Content-Type: multipart/mixed; boundary="-------------0309121729118" This is a multi-part message in MIME format. ---------------0309121729118 Content-Type: text/plain; name="03-416.comments" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="03-416.comments" Distilled from unpublished aspects of my PhD thesis from Kings College under the tutilage of Professor David Robinson ---------------0309121729118 Content-Type: text/plain; name="03-416.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="03-416.keywords" two spinors, differential forms, Noether charge, symplectic, pre-symplectic form, Witten-Nester form, Einstein-Maxwell Lagrangian, Grassman,Weyl spinor, chiral, Ashtekar variables ---------------0309121729118 Content-Type: application/x-tex; name="ARTICLE2.TEX" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="ARTICLE2.TEX" % ---------------------------------------------------------------- % AMS-LaTeX Paper ************************************************ % **** ----------------------------------------------------------- \documentclass{amsart} \usepackage[active]{srcltx} % SRC Specials: DVI [Inverse] Search % ---------------------------------------------------------------- \vfuzz2pt % Don't report over-full v-boxes if over-edge is small \hfuzz2pt % Don't report over-full h-boxes if over-edge is small % THEOREMS ------------------------------------------------------- \newtheorem{thm}{Theorem}[section] \newtheorem{cor}[thm]{Corollary} \newtheorem{lem}[thm]{Lemma} \newtheorem{prop}[thm]{Proposition} \theoremstyle{definition} \newtheorem{defn}[thm]{Definition} \theoremstyle{remark} \newtheorem{rem}[thm]{Remark} \numberwithin{equation}{section} % MATH ----------------------------------------------------------- \newcommand{\norm}[1]{\left\Vert#1\right\Vert} \newcommand{\abs}[1]{\left\vert#1\right\vert} \newcommand{\set}[1]{\left\{#1\right\}} \newcommand{\Real}{\mathbb R} \newcommand{\eps}{\varepsilon} \newcommand{\To}{\longrightarrow} \newcommand{\BX}{\mathbf{B}(X)} \newcommand{\A}{\mathcal{A}} % ---------------------------------------------------------------- \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}% %\date{}% %\dedicatory{}% %\commby{}% % ---------------------------------------------------------------- \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 $\bar{\nu}^{A'}$. 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