Supersymmetric Cubic Interactions For Lower Spins From “Higher Spin” Approach
I.L. Buchbindera***e-mail<EMAIL_ADDRESS>, V.A. Krykhtina†††e-mail:
<EMAIL_ADDRESS>,
M. Tsulaiab‡‡‡e-mail<EMAIL_ADDRESS>, D. Weissmanb§§§e-mail:
<EMAIL_ADDRESS>
a Center of Theoretical Physics, Tomsk State Pedagogical University,
634041 Tomsk, Russia

c Okinawa Institute of Science and Technology,
1919-1 Tancha, Onna-son, Okinawa 904-0495, Japan

We demonstrate how to reconstruct standard cubic vertices for $N=1$
supersymmetric Yang-Mills and Supergravities using various techniques adopted
for the description of cubic interactions between higher spin fields.

_
Based on talks given at Quarks 2020 and Aspects of Symmetry 2021_

##  1 Introduction

The purpose of these notes is to demonstrate in detail various approaches of
constructing supersymmetric cubic interaction vertices for higher spin fields.
To this end we reconstruct well known cubic vertices for $N=1$ Super Yang-
Mills and for linearized $N=1$ Supergravities. In particular, we consider the
field theoretic limits of the pure $N=1$, ${\cal D}=10$ supergravity [1]–[2],
of the $N=1$, ${\cal D}=4$ supergravity coupled with one chiral supermultiplet
[3], and of the $N=1$, ${\cal D}=6$ supergravity coupled to one $(1,0)$ tensor
supermultiplet [4]– [5]. The reason for choosing these particular types of
supergravity will become clear in the following. We shall also comment on a
generalization of these vertices to higher spin “Yang-Mills-like” and
“Supergravity- like” vertices [6], and on the higher spin generalizations of
the corresponding free Lagrangians [7].

First, we shall describe the covariant BRST approach,111See [8] for a review
of the BRST approach and [9] –[22] for reviews of different approaches to
higher spin theories. which is similar to the one of Open String Field
Theory.222See e.g. [23] for a recent review. However, unlike String Field
Theory, the BRST approach to higher spin fields is essentially a method for
the construction of free and interacting Lagrangians using gauge invariance as
the only guiding principle, without any recourse to a world-sheet description.

As the first step in this approach, one constructs free Lagrangians invariant
under linear gauge transformations. Because of the presence of gauge symmetry,
these Lagrangians contain both physical and non-physical degrees of freedom.
Some of the non-physical degrees of freedom are removed using the equations of
motion, the others are gauged away, and in the end one is left only with
physical polarizations. In general, these systems can contain bosonic and
fermionic fields, which are described by Young tableaux with mixed symmetries.
However, at the free level one can consider Lagrangians for just one or
several (a finite number of) representations of the Poincaré group [24]– [25].
A further requirement of supersymmetry singles out some particular
representations of the Poincaré group, both in bosonic and fermionic sectors,
so the corresponding bosonic and fermionic Lagrangians are related by
supersymmetry transformations.

As the second step in the BRST approach, one promotes the original gauge
symmetry to an interacting level by deforming the Lagrangian and gauge
transformation rules with nonlinear terms, in such a way that the gauge
invariance is kept order by order in the coupling constant. As in the case of
free Lagrangians, supersymmetry singles out some particular subclass of the
cubic vertices, which were found for non-supersymmetric systems [26]–[31].

To describe how this approach works on the examples of $N=1$ Super Yang-Mills
and linearized Supergravities, we start in Section 2 with a description of
gauge invariant free Lagrangians. For the massless vector field the
corresponding Lagrangian is the standard (Maxwell) one. A gauge invariant
Lagrangian which contains the second rank symmetric massless tensor field as a
physical component describes spins $2$ and $0$ simultaneously. Then we present
a similar gauge invariant description for a massless spin-vector field which
contains irreducible representations of the Poincaré group with spins
$\frac{3}{2}$ and $\frac{1}{2}$.

In Section 3 we reformulate the results of Section 2 in the BRST approach.
Then we impose an additional requirement of $N=1$ supersymmetry on these
systems. Using the technique developed in the Open Superstring Field Theory
[32] we show that $N=1$ supersymmetry requires some extra fields (both
physical and auxiliary) in the bosonic sector [7]. The obtained Lagrangian
provides a “unified” description of the above mentioned $N=1$ linearized
supergravities in ${\cal D}=4,6$ and $10$ dimensions.

In Section 4 we turn to cubic interactions in the covariant formalism and
present generic equations which determine the cubic vertices, as well as
solutions to these equations for three massless bosonic fields with arbitrary
spins [30], [33]. We also present the generic equations for determining cubic
vertices for two massless fermions and one massless boson [6].

In Section 5 we describe $N=1$ Super Yang-Mills in this formalism and in
Section 6 we consider cubic vertices for linearized supergravities. We comment
on the higher spin generalization of the vertices given in Sections 5 and 6.
These cubic vertices [6] are covariant versions of the vertices for two
fermionic and one bosonic higher spin fields in arbitrary dimensions, first
derived in the light cone formalism [27].333Covariant cubic vertices with two
fermions and electromagnetic and gravitational fields are given in [34]–[35].
Supersymmetric cubic interactions on flat and $AdS$ backgrounds are given in
[36]–[43].

Finally, in Section 7 we describe the light cone approach to the construction
of the cubic vertices. In this approach one splits the generators of the
Poincaré (super)group into dynamical and kinematical operators. When using the
field theoretic realization of the generators one takes the kinematical
operators to be quadratic in terms of superfields, whereas the dynamical
generators contain also cubic and higher order terms, i.e., cubic and higher
order vertices. The requirement that the Poincaré superalgebra stays intact
after the nonlinear deformation of the dynamical operators determines these
vertices order by order in the coupling constant. We shall briefly review the
construction of [47] (see [48],[49] for the case of an arbitrary $N$) for
arbitrary spin supermultiplets in $N=1$, ${\cal D}=1$, and show how to obtain
cubic vertices for four dimensional $N=1$ super Yang-Mills and $N=1$
supergravity in the light cone gauge444The cubic vertices for ${\cal D}=4$,
$N=4$ super Yang-Mills in the light cone formulation are given in [50]–[52].
as a particular example.

##  2 Free Lagrangians

###  2.1 $s=1$

Let us start with a massless vector field $\phi_{\mu}(x)$ with a standard
gauge transformation rule

$\delta\phi_{\mu}(x)=\partial_{\mu}\lambda(x)$ (2.1)

The gauge invariant Klein -Gordon and transversality equations for the
massless vector field can be written as

$\Box\phi_{\mu}(x)=\partial_{\mu}C(x),\quad C(x)=\partial^{\mu}\phi_{\mu}(x)$
(2.2)

where we introduced an auxiliary field $C(x)$, which transforms as

$\delta C(x)=\Box\lambda(x)$ (2.3)

The equations (2.2) can be obtained from the Lagrangian

${\cal
L}=-\frac{1}{2}(\partial^{\mu}\phi^{\nu})(\partial_{\mu}\phi_{\nu})+C\partial^{\mu}\phi_{\mu}-\frac{1}{2}C^{2}$
(2.4)

After eliminating the field $C(x)$ via its own equation of motion one obtains
the Maxwell Lagrangian for the vector field $\phi_{\mu}(x)$.

###  2.2 $s=2$ and $s=0$

One can repeat a similar consideration for a second rank symmetric tensor
field $\phi_{\mu\nu}(x)$. Now the gauge invariant Klein-Gordon equation reads,

$\Box\phi_{\mu\nu}(x)=\partial_{\mu}C_{\nu}(x)+\partial_{\nu}C_{\mu}(x)$ (2.5)

where the physical field $\phi_{\mu\nu}(x)$ and the auxiliary field
$C_{\mu}(x)$ transform as

$\delta\phi_{\mu\nu}(x)=\partial_{\mu}\lambda_{\nu}(x)+\partial_{\nu}\lambda_{\mu}(x),\quad\delta
C_{\mu}(x)=\Box\lambda_{\mu}(x)$ (2.6)

In order to write gauge invariant transversality conditions one needs one more
auxiliary field $D(x)$, which transforms as

$\delta D(x)=\partial^{\mu}\lambda_{\mu}(x)$ (2.7)

Then the gauge invariant transversality equation is

$\partial^{\nu}\phi_{\mu\nu}(x)-\partial_{\mu}D(x)=C_{\mu}(x)$ (2.8)

Finally, one can write a gauge invariant Klein-Gordon equation for the field
$D(x)$

$\Box D(x)=\partial^{\mu}C_{\mu}(x)$ (2.9)

and “integrate” the equations (2.5), (2.8) and (2.9) back into the Lagrangian

${\cal
L}=-\frac{1}{2}(\partial^{\mu}\phi^{\nu\rho})(\partial_{\mu}\phi_{\nu\rho})+2C^{\mu}\partial^{\nu}\phi_{\mu\nu}-C^{\mu}C_{\mu}+(\partial^{\mu}D)(\partial_{\mu}D)+2D\partial^{\mu}C_{\mu}$
(2.10)

Again, the only propagating degrees of freedom are the physical components of
the field $\phi_{\mu\nu}(x)$. Its longitudinal components and the fields
$C_{\mu}(x)$ and $D(x)$ are either pure gauge or zero on shell. Finally, since
there is no zero trace condition involved, one obtains a gauge invariant
description simultaneously for a spin $2$ field $g_{\mu\nu}(x)$ and for a
scalar $\phi(x)$, both packed in the field $\phi_{\mu\nu}(x)$.

###  2.3 $s=\frac{3}{2}$ and $s=\frac{1}{2}$

The Lagrangian describing only spin $\frac{1}{2}$ field is simply

${\cal L}=-i\,\bar{\Psi}\gamma^{\mu}\partial_{\mu}\Psi$ (2.11)

The next simplest example is a spin-vector field $\Psi_{\mu}^{a}(x)$, where
$"a"$ is a spinorial index (see appendix A for the present conventions). Gauge
invariant Dirac and transversality equations can be written by introducing one
auxiliary field $\chi^{a}(x)$ as

$\gamma^{\nu}\partial_{\nu}\Psi_{\mu}(x)+\partial_{\mu}\chi(x)=0$ (2.12)
$\partial^{\mu}\Psi_{\mu}(x)+\gamma^{\nu}\partial_{\nu}\chi(x)=0$ (2.13)

The equations (2.12)– (2.12) are invariant under gauge transformations

$\delta\Psi_{\mu}(x)=\partial_{\mu}\,\lambda^{\prime}(x),\quad\delta\chi(x)=-\gamma^{\nu}\partial_{\nu}\lambda^{\prime}(x)$
(2.14)

and can be obtained from the Lagrangian

${\cal
L}=-i\,\bar{\Psi}^{\nu}\gamma^{\mu}\partial_{\mu}\Psi_{\nu}-i\,\bar{\Psi}^{\mu}\partial_{\mu}\chi+i\,\bar{\chi}\partial^{\mu}\Psi_{\mu}+i\,\bar{\chi}\gamma^{\mu}\partial_{\mu}\chi$
(2.15)

Similarly to the previous case, one can gauge away the auxiliary field
$\chi^{a}(x)$ and the non-physical polarizations of $\Psi^{a}_{\mu}(x)$. Then,
one has a gauge invariant description simultaneously of spins $\frac{3}{2}$
and $\frac{1}{2}$, the latter being the gamma-trace of the field
$\Psi_{\mu}^{a}(x)$.

##  3 BRST invariant formulation

###  3.1 Set Up

A systematic way to obtain the systems described above is to use the BRST
approach. Let us introduce an auxiliary Fock space spanned by one set of
creation and annihilation operators. The commutation relations and the vacuum
are defined in the usual way

$[\alpha_{\mu},\alpha_{\nu}^{+}]=\eta_{\mu\nu},\quad\alpha_{\mu}|0\rangle_{\alpha}=0$
(3.1)

A vector $|\Phi\rangle$ in this Fock space is a series expansion in terms of
the creation operators $\alpha_{\mu}^{+}$. In the rest of this paper we shall
take the maximal number of these oscillators to be equal to two. Using more
than two oscillators will result in components of higher spin. The divergence,
gradient and d’Alembertian operators are realized as

$l=p\cdot\alpha,\quad l^{+}=p\cdot\alpha^{+},\quad l_{0}=p\cdot p,$ (3.2)

with $p_{\mu}=-i\partial_{\mu}$ and $A\cdot
B\equiv\eta_{\mu\nu}A^{\mu}B^{\nu}$. The operators (3.2) form a simple algebra
with only non-zero commutator, being

$[l,l^{+}]=l_{0}$ (3.3)

Following a standard procedure (see [8] for a review) for each operator
$l^{+},l$ and $l_{0}$ one introduces ghosts $c,c^{+},c_{0}$ of ghost number
$+1$ and the corresponding momenta $b^{+},b,b_{0}$ which have ghost number
$-1$ . These operators obey the anticommutation relations

$\\{b,c^{+}\\}=\\{b^{+},c\\}=\\{b_{0},c_{0}\\}=1$ (3.4)

We define the ghost vacuum as

$c|0\rangle_{gh.}=b|0\rangle_{gh.}=b_{0}|0\rangle_{gh.}=0,$ (3.5)

the total vacuum being $|0\rangle=|0\rangle_{\alpha}\otimes|0\rangle_{gh.}$.
Using the corresponding nilpotent BRST charge

$Q=c_{0}l_{0}+cl^{+}+c^{+}l-c^{+}cb_{0},\quad Q^{2}=0$ (3.6)

this set up allows one to build gauge invariant free Lagrangians in a compact
form,

${\cal L}=\int dc_{0}\langle\Phi|Q|\Phi\rangle$ (3.7)

where the gauge transformation is given by

$\delta|\Phi\rangle=Q|\Lambda\rangle$ (3.8)

The Grassmann integration is carried out using the standard rule of

$\int dc_{0}\,c_{0}=1$ (3.9)

The requirement that the Lagrangian (3.7) has zero ghost number uniquely fixes
the expansion of an arbitrary vector $|\Phi\rangle$ in terms of the ghost
variables. Noticing that the number operator

$N=\alpha^{+}\cdot\alpha+c^{+}b+b^{+}c$ (3.10)

commutes with the BRST operator (3.6), one has the following expansion for the
case of the vector field

$|\Phi\rangle=(\phi^{\mu}(x)\alpha^{+}_{\mu}-iC(x)c_{0}b^{+})|0\rangle$ (3.11)

Since the BRST charge has the ghost number $+1$ and the vector $|\Phi\rangle$
has the ghost number zero, then due to (3.7) the parameter of gauge
transformations has the ghost number $-1$. Therefore

$|\Lambda\rangle=ib^{+}\lambda(x)|0\rangle$ (3.12)

Using the equations (3.7), (3.6) and (3.11) one can recover the Lagrangian
(2.4) after performing the normal ordering and integrating over $c_{0}$.
Similarly, using (3.6), (3.11) and (3.12) one recovers the gauge
transformation rules (2.1) and (2.3).

One can repeat the same procedure for the system described in the subsection
2.2. In particular, the expansion of the vector $|\Phi\rangle$ and of the
parameter of gauge transformations $|\Lambda\rangle$ have the form

$|\Phi\rangle=(\phi^{\mu\nu}(x)\alpha^{+}_{\mu}\alpha^{+}_{\nu}-ic_{0}b^{+}C^{\mu}(x)\alpha^{+}_{\mu}+c^{+}b^{+}D(x))|0\rangle$
(3.13) $|\Lambda\rangle=ib^{+}\lambda^{\mu}(x)\alpha^{+}_{\mu}|0\rangle$
(3.14)

Using this expansion one obtains the Lagrangian (2.10) and the gauge
transformation rules (2.6) – (2.7).

The BRST formulation for fermions is slightly more complicated, because of the
anticommuting nature of the Dirac operator

$g_{0}=p\cdot\gamma$ (3.15)

As a result, one has to introduce a commuting ghost variable, which in turn
leads to an infinite expansion in its powers. One can, however, partially fix
the BRST gauge to truncate the expansion of the vector in the Fock space to a
finite form and write the Lagrangian

$\displaystyle{\cal L}_{F}$ $\displaystyle=$
$\displaystyle\frac{1}{{\sqrt{2}}}\,\,{}_{a}\langle\Psi_{1}|(g_{0})^{a}{}_{b}|\Psi_{1}\rangle^{b}+{}_{a}\langle\Psi_{2}|\tilde{Q}_{F}|\Psi_{1}\rangle^{a}+$
$\displaystyle+$
$\displaystyle{}_{a}\langle\Psi_{1}|\tilde{Q}_{F}|\Psi_{2}\rangle^{a}+\sqrt{2}\,\,{}_{a}\langle\Psi_{2}|c^{+}c(g_{0})^{a}{}_{b}|\Psi_{2}\rangle^{b}\,$

where

$\tilde{Q}_{F}=c^{+}_{1}\,l_{1}+c_{1}\,l^{+}_{1}$ (3.17)

One can check, that the Lagrangian (3.1) is invariant under the gauge
transformations

$\displaystyle\delta\,|\Psi_{1}\rangle^{a}\ =\
\tilde{Q}_{F}|\Lambda^{\prime}\rangle^{a}$
$\displaystyle\delta\,|\Psi_{2}\rangle^{a}\ =\
-\frac{1}{\sqrt{2}}(g_{0})^{a}{}_{b}\,|\Lambda^{\prime}\rangle^{b}\ $ (3.18)

and is equivalent to (2.15) with

$|\Psi_{1}\rangle^{a}=\Psi^{\mu,a}(x)\alpha_{\mu}^{+}|0\rangle,\quad|\Psi_{2}\rangle^{a}=b^{+}\chi^{a}(x)|0\rangle$
(3.19)

The gauge transformations are obtained by taking the gauge parameter
$|\Lambda^{\prime}\rangle$ to be of the form

$|\Lambda^{\prime}\rangle^{a}=ib^{+}\lambda^{\prime a}(x)|0\rangle$ (3.20)

As in the case of the bosonic fields, the dependence on $\alpha_{\mu}^{+}$ and
on ghost variables is uniquely fixed by the choice that the field
$|\Psi_{1}\rangle^{a}$ contains a maximal spin equal to $\frac{3}{2}$ and the
requirement that the Lagrangian (3.1) has zero ghost number.

###  3.2 Supersymmetry. Linearized Supergravities

Let us notice that the systems considered in the subsections 2.2 and 2.3 can
not be connected by supersymmetry transformations, because the fields
$\phi_{\mu\nu}(x)$ and $\Psi_{\mu}(x)$ have different numbers of physical
degrees of freedom on-shell.

In order to establish $N=1$ supersymmetry, one can take a formulation of the
Open Superstring Field Theory [32] as a hint and proceed as follows [7].
Consider two independent sets of $\alpha$-oscillators

$[\alpha_{\mu,m},\alpha_{\nu,n}^{+}]=\eta_{\mu\nu}\delta_{mn},\quad m,n=1,2$
(3.21)

The corresponding divergence and gradient operators, as well as the ghost
variables $c^{\pm}_{m}$ and $b^{\mp}_{m}$ will get the index $"m"$ as well.
Therefore, we have the algebra

$[l_{m},l^{+}_{n}]=\delta_{mn}\,l_{0}$ (3.22)
$\\{b_{m},c^{+}_{n}\\}=\\{b^{+}_{m},c_{n}\\}=\delta_{mn},\quad\\{b_{0},c_{0}\\}=1$
(3.23)

We take the fields in the fermionic sector to contain only the first set of
oscillators. In other words, we consider the system described in the
Subsection 2.3 without changes and in the corresponding BRST formulation in
the Section 3 we assume that all oscillators belong to the first set ($m=1$).

On the other hand, the vectors in the Fock space in the bosonic sector contain
both types of oscillators. Taking physical component of the field
$|\Phi\rangle$ to contain one oscillator of each type, we get the following
expansions

$\displaystyle|\Phi\rangle$ $\displaystyle=$
$\displaystyle(\phi_{\mu,\nu}(x)\alpha^{\mu,+}_{1}\alpha^{\nu,+}_{2}-A(x)c^{+}_{1}b^{+}_{2}-B(x)c_{2}^{+}b^{+}_{1}$
$\displaystyle+$ $\displaystyle
ic_{0}b^{+}_{1}C_{\mu}(x)\alpha^{\mu,+}_{2}+ic_{0}b^{+}_{2}E_{\mu}(x)\alpha^{\mu,+}_{1})|0\rangle.$

and

$\displaystyle|\Lambda\rangle$ $\displaystyle=$
$\displaystyle(ib_{2}^{+}\lambda_{\mu}(x)\alpha^{\mu,+}_{1}+ib_{1}^{+}\rho_{\mu}(x)\alpha^{\mu,+}_{2}-c_{0}b_{1}^{+}b_{2}^{+}\tau(x))|0\rangle.$
(3.25)

Using the corresponding nilpotent BRST charge

$Q=c_{0}l_{0}+\sum_{m=1,2}(c_{m}l_{m}^{+}+c^{+}_{m}l_{m}-c^{+}_{m}c_{m}b_{0}),\quad
Q^{2}=0$ (3.26)

it is straightforward to obtain the Lagrangian

$\displaystyle L_{B}$ $\displaystyle=-\phi^{\mu,\nu}\Box\phi_{\mu,\nu}+B\Box
A+A\Box B$
$\displaystyle+E^{\mu}\partial_{\mu}B+C^{\nu}\partial^{\mu}\phi_{\nu,\mu}+C^{\nu}\partial_{\nu}A+E^{\mu}\partial^{\nu}\phi_{\nu,\mu}$
$\displaystyle-B\partial_{\mu}E^{\mu}-\phi^{\nu,\mu}\partial_{\mu}C_{\nu}-A\partial_{\mu}C^{\mu}-\phi^{\mu,\nu}\partial_{\mu}E_{\nu}$
$\displaystyle+C^{\mu}C_{\mu}+E^{\mu}E_{\mu}\,.$

by plugging the expressions (3.2) and (3.26) into (3.7), performing the normal
ordering of oscillators and integrating over $c_{0}$ according to (3.9).
Similarly, one can find, that Lagrangian (3.2) is invariant under the gauge
transformations

$\displaystyle\delta\phi_{\nu,\mu}(x)=\partial_{\mu}\lambda_{\nu}(x)+\partial_{\nu}\rho_{\mu}(x),$
$\displaystyle\delta A(x)=-\partial^{\mu}\rho_{\mu}(x)-\tau(x),$
$\displaystyle\delta B(x)=-\partial^{\mu}\lambda_{\mu}(x)+\tau(x),$
$\displaystyle\delta C_{\mu}(x)=-\Box\lambda_{\mu}(x)+\partial_{\mu}\tau(x),$
(3.28) $\displaystyle\delta
E_{\mu}(x)=-\Box\rho_{\mu}(x)-\partial_{\mu}\tau(x).$

The Lagrangian (3.2) is analogous to the one given in the equation (2.10).
However, the present Lagrangian describes a physical field $\phi_{\mu,\nu}(x)$
with no symmetry between the indices $\mu$ and $\nu$. As a result, the
Lagrangian contains more auxiliary fields. In particular, the fields
$C_{\mu}(x)$ and $E_{\mu}(x)$ in (3.2) are analogous to the field $C_{\mu}(x)$
in (2.10), and the fields $A(x)$ and $B(x)$ are analogous to the field $D(x)$.
Again, after eliminating the auxiliary fields after gauge fixing and using the
equations of motion one is left with only physical polarizations in the field
$\phi_{\mu,\nu}(x)$. This means, that we have a description of a spin $2$
field $g_{\mu\nu}(x)$, of an antisymmetric second rank tensor $B_{\mu\nu}(x)$
and of a scalar $\phi(x)$, all contained in the field $\phi_{\mu,\nu}(x)$.

Finally, one can check that the total Lagrangian

$\displaystyle L_{tot.}$
$\displaystyle=-\phi^{\mu,\nu}\Box\phi_{\mu,\nu}+B\Box A+A\Box B$
$\displaystyle+E^{\mu}\partial_{\mu}B+C^{\nu}\partial^{\mu}\phi_{\nu,\mu}+C^{\nu}\partial_{\nu}A+E^{\mu}\partial^{\nu}\phi_{\nu,\mu}$
$\displaystyle-B\partial_{\mu}E^{\mu}-\phi^{\nu,\mu}\partial_{\mu}C_{\nu}-A\partial_{\mu}C^{\mu}-\phi^{\mu,\nu}\partial_{\mu}E_{\nu}$
$\displaystyle+C^{\mu}C_{\mu}+E^{\mu}E_{\mu}$
$\displaystyle-i\,\bar{\Psi}^{\nu}\gamma^{\mu}\partial_{\mu}\Psi_{\nu}-i\,\bar{\Psi}^{\mu}\partial_{\mu}\chi+i\,\bar{\chi}\partial^{\mu}\Psi_{\mu}+i\,\bar{\chi}\gamma^{\mu}\partial_{\mu}\chi$

being a sum of the Lagrangians (2.15) and (3.2), is invariant under the
supersymmetry transformations [7]

$\delta\phi_{\nu,\mu}(x)=i\,\bar{\Psi}_{\mu}(x)\gamma_{\nu}\,\epsilon,\quad\delta
C_{\nu}(x)=-i\,\partial_{\mu}\bar{\chi}(x)\gamma^{\mu}\gamma_{\nu}\,\epsilon,\quad\delta
B(x)=-i\,\bar{\chi}(x)\,\epsilon,$ (3.30)
$\delta\Psi_{\mu}(x)=-\gamma^{\nu}\gamma^{\rho}\epsilon\,\partial_{\nu}\phi_{\rho,\mu}(x)-\epsilon
E_{\mu}(x),\quad\delta\chi(x)=-\gamma^{\nu}\epsilon\,C_{\nu}(x)\,.$ (3.31)

Let us note that we have not encountered any restriction on the number of
space-time dimensions until now. The requirement that the algebra of
supersymmetry transformations (3.30)–(3.31) closes on shell singles out the
number of space-time dimensions to be ${\cal D}=3$, 4, 6, or 10.

Decomposing the fields into irreducible representations of the Poincaré group
as

$\phi_{\mu,\nu}=\left(\phi_{(\mu,\nu)}-\eta_{\mu\nu}\frac{1}{{\cal
D}}\phi^{\rho}{}_{\rho}\right)+\phi_{[\mu,\nu]}+\eta_{\mu\nu}\frac{1}{{\cal
D}}\phi^{\rho}{}_{\rho}\equiv h_{\mu\nu}+B_{\mu\nu}+\frac{1}{{\cal
D}}\eta_{\mu\nu}\varphi$ (3.32)

and

$\psi_{\mu}^{a}=\Psi_{\mu}^{a}+\frac{1}{{\cal
D}}(\gamma_{\mu})^{ab}(\gamma^{\nu})_{bc}\psi_{\nu}^{c}\equiv\Psi_{\mu}^{a}+\frac{1}{{\cal
D}}(\gamma_{\mu})^{ab}\Xi_{b}$ (3.33)

one obtains the following $N=1$ supermultiplets:

  * •

In ${\cal D}=4$: a supergravity multiplet
$\left(g_{\mu\nu}(x),\psi_{\mu}^{a}(x)\right)$ and a chiral multiplet
$\left(\phi(x),a(x),\Xi(x)\right)$ where
$\partial_{\mu}a(x)=\frac{1}{3!}\epsilon_{\mu\nu\rho\sigma}\partial^{\nu}B^{\rho\sigma}(x)$.

  * •

In ${\cal D}=6$: a supergravity multiplet
$\left(g_{\mu\nu}(x),B_{\mu\nu}^{+}(x),\psi_{\mu}^{a}(x)\right)$ and a $(1,0)$
tensor multiplet $\left(\phi(x),B_{\mu\nu}^{-}(x),\Xi(x)\right)$, where we
decomposed $B_{\mu\nu}(x)$ into self-dual and anti self-dual parts.

  * •

In ${\cal D}=10$: one irreducible supergravity multiplet.

Therefore, one can say, that the Lagrangian (3.2) gives an “uniform”
description of various linearized $N=1$ supergravities. The case of ${\cal
D}=3$ contains no massless propagating degrees of freedom with spin $2$, so we
shall not consider it here.

Writing the supersymmetry transformations (3.30)–(3.31) in terms of auxiliary
oscillators,

$\displaystyle\langle
0|\delta\,\phi_{\mu,\nu}(x)\,\alpha^{\mu}_{1}\,\alpha^{\nu}_{2}=i\,\langle
0|{\overline{\Psi}}_{\mu}(x)\,\alpha_{1}^{\mu}\,(\gamma\cdot\alpha_{2})\,\epsilon$
(3.34) $\displaystyle\langle
0|\delta\,C_{\mu}(x)\,\alpha^{\mu}_{2}\,b_{1}=-\langle
0|{\overline{\chi}(x)}\,g_{0}\,(\gamma\cdot\alpha_{2})\,\epsilon\,b_{1}$
$\displaystyle\langle 0|\delta\,B(x)\,b_{1}\,c_{2}=-i\,\langle
0|{\overline{\chi}(x)}\,b_{1}\,c_{2}\,\epsilon$
$\displaystyle\delta\,\Psi_{\mu}(x)\,\alpha^{\mu,+}_{1}|0\rangle=(-i\,g_{0}\,(\gamma\cdot\alpha_{2})\,\epsilon\,\phi_{\mu,\nu}(x)\,\alpha^{\mu,+}_{1}\,\alpha^{\nu,+}_{2}-\epsilon\,E_{\mu}(x)\,\alpha^{\mu,+}_{1})|0\rangle$
$\displaystyle\delta\,\chi(x)\,b^{+}_{1}|0\rangle=-(\gamma\cdot\alpha_{2})\,\epsilon\,b^{+}_{1}\,C^{\mu}(x)\,\alpha_{\mu,2}^{+}|0\rangle$
(3.35)

one can see that supersymmetry is ”generated” by the second set of oscillators
($m=2$). In other words, to obtain the $N=1$ supermultiplets one can start
with the fermionic sector, which contains only the first set ($m=1$), then
apply the transformations (3.34)–(3.2) and require the closure of SUSY
algebra.555The same pattern persists for the higher spin supermultiplets: the
fermionic sector contains only the first set of the oscillators, while the
bosonic sector contains the oscillators from the first set and at most one
oscillator from the second set, see [7] for the details.

The description for the $N=1$ supersymmetric vector multiplet is similar.
Taking the fields in the bosonic and the fermionic sectors as

$|\Phi\rangle=(\phi_{\mu}(x)\alpha_{2}^{\mu,+}-ic_{0}b^{+}_{2}E(x))|0\rangle,\quad|\Psi\rangle=\Psi(x)|0\rangle,$
(3.36)

one can check that the corresponding Lagrangian

${\cal
L}=(\partial_{\mu}\phi^{\nu})(\partial_{\mu}\phi_{\nu})-2E\partial^{\mu}\phi_{\mu}+E^{2}-i\,\bar{\Psi}\gamma^{\mu}\partial_{\mu}\Psi$
(3.37)

is invariant under the supersymmetry transformations

$\displaystyle\langle 0|\delta\,\phi_{\mu}(x)\,\alpha^{\mu}_{2}$
$\displaystyle=$ $\displaystyle i\,\langle
0|{\overline{\Psi}}(x)\,(\gamma\cdot\alpha_{2})\,\epsilon$ (3.38)
$\displaystyle\delta\,\Psi(x)|0\rangle$ $\displaystyle=$
$\displaystyle(-i\,g_{0}\,(\gamma\cdot\alpha_{2})\,\epsilon\,\phi_{\mu}(x)\,\alpha^{\mu,+}_{2}-\epsilon\,E(x))|0\rangle$

After eliminating the auxiliary field $E(x)$ via its own equations of motion
one obtains the standard formulation of $N=1$ vector supermultiplet in ${\cal
D}=4$, 6, or 10 with an on-shell supersymmetry.

##  4 Cubic Interactions

###  4.1 Three bosons

In order to construct cubic interactions for the fields considered in the
previous sections, 666See [44], [45] for the details of construction for
higher spin fields and [46] for the analogous construction in Open Bosonic
String Field Theory. we take three copies of the auxiliary Fock space and
corresponding operators. The oscillators now obey the commutation relations

$[\alpha_{\mu,m}^{(i)},\alpha_{\nu,n}^{(j),+}]=\delta^{ij}\delta_{mn}\eta_{\mu\nu},$
(4.1)
$\\{c^{(i),+}_{m},b^{(j)}_{n}\\}=\\{c^{(i)}_{m},b^{(j),+}_{n}\\}=\\{c_{0,m}^{(i)},b_{0,n}^{(j)}\\}=\delta^{ij}\delta_{mn}\,,$
(4.2) $i,j=1,2,3,\quad m,n=1,2,\quad\mu,\nu=0,...,{\cal D}-1$

Then, we can consider the cubic Lagrangian

$\displaystyle{\cal L}_{3B,\text{int}}$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{3}\int
dc_{0}^{(i)}\langle\Phi^{(i)}|Q^{(i)}|\Phi^{(i)}\rangle+$ $\displaystyle+$
$\displaystyle g\left(\int
dc_{0}^{(1)}dc_{0}^{(2)}dc_{0}^{(3)}\langle\Phi^{(1)}|\langle\Phi^{(2)}|\langle\Phi^{(3)}||V\rangle+h.c.\right)$

where $g$ is a coupling constant and

$|V\rangle=V(p_{\mu}^{(i)},\alpha_{\mu,m}^{(i),+},c^{(i),+}_{m},b^{(i),+}_{m},b^{(i)}_{0,m})\,\,c_{0}^{(1)}c_{0}^{(2)}c_{0}^{(3)}\,\,|0^{(1)}\rangle\otimes|0^{(2)}\rangle\otimes|0^{(3)}\rangle$
(4.4)

where $V$ is a function of the creation operators that is restricted as
follows. An obvious requirement is that $V$ must be Lorentz invariant. In
order the Lagrangian to have the ghost number zero, the function $V$ must have
the ghost number equal to zero, and finally, the requirement of the invariance
of (4.1) under the non-linear gauge transformations

$\displaystyle\delta|\Phi^{(i)}\rangle$ $\displaystyle=$ $\displaystyle
Q^{(i)}|\Lambda^{(i)}\rangle-$ $\displaystyle-$ $\displaystyle g\int
dc_{0}^{(i+1)}dc_{0}^{(i+2)}\left((\langle\Phi^{(i+1)}|\langle\Lambda^{(i+2)}|+\langle\Phi^{(i+2)}|\langle\Lambda^{(i+1)}|)|V\rangle\right)$

up to the first power in $g$, implies that the vertex $|V\rangle$ is BRST
invariant:

$(Q^{(1)}+Q^{(2)}+Q^{(3)})|V\rangle=0$ (4.6)

The same condition guarantees that the group structure of the gauge
transformations is preserved up to the first order in $g$. Using momentum
conservation

$p_{\mu}^{(1)}+p_{\mu}^{(2)}+p_{\mu}^{(3)}=0$ (4.7)

and the commutation relations (4.1), one can show that that the following
expressions are BRST invariant for any values of the spins entering the cubic
vertex [30], [33]

${\cal
K}^{(i)}_{m}=(p^{(i+1)}-p^{(i+2)})\cdot\alpha^{(i),+}_{m}+(b_{0}^{(i+1)}-b_{0}^{(i+2)})\,c^{(i),+}_{m},$
(4.8)

${\cal
O}^{(i,i)}_{mn}=\alpha^{(i),+}_{m}\cdot\alpha^{(i),+}_{n}+c^{(i),+}_{m}b^{(i),+}_{n}+c^{(i),+}_{n}b^{(i),+}_{m},$
(4.9)

${\cal Z}_{mnp}={\cal Q}^{(1,2)}_{mn}{\cal K}^{(3)}_{p}+{\cal
Q}^{(2,3)}_{np}{\cal K}^{(1)}_{m}+{\cal Q}^{(3,1)}_{pm}{\cal K}^{(2)}_{n},$
(4.10)

where

${\cal
Q}^{(i,i+1)}_{mn}=\alpha^{(i),+}_{m}\cdot\alpha^{(i+1),+}_{n}+\frac{1}{2}b^{(i),+}_{m}c^{(i+1),+}_{n}+\frac{1}{2}b^{(i+1),+}_{n}c^{(i),+}_{m}.$
(4.11)

Before turning to a description of cubic vertices between bosonic and
fermionic fields, let us note that one can consider the cubic vertices between
three bosonic fields obeying some off-shell constraints. In particular, for
the fields considered in Subsection 3.2 we shall impose off-shell
transversality conditions

$\partial^{\mu}\phi_{\mu,\nu}(x)=\partial^{\nu}\phi_{\mu,\nu}(x)=0$ (4.12)

These conditions in turn restrict the parameters of gauge transformations

$\partial^{\mu}\lambda_{\mu}(x)=\partial^{\mu}\rho_{\mu}(x)=0,\quad\Box\lambda_{\mu}(x)=\Box\rho_{\mu}(x)=0,\quad\tau(x)=0.$
(4.13)

The constraints can be rewritten as

$l^{(i)}_{1}|\phi^{(i)}\rangle=l^{(i)}_{2}|\phi^{(i)}\rangle=0\,,\quad
l_{1}^{(i)}|\Lambda^{(i)}\rangle=l_{2}^{(i)}|\Lambda^{(i)}\rangle=l_{0}^{(i)}|\Lambda^{(i)}\rangle=0$
(4.14)

As a result of these constraints, all auxiliary fields and the ghost
dependence disappears in $|\Phi^{(i)}\rangle$ and the Lagrangian (4.1) reduces
to

${\cal
L}_{\text{3B,int}}=\sum_{i=1,2,3}\langle\phi^{(i)}|l_{0}^{(i)}|\phi^{(i)}\rangle+g\left(\langle\phi^{(1)}|\,\,\langle\phi^{(2)}|\,\,\langle\phi^{(3)}||{V}\rangle+h.c\right)$
(4.15)

The formulation in terms of the constrained fields considerably simplifies the
consideration of supersymmetry, as we shall see below.

###  4.2 Two fermions and one boson

For cubic interactions between two fermionic and one bosonic fields the
procedure is similar. Again, in order to simplify the consideration one
imposes an off-shell transversality constraint on the physical field,

$\partial^{\mu}\Psi_{\mu}^{a}(x)=0\quad\Leftrightarrow\quad
l_{1}|\Psi\rangle=0$ (4.16)

thus putting to zero the auxiliary field $\chi(x)$ (see subsection 2.3). This
constraint, in turn, restricts the parameter of gauge transformations to

$\gamma^{\mu}\partial_{\mu}\lambda^{\prime}(x)=0\quad\Leftrightarrow\quad
g_{0}|\Lambda\rangle=0$ (4.17)

The corresponding cubic Lagrangian which describes interactions between two
fermionic and one bosonic fields has the form

$\displaystyle{\cal L}_{\text{2F1B,int}}$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{2}{}_{a}\langle\Psi^{(i)}|(g_{0}^{(i)})^{a}{}_{b}|\Psi^{(a)}\rangle^{b}+{}\langle\phi^{(3)}|l_{0}^{(3)}|\phi^{(3)}\rangle+$
$\displaystyle+g\left(\,\,\langle\phi^{(3)}|\,\,{}_{a}\langle\Psi^{(1)}|\,\,{}_{b}\langle\Psi^{(2)}||{\cal
V}\rangle^{ab}+h.c\right).$

The requirement that the Lagrangian (4.2) is invariant under the non-linear
gauge transformations

$\displaystyle\delta|\phi^{(3)}\rangle$ $\displaystyle=$
$\displaystyle\tilde{Q}_{B}^{(3)}|\Lambda_{B}^{(3)}\rangle+$ $\displaystyle+$
$\displaystyle
g({}_{a}\langle{\Psi}^{(1)}|\,\,{}_{b}\langle{\Lambda_{F}}^{(2)}||{\cal
W}_{3}^{1,2}\rangle^{ab}+{}_{a}\langle\Psi^{(2)}|\,\,{}_{b}\langle{\Lambda}_{F}^{(1)}|\
|{\cal W}_{3}^{2,1}\rangle^{ab}),$ $\displaystyle\delta|\Psi^{(1)}\rangle^{a}$
$\displaystyle=$
$\displaystyle{\tilde{Q}}_{F}^{(1)}|{\Lambda_{F}}^{(1)}\rangle^{a}+$
$\displaystyle+$ $\displaystyle
g({}_{b}\langle\Psi^{(2)}|\,\,{}\langle\Lambda_{B}^{(3)}||{\cal
W}_{1}^{2,3}\rangle^{ab}+{{}\langle\phi^{(3)}|\,\,{}_{b}\langle\Lambda}_{F}^{(2)}||{\cal
W}^{3,2}_{1}\rangle^{ab}),$ $\displaystyle\delta|\Psi^{(2)}\rangle^{a}$
$\displaystyle=$
$\displaystyle{\tilde{Q}}_{F}^{(2)}|{\Lambda_{F}}^{(2)}\rangle^{a}+$
$\displaystyle+$ $\displaystyle
g({}^{A}\langle\phi^{(3)}|\,\,{}_{b}^{B}\langle\Lambda_{F}^{(1)}||{\cal
W}_{2}^{3,1}\rangle^{ab}+{}_{b}\langle\Psi^{(1)}|\,\,{}\langle{\Lambda}_{B}^{(3)}||{\cal
W}_{2}^{1,3}\rangle^{ab}),$

as well as the requirement of preservation of the group structure for the
gauge transformations up to the first power in the coupling constant $g$
imposes conditions on the vertices $|{\cal V}\rangle^{ab}$ and $|{\cal
W}\rangle^{ab}$ [6], which are similar to (4.6).

The vertex $|{\cal V}\rangle$ is again defined by a Lorentz invariant, ghost
number zero function of the creation operators. Given $|{\cal V}\rangle$,
gauge invariance of the Lagrangian (4.2) holds provided one can find
transformation vertices $|{\cal W}\rangle$ such that

$({g}_{0}^{(1)})^{a}{}_{b}|{\cal
W}_{1}^{2,3}\rangle^{bc}-({g}_{0}^{(2)})^{c}{}_{b}|{\cal
W}_{2}^{1,3}\rangle^{ba}+{\tilde{Q}}_{B}^{(3)}|{\cal V}\rangle^{ac}=0$ (4.22)
$({g}_{0}^{(1)})^{a}{}_{b}|{\cal W}_{1}^{3,2}\rangle^{bc}+l_{0}^{(3)}|{\cal
W}_{3}^{1,2}\rangle^{ac}+{\tilde{Q}}_{F}^{(2)}|{\cal V}\rangle^{ac}=0$ (4.23)
$({g}_{0}^{(2)})^{a}{}_{b}|{\cal W}_{2}^{3,1}\rangle^{bc}+l_{0}^{(3)}|{\cal
W}_{3}^{2,1}\rangle^{ac}-{\tilde{Q}}_{F}^{(1)}|{\cal V}\rangle^{ca}=0$ (4.24)

The preservation of the group structure leads to another set of equations. For
consistency, there must be some functions $|{\cal X}_{i}\rangle$ such that

${\tilde{Q}}_{F}^{(2)}|{\cal
W}_{1}^{2,3}\rangle^{ab}+\tilde{Q}_{B}^{(3)}|{\cal
W}_{1}^{3,2}\rangle^{ab}-{\tilde{Q}}_{F}^{(1)}|{\cal X}_{1}\rangle^{ab}=0$
(4.25) ${\tilde{Q}}_{F}^{(1)}|{\cal
W}_{2}^{1,3}\rangle^{ab}+\tilde{Q}_{B}^{(3)}|{\cal
W}_{2}^{3,1}\rangle^{ab}-{\tilde{Q}}_{F}^{(2)}|{\cal X}_{2}\rangle^{ab}=0$
(4.26) ${\tilde{Q}}_{F}^{(1)}|{\cal
W}_{3}^{1,2}\rangle^{ab}-{\tilde{Q}}_{F}^{(2)}|{\cal
W}_{3}^{2,1}\rangle^{ba}-\tilde{Q}_{B}^{(3)}|{\cal X}_{3}\rangle^{ab}=0$
(4.27)

Note that the equations (4.22)-(4.27) should hold only when acting on fields
and transformations satisfying the constraints of equations (4.14), (4.16),
and (4.17). The generalization to the unconstrained case was written in [6].

##  5 $N=1$ Super Yang-Mills

Let us turn to particular examples.

For the case of Super Yang-Mills one introduces colour indices in the
equations (4.1)–(4.1) and in (4.2)–(4.2) and takes the fields in the bosonic
and the fermionic sectors as

$|\Phi^{(i)}\rangle^{A}=({\phi}_{\mu}^{A}(x)\alpha_{2}^{\mu(i),+}-i{E}^{A}(x)c_{0}^{(i),+}b_{2}^{(i),+})|0^{(i)}\rangle,$
(5.1) $|\Psi^{(i)}\rangle^{a,A}=\Psi^{a,A}(x)|0^{(i)}\rangle$ (5.2)

the only non-zero parameter of gauge transformations being

$|\Lambda^{(i)}\rangle^{A}=ib_{2}^{(i),+}\lambda^{A}(x)|0^{(i)}\rangle$ (5.3)

The full interacting cubic Lagrangian is a sum of (4.15) and of777Note that in
Section 4.2 we placed the boson in the third Fock space, whereas now we have
three copies of the boson’s Fock space in addition to three copies of the
fermion’s. To write the interaction in a symmetric way, we introduce a cyclic
sum over the Fock space indices.

${\cal
L}_{\text{int}}=\sum_{i=1}^{3}{}^{A}\langle\Psi^{(i)}|g_{0}^{(i)}|\Psi^{(i)}\rangle_{A}+g\left({}^{A}\langle\Psi^{(1)}|{}^{B}\langle\Psi^{(2)}|{}^{C}\langle\Phi^{(3)}||{\cal
V}\rangle_{ABC}+\,\,\text{cyclic}\right)$ (5.4)

The cubic interaction vertex between three bosons is given by the expression
(4.10) with added colour indices

$|V\rangle_{ABC}=-\frac{i}{12}f_{ABC}{\cal Z}_{222}\times
c_{0}^{(1)}c_{0}^{(2)}c_{0}^{(3)}\,\,|0^{(1)}\rangle\otimes|0^{(2)}\rangle\otimes|0^{(3)}\rangle\,\,$
(5.5)

The cubic interaction vertex between two fermions with spins one-half and one
boson with spin one is

$|{\cal
V}\rangle^{ab}_{ABC}=\frac{i}{3}f_{ABC}(\alpha_{2}^{(3),+}\cdot\gamma)^{ab}\,\,|0^{(1)}\rangle\otimes|0^{(2)}\rangle\otimes|0^{(3)}\rangle+\,\,\text{cyclic}$
(5.6)

The only non-trivial $|{\cal W}\rangle$ vertices in the solution of equations
(4.22)–(4.24) are given by

$|{\cal W}_{1}^{2,3}\rangle^{ab}_{ABC}=|{\cal
W}_{2}^{1,3}\rangle^{ab}_{ABC}=f_{ABC}\,C^{ab}c_{2}^{+}\,|0^{(1)}\rangle\otimes|0^{(2)}\rangle\otimes|0^{(3)}\rangle$
(5.7)

The vertex (5.6) enters the Lagrangian and determines the interaction between
the vector field and two fermions, whereas the vertices (5.7) express the
nonlinear part of the gauge transformations.

After eliminating the auxiliary field $E^{A}(x)$ via its own equations of
motion one obtains an action for $N=1$ Super Yang-Mills up to the cubic order.
Alternatively, one could have imposed an off-shell transversality constraint
on the physical field $\phi_{\mu}^{A}(x)$ similarly to how it was done for the
supergravity multiplets (see the discussion around the equations (4.12) –
(4.15)). This would have put an auxiliary field $E^{A}(x)$ equal to zero and
restricted the parameter of gauge transformations as $\Box\lambda(x)=0$.

A higher spin generalization is given in [6], by multiplying the vertex (5.6)
with an arbitrary function of the BRST invariant expressions (4.8)–(4.10) and
then finding corresponding $|{\cal W}\rangle$ and $|{\cal X}\rangle$ vertices.
These solutions are covariant versions of the vertices found in the light-cone
formalism [27].

Because of the cubic interactions, the supersymmetry transformations for the
fermion in (3.38) will be deformed with a nonlinear term

$\displaystyle\delta^{\prime}|\Psi^{(i)}\rangle^{a}_{A}$ $\displaystyle=$
$\displaystyle g\,f_{ABC}\,\times$ $\displaystyle\times$
$\displaystyle{}_{B}\langle\phi^{(i+1)}|{}_{C}\langle\phi^{(i+2)}|(\gamma^{\mu\nu})^{a}{}_{b}\alpha^{(i+1),+}_{2,\mu}\alpha^{(i+2),+}_{2,\nu}\epsilon^{b}\,\,|0^{(i+1)}\rangle\otimes|0^{(i+2)}\rangle\otimes|0^{(i)}\rangle$

being the standard supersymmetry transformations for the $N=1$ Yang-Mills
supermultiplet. Let us notice, however, that if one imposes the off-shell
transversality condition on the vector field, the supersymmetry
transformations put the fields completely on shell. In this way one considers
on-shell cubic vertices, which transform into each other under linear
supersymmetry transformations (3.38).

##  6 $N=1$ Supergravities

In the following we show how we can write the cubic interaction vertices of
$N=1$ supergravity in ${\cal D}=4,6$ and $10$ dimensions. These can be
compared with the full Lagrangians of the theories which we include in
Appendix B.

The cubic vertices for $N=1$ supergravities, which describe a nonlinear
deformations of the Lagrangian (3.2) can be divided into two types. The first
type is universal in the same sense as is the Lagrangian (3.2), i.e., the
vertices have the same form in ${\cal D}=4,6$ and $10$. The second type of the
vertices are specific to particular dimensions.

In this section we consider the cubic interaction vertices of supergravities
whose free versions were written in Section 3.2. Namely, these are
supergravity in ${\cal D}=4$ coupled to a chiral multiplet, supergravity in
${\cal D}=6$ coupled to a $(1,0)$ tensor multiplet, and pure supergravity in
${\cal D}=10$.

In Subsection 6.1 we write the “universal” vertices, which are present in
${\cal D}=4$, 6, and 10. In Subsections 6.2 and 6.3 we write specific boson-
fermion-fermion vertices for ${\cal D}=10$ and 6, respectively.

###  6.1 Universal vertices

Let us start with the first type of cubic vertices. We impose off-shell
transversality constraints on the physical fields $\phi_{\mu,\nu}(x)$ and
$\Psi_{\mu}^{a}(x)$. Therefore, we have

$|\phi^{(i)}\rangle=\phi_{\mu,\nu}(x)\,\alpha_{1}^{\mu(i),+}\alpha_{2}^{\nu(i),+}|0^{(i)}\rangle,\qquad|\Psi^{(i)}\rangle^{a}=\Psi_{\mu}^{a}(x)\,\alpha_{1}^{\mu(i),+}|0^{(i)}\rangle$
(6.1)

We take the cubic vertices for three bosons

$-2\,g\,\langle\phi^{(1)}|\,\,\langle\phi^{(2)}|\,\,\langle\phi^{(3)}|{\cal
Z}_{111}{\cal
Z}_{222}|0^{(1)}\rangle\otimes|0^{(2)}\rangle\otimes|0^{(3)}\rangle$ (6.2)

where the expressions for ${\cal Z}_{mnp}$ are given in (4.10). For two
fermions and one boson we take the vertex

$g\,\,\langle\phi^{(3)}|\,\,{}_{a}\langle\Psi^{(1)}|\,\,{}_{b}\langle\Psi^{(2)}|{\cal
Z}_{111}(\gamma\cdot\alpha_{2}^{(3),+})^{ab}|0^{(1)}\rangle\otimes|0^{(2)}\rangle\otimes|0^{(3)}\rangle+\,\,\text{cyclic}.$
(6.3)

which solves the equations (4.22)–(4.24) and (4.25)–(4.27) with

$\displaystyle({\cal W}_{3}^{1,2})^{ab}$ $\displaystyle=$
$\displaystyle\,c^{(2),+}_{1}(\gamma\cdot\alpha_{2}^{+})^{ab}(\alpha^{(3),+}_{1}\cdot\alpha^{(1),+}_{1})$
(6.4) $\displaystyle({\cal W}_{3}^{2,1})^{ab}$ $\displaystyle=$ $\displaystyle
c^{(1),+}_{1}(\gamma\cdot\alpha_{2}^{+})^{ab}(\alpha^{(2),+}_{1}\cdot\alpha^{(3),+}_{1})$
$\displaystyle({\cal W}_{1}^{3,2})^{ab}$ $\displaystyle=$
$\displaystyle-\,c^{(2),+}_{1}(p^{(1)}\cdot\gamma)^{a}{}_{c}(\gamma\cdot\alpha_{2}^{+})^{cb}(\alpha^{(3),+}_{1}\cdot\alpha^{(1),+}_{1})$
$\displaystyle({\cal W}_{2}^{3,1})^{ab}$ $\displaystyle=$
$\displaystyle-c^{(1),+}_{1}(p^{(2)}\cdot\gamma)^{a}{}_{c}(\gamma\cdot\alpha_{2}^{+})^{cb}(\alpha^{(2),+}_{1}\cdot\alpha^{(3),+}_{1})$
$\displaystyle({\cal W}_{1}^{2,3})^{ab}$ $\displaystyle=$
$\displaystyle-c^{+}_{2}C^{ab}{\cal
Z}_{111}+c_{1}^{(3),+}(p^{(1)}\cdot\gamma)^{a}{}_{c}(\gamma\cdot\alpha_{2}^{+})^{cb}(\alpha^{(1),+}_{1}\cdot\alpha^{(2),+}_{1})$
$\displaystyle({\cal W}_{2}^{1,3})^{ab}$ $\displaystyle=$ $\displaystyle
c_{2}^{+}C^{ab}{\cal
Z}_{111}+c_{1}^{(3),+}(p^{(2)}\cdot\gamma)^{a}{}_{c}(\gamma\cdot\alpha_{2}^{+})^{cb}(\alpha^{(1),+}_{1}\cdot\alpha^{(2),+}_{1})$

and

$\displaystyle{\cal X}_{1}^{ab}$ $\displaystyle=$ $\displaystyle
c_{1}^{(2),+}c_{2}^{+}b_{1}^{(1),+}C^{ab}(p^{(1)}\cdot\alpha_{1}^{(3),+})-c_{1}^{(2),+}c_{1}^{(3),+}b_{1}^{(1),+}(p^{(1)}\cdot\gamma)^{a}{}_{c}(\gamma\cdot\alpha_{2}^{+})^{cb}$
$\displaystyle{\cal X}_{2}^{ab}$ $\displaystyle=$ $\displaystyle-
c_{2}^{+}c_{1}^{(1),+}b_{1}^{(2),+}C^{ab}(p^{(2)}\cdot\alpha_{1}^{(3),+})+c_{1}^{(3),+}c_{1}^{(1),+}b_{1}^{(2),+}(p^{(2)}\cdot\gamma)^{a}{}_{c}(\gamma\cdot\alpha_{2}^{+})^{cb}$
$\displaystyle{\cal X}_{3}^{ab}$ $\displaystyle=$ $\displaystyle-
c_{1}^{(1),+}c_{1}^{(2),+}b_{1}^{(3)+}(\gamma\cdot\alpha_{2}^{+})^{ab}$ (6.5)

The total Lagrangian, which is the sum of (4.15) and of

${\cal
L}_{\text{int}}=\sum_{i=1}^{3}\langle\Psi^{(i)}|g_{0}^{(i)}|\Psi^{(i)}\rangle+g\left(\langle\Psi^{(1)}|\langle\Psi^{(2)}|\langle\Phi^{(3)}||{\cal
V}\rangle+\,\,\text{cyclic}\right)$ (6.6)

has the form

$\displaystyle{\cal L}$ $\displaystyle=$
$\displaystyle-\phi^{\mu,\nu}\Box\phi_{\mu,\nu}+48g(\partial_{\rho}\partial_{\tau}\phi_{\mu,\nu})\phi^{\mu,\nu}\phi^{\rho,\tau}-96g(\partial_{\rho}\partial_{\tau}\phi_{\mu,\nu})\phi^{\mu,\tau}\phi^{\rho,\nu}$
$\displaystyle-$
$\displaystyle\frac{1}{2}{\bar{\Psi}}^{\mu}\gamma^{\nu}\partial_{\nu}\Psi_{\mu}+12ig\phi^{\mu,\nu}{\bar{\Psi}}^{\alpha}\gamma_{\nu}\partial_{\alpha}\Psi_{\mu}-6ig\phi^{\mu,\nu}{\bar{\Psi}}^{\alpha}\gamma_{\nu}\partial_{\mu}\Psi_{\alpha}$

Let us note that the overall coefficients in the cubic vertices (6.2) and
(6.3) are not fixed by the requirement of the gauge invariance and this
particular choice is dictated by the supersymmetry transformations. However,
due to the off-shell transversality conditions the supersymmetry
transformations

$\phi_{\nu,\mu}(x)=i\,\bar{\Psi}_{\mu}(x)\gamma_{\nu}\,\epsilon,\quad\delta\Psi_{\mu}(x)=-\gamma^{\nu}\gamma^{\rho}\epsilon\,\partial_{\nu}\phi_{\rho,\mu}(x)$
(6.8)

put the fields completely on shell.

Similarly to the case of cubic vertices in Super Yang-Mills, the supergravity
vertices of the first type (6.3) can be generalized to higher spins [6] by
multiplying them by an arbitrary function of the BRST invariant expressions
(4.8)–(4.10) and finding corresponding $|{\cal W}\rangle$ and $|{\cal
X}\rangle$ vertices.

###  6.2 Vertices of ${\cal D}=10$ supergravity

In order to consider the cubic vertices of the second type it is easier to
decompose the fermionic fields into irreducible representations of Poincaré
group according to (3.33). Then in ten dimensions we have the vertex

${}\langle\phi^{(3)}|\,\,{}_{a}\langle\psi^{(1)}|\,\,{}_{b}\langle\psi^{(2)}||{\cal
V}_{{\cal J}}\rangle^{ab},$ (6.9)

where

$\displaystyle{\cal V}^{ab}_{{\cal J}}$ $\displaystyle=$
$\displaystyle(\gamma_{\mu\tau\sigma\lambda\nu})^{ab}\alpha_{1}^{\mu(1),+}\alpha_{1}^{\nu(2),+}p^{\tau,(3)}\alpha_{1}^{\sigma(3),+}\alpha_{2}^{\lambda(3),+}+$
$\displaystyle+$
$\displaystyle(\alpha_{1}^{(3),+}\cdot\gamma)^{ab}[(p^{(3)}\cdot\alpha_{1}^{(1),+})(\alpha_{1}^{(2),+}\cdot\alpha_{2}^{(3),+})-(p^{(3)}\cdot\alpha_{1}^{(2),+})(\alpha_{1}^{(1),+}\cdot\alpha_{2}^{(3),+})]-$
$\displaystyle-$
$\displaystyle(\alpha_{2}^{(3),+}\cdot\gamma)^{ab}[(p^{(3)}\cdot\alpha_{1}^{(1),+})(\alpha_{1}^{(2),+}\cdot\alpha_{1}^{(3),+})-(p^{(3)}\cdot\alpha_{1}^{(2),+})(\alpha_{1}^{(1),+}\cdot\alpha_{1}^{(3),+})]+$
$\displaystyle+$
$\displaystyle(p^{(3)}\cdot\gamma)^{ab}[(\alpha_{1}^{(1),+}\cdot\alpha_{2}^{(3),+})(\alpha_{1}^{(2),+}\cdot\alpha_{1}^{(3),+})-(\alpha_{1}^{(2),+}\cdot\alpha_{2}^{(3),+})(\alpha_{1}^{(1),+}\cdot\alpha_{1}^{(3),+})]$

This vertex corresponds to the coupling of the field $B_{\mu\nu}(x)$ with two
gravitini. The corresponding ${\cal W}$ vertices have the form

$\displaystyle({\cal W}_{3}^{2,1})^{ab}$ $\displaystyle=$
$\displaystyle\frac{1}{2}c^{(1),+}_{1}\alpha_{1}^{\nu(2),+}\alpha_{1}^{\sigma(3),+}\alpha_{2}^{\lambda(3),+}(\gamma_{\sigma\lambda}\gamma_{\nu})^{ab}$
(6.11) $\displaystyle({\cal W}_{3}^{1,2})^{ab}$ $\displaystyle=$
$\displaystyle\frac{1}{2}c^{(2),+}_{1}\alpha_{1}^{\nu(1),+}\alpha_{1}^{\sigma(3),+}\alpha_{2}^{\lambda(3),+}(\gamma_{\sigma\lambda}\gamma_{\nu})^{ab}$
$\displaystyle({\cal W}_{2}^{3,1})^{ab}$ $\displaystyle=$
$\displaystyle-c^{(1),+}_{1}[\frac{1}{2}(p^{(2)}\cdot\gamma)^{a}{}_{c}\alpha_{1}^{\nu(2),+}\alpha_{1}^{\sigma(3),+}\alpha_{2}^{\lambda(3),+}(\gamma_{\sigma\lambda}\gamma_{\nu})^{cb}+$
$\displaystyle\quad+p^{(1),\mu}\alpha_{1}^{\nu(2),+}\alpha_{1}^{\sigma(3),+}\alpha_{2}^{\lambda(3),+}(\eta_{\mu\nu}\gamma_{\lambda}\gamma_{\nu}-\eta_{\mu\lambda}\gamma_{\sigma}\gamma_{\nu}+\eta_{\mu\nu}\gamma_{\sigma\lambda})^{ab}]$
$\displaystyle({\cal W}_{1}^{3,2})^{ab}$ $\displaystyle=$ $\displaystyle
c^{(2),+}_{1}[\frac{1}{2}(p^{(2)}\cdot\gamma)^{b}{}_{c}\alpha_{1}^{\nu(1),+}\alpha_{1}^{\sigma(3),+}\alpha_{2}^{\lambda(3),+}(\gamma_{\sigma\lambda}\gamma_{\nu})^{ca}+$
$\displaystyle\quad+p^{(2),\mu}\alpha_{1}^{\nu(1),+}\alpha_{1}^{\sigma(3),+}\alpha_{2}^{\lambda(3),+}(\eta_{\mu\nu}\gamma_{\lambda}\gamma_{\nu}-\eta_{\mu\lambda}\gamma_{\sigma}\gamma_{\nu}+\eta_{\mu\nu}\gamma_{\sigma\lambda})^{ba}]$

The solution for the group structure equations includes

$\displaystyle{\cal X}_{1}^{ab}$ $\displaystyle=$ $\displaystyle-
b_{1}^{(1),+}[c_{1}^{(3),+}c_{1}^{(2),+}(\gamma\cdot\alpha_{2}^{(3),+})^{a}{}_{c}(\gamma\cdot
p^{(1)})^{cb}-c_{2}^{(3),+}c_{1}^{(2),+}(\gamma\cdot\alpha_{1}^{(3),+})^{a}{}_{c}(\gamma\cdot
p^{(1)})^{cb}]$ $\displaystyle{\cal X}_{2}^{ab}$ $\displaystyle=$
$\displaystyle
b_{1}^{(2),+}[c_{1}^{(3),+}c_{1}^{(1),+}(\gamma\cdot\alpha_{2}^{(3),+})^{a}{}_{c}(\gamma\cdot
p^{(2)})^{cb}-c_{2}^{(3),+}c_{1}^{(1),+}(\gamma\cdot\alpha_{1}^{(3),+})^{a}{}_{c}(\gamma\cdot
p^{(2)})^{cb}]$ $\displaystyle{\cal X}_{3}^{ab}$ $\displaystyle=$
$\displaystyle
b_{1}^{(3),+}c_{1}^{(3),+}c_{1}^{(1),+}(\gamma\cdot\alpha_{2}^{(3),+})^{ab}-b_{2}^{(3),+}c_{2}^{(3),+}c_{1}^{(1),+}(\gamma\cdot\alpha_{1}^{(3),+})^{ab}$
(6.12)

The remaining cubic vertices in ${\cal D}=10$ are those which correspond to
the coupling of the $B_{\mu\nu}(x)$ to one gravitino and one dilatino,

${}\langle\phi^{(3)}|\,\,{}_{a}\langle\psi^{(1)}|\,\,{}^{b}\langle\Xi^{(2)}||{\cal
V}_{{\cal L}}\rangle^{a}_{b},$ (6.13)

There are two such couplings. The first is

$\displaystyle({\cal V}_{{\cal
L}_{1}})^{a}{}_{b}=(\gamma_{\mu\nu\tau})^{ac}p^{\mu,(3)}\alpha_{1}^{\nu(3),+}\alpha_{2}^{\tau(3),+}(\alpha_{1}^{(1),+}\cdot\gamma)_{cb}$
(6.14)

The non-trivial ${\cal W}$ vertices are

$\displaystyle({\cal W}_{3}^{2,1})^{ab}$ $\displaystyle=$
$\displaystyle-c^{(1),+}_{1}(\gamma_{\nu\tau})^{ab}\alpha_{1}^{\nu(3),+}\alpha_{2}^{\tau(3),+}$
(6.15) $\displaystyle({\cal W}_{2}^{3,1})^{ab}$ $\displaystyle=$
$\displaystyle
c^{(1),+}_{1}\big{[}(\gamma_{\nu\tau})^{a}{}_{c}\alpha_{1}^{\nu(3),+}\alpha_{2}^{\tau(3),+}(p_{2}\cdot\gamma)^{cb}\big{]}$

The second vertex of this type is

$\displaystyle({\cal V}_{{\cal L}_{2}})^{a}{}_{b}$ $\displaystyle=$
$\displaystyle\delta^{a}{}_{b}(\alpha_{1}^{(3),+}\cdot\alpha_{2}^{(3),+})(\alpha_{1}^{(1),+}\cdot
p^{(3)})$ (6.16)

for which

$\displaystyle({\cal W}_{3}^{2,1})^{ab}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}c^{(1),+}_{1}C^{ab}\alpha_{1}^{(3),+}\cdot\alpha_{2}^{(3),+}$
(6.17) $\displaystyle({\cal W}_{2}^{3,1})^{ab}$ $\displaystyle=$
$\displaystyle\frac{1}{2}c^{(1),+}_{1}(p_{2}\cdot\gamma)^{ab}\alpha_{1}^{(3),+}\cdot\alpha_{2}^{(3),+}$

For the last two vertices the solutions for the group structure equations are
with $|{\cal X}_{i}\rangle=0$.

###  6.3 Vertices of ${\cal D}=6$ supergravity

Most of the vertices of ${\cal D}=6$ supergravity have already been described
above. They include the universal vertices of Subsection 6.1, as well as the
vertices (6.14) and (6.16), which are both present in ${\cal D}=6$.

The vertex that has a different form is the coupling of the $B$-field to two
gravitini, which is now given by

${\cal
V}^{ab}=(\gamma_{\tau}\gamma_{\mu\nu\rho}\gamma_{\lambda})^{ab}\alpha_{1}^{\lambda(1),+}\alpha_{1}^{\tau(2),+}p^{\mu(3)}\alpha_{1}^{\mu(3),+}\alpha_{2}^{\nu(3),+}$
(6.18)

The corresponding ${\cal W}$ vertices have the form

$\displaystyle({\cal W}_{3}^{2,1})^{ab}$ $\displaystyle=$ $\displaystyle
c_{1}^{(1),+}\alpha_{2}^{\mu(2),+}\alpha_{1}^{\nu(3),+}\alpha_{1}^{\rho(3),+}(\gamma_{\nu\rho}\gamma_{\mu})^{ab}$
(6.19) $\displaystyle({\cal W}_{3}^{1,2})^{ab}$ $\displaystyle=$
$\displaystyle
c_{1}^{(2),+}\alpha_{1}^{\mu(2),+}\alpha_{1}^{\nu(3),+}\alpha_{1}^{\rho(3),+}(\gamma_{\nu\rho}\gamma_{\mu})^{ab}$
$\displaystyle({\cal W}_{2}^{3,1})^{ab}$ $\displaystyle=$ $\displaystyle-
c_{1}^{(1),+}\big{[}\alpha_{1}^{\nu(3),+}\alpha_{2}^{\rho(3),+}(\gamma_{\nu\rho})^{a}{}_{c}p^{\mu(2)}\alpha_{1}^{\sigma(2),+}(\gamma_{\mu}\gamma_{\sigma})^{cb}+$
$\displaystyle\qquad\qquad+2(p^{(1)}\cdot\alpha_{1}^{(2),+})\alpha_{1}^{\nu(3),+}\alpha_{2}^{\rho(3),+}(\gamma_{\nu\rho})^{ab}\big{]}$
$\displaystyle({\cal W}_{1}^{3,2})^{ab}$ $\displaystyle=$ $\displaystyle
c_{1}^{(2),+}\big{[}\alpha_{1}^{\nu(3),+}\alpha_{2}^{\rho(3),+}(\gamma_{\nu\rho})^{a}{}_{c}p^{\mu(1)}\alpha_{1}^{\sigma(1),+}(\gamma_{\mu}\gamma_{\sigma})^{cb}+$
$\displaystyle\qquad\qquad+2(p^{(2)}\cdot\alpha_{1}^{(1),+})\alpha_{1}^{\nu(3),+}\alpha_{2}^{\rho(3),+}(\gamma_{\nu\rho})^{ab}\big{]}$

with

$\displaystyle{\cal X}_{1}^{ab}$ $\displaystyle=$ $\displaystyle-
b_{1}^{(1),+}[c_{1}^{(3),+}c_{1}^{(2),+}(\gamma\cdot\alpha_{2}^{(3),+})^{a}{}_{c}(\gamma\cdot
p^{(1)})^{cb}-c_{2}^{(3),+}c_{1}^{(2),+}(\gamma\cdot\alpha_{1}^{(3),+})^{a}{}_{c}(\gamma\cdot
p^{(1)})^{cb}]$ $\displaystyle{\cal X}_{2}^{ab}$ $\displaystyle=$
$\displaystyle
b_{1}^{(2),+}[c_{1}^{(3),+}c_{1}^{(1),+}(\gamma\cdot\alpha_{2}^{(3),+})^{a}{}_{c}(\gamma\cdot
p^{(2)})^{cb}-c_{2}^{(3),+}c_{1}^{(1),+}(\gamma\cdot\alpha_{1}^{(3),+})^{a}{}_{c}(\gamma\cdot
p^{(2)})^{cb}]$ $\displaystyle{\cal X}_{3}^{ab}$ $\displaystyle=$
$\displaystyle
2b_{1}^{(3),+}c_{1}^{(1),+}c_{1}^{(2),+}(\gamma\cdot\alpha_{2}^{(3),+})^{ab}-2b_{2}^{(3),+}c_{1}^{(1),+}c_{1}^{(2),+}(\gamma\cdot\alpha_{1}^{(3),+})^{ab}$
(6.20)

solving the group structure equations.

There is also the coupling of $B_{\mu\nu}$ to two dilatini,

${}\langle\phi^{(3)}|\,\,{}^{a}\langle\Xi^{(1)}|\,\,{}^{b}\langle\Xi^{(2)}||{\cal
V}_{{\cal X}}\rangle_{ab},$ (6.21)

which is of the form

$({\cal V}_{\cal
X})_{ab}=(\gamma_{\mu\nu\rho})_{ab}p^{\mu(3)}\alpha_{1}^{\nu(3),+}\alpha_{2}^{\rho(3),+}$
(6.22)

In this case all the ${\cal W}$ vertices are trivial.

##  7 Light Cone Formalism

In this Section we describe how to construct cubic vertices of ${\cal D}=4$
$N=1$ super Yang-Mills and Supergravity in the light cone approach following
[47] (see also [53]–[54] for a brief review of the light cone approach).

###  7.1 Set Up

To construct cubic interaction vertices in the light cone approach let us
consider a field theoretic realization of the ${\cal D}=4$ $N=1$ super
Poincaré algebra

$\displaystyle[Q^{a},Q^{b}]$
$\displaystyle=\frac{1}{2}(\gamma^{\mu})^{ab}P_{\mu}\,,$ (7.1)
$\displaystyle[Q^{a},J^{\mu\nu}]$
$\displaystyle=\frac{1}{2}(\gamma^{\mu\nu})^{a}{}_{b}Q^{b}\,,$ (7.2)
$\displaystyle[J^{\mu\nu},P^{\rho}]$
$\displaystyle=P^{\mu}\eta^{\nu\rho}-P^{\nu}\eta^{\mu\rho}\,,$ (7.3)
$\displaystyle[J^{\mu\nu},J^{\rho\sigma}]$
$\displaystyle=J^{\mu\sigma}\eta^{\nu\rho}-J^{\nu\sigma}\eta^{\mu\rho}-J^{\mu\rho}\eta^{\nu\sigma}+J^{\nu\rho}\eta^{\mu\sigma}\,.$
(7.4)

Here $J^{\mu\nu}$ are generators of Lorentz transformations, $P^{\mu}$ are
generators of translations, and $Q^{a}$ are generators of Supersymmetry
transformations. These generators are split into kinematical and dynamical
generators. Kinematical generators preserve the Cauchy surface (the light
cone) and are quadratic in fields both on free and interacting levels. The
other generators are dynamical and they receive higher order corrections in
fields. These corrections are determined from the requirement that the
Poincaré algebra is preserved at the interacting level.

We choose the four dimensional coordinates as

$x^{\pm}=\frac{1}{\sqrt{2}}(x^{3}\pm x^{0}),\quad
z=\frac{1}{\sqrt{2}}(x^{1}+ix^{2}),\quad\bar{z}=\frac{1}{\sqrt{2}}(x^{1}-ix^{2})$
(7.5)

The coordinate $x^{+}$ is treated as the time direction and $H=P^{-}$ is the
Hamiltonian888 Note that $\beta$ is used instead of $p^{+}$ in order to
simplify the form of the equations.We shall also put $x^{+}=0$ for now on..
The generators of the super Poincaré algebra are split according to

kinematical $\displaystyle:$ $\displaystyle
P^{+},P^{z},{P}^{\bar{z}},J^{z+},{J}^{\bar{z}+},J^{+-},J^{z\bar{z}},Q^{+},{\bar{Q}}^{+},$
$\displaystyle:9$ (7.6) dynamical $\displaystyle:$ $\displaystyle
P^{-},J^{z-},{J}^{\bar{z}-},Q^{-},{\bar{Q}}^{-}$ $\displaystyle:5$ (7.7)

It is sufficient to construct the Poincare algebra at $x^{+}=0$ and then
evolve all the generators according to $\dot{G}=i[H,G]$. The equations to be
solved are

$[Q^{-},P^{-}]=[\bar{Q}^{-},P^{-}]=0\,,\quad[J^{z,-},P^{-}]=[J^{\bar{z},-},P^{-}]=0\,.$
(7.8)

The spectrum consists of bosonic $\phi_{\lambda}(x)$ and fermionic
$\psi_{\lambda}(x)$ fields999In this Section, unlike the previous ones, the
index $\lambda$ denotes a helicity of a field, rather than its Lorentz index.
with the helicities $\lambda=\pm 1,\pm 2$ for bosons,
$\lambda=\pm\frac{1}{2},\pm\frac{3}{2}$ for fermions. It is convenient to work
with partial Fourier transforms

$\displaystyle\phi_{\lambda}(x)$ $\displaystyle=(2\pi)^{-\tfrac{3}{2}}\int
e^{+i(x^{-}\beta+z\bar{p}+\bar{z}p)}\phi_{\lambda}(\vec{p})\,d^{3}p\,,$ (7.9)
$\displaystyle\psi_{\lambda}(x)$ $\displaystyle=(2\pi)^{-\tfrac{3}{2}}\int
e^{+i(x^{-}\beta+z\bar{p}+\bar{z}p)}\psi_{\lambda}(\vec{p})\,d^{3}p\,$ (7.10)

with $d^{3}p=d\beta\,dp\,d{\bar{p}}$. The fields obey the following
conjugation rules

$\phi^{\dagger}_{\lambda}(\vec{p})=\phi_{-\lambda}(-\vec{p}),\quad\psi^{\dagger}_{\lambda}(\vec{p})=\psi_{-\lambda}(-\vec{p})$
(7.11)

Introducing a Grassman momentum $p_{\theta}$, one can combine the bosonic and
fermionic fields into superfields

$\Phi_{\lambda}=\phi_{\lambda}+\frac{p_{\theta}}{\beta}\psi_{\lambda-\frac{1}{2}},\quad\Phi_{-\lambda+\frac{1}{2}}=\psi_{-\lambda+\frac{1}{2}}+{p_{\theta}}\phi_{-\lambda},$
(7.12)

with conjugation properties

$\overline{\Phi}_{-\lambda}=\phi_{-\lambda}^{\dagger}+\frac{p_{\theta}}{\beta}\psi^{\dagger}_{-\lambda+\frac{1}{2}},\quad\overline{\Phi}_{\lambda-\frac{1}{2}}=-\psi_{\lambda-\frac{1}{2}}^{\dagger}+{p_{\theta}}\phi_{\lambda}^{\dagger},$
(7.13)

The equal time Poisson brackets between the fields

$[\phi_{\lambda}(\vec{p}),\phi_{\lambda^{\prime}}^{\dagger}(\vec{p}^{\prime})]=\delta_{\lambda,\lambda^{\prime}}\frac{\delta^{3}(\vec{p}-\vec{p}^{\prime})}{2\beta},\quad[\psi_{\lambda}(\vec{p}),\psi_{\lambda^{\prime}}^{\dagger}(\vec{p}^{\prime})]=\delta_{\lambda,\lambda^{\prime}}\frac{\delta^{3}(\vec{p}-\vec{p}^{\prime})}{2}$
(7.14)

read in terms of the superfields as

$[\Phi_{\lambda}(\vec{p},p_{\theta}),\overline{\Phi}_{\lambda^{\prime}}(\vec{p}^{\prime},p_{\theta}^{\prime})]=(-)^{\epsilon_{\lambda+\frac{1}{2}}}\delta_{\lambda,\lambda^{\prime}+\frac{1}{2}}\frac{\delta^{3}(\vec{p}-\vec{p}^{\prime})\,\delta(p_{\theta}-p_{\theta}^{\prime})}{2\beta}$
(7.15)

where $\epsilon_{\lambda}$ is $0$ for integer $\lambda$ and is $1$ for half-
integer $\lambda$.

The kinematical generators, which are the same both on free and interacting
levels have the form

${P}^{+}=\beta\,,\quad{P}^{z}=p,\quad
P^{\bar{z}}={\bar{p}},\quad{J}^{z+}=-\beta{\frac{\partial}{\partial{\bar{p}}}},\quad{J}^{{\bar{z}}+}=-\beta{\frac{\partial}{\partial
p}},$ (7.16)
${J}^{-+}=-\frac{\partial}{\partial\beta}\beta-\frac{1}{2}p_{\theta}\frac{\partial}{\partial
p_{\theta}}+\frac{1}{2}\epsilon_{\lambda},\quad{J}^{z{\bar{z}}}=p\partial_{p}-\bar{p}\frac{\partial}{\partial\bar{p}}+\lambda-
p_{\theta}\frac{\partial}{\partial p_{\theta}}$
$Q^{+}=(-)^{\epsilon_{\lambda}}\beta\frac{\partial}{\partial
p_{\theta}},\quad\bar{Q}^{+}=(-)^{\epsilon_{\lambda}}p_{\theta}$

The dynamical generators at the free level are

$\displaystyle H_{2}$ $\displaystyle=$
$\displaystyle-\frac{p{\bar{p}}}{\beta}\,,$ (7.17) $\displaystyle{J}^{z-}_{2}$
$\displaystyle=$
$\displaystyle-{\frac{\partial}{\partial{\bar{p}}}}\frac{p{\bar{p}}}{\beta}+p{\frac{\partial}{\partial\beta}}-\left(\lambda-\frac{1}{2}p_{\theta}\frac{\partial}{\partial
p_{\theta}}\right)\frac{p}{\beta}+\left(\frac{1}{2}p_{\theta}\frac{\partial}{\partial
p_{\theta}}-\frac{1}{2}\epsilon_{\lambda}\right)\frac{p}{\beta}$
$\displaystyle{J}^{{\bar{z}}-}_{2}$ $\displaystyle=$
$\displaystyle-{\frac{\partial}{\partial
p}}\frac{p{\bar{p}}}{\beta}+{\bar{p}}{\frac{\partial}{\partial\beta}}+\left(\lambda-\frac{1}{2}p_{\theta}\frac{\partial}{\partial
p_{\theta}}\right)\frac{\bar{p}}{\beta}+\left(\frac{1}{2}p_{\theta}\frac{\partial}{\partial
p_{\theta}}-\frac{1}{2}\epsilon_{\lambda}\right)\frac{p}{\beta}$
$\displaystyle Q_{2}^{-}$ $\displaystyle=$
$\displaystyle(-)^{\epsilon_{\lambda}}\frac{p}{\beta}p_{\theta}$
$\displaystyle\bar{Q}_{2}^{-}$ $\displaystyle=$
$\displaystyle(-)^{\epsilon_{\lambda}}\bar{p}\frac{\partial}{\partial
p_{\theta}}$

At the level of cubic interactions one assumes the following expansion for the
dynamical generators

$\displaystyle H_{3}$ $\displaystyle=$ $\displaystyle H_{2}+\int
d\Gamma_{[3]}\,\overline{\Phi}^{\lambda_{1}\lambda_{2}\lambda_{3}}_{q_{1}q_{2}q_{3}}\,h_{\lambda_{1}\lambda_{2}\lambda_{3}}^{q_{1}q_{2}q_{3}}$
(7.18) $\displaystyle Q_{3}^{-}$ $\displaystyle=$ $\displaystyle
Q_{2}^{-}+\int
d\Gamma_{[3]}\,\overline{\Phi}^{\lambda_{1}\lambda_{2}\lambda_{3}}_{q_{1}q_{2}q_{3}}\,q_{\lambda_{1}\lambda_{2}\lambda_{3}}^{q_{1}q_{2}q_{3}}$
$\displaystyle{\bar{Q}}_{3}^{-}$ $\displaystyle=$
$\displaystyle{\bar{Q}}_{2}^{-}+\int
d\Gamma_{[3]}\,\overline{\Phi}^{\lambda_{1}\lambda_{2}\lambda_{3}}_{q_{1}q_{2}q_{3}}\,{\bar{q}}_{\lambda_{1}\lambda_{2}\lambda_{3}}^{q_{1}q_{2}q_{3}}$
$\displaystyle J^{z-}_{3}$ $\displaystyle=$ $\displaystyle J^{z-}_{2}+\int
d\Gamma_{[3]}\times$ $\displaystyle\times$
$\displaystyle\left[\overline{\Phi}^{\lambda_{1}\lambda_{2}\lambda_{3}}_{q_{1}q_{2}q_{3}}\,\,j_{\lambda_{1}\lambda_{2}\lambda_{3}}^{q_{1}q_{2}q_{3}}-\frac{1}{3}\left(\sum_{k=1}^{3}\frac{\partial\overline{\Phi}^{\lambda_{1}\lambda_{2}\lambda_{3}}_{q_{1}q_{2}q_{3}}}{\partial\bar{q}_{k}}\right)h_{\lambda_{1}\lambda_{2}\lambda_{3}}^{q_{1}q_{2}q_{3}}-\frac{1}{3}\left(\sum_{k=1}^{3}\frac{\partial\overline{\Phi}^{\lambda_{1}\lambda_{2}\lambda_{3}}_{q_{1}q_{2}q_{3}}}{\partial{q}_{\theta,k}}\right)q_{\lambda_{1}\lambda_{2}\lambda_{3}}^{q_{1}q_{2}q_{3}}\right]\,$
$\displaystyle J^{{\bar{z}}-}_{3}$ $\displaystyle=$ $\displaystyle
J^{\bar{z}-}_{2}+\int d\Gamma_{[3]}\times$ $\displaystyle\times$
$\displaystyle\left[\overline{\Phi}^{\lambda_{1}\lambda_{2}\lambda_{3}}_{q_{1}q_{2}q_{3}}\,\,{\bar{j}}_{\lambda_{1}\lambda_{2}\lambda_{3}}^{q_{1}q_{2}q_{3}}-\frac{1}{3}\left(\sum_{k=1}^{3}\frac{\partial\overline{\Phi}^{\lambda_{1}\lambda_{2}\lambda_{3}}_{q_{1}q_{2}q_{3}}}{\partial
q_{k}}\right)h_{\lambda_{1}\lambda_{2}\lambda_{3}}^{q_{1}q_{2}q_{3}}+\frac{1}{3}\left(\sum_{k=1}^{3}\frac{q_{\theta,k}}{\beta_{k}}\right)\overline{\Phi}^{\lambda_{1}\lambda_{2}\lambda_{3}}_{q_{1}q_{2}q_{3}}{\bar{q}}_{\lambda_{1}\lambda_{2}\lambda_{3}}^{q_{1}q_{2}q_{3}}\right]$

where
$\overline{\Phi}^{\lambda_{1}\lambda_{2}\lambda_{3}}_{q_{1}q_{2}q_{3}}\equiv\overline{\Phi}^{\lambda_{1}}_{q_{1}}\overline{\Phi}^{\lambda_{2}}_{q_{2}}\overline{\Phi}^{\lambda_{3}}_{q_{3}}$
and

$d\Gamma_{[3]}=(2\pi)^{3}\prod_{k=1}^{3}\frac{d^{3}q_{k}}{(2\pi)^{\frac{3}{2}}}\delta^{3}\left(\sum_{i=1}^{3}q_{i}\right)\prod_{l=1}^{3}dq_{\theta,l}\,\delta\left(\sum_{j=1}^{3}q_{\theta,j}\right)$
(7.19)

is an integration measure.

###  7.2 ${\cal D}=4$, $N=1$ Super Yang Mills and Pure Supergravity

The cubic vertices which are present in the interaction part of dynamical
generators (7.18) are determined from the requirement of preservation of the
algebra (7.1). A solution which contains the $N=1$ Super Yang Mills and
Supergravity vertices has the form [47]

$h_{\lambda_{1}\lambda_{2}\lambda_{3}}^{q_{1}q_{2}q_{3}}=C^{\lambda_{1}\lambda_{2}\lambda_{3}}({\overline{\mathbb{P}}})^{M_{\lambda}+1}\prod_{i=1}^{3}\beta_{i}^{-\lambda_{i}-\frac{1}{2}\epsilon_{\lambda_{i}}}+\overline{C}^{\lambda_{1}\lambda_{2}\lambda_{3}}({\mathbb{P}})^{-M_{\lambda}-\frac{1}{2}}{\mathbb{P}}_{\theta}\prod_{i=1}^{3}\beta_{i}^{\lambda_{i}-\frac{1}{2}\epsilon_{\lambda_{i}}}$
(7.20)
$q_{\lambda_{1}\lambda_{2}\lambda_{3}}^{q_{1}q_{2}q_{3}}=-C^{\lambda_{1}\lambda_{2}\lambda_{3}}({\overline{\mathbb{P}}})^{M_{\lambda}}{\mathbb{P}}_{\theta}\prod_{i=1}^{3}\beta_{i}^{-\lambda_{i}-\frac{1}{2}\epsilon_{\lambda_{i}}}$
(7.21)
$j_{\lambda_{1}\lambda_{2}\lambda_{3}}^{q_{1}q_{2}q_{3}}=2C^{\lambda_{1}\lambda_{2}\lambda_{3}}({\overline{\mathbb{P}}})^{M_{\lambda}}\,\chi\prod_{i=1}^{3}\beta_{i}^{-\lambda_{i}-\frac{1}{2}\epsilon_{\lambda_{i}}}$
(7.22)

and

$M_{\lambda}=\lambda_{1}+\lambda_{2}+\lambda_{3},\quad\lambda_{1}=s_{1}-\frac{1}{2},\quad\lambda_{2}=s_{2}-\frac{1}{2},\quad\lambda_{3}=-s_{3}$
(7.23)

In these equations $C^{\lambda_{1}\lambda_{2}\lambda_{3}}$,
$\overline{C}^{\lambda_{1}\lambda_{2}\lambda_{3}}$ are coupling constants and

${\mathbb{P}}=\frac{1}{3}\left[(\beta_{1}-\beta_{2})p_{3}+(\beta_{2}-\beta_{3})p_{1}+(\beta_{3}-\beta_{1})p_{2}\right]\,,\\\
$ (7.24)
${\mathbb{P}}_{\theta}=\frac{1}{3}\left[(\beta_{1}-\beta_{2})p_{\theta,3}+(\beta_{2}-\beta_{3})p_{\theta,1}+(\beta_{3}-\beta_{1})p_{\theta,2}\right]\,,\\\
$ (7.25)
$\chi=\beta_{1}(\lambda_{2}-\lambda_{3})+\beta_{2}(\lambda_{3}-\lambda_{1})+\beta_{3}(\lambda_{1}-\lambda_{2})\,.$
(7.26)

The momenta $\beta_{i},p_{i},\bar{p}_{i}$ and $p_{\theta,i}$ obey the
conservation properties as in (4.7). The vertices
${\bar{q}}_{\lambda_{1}\lambda_{2}\lambda_{3}}^{q_{1}q_{2}q_{3}}$ and
${\bar{j}}_{\lambda_{1}\lambda_{2}\lambda_{3}}^{q_{1}q_{2}q_{3}}$ can be
obtained from (7.21)–(7.22) by relevant hermitean conjugation.

The cubic vertices for $N=1$ Super Yang Mills can be recovered by choosing
$s_{1}=s_{2}=s_{3}=1$ in the equations above. Similarly, pure $N=1$
Supergravity vertices can be recovered by putting $s_{1}=s_{2}=s_{3}=2$.

Acknowledgments. M.T. would like to thank the organizers of the online
conference “Quarks 2020” and of the online workshop“Aspects of Symmetry” for
the invitation to give talks. M.T. would like to thank the Department of
Mathematics, the University of Auckland for the hospitality during the final
stage of the project. D.W. would like to thank the Yukawa Institute of
Theoretical Physics (YITP) for its support in the period when this work was
completed. The work of I.L.B. and V.A.K. was partially supported by the
Ministry of Education of Russian Federation, project FEWF-2020-0003. The work
of M.T. and D.W. was supported by the Quantum Gravity Unit of the Okinawa
Institute of Science and Technology Graduate University (OIST).

##  Appendix A Conventions

We mainly follow the notations of [55].

The Latin letters $a,b\ldots$ label spinorial indices. The Greek letters
$\mu,\nu,\ldots$ label flat space-time vector indices and Greek letters with
“hat” ${\hat{\mu}},{\hat{\nu}},\ldots$ label vector indices in curved space-
time.

We choose a real representation for Majorana spinors

$(\lambda^{a})^{\star}=\lambda^{a},\quad\bar{\lambda}_{a}=\lambda^{b}C_{ba}$
(A.1)

The spinor indices can be raised and lowered by anti-symmetric charge
conjugation matrices $C_{ab}$ and $C^{ab}$ as

$\lambda^{a}=C^{ab}\lambda_{b},\quad\lambda_{a}=\lambda^{b}C_{ba},\quad
C^{ab}C_{bc}=-\delta^{a}_{c}.$ (A.2)

The $\gamma$–matrices satisfy the following anti-commutation relations

$(\gamma^{\mu})^{a}{}_{c}(\gamma^{\nu})^{c}{}_{b}+(\gamma^{\nu})^{a}{}_{c}(\gamma^{\mu})^{c}{}_{b}=2\eta^{\mu\nu}\delta^{a}_{b}.$
(A.3)

In ${\cal D}=4$ the matrices $\gamma_{\mu}$ and $\gamma_{\mu\nu}$ with both
spinorial indices up (down) are symmetric and the matrices $C$, $\gamma_{5}$
and $\gamma_{5}\gamma_{\mu}$ are anisymmetric. In ${\cal D}=10$ the matrices
$\gamma_{\mu}$ and $\gamma_{\mu_{1},...,\mu_{5}}$ with both spinorial indices
up (down) are symmetric, and the matrices $\gamma_{\mu_{1}\mu_{2}\mu_{3}}$ are
antisymmetric.

For checking the on-shell closure of the supersymmetry algebra and of the
supersymmetry of the vertices we have used the following gamma-matrix
identities

$(\gamma^{\nu})_{ab}{(\gamma_{\nu})}_{cd}+(\gamma^{\nu})_{ac}{(\gamma_{\nu})}_{db}+(\gamma^{\nu})_{ad}{(\gamma_{\nu})}_{bc}=0,$
(A.4)
$\gamma^{\mu}\gamma^{\nu_{1},\nu_{2},...\nu_{r}}\gamma_{\mu}=(-1)^{r}(D-2r)\gamma^{\nu_{1},\nu_{2},...\nu_{r}}.$
(A.5)

For a product of gamma matrices we have

$\gamma^{\nu_{1},...,\nu_{i}}\gamma_{\mu_{1},...,\mu_{j}}=\sum_{k=0}^{k=min(i,j)}\frac{i!j!}{(i-k)!(j-k)!k!}\gamma^{[\nu_{1},...,\nu_{i-k}}{}_{[\mu_{k+1},...,\mu_{j}}\delta^{\nu_{i}}_{\mu_{1}}\delta^{\nu_{i-1}}_{\mu_{2}}...\delta^{\nu_{n-k+1}]}_{\mu_{k}]}$
(A.6)

and in particular

$\gamma_{\mu}\gamma_{\nu_{1},\ldots\nu_{k}}=\gamma_{\mu,\nu_{1},\ldots,\nu_{k}}+k\eta_{\mu[\nu_{1}}\gamma_{\nu_{2},\ldots,\nu_{k}]}$
(A.7)

##  Appendix B $N=1$ Supergravities in ${\cal D}=4,6,10$

The Lagrangian for ${\cal D}=10$ $N=1$ Supergravity is [1] – [2]

$\displaystyle L$ $\displaystyle=$
$\displaystyle-\frac{1}{2}R-\frac{1}{2}\bar{\psi}_{\hat{\mu}}\gamma^{\hat{\mu}\hat{\nu}\hat{\rho}}{\cal
D}_{\hat{\nu}}{\psi}_{\hat{\rho}}-\frac{3}{4}\phi^{-\frac{3}{2}}H^{\hat{\mu}\hat{\nu}\hat{\rho}}H_{\hat{\mu}\hat{\nu}\hat{\rho}}-$
$\displaystyle-$ $\displaystyle\frac{1}{2}\bar{\Xi}\gamma^{\hat{\mu}}{\cal
D}_{\hat{\mu}}\Xi-\frac{9}{16}\frac{\partial^{\hat{\mu}}\phi\,\partial_{\hat{\mu}}\phi}{\phi^{2}}-\frac{3\sqrt{2}}{8}\bar{\psi}_{\hat{\mu}}\gamma^{\hat{\nu}}\gamma^{\hat{\mu}}\Xi\,\frac{\partial_{\hat{\nu}}\phi}{\phi}+$
$\displaystyle+$
$\displaystyle\frac{\sqrt{2}}{16}\phi^{-\frac{3}{4}}H_{\hat{\nu}\hat{\rho}\hat{\tau}}(\bar{\psi}_{\hat{\mu}}\gamma^{\hat{\mu}\hat{\nu}\hat{\rho}\hat{\tau}\hat{\lambda}}{\psi}_{\hat{\lambda}}+6\bar{\psi}^{\hat{\nu}}\gamma^{\hat{\rho}}{\psi}^{\hat{\tau}}-\sqrt{2}\bar{\psi}_{\hat{\mu}}\gamma^{\hat{\nu}\hat{\rho}\hat{\tau}}\gamma^{\hat{\mu}}\Xi)+$
$\displaystyle+$ $\displaystyle(\textrm{fermion})^{4}$

where
$H_{\hat{\mu}\hat{\nu}\hat{\rho}}=\partial_{[\hat{\mu}}B_{\hat{\nu}\hat{\rho}]}$
and ${\cal D}_{\hat{\mu}}$ is a covariant derivative

${\cal
D}_{{\hat{\mu}}}\Psi_{{\hat{\nu}}}^{a}=\partial_{{\hat{\mu}}}\Psi_{{\hat{\nu}}}^{a}+\frac{1}{4}\omega_{{\hat{\mu}}}{}^{\rho\sigma}(\gamma_{\rho\sigma}\Psi_{{\hat{\nu}}})^{a}$
(B.1)

The supersymmetry transformations with a local parameter $\epsilon^{a}(x)$ are

$\displaystyle\delta
e_{\hat{\mu}}^{\mu}=\frac{1}{2}{\bar{\epsilon}}\,\gamma^{\mu}\,\psi_{\hat{\mu}}$
(B.2) $\displaystyle\delta\phi=-\frac{\sqrt{2}}{3}\phi\,{\bar{\epsilon}}\,\Xi$
$\displaystyle\delta
B_{\hat{\mu}\hat{\nu}}=\frac{\sqrt{2}}{4}\phi^{\frac{3}{4}}\,({\bar{\epsilon}}\,\gamma_{\hat{\mu}}\,\psi_{\hat{\nu}}-{\bar{\epsilon}}\,\gamma_{\hat{\nu}}\,\psi_{\hat{\mu}}-\frac{\sqrt{2}}{2}\,{\bar{\epsilon}}\,\gamma_{\hat{\mu}\hat{\nu}}\Xi)$
$\displaystyle\delta\psi_{\hat{\mu}}=\left({\cal
D}_{\hat{\mu}}+\frac{\sqrt{2}}{32}\phi^{-\frac{3}{4}}(\gamma_{\hat{\mu}}{}^{\hat{\nu}\hat{\rho}\hat{\sigma}}-9\,\delta_{\hat{\mu}}^{\hat{\nu}}\,\gamma^{\hat{\rho}\hat{\sigma}})\,H_{\hat{\nu}\hat{\rho}\hat{\sigma}}\right)\epsilon+(\textrm{fermion})^{2}$
$\displaystyle\delta\Xi=-\frac{3\sqrt{2}}{8}\phi^{-1}\,\gamma^{\hat{\mu}}\,\partial_{\hat{\mu}}\phi\,\epsilon+\frac{1}{8}\phi^{-\frac{3}{4}}\,\gamma^{\hat{\mu}\hat{\nu}\hat{\rho}}\,\epsilon\,H_{\hat{\mu}\hat{\nu}\hat{\rho}}+(\textrm{fermion})^{2}$
(B.3)

The Lagrangian for ${\cal D}=4$ $N=1$ Supergravity coupled with one chiral
supermultiplet, with no superpotential and the cannonical kinetic term for the
scalars is [3]

$\displaystyle L$ $\displaystyle=$
$\displaystyle-\frac{1}{2}R-\frac{1}{2}\bar{\psi}_{\hat{\mu}}\gamma^{\hat{\mu}\hat{\nu}\hat{\rho}}\left({\cal
D}_{\hat{\nu}}+\frac{1}{8}((\partial_{\hat{\nu}}z)z^{\star}-(\partial_{\hat{\nu}}z^{\star})z)\gamma_{5}\right){\psi}_{\hat{\rho}}-$
$\displaystyle-$
$\displaystyle\frac{1}{2}(\partial_{\mu}z)(\partial_{\mu}z^{\star})-\frac{1}{2}\bar{\Xi}\left(\gamma^{\hat{\mu}}{\cal
D}_{\hat{\mu}}-\frac{1}{8}((\partial_{\hat{\nu}}z)z^{\star}-(\partial_{\hat{\nu}}z^{\star})z)\right)\Xi$
$\displaystyle+$ $\displaystyle(\textrm{fermion})^{4}$

which is invariant under supersymmetry transformations

$\displaystyle\delta
e_{\hat{\mu}}^{\mu}=\frac{1}{2}{\bar{\epsilon}}\,\gamma^{\mu}\,\psi_{\hat{\mu}}$
(B.4) $\displaystyle\delta z=\frac{1}{2}\,{\bar{\epsilon}}\,\Xi$
$\displaystyle\delta\psi_{\hat{\mu}}=\left({\cal
D}_{\hat{\mu}}+\frac{1}{8}((\partial_{\hat{\nu}}z)z^{\star}-(\partial_{\hat{\nu}}z^{\star})z)\right)\epsilon+(\textrm{fermion})^{2}$
$\displaystyle\delta\Xi=\frac{1}{2}(1+\gamma_{5})\,\gamma^{\hat{\mu}}(\partial_{\hat{\mu}}z)\epsilon+(\textrm{fermion})^{2}$

with $\gamma_{5}=-\gamma_{0}\gamma_{1}\gamma_{2}\gamma_{3}$.

The Lagrangian for ${\cal D}=6$ $N=1$ Supergravity coupled to one $(1,0)$
tensor multiplet is

$\displaystyle L$ $\displaystyle=$
$\displaystyle-\frac{1}{2}R-\frac{i}{2}\bar{\psi}_{\hat{\mu}}\gamma^{\hat{\mu}\hat{\nu}\hat{\rho}}{\cal
D}_{\hat{\nu}}{\psi}_{\hat{\rho}}+\frac{1}{12}e^{2\sqrt{2}\phi}\,H^{\hat{\mu}\hat{\nu}\hat{\rho}}H_{\hat{\mu}\hat{\nu}\hat{\rho}}+$
$\displaystyle+$ $\displaystyle\frac{i}{2}\bar{\Xi}\gamma^{\hat{\mu}}{\cal
D}_{\hat{\mu}}\Xi+\frac{1}{2}{\partial^{\hat{\mu}}\phi\,\partial_{\hat{\mu}}\phi}-\frac{1}{\sqrt{2}}\bar{\Xi}\gamma^{\hat{\nu}}\gamma^{\hat{\mu}}\psi_{\hat{\nu}}\,\partial_{\hat{\nu}}\phi+$
$\displaystyle-$
$\displaystyle\frac{i}{24}e^{\sqrt{2}\phi}H_{\hat{\mu}\hat{\nu}\hat{\rho}}(-\bar{\psi}^{\hat{\lambda}}\gamma_{[\hat{\lambda}}\gamma^{\hat{\mu}\hat{\nu}\hat{\rho}}\gamma_{\hat{\tau}]}{\psi}^{\hat{\tau}}+2i\bar{\psi}_{\hat{\lambda}}\gamma^{\hat{\mu}\hat{\nu}\hat{\rho}}\gamma^{\hat{\lambda}}\Xi-\bar{\Xi}\gamma^{\hat{\mu}\hat{\nu}\hat{\rho}}\Xi)+$
$\displaystyle+$ $\displaystyle(\textrm{fermion})^{4}$

where
$H_{\hat{\mu}\hat{\nu}\hat{\rho}}=\partial_{[\hat{\mu}}B_{\hat{\nu}\hat{\rho}]}$
and ${\cal D}_{\hat{\mu}}$ is a covariant derivative

${\cal
D}_{{\hat{\mu}}}\Psi_{{\hat{\nu}}}^{a}=\partial_{{\hat{\mu}}}\Psi_{{\hat{\nu}}}^{a}+\frac{1}{4}\omega_{{\hat{\mu}}}{}^{\rho\sigma}(\gamma_{\rho\sigma}\Psi_{{\hat{\nu}}})^{a}$
(B.5)

The supersymmetry transformations with a local parameter $\epsilon^{a}(x)$ are

$\displaystyle\delta
e_{\hat{\mu}}^{\mu}=-i{\bar{\epsilon}}\,\gamma^{\mu}\,\psi_{\hat{\mu}}$ (B.6)
$\displaystyle\delta\phi=\frac{1}{\sqrt{2}}\,{\bar{\epsilon}}\,\Xi$
$\displaystyle\delta
B_{\hat{\mu}\hat{\nu}}=-\frac{i}{2}e^{-\sqrt{2}\phi}\,({\bar{\epsilon}}\,\gamma_{\hat{\mu}}\,\psi_{\hat{\nu}}-{\bar{\epsilon}}\,\gamma_{\hat{\nu}}\,\psi_{\hat{\mu}}-i\,{\bar{\epsilon}}\,\gamma_{\hat{\mu}\hat{\nu}}\Xi)$
$\displaystyle\delta\psi_{\hat{\mu}}=\left({\cal
D}_{\hat{\mu}}-\frac{1}{24}e^{\sqrt{2}\phi}\gamma^{\hat{\rho}\hat{\sigma}\hat{\tau}}\gamma_{\hat{\mu}}\,H_{\hat{\rho}\hat{\sigma}\hat{\tau}}\right)\epsilon+(\textrm{fermion})^{2}$
$\displaystyle\delta\Xi=-\frac{i}{\sqrt{2}}\gamma^{\hat{\mu}}\epsilon\,\partial_{\hat{\mu}}\,\phi-\frac{i}{12}e^{\sqrt{2}\phi}\gamma^{\hat{\mu}\hat{\nu}\hat{\rho}}\epsilon
H_{\hat{\mu}\hat{\nu}\hat{\rho}}+(\textrm{fermion})^{2}$ (B.7)

We linearize around a flat background

$e_{{\hat{\mu}}}^{\mu}(x)=\delta_{\hat{\mu}}^{\mu}+\frac{1}{2}h_{{\hat{\mu}}}^{\mu}(x),\quad
e^{{\hat{\mu}}}_{\mu}(x)=\delta^{\hat{\mu}}_{\mu}-\frac{1}{2}h^{{\hat{\mu}}}_{\mu}(x),$
(B.8)

and consider a cubic Lagrangian with global supersymmetry.

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