§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§**University
of Maryland * Center for String and Particle Theory* Physics
Department***University of Maryland *Center for String and Particle Theory**
**University of Maryland * Center for String and Particle Theory* Physics
Department***University of Maryland *Center for String and Particle Theory**

July 2009 UMDEPP 09-043

hep-th/0907.5264

Ectoplasm & Superspace Integration Measure for
$2$D Supergravity with Four Spinorial Supercurrents111Supported in part by
National Science Foundation Grant PHY-0354401.

S. James Gates<EMAIL_ADDRESS>and Gabriele Tartaglino-
<EMAIL_ADDRESS>
Center for String and Particle Theory
Department of Physics, University of Maryland
College Park, MD 20742-4111 USA

ABSTRACT

> Building on a previous derivation of the local chiral projector for a two
> dimensional superspace with eight real supercharges, we provide the complete
> density projection formula required for locally supersymmetrical theories in
> this context. The derivation of this result is shown to be very efficient
> using techniques based on the Ectoplasmic construction of local measures in
> superspace.
>
>
> PACS: 04.65.+e, 11.30.Pb

##  1 Introduction

Some years ago, a formulation of a 2D supergravity theory which included off-
shell closure of the local supersymmetry algebra with four real spinorial
supercharges and a necessary set of auxiliary fields was introduced into the
literature [1]. In a subsequent development there was made a proposal (called
‘Ectoplasm’ [2]) for a conceptual framework leading to efficient derivations
of local superspace integration measures (density projection operators)444 A
mathematical construction giving the formal bases for the Ectoplasm methods
can be found
in the theory of integration over surfaces in supermanifolds developed in [3,
4, 5].. In addition, about the same time there was put forward an alternative
general framework for the derivation of density projection operators based on
the use of superspace normal coordinate expansions first introduced in [6] and
rediscovered555Previous approaches for component reduction, ultimately related
to normal coordinates
expansions, can be found in [8, 9, 10, 11, 12, 13, 14]. in [7]; see [15] for
recent reformulations and improvements of the normal coordinates techniques.
The ectoplasm and normal coordinates frameworks have been found to be closely
related [16, 17].

Prior to the introduction of the ectoplasmic and normal coordinate approaches,
the question of how to construct local superspace supergravity densities had
been approached by two other and more cumbersome methods. Both of these can be
seen in two books on the subject. In the first, Superspace [18], an approach
was taken to reproduce, at the level of superfields, a Noether approach thus
leading to the density projector. In the second Ideas [19], an approach that
was taken to utilize the prepotential formulation of supergravity theory to
derive the density projector.

It has been argued from its inception that the ectoplasmic concept is not only
extremely efficient but also likely applies to even more complicated theories
such as string theory. Though there was no such evidence at the time of the
introduction of the ectoplasm approach, later it was shown that integration
measures in the ‘pure spinor formulation’ of superstrings follow precisely
from the extension of the ectoplasmic concept to this realm of theories [20].

The off-shell formulation of a 2D, $\cal N$ = 4 supergravity theory implies
the existence of a straightforward way to completely develop an efficient
local integration theory for the associated local Salam-Strathdee superspace.
We will complete such a construction in the current work by use of the
ectoplasmic suggestion.

This paper is organized as follows. In section 2 we review the 2D, $\cal N$=4
supergravity formulation of [1]. Section 3 is devoted to the presentation of a
new super 2-form multiplet. In section 4 we make use of the ectoplasmic
approach to build the density projector for the 2D, $\cal N$=4 supergravity of
[1]; this is the main result of the paper. Section 5 collects some
conclusions. The paper includes two appendices. Appendix A contains the
derivation of the result of section 3. Then, the appendix B is a collection of
formulas used in the paper.

##  2 An Off-Shell 2D Supergravity Geometry With Eight Real Local
Supersymmetries

In this section we review some aspects of the off-shell 2D, ${\cal N}=4$
minimal supergravity multiplet first introduced in [1]. We focus on the curved
superspace geometry underlining the minimal supergravity that will be used in
the computations of this paper.

The work in [1] showed there exists component fields
$(e_{a}{}^{m},~{}\psi_{a}{}^{\alpha i},~{}A_{ai}{}^{j},~{}B,~{}G,~{}H)$ which
describe an off-shell 2D supergravity theory possessing eight real local (or
four real spinorial) supercharges. The previous list of component fields
contains the graviton, the gravitini, SU(2) connection, a complex scalar $B$,
one real scalar $G$ and one real pseudoscalar $H$. These are the components
associated with the following constraints on the 2D, ${\cal N}$ = 4 superspace
supergravity covariant derivative algebra666In the present paper we adopt the
Lorentz and SU(2) notations collected in Appendix
A of [21] and consistent with the conventions of [18].

$\displaystyle\\{\,\nabla{}_{\alpha i}~{},~{}\nabla{}_{\beta j}\,\\}$
$\displaystyle=$ $\displaystyle 2\overline{B}[\,C{}_{\alpha\beta}C{}_{ij}{\cal
M}~{}-~{}(\gamma{}^{3}){}_{\alpha\beta}{\cal
Y}{}_{ij}\,]\,~{}~{},{~{}~{}~{}~{}~{}~{}~{}~{}~{}}{~{}~{}~{}~{}~{}~{}~{}~{}~{}}{~{}~{}~{}~{}~{}~{}~{}~{}~{}}{~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}}$
(1)
$\displaystyle\\{\,{\overline{\nabla}}_{\alpha}{}^{i}~{},~{}{\overline{\nabla}}_{\beta}{}^{j}\,\,\\}$
$\displaystyle=$ $\displaystyle 2B{[}\,C_{\alpha\beta}C^{ij}{\cal
M}~{}-~{}(\gamma{}^{3})_{\alpha\beta}{\cal
Y}^{ij}\,{]}\,~{}~{},{~{}~{}~{}~{}~{}~{}~{}~{}~{}}{~{}~{}~{}~{}~{}~{}~{}~{}~{}}{~{}~{}~{}~{}~{}~{}~{}~{}~{}}{~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}}$
(2) $\displaystyle\\{\,\nabla{}_{\alpha
i}~{},~{}\overline{\nabla}{}_{\beta}{}^{j}\,\\}$ $\displaystyle=$
$\displaystyle 2{\rm
i}\,\delta{}_{i}{}^{j}(\gamma{}^{c}){}_{\alpha\beta}\nabla{}_{c}~{}+~{}2\delta{}_{i}{}^{j}\phi{}_{\alpha}{}^{\gamma}(\gamma^{3})_{\gamma\beta}{\cal
M}~{}-~{}2\phi{}_{\alpha\beta}{\cal
Y}{}_{i}{}^{j}~{}~{},{~{}~{}~{}~{}~{}~{}~{}~{}}{~{}~{}~{}~{}~{}~{}~{}~{}}$ (3)
$\displaystyle{[}\,\nabla{}_{\alpha i}~{},~{}\nabla{}_{b}\,{]}$
$\displaystyle=$ $\displaystyle\hbox{\large{${{\textstyle{{{\rm
i}}\over\vphantom{2}\smash{\raise
0.72331pt\hbox{$\scriptstyle{{2}}$}}}}}$}}\phi{}_{\alpha}{}^{\gamma}(\gamma{}_{b}){}_{\gamma}{}^{\beta}\nabla{}_{\beta
i}~{}+~{}\hbox{\large{${{\textstyle{{{\rm i}}\over\vphantom{2}\smash{\raise
0.72331pt\hbox{$\scriptstyle{{2}}$}}}}}$}}(\gamma^{3}\gamma{}_{b}){}_{\alpha}{}^{\beta}\overline{B}C{}_{ij}\overline{\nabla}{}_{\beta}{}^{j}$
(4) $\displaystyle~{}-~{}{\rm
i}(\gamma^{3}\gamma_{b})_{\alpha\beta}\bar{\Sigma}^{\beta}{}_{i}{\cal
M}~{}+~{}{\rm i}(\gamma{}_{b})_{\alpha\beta}\bar{\Sigma}^{\beta}{}_{j}{\cal
Y}_{i}{}^{j}~{}~{},{~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}}{~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}}{~{}~{}~{}~{}~{}~{}~{}~{}}$
$\displaystyle~{}{[}\,{\overline{\nabla}}_{\alpha}{}^{i}~{},~{}\nabla{}_{b}\,{]}$
$\displaystyle=$ $\displaystyle-~{}\hbox{\large{${{\textstyle{{{\rm
i}}\over\vphantom{2}\smash{\raise
0.72331pt\hbox{$\scriptstyle{{2}}$}}}}}$}}\bar{\phi}_{\alpha}{}^{\gamma}(\gamma_{b})_{\gamma}{}^{\beta}{\overline{\nabla}}_{\beta}{}^{i}~{}+~{}\hbox{\large{${{\textstyle{{{\rm
i}}\over\vphantom{2}\smash{\raise
0.72331pt\hbox{$\scriptstyle{{2}}$}}}}}$}}(\gamma^{3}\gamma_{b})_{\alpha}{}^{\beta}BC^{ij}\nabla_{\beta
j}$ (5) $\displaystyle~{}~{}-~{}{\rm
i}(\gamma^{3}\gamma_{b})_{\alpha\beta}\Sigma^{\beta i}{\cal M}~{}-~{}{\rm
i}(\gamma{}_{b})_{\alpha\beta}\Sigma^{\beta j}{\cal
Y}^{i}{}_{j}~{}~{},{~{}~{}~{}~{}~{}~{}~{}~{}}{~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}}{~{}~{}~{}~{}~{}~{}~{}~{}}$
$\displaystyle~{}{[}\,\nabla{}_{a}~{},~{}\nabla{}_{b}\,{]}$ $\displaystyle=$
$\displaystyle-~{}\hbox{\large{${{\textstyle{{1}\over\vphantom{2}\smash{\raise
0.72331pt\hbox{$\scriptstyle{{2}}$}}}}}$}}\varepsilon_{ab}[(\gamma{}^{3}){}_{\alpha}{}^{\beta}\Sigma{}^{\alpha
i}\nabla{}_{\beta
i}~{}+~{}(\gamma{}^{3}){}_{\alpha}{}^{\beta}\overline{\Sigma}{}^{\alpha}{}_{i}\overline{\nabla}{}_{\beta}{}^{i}~{}+~{}{\cal
R}{\cal M}~{}+~{}{\rm i}{\cal F}{}_{i}{}^{j}{\cal
Y}{}_{j}{}^{i}]~{}.~{}~{}~{}~{}$ (6)

where

$\displaystyle(B)^{*}$ $\displaystyle=$
$\displaystyle\bar{B}~{},~{}~{}~{}(G)^{*}~{}=\,G~{},~{}~{}~{}(H)^{*}~{}=\,H~{},~{}~{}~{}(\Sigma_{\alpha}{}^{i})^{*}~{}=\,\bar{\Sigma}_{\alpha
i}~{},$ (7) $\displaystyle\phi_{\alpha\,\beta}$ $\displaystyle=$
$\displaystyle C_{\alpha\beta}G+{\rm i}(\gamma^{3})_{\alpha\beta}H~{},$ (8)
$\displaystyle\bar{\phi}_{\alpha\,\beta}$ $\displaystyle=$
$\displaystyle(\phi_{\alpha\beta})^{*}=-C_{\alpha\beta}G+{\rm
i}(\gamma^{3})_{\alpha\beta}H=\phi_{\beta\alpha}~{}.$ (9)

In writing these, we have corrected some coefficients that appear in the work
of [21] in the terms that appear in (3) - (5). These corrected coefficients do
not affect (1) and (2). Thus the result in the work of [21] is unaffected by
this change.

In the previous algebra, the covariant derivatives are
$\nabla_{A}=(\nabla_{a},\nabla_{\alpha i},{\overline{\nabla}}_{\alpha}{}^{i})$

$\displaystyle\nabla_{A}~{}=~{}E_{A}{}^{M}\partial_{M}+\omega_{A}{\cal M}+{\rm
i}\,\Gamma_{A}{}_{k}{}^{l}{\cal Y}_{l}{}^{k}~{}.$ (10)

The 2D, ${\cal N}=4$ curved superspace is locally parametrized by the
coordinates $z^{M}=(x^{m},\theta^{\mu i},\bar{\theta}^{\mu}{}_{i})$ with the
Grassmann variables $\theta^{\mu i}$ and $\bar{\theta}^{\mu}{}_{i}$ related by
complex conjugation $\bar{\theta}^{\mu}{}_{i}=(\theta^{\mu i})^{*}$; the
bosonic coordinates will be also denoted as $x^{m}=(\tau,\sigma)$. In (10),
$E_{A}{}^{M}$ is the inverse of the vielbein $E_{M}{}^{A}$
($E_{M}{}^{A}E_{A}{}^{N}=\delta_{M}^{N}$,
$E_{A}{}^{M}E_{M}{}^{B}=\delta_{A}^{B}$) with $\partial_{M}=\partial/\partial
z^{M}$, $\omega_{A}$ the 2D Lorentz connection and $\Gamma_{A}{}_{k}{}^{l}$ is
the SU(2) connection. The torsion $T_{AB}{}^{C}$, Lorentz curvature $R_{AB}$
and SU(2) curvature $R_{AB}{}_{k}{}^{l}$ superfields are defined by (1) - (6)
and

$\displaystyle[\nabla_{A},\nabla_{B}\\}~{}=~{}T_{AB}{}^{C}\nabla_{C}+R_{AB}{\cal
M}+{\rm i}\,R_{AB}{}_{k}{}^{l}{\cal Y}_{l}{}^{k}~{}.$ (11)

The action of the local 2D Lorentz generator ${\cal M}$ and of the local SU(2)
generator ${\cal Y}_{k}{}^{l}$ on the spinor covariant derivatives are the
following (${\cal Y}_{kl}={\cal Y}_{k}{}^{p}C_{pl}$)

$\displaystyle{[}{\cal M},\nabla_{\alpha i}{]}$ $\displaystyle=$
$\displaystyle{\textstyle{1\over\vphantom{2}\smash{\raise
0.60275pt\hbox{$\scriptstyle{2}$}}}}(\gamma^{3})_{\alpha}{}^{\beta}\nabla_{\beta
i}~{},~{}~{}~{}{[}{\cal
M},{\overline{\nabla}}_{\alpha}{}^{i}{]}={\textstyle{1\over\vphantom{2}\smash{\raise
0.60275pt\hbox{$\scriptstyle{2}$}}}}(\gamma^{3})_{\alpha}{}^{\beta}{\overline{\nabla}}_{\beta}{}^{i}~{},~{}~{}~{}$
(12) $\displaystyle{[}{\cal Y}_{kl},\nabla_{\alpha i}{]}$ $\displaystyle=$
$\displaystyle{\textstyle{1\over\vphantom{2}\smash{\raise
0.60275pt\hbox{$\scriptstyle{2}$}}}}C_{i(k}\nabla_{\beta
l)}~{},~{}~{}~{}{[}{\cal
Y}_{kl},{\overline{\nabla}}_{\alpha}{}^{i}{]}=-{\textstyle{1\over\vphantom{2}\smash{\raise
0.60275pt\hbox{$\scriptstyle{2}$}}}}\delta^{i}_{(k}{\overline{\nabla}}_{\beta
l)}~{}.$ (13)

It is worthy to recall that the consistency of the Bianchi identities
constructed from the commutator algebra above requires the conditions [1]

$\displaystyle\overline{\nabla}{}_{\alpha}{}^{i}B$ $\displaystyle=$
$\displaystyle 0~{}~{}~{}~{}~{}~{},~{}~{}~{}\nabla{}_{\alpha
i}B~{}=\,-2C{}_{ij}(\gamma{}^{3}){}_{\alpha\beta}\Sigma{}^{\beta j}~{}~{},$
(14) $\displaystyle\nabla{}_{\alpha i}G$ $\displaystyle=$
$\displaystyle\overline{\Sigma}{}_{\alpha
i}~{}~{}~{},~{}~{}~{}\nabla{}_{\alpha i}H~{}=\,{\rm
i}(\gamma{}^{3}){}_{\alpha}{}^{\beta}\overline{\Sigma}{}_{\beta i},~{}~{}~{},$
(15) $\displaystyle\overline{\nabla}{}_{\alpha}{}^{i}\Sigma{}^{\beta j}$
$\displaystyle=$ $\displaystyle{\rm
i}C{}^{ij}(\gamma{}^{3}\gamma{}^{a}){}_{\alpha}{}^{\beta}\nabla{}_{a}B~{}~{},$
(16) $\displaystyle\nabla{}_{\alpha i}\Sigma{}^{\beta j}$ $\displaystyle=$
$\displaystyle{1\over 2}\delta{}_{\alpha}{}^{\beta}\delta{}_{i}{}^{j}[{\cal
R}~{}-~{}2G{}^{2}~{}-~{}2H{}^{2}~{}-~{}2B\overline{B}]~{}+~{}{\rm
i}(\gamma{}^{3}){}_{\alpha}{}^{\beta}{\cal F}{}_{i}{}^{j}$ (18)
$\displaystyle+~{}{\rm
i}\delta{}_{i}{}^{j}(\gamma{}^{a}){}_{\alpha}{}^{\beta}(\nabla{}_{a}G)-~{}\delta{}_{i}{}^{j}(\gamma{}^{3}\gamma{}^{a}){}_{\alpha}{}^{\beta}(\nabla{}_{a}H)~{}~{}~{}.$

The component gauge fields occur in the above supertensors in the following
manner777Given a superfield $\Psi(\tau,\sigma,\theta,\bar{\theta})$, we denote
as usual with $\Psi|:=\Psi|_{\theta=0}$ the field obtained by
setting to zero all the Grassmanian coordinates.

$\begin{array}[]{lll}{\cal R}{\big{|}}&=&\varepsilon^{ab}{\large\\{}~{}{{\cal
R}}_{ab}(\hat{\omega})~{}+~{}[~{}2{\rm
i}(\gamma^{3}\gamma_{a})_{\alpha\beta}\psi_{b}{}^{\alpha
i}\overline{\Sigma}^{\beta}{}_{i}~{}+~{}{\rm{h.c.}}~{}]\\\
&&~{}~{}~{}~{}~{}~{}+~{}4\phi_{\alpha}{}^{\gamma}(\gamma^{3})_{\gamma\beta}\psi_{a}{}^{\alpha
i}{\overline{\psi}}_{b}{}^{\beta}{}_{i}~{}-~{}2[~{}C_{ij}\overline{B}\psi_{a}{}^{\alpha
i}\psi_{b\alpha}{}^{j}~{}+~{}{\rm{h.c.}}~{}]~{}{\large\\}}~{}~{},\\\ &&\\\
\Sigma^{\alpha i}{\big{|}}&=&\varepsilon^{ab}{\large\\{}~{}\psi_{ab}{}^{\beta
i}(\gamma^{3})_{\beta}{}^{\alpha}~{}-~{}{\rm i}\psi_{a}{}^{\beta
i}{\phi}_{\beta}{}^{\gamma}(\gamma^{3}\gamma_{b})_{\gamma}{}^{\alpha}~{}+~{}{\rm
i}C^{ij}B\overline{\psi}_{a}{}^{\beta}{}_{j}(\gamma_{b})_{\beta}{}^{\alpha}~{}{\large\\}}~{}~{},\\\
&&\\\ {\cal F}_{i}{}^{j}{\big{|}}&=&\varepsilon^{ab}{\large\\{}~{}{\rm
F}_{ab}(A)_{i}{}^{j}~{}-~{}2{\rm
i}(\gamma_{a})_{\alpha\beta}[~{}\psi_{b}{}^{\alpha
j}\overline{\Sigma}^{\beta}{}_{i}~{}+~{}\overline{\psi}_{b}{}^{\alpha}{}_{i}\Sigma^{\beta
j}\\\
&&~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}-~{}{\textstyle{1\over\vphantom{2}\smash{\raise
0.60275pt\hbox{$\scriptstyle{2}$}}}}\delta_{i}^{j}(\psi_{b}{}^{\alpha
k}\overline{\Sigma}^{\beta}{}_{k}~{}+~{}\overline{\psi}_{b}{}^{\alpha}{}_{k}\Sigma^{\beta
k})~{}]\\\
&&~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}-~{}4\phi_{\alpha\beta}[\psi_{a}{}^{\alpha
j}\overline{\psi}_{b}{}^{\beta}{}_{i}~{}-~{}{\textstyle{1\over\vphantom{2}\smash{\raise
0.60275pt\hbox{$\scriptstyle{2}$}}}}\delta_{i}^{j}\psi_{a}{}^{\alpha
k}\overline{\psi}_{b}{}^{\beta}{}_{k}]\\\
&&~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}-~{}2(\gamma^{3})_{\alpha\beta}[~{}\overline{B}(C_{ik}\psi_{a}{}^{\alpha
k}\psi_{b}{}^{\beta}{}^{k}~{}-~{}{\textstyle{1\over\vphantom{2}\smash{\raise
0.60275pt\hbox{$\scriptstyle{2}$}}}}\delta_{i}^{j}C_{kl}\psi_{a}{}^{\alpha
k}\psi_{b}{}^{\beta l})\\\
&&~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+~{}B(C^{jk}\overline{\psi}_{a}{}^{\alpha}{}_{i}\overline{\psi}_{b}{}^{\beta}{}_{k}~{}-~{}{\textstyle{1\over\vphantom{2}\smash{\raise
0.60275pt\hbox{$\scriptstyle{2}$}}}}\delta_{i}^{j}C^{kl}\overline{\psi}_{a}{}^{\alpha}{}_{k}\overline{\psi}_{b}{}^{\beta}{}_{l})~{}]~{}{\large\\}}~{}~{},\end{array}$
(19)

where $\varepsilon^{ab}\,{{\cal R}}_{ab}(\hat{\omega})$ is the usual two-
dimensional curvature in terms of the inverse of the vielbein ${e}_{a}{}^{m}$
and of the Lorentz connection $\hat{\omega}_{a}$;
$\varepsilon^{ab}\psi_{ab}{}^{\beta i}$ is the gravitini field strength;
$\varepsilon^{ab}F_{ab}(A)$ is the SU(2) field strength function of
${e}_{a}{}^{m}$ and of the SU(2) connection $A_{a}{}_{k}{}^{l}$ [1]. The
component gauge fields $e_{a}{}^{m},\,\hat{\omega}_{a},\,A_{a}{}_{k}{}^{l}$
are easily related to the gauge superfields
$E_{A}{}^{M},\,\omega_{A},\,\Gamma_{A}{}_{k}{}^{l}$ in (10) by using standard
Wess-Zumino gauge reduction techniques [18, 19].

##  3 Defining A Closed 2D, ${\cal N}=4$ Super Two-Form

In this section we are going to present a new closed 2D, ${\cal N}=4$ super
two-form defined in terms of an unconstrained scalar chiral superfield. The
result contained in the Theorem 1 is crucial to build the measure of the local
superspace integration theory for 2D, ${\cal N}=4$ supergravity theories as we
will see in section 4.

The work in [21] established that the fourth-order spinorial derivatives
operator ${\cal{\overline{D}}}{}^{(4)}$, defined by

$\displaystyle{{\cal{\overline{D}}}{}^{(4)}~{}=~{}\left[\,{\overline{\nabla}}{}^{(2)\,\alpha\,\beta}~{}-~{}2\,B\,(\gamma^{3}){}^{\alpha\,\beta}\,\right]\,{\overline{\nabla}}{}^{(2)}_{\,\alpha\,\beta}~{}~{},}$
---
(20)

is the chiral projection operator satisfying

$\displaystyle{{\overline{\nabla}}{}_{\gamma}^{i}\,{\cal{\overline{D}}}{}^{(4)}\,\Psi~{}=~{}{\overline{\nabla}}{}_{\gamma}^{i}\,\left[\,{\overline{\nabla}}{}^{(2)\,\alpha\,\beta}~{}-~{}2\,B\,(\gamma^{3}){}^{\alpha\,\beta}\,\right]\,{\overline{\nabla}}{}^{(2)}_{\,\alpha\,\beta}\,\Psi~{}=~{}0}$
---
(21)

for any general scalar superfield $\Psi$. We note the derivation of (20) and
(21) given in [21] follows solely from algebraic manipulations of the
derivatives that appear in (2).

In a later section we will exploit the fact that a closed 2D, $\cal N$ = 4
super two-form is sufficient to determine the local integration measure for an
appropriate curved superspace. For this purpose it is necessary to define the
components of a 2D, $\cal N$ = 4 super two-form. The general framework for the
construction of such forms was presented some time ago [22] which implies for
the present consideration we should introduce a super 2-form whose component
superfields may be written in the form $J_{A}{}_{B}~{}=~{}\big{(}\,J_{\alpha
i}{}_{\beta j},~{}J_{\alpha
i}{}_{\beta}{}^{j},~{}J_{\alpha}{}^{i}{}_{\beta}{}^{j},~{}J_{\gamma
k}{}_{a},~{}J_{\gamma}{}^{k}{}_{a},~{}J_{ab}\,\big{)}$. We refer the reader to
[22, 18] for the notations we adopt in the use of super p-forms. In general,
given a super p-form $\Omega$, described by the component superfields
$\Omega_{A_{1}\cdots A_{p}}$, its exterior derivative $F=d\Omega$ has
components $F_{A_{1}\cdots A_{p+1}}$ given by 888With $[\cdots)$ we denote the
complete graded symmetrization of indeces.

$\displaystyle F_{A_{1}\cdots A_{p}A_{p+1}}={1\over
p!}\nabla_{[A_{1}}\Omega_{A_{2}\cdots A_{p+1})}-{1\over
2((p-1)!)}T_{[A_{1}A_{2}|}{}^{B}\Omega_{B|A_{3}\cdots A_{p+1})}~{}.$ (22)

The superform $\Omega$ is closed if $F_{A_{1}\cdots A_{p+1}}=0$. We can now
state a theorem.

Theorem 1
If $U$ is a chiral superfield, i.e. satisfies
$\,{\overline{\nabla}}{}_{\alpha}^{i}\,U\,=\,0\,,$ the components defined by

$\displaystyle{J_{\alpha i}{}_{\beta}{}^{j}}$ | $\displaystyle{{}=~{}0~{}~{}~{},}$
---|---
$\displaystyle{J_{\alpha i}{}_{\beta j}}$ | $\displaystyle{{}=~{}2(\gamma^{3})_{\alpha\beta}{\overline{\nabla}}^{(2)}_{ij}\overline{U}-\,C_{\alpha\beta}C_{ij}(\gamma^{3})^{\gamma\delta}{\overline{\nabla}}^{(2)}_{\gamma\delta}\overline{U}~{}~{}~{},}$
$\displaystyle{J_{\alpha}{}^{i}{}_{\beta}{}^{j}}$ | $\displaystyle{{}=~{}\,2\,(\gamma^{3})_{\alpha\beta}\nabla^{(2)\,ij}{U}~{}-~{}\,C_{\alpha\beta}C^{ij}(\gamma^{3})^{\gamma\delta}\nabla^{(2)}_{\gamma\delta}{U}~{}~{}~{},}$
$\displaystyle{J_{\gamma k}{}_{a}}$ | $\displaystyle{{}=~{}-~{}\hbox{\large{${{\textstyle{{{\rm i}}\over\vphantom{2}\smash{\raise 0.72331pt\hbox{$\scriptstyle{{3}}$}}}}}$}}\varepsilon_{ab}(\gamma^{b})_{\gamma}{}^{\delta}{\overline{\nabla}}_{\delta}{}^{p}{\overline{\nabla}}^{(2)}_{kp}\overline{U}~{}~{}~{},}$
$\displaystyle{J_{\gamma}{}^{k}{}_{a}}$ | $\displaystyle{{}=~{}-~{}\hbox{\large{${{\textstyle{{{\rm i}}\over\vphantom{2}\smash{\raise 0.72331pt\hbox{$\scriptstyle{{3}}$}}}}}$}}\varepsilon_{ab}(\gamma^{b})_{\gamma}{}^{\delta}\nabla_{\delta p}\nabla^{(2)\,kp}{U}~{}~{}~{},}$
$\displaystyle{J_{ab}}$ | $\displaystyle{{}=~{}-~{}\hbox{\large{${{\textstyle{{1}\over\vphantom{2}\smash{\raise 0.72331pt\hbox{$\scriptstyle{{8}}$}}}}}$}}\,\varepsilon_{ab}\,\Big{[}\,\Big{(}\nabla^{(4)}~{}-~{}2\,\bar{B}(\gamma^{3})^{\alpha\beta}\nabla^{(2)}_{\alpha\beta}\Big{)}{U}~{}+~{}\Big{(}{\overline{\nabla}}^{(4)}~{}-~{}2\,B(\gamma^{3})^{\alpha\beta}{\overline{\nabla}}^{(2)}_{\alpha\beta}\Big{)}\overline{U}\,\Big{]}~{},}$
(23)

describe a closed 2D, $\cal N$ = 4 super two-form with respect to the
supergravity
commutator algebra in Eq. (1) - Eq. (6). The superfield $\bar{U}:=(U)^{*}$ is
antichiral $\nabla_{\alpha i}\bar{U}=0$. In writing these results, we have
introduced second and fourth order spinorial derivative operators via the
equations

|
$\displaystyle{{}\nabla{}^{(2)}_{\alpha\,\beta}~{}=~{}\hbox{\large{${{\textstyle{{1}\over\vphantom{2}\smash{\raise
0.72331pt\hbox{$\scriptstyle{{2}}$}}}}}$}}\,C^{i\,j}\left[\,\nabla_{\alpha\,i}\,\nabla_{\beta\,j}~{}+~{}\nabla_{\beta\,i}\,\nabla_{\alpha\,j}\,\right]~{},~{}\nabla{}^{(2)}_{i\,j}~{}=~{}\hbox{\large{${{\textstyle{{1}\over\vphantom{2}\smash{\raise
0.72331pt\hbox{$\scriptstyle{{2}}$}}}}}$}}\,C^{\alpha\,\beta}\left[\,\nabla_{\alpha\,i}\,\nabla_{\beta\,j}~{}+~{}\nabla_{\alpha\,j}\,\nabla_{\beta\,i}\,\right]~{},}$
---|---
|
$\displaystyle{{}{\overline{\nabla}}{}^{(2)}_{\alpha\,\beta}~{}=~{}\hbox{\large{${{\textstyle{{1}\over\vphantom{2}\smash{\raise
0.72331pt\hbox{$\scriptstyle{{2}}$}}}}}$}}\,C_{i\,j}\left[\,{\overline{\nabla}}_{\alpha}{}^{i}\,{\overline{\nabla}}_{\beta}{}^{j}~{}+~{}{\overline{\nabla}}_{\beta}{}^{i}\,{\overline{\nabla}}_{\alpha}{}^{j}\,\right]~{},~{}{\overline{\nabla}}{}^{(2)}{}^{i\,j}~{}=~{}\hbox{\large{${{\textstyle{{1}\over\vphantom{2}\smash{\raise
0.72331pt\hbox{$\scriptstyle{{2}}$}}}}}$}}\,C^{\alpha\,\beta}\left[\,{\overline{\nabla}}_{\alpha}{}^{i}\,{\overline{\nabla}}_{\beta}{}^{j}~{}+~{}{\overline{\nabla}}_{\alpha}{}^{j}\,{\overline{\nabla}}_{\beta}{}^{i}\,\right]~{},}$
|
$\displaystyle{{}\nabla{}^{(4)}~{}=~{}\hbox{\large{${{\textstyle{{1}\over\vphantom{2}\smash{\raise
0.72331pt\hbox{$\scriptstyle{{3}}$}}}}}$}}\nabla{}^{(2)}{}^{k\,l}\,\nabla{}^{(2)}_{k\,l}~{}~{}~{},~{}~{}~{}~{}{\overline{\nabla}}{}^{(4)}~{}=~{}\hbox{\large{${{\textstyle{{1}\over\vphantom{2}\smash{\raise
0.72331pt\hbox{$\scriptstyle{{3}}$}}}}}$}}{\overline{\nabla}}{}^{(2)}{}^{k\,l}\,{\overline{\nabla}}{}^{(2)}_{k\,l}~{}~{}~{}.}$
(24)

The proof of the theorem involves using the above equations to show that the
Bianchi identities for this two-form vanish. This is relegated to an appendix.
We next note that the chiral superfield $U$ above may be replaced using the
result from (21) according to $U=\bar{\cal D}^{(4)}{\cal L}$ ($\bar{U}={\cal
D}^{(4)}\bar{\cal L}=(\bar{\cal D}^{(4)}{\cal L})^{*}$) where the 2D, $\cal N$
= 4 superfield $\cal L$, ($\bar{\cal L}:=({\cal L})^{*}$), is not subject to
any algebraic nor differential restrictions. Stated another way, this implies
that an arbitrary 2D, $\cal N$ = 4 superfield $\cal L$ can be used to create a
closed super 2-form whose components are defined by $J_{A\,B}$ above. We
conclude with a result that will be needed in the next section. Defining the
component vierbein $E_{m}{}^{a}|=e_{m}{}^{a}$
($e_{m}{}^{a}e_{a}{}^{n}=\delta_{m}^{n}$,
$e_{a}{}^{m}e_{m}{}^{b}=\delta_{a}^{b}$), and the gravitini $E_{m}{}^{\alpha
i}|=-\psi_{m}{}^{\alpha i}$ ($\psi_{a}{}^{\alpha
i}=e_{a}{}^{m}\psi_{m}{}^{\alpha i}$),
$E_{m}{}^{\alpha}_{i}|=-\bar{\psi}_{m}{}^{\alpha}_{i}$
($\bar{\psi}_{a}{}^{\alpha}_{i}=e_{a}{}^{m}\bar{\psi}_{m}{}^{\alpha}_{i}$), by
a general result given in [18, 2] taking the limit as all Grassmann
coordinates go to zero one obtains

$\begin{array}[]{lll}~{}~{}\varepsilon^{ab}{J}_{ab}{\Big{|}}&=&\varepsilon^{ab}{\cal
J}_{ab}{\Big{|}}~{}+~{}2\,\varepsilon^{ab}(\psi_{a}{}^{\alpha i}{J}_{\alpha
i~{}b}{\Big{|}}~{}+~{}{\bar{\psi}}_{a}{}^{\alpha}{}_{i}{J}_{\alpha}{}^{i}{}_{~{}b}{\Big{|}})~{}+~{}2\,\varepsilon^{ab}\psi_{a}{}^{\alpha
i}{\bar{\psi}}_{b}{}^{\beta}{}_{j}{J}_{\alpha}{}_{i}{}_{\beta}{}^{j}{\Big{|}}\\\
&&\\\ &}{&+~{}\varepsilon^{ab}\psi_{a}{}^{\alpha i}\psi_{b}{}^{\beta
j}{J}_{\alpha\,i\,\beta\,j}{\Big{|}}~{}+~{}\varepsilon^{ab}{\bar{\psi}}_{a}{}^{\alpha}{}_{i}{\bar{\psi}}_{b}{}^{\beta}{}_{j}{J}_{\alpha}{}^{i}{}_{\beta}{}^{j}{\Big{|}}~{}~{}~{},\end{array}$
(25)

where ${\cal J}_{ab}{\Big{|}}$ describe an ordinary space closed 2-form.

##  4 A 2D, ${\cal N}=4$ Density Projection Operator

It remains for us to calculate the explicit form of the density projection
operator (that we will denote by ${\Delta}^{(4)}$) which is the main purpose
of this work. As we are going to describe in this section, by using
${\Delta}^{(4)}$ and the chiral projector $\overline{{\cal D}}^{(4)}$, we can
build the integration measure of component actions in 2D, ${\cal N}=4$ minimal
supergravity.

As noted by Siegel [17], the ‘secret’ to the ectoplasmic approach is to
realize that the integration theory of superspace can be totally cast into the
language of closed super-forms. Indeed it was argued in the work of [2] that
the requirement that the topology of a superspace be totally determined by the
topology of its purely bosonic sub-manifold naturally provides a reason for
the appearance of super-forms in constructing integration measures of
superspace.

In the work of Ref. [18] it was noted that the derivation of component results
follows efficiently from replacing the integration of fermionic coordinates by
a process using first application of the superspace covariant derivative
followed by taking the limit as the Grassmann coordinates are taken to zero.
In the 2D, $\cal N$=4 case, this is described in the form of an equation

$\displaystyle{\int d^{2}\sigma\,d^{4}\theta\,d^{4}{\overline{\theta}\ }{\rm E}^{-1}~{}{\cal L}}$ | $\displaystyle{{}\to~{}\int d^{2}\sigma\ \,\hbox{\large{${{\textstyle{{1}\over\vphantom{2}\smash{\raise 0.72331pt\hbox{$\scriptstyle{{2}}$}}}}}$}}\,{\rm e}{}^{-1}\,{\Big{[}}\,{\Delta}{}^{(4)}\,{{\cal{\overline{D}}}{}^{(4)}}\,\,{\cal L}~{}+~{}{\rm h}.\,{\rm c}.\,{\Big{]}}\,{\Big{|}}}$
---|---
(26)

in terms of two differential operators, ${{\Delta}}^{(4)}$ (the density
projection operator) and ${\cal{\overline{D}}}{}^{(4)}$ (the chiral projection
operator) which may be expanded as

${{\Delta}}{}^{(4)}~{}=~{}\sum_{i=0}^{4}\,b_{(4-i)}\,\cdot\,\,\left[\,(\nabla)\,\times\,\,\cdots\,\,\times\,(\nabla)^{4-i}\,\right]~{}~{}~{},$
(27)
${\cal{\overline{D}}}{}^{(4)}~{}=~{}\sum_{i=0}^{4}\,a_{(4-i)}\,\cdot\,\,\left[\,(\overline{\nabla})\,\times\,\cdots\,\,\times\,({\overline{\nabla}})^{4-i}\,\right]~{}~{}~{},$
(28)

in terms of some field-dependent coefficients $a_{(4-i)}$ and $b_{(4-i)}$ and
powers of the spinorial superspace supergravity covariant derivatives
$\nabla_{\alpha\,i}$ and ${\overline{\nabla}}{}_{\alpha}{}^{i}$. In (26) we
have the expressions ${\rm E}^{-1}=[{\rm Ber}\,{E_{A}{}^{M}}]^{-1}$ and ${\rm
e}^{-1}=[\det{e_{a}{}^{m}}]^{-1}$ which are functions respectively of the
supervielbein and the component vielbein and $d^{2}\sigma$ denotes the measure
over the two-dimensional bosonic space. A further consequence of (26) - (28)
is that the superfield Lagrangian $\cal L$ need not be hermitian as it is the
linear combination of terms that appear in the action formula that must
satisfy this requirement. In the present context these spinorial superspace
supergravity covariant derivatives satisfy the relations given in section 2.

The basis for the ectoplasmic derivations of local supergravity measures and
projections operators, lies in a proposition for how to integrate an arbitrary
super $p$-form. This was proposed in the work of [2]. Given a curved
superspace with ${\rm N}_{B}$ bosonic coordinates (labelled by $\underline{m}$
indeces) and ${\rm N}_{F}$ fermionic coordinates (labelled by
$\underline{\mu}$ indices), we have Proposition 1
If $J{}_{{\underline{A}}_{1}\dots{\underline{A}}_{p}}$ is a closed super
$p$-form superfield whose Bianchi identities
vanish and $d\Omega{}^{{\underline{m}}_{1}\dots{\underline{m}}_{p}}$ is a co-
chain of dimension $p$ $\leq$ NB (where NB is
the dimensionality of the bosonic subspace), then the integral of the su-
per $p$-form over the co-chain is given by

$\displaystyle{{\cal S}(d\Omega{\,|}J)~{}}$ | $\displaystyle{{}\equiv~{}(p!)^{-1}\,\int~{}d\Omega^{{\@@underline a}_{1}\cdots{\@@underline a}_{p}}\,{\cal J}^{(p)}_{{\@@underline a}_{1}\cdots{\@@underline a}_{p}}{\Big{|}}~{}~{}~{}.}$
---|---
(29)

and this is a supersymmetrical invariant.  In (29) we note the quantity ${\cal
J}^{(p)}_{{\@@underline a}_{1}\cdots{\@@underline a}_{p}}{\Big{|}}$ is related
to the super $p$-form $J{}_{{\underline{A}}_{1}\dots{\underline{A}}_{p}}$ via

$\displaystyle{\Big{(}\,J_{{\@@underline a}_{1}\cdots{\@@underline a}_{p}}{\Big{|}}\,\Big{)}~{}\equiv~{}\Big{[}}$ | $\displaystyle{{}~{}{\cal J}^{(p)}_{{\@@underline a}_{1}\cdots{\@@underline a}_{p}}{\Big{|}}\,+\,\lambda^{(p,1)}\psi_{[\,{\@@underline a}_{1}|}{}^{{\@@underline\alpha}_{1}}\Big{(}\,J_{{\@@underline\alpha}_{1}|\,{\@@underline a}_{2}\cdots{\@@underline a}_{p}\,]}{\Big{|}}\,\Big{)}}$
---|---
| $\displaystyle{{}+\,\lambda^{(p,2)}\psi_{[\,{\@@underline
a}_{1}|}{}^{{\@@underline\alpha}_{1}}\psi_{|{\@@underline
a}_{2}|}{}^{{\@@underline\alpha}_{2}}\Big{(}\,J_{{\@@underline\alpha}_{1}{\@@underline\alpha}_{2}|\,{\@@underline
a}_{3}\cdots{\@@underline a}_{p}\,]}{\Big{|}}\,\Big{)}\cdots}$
| $\displaystyle{{}+\,\lambda^{(p,p)}\,[\,\psi_{{\@@underline
a}_{1}}{}^{{\@@underline\alpha}_{1}}\cdots\psi_{{\@@underline
a}_{p}}{}^{{\@@underline\alpha}_{p}}\,]\Big{(}\,J_{{\@@underline\alpha}_{1}{\@@underline\alpha}_{2}\cdots{\@@underline\alpha}_{p}}{\Big{|}}\,\Big{)}~{}\Big{]}~{}~{}~{}.}$
(30)

where $\psi_{{\@@underline a}}{}^{{\@@underline\alpha}}$ denotes the
gravitino. The quantities ${\cal J}^{(p)}_{{\@@underline
a}_{1}\cdots{\@@underline a}_{p}}{\Big{|}}$ and coefficients
$\lambda^{(p,1)}\,$ $\cdots$ $\lambda^{(p,p)}\,$ are determined by taking the
limit as the Grassmann coordinates go to zero in $J_{{\@@underline
a}_{1}\cdots{\@@underline a}_{p}}$. In the 2D, ${\cal N}=4$ case with $J_{ab}$
the component of a super 2-form, the equation (25) informs us about the
$\lambda$-coefficients.”

We next observe that upon setting $p$ = NB the proposition takes the form

$\displaystyle{{\cal S}(d\Omega{\,|}J)~{}}$ | $\displaystyle{{}=~{}\int~{}d^{N_{B}}x\,{\rm e}^{-1}\,\hbox{\large{${{\textstyle{{1}\over\vphantom{2}\smash{\raise 0.72331pt\hbox{$\scriptstyle{{~{}{\rm N}{}_{B}!~{}}}$}}}}}$}}\,\varepsilon^{{\@@underline a}_{1}\cdots{\@@underline a}_{N_{B}}}\,{\cal J}^{({\rm N}{}_{B})}_{{\@@underline a}_{1}\cdots{\@@underline a}_{{\rm N}{}_{B}}}{\Big{|}}~{}~{}~{},}$
---|---
(31)

where e-1 denotes the determinant of the vielbein for the bosonic subspace. In
the case considered in this paper, we thus reach the result

$\displaystyle{{~{}~{}~{}~{}~{}~{}}{\cal S}(d\Omega{\,|}J)~{}}$ | $\displaystyle{{}=~{}\int~{}d^{2}\sigma\,{\rm e}^{-1}\,\hbox{\large{${{\textstyle{{1}\over\vphantom{2}\smash{\raise 0.72331pt\hbox{$\scriptstyle{{2}}$}}}}}$}}\,\varepsilon^{a\,b}\,{\cal J}^{(2)}_{a\,b}{\Big{|}}}$
---|---
| $\displaystyle{{}=~{}\int~{}d^{2}\sigma\,{\rm
e}^{-1}\,{\Big{[}}~{}\hbox{\large{${{\textstyle{{1}\over\vphantom{2}\smash{\raise
0.72331pt\hbox{$\scriptstyle{{2}}$}}}}}$}}\,\varepsilon^{a\,b}\,{J}_{a\,b}{\Big{|}}~{}-~{}\varepsilon^{ab}(\psi_{a}{}^{\alpha
i}{J}_{\alpha
i~{}b}{\Big{|}}~{}+~{}{\bar{\psi}}_{a}{}^{\alpha}{}_{i}{J}_{\alpha}{}^{i}{}_{~{}b}{\Big{|}})}$
|
$\displaystyle{{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}-~{}\,\varepsilon^{ab}\psi_{a}{}^{\alpha
i}{\bar{\psi}}_{b}{}^{\beta}{}_{j}{J}_{\alpha}{}_{i}{}_{\beta}{}^{j}{\Big{|}}}$
|
$\displaystyle{{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}-~{}{\textstyle{1\over\vphantom{2}\smash{\raise
0.60275pt\hbox{$\scriptstyle{2}$}}}}\,\varepsilon^{ab}\psi_{a}{}^{\alpha
i}\psi_{b}{}^{\beta
j}{J}_{\alpha\,i\,\beta\,j}{\Big{|}}~{}-~{}{\textstyle{1\over\vphantom{2}\smash{\raise
0.60275pt\hbox{$\scriptstyle{2}$}}}}\,\varepsilon^{ab}{\bar{\psi}}_{a}{}^{\alpha}{}_{i}{\bar{\psi}}_{b}{}^{\beta}{}_{j}{J}_{\alpha}{}^{i}{}_{\beta}{}^{j}{\Big{|}}~{}{\Big{]}}~{}~{}~{}.}$
(32)

More explicitly the equations in (23) are expressed as

$\displaystyle{{~{}~{}}J_{\alpha i}{}_{\beta}{}^{j}}$ | $\displaystyle{{}=~{}0~{}~{}~{},}$
---|---
$\displaystyle{J_{\alpha i}{}_{\beta j}}$ | $\displaystyle{{}=~{}2(\gamma^{3})_{\alpha\beta}{\overline{\nabla}}^{(2)}_{ij}\left[\,{{\nabla}}{}^{(2)\,\epsilon\,\kappa}~{}-~{}2\,{\bar{B}}\,(\gamma^{3}){}^{\epsilon\,\kappa}\,\right]\,{{\nabla}}{}^{(2)}_{\,\epsilon\,\kappa}\,{\overline{\cal L}}}$
|
$\displaystyle{{}~{}~{}~{}~{}~{}-\,C_{\alpha\beta}C_{ij}(\gamma^{3})^{\gamma\delta}{\overline{\nabla}}^{(2)}_{\gamma\delta}\left[\,{{\nabla}}{}^{(2)\,\epsilon\,\kappa}~{}-~{}2\,{\bar{B}}\,(\gamma^{3}){}^{\epsilon\,\kappa}\,\right]\,{{\nabla}}{}^{(2)}_{\,\epsilon\,\kappa}\,{\overline{\cal
L}}~{}~{}~{},}$
$\displaystyle{J_{\alpha}{}^{i}{}_{\beta}{}^{j}}$ | $\displaystyle{{}=~{}\,2\,(\gamma^{3})_{\alpha\beta}\nabla^{(2)\,ij}\left[\,{\overline{\nabla}}{}^{(2)\,\epsilon\,\kappa}~{}-~{}2\,B\,(\gamma^{3}){}^{\epsilon\,\kappa}\,\right]\,{\overline{\nabla}}{}^{(2)}_{\,\epsilon\,\kappa}\,{\cal L}}$
|
$\displaystyle{{}~{}~{}~{}~{}~{}~{}-~{}\,C_{\alpha\beta}C^{ij}(\gamma^{3})^{\gamma\delta}\nabla^{(2)}_{\gamma\delta}\left[\,{\overline{\nabla}}{}^{(2)\,\epsilon\,\kappa}~{}-~{}2\,B\,(\gamma^{3}){}^{\epsilon\,\kappa}\,\right]\,{\overline{\nabla}}{}^{(2)}_{\,\epsilon\,\kappa}\,{\cal
L}~{}~{}~{},}$
$\displaystyle{J_{\gamma k}{}_{a}}$ | $\displaystyle{{}=~{}-~{}\hbox{\large{${{\textstyle{{{\rm i}}\over\vphantom{2}\smash{\raise 0.72331pt\hbox{$\scriptstyle{{3}}$}}}}}$}}\varepsilon_{ab}(\gamma^{b})_{\gamma}{}^{\delta}{\overline{\nabla}}_{\delta}{}^{p}{\overline{\nabla}}^{(2)}_{kp}\left[\,{{\nabla}}{}^{(2)\,\epsilon\,\kappa}~{}-~{}2\,{\bar{B}}\,(\gamma^{3}){}^{\epsilon\,\kappa}\,\right]\,{{\nabla}}{}^{(2)}_{\,\epsilon\,\kappa}\,{\overline{\cal L}}~{}~{}~{},}$
$\displaystyle{J_{\gamma}{}^{k}{}_{a}}$ | $\displaystyle{{}=~{}-~{}\hbox{\large{${{\textstyle{{{\rm i}}\over\vphantom{2}\smash{\raise 0.72331pt\hbox{$\scriptstyle{{3}}$}}}}}$}}\varepsilon_{ab}(\gamma^{b})_{\gamma}{}^{\delta}\nabla_{\delta p}\nabla^{(2)\,kp}\left[\,{\overline{\nabla}}{}^{(2)\,\epsilon\,\kappa}~{}-~{}2\,B\,(\gamma^{3}){}^{\epsilon\,\kappa}\,\right]\,{\overline{\nabla}}{}^{(2)}_{\,\epsilon\,\kappa}\,{\cal L}~{}~{}~{},}$
$\displaystyle{J_{ab}}$ | $\displaystyle{{}=~{}-~{}\hbox{\large{${{\textstyle{{1}\over\vphantom{2}\smash{\raise 0.72331pt\hbox{$\scriptstyle{{8}}$}}}}}$}}\,\varepsilon_{ab}\,\Big{[}\nabla^{(4)}~{}-~{}2\,\bar{B}(\gamma^{3})^{\alpha\beta}\nabla^{(2)}_{\alpha\beta}\Big{]}\left[\,{\overline{\nabla}}{}^{(2)\,\epsilon\,\kappa}~{}-~{}2\,B\,(\gamma^{3}){}^{\epsilon\,\kappa}\,\right]\,{\overline{\nabla}}{}^{(2)}_{\,\epsilon\,\kappa}\,{\cal L}}$
|
$\displaystyle{{}~{}~{}~{}~{}~{}-~{}\hbox{\large{${{\textstyle{{1}\over\vphantom{2}\smash{\raise
0.72331pt\hbox{$\scriptstyle{{8}}$}}}}}$}}\,\varepsilon_{ab}\,\Big{[}~{}{\overline{\nabla}}^{(4)}~{}-~{}2\,B(\gamma^{3})^{\alpha\beta}{\overline{\nabla}}^{(2)}_{\alpha\beta}\Big{]}\left[\,{{\nabla}}{}^{(2)\,\epsilon\,\kappa}~{}-~{}2\,{\bar{B}}\,(\gamma^{3}){}^{\epsilon\,\kappa}\,\right]\,{{\nabla}}{}^{(2)}_{\,\epsilon\,\kappa}\,{\overline{\cal
L}}~{}~{}~{}.}$
(33)

Finally, the results in (33) can be substituted into equation (32) to reach
the main result of this presentation. Given an arbitrary 2D, $\cal N$ = 4
superfield Lagrangian $\cal L$, a local supersymmetrical invariant is given by

$\displaystyle{{\cal S}}$ | $\displaystyle{{}=~{}\int~{}d^{2}\sigma\,{\rm e}^{-1}\,\Delta^{(4)}\,\bar{\cal D}^{(4)}\,{\cal L}{\Big{|}}~{}+~{}{\rm h.~{}c.}}$
---|---
| $\displaystyle{{}=~{}\int~{}d^{2}\sigma\,{\rm
e}^{-1}\,{\Big{\\{}}~{}\hbox{\large{${{\textstyle{{1}\over\vphantom{2}\smash{\raise
0.72331pt\hbox{$\scriptstyle{{8}}$}}}}}$}}\nabla^{(4)}~{}-~{}\hbox{\large{${{\textstyle{{1}\over\vphantom{2}\smash{\raise
0.72331pt\hbox{$\scriptstyle{{4}}$}}}}}$}}\,\bar{B}(\gamma^{3})^{\alpha\beta}\nabla^{(2)}_{\alpha\beta}~{}+~{}\hbox{\large{${{\textstyle{{{\rm
i}}\over\vphantom{2}\smash{\raise
0.72331pt\hbox{$\scriptstyle{{3}}$}}}}}$}}{\bar{\psi}}_{a}{}^{\gamma}{}_{i}(\gamma^{a})_{\gamma}{}^{\delta}\nabla_{\delta
j}\nabla^{(2)\,ij}}$
|
$\displaystyle{{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}-~{}\varepsilon^{ab}{\bar{\psi}}_{a}{}^{\alpha}{}_{i}{\bar{\psi}}_{b}{}^{\beta}{}_{j}(\gamma^{3})_{\alpha\beta}\nabla^{(2)\,ij}~{}+~{}\hbox{\large{${{\textstyle{{1}\over\vphantom{2}\smash{\raise
0.72331pt\hbox{$\scriptstyle{{2}}$}}}}}$}}\,\varepsilon^{ab}{\bar{\psi}}_{a}{}^{\alpha}{}_{i}{\bar{\psi}}_{b}{}^{\beta}{}_{j}C_{\alpha\beta}C^{ij}(\gamma^{3})^{\gamma\delta}\nabla^{(2)}_{\gamma\delta}~{}{\Big{\\}}}\times}$
|
$\displaystyle{{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\times\left[\,{\overline{\nabla}}{}^{(2)\,\epsilon\,\kappa}~{}-~{}2\,B\,(\gamma^{3}){}^{\epsilon\,\kappa}\,\right]\,{\overline{\nabla}}{}^{(2)}_{\,\epsilon\,\kappa}\,{\cal
L}{\Big{|}}~{}+~{}{\rm h.~{}c.}}$
(34)

in the presence of the off-shell supergravity theory described in section 2.

##  5 Conclusion

With this present work, we have completed the task of developing an efficient
local superspace integration theory for two dimensional theories that possess
eight real supercharges. We believe that the result given in (34) is
unexpectedly elegant and simple given that the general form of the eigth-order
spinorial differential operator defined by (26), (27) and (28) could, in
principle, take a more complicated form. Perhaps one of most surprising
features of this derivation has been the use of the closed 2D, $\cal N$ = 4
super 2-form used in Theorem 1. The superfield $U$ that appeared in equation
(23) is not required to describe any irreducible supermultiplet. The only
requirement imposed on the superfield $U$ is its chirality.

As proved in [21], the chiral superfield $U$ can be expressed in terms of the
chiral projector $\overline{\cal{D}}^{(4)}$ and an unconstrained superfield
${\cal{L}}$ as $U=\overline{\cal{D}}^{(4)}{\cal{L}}$. This result has been
used in sections 3 and 4. According to the discussion of section 4, the main
result of this paper is the computation of the density projector operator
$\Delta^{(4)}$ of (26), (27) and (34), which, together with
$\overline{\cal{D}}^{(4)}$, allows to define the component supergravity
integration measure (34). In deriving for the first time $\Delta^{(4)}$ we
used the Ectoplasmic techniques and the new super 2-form of Theorem 1 (23).

One other point we wish to emphasize is the efficiency of the Ectoplasmic
approach in the case we considered here. It would be interesting to re-derive
the integration measure (34) via the normal coordinate expansion technique [7,
16, 15] (in particular using its last version [15]) even if we do not expect
that the latter approach would require shorter computations. This is
especially true considering that in 2D the number of Bianchi identities to be
solved for a closed super 2-form is relatively low. This once more emphasizes
the important role of forms as a basis for superspace integration theory as
advocated in the ectoplasmic approach. The success of this also points to the
generality of using this as a tool in all cases to derive superspace local
integration measures.

Note also that here we focused on the 2D, ${\cal N}=4$ minimal superspace
geometry of [1] as described in section 2. In general, it is known that there
could exist different off-shell superspace supergravities. We expect that the
Ectoplasm paradigm and the results of our paper can be extended to any
covariant superspace formulation of 2D, ${\cal N}=4$ supergravity. For
example, in the first paper of [1] a variant central charge formulation of the
minimal multiplet was given; once noticed that the Lagrangian ${\cal L}$ in
(34) has to be neutral for the central charges, one can see that the results
of our paper apply without modifications to the variant formulation. Moreover,
recently a new extended covariant formulation of 2D, ${\cal N}=4$ supergravity
in superspace was given [24]. The ectoplasm techniques to compute the chiral
action in components apply straightforward if one consider the geometry of
[24] even if in this case longer computations are expected due to the more
involved structure of the torsion multiplet. Other superspace formulations of
2D, ${\cal N}=4$ supergravity [25] are known in the bi-harmonic superspace of
[26]. Being those superspace supergravities based on a prepotential approach
the definition of a covariant components reduction is not clear. However, on
the ground of the related bi-projective formalism [27], recently extended to
covariantly study 2D, ${\cal N}=4$ matter-couplet supergravity, it would be of
interest and well defined to find by using Ectoplasm techniques the bi-
projective density operator analogously to the chiral action studied here.

“Where the senses fail us, reason must step in.”
\- Galileo Galilei

Acknowledgements
This research was supported in part by the endowment of the John S. Toll
Professorship, the University of Maryland Center for String & Particle Theory,
National Science Foundation Grant PHY-0354401.

Appendix A: Consistency of Bianchi Identities & Constraints For 2-Form

In this appendix, we will present the explicit proof that the Bianchi
identities associated with the results in (23) imply that it is a closed 2D,
$\cal N$ = 4 super 2-form. We begin by writing an ansatz for the lowest
components of a 2D, $\cal N$ = 4 super 2-form under the assumption that these
component should:
(a.) be linear in a (anti)chiral superfield $U$ $(\bar{U})$;
${\overline{\nabla}}_{\alpha}{}^{i}U=0$ $(\nabla_{\alpha i}\bar{U}=0)$,
(b.) depend on the superspace supergravity covariant derivative,
(c.) and are local functions of the superspace supergravity field strengths
$B$, $\overline{B}$, $G$ and $H.$
Under the previous assumptions we will begin with an ansatz given by 999The
ansatz we are using can also be guessed by: (i) consider the flat 4D, $\cal
N$=2 “chiral” closed
super 4-form introduced in [23]; (ii) perform a dimensional reduction of the
4D, $\cal N$=2 super
4-form to derive a 2D, $\cal N$=4 closed super 2-form; (iii) extend the
resulting dimension-1
components of the flat 2D, $\cal N$=4 2-form to the curved case by modifying
the flat derivatives to
the curved covariant derivatives and by adding torsion dependent terms.

$\displaystyle{J_{\alpha i}{}_{\beta
j}=a(\gamma^{3})_{\alpha\beta}{\overline{\nabla}}^{(2)}_{ij}\bar{U}+b\,C_{\alpha\beta}C_{ij}(\gamma^{3})^{\gamma\delta}{\overline{\nabla}}^{(2)}_{\gamma\delta}\bar{U}+C_{\alpha\beta}C_{ij}F\bar{U}~{},~{}~{}~{}}$
---
$None$ $\displaystyle{{J}_{\alpha
i}{}_{\beta}{}^{j}=0~{},~{}~{}~{}~{}~{}~{}{J}_{\alpha}{}^{i}{}_{\beta}{}^{j}=-(J_{\alpha
i}{}_{\beta j})^{*}~{},}$
---
$None$

where

$\displaystyle{F=F(B,\bar{B},G,H)=b_{1}B+b_{2}\bar{B}+gG+hH~{},}$
---
$None$

and $a,\,b,\,b_{1},\,b_{2},\,g,\,h$ are constants to be fixed.

The task is to study the Bianchi identities that derive from the closure of
the 2-form $J$

$dJ=0~{},~{}~{}~{}~{}~{}~{}\Longleftrightarrow~{}~{}~{}~{}~{}~{}0={1\over
2}\nabla_{[A}J_{BC)}-{1\over 2}T_{[AB|}{}^{D}J_{D|C)}~{},$ $None$

with $J_{A}{}_{B}=\big{(}\,J_{\alpha i}{}_{\beta j},~{}J_{\alpha
i}{}_{\beta}{}^{j},~{}J_{\alpha}{}^{i}{}_{\beta}{}^{j},~{}J_{\gamma
k}{}_{a},~{}J_{\gamma}{}^{k}{}_{a},~{}J_{ab}\,\big{)}$ and the lowest
components satisfying (A.1) - (A.3).

Substituting the results of (A.1), (A.2) into the identity (A.4) with
$A=\alpha i,\,B=\beta j,\,C=\gamma k$ one obtains

$\displaystyle{{~{}~{}~{}~{}~{}~{}~{}}0}$ | $\displaystyle{{}=~{}a(\gamma^{3})_{\beta\gamma}[\nabla_{\alpha i},{\overline{\nabla}}^{(2)}_{jk}]\overline{U}+a(\gamma^{3})_{\gamma\alpha}[\nabla_{\beta j},{\overline{\nabla}}^{(2)}_{ki}]\overline{U}+a(\gamma^{3})_{\alpha\beta}[\nabla_{\gamma k},{\overline{\nabla}}^{(2)}_{ij}]\overline{U}}$
---|---
|
$\displaystyle{{}~{}~{}~{}~{}+b\,C_{\beta\gamma}C_{jk}(\gamma^{3})^{\delta\rho}[\nabla_{\alpha
i},{\overline{\nabla}}^{(2)}_{\delta\rho}]\overline{U}+b\,C_{\gamma\alpha}C_{ki}(\gamma^{3})^{\delta\rho}[\nabla_{\beta
j},{\overline{\nabla}}^{(2)}_{\delta\rho}]\overline{U}}$
|
$\displaystyle{{}~{}~{}~{}~{}+b\,C_{\alpha\beta}C_{ij}(\gamma^{3})^{\delta\rho}[\nabla_{\gamma
k},{\overline{\nabla}}^{(2)}_{\delta\rho}]\overline{U}+C_{\beta\gamma}C_{jk}\,(\nabla_{\alpha
i}F)\overline{U}+C_{\gamma\alpha}C_{ki}\,(\nabla_{\beta j}F)\overline{U}}$
| $\displaystyle{{}~{}~{}~{}~{}+C_{\alpha\beta}C_{ij}\,(\nabla_{\gamma
k}F)\overline{U}~{},~{}~{}~{}~{}~{}~{}}$
$None$

where we have used the fact that $\bar{U}$ is antichiral to write this. At
this point, there are two useful identities to note

${[}\nabla_{\alpha
i},{\overline{\nabla}}^{(2)}_{ij}{]}\overline{U}~{}=~{}\Big{(}-2{\rm
i}C_{i(j}(\gamma^{c})_{\alpha}{}^{\delta}\nabla_{c}{\overline{\nabla}}_{\delta
k)}\Big{)}\overline{U}$ $None$ ${[}\nabla_{\alpha
i},(\gamma^{3})^{\delta\rho}{\overline{\nabla}}^{(2)}_{\delta\rho}{]}~{}=~{}\Big{(}-4{\rm
i}\varepsilon^{bc}(\gamma_{b})_{\alpha}{}^{\beta}\nabla_{c}{\overline{\nabla}}_{\beta
i}\Big{)}\overline{U}$ $None$

which shows that in principle there are terms containing spacetime derivatives
in (A.5). In order to satisfy the Bianchi identity, two sets of conditions are
required:

| $\displaystyle{{}(a.)~{}~{}a~{}=~{}-2b~{}~{}{\rm{and}}}$
---|---
|
$\displaystyle{{}(b.)~{}~{}b_{1}~{}=~{}b_{2}~{}=~{}g~{}=~{}h~{}=~{}0~{}~{}.}$
$None$

For simplicity we also set

$a=1~{}.$ $None$

The next Bianchi identity encountered takes the form

$\displaystyle{0}$ | $\displaystyle{{}=~{}{\overline{\nabla}}_{\alpha}^{i}J_{\beta j}{}_{\gamma k}+T_{\alpha}{}^{i}{}_{\beta j}{}^{a}J_{\gamma k}{}_{a}+T_{\alpha}{}^{i}{}_{\gamma k}{}^{a}J_{\beta j}{}_{a}~{}~{}~{}.}$
---|---
$None$

The result in (A.1), subject to (A.8), (A.9), can be substituted into this
equation. To satisfy this, it is useful to use the following identities

$\displaystyle{{\overline{\nabla}}_{\alpha i}{\overline{\nabla}}^{(2)}_{jk}\bar{U}}$ | $\displaystyle{{}=~{}-\,\hbox{\large{${{\textstyle{{1}\over\vphantom{2}\smash{\raise 0.72331pt\hbox{$\scriptstyle{{3}}$}}}}}$}}C_{i(j}{\overline{\nabla}}_{\alpha}^{p}{\overline{\nabla}}^{(2)}_{k)p}\overline{U}~{}~{},}$
---|---
$\displaystyle{{\overline{\nabla}}_{\alpha i}{\overline{\nabla}}^{(2)}_{\beta\gamma}\bar{U}}$ | $\displaystyle{{}=~{}\hbox{\large{${{\textstyle{{1}\over\vphantom{2}\smash{\raise 0.72331pt\hbox{$\scriptstyle{{3}}$}}}}}$}}C_{\alpha(\beta}{\overline{\nabla}}_{\gamma)}^{p}{\overline{\nabla}}^{(2)}_{ip}\overline{U}~{}-~{}\hbox{\large{${{\textstyle{{4}\over\vphantom{2}\smash{\raise 0.72331pt\hbox{$\scriptstyle{{3}}$}}}}}$}}\,BC_{\alpha(\beta}(\gamma^{3})_{\gamma)}{}^{\delta}{\overline{\nabla}}_{\delta i}\overline{U}~{}+~{}\hbox{\large{${{\textstyle{{1}\over\vphantom{2}\smash{\raise 0.72331pt\hbox{$\scriptstyle{{3}}$}}}}}$}}\,B(\gamma^{3})_{(\alpha\beta}{\overline{\nabla}}_{\gamma)i}\overline{U}~{}~{},}$
$\displaystyle{(\gamma^{3})^{\beta\gamma}{\overline{\nabla}}_{\alpha i}{\overline{\nabla}}^{(2)}_{\beta\gamma}\bar{U}}$ | $\displaystyle{{}=~{}-\,\hbox{\large{${{\textstyle{{2}\over\vphantom{2}\smash{\raise 0.72331pt\hbox{$\scriptstyle{{3}}$}}}}}$}}\,(\gamma^{3})_{\alpha}{}^{\gamma}{\overline{\nabla}}_{\gamma}^{p}{\overline{\nabla}}^{(2)}_{ip}\overline{U}~{}~{}~{}.}$
$None$

Then, to completely satisfy (A.10) one has to impose

$J{}_{\gamma k}{}_{a}~{}=~{}-~{}\hbox{\large{${{\textstyle{{{\rm
i}}\over\vphantom{2}\smash{\raise
0.72331pt\hbox{$\scriptstyle{{3}}$}}}}}$}}\,\varepsilon_{ab}(\gamma^{b})_{\gamma}{}^{\rho}{\overline{\nabla}}_{\rho}{}^{p}{\overline{\nabla}}^{(2)}_{kp}\bar{U}~{}~{}~{}.$
$None$

Note that it holds

$J{}_{\gamma}{}^{k}{}_{a}~{}=~{}-(J{}_{\gamma
k}{}_{a})^{*}~{}=~{}-~{}\hbox{\large{${{\textstyle{{{\rm
i}}\over\vphantom{2}\smash{\raise
0.72331pt\hbox{$\scriptstyle{{3}}$}}}}}$}}\,\varepsilon_{ab}(\gamma^{b})_{\gamma}{}^{\rho}\nabla_{\rho
p}\nabla^{(2)}{}^{kp}{U}~{}~{}~{}.$ $None$

We can continue our deliberations by considering the Bianchi identity given by

$0~{}=~{}\nabla_{a}J_{\beta j\gamma k}~{}+~{}\nabla_{\beta j}J_{\gamma
k}{}_{a}~{}+~{}\nabla_{\gamma k}J_{\beta j}{}_{a}~{}-~{}T_{a\beta j}{}^{\delta
l}J_{\delta l\gamma k}~{}-~{}T_{a\gamma k}{}^{\delta l}J_{\delta l\beta
j}~{}~{}~{},$ $None$

and into this are substituted the results (A.1), (A.8), (A.9) and (A.12). When
this is done, a differential equation on $\overline{U}$ of the form

$\displaystyle{{~{}~{}~{}~{}~{}~{}~{}}0~{}=}$ | $\displaystyle{{}~{}\nabla_{a}\Big{(}2(\gamma^{3})_{\beta\gamma}{\overline{\nabla}}^{(2)}_{jk}-\,C_{\beta\gamma}C_{jk}(\gamma^{3})^{\delta\rho}{\overline{\nabla}}^{(2)}_{\delta\rho}\Big{)}\overline{U}}$
---|---
| $\displaystyle{{}~{}-~{}\hbox{\large{${{\textstyle{{{\rm
i}}\over\vphantom{2}\smash{\raise
0.72331pt\hbox{$\scriptstyle{{3}}$}}}}}$}}\,\nabla_{\beta
j}\Big{(}\varepsilon_{ab}(\gamma^{b})_{\gamma}{}^{\delta}{\overline{\nabla}}_{\delta}{}^{p}{\overline{\nabla}}^{(2)}_{kp}\overline{U}\Big{)}~{}-~{}\hbox{\large{${{\textstyle{{{\rm
i}}\over\vphantom{2}\smash{\raise
0.72331pt\hbox{$\scriptstyle{{3}}$}}}}}$}}\,\nabla_{\gamma
k}\Big{(}\varepsilon_{ab}(\gamma^{b})_{\beta}{}^{\delta}{\overline{\nabla}}_{\delta}{}^{p}{\overline{\nabla}}^{(2)}_{jp}\overline{U}{\Big{)}}}$
| $\displaystyle{{}~{}+~{}\hbox{\large{${{\textstyle{{{\rm
i}}\over\vphantom{2}\smash{\raise
0.72331pt\hbox{$\scriptstyle{{2}}$}}}}}$}}\,\delta_{j}^{l}\phi_{\beta}{}^{\rho}(\gamma_{a})_{\rho}{}^{\delta}\Big{(}2(\gamma^{3})_{\delta\gamma}{\overline{\nabla}}^{(2)}_{lk}-\,C_{\delta\gamma}C_{lk}(\gamma^{3})^{\rho\tau}{\overline{\nabla}}^{(2)}_{\rho\tau}\Big{)}\overline{U}}$
| $\displaystyle{{}~{}+~{}\hbox{\large{${{\textstyle{{{\rm
i}}\over\vphantom{2}\smash{\raise
0.72331pt\hbox{$\scriptstyle{{2}}$}}}}}$}}\,\delta_{k}^{l}\phi_{\gamma}{}^{\rho}(\gamma_{a})_{\rho}{}^{\delta}\Big{(}2(\gamma^{3})_{\delta\beta}{\overline{\nabla}}^{(2)}_{lj}-\,C_{\delta\beta}C_{lj}(\gamma^{3})^{\rho\tau}{\overline{\nabla}}^{(2)}_{\rho\tau}\Big{)}\overline{U}}$
$None$

emerges. Further progress is possible by using the identity

$\displaystyle{{~{}~{}}\\{\nabla_{\alpha i},{\overline{\nabla}}_{\delta}^{p}{\overline{\nabla}}^{(2)}_{kp}\\}\overline{U}}$ | $\displaystyle{{}=~{}\Bigg{(}3{\rm i}(\gamma^{a})_{\alpha\delta}\nabla_{a}{\overline{\nabla}}^{(2)}_{ik}-3{\rm i}C_{ik}(\gamma^{a})_{\alpha}{}^{\rho}\nabla_{a}{\overline{\nabla}}^{(2)}_{\delta\rho}}$
---|---
|
$\displaystyle{{}{~{}~{}~{}~{}~{}~{}~{}~{}~{}}+~{}\hbox{\large{${{\textstyle{{3}\over\vphantom{2}\smash{\raise
0.72331pt\hbox{$\scriptstyle{{2}}$}}}}}$}}C_{ik}{\phi}^{\tau}{}_{\delta}(\gamma^{3})_{\alpha\tau}(\gamma^{3})^{\rho\beta}{\overline{\nabla}}^{(2)}_{\beta\rho}-3\phi_{\alpha\delta}{\overline{\nabla}}^{(2)}_{ik}}$
|
$\displaystyle{{}{~{}~{}~{}~{}~{}~{}~{}~{}}~{}+~{}6C_{ik}C_{\alpha\delta}\Sigma^{\beta
p}{\overline{\nabla}}_{\beta
p}~{}+~{}6C_{ik}(\gamma^{3})_{\alpha\delta}(\gamma^{3})^{\beta\rho}\Sigma_{\beta}{}^{p}{\overline{\nabla}}_{\rho
p}\Bigg{)}\overline{U}~{}~{}.}$
$None$

This result is substituted into (A.15) and after some algebra, the
$\Sigma$-dependent terms are seen to cancel leaving

$\displaystyle{{~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}}0\,}$ | $\displaystyle{{}=\Bigg{(}2(\gamma^{3})_{\alpha\gamma}\nabla_{a}{\overline{\nabla}}^{(2)}_{ik}~{}-~{}C_{\alpha\gamma}C_{ik}(\gamma^{3})^{\delta\rho}\nabla_{a}{\overline{\nabla}}^{(2)}_{\delta\rho}}$
---|---
|
$\displaystyle{{}~{}~{}~{}~{}~{}~{}~{}-2(\gamma^{3})_{\alpha\gamma}\nabla_{a}{\overline{\nabla}}^{(2)}_{ik}+C_{ik}C_{\alpha\gamma}(\gamma^{3})^{\delta\rho}\nabla_{a}{\overline{\nabla}}^{(2)}_{\delta\rho}}$
| $\displaystyle{{}~{}~{}~{}~{}~{}~{}~{}-~{}{\rm
i}\varepsilon_{ab}\phi_{\beta\delta}(\gamma^{b})_{\alpha\gamma}C^{\beta\delta}{\overline{\nabla}}^{(2)}_{ik}~{}-~{}{\rm
i}\phi_{\beta\delta}(\gamma^{3})^{\beta\delta}(\gamma_{a})_{\alpha\gamma}{\overline{\nabla}}^{(2)}_{ik}}$
| $\displaystyle{{}~{}~{}~{}~{}~{}~{}~{}+~{}{\rm
i}\varepsilon_{ab}\phi_{\alpha^{\prime}\delta}(\gamma^{b})_{\alpha\delta}C^{\alpha^{\prime}\delta}{\overline{\nabla}}^{(2)}_{ik}~{}+~{}{\rm
i}\phi_{\alpha^{\prime}\delta}(\gamma_{c})_{\alpha\delta}(\gamma^{3})^{\alpha^{\prime}\delta}{\overline{\nabla}}^{(2)}_{ik}\Bigg{)}\overline{U}~{}~{}}$
$None$

which is clearly identically satisfied.

There is a second dimension-2 Bianchi identity of the form

$\displaystyle{{~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}}0\,}$ | $\displaystyle{{}=~{}-~{}{\overline{\nabla}}_{\alpha}{}^{i}J_{\gamma k}{}_{b}~{}-~{}\nabla_{\gamma k}J_{\alpha}{}^{i}{}_{b}~{}+~{}T_{b}{}_{\gamma k}{}^{\delta}{}_{l}J_{\delta}{}^{l}{}_{\alpha}{}^{i}~{}+~{}T_{b}{}_{\alpha}{}^{i}{}^{\delta l}J_{\delta l}{}_{\gamma k}~{}+~{}T_{\gamma k}{}_{\alpha}{}^{i}{}^{c}J_{cb}~{}~{}.}$
---|---
$None$

One may substitute from results derived previously to cast this into the form
of

$\displaystyle{2{\rm i}\delta^{i}_{k}(\gamma^{c})_{\alpha\gamma}J_{bc}}$ | $\displaystyle{{}=~{}{\rm i}\Big{(}\hbox{\large{${{\textstyle{{1}\over\vphantom{2}\smash{\raise 0.72331pt\hbox{$\scriptstyle{{3}}$}}}}}$}}\varepsilon_{bc}(\gamma^{c})_{\gamma}{}^{\rho}{\overline{\nabla}}_{\alpha}{}^{i}{\overline{\nabla}}_{\rho}{}^{p}{\overline{\nabla}}^{(2)}_{kp}\,+\,B(\gamma_{b})_{\alpha\gamma}{\overline{\nabla}}^{(2)}{}^{i}{}_{k}\,-\,\hbox{\large{${{\textstyle{{1}\over\vphantom{2}\smash{\raise 0.72331pt\hbox{$\scriptstyle{{2}}$}}}}}$}}B\delta^{i}_{k}(\gamma^{3}\gamma_{b})_{\alpha\gamma}(\gamma^{3})^{\rho\tau}{\overline{\nabla}}^{(2)}_{\rho\tau}\Big{)}\overline{U}}$
---|---
| $\displaystyle{{}~{}~{}~{}~{}-~{}\nabla_{\gamma
k}J_{\alpha}{}^{i}{}_{b}~{}+~{}T_{b}{}_{\gamma
k}{}^{\delta}{}_{l}J_{\delta}{}^{l}{}_{\alpha}{}^{i}~{}~{},}$
$None$

and progress is achieved in analyzing this identity by noting that it holds

$\displaystyle{{\overline{\nabla}}_{\alpha}{}^{i}{\overline{\nabla}}_{\beta}{}^{k}{\overline{\nabla}}^{(2)}_{jk}\overline{U}~{}=~{}\Big{(}{\textstyle{1\over\vphantom{2}\smash{\raise
0.60275pt\hbox{$\scriptstyle{2}$}}}}C_{\alpha\beta}{\overline{\nabla}}^{(2)}{}^{ik}{\overline{\nabla}}^{(2)}_{jk}~{}+~{}{\textstyle{1\over\vphantom{2}\smash{\raise
0.60275pt\hbox{$\scriptstyle{2}$}}}}C^{ik}{\overline{\nabla}}^{(2)}_{\alpha\beta}{\overline{\nabla}}^{(2)}_{jk}~{}-~{}2B(\gamma^{3})_{\alpha\beta}C^{ip}{\overline{\nabla}}^{(2)}_{pj}\Big{)}\bar{U}~{}~{}~{}.}$
---
$None$

One other identity tells us

$\displaystyle{{\overline{\nabla}}^{(2)}_{\alpha\beta}{\overline{\nabla}}^{(2)}_{ij}\bar{U}~{}=~{}-2B(\gamma^{3})_{\alpha\beta}{\overline{\nabla}}^{(2)}_{ij}\overline{U}~{}~{}~{},}$
---
$None$

so that (A.20) becomes

$\displaystyle{{~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}}{\overline{\nabla}}_{\alpha}{}^{i}{\overline{\nabla}}_{\beta}{}^{k}{\overline{\nabla}}^{(2)}_{jk}\overline{U}~{}}$ | $\displaystyle{{}=~{}\Big{(}{\textstyle{1\over\vphantom{2}\smash{\raise 0.60275pt\hbox{$\scriptstyle{2}$}}}}C_{\alpha\beta}{\overline{\nabla}}^{(2)}{}^{ik}{\overline{\nabla}}^{(2)}_{jk}~{}-~{}3B(\gamma^{3})_{\alpha\beta}C^{ip}{\overline{\nabla}}^{(2)}_{pj}\Big{)}\overline{U}}$
---|---
|
$\displaystyle{{}=~{}\Big{(}\hbox{\large{${{\textstyle{{1}\over\vphantom{2}\smash{\raise
0.72331pt\hbox{$\scriptstyle{{4}}$}}}}}$}}\,\delta_{j}{}^{i}\,C_{\alpha\beta}{\overline{\nabla}}^{(2)}{}^{kl}{\overline{\nabla}}^{(2)}_{kl}~{}-~{}3B(\gamma^{3})_{\alpha\beta}C^{ip}{\overline{\nabla}}^{(2)}_{pj}\Big{)}\overline{U}}$
|
$\displaystyle{{}=~{}\Big{(}\hbox{\large{${{\textstyle{{3}\over\vphantom{2}\smash{\raise
0.72331pt\hbox{$\scriptstyle{{4}}$}}}}}$}}\,\delta_{j}{}^{i}\,C_{\alpha\beta}\,{\overline{\nabla}}^{(4)}~{}-~{}3B(\gamma^{3})_{\alpha\beta}C^{ip}{\overline{\nabla}}^{(2)}_{pj}\Big{)}\overline{U}~{}~{}~{}.}$
$None$

where on the first term we have used a sequence of identities (see also the
final appendix). The final line of (A.22) can now be substituted into (A.19)
to yield after a bit of algebra

$\displaystyle{{~{}~{}~{}~{}~{}~{}~{}}2{\rm i}\delta^{i}_{k}(\gamma^{c})_{\alpha\gamma}J_{bc}~{}}$ | $\displaystyle{{}=~{}{\rm i}\Big{(}\,-\hbox{\large{${{\textstyle{{1}\over\vphantom{2}\smash{\raise 0.72331pt\hbox{$\scriptstyle{{4}}$}}}}}$}}\varepsilon_{bc}(\gamma^{c})_{\alpha\gamma}\delta^{i}_{k}{\overline{\nabla}}^{(4)}~{}+~{}\hbox{\large{${{\textstyle{{1}\over\vphantom{2}\smash{\raise 0.72331pt\hbox{$\scriptstyle{{2}}$}}}}}$}}B\varepsilon_{bc}\delta^{i}_{k}(\gamma^{c})_{\alpha\gamma}(\gamma^{3})^{\rho\tau}{\overline{\nabla}}^{(2)}_{\rho\tau}\,\Big{)}\overline{U}}$
---|---
| $\displaystyle{{}~{}~{}~{}~{}-\nabla_{\gamma
k}J_{\alpha}{}^{i}{}_{b}+T_{b}{}_{\gamma
k}{}^{\delta}{}_{l}J_{\delta}{}^{l}{}_{\alpha}{}^{i}~{}~{}~{}.}$
$None$

Finally this equation informs us that

$\displaystyle{J_{ab}~{}=~{}\varepsilon_{ab}\Big{(}\,-\hbox{\large{${{\textstyle{{1}\over\vphantom{2}\smash{\raise
0.72331pt\hbox{$\scriptstyle{{8}}$}}}}}$}}{\overline{\nabla}}^{(4)}~{}+~{}\hbox{\large{${{\textstyle{{1}\over\vphantom{2}\smash{\raise
0.72331pt\hbox{$\scriptstyle{{4}}$}}}}}$}}B(\gamma^{3})^{\alpha\beta}{\overline{\nabla}}^{(2)}_{\alpha\beta}\,\Big{)}\overline{U}~{}+~{}{\rm
h.c.}}$
---
$None$

There remains one final Bianchi Identity of the form

$\displaystyle{{~{}~{}~{}}0~{}}$ | $\displaystyle{{}=~{}\nabla_{\alpha i}J_{bc}-\nabla_{b}J_{\alpha i}{}_{c}~{}+~{}\nabla_{c}J_{\alpha i}{}_{b}~{}+~{}T_{\alpha ib}{}^{D}J_{Dc}~{}+~{}T_{\alpha ic}{}^{D}J_{Db}~{}-~{}T_{bc}{}^{\delta l}J_{\delta l\alpha i}~{}~{}~{},}$
---|---
$\displaystyle{0~{}}$ | $\displaystyle{{}=~{}\varepsilon^{ab}\Big{(}\nabla_{\alpha i}J_{ab}~{}+~{}2\nabla_{a}J_{b\alpha i}~{}-~{}2T_{\alpha ia}{}^{\delta l}J_{\delta l}{}_{b}~{}-~{}2T_{\alpha ia}{}^{\delta}{}_{l}J_{\delta}{}^{l}{}_{b}~{}-~{}T_{ab}{}^{\delta l}J_{\delta l\alpha i}\Big{)}~{}~{}~{}.}$
$None$

To prove this identity is satisfied requires a calculation of some length. The
key to its satisfaction requires one final identity

$\displaystyle{{~{}~{}~{}~{}~{}~{}~{}~{}~{}}[\nabla_{\alpha i},{\overline{\nabla}}^{(4)}]\bar{U}}$ | $\displaystyle{{}=~{}\Big{(}~{}-~{}\hbox{\large{${{\textstyle{{8{\rm i}}\over\vphantom{2}\smash{\raise 0.72331pt\hbox{$\scriptstyle{{3}}$}}}}}$}}\,(\gamma^{a})_{\alpha}{}^{\rho}\nabla_{a}{\overline{\nabla}}_{\rho}{}^{p}{\overline{\nabla}}^{(2)}_{ip}~{}-~{}8{\rm i}B\varepsilon_{bc}(\gamma^{b})_{\alpha}{}^{\beta}\nabla^{c}{\overline{\nabla}}_{\beta i}}$
---|---
|
$\displaystyle{{}~{}~{}~{}~{}~{}~{}~{}+\hbox{\large{${{\textstyle{{8}\over\vphantom{2}\smash{\raise
0.72331pt\hbox{$\scriptstyle{{3}}$}}}}}$}}\,\phi_{\alpha}{}^{\gamma}{\overline{\nabla}}_{\gamma}{}^{p}{\overline{\nabla}}^{(2)}_{ip}~{}+~{}8\Sigma_{\alpha}{}^{l}{\overline{\nabla}}^{(2)}_{il}~{}\Big{)}\overline{U}}$
$None$

that is valid for the supergravity covariant derivative acting on a anti-
chiral scalar superfield such as $\bar{U}$.

Other Bianchi identities, not explicitly mentioned here, are identically
solved by complex conjugation of the results obtained in this section.

Appendix B: Miscellaneous Identities

For the reader convenience, here we also collect some useful formulas used in
the derivations provided in this paper and especially in Appendix A (we remind
that $\bar{U}$ is anti-chiral)

$\displaystyle{{~{}~{}~{}~{}~{}~{}}{\overline{\nabla}}_{\alpha}{}^{i}{\overline{\nabla}}_{\beta}{}^{j}\,}$ | $\displaystyle{{}=~{}\hbox{\large{${{\textstyle{{1}\over\vphantom{2}\smash{\raise 0.72331pt\hbox{$\scriptstyle{{2}}$}}}}}$}}C_{\alpha\beta}{\overline{\nabla}}^{(2)}{}^{ij}~{}+~{}\hbox{\large{${{\textstyle{{1}\over\vphantom{2}\smash{\raise 0.72331pt\hbox{$\scriptstyle{{2}}$}}}}}$}}C^{ij}{\overline{\nabla}}^{(2)}_{\alpha\beta}~{}+~{}BC_{\alpha\beta}C^{ij}{\cal M}~{}-~{}B(\gamma^{3})_{\alpha\beta}{\cal Y}^{ij}~{}~{}~{},}$
---|---
$None$ $\displaystyle{{[}\nabla_{\alpha i},{\overline{\nabla}}^{(2)}_{jk}{]}\overline{U}\,}$ | $\displaystyle{{}=~{}-2{\rm i}C_{i(j}(\gamma^{c})_{\alpha}{}^{\delta}\nabla_{c}{\overline{\nabla}}_{\delta k)}\overline{U}~{}~{}~{},{~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}}}$
---|---
$None$ $\displaystyle{{[}\nabla_{\alpha i},{\overline{\nabla}}^{(2)}_{\delta\rho}{]}\overline{U}\,}$ | $\displaystyle{{}=~{}\Big{(}-2{\rm i}(\gamma^{c})_{\alpha(\delta}\nabla_{c}{\overline{\nabla}}_{\rho)i}-G(\gamma^{3})_{\alpha(\delta}(\gamma^{3})_{\rho)}{}^{\gamma}{\overline{\nabla}}_{\gamma i}+GC_{\alpha(\delta}{\overline{\nabla}}_{\rho)i}}$
---|---
| $\displaystyle{{}{~{}~{}~{}~{}~{}~{}\,}-\hbox{\large{${{\textstyle{{{\rm
i}}\over\vphantom{2}\smash{\raise
0.72331pt\hbox{$\scriptstyle{{2}}$}}}}}$}}HC_{\alpha(\delta}(\gamma^{3})_{\rho)}{}^{\tau}{\overline{\nabla}}_{\tau
i}~{}+~{}\hbox{\large{${{\textstyle{{{\rm i}}\over\vphantom{2}\smash{\raise
0.72331pt\hbox{$\scriptstyle{{2}}$}}}}}$}}H(\gamma^{3})_{\alpha(\delta}{\overline{\nabla}}_{\rho)i}-{\rm
i}H(\gamma^{3})_{\delta\rho}{\overline{\nabla}}_{\alpha
i}\Big{)}\overline{U}~{},}$
$None$ $\displaystyle{{[}\nabla_{\alpha i},(\gamma^{3})^{\delta\rho}{\overline{\nabla}}^{(2)}_{\delta\rho}{]}\overline{U}\,}$ | $\displaystyle{{}=~{}-4{\rm i}\varepsilon^{bc}(\gamma_{b})_{\alpha}{}^{\beta}\nabla_{c}{\overline{\nabla}}_{\beta i}\overline{U}~{}~{}~{},{~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}}}$
---|---
$None$ $\displaystyle{{\overline{\nabla}}_{\alpha i}{\overline{\nabla}}^{(2)}_{jk}\overline{U}\,}$ | $\displaystyle{{}=~{}-\hbox{\large{${{\textstyle{{1}\over\vphantom{2}\smash{\raise 0.72331pt\hbox{$\scriptstyle{{3}}$}}}}}$}}C_{i(j}{\overline{\nabla}}_{\alpha}{}^{p}{\overline{\nabla}}^{(2)}_{k)p}\overline{U}~{}~{}~{},{~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}}}$
---|---
$None$ $\displaystyle{{\overline{\nabla}}_{\alpha i}{\overline{\nabla}}^{(2)}_{\beta\gamma}\overline{U}\,}$ | $\displaystyle{{}=~{}{1\over 3}C_{\alpha(\beta}{\overline{\nabla}}_{\gamma)}{}^{p}{\overline{\nabla}}^{(2)}_{ip}\overline{U}-\hbox{\large{${{\textstyle{{4}\over\vphantom{2}\smash{\raise 0.72331pt\hbox{$\scriptstyle{{3}}$}}}}}$}}BC_{\alpha(\beta}(\gamma^{3})_{\gamma)}{}^{\delta}{\overline{\nabla}}_{\delta i}\overline{U}+\hbox{\large{${{\textstyle{{1}\over\vphantom{2}\smash{\raise 0.72331pt\hbox{$\scriptstyle{{3}}$}}}}}$}}B(\gamma^{3})_{(\alpha\beta}{\overline{\nabla}}_{\gamma)i}\overline{U}~{},}$
---|---
$None$ $\displaystyle{(\gamma^{3})^{\beta\gamma}{\overline{\nabla}}_{\alpha i}{\overline{\nabla}}^{(2)}_{\beta\gamma}\overline{U}\,}$ | $\displaystyle{{}=~{}-\hbox{\large{${{\textstyle{{2}\over\vphantom{2}\smash{\raise 0.72331pt\hbox{$\scriptstyle{{3}}$}}}}}$}}(\gamma^{3})_{\alpha}{}^{\gamma}{\overline{\nabla}}_{\gamma}{}^{p}{\overline{\nabla}}^{(2)}_{ip}\overline{U}~{}~{}~{},{~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}}}$
---|---
$None$ $\displaystyle{{\overline{\nabla}}^{\gamma}{}_{i}{\overline{\nabla}}^{(2)}_{\alpha\gamma}\overline{U}\,}$ | $\displaystyle{{}=~{}-{\overline{\nabla}}_{\alpha}{}^{p}{\overline{\nabla}}^{(2)}_{ip}\overline{U}+4B(\gamma^{3})_{\alpha}{}^{\delta}{\overline{\nabla}}_{\delta i}\overline{U}~{}~{}~{},{~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}}}$
---|---
$None$ $\displaystyle{{}\\{\nabla_{\alpha
i},{\overline{\nabla}}_{\delta}{}^{p}{\overline{\nabla}}^{(2)}_{kp}\\}\overline{U}=\Bigg{(}3{\rm
i}(\gamma^{a})_{\alpha\delta}\nabla_{a}{\overline{\nabla}}^{(2)}_{ik}-3{\rm
i}C_{ik}(\gamma^{a})_{\alpha}{}^{\rho}\nabla_{a}{\overline{\nabla}}^{(2)}_{\delta\rho}}$
---
$\displaystyle{{}{~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\,}+\hbox{\large{${{\textstyle{{3}\over\vphantom{2}\smash{\raise
0.72331pt\hbox{$\scriptstyle{{2}}$}}}}}$}}C_{ik}{\phi}^{\tau}{}_{\delta}(\gamma^{3})_{\alpha\tau}(\gamma^{3})^{\rho\beta}{\overline{\nabla}}^{(2)}_{\beta\rho}-3\phi_{\alpha\delta}{\overline{\nabla}}^{(2)}_{ik}}$
$\displaystyle{{}{~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\,}+6C_{ik}C_{\alpha\delta}\Sigma^{\beta
p}{\overline{\nabla}}_{\beta
p}+6C_{ik}(\gamma^{3})_{\alpha\delta}(\gamma^{3})^{\beta\gamma}\Sigma_{\beta}{}^{p}{\overline{\nabla}}_{\gamma
p}\Bigg{)}\overline{U}~{},}$
$None$ $\displaystyle{{~{}}{\overline{\nabla}}^{(2)}_{\alpha\beta}{\overline{\nabla}}^{(2)}_{ij}\overline{U}\,}$ | $\displaystyle{{}=~{}-2B(\gamma^{3})_{\alpha\beta}{\overline{\nabla}}^{(2)}_{ij}\overline{U}~{}~{}~{},{~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\,}{~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\,}}$
---|---
$None$ $\displaystyle{{\overline{\nabla}}^{(2)}{}^{\alpha\beta}{\overline{\nabla}}^{(2)}_{\alpha\beta}\overline{U}\,}$ | $\displaystyle{{}=~{}-{\overline{\nabla}}^{(2)}{}^{ij}{\overline{\nabla}}^{(2)}_{ij}\overline{U}-4B(\gamma^{3})^{\alpha\beta}{\overline{\nabla}}^{(2)}_{\alpha\beta}\overline{U}~{}~{}~{},{~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\,}{~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\,}}$
---|---
$None$ $\displaystyle{{\overline{\nabla}}^{(4)}\overline{U}\,}$ | $\displaystyle{{}:=~{}-\hbox{\large{${{\textstyle{{1}\over\vphantom{2}\smash{\raise 0.72331pt\hbox{$\scriptstyle{{3}}$}}}}}$}}{\overline{\nabla}}^{(2)}{}^{kl}{\overline{\nabla}}^{(2)}_{kl}\overline{U}~{}~{}~{},{~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\,}{~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\,}}$
---|---
$None$ $\displaystyle{{\overline{\nabla}}_{\alpha}{}^{i}{\overline{\nabla}}_{\beta}{}^{k}{\overline{\nabla}}^{(2)}_{jk}\overline{U}\,}$ | $\displaystyle{{}=~{}\Big{(}\,\hbox{\large{${{\textstyle{{3}\over\vphantom{2}\smash{\raise 0.72331pt\hbox{$\scriptstyle{{4}}$}}}}}$}}C_{\alpha\beta}\delta^{i}_{j}{\overline{\nabla}}^{(4)}-3B(\gamma^{3})_{\alpha\beta}{\overline{\nabla}}^{(2)}{}^{i}{}_{j}\Big{)}\overline{U}~{}~{}~{},{~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\,}{~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\,}}$
---|---
$None$
$\displaystyle{{\overline{\nabla}}_{\alpha}{}^{i}\big{(}{\overline{\nabla}}^{(2)}{}^{\gamma\delta}-2{B}(\gamma^{3})^{\gamma\delta}\big{)}{\overline{\nabla}}^{(2)}_{\gamma\delta}\overline{U}\,~{}=~{}0~{},}$
---
$None$ $\displaystyle{[\nabla_{\alpha i},{\overline{\nabla}}^{(4)}]\overline{U}\,}$ | $\displaystyle{{}=~{}\Bigg{(}-\hbox{\large{${{\textstyle{{8{\rm i}}\over\vphantom{2}\smash{\raise 0.72331pt\hbox{$\scriptstyle{{3}}$}}}}}$}}(\gamma^{c})_{\alpha}{}^{\beta}\nabla_{c}{\overline{\nabla}}_{\beta}{}^{k}{\overline{\nabla}}^{(2)}_{ik}-8{\rm i}B\varepsilon^{ab}(\gamma_{a})_{\alpha}{}^{\delta}\nabla_{b}{\overline{\nabla}}_{\delta i}}$
---|---
|
$\displaystyle{{}{~{}~{}~{}~{}~{}~{}~{}~{}}+8\Sigma_{\alpha}{}^{j}{\overline{\nabla}}^{(2)}_{ij}+\hbox{\large{${{\textstyle{{8}\over\vphantom{2}\smash{\raise
0.72331pt\hbox{$\scriptstyle{{3}}$}}}}}$}}\phi_{\alpha}{}^{\gamma}{\overline{\nabla}}_{\gamma}{}^{k}{\overline{\nabla}}^{(2)}_{ik}\Bigg{)}\overline{U}~{}.~{}~{}~{}~{}~{}~{}~{}~{}~{}}$
$None$

By complex conjugation, the reader can derive an analogue set of equations for
the chiral superfield $U$.

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