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Distilled from unpublished aspects of my PhD thesis from Kings College under the tutilage of Professor David Robinson
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two spinors, differential forms, Noether charge, symplectic, pre-symplectic form, Witten-Nester form, Einstein-Maxwell Lagrangian, Grassman,Weyl spinor, chiral, Ashtekar variables
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\begin{document}
\title{Symplectic structures and chiral formulations
of Einstein-matter equations}%
\author{Dr Lee McCulloch-James}%
\address{Senior Support Analyst, Barra International}
\address{29 South Hill Park, Hampstead, London NW3 2ST, UK}%
\email{lee.mcculloch@barra.com}%
\thanks{}%
\subjclass{}%
\keywords{two spinors, Noether charge, symplectic, Witten-Nester form, BF theory, Einstein-Maxwell Lagrangian, Grassman, Plebanski,Weyl spinor, chiral}%
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\begin{abstract}
Equivalent chiral Lagrangians for (complex) Einstein
equations
are collected and related by their
identical (infinite dimensional) symplectic structures. Viewing
Plebanski's Lagrangian, which employs an $sl(2,C)$-valued field
variable alternative to the tetrad as a constrained BF theory,
extensions to other (super) gauge groups follow naturally.
Accordingly, scalar and spinor matter field
couplings to an
Einstein-Maxwell Lagrangian employing $gl(2,C)$-valued field variables are investigated.
Employing instead, spin $\thalf$ fields
as dynamical variables and symplectic techniques
a succinct derivation of the Witten-Nester two-form as a Noether
charge is realised.\\
\end{abstract}
\maketitle
% ----------------------------------------------------------------
\section{Introduction}
The renewed interest in re-formulating Einstein's field equations
has been stimulated by influences
such as the developments in the study of gauge theories, the
construction of half-flat solutions in the 1980's by, for example,
Penrose, Newman and Plebanski \cite{Pleb1} and the recasting of
the Hamiltonian formulation of general relativity in terms of new
variables by Ashtekar \cite{Ash1}. The latter, itself a response
to the first two influences, reintroduced the idea of regarding
the connection and bases of two forms as primary dynamical
variables with the metric a secondary derived variable. The
novelty of such schemes is the focus on so-called complex chiral
actions and complex versions of Einstein's equations.
Of these investigations Plebanski's, \cite{Pleb1} formulation in
the mid 1970's of a chiral first order variational principle for
general relativity in which the basic field variables are
$sl(2,C)$ valued two-form, connection form and a spinor-valued
zero-form is of particular interest. The chiral nature of the
formulation is in the sense that local Lorentz representations
involve only $SL(2,C)$ and not its conjugate,
$\overline{SL(2,C)}$. Viewed as a `constrained' BF theory this
lends itself to natural generalisations as in the Einstein-Maxwell
theory of Robinson, \cite{DC1} that employs $sl(2,C)\oplus C$
field variables. Constrained here is in the sense that a
multiplier term is needed (spoiling the precise BF form of the
Lagrangian)
to ensure that the basic two form field variable is derived from
a co-frame. Chiral N=1 and N=2 Supergravity theories, \cite{Kun},
\cite{Eza}, \cite{JaSD}
amongst others have been formulated. Such chiral formulations lead to complex vacuum
field equations for a complex metric with the real theory
recovered only upon imposing (by hand) reality conditions.
Included in subsequent developments have been chiral Lagrangian
formulations, $\lL_{\thalf}$
in which the
primary variables are spin $3\over 2$ fields and $sl(2,C)$
valued connections, \cite{DC4, Wall1, Tu}. Goldberg, \cite{Goldberg}
using a Witten-Nester two form associated to the
anti-self-dual part of the Levi-Civita connection derived
also semi-chiral Lagrangian.\\
Einstein's field equations as Euler-Lagrange equations
for some Lagrangian field theory have associated Noether
identities so nothing is gained by looking at their
integrability conditions that are automatically satisfied
due to their associated Bianchi identities.
Accordingly in a complementary development it has proven useful to view Einsteins
equations in another way asking for equations that
have Einstein's field equations themselves as integrability conditions.
Penrose, \cite{Pen5} and his collaborators' twistorial approach to Einstein's
vacuum field equations has focused on the spin ${3\over 2}$ zero
rest-mass field equations which can have the Einstein vacuum
equations, with or without cosmological constant, as integrability
conditions.\\
The structure of this article explores certain aspects of these
investigations by deploying a phase space construction of Crnkovic
and Witten
\cite{Wit2}. Using
their invariant symplectic 2-form associated to the phase space of
solutions to the equations of motion one may readily relate the
various chiral Lagrangian formulations. In this scheme, the (pre)
symplectic
potential is transcribed
as the boundary term
that results from the `integration by parts'
when implementing the variation of the Lagrangian,
while the (pre)symplectic
2-form structure is its functional exterior derivative.
In general this two-form
is degenerate and motions along these degenerate directions correspond to
gauge transformations of the theory, \cite{Bomb}. It is possible however to
quotient this phase space by the
integral manifolds of the degenerate directions and obtain a natural
(non-degenerate) symplectic structure on the resulting phase space,
\cite{Luo}.\\
In Chapter 2 a
presentation of the Einstein-Cartan equations in terms of two
spinor valued three form equations introduces the notation.
Further a clarification of the semi-chiral Lagrangian formulation
of the Einstein-Weyl equations, \cite{JaF} is made with a view to
applying the results to the work of Robinson, \cite{DC1}.\\
The recent proliferation of formulations of vaccuum
Einstein equations has been reviewed by Peldan, \cite{Peldanrev}
emphasising the Hamiltonian aspects of the theories. In Chapter 3
the Lagrangian formulations are collected and related by their
identical symplectic structures, the general theory of which is
also reviewed.\\
Work on chiral variational principles has been extended to include
various matter fields by Capovilla et al \cite{JaSD} and Pillin,
\cite{Pillin}. Following their prescriptions, the effects of
coupling (charged) Higgs and fermionic matter fields to the
$GL(2,C)$ formulation of electromagnetism and gravity by Robinson,
\cite{DC1} are presented in Chapter 4. In doing so the necessity
to restrict the gauge group to $SL(2,C)\otimes U(1)$ in order to
recover the real theory is clarified, as is how the formulation is
distinct from those theories which use a linear connection to
incorporate Maxwell, \cite{Ma}. The notion of a spinor density is
employed to describe the charged complex (Higgs) scalar field and
the theory as developed previously by Plebanski, \cite{Pleb2} is
summarised in the appendix.
\\
In Chapter 5, by employing symplectic techniques and the chiral
Lagrangian of Jacobson et al, \cite{Tu} its associated quasi-local
charges are derived and their relation to the Witten-Nester forms,
\cite{PenW2}
is clarified thus providing an alternative derivation to that in \cite{Tu} that used a canonical
approach and the ambiguously defined notion of a Lie deriviative (with respect to a timelike vector field)
of spinor-valued forms.
\\
Included in the appendices are some useful spinor decompositions. In the context of BF-like
Lagrangian formulations boundary condition
considerations and N=2 Supergravity are also discussed.
Further illustration of the symplectic techniques and the
formulation of chiral Lagrangians is provided in the presentation of a
chiral Lagrangian for
spin $\thalf$ fields propogating on a (fixed) curved backgroung
spacetime. The solution
space of the Lagrangian for the Rarita-Schwinger equations is
studied, \cite{Fraud1},
in which `charges' \cite{Pen1,Pen2} are obtained in a way
consistent with Einstein's equations.
\section{Two Spinor-valued formulations of Einstein-Cartan Equations }
\input{defin}
Derived from the usual Palatini formalism where an orthonormal
frame $\theta^a$ is used, the field equations of a metric theory
(in which
the constraint that
the dynamical connection is a metric compatible connection
$Q_{ab}=-\ ^\Gamma\nabla g_{ab}=0$ is put in by hand) read
\eqa
\ ^\star\FF^a{}_{b}\we\theta^b&=&-8\pi{} T^a,\label{eq:ec1}\\
\
^\Gamma\nabla\eta^{ab}&=&\eta^{abc}\we\Theta_c=-8\pi{}\tau^{ab}\label{eq:ec2},
\eeqa
where the components of the energy-momentum, $T_{ab}$ and spin
tensors, $\tau_{abc}$ are given in terms of the three forms
$T_{a}=T_{ab}\eta^b$, $\tau_{ab}=\tau_{abc}\eta^c$ with $\eta^a=\
^*\theta^a$. As Euler-Lagrange equations of a Lagrangian field
theory, these Einstein-Cartan equations lend themselves to many
alternative kinematical descriptions. Indeed they may be
considered as a conservation law for a certain
Sparling 3-form
defined on the bundle of orthonormal frames over spacetime, $M$. They
are most elegantly presented as
two spinor-valued differential 3-form equations,
\eqa \
^\Gamma\nabla(\theta^A{}_{A'}\we\theta^{BA'})&=&{\sigma}^{AB},
\label{eq:the2}\\ \fF^A{}_B\we\theta^{BA'}&=&{
S}^{AA'},\label{eq:the3} \eeqa where ${\sigma}^{AB}$ and ${
S}^{AA'}$ are known source quantities\footnote{Defining the dual
to $\theta^{AA'}$ as
$\eta^{AA'}&=\frac{i}{3}(\theta^{AB'}\we\theta^{BA'}\we\theta_{BB'})$
explicitly the terms read
\begin{align} \sigma^{AB}&=-4\pi
i\tau^{AB}{}_{CC'}\eta^{CC'}\quad S^{AA'}=-4\pi
iT^{AA'}{}_{CC'}\eta^{CC'}+\half\ ^\Gamma\nabla\Theta^{CC'}.
\end{align}}
and $\theta^{AA'}=\theta^{AA'}{}_\mu dx^\mu$ is a Hermitian
matrix-valued one-form so that the (real Lorentian) metric is
given by
\eqa
ds^2=\epsilon_{AB}\epsilon_{A'B'}\theta^{AA'}\otimes\theta^{BB'}\label{eq:compmetric}
\eeqa and where for real general relativity the soldering functor
is required to be real,
$\overline{\theta_{\mu}{}^{AA'}}=\theta_{\mu}{}^{AA'}$.
With $\Gamma^A{}_B$ and $\bar{\Gamma}^{A'}{}_{B'}$ (complex
conjugate) $sl(2,\CC)$-valued connection one-forms and the torsion
two form denoted as $\Theta^{AA'}$, the first Cartan structure
equation reads
\eqa
\Theta^{AA'}&:=&d\thet^{AA'}-\thu{AB'}\we\bar{\Gamma}^{A'}{}_{B'}-\thu{BA'}\we\Gamma^A{}_B,\nn\\
&=&\nabla\thu{AA'},\label{eq:C1} \eeqa where $\nabla\equiv\
^\Gamma\nabla$ denotes the exterior covariant derivative with
respect to the $sl(2,\CC)$-valued connection(s). Defining the
basis of anti-self dual two-forms as \eqa
\Sigma^{AB}:={1\over2}\theta^A_{\
A^\prime}\wedge\theta^{BA^\prime}\ , \label{eq:sift} \eeqa
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
the second Cartan structure equations, \cite{PenTor} take the
complex form \eqa { F}^A_{\ B}&:=&d\Gamma^A_{\ B}+\Gamma^A_{\
C}\wedge\Gamma^C_{\ B}\nn\\ &=&{\Psi}^A_{\
BCD}\Sigma^{CD}+{\Phi}^A_{\ BC^\prime
D^\prime}\bar\Sigma^{C^\prime D^\prime }+2\Lambda\Sigma^A_{\ B}
+(\chi_{D}{}^{A}\Si_{B}{}^{D}+\chi_{DB}\Si^{AD}),\label{eq:sifC2}
\eeqa
where the curvature two-form, ${ F}^A_{\ B}$, has been
decomposed into spinor fields of dimension 5,9,1 and 3
respectively , corresponding to the anti-self dual part of the
Weyl conformal spinor, $\Psi^A_{\ BCD}$, the spinor representation
of the trace-free part of the Ricci tensor, $-2\Phi^A_{\ B
C^\prime D^\prime}$ and the Ricci scalar $24\Lambda$, - all with
respect to the curvature of the $SL(2,\CC)$ connection and
$\chi^{AB}$ arising from the presence of non-zero torsion.
%%%%%%%%%%%%%%%%%%%
\subsection{Semi-Chiral Lagrangian formulation} Equations
(\ref{eq:the2}) and (\ref{eq:the3}) are two Euler-Lagrange
equations arising from Lagrangians of the general form \eqa
\lL_{Tot}&=&(\Gamma^A{}_B,\theta^{AA'},\rho..),\label{eq:the1}
\eeqa with $\theta^{AA'}$ hermitian and $\rho$ representing some
matter fields. Note that \`{a} priori there is no relation between
the tangent space and the internal space of the vector bundle $B$
associated to the spinor structure\footnote{If a (non compact)
spacetime manifold, $M$ admits a global null tetrad it has a
spinor-structure ${PB}$ defined on it. A spinor structure, $PB$ is
a principal fibre bundle with structure group $SL(2,\CC)$, the
gauge group for spinor dyads. The (real) space-time manifold
carries a $SL(2,\CC)$ spin (trivial vector) bundle, B associated
to $PB$ and its conjugate on it. The tensor product of these two
bundles can be identified with the complexified tangent bundle.
Each fibre, $S\equiv \CC^2$ of B consists of a 2-complex
dimensional
vector space equipped with a symplectic metric, $\epsilon_{AB}$.} over space time, M.
The internal indices $AA'$
only acquire the interpretation as spinor indices through the
dynamical soldering form, $\theta^{AA'}{}_\mu$ and the internal
`symplectic metric', $\epd{AB}$ is given as fixed so that the
internal $SL(2,\CC)$ connection is then traceless
$\Gamma_{AB}=\Gamma_{BA}$ due to $ \nabla\epd{AB}=0.$ The internal
$SO(1,3)_{\CC}^-\cong SL(2,\CC)$ connection is not associated to
the tangent bundle and is thus not a linear connection but a
spinor connection. The variation of the Lagrangian with respect to
$\Gamma^A{}_B$ will determine this connection in terms of the
co-frame so that the bundle $B$ can then be considered soldered to
$M$. The co-frame variation evaluated at the particular value of
the connection just determined gives equations for the co-frames
only. There exists a unique Levi Civita connection, $\omega$ (with
curvature $\Omega$) so the $sl(2,C)$ connection, $\Gamma$ can be
decomposed according to \eqa
\Gamma^A{}_B&=&\omega^A{}_B+K^A{}_B,\\ \fF^A{}_B&=&\Omega^A{}_B+\
^\omega\nabla K^A{}_B+ K^A{}_C\we K_{B}{}^{C},\label{eq:the6}
\eeqa
where
$K^A{}_B$ is the contorsion one form, irreducibly written (as
described in the appendix) in terms of totally symmetric and
`axial' parts as
\begin{align}
K_{AB}&=-\frac{1}{2}\sigma_{ABCC'}\theta^{CC'}+
2\AT_{(A|C'|}\theta_{B)}{}^{C'}.
\label{eq:contort}
\end{align}
The Einstein-Matter equations
owing to the triviality of the Bianchi Identity
\eqa
^\omega\nabla\Theta^{AA'}=\Omega^A{}_B\we\theta^{BA'}+\Omega^{A'}{}_{B'}\we\theta^{AB'}=0,
\label{eq:Bianchi} \eeqa
have the simpler form, \eqa
^\omega\nabla(\theta^A{}_{A'}\we\theta^{BA'})&=&0,\label{eq:the4}\\
-2i\Omega^A{}_B\we\theta^{BA'}&=&-8\pi{T}^{AA'}.\label{eq:the5}
\eeqa It is possible to rewrite (\ref{eq:the2}) and
(\ref{eq:the3}) in terms of (\ref{eq:the4}) and (\ref{eq:the5})
where the source term, ${T}^{AA'}$ is then determined by
${\sigma}^{AB}$, ${ S}^{AA'}$ and $\Gamma^A{}_B$. That is, solve
for $K^A{}_B$ from (\ref{eq:the2}) and (\ref{eq:the4}) and
substitute (\ref{eq:the6}) in (\ref{eq:the3}) and replace any
$\Gamma^A{}_B$ in ${ S}^{AA'}$ by $\omega^A{}_B+K^A{}_B$.
\subsection{Semi-Chiral Lagrangian for fermionic fields}
The Lagrangian for fermionic matter has a dependence on the
connection so admits torsion contributions but nevertheless can be
written as the sum of a semi-chiral complex Lagrangian for vaccuum
General Relativity, $\Ll_{SC}(\theta,\Gamma)$, a complex (semi)
chiral fermionic matter Lagrangian, $\lL_{\half}$ and a term,
$\lL_{J^2}$ that ensures the standard Einstein-Weyl form of the
field equations,
\eqa \lL_{SC}(\theta,\Gamma)&=&
i\theta^{A}{}_{A'}\we\theta^{BA'}\we\FF_{AB},\label{eq:lagssj} \\
{ L}_{\half}(\theta,\Gamma,\lambda,\tilde{\lambda})&=&+\eta^{AA'}
\we\tilde{\lambda}_{A'}\DD\lambda_{A}\label{eq:1dd1}\ ,\\
{
L}_{J^2}(\lambda,\tilde{\lambda})&=&\frac{3}{16}
\lambda_{A}\tilde{\lambda}_{A'}\lambda^A\tilde{\lambda}^{A'},\\
\Ll_{Tot}&=&\Ll_{SC}+{ L}_{\half}+ L_{J^2}.
\eeqa
The $\lambda_A(\tilde{\lambda}_{A'})$ are the left (resp.
right)-handed zero forms. The theory uses only the anti-self dual
connection, ${ D}$ (which does not act on tensors, e.g.
$D\theta^{AA'}=d\theta^{AA'}-\theta^{BA'}\we\Gamma^A{}_B$) but is
complete and it turns out, (by varying $K^A{}_B$), that the real
source current, $J_{AA'}=\lambda_A\lambda_{A'}=-J_{A'A}$ supports
only the axial part of the torsion of $\Gamma^A{}_B$,
\begin{align}
K_{AB}&=-\frac{1}{4}J_{C'(A}\theta_{B)}{}^{C'}.
% \intertext{and the imaginary part of $\lL_{\half}$ is (modulo exact forms)}
% \Im(\lL_{\half})
%=-6\hat{\Theta}{}^{AA'}J_{AA'}&
%=-\frac{3}{4}J_{AA'}J^{AA'}=:-\frac{3}{4}J^2.
\end{align}
Because ultimately the real
theory is of interest (where
$\tilde{\lambda}{}_{A'}=\overline{\lambda_A}$ and $\theta$ is
hermitian) it proves useful to extend ${ D}$ to $\nabla$. Although
it is argued that the spin $\half$ field variables can
be taken to be either Grassman [or complex]-valued,
\cite{JaF} in fact the use of complex spin $\half$ fields leads
to a non-standard energy-momentum tensor which includes quartic
spin $\half$ fields and the details of which are included in the
appendix.
%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%Introducing Lags in terms of theta%%%%%%%%
\section{Presymplectic forms and Chiral Vaccuum Lagrangians}\\
\\
Manifestly covariant dynamical descriptions of chiral first order
Lagrangians for gravity are presented in this section by
exploiting the symplectic structure, $\varpi$ on the phase space
of solutions to the equations of motion, ${\varphi}$.\\
Consider a one
parameter family of field configurations, $\phi$, with some
possible internal and tangent space indices suppressed.
A variation of some local function $f[\phi(\lambda)]$ (at a given space-time
point) is denoted by $\delta_\lambda f=\frac{d}{d\lambda}
f[\phi(\lambda)]$.
A (first order) action functional, $S$ is viewed as a scalar
function on the space of all field variables and their first
derivatives, $\Delta$ so that $S:\Delta\rightarrow \Re$. The
variation of a field $\delta\phi$ is a tangent vector
to this space
and the first variation of the action is viewed as an exact form
on the space of all histories
$\delta\phi$
of the exterior derivative of S
\eqa \delta S=\int_M\delta{\lL}(\phi,\nabla\phi)=dS(\delta\phi).\label{eq:act}
\eeqa The presymplectic potential $\vartheta_\Si$ is a 1 form on
${\varphi}$ and defined in terms of the symplectic current $j^a$
as ,
\begin{align}
\vartheta_\Si(\delta\phi)&=\int_\Si j^a
(\delta\phi)\eta_a,\label{eq:varpib}\\ \intertext{so that
(\ref{eq:act}) reads}dS= \int_{\Si}\delta\lL&=\int_{\Si}
E\we\delta\phi+d\vartheta,
\end{align}
for some Cauchy surface, $\Si$ in M in which $E=0$ in the
submanifold, $\varphi\in\Delta$ of solutions to the equations of
motion. The (pre)symplectic 2 form $\varpi$ on $\varphi$ is
defined as the functional exterior derivative of $\vartheta$ \eqa
\varpi(\phi,\delta_\lam\phi,\delta_\eps\phi)=
\delta_\eps\vartheta(\phi,\delta_\lam\phi)-\delta_\lam\vartheta
(\phi,\delta_\eps\phi),\label{eq:syd1}
\eeqa which can be written succinctly as
\eqa
\varpi(\delta\phi)=\delta\vartheta(\delta\phi),
\eeqa
by observing that the closed two form, $\varpi$ is an antisymmetric
tensor field.\footnote{That is, by being additive
and anti-symmetric in its dependence on each of the perturbed fields
(where $\alp$ here represents some tangent and/or internal index),
means that the bilinear product
$\varpi(\phi^\alp,\delta_\lam\phi^\alp,\delta_\eps\phi^{ B})=
\varpi_{\alp{ B}}\delta_\lam\phi^\alp\delta_\eps\phi^{ B}$
is necessarily antisymmetric in $[\alp,{ B}]$.}
From this viewpoint, the linearised solutions of $\phi$ can be regarded as anticommuting (or Grassman valued)
one forms
$\delta\phi(x)$ on the solution space.
For fields satisfying the linearised equations of motion $\delta
E=0$ the symplectic form is closed, $d\varpi=0,$ although the
fields, $\phi$ need not be a solution of the equation of motion
$E(\phi)=0$ for this closure condition to hold.\\
Alternative
chiral Lagrangian formulations to (\ref{eq:lagssj}) are apparent
by observing that, since $\delta^2\mu=0$, the symplectic two form
remains invariant under the addition of a boundary term, $\mu$
(locally constructed from the field variables)
to a Lagrangian $ L$,
\begin{align}
\hat{{\lL}}&={\lL}+d\mu,\label{eq:1bound} \\
\hat{\vartheta}&=\vartheta+\delta\mu,\nn\\
\hat{\varpi}&=\delta\hat{\vartheta}=\delta\vartheta.\nn
\end{align}
Starting then from the Trautman form of the Palatini Lagrangian $
\lL_{EC}=-\half\fF_{ab}\we\eta^{ab}$ with an associated
symplectic potential, $ -\delta\omega_{ab}\we\eta^{ab}$ the alternative
chiral formulations are collected in the table below.
\eqan
\begin{array}{||rl|l||} \hline
\mbox{ Vaccuum Einstein-Cartan Lagrangian}&&\mbox{Boundary Term, $\vartheta$}\\
&& \\ \hline
\lL_{SC}=\lL_{EC}-
\frac{i}{2}d(\theta^a\we\Theta_a)+\frac{i}{2}\Theta^a\we\Theta_a&&
i\delta\omega_{AB}\we\theta^A{}_{A'}\we\theta^{BA'} \\
&& \\ \hline
\lL_{QS}=\lL_{SC}+id(\theta^{AA'}\we{ D}\theta_{AA'})&&i\delta\theta_{AA'} \we{ D}\theta^{AA'} \\ &&\\ \hline
\lL_{CG}=\lL_{SC}+d(\theta_{AA'}\we\ ^\Gamma\bar\sigma^{AA'})&&
2i\delta\omega_{AB}\we\theta^{A}{}_{A'}\we\theta^{BA'}+\delta\theta_{AA'}\we\ ^\omega{}\bar \sigma^{AA'} \\ &&\\ \hline
\end{array}
\eeqan The Lagrangians of \cite{JaSD} and \cite{Goldberg}
respectively here are
\begin{align}
L_{QS}&=iD\theta^{AA'}\we D\theta_{AA'},\\
L_{CG}&=-i\theta^{}_{A'}\we\theta^{CA'}\we\omega^A{}_C\we\omega_{AB},
\end{align}
and the 'superpotential'\footnote{Einstein's field equations
(\ref{eq:the4}), upon pulling out an exact form and using the
first structural equation read
\begin{align}
-id^\omega\bar{\sigma}^{AA'}&=-4\pi i\Xi^{AA'},\nn\\
\intertext{where $\Xi^{AA'}:=T^{AA'}+t^{AA'}$ is the total
(matter+field) energy-momentum 3-form and the Sparling 3-form is}
t^{AA'}&:=\frac{i}{4\pi}(\omega^A{}_B\we\bar{\omega}^{A'}{}_{B'}
\we\theta^{BB'}),
\end{align}
having components of a pseudo-tensor and being exact only for
torsion-free connections that are Ricci flat so that $dt^{AA'}=0$
holds for vaccuum Einstein. Such constructions are peculiar to the
torsion-free connection case because of the Bianchi
identity (\ref{eq:Bianchi}).}
and Witten-Nester type two-form
constructed from the anti-self-dual part of an
$so(1,3)$ connection are defined (with
$\delta^A{}_B=\epsilon_B{}^A$) as
\begin{align}
^\omega{}\bar{\sigma}^{AA'}&:={i}\omega^A{}_B\delta^{A'}{}_{B'}
\theta^{BB'},\label{eq:superpot}\\
^\Gamma\bar{\sigma}^{AA'}&:={i}\Gamma^A{}_B\delta^{A'}{}_{B'}
\we\theta^{BB'}.\label{eq:SparlinG}
\end{align}
The equivalence modulo divergences of these Lagrangians
implies equivalence modulo expressions
for quasilocal conserved charges and choice
of appropriate boundary conditions.
In fact, according to the above argument the Lagrangian,
$\lL_{SC}$ happens to be cohomologous to
\begin{align}
\lL=i\theta^A{}_{B'}\we\theta^{BB'}\we\FF_{AB}-\frac{i}{2}
\Theta^{AA'}\we \Theta_{AA'}\label{eq:CSL}
\end{align}
both yielding vanishing torsion as the chiral breaking connection
field equation is equated to zero. When coupling chiral spinor
matter to the theory, a consistent set of equations fails to
result if the quadratic torsion term is not added. Note also that
Goldberg's Lagrangian $L_{CG}$ may also be viewed as a
teleparallel version of $\lL_{SC}$ with the curvature of the
$sl(2,\CC)$ connection decomposing according to (\ref{eq:the6})
the teleparallel condition,
$\FF^A{}_B\delta^{A'}{}_{B'}+\FF^{A'}{}_{B'}\delta^A{}_B=0$
applied to $\lL_{SC}$ gives accordingly
$\Gamma^A{}_B=0, \ ^{{\Gamma}}\Theta^{AA'}=d\theta^{AA'}$
and $K^A{}_B=-\omega^A{}_B$.
\section{Constrained BF theory Lagrangians}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\label{sec:Bfs}
An alternative chiral Lagrangian for
complex
vaccuum
general relativity is that of
Plenbanksi, \cite{Pleb1} that employs (\ref{eq:ec1}) as the basic variables,
\eqa
\frac{i}{2}\lL_{\Si}(\Si,\Psi,\Gamma)= \{\Si^{AB}\we \FF_{AB}-
\frac{1}{2}\Psi_{ABCD}\Si^{AB}\we\Si^{CD}\}.\label{eq:CDJM}
\eeqa
\\
The latter term, $-\half\Psi_{ABCD}\Si^{AB}\we\Si^{CD}$ forces the
Ricci part of the Curvature two-form to vanish and crucially the
constraint arising from the variation of $\Psi$ dictates that
$\Si^{AB}$ is determined by a tetrad \cite{JaSD}, according to
(\ref{eq:sift})
up to $\overline{SL(2,\CC)}$
% \footnote{In fact up to conformal
%$\overline{GL(2,\CC)}$ transformations.}
transformations
on primed indices. This Lagrangian is incomplete for real General
Relativity in that reality conditions on the $SL(2,C)$ valued two
forms, $\Si^{AB}$ need to be put in by hand, to ensure a real Lorentzian space-time,
\eqa
\Si^{AB}\we\overline{\Si}^{A'B'}=0,\label{eq:p4}
\eeqa
and
\eqa
\Si^{AB}\we\Si_{AB}+
\overline{\Si}^{A'B'}\we\overline{\Si}_{A'B'}=0.\label{eq:p5}
\eeqa
The Lagrangian, (\ref{eq:CDJM}) as such is of the constrained BF
theory form
\begin{align}
S_{BF}({\bf B},\alpha,{\bf F})&=\int Tr({\bf B}\we{\bf F}+\half
Tr[{\bf B}\we \alpha({\bf B})]. \label{eq:BFtr} \end{align} Such a
functional of a $g$-valued ${\bf B}$ field, connection form ${\bf
A}$ and a Lagrange multiplier field, $\alpha$\footnote{Here a
$\sim$ under(over) a kernel
denotes a density of weight-1(+1)}
as
\eqa \alpha^B{}_{A\mu\nu}({\bf
B}):=\ualp^B{}_A{}^C{}_{D\mu\nu\rho\sigma} \tilde{{\bf
B}}{}^{\rho\sigma}{}_{C}{}^D, \eeqa (that will posess symmetries
inherited from its internal indices being saturated by the indices
of commuting 2-forms, ${\bf B}={\bf B}_{\mu\nu} dx^{\mu}\we
dx^\nu}$ and have some imposed by hand in order that the correct
equations of motion are obtained) lends itself to extensions to
other (super) gauge groups.
\subsection{Chiral BF type Lagrangian for
Einstein Maxwell} Ii is in this context that Robinson, \cite{DC1}
has studied the gauge group $g=GL(2,\CC)\cong SL(2,\CC)\otimes\CC$
in order to obtain the complex Einstein-Maxwell equations. With
$GL(2,\CC)$-valued $S^A{}_B$-forms as
the primary field variable derived from the tetrad and determined up to
$\overline{GL(2,\CC)}$
transformations on primed indices and a
$GL(2,\CC)$-valued connection, $\gamma^A{}_B$ the chiral Lagrangian for
Einstein-Maxwell reads
\begin{align}
\frac{i}{2}\lL_S(S,\alpha,\gamma)&=f^A{}_B\we S^B{}_A+
\frac{1}{2}\al_B{}^A{}_D{}^C S^B{}_A\we S^D{}_C.
\end{align}
To ensure, that the metric is
both real and Lorentian further conditions, as in Plebanski's
formulation, (\ref{eq:p4}) and (\ref{eq:p5}) have to be put in by
hand. Note that this work is distinct from that which has tried to
relate Maxwell to linear connections which were non-metric or have
torsion, \cite{Ma} as can be seen by using the soldering form
determined by the metric to construct, from the $GL(2,\CC)$
connection a real linear connection,
\begin{align}
\gamma^a{}_b=\omega^a{}_b+\delta^a{}_b(A+\bar{A}).\label{eq:realtor}
\end{align}
If one does not assume that $A$ is $u(1)$-valued the real part
is non-zero so the real linear connection
above will not be metric and will have torsion.\\
\subsection{Coupling of Charged fermions}
The reality of the Maxwell field arises from restricting the gauge
group to $SL(2,\CC)\otimes U(1)$ as the latter term in
(\ref{eq:realtor}) vanishes when $A$ is assumed to be
$u(1)$-valued\footnote{The Maurer-Cartan form is pure imaginary
for real scalar potential $a$ as $ A(a)=e^{-ia}de^{ia}=ida=:i{\bf
A}$ and $\bar{A}=-i\bf{A}$} and the corresponding real linear
connection, $\gamma^a{}_b$ is just the Levi-Civita connection.
Interestingly, as the following illustrates, this becomes a
necessary condition when coupling charged fermionic matter
according to the prescription of {\cite{JaSD} and \cite{JaPC}.
Consider then
the $GL(2,\CC)$-valued two-form chiral Lagrangian for Einstein-Maxwell with chiral spinor
field source, \eqa {
L}_{S\rho}(S,\gamma,\rho,\tau,\alpha)&=&-2i\lL_S+(S^B{}_A+
\frac{1}{2}\delta^B{}_A S^E{}_E) \we\rho^A\we\tna \lambda_B\nn\\ &
&\mbox{}+\tau_A{}^B{}_C \we S^A{}_B\we
\rho^C+\frac{3}{32}\lambda_C\lambda^C\rho_A\we\rho^B\we
S^A{}_B.\label{eq:sfspi} \eeqa
%__________________________________________
Here $\tna$
represents the exterior covariant derivative associated to the $gl(2,\CC)$-valued
connection $\gamma^A{}_B$. The spin $\half$ fields are Grassman-odd objects and the
chiral spin $\half$ field quartic term
has been included in order that the Einstein-Maxwell-Weyl field equations can be
obtained from a first order Lagrangian. The field equation
arising from the variation of one form $\tau_{ABC}=\tau_{(ABC)}$
means that the right-handed fermion may be represented as a left
handed one form according to
$\rho^C=\theta^{CC'}\tilde{\lambda}_{C'}$. These issued are
discussed extensively in \cite{JaSD} and a related discussion for spin $\thalf$ fields is
included in the appendix.
Now as before since the real theory is ultimately of interest
$\DD$ is extended to $\ ^\Gamma\nabla$ as well as $\ ^\gamma\DD$
to $\ ^\gamma\nabla$ to act on primed and unprimed spinors. The
field equation
resulting from the variation of $\gamma$ is
\begin{align}
\ ^\gamma\nabla\eta^{AA'}&=\
^\Gamma\nabla\eta^{AA'}-\eta^{AA'}(A+\bar{A}),\label{eq:sl2cc}\\
\intertext{where}
\
^\Gamma\nabla\eta^{AA'}&\equiv\frac{3}{4}J^{AA'}\quad\text{and}\quad
K_{ABCC'}=\frac{1}{4}\epd{C(A}J_{B)C'}.
\end{align}
It turns out that the correct charged Weyl equations written in
terms of the connection
$\tilde{\Gamma}^{A}{}_B:=\omega^A{}_B+\delta^A{}_BA$ (and its
complex conjugate) are obtained from the variations of $\lambda$
and $\bar{\lambda}$,
\begin{align}
\ ^{\tilde{\Gamma}}\nabla^{BA'}\bla_{A'}&=0\quad\text{and}\quad \
^{\tilde{\Gamma}} \nabla^C{}_{D'}\lambda_C=0,
\end{align}
only when the gauge group is restricted to $SL(2,\CC)\otimes U(1)$
so that the the latter term in (\ref{eq:sl2cc}) vanishes. The
(real) linear connection, associated to $\ ^\gamma\nabla$ is then
the $SO(1,3)$ connection, $\Gamma^a{}_b$.
% The Maxwell equation with source reads \eqa
%\nabla^{BB'}\phi_{A'B'}=\frac{3i}{2}J^B{}_{A'}. \eeqa\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Coupling of Higgs fields}
The Higgs
field source needs to be a spinorial-density\footnote{A densitised
spinorial object that transforms under $GL(2,\CC)$ with a
characteristic
weighting with the usual $SL(2,\CC)$-spinor object transforming with zero
weight.\cite{Pleb2}} valued zero form when
coupled to this Einstein-Maxwell formulation. Only then can the
covariant derivative $^\gamma{D}$ associated to the $GL(2,\CC)$
connection $\gamma^A{}_B$, defined by (\ref{eq:tildecon}) act on
such a complex scalar field (with no spinorial indices upon which
to contract) according to $\ ^\gamma{D}_\mu\phi\equiv\
^\Gamma{D}_\mu\phi+A\phi$. Consult the appendix for details. The
fields $\pi,\phi$ have weights $(w,\bar{w})=(1,0)$ with complex
conjugates
$\bar{\pi}$ and $\bar{\phi}$ weighted as $(0,1)$ and the Lagrangian reads
\eqa
\Ll_{H}&=&\sqrt{g}\{\bar{\pi}^\mu(^\gamma{D}_\mu\phi)-\bar{\pi}^\mu\pi^\nu
g_{\mu\nu} +\pi^\mu(^\gamma{D}_\mu\bar{\phi})\}.\label{eq:lageph}
\eeqa
\section{Quasi-local charges}
In this section, by deploying covariant
symplectic techniques, the Witten-Nester expression for quasi-local
energy is obtained succinctly from the Lagrangian $\lL_{QS}$ of
\cite{Wall2}
rewritten \cite{Tu} that can be written in terms of two spin $3\over 2$ dynamic
fields\footnote{These fields are subject to the condition that
they need to define a volume
\begin{align}
&\alpha^{(A}\we\beta^{B)}\we\alpha_{A}\we\beta_B\neq 0.
%\intertext{that is} &\alpha^0\we\alpha^1\we\beta^0\we\beta^1\neq
%0.
\end{align}}, $\alpha^A, \beta^A$ as
\eqa \lL_{QS}
&=&2i \DD\beta^A\we\DD\alpha_A.\label{eq:rohslag2}
\eeqa
The spinor potential fields, $\alpha^A,\beta^A$ determine a
conformal class of complex metrics (\ref{eq:compmetric}) where,
\begin{align}
&\theta^{AA'}=o^{A'}\alpha^A+\iota^{A'}\beta^A,\label{eq:thetab}
\end{align}
and the spin dyad satisfies $o_{A'}\iota^{A'}=\chi$. Upon choosing
a spin frame by replacing $\iota^{A'}$ with $\iota^{A'}\chi^{-1}$
they are
\begin{align}
\alpha^A=-\theta^{AA'}\iota_{A'}=\theta^{A0'}\quad\text{and}
\quad\beta^A=\theta^{AA'}o_{A'}=\theta^{A1'}.
\end{align}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Currents \& Noether Charges}\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\label{sec:Noethery}
The symplectic potential is useful for calculating a Hamiltonian that generates
a given infinitesimal canonical transformation,
$\delta_*{\phi}{}$, \cite{Bomb}. The Noether charge arises
out of variations of the field variables, $\delta_*\phi$.
By definition, the
Lie derivative of $\varpi$ by $\delta_*{\phi}{}$ vanishes
while finding a symplectic
potential $\vartheta$ which is indeed Lie-dragged by
$\delta_*{\phi}$ through the identity
\eqa
\pounds_{\delta_*{\phi}}\vartheta\equiv\delta_*{\phi}\hook\varpi+
\delta[\vartheta(\delta_*{\phi})]=0,\label{eq:canLie}
\eeqa
implies that (modulo an additive constant) $H=\vartheta(\delta_*{\phi}{})$ is the unique
Hamiltonian generating the canonical transformation.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\\
Quite generally, without using the equations of motion,
a variation $\delta_*\phi$ implemented by a symmetry transformation
means that $\delta_*\lL=d\tau$, so that
\begin{align}
E\delta_*\phi&=-d(\vartheta_*-\tau),\label{eq:taueqn}
\end{align}
where $\vartheta_*=\vartheta(\phi,\delta_*\phi)$. A Noether
current $J$ corresponding to a weak conservation law is defined as
\begin{align}
J(\phi,\delta_*\phi )&:=\vartheta_* -\tau,\nn \end{align} such
that `on shell' ($E=0$) it is conserved, $dJ\approx 0$ meaning
that for local symmetries there exists a superpotential (charge),
$H$ for which $J\approx dH.$ For the particular case
$\tau=0$ in (\ref{eq:taueqn}), so that $\delta_*\lL=0$ the associated charge
is the `superpotential' (Witten-Nester form), $\sigma$.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Presymplectic Form and Witten-Nester Charge}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\label{sec:quasi}
The symplectic potential associated to
(\ref{eq:rohslag2}) is
\begin{align}
\vartheta_{QS}(\delta\beta^A,\delta\alpha^A)&=
2i\int_\Si(\delta\beta^A\we{ D}\alpha_A+\delta\alpha_A\we{
D}\beta^A).
%\varpi_{QS}(\delta\beta^A,\delta\alpha^A)&=2i\int_\Si\delta\beta_A\we[{
%D} \delta\alpha^A-\alpha^B\we\delta\omega^A{}_B]
%-\delta\alpha_A\we[{
%D}\delta\beta^A-\beta^B\we\delta\omega^A{}_B].
\end{align}
Symmetries are
generated by $
\delta\alpha^A={ D}\xi^A $ and $\delta\beta^A={ D}\eta^A$ for some
spinor-valued zero-forms, $\eta^A,\xi^A$ so that $\sigma_{QS}$ can
be defined as
\begin{align}
\vartheta_{QS}({ D}\eta^A,{ D}\xi^A)&=d\sigma_{QS}.
\end{align}
The charge has the form (extending the action of
$\dd$ to $\nabla$ by conjugation)
\begin{align}
\sigma_{QS}
%&=2i\int_{\partial\Si}(\eta^A{ D}\alpha_A+\xi_A{
%D}\beta^A),\nn\\
&=2i\int_{\partial\Si}-\eta_A\nabla[-\theta^{AA'}\iota_{A'}]+\xi_A\nabla[\theta^{AA'}o_{A'}],\nn\\
&=2i\int_{\partial\Si}(\eta_A\iota_{A'}+\xi_Ao_{A'})\nabla\theta^{AA'}
+(\eta_A\nabla\iota_{A'}+\xi_A\nabla o_{A'})\we\theta^{AA'},\\
\intertext{where since $\vartheta$ is defined `on shell' reads}
\sigma_{QS}&=2i\int_{\partial\Si}(\eta_A\nabla\iota_{A'}+
\xi_A\nabla o_{A'})\we\theta^{AA'}.\label{eq:tungcharge}
\end{align}
Now the covariant form of the Witten-Nester two form
(\ref{eq:superpot}) is given (for some spinor field $\pi_A$) by
\begin{align}
\ ^\omega\sigma^{A'}&\equiv\nabla\pi_A\we\theta^{AA'},\nn\\
&=(d\pi_A-\pi_B\omega^B{}_A)\we\theta^{AA'}.\label{eq:Witcov}
\end{align}
For constant $\pi_A$, equation (\ref{eq:Witcov}) is then given by
(\ref{eq:superpot}). Putting $\eta_A=0$ and $\xi^A=o^A$ in
(\ref{eq:tungcharge}) reproduces the usual spinorial quasilocal
energy expression, \cite{PenW2} \eqa
\sigma_{QS}&=&2i\int_{\partial\Si} o_A\nabla
o_{A'}\we\theta^{AA'}.\label{eq:covSpar} \eeqa Equation
(\ref{eq:covSpar}) is the local covariant expression
for the Witten-Nester two form that depends on the choice
of spin-frame. It is in fact the local expression of the pullback
of the Sparling form to the spin-frame bundle from the bundle of
null tetrads, \cite{Fraud3}.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{Appendices}
%%%%%%%%%%%%%%%%%%%%
\section{$SL(2,\CC)$ Spinor Decompositions}
\subsection{Decomposition of Generalised Weyl Spinor}
\label{sec:genlag} The spinor with the interchange symmetry
$\alpha_{ABCD}=\alpha_{CDAB}$ used as a multiplier field in the $GL(2,\CC)$-form formulation
of complex Einstein-Maxwell, \cite{DC1} has a partial decomposition of
\eqan
\alpha_{ABCD}&=&\alpha_{(AB)(CD)}+
\half\{\epd{AB}\alpha_E{}^E{}_{(CD)}+\alpha_{(AB)E}{}^E\epd{CD}\}
+\frac{1}{4}\epd{AB}\epd{CD}\alpha_E{}^E{}_F{}^F.
\eeqan
The spinor $a_{ABCD}:=\alpha_{(AB)(CD)}$ possesses the symmetries
of $X_{ABCD}$, the curvature spinor \cite{Pen4} for an
$sl(2,\CC)$-valued connection
\begin{align}
X_{ABCD}&={\Psi}_{ABCD}+(\epd{BC}\epd{AD}+\epd{BD}\epd{AC}){\Lambda}
+(\epd{CB}\chi_{AD}+\epd{BA}\chi_{CD}+\epd{DB}\chi_{AC}),\nn
\end{align}
with the additional interchange symmetry $a_{ABCD}=a_{CDAB}$ that
means $\chi_{AD}:=\frac{1}{6}a_{H(AD)}{}^H=0$ so that $a_{ABCD}$
decomposes,
\begin{align}
a_{ABCD}&=\Psi_{ABCD}+2\epd{B(C}\epd{|A|D)}\lambda+
\epd{AB}\phi_{CD}+\phi_{AB}\epd{CD}+2\epd{AB}\epd{CD}k,\nn\\
\Psi_{ABCD}&
:=a_{(ABCD)},\quad \lambda :=\frac{1}{6}a^{HE}{}_{EH},\quad
k:=\frac{1}{8}\alpha_E{}^E{}_F{}^F,\quad
\phi_{AB}:=\alpha_{(AB)E}{}^E.\nn
\end{align}
Further, for $\alpha_E{}^E{}_F{}^F=-\frac{2}{3}\alpha^{AB}{}_{AB}$,
\eqa
\alpha_{ABCD}&=&\Psi_{ABCD}+
\epd{AB}\phi_{CD}+\phi_{AB}\epd{CD}.\nn
\eeqa
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Irreducible Spinor decomposition of Torsion and Contorsion}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\label{sec-asdec}Presented here are two-spinor decompositions of
torsion and cortorsion comparable to Hehl et al, \cite{McR,Milk}
from which the terms `axial Torsion' are imported, \cite{Hehl}.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The following analysis of the spinor form of the $SL(2,\CC)$
connection and its contorsion parts follows \cite{Pen4} although
different conventions are used such that in a holonomic basis for
which $C^a{}_{bc}e_a=[e_b,e_c]=0$,
\begin{align}
\Theta^a&=\half\Theta^a{}_{bc}\theta^b\we\theta^c=d\theta^a+\Gamma^a{}_b\we\theta^b,\nn\\
\Theta^a{}_{bc}e_a&=(\
^\Gamma\nabla_ce_b-^\Gamma\nabla_be_c)=(-\Gamma^a{}_{bc}
+\Gamma^a{}_{cb})e_a.\nn
\end{align}
The spinorial decomposition of the (vector-valued) Torsion 2-form
reads
\begin{align}
\Theta_a &\leftrightarrow
\T_{AA'BC}\Si^{BC}+\bT{}_{AA'B'C'}\bSi^{B'C'} \T_{ABCA'},\nn
\end{align}
where the spinor $\T_{ABCA'}$ decomposes respectively into totally
symmetric,
and `axial', $\AT_{CA'}$
parts
\begin{align}
\T_{AA'BC}&:=\half\T_{ABCA'P'}{}^{P'}=\sigma_{ABCA'}+2\epd{A(B}\AT_{C)A'}=\T_{A(BC)A'},\\
\sigma_{ABCA'}&:=\frac{1}{2}\T_{(A|A'|C|P'|B)}{}^{P'},\quad
\AT_{CA'}:=-\frac{1}{6}\T_{EA'CP'}{}^{EP'}=\frac{1}{3}\Theta^D{}_{CD}{}_{A'},\nn
\end{align}
which transform according to the $(\thalf,\half)$ and
$(\half,\half)$ representations \footnote{Here (i,j) denotes the
finite dimensional representations of $sl(2,C)$}
with dimensions
8 and 4 respectively.\\
Consider now, two metric compatible exterior
covariant derivatives $^\Gamma\nabla$ and $^\omega\nabla$
associated to the metric connections, $\Gamma_{ab}$ and
$\omega_{ab}$ ($\ ^\omega\Theta^a=0$). The difference of their
action on a vector $U^b$ is given in terms of spinors as
\begin{align}
(\ ^\Gamma\nabla_a-^\omega\nabla_a) U^{BB'}&=U^{CB'}K^B{}_{Ca}+
U^{BC'}\tilde{K}^{B'}{}_{C'a},\nn\\
&=U^{CC'}(\epd{C'}{}^{B'}K^B{}_{CAA'}+\epd{C}{}^B\tilde{K}^{B'}{}_{C'AA'}),\nn\\
\intertext{so that upon adopting the useful notation
$\delta^A{}_B:=\epd{A}{}^B$ the contorsion tensor has the spinor
form} K^b{}_{ca}&\leftrightarrow\delta^{B'}{}_{C'}K^B{}_{CAA'}+
\delta^{B}{}_C\tilde{K}^{B'}{}_{C'AA'}.\label{eq:contortspin}
\end{align}
Further, the metricity conditions $^\Gamma\nabla
g_{ab}=^\omega\nabla g_{ab}=0$ implies
\begin{align}
\
^\Gamma\nabla_a\epd{BC}&=^\omega\nabla_a\epd{BC}-\epd{DC}K^D{}_{BAA'}-
\epd{BD}K^D{}_{CAA'},
\end{align}
so imposing a symmetry on the contorsion one form
$K_{BCAA'}=K_{CBAA'}$, that decomposes then as
\begin{align}
K_{ABCC'}&=-\frac{1}{2}\sigma_{ABCC'}+2\epd{C(A}\AT_{B)C'}.\label{eq:contort}
\end{align}
\section{BF-like chiral formulations}
\subsection{Spatial Boundary Considerations}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Consider now spacetime as $3+1$ dimensional, $M=\Re\times\Si$ with
the spatial part being a manifold with boundary $\partial\Si$ so
that the boundary of the spacetime is $\partial
M\equiv\Re\times\partial\Si$. Functional differentiability of
Lagrangians in the prescence of spatial boundaries requires the
action be augmented with certain topological terms, \cite{Sm2}.
The boundary term, $\vartheta$ needs to vanish by
imposing suitable conditions
on the field variables at the boundary so that the equations of motion are well
defined. This is achieved in a covariant manner by
adding the invariant four form, $I_{EP}$ comprised of the
Euler, $E_4$ and Chern-Pontryagin, $P_4$
invariants
\begin{align}
I_{EP}&=2F_{AB}\we F^{AB}\leftrightarrow F_{ab}\we F^{ab}+
i F_{ab}\we^\star F^{ab}\\
\intertext{whose variation contributes a boundary term}
\vartheta_{EP}&=\delta\Gamma_{AB}\we F^{AB}.
\end{align}
The resulting total boundary term (for constant $l$ of dimension length,
$|\Lambda|=2l^{-2}$) of the Lagrangian, (\ref{eq:CDJM}) reads
\eqa
\vartheta_{\Sigma}=2i\delta\Gamma_{AB}\we(\Si^{AB}-\frac{1}{l^2}\fF^{AB}),
\eeqa
vanishing if a self dual condition on the boundary is imposed \eqa
\fF^{AB}=\frac{1}{l^2}\Si^{AB}|_{\partial M}.\label{eq:selfdualb}
\eeqa
\end{enumerate}
Such a self-dual boundary condition (\ref{eq:selfdualb}) is the
condition of vanishing anti-De Sitter curvature at the boundary
for a chiral gauge theory of gravity based on the anti-De Sitter
group, \cite{Nieto}
\begin{align}
\lL_{DS}&=-\frac{1}{2} \ef_{\star ab}\we\ef^{ab-}\leftrightarrow
{i} \ef_{AB}\we\ef^{AB},\nn\\
&\leftrightarrow + iF_{AB}\we F^{AB}-\frac{2i}{l^2}\Sigma^{AB}\we F_{AB} +
\frac{i}{l^4}\Si_{AB}\we\Si^{AB}.\nn
\end{align}
where the anti-De Sitter curvature $R_{ab}$ is
\begin{align}
\ef_{ab}&=\FF_{ab}-\frac{1}{l^2} \theta_a\wedge\theta_b,\nn\\
\intertext{and the anti self-dual part of the De Sitter curvature
$R^{ab-} $ is defined according to}
\ef_{ab} &=
\ef_{ab}^{+}+\ef_{ab}^{-}\quad\text{where}\quad \ef_{ab}^{-} =
\frac{1}{2} (\ef_{ab}+i^{\star}\ef_{ab})\leftrightarrow
\ef_{AB}\epsilon_{A'B'}.\n
\end{align}
\subsection{BF-like formulation of N=2 Supergravity}
BF-like chiral formulations of N=1,2 Supergravity
theories that use a super gauge group\footnote{The constraints
$\Si^{(AB}\we\lambda^{C)}=0$ and
$\Si^{(AB}\we\Si^{CD)}=0$
need to be enforced by adding suitable multiplier terms to
(\ref{eq:superBF}). The right-handed (Grassman odd)
Rarita-Schwinger field is then represented by the left-handed
(Grassmann odd) $\lambda^A=\theta^A{}_{A'}\we\kappa^{A'}$ and
$\Si^{AB}=\half\theta^A{}_{A'}\we\theta^{BA'}.$ The theories as
such, are then `constrained' BF (-like) theories.} and $I_{EP}$
like terms are of the form
\eqa \lL_{BF}=STr({\bf B}\we{\bf F}-l^{-2}{\bf
B}\we{\bf B}+{\bf F}\we{\bf F}),\label{eq:superBF} \eeqa where the
Super trace, $STr$ are Super $SL(2,\CC)$ and Super $GL(2,\CC)$
invariant bilinear forms respectively.
The Lagrangian for N=2
Supergravity employs a ${\bf B}$ field and connection that
decompose according to the Super Lie algebra structure ($\BA,\BB$
are $SU(2)$ indices) and has been formulated by \cite{Eza}, \eqa
{\bf B}&=&\Si^{AB}J_{AB}+\frac{l^2}{4}{\Si}J-
\frac{l}{2}(\tau^3)^{\BB}{}_{\BA}\lambda_{\BB}{}^AQ_A^{\BA},\nn\\
{\bf A}&=&\Gamma^{AB}J_{AB}+AJ+\kappa^A_{\BA}Q^{\BA}_A.\nn \eeqa
The
$GL(2,\CC)$-valued $S^A{}_B$-form and connection form
$\gamma^A{}_B$ are the bosonic parts of these $BF$ fields and the
graded `Super $GL(2,\CC)$' Lie algebra is provided by
\begin{align}
[J_i,J_j]&=\epd{ijk}J_i,\quad
[J_{AB},Q^\BB_C]=\epd{C(A}Q^\BB_{B)},\nn\\ [J_i,J]&=0,\quad
[J,Q_A^\BA]=\frac{1}{l}(\tau^3)^\BA{}_{\BB}Q^\BB_A,\quad[J,J]=0,\nn\\
\{Q_{A}^{\BA},Q_B^{\BB}\}&=-\epu{\BA\BB}\epd{AB}J+\frac{4}{l}(\tau^3)^{\BA\BB}J_{AB}.\nn
\end{align}
The invariant bilinear form, $STr$ is given by
\begin{align}
STr(J_{AB}J_{CD})&=\epd{C(A}\epd{|B|D)},\quad STr (Q_A^\BA
Q^\BB_B)=\frac{4}{l} \epd{AB}(\tau^3)^{\BA\BB}, \quad STr
(JJ)=\frac{4}{l^2}.\nn
\end{align}
Paranthetically, this suggests that the breaking of the `Super
$GL(2,\CC)$' invariance of $STr$ in a suitable manner will produce
a Supersymmetric Gauge theory of gravity with coupled Maxwell
fields.
\section{Spinor Densities and the $GL(2,\CC)$ connection}
In order to couple charged scalar matter to a Lagrangian employing the chiral
$gl(2,C)$-valued $S$-forms as basic variables, expressions for the
metric density and determinant of the metric are required
analogous to those of Urbankte, \cite{Ob3}. They read
\begin{align}
\sqrt{g}g_{\alpha\beta}&=\frac{2i}{3}\epu{\mu\rho\sigma\nu}
\{S^A{}_{B\alpha\mu}S^B{}_{C\rho\sigma}
S^C{}_{A\beta\nu}-\frac{1}{4}S_{\alpha\mu}S_{\rho\sigma}S_{\beta\nu}\nn\\
&\mbox{}-\half(S^A{}_{B\alpha\mu}S^B{}_{A\rho\sigma}S_{\beta\nu}+
S^A{}_{B\alpha\mu} S_{\rho\sigma}S^B{}_{A\beta\nu}
+S_{\alpha\mu}S^A{}_{C\rho\sigma}S^C{}_{A\beta\nu})\},\nn\\
\sqrt{g}&=\frac{i}{3}\{S^A{}_{B\alpha\beta}S^B{}_{A\gamma\delta}+
\frac{1}{2}S_{\alpha\beta}S_{\gamma\delta}\}\epu{\alpha\beta\gamma\delta}.
\end{align}
Moreover, the action of an exterior covariant
derivative on
a complex scalar field with no spinorial indices upon which to contract needs to
be defined. The Higgs fields are thus spinor density-valued 0-forms, with $\phi$ having
weight $w=(1,0)$ and its
complex conjugate, $\bar{\phi}$ having $w=(0,1)$. The notion of a
spinor-density valued object has been developed in unpublished
notes of Plebanski, \cite{Pleb2}, the salient features of which
follow. The group $SL(2,\CC)$
is a six parameter
Lie subgroup of $GL(2,\CC):=\{\xi\in GL(2,\CC)|\xi^A{}_B=fL^A{}_B,\
f^2=det(\xi^A{}_B)\}$, where
$GL(2,\CC)$ transformations
can always be decomposed into
$SL(2,\CC)\otimes\CC$ according to
\begin{align}
\xi^A{}_B&=fL^A{}_B\quad\text{and}\quad(\xi^{-1})^A{}_B=f^{-1}(L^{-1})^A{}_B.
\end{align}
The transition functions of the $GL(2,\CC)$ bundle over M are
represented by complex non-singular $2\times 2$, matrices
$(\gl^A{}_B)$ and contain 8 real parameters. The inverse and
conjugates are denoted by, $(\gl^{-1})^A{}_B,\ \gl^{{A'}}{}_{B'}$
and $(\gl^{-1})^{{A'}}{}_{B'}$, determinants by
$f^2:=det(\gl^A{}_B),\bar{f}{}^2= det(\gl^{{A'}}{}_{B'})$. A
generic spinor density transforms under $GL(2,\CC)$ into a new
dyad according
to, for example
\eqa \tilde{\psi}^{AB'}{}_{CD'}=(f)^{2w}(f')^{2\overline{w}}
(\xi^{-1})^A{}_B(\xi^{-1})^{B'}{}_{C'}
\xi^D{}_C\xi^{E'}{}_{D'}\psi^{BC'}{}_{DE'},\label{eq:sp1} \eeqa
where the independent complex numbers $(w,\overline{w})$ are the
weight and anti-weight that characterise $\psi$. The complex
conjugate of $\psi^{AB'}{}_{CD'}$ is $\bar{\psi}^{A'B}{}_{C'D}$
and carries weights $(\bar{w}',w')$. The symplectic metrics are
thus numerical spinor densities with $\epsilon_{AB}$, for example,
having weight $w=+1$
\begin{align}
\tilde{\epsilon}{}_{AB}&=\xi^C{}_A\xi^D{}_B\epd{CD},\nn\\
&=(f^2)L^C{}_AL^D{}_B\epd{CD},\nn\\
&=(f^2)\epd{AB},\label{eq:reduc}\nn\\ &=(det\xi)\epd{AB}.
\end{align}
%%%%%%%%%%%%%%%%%%%%%%%%%%%
The hermitian matrix of 1-forms,
$\theta^{AB'}=\theta^{AB'}{}_a\theta^a$
as a
spinorial density under $GL(2,\CC)$, has weights assigned such
that $w=\bar{w}$. Under a transformation (recall the $SL(2,\CC)$
transformation law, $ \theta^{AA^\prime}\mapsto(L^{-1})^A_{\
B}\theta^{BB^\prime}{(L^{-1})}{}^{A^\prime}_{\ B^ \prime}$)to a
new spinorial frame
\begin{align}
\tilde{\theta}^{AB'}&=(\xi^{-1})^A{}_C\theta^{CD'}(\xi^{-1})^{B'}{}_{D'},\nn\\
&=(f^{-1})(\bar{f}^{-1})(L^{-1})^A{}_C(L^{-1})^{B'}{}_{D'}\theta^{CD'},
\end{align}
the weights are read off as
$\theta^{AB'}$ are $w=-\half,\ \bar{w}=-\half$.
Now according to
the decomposition of the weightless connection \eqa
\gamma^A{}_{B}=\Gamma^A{}_B+\delta^A{}_BA,
\label{eq:gldecomp}
\eeqa and applying the natural axioms \cite{genstew} of a
covariant exterior derivative one has \eqa
\gna\epsilon_{AB}=-Q\epsilon_{AB},\nn\\
\gna\epsilon^{AB}=Q\epsilon^{AB}, \eeqa for $Q$ a spin-weightless
complex covariant vector, so that
\begin{align}
\tna\epd{AB}&=\DD\epd{AB}-2\epd{AB}A,
\end{align}
for $A=\half Q$.\footnote{ The point of view taken here is that
the symplectic metric $\epsilon_{AB}$ is fixed once and for all
and that the soldering form, $\sigma^{AA'}{}_{\mu}$
contain all the information pertaining
to the metric $g_{\mu\nu}$.
See alternatively \cite{Bai1}, \cite{PenTor}.} The
$gl(2,\CC)$-valued spinorial connection transforms according to,
\begin{align}
\tilde{\gamma}^A{}_B&=(\gi)^A{}_C\gamma^C{}_D\xi^D{}_B+
(\gi)^A{}_Cd\xi^C{}_B,\nn\\
&=(L^{-1})^A{}_CdL^C{}_B+f^{-1}df\delta^A{}_B+(L^{-1})^A{}_C
\gamma^C{}_DL^D{}_B,\\ \intertext{so that the trace transforms as}
\tilde{\gamma}^A{}_A&=
(L^{-1})^A{}_CdL^C{}_A+2f^{-1}df+(L^{-1})^A{}_C\gamma^C{}_DL^D{}_A,\nn\\
&=2f^{-1}df+\gamma^A{}_A,\nn\\
&=dlnf^2+\gamma^A{}_A=dln(det\xi)+\gamma^A{}_A.\label{eq:tr2}
\end{align}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
According to (\ref{eq:tr2}), a $GL(2,\CC)$ transformation to a new
dyad implies that
\begin{align}
\tilde{\gamma}^A{}_A&=2\tilde{A}\quad\text{and}\quad\gamma^A{}_A=2A,\\
\intertext{so that $A$ changes according to complex gauge
transformations} \tilde{A}&=A+\half d{\rm ln(det}\gl).
\end{align}
Therefore $A_\mu$ is a complex covariant vector defined up to some
complex gauge; the forms $\Gamma^A{}_B$ are defined up to
$SL(2,\CC)$ transformations induced by $GL(2,\CC)$ according to
the affine representation \eqa \Gamma^A_{\
B}&\mapsto&(L^{-1})^A_{\ C}dL^C_{\ B}+(L^{-1})^A_{\ C}\Gamma^C_{\
D}(\epsilon )L^D_{\ B}\ ,\label{eq:wgt}\eeqa where $L^A_{\
B}(\epsilon)$ belongs to $SL(2,C)$ and $L^A_{\ B}(0)=\delta^A_{\
B}$. With the decomposition (\ref{eq:gldecomp}) the exterior
covariant derivative for an arbitrary spinor density, $\psi$
endowed with complex weights $(w,\bar{w})$ reads, \eqa \
^\gamma{\nabla}\psi^{A_1..A_kB_1'..B'_l}_{C_1..C_pD_1'..D'_q}&:=&
\{\ ^\Gamma\nabla+[w+\half (k-p)]A+[\bar{w}+\half(l-q)]\bar{A}\}
\psi^{A_1..A_kB'_1...B'_l}_{C_1..C_pD'_1..D'_q}.\label{eq:tildecon}\nn\\
&& \eeqa
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Non-standard Energy-momentum }
For Grassman odd [even] fields, $\lambda$,
\cite{JaF} the Lagrangian $\lL_{\half}$ has the real [imaginary]
part
\begin{align}
\lL_{\half}+\bar{\lL}_{\half}&=-\eta^{AA'}\we(\bar{\lambda}_{A'}\nabla\lambda_A
-\lambda_A\nabla\bar{\lambda}_{A'})[=\Im(\lL_\half)].\\
\intertext{Its imaginary [real] part is}
\lL_\half-\bar{\lL}_\half&=\nabla(\eta^{AA'}\bar{\lambda}_{A'}\lambda_A)-
(\nabla\eta^{AA'})J_{AA'}[=\Re(\lL_\half)].
\end{align}
Now, the equation of motion obtained by varying $\Gamma^A{}_B$ (or
$K^A{}_B$) shows that the Grassman odd [even] spin $\half$ field
current, $J_{AA'}=\lambda_A\bar{\lambda}_{A'}=\bar{J}_{AA'}$
supports only the axial part of the torsion\footnote{A useful
identity involving the axial part of torsion and the 3-form dual
to $\theta^{AA'}$ is
\begin{align}
\
^\Gamma\nabla\eta^a&=^\Gamma\Theta^c\we\eta^a{}_c=^\Gamma\Theta^{ca}{}_c\eta
=-(e^a\hook e_c\hook\ ^\Gamma\Theta^c)\eta=-(e^a\hook\Theta)\eta
\leftrightarrow 6\AT^{AA'}\eta.\nn
\end{align}} of $^\Gamma\nabla$,
\eqa \hat{\Theta}{}^{AA'}=\frac{1}{8}J^{AA'}. \eeqa
This means
that the imaginary [real] part of $\lL_\half$ is (modulo exact
forms) \eqa \Im(\lL_{\half})
=-6\hat{\Theta}{}^{AA'}J_{AA'}=-\frac{3}{4}J_{AA'}J^{AA'}=:-\frac{3}{4}J^2.
\eeqa
Now, with the use of Grassman even spin $\half$ fields
$\lambda$, `on shell' $\Re(\lL_\half)$ vanishes since
$\lambda_A\lambda^A=0$. Therefore the equations of motion
resulting from the complex Lagrangian $\lL_{SSJ+\half}$ are
identical, for real $\theta^{AA'}$, to those which arise from
keeping only its imaginary part but the trace-free part of the
Ricci tensor is
\eqa
\Phi_{ABC'D'}=\frac{1}{4}\{\lambda_{(C'}{}^\omega\nabla_{D')(A}\lambda_{B)}+
\lambda_{(A}{}^\omega\nabla_{B)(C'}\bla{}_{D')}\}-
\frac{1}{8}\bla{}_{(C'}\bla{}_{D')}\lambda_{(A}\lambda_{B)}.\nn
\eeqa
So although the equations of motion derived from the
variation of $\lambda$ and $\bar{\lambda}$ are the standard Weyl
equations (not
be so for the Grassman odd case), there are no quartic terms
comprising the spin $\half$ fields that may be added in order to
eliminate the above quartic spin $\half$ term.\\
\section{Chiral
Lagrangian for spin $\thalf$ fields}
%%%%%%%%%%%%%
This appendix serves to illustrate both how chiral Lagrangians are
constructed and how the symplectic techniques developed can be
used to define a charge. The complex Lagrangian for
spin $\thalf$ fields
propogating on a (fixed) curved background space-time with
symmetric, metric connection employed by Frauendiener et al,
\cite{Fraud1}
is\eqa
\lL_{\thalf}(\theta,\kappa,\bar{\kappa})=i\bar{\kappa}^{(A'B')}_A
\nabla \kappa^{(AB)}{}_{A'}\eta_{BB'}\nn \eeqa and may usefully be
written as a chiral Lagrangian \eqa
\lL_{\thalf}(\lambda,\bar{\lambda},\mu,\psi)=\lambda^A\we
\nabla\kappa_A
-\Si^{AB}\we(\psi_{ABC}+\epd{C(A}\mu_{B)})\lambda^C,\label{eq:realDi}
\eeqa with $\lambda_{AA'B'}$ chosen to be the complex conjugate
of $\kappa_{A'AB}$ \cite{DC4},
\eqa
\lambda_{A(A'B')}=\overline{\kappa_{A'(AB)}}.\nn
\eeqa
Here $\lambda^A$ is a two form defined in terms of the $\Si$-basis,
\eqa
\lambda^C&=&\lambda^{CEF}\Si_{EF}+\epu{C(E}r^{F)}\Si_{EF}+\lambda^{CB'C'}\Si_{B'C'},\nn\\
&\in&(\thalf,0)\oplus(\half,0)\oplus(\half,1),\nn \eeqa where
$r^F:=\frac{2}{3}\lambda^D{}_D{}^F$ and the field equations
arising from the variation of $\psi$ and $\mu$ are
\begin{align}
\Si^{(AB}\we\lambda^{C)}&=0,\label{eq:lambda1}\\
\quad\Si^{AB}\we\lambda_B&=0.\label{eq:lambda2}
\end{align}
Equation (\ref{eq:lambda1}) on its own determines the left-handed
two-form as \eqa \lambda^C&=&
r^F\Si^C{}_F+\lambda^{CE'F'}\tiSi_{E'F'},\nn\\
&=&\theta^{CE'}\we\{-\half\theta_{FE'}-\theta_F{}^{F'}r^F{}_{E'F'}\},\nn\\
&:=&\theta^{CA'}\we\tilde{\kappa}_{A'},\nn \eeqa while
(\ref{eq:lambda2}) by eliminating $r^F$ gives the two-form
representation of the potential
\eqa
\lambda^A=\lambda^A{}_{(A'B')}\Si^{A'B'},\eeqa thus
providing a
Dirac form of the Rarita-Schwinger equations, $\nabla\lambda^A=0$.
Now, the potentials, $\kappa^A$ possess the
gauge
freedom
\eqa
\delta_\nu\kappa^A&=&\nabla\nu^A=-\nabla^{(B}{}_{B'}\nu^{A)}\theta_B{}^{B'},\nn\\
\delta_\rho\lambda^A&=&\nabla\rho^A=\nabla^A_{(B'}\bar{\nu}_{A')}\bar{\Si}^{A'B'},\nn
\eeqa
so that the Lagrangian, (\ref{eq:realDi}) possesses a symmetry
if
the condition, $\Phi_{(AB)(A'B')}=0$ is satisfied on the
background space-time. This symmetry is generated by an
infinitesimal (real) spinor parameter, $\nu^A$. Denoting the
solution space of the Dirac form of
the equations for the potentials by ${\varphi}_{\thalf}$,
solutions of ${\varphi}_{\thalf}$ can be obtained by taking a
symmetrised derivative,
\begin{align}
\nabla^{B'(A}\nu^{B)}\theta_{BB'}&=0,\\ \intertext{for all Weyl
spinors, $\nu\in{\varphi}_{\half}$ in the solution space of the
Weyl equation} \nabla_{CC'}\nu^C&=0.
\end{align}
The Hamiltonian that generates the transformations is found by
finding the symplectic potential whose Lie derivative by
$\delta_\nu\kappa$ vanishes (\ref{eq:canLie}), so that for all
$\kappa\in{ \varphi}_{\thalf}$ one has
$\varpi(\delta\kappa,\delta_\nu\kappa)=-\delta H_\nu(\kappa)$ and
accordingly
\begin{align}
\varpi(\delta\kappa_A,\delta_\nu\kappa_A)&=\delta\vartheta(\kappa_A,\delta_\nu\kappa_A)-
\delta_\nu\vartheta(\kappa_A,\delta\kappa_A)=\delta\lambda^A\we
\nabla\nu_A-\delta\kappa_A\we \nabla\rho^A,\nn \intertext{so that}
\delta H_\nu&=\delta\int_\Si j_\nu, \quad
j_\nu=d\sigma_\nu=-\lambda^A\we \nabla\nu_A+\kappa_A\we
\nabla\rho^A.\label{eq:twistier}
\end{align}
The Hamiltonian, \cite{Fraud1} is then
the surface integral
\eqa
H_\nu(\kappa_A)=\int_{\partial\Si}\sigma_\nu.\label{eq:symcharge}
\eeqa
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
This result is to be contrasted with that flat space expression
for the charge obtained by considering the complex potential
\cite{Esp1}, $^+{ A}$ defined as \eqa ^+{
A}:=\lambda_{A'B'A}\mu^{(B'}\theta^{A')A}. \nn\eeqa
With $\psi_{A'B'C'}=:\partial_{A(A'}\lambda^A{}_{B'C')}$ the helicity ($\thalf$)
field strength and $\mu^{A'}$ is to be interpreted as the
primary part of a dual twistor,
$(\mu^{A'},\gamma_A)$, where
\begin{align}
\partial_{AA'}\mu^{B'}&=i\epd{A'}{}^{B'}\gamma_A\quad
\text{and }\partial_{AA'}\gamma_B=0.\nn
\end{align}
The field equation of the spin potential, $\kappa$ determes $^+{
F}$ as self-dual
\begin{align}
\ ^+{
F}&=[\psi_{A'B'C'}\mu^{C'}+i\lambda_{A'B'A}\gamma^A]\Si^{A'B'},\nn
\end{align}
having an associated electric charge
\begin{align}
Q^E&=-\int_{\partial\Si}\ ^+{ F}=
-\int_{\partial\Si}\{\mu^{C'}
\psi_{A'B'C'}+i\lambda_{A'B'A}\gamma^A\}\bar{\Si}^{A'B'}.\nn\\
\intertext{The imaginary part of the flat space expression
representing the electric charge corresponds then to the symplectic
expression (\ref{eq:symcharge})}
H_\nu&=\int_{\partial\Si}\lambda_{AA'B'}\gamma^A\Si^{A'B'}
-\kappa_{ABA'}\pi^{A'}\Si^{AB},
\end{align}
if $\gamma^A$ is identified with $\nu^A$ and $\pi^{A'}$ is identified
with $\bar{\nu}^{A'}$. This agrees with the comparable calculation
given by Frauendiener, \cite{Fraud1} which employed a non-chiral Lagrangian.
\end{appendices}
% ----------------------------------------------------------------
\bibliographystyle{amsplain}
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