\theta }}_{+}\Gamma _{11}\Gamma _{\mu }\theta _{-})\nonumber \\&\times (\partial _{j}{\overline{\theta }}_{-}\Gamma _{11}\Gamma ^{\mu lmn}\theta _{-})-\frac{1}{32}({\overline{\theta }}_{-}\Gamma _{11}\Gamma _{d}\partial _{k}\theta _{-})\nonumber \\&({\overline{\theta }}_{-}\Gamma _{11}\Gamma ^{lmnd}\partial _{j} \theta _{+})+\frac{1}{32}({\overline{\theta }}_{-}\Gamma _{11}\partial _{k}\theta _{+})\nonumber \\&\times ({\overline{\theta }}_{-}\Gamma ^{lmn}\Gamma _{11}\partial _{j}\theta _{-})-\frac{1}{32}(\partial _{k}{\overline{\theta }}_{+}\Gamma ^{df}\Gamma _{11}\theta _{-}) \nonumber \\&\times (\partial _{j}{\overline{\theta }}_{-}\Gamma _{11}\Gamma _{d}\Gamma ^{lmn}\Gamma _{f}\theta _{-})+\frac{1}{16}(\partial _{j}{\overline{\theta }}_{-}\Gamma ^{df}\Gamma _{11} \theta _{+})\nonumber \\&\times (\partial _{k}{\overline{\theta }}_{-}\Gamma _{11}\Gamma _{d}\Gamma ^{lmn}\Gamma _{f}\theta _{-})-\frac{3}{32}({\overline{\theta }}_{-}\Gamma _{11}\Gamma ^{l} \partial _{k}\theta _{-})\nonumber \\&\times ({\overline{\theta }}_{-}\Gamma _{11}\Gamma ^{mn}\partial _{j}\theta _{+}-4({\overline{\theta }}_{+}\Gamma _{11}\Gamma ^{mn}\partial _{j}\theta _{-})) \nonumber \\&-\frac{1}{32}(({\overline{\theta }}_{-}\Gamma _{11}\Gamma ^{d\mu }\partial _{j}\theta _{-})(\partial _{k}{\overline{\theta }}_{+}\Gamma _{11}\Gamma ^{lmn}\Gamma _{d\mu }\theta _{-}) \end{aligned}$$

$$\begin{aligned}&+\, ({\overline{\theta }}_{-}\Gamma _{df}\partial _{j}\theta _{-})(\partial _{k}{\overline{\theta }}_{+}\Gamma ^{d}\Gamma ^{lmn}\Gamma ^{f}\theta _{-})\nonumber \\&+({\overline{\theta }}_{-}\partial _{j}\theta _{-})(\partial _{k}{\overline{\theta }}_{+}\Gamma ^{lmn}\theta _{-}))\nonumber \\&-\frac{1}{16}({\overline{\theta }}_{-} \Gamma ^{df}\partial _{j}\theta _{-})({\overline{\theta }}_{+}\Gamma _{d}\Gamma ^{lmn}\Gamma _{f}\partial _{k}\theta _{-})]\nonumber \\&-\frac{1}{96\times 4!\times 5!}\epsilon _{defglmn}\delta ^{[abcd]}_{defg}[-5({\overline{\theta }}_{-}\partial _{j}\theta _{-})\nonumber \\&(\partial _{k}{\overline{\theta }}_{+}\Gamma _{11} \Gamma ^{abcd}\theta _{-})+3(\partial _{j}{\overline{\theta }}_{-}\Gamma _{\mu }\theta _{-})\nonumber \\&\times ({\overline{\theta }}_{-}\Gamma _{11}\Gamma ^{\mu abcd}\partial _{k}\theta _{+})+(\partial _{j}\theta _{-}\Gamma _{11}\Gamma ^{f}\theta _{-})\nonumber \\&\times (7({\overline{\theta }}_{-}\Gamma ^{abcd}\Gamma _{f}\partial _{k}\theta _{+})- 2({\overline{\theta }}_{+}(\Gamma ^{abcd}\Gamma _{f}\nonumber \\&+\Gamma _{f}\Gamma ^{abcd})\partial _{k}\theta _{-}))-(\partial _{k}{\overline{\theta }}_{+}\Gamma _{f}\theta _{-})({\overline{\theta }}_{-}\Gamma ^{abcd}\Gamma _{11} \Gamma ^{f}\partial _{j}\theta _{-}\nonumber \\&+6({\overline{\theta }}_{-}\Gamma ^{f}\Gamma ^{abcd}\Gamma _{11}\partial _{j}\theta _{-}))\nonumber \\&-8(\partial _{k}{\overline{\theta }}_{-}\Gamma _{f}\theta _{+})({\overline{\theta }}_{-}(\Gamma ^{abcd}\Gamma ^{f}+\Gamma ^{f}\Gamma ^{abcd})\Gamma _{11}\partial _{j} \theta _{-})\nonumber \\&+({\overline{\theta }}_{-}\Gamma _{\mu f}\partial _{j}\theta _{+})(\partial _{k}{\overline{\theta }}_{-}\Gamma _{11}\Gamma ^{abcd}\Gamma ^{\mu f}\theta _{-}\nonumber \\&+2(\partial _{k}{\overline{\theta }}_{-}\Gamma ^{\mu f}\Gamma ^{abcd}\Gamma _{11}\theta _{-}))\nonumber \\&-({\overline{\theta }}_{-}\Gamma _{11}\Gamma _{\mu f}\partial _{j}\theta _{-})(\partial _{k}{\overline{\theta }}_{+}\Gamma ^{abcd}\Gamma ^{\mu f}\theta _{-}\nonumber \\&+2(\partial _{k}{\overline{\theta }}_{+}\Gamma ^{\mu f}\Gamma ^{abcd}\theta _{-}))\nonumber \\&+(\partial _{j}{\overline{\theta }}_{-}\Gamma ^{f}\Gamma ^{abcd}\Gamma ^{e}\theta _{-})(\partial _{k}{\overline{\theta }}_{+}\Gamma _{fe}\Gamma _{11}\theta _{-}\nonumber \\&+ 2(\partial _{k}{\overline{\theta }}_{-}\Gamma _{fe}\Gamma _{11}\theta _{+}))-5({\overline{\theta }}_{-}\Gamma ^{abcd}\partial _{j}\theta _{-})\nonumber \\&\times ({\overline{\theta }}_{-}\Gamma _{11}\partial _{k}\theta _{+})-({\overline{\theta }}_{-}\Gamma _{ef}\partial _{k}\theta _{-})({\overline{\theta }}_{-} \Gamma ^{e}\Gamma ^{abcd}\Gamma ^{f}\partial _{j}\theta _{+}\nonumber \\&+2({\overline{\theta }}_{+}\Gamma ^{e}\Gamma ^{abcd}\Gamma ^{f}\Gamma _{11}\partial _{j}\theta _{-}))\nonumber \\&+3(\partial _{j}{\overline{\theta }}_{-}\Gamma ^{\mu abcd}\theta _{-})(\partial _{k}{\overline{\theta }}_{+}\Gamma _{11}\Gamma _{\mu }\theta _{-})] \end{aligned}$$

(5.9)

$$\begin{aligned} {\overline{F}}_{\nu abc}= & {} \frac{1}{288}T_{NR}\epsilon _{0jk}[({\overline{\theta }}_{-}\Gamma ^{\mu }\partial _{j}\theta _{-})(3({\overline{\theta }}_{-}\Gamma _{\mu \nu abc }\partial _{k}\theta _{+})\nonumber \\&-23({\overline{\theta }}_{-}\Gamma _{abc}\partial _{k}\theta _{+})\eta _{\mu \nu })\nonumber \\&+({\overline{\theta }}_{-}\Gamma ^{d}\partial _{j}\theta _{+})(5({\overline{\theta }}_{-}\Gamma _{\nu abcd }\partial _{k}\theta _{-})\nonumber \\&+3({\overline{\theta }}_{-}\Gamma _{\nu bc}\partial _{k}\theta _{-})\eta _{ad})-24({\overline{\theta }}_{+}\Gamma _{a}\partial _{j}\theta _{-})\nonumber \\&\times ({\overline{\theta }}_{-}\Gamma _{\nu bc}\partial _{k}\theta _{-})+({\overline{\theta }}_{-}\Gamma _{11}\Gamma ^{\mu }\partial _{k}\theta _{+})\nonumber \\&\times (-3({\overline{\theta }}_{-}\Gamma _{11}\Gamma _{abc}\partial _{j} \theta _{-})\eta _{\mu \nu }+5(\partial _{j}{\overline{\theta }}_{-}\Gamma _{11}\Gamma _{\mu \nu abc }\theta _{-}))\nonumber \\&+({\overline{\theta }}_{-}\Gamma _{11}\Gamma ^{d}\partial _{j}\theta _{-})({\overline{\theta }}_{-}\Gamma _{11}\Gamma _{\nu abc }\Gamma _{d}\partial _{k}\theta _{+}\nonumber \\&-4({\overline{\theta }}_{-}\Gamma _{d}\Gamma _{\nu abc }\Gamma _{11}\partial _{k}\theta _{+})\nonumber \\&+12({\overline{\theta }}_{+}\Gamma _{\nu bc }\Gamma _{11}\partial _{k}\theta _{-})\eta _{ad})+({\overline{\theta }}_{-}\Gamma ^{\mu d}\partial _{k}\theta _{+})\nonumber \\&\times (\partial _{j}{\overline{\theta }}_{-}\Gamma _{\mu }\Gamma _{\nu abc }\Gamma _{d}\theta _{-}-4(\partial _{j}{\overline{\theta }}_{-}\Gamma _{d}\Gamma _{\nu abc }\Gamma _{\mu }\theta _{-}))\nonumber \\&-({\overline{\theta }}_{-}\Gamma ^{df}\partial _{k}\theta _{-})({\overline{\theta }}_{-}\Gamma _{d}\Gamma _{\nu abc }\Gamma _{f}\partial _{j}\theta _{+}\nonumber \\&+2({\overline{\theta }}_{+}\Gamma _{d}\Gamma _{\nu abc }\Gamma _{f}\partial _{j}\theta _{-}))+(\partial _{k}{\overline{\theta }}_{-}\Gamma _{d}\Gamma _{11}\Gamma _{\nu abc }\Gamma _{f}\theta _{-})\nonumber \\&\times (3(\partial _{j}{\overline{\theta }}_{+}\Gamma ^{df}\Gamma _{11}\theta _{-})+2(\partial _{j}{\overline{\theta }}_{-}\Gamma ^{df}\Gamma _{11}\theta _{+}))\nonumber \\&+5( \partial _{j}{\overline{\theta }}_{+}\Gamma _{11}\theta _{-})(\partial _{k}{\overline{\theta }}_{-}\Gamma _{11}\Gamma _{\nu abc}\theta _{-})\nonumber \\&+3({\overline{\theta }}_{-}\Gamma _{11}\Gamma _{\mu d}\partial _{j}\theta _{-})(\partial _{k}{\overline{\theta }}_{+}\Gamma ^{\mu }\Gamma _{11}\Gamma _{\nu abc }\Gamma ^{d}\theta _{-})\nonumber \\&-3({\overline{\theta }}_{-}\partial _{j}\theta _{-})(\partial _{k}{\overline{\theta }}_{+}\Gamma _{\nu abc }\theta _{-})] \end{aligned}$$

(5.10)

$$\begin{aligned} F^{\nu abdf}= & {} \frac{1}{96}T_{NR}\epsilon _{0jk}[6({\overline{\theta }}_{-}\Gamma ^{c}\partial _{j}\theta _{+})({\overline{\theta }}_{-} \Gamma ^{\nu abdf}\Gamma _{c}\partial _{k}\theta _{+} \nonumber \\&-{\overline{\theta }}_{-}\Gamma _{c}\Gamma ^{\nu abdf }\partial _{k}\theta _{+}) \nonumber \\&-2({\overline{\theta }}_{-}\Gamma _{11}\Gamma _{\mu }\partial _{k}\theta _{+})(\partial _{j}{\overline{\theta }}_{+}\Gamma _{11}\Gamma ^{\mu }\Gamma ^{\nu abdf }\theta _{-} \nonumber \\&+\partial _{j}{\overline{\theta }}_{+}\Gamma _{11}\Gamma ^{\nu abdf }\Gamma ^{\mu }\theta _{-})-2({\overline{\theta }}_{-}\Gamma _{11}\Gamma _{\rho \sigma }\partial _{k}\theta _{+}) \nonumber \\&\times (\partial _{j}{\overline{\theta }}_{+}\Gamma _{11}\Gamma ^{\rho }\Gamma ^{\nu abdf }\Gamma ^{\sigma }\theta _{-})+2({\overline{\theta }}_{-}\Gamma ^{c}\partial _{j}\theta _{+}) \nonumber \\&\times (\partial _{k}{\overline{\theta }}_{-}\Gamma ^{\nu adf }\Gamma _{c}\theta _{+}+\partial _{k}{\overline{\theta }}_{-}\Gamma _{c}\Gamma ^{\nu abdf }\theta _{+}) \nonumber \\&+3({\overline{\theta }}_{+}\Gamma ^{c}\partial _{j}\theta _{-})(-2(\partial _{k}{\overline{\theta }}_{+}\Gamma ^{\nu abdf }\Gamma _{c}\theta _{-}) \nonumber \\&+{\overline{\theta }}_{-}\Gamma ^{\nu abdf}\Gamma _{c}\partial _{k}\theta _{+})-4({\overline{\theta }}_{-}\Gamma _{11}\Gamma _{\mu }\partial _{k}\theta _{+}) \nonumber \\&\times (\partial _{j}{\overline{\theta }}_{-}\Gamma _{11}\Gamma ^{\mu }\Gamma ^{\nu abdf }\theta _{+}) \nonumber \\&+({\overline{\theta }}_{+}\Gamma _{11}\Gamma _{\mu }\partial _{k}\theta _{-})(\partial _{j}{\overline{\theta }}_{+}\Gamma _{11}\Gamma ^{\mu } \Gamma ^{\nu abdf}\theta _{-} \nonumber \\&-2(\partial _{j}{\overline{\theta }}_{+}\Gamma _{11}\Gamma ^{\nu abdf }\Gamma ^{\mu }\theta _{-})) \nonumber \\&-2(\partial _{j}{\overline{\theta }}_{-}\Gamma ^{\mu }\theta _{-})(\partial _{k}{\overline{\theta }}_{+}\Gamma _{\mu }\Gamma ^{\nu abdf }\theta _{+}) \end{aligned}$$

$$\begin{aligned}&+2(\partial _{k}{\overline{\theta }}_{+}\Gamma _{\mu c}\theta _{-}) ({\overline{\theta }}_{+}\Gamma ^{c}\Gamma ^{\nu abdf}\Gamma ^{\mu }\partial _{j}\theta _{-}) \nonumber \\&+2({\overline{\theta }}_{-}\Gamma ^{c}\Gamma _{11}\partial _{j}\theta _{-})(\partial _{k}{\overline{\theta }}_{+}\Gamma _{11} \Gamma ^{\nu abdf}\Gamma _{c}\theta _{+} \nonumber \\&-2(\partial _{k}{\overline{\theta }}_{+}\Gamma _{11}\Gamma _{c}\Gamma ^{\nu abdf }\theta _{+})) \nonumber \\&+2(\partial _{k}{\overline{\theta }}_{+}\Gamma _{11}\theta _{-})({\overline{\theta }}_{+}\Gamma _{11}\Gamma ^{\nu abdf }\partial _{j}\theta _{-})+({\overline{\theta }}_{+}\Gamma ^{\mu }\partial _{k}\theta _{+}) \nonumber \\&\times (6(\partial _{j}{\overline{\theta }}_{-}\Gamma _{\mu }\Gamma ^{\nu abdf }\theta _{-})+5(\partial _{j}\theta _{-}\Gamma ^{\nu abdf}\Gamma _{\mu }\theta _{-})) \nonumber \\&+7({\overline{\theta }}_{+}\Gamma _{\mu \sigma }\partial _{k}\theta _{+})(\partial _{j}{\overline{\theta }}_{-}\Gamma ^{\sigma }\Gamma ^{\nu abdf}\Gamma ^{\mu }\theta _{-}) \nonumber \\&-3({\overline{\theta }}_{+}\Gamma _{\mu c}\partial _{j}\theta _{-})(\partial _{k}{\overline{\theta }}_{+}\Gamma ^{c}\Gamma ^{\nu abdf}\Gamma ^{\mu }\theta _{-}) \nonumber \\&-4({\overline{\theta }}_{+}\Gamma _{11}\partial _{j}\theta _{-})(\partial _{k}{\overline{\theta }}_{+}\Gamma _{11}\Gamma ^{\nu abdf}\theta _{-}) \nonumber \\&+(\partial _{k}{\overline{\theta }}_{+}\Gamma _{\mu c}\Gamma _{11}\theta _{+})(\partial _{j}{\overline{\theta }}_{-}\Gamma ^{c}\Gamma _{11}\Gamma ^{\nu abdf}\Gamma ^{\mu }\theta _{-}) \nonumber \\&+(\partial _{k}{\overline{\theta }}_{+}\Gamma _{\mu }\Gamma _{\sigma }\theta _{+}) \nonumber \\&\times (\partial _{j}{\overline{\theta }}_{-}\Gamma ^{\sigma }\Gamma ^{\nu abdf}\Gamma ^{\mu }\theta _{-})+({\overline{\theta }}_{+}\Gamma _{11}\Gamma _{\sigma }\Gamma _{\mu }\partial _{j}\theta _{-}) \nonumber \\&\times (\partial _{k}{\overline{\theta }}_{+} \Gamma ^{\sigma }\Gamma _{11}\Gamma ^{\nu abdf}\Gamma ^{\mu }\theta _{-}) \nonumber \\&-(\partial _{k}{\overline{\theta }}_{+}\Gamma _{11}\Gamma _{c}\theta _{+})(\partial _{j}{\overline{\theta }}_{-}\Gamma _{11}\Gamma ^{c}\Gamma ^{\nu abdf}\theta _{-}) \nonumber \\&+2({\overline{\theta }}_{-}\Gamma _{ce}\partial _{k}\theta _{-})({\overline{\theta }}_{+}\Gamma ^{c}\Gamma ^{\nu abdf}\Gamma ^{e}\partial _{j}\theta _{+}) \nonumber \\&-\frac{1}{4!}\epsilon _{cgeabdf}\delta ^{[lmn]}_{cge}\frac{T_{NR}}{576}[-36({\overline{\theta }}_{-}\Gamma ^{l}\partial _{j}\theta _{+}) \nonumber \\&\times (\partial _{k}{\overline{\theta }} _{+}\Gamma ^{\nu mn}\Gamma _{11}\theta _{-})+2({\overline{\theta }}_{-}\Gamma ^{r}\partial _{j}\theta _{+}) \nonumber \\&\times (\partial _{k}{\overline{\theta }}_{-}\Gamma _{11}\Gamma ^{\nu lmn}\Gamma _{r}\theta _{+}+\partial _{k}{\overline{\theta }}_{-}\Gamma _{11}\Gamma _{r}\Gamma ^{\nu lmn}\theta _{+}) \nonumber \\&+3({\overline{\theta }}_{+}\Gamma ^{r}\partial _{j}\theta _{-}) \nonumber \\&\times (\partial _{k}{\overline{\theta }}_{+}\Gamma _{r}\Gamma ^{\nu lmn}\Gamma _{11}\theta _{-}+2(\partial _{k}{\overline{\theta }}_{+}\Gamma _{11}\Gamma ^{\nu lmn}\Gamma _{r}\theta _{-})) \nonumber \\&-4({\overline{\theta }}_{-}\Gamma _{11}\Gamma _{\mu }\partial _{k}\theta _{+}) \nonumber \\&\times (\partial _{j}{\overline{\theta }}_{-}\Gamma ^{\mu }\Gamma ^{\nu lmn}\theta _{+})-({\overline{\theta }}_{+}\Gamma _{11}\Gamma _{\mu }\partial _{k}\theta _{-}) \nonumber \\&\times (2(\partial _{j}{\overline{\theta }}_{+}\Gamma ^{\nu lmn}\Gamma ^{\mu }\theta _{-})+\partial _{j}{\overline{\theta }}_{+}\Gamma ^{\mu }\Gamma ^{\nu lmn}\theta _{-}) \nonumber \\&+2({\overline{\theta }}_{-}\Gamma ^{\mu }\partial _{j}\theta _{-})(\partial _{k}{\overline{\theta }}_{+}\Gamma _{11}\Gamma _{\mu }\Gamma ^{\nu lmn}\theta _{+}) \nonumber \\&-2( \partial _{k}{\overline{\theta }}_{+}\Gamma _{r\mu }\theta _{-})({\overline{\theta }}_{+}\Gamma ^{r}\Gamma ^{\nu lmn}\Gamma _{11}\Gamma ^{\mu }\partial _{j}\theta _{-}) \nonumber \\&+2({\overline{\theta }}_{-}\Gamma ^{r}\Gamma _{11}\partial _{j}\theta _{-})(-2(\partial _{k}{\overline{\theta }}_{+}\Gamma _{r}\Gamma ^{\nu lmn}\theta _{+}) \nonumber \\&+ \partial _{k}{\overline{\theta }}_{+}\Gamma ^{\nu lmn}\Gamma _{r}\theta _{+})+2(\partial _{k}{\overline{\theta }}_{+}\Gamma _{11}\theta _{-}) \nonumber \\&\times ({\overline{\theta }}_{+}\Gamma ^{\nu lmn}\partial _{j}\theta _{-})+6({\overline{\theta }}_{+}\Gamma _{\sigma \mu }\partial _{k}\theta _{+}) \nonumber \\&\times (\partial _{j}{\overline{\theta }}_{-}\Gamma ^{\mu }\Gamma ^{\nu lmn}\Gamma _{11}\Gamma ^{\sigma }\theta _{-})-(\partial _{k}{\overline{\theta }}_{+}\theta _{+}) \nonumber \\&\times (\partial _{j}{\overline{\theta }}_{-}\Gamma ^{\nu lmn}\Gamma _{11}\theta _{-})+(\partial _{k}{\overline{\theta }}_{+}\Gamma _{\mu r}\Gamma _{11}\theta _{+}) \nonumber \\&( \partial _{j}{\overline{\theta }}_{-}\Gamma ^{r}\Gamma ^{\nu lmn}\Gamma ^{\mu }\theta _{-})+(\partial _{j}{\overline{\theta }}_{-}\Gamma _{11}\Gamma _{\mu \sigma }\theta _{+}) \nonumber \\&\times (\partial _{k}{\overline{\theta }}_{+}\Gamma ^{\mu }\Gamma ^{\nu lmn}\Gamma ^{\sigma }\theta _{-})+3(\partial _{j}{\overline{\theta }}_{-}\Gamma _{11}\theta _{+})\nonumber \\&\times (\partial _{k}{\overline{\theta }}_{+}\Gamma ^{\nu lmn}\theta _{-})+(\partial _{k}{\overline{\theta }}_{+}\Gamma _{11}\Gamma ^{r}\theta _{+})\nonumber \nonumber \\&\times (\partial _{j}{\overline{\theta }}_{-}\Gamma _{r}\Gamma ^{\nu lmn}\theta _{-})\nonumber \nonumber \\&+3({\overline{\theta }}_{+}\Gamma _{r\sigma }\partial _{j}\theta _{-})(\partial _{k}{\overline{\theta }}_{+}\Gamma ^{r}\Gamma ^{\nu lmn}\Gamma _{11}\Gamma ^{\sigma }\theta _{-})\nonumber \nonumber \\&+({\overline{\theta }}_{+}\Gamma ^{\mu }\partial _{k}\theta _{+}) (6(\partial _{j}{\overline{\theta }}_{-}\Gamma _{\mu }\Gamma ^{\nu lmn}\Gamma _{11}\theta _{-})\nonumber \nonumber \\&+5(\partial _{j}{\overline{\theta }}_{-}\Gamma ^{\nu lmn}\Gamma _{11}\Gamma _{\mu }\theta _{-}))+2({\overline{\theta }}_{-}\Gamma _{rs}\partial _{k}\theta _{-})\nonumber \nonumber \\&\times ({\overline{\theta }}_{+}\Gamma ^{r}\Gamma ^{\nu lmn}\Gamma _{11}\Gamma ^{s}\partial _{j}\theta _{+})] \end{aligned}$$

(5.11)

$$\begin{aligned} {\overline{F}}_{fglmn}= & {} \frac{1}{96\times 5!}T_{NR}\epsilon _{0jk}[3({\overline{\theta }}_{-}\Gamma ^{\mu }\partial _{j}\theta _{-})(-5({\overline{\theta }}_{-}\Gamma _{\mu fglmn }\partial _{k}\theta _{+}) \nonumber \\&+4({\overline{\theta }}_{+}\Gamma _{\mu fglmn}\partial _{k}\theta _{-}))+({\overline{\theta }}_{-}\Gamma ^{a}\partial _{j}\theta _{+}) \nonumber \\&\times (17({\overline{\theta }}_{-}\Gamma _{fglmn}\Gamma _{a}\partial _{k}\theta _{-})-22({\overline{\theta }}_{-}\Gamma _{a}\Gamma _{fglmn}\partial _{k}\theta _{-})) \nonumber \\&+4({\overline{\theta }}_{+}\Gamma ^{a}\partial _{j}\theta _{-})({\overline{\theta }}_{-}\Gamma _{fglmn}\Gamma _{a}\partial _{k}\theta _{-} \nonumber \\&-5({\overline{\theta }}_{-} \Gamma _{a}\Gamma _{fglmn}\partial _{k}\theta _{-}))+9({\overline{\theta }}_{-}\Gamma _{11}\Gamma ^{\mu }\partial _{k}\theta _{+}) \nonumber \\&\times (\partial _{j}{\overline{\theta }}_{-}\Gamma _{11}\Gamma _{\mu fglmn}\theta _{-}) \nonumber \\&+12({\overline{\theta }}_{+}\Gamma _{11}\Gamma ^{\mu }\partial _{k}\theta _{-})(\partial _{j}{\overline{\theta }}_{-}\Gamma _{\mu fglmn }\Gamma _{11}\theta _{-}) \nonumber \\&+(\partial _{j}{\overline{\theta }}_{-}\Gamma _{11}\Gamma ^{a}\theta _{-}) \nonumber \\&\times (-4(\partial _{k}{\overline{\theta }}_{+}\Gamma _{11}\Gamma _{fglmn}\Gamma _{a}\theta _{-}) \nonumber \\&+13(\partial _{k}{\overline{\theta }}_{+}\Gamma _{11}\Gamma _{a} \Gamma _{fglmn}\theta _{-})+6(\partial _{k}{\overline{\theta }}_{-}\Gamma _{11}\Gamma _{a}\Gamma _{fglmn}\theta _{+}) \end{aligned}$$

$$\begin{aligned}&+2(\partial _{k}{\overline{\theta }}_{-}\Gamma _{11}\Gamma _{fglmn}\Gamma _{a}\theta _{+})) \nonumber \\&-(\partial _{k}{\overline{\theta }}_{-}\Gamma _{11}\Gamma ^{a} \Gamma _{fglmn}\Gamma ^{b}\theta _{-})(4(\partial _{j}{\overline{\theta }}_{-}\Gamma _{b}\Gamma _{a}\Gamma _{11}\theta _{+}) \nonumber \\&+3(\partial _{j}{\overline{\theta }}_{+}\Gamma _{b}\Gamma _{a}\Gamma _{11}\theta _{-}))+3({\overline{\theta }}_{-}\Gamma _{b}\Gamma _{a}\partial _{j}\theta _{-} ) \nonumber \\&\times (\partial _{k}{\overline{\theta }}_{+}\Gamma ^{b}\Gamma _{fglmn}\Gamma ^{a}\theta _{-})+2({\overline{\theta }}_{-}\Gamma ^{ab}\partial _{k}\theta _{-}) \nonumber \\&\times ({\overline{\theta }}_{-}\Gamma _{a}\Gamma _{fglmn}\Gamma _{b}\partial _{j}\theta _{+}+{\overline{\theta }}_{+}\Gamma _{a}\Gamma _{fglmn}\Gamma _{b}\partial _{j} \theta _{-}) \nonumber \\&-6({\overline{\theta }}_{-}\Gamma _{11}\partial _{k}\theta _{+})(\partial _{j}{\overline{\theta }}_{-}\Gamma _{11}\Gamma _{fglmn}\theta _{-}) \nonumber \\&+12(\partial _{j}{\overline{\theta }}_{-}\Gamma _{11}\theta _{+})(\partial _{k}{\overline{\theta }}_{-}\Gamma _{11}\Gamma _{fglmn}\theta _{-}) \nonumber \\&-3(\partial _{k} {\overline{\theta }}_{+}\Gamma _{11}\Gamma _{fglmn}\Gamma ^{\mu a}\theta _{-})({\overline{\theta }}_{-}\Gamma _{11}\Gamma _{\mu a}\partial _{j}\theta _{-}) \nonumber \\&-({\overline{\theta }}_{-}\Gamma ^{\mu a}\partial _{k}\theta _{+})(4(\partial _{j}{\overline{\theta }}_{-}\Gamma _{\mu a }\Gamma _{fglmn}\theta _{-}) \nonumber \\&-\partial _{j}{\overline{\theta }}_{-}\Gamma _{fglmn}\Gamma _{\mu a}\theta _{-})-2({\overline{\theta }}_{+}\Gamma ^{\mu a}\partial _{k}\theta _{-}) \nonumber \\&\times ({\overline{\theta }}_{-}\Gamma _{\mu a}\Gamma _{fglmn}\partial _{j}\theta _{-}+\partial _{j}\Gamma _{\mu a}\Gamma _{fglmn}\theta _{-})] \nonumber \\&-\frac{1}{192\times 5!}\epsilon _{0jk}\epsilon _{defglmn}\delta ^{ba}_{de}[(16({\overline{\theta }}_{+}\Gamma _{c}\partial _{j}\theta _{-}) \nonumber \\&+21({\overline{\theta }}_{-}\Gamma _{c}\partial _{j}\theta _{+}))({\overline{\theta }}_{-}\Gamma ^{abc}\Gamma _{11}\partial _{k}\theta _{-}) \nonumber \\&+8(7({\overline{\theta }}_{+}\Gamma ^{a}\partial _{j}\theta _{-})+3({\overline{\theta }}_{-}\Gamma ^{a}\partial _{j}\theta _{+}))({\overline{\theta }}_{-} \Gamma ^{b}\Gamma _{11}\partial _{k}\theta _{-}) \nonumber \\&-({\overline{\theta }}_{-}\Gamma _{\mu }\partial _{j}\theta _{-}) \nonumber \\&\times (12({\overline{\theta }}_{+}\Gamma ^{\mu ab}\Gamma _{11}\partial _{k}\theta _{-})+11({\overline{\theta }}_{-}\Gamma _{11}\Gamma ^{\mu ab}\partial _{k}\theta _{+})) \nonumber \\&+8({\overline{\theta }}_{-}\Gamma _{\mu }\Gamma ^{b}\Gamma _{11}\partial _{j}\theta _{-})({\overline{\theta }}_{+}\Gamma ^{\mu a} \partial _{k}\theta _{-} \nonumber \\&+{\overline{\theta }}_{-}\Gamma ^{\mu a}\partial _{k}\theta _{+})+(\partial _{k}{\overline{\theta }}_{+}\Gamma _{\mu c}\theta _{-})(\partial _{j}{\overline{\theta }}_{-}\Gamma ^{\mu abc}\Gamma _{11}\theta _{-} \nonumber \\&+2({\overline{\theta }}_{-}\Gamma ^{\mu b}\Gamma _{11}\partial _{j}\theta _{-})\eta ^{ac}) \nonumber \\&+({\overline{\theta }}_{-}\Gamma _{11}\Gamma _{c}\partial _{k}\theta _{-})(-8({\overline{\theta }}_{+}\Gamma ^{abc}\partial _{j}\theta _{-}) \nonumber \\&+3({\overline{\theta }}_{-} \Gamma ^{abc}\partial _{j}\theta _{+}))+({\overline{\theta }}_{-}\Gamma ^{\mu ab}\partial _{j}\theta _{-}) \nonumber \\&\times (12({\overline{\theta }}_{+}\Gamma _{11}\Gamma _{\mu }\partial _{k}\theta _{-})-17({\overline{\theta }}_{-}\Gamma _{11}\Gamma _{\mu } \partial _{k}\theta _{+})) \nonumber \\&-4(\partial _{j}{\overline{\theta }}_{-}\Gamma _{cd}\Gamma _{11}\theta _{+})(\partial _{k}{\overline{\theta }}_{-}\Gamma ^{c}\Gamma ^{ab} \Gamma ^{d}\theta _{-}) \nonumber \\&+(\partial _{j}{\overline{\theta }}_{+}\Gamma _{cd}\Gamma _{11}\theta _{-})(2(\partial _{k}{\overline{\theta }}_{-}\Gamma ^{cd}\Gamma ^{ab}\theta _{-})\nonumber \\&-17( {\overline{\theta }}_{-}\partial _{k}\theta _{-})\eta ^{ac}\eta ^{bd})+({\overline{\theta }}_{-}\Gamma _{cd}\partial _{k}\theta _{-})\nonumber \nonumber \\&\times (11(\partial _{j}{\overline{\theta }}_{+}\Gamma _{11}\theta _{-})\eta ^{ac}\eta ^{bd}+2(-\partial _{j}{\overline{\theta }}_{+}\Gamma ^{cd}\Gamma ^{ab}\Gamma _{11} \theta _{-}\nonumber \nonumber \\&+{\overline{\theta }}_{-}\Gamma ^{c}\Gamma ^{ab}\Gamma ^{d}\Gamma _{11}\partial _{j}\theta _{+}\nonumber \nonumber \\&-{\overline{\theta }}_{+}\Gamma ^{c}\Gamma ^{ab} \Gamma ^{d}\Gamma _{11}\partial _{j}\theta _{-}))+3({\overline{\theta }}_{-}\Gamma _{\mu c}\Gamma _{11}\partial _{j}\theta _{-})\nonumber \nonumber \\&\times (\partial _{k}{\overline{\theta }}_{+}\Gamma ^{\mu abc}\theta _{-}+2(\partial _{k}{\overline{\theta }}_{+}\Gamma ^{\mu b}\theta _{-})\eta ^{ac})] \end{aligned}$$

(5.12)