Four-dimensional real extended super-Poincare algebra

Use the complex field $\mathbb{C}$ with conjugation $z\mapsto \bar{z}$. Fix an integer $N\ge 1$. Let $S={\mathbb{C}}^2$ have standard basis $s_1,s_2$. For a complex vector space $W$, its conjugate vector space $\bar{W}$ consists of symbols $\bar{w}$ for $w\in W$, with addition $\bar{w}+\bar{v}=\overline{w+v}$ and scalar action $\lambda \bar{w}=\overline{\stackrel{‾}{\lambda }w}$. Set $\dot{S}=\bar{S}$ with dotted basis ${\dot{s}}_{\dot{1}}=\overline{s_1}$ and ${\dot{s}}_{\dot{2}}=\overline{s_2}$. Let $F={\mathbb{C}}^N$ have standard basis $f_1,\dots ,f_N$, and let $F^{\ast }$ be its complex-linear dual with dual basis $f^1,\dots ,f^N$ and scalar action $(\lambda \mathrm{\ell })(v)=\lambda \mathrm{\ell }(v)$. All tensor products are over $\mathbb{C}$, with scalar action $\lambda (v\otimes w)=(\lambda v)\otimes w=v\otimes (\lambda w)$ and product bases formed from the displayed bases. Define ${\mathrm{\Lambda }}^{2}F=(F{\otimes }_{\mathbb{C}}F)\mathrm{/}{\mathrm{span}}_{\mathbb{C}}\{v\otimes v:v\in F\}$; its basis is $f_I\wedge f_J$ for $1\le I<J\le N$, with quotient scalar action. Its conjugate $\overline{{\mathrm{\Lambda }}^{2}F}$ has basis $\overline{f_I\wedge f_J}$ and the conjugate scalar action just defined. For $N=1$ these exterior-square spaces are zero. In the displayed bases, $\mathfrak{s}\mathfrak{l}(S)=\{A\in {\mathrm{End}}_{\mathbb{C}}(S):\mathrm{tr}A=0\}$ and $\mathfrak{s}\mathfrak{l}(\dot{S})=\{B\in {\mathrm{End}}_{\mathbb{C}}(\dot{S}):\mathrm{tr}B=0\}$. Put ${ϵ}_{12}={ϵ}_{\dot{1}\dot{2}}=1$ and ${ϵ}_{21}={ϵ}_{\dot{2}\dot{1}}=-1$, with zero diagonal. The complex even space is

${\mathfrak{g}}_{\bar{0}}=(\mathfrak{s}\mathfrak{l}(S)\oplus \mathfrak{s}\mathfrak{l}(\dot{S}))\oplus V\oplus C_N,V=S\otimes \dot{S},C_N={\mathrm{\Lambda }}^{2}F\oplus \overline{{\mathrm{\Lambda }}^{2}F}\mathrm{.}$

Write $T_{\alpha \dot{\beta }}=s_{\alpha }\otimes {\dot{s}}_{\dot{\beta }}$ for the basis of $V$. The listed central generators form a basis of $C_N$: $Z^{IJ}$ corresponds to $f_I\wedge f_J$ and ${\bar{Z}}_{IJ}$ to $\overline{f_I\wedge f_J}$ for $1\le I<J\le N$; extend them by antisymmetry in the flavor indices and set their diagonal entries to zero. The complex odd space is

${\mathfrak{g}}_{\bar{1}}=(S\otimes F)\oplus (\dot{S}\otimes F^{\ast }),$

with basis $Q_{\alpha }^I=s_{\alpha }\otimes f_I$ and ${\bar{Q}}_{\dot{\alpha }I}={\dot{s}}_{\dot{\alpha }}\otimes f^I$. A homogeneous vector has parity $\mathrm{\mid }x\mathrm{\mid }\in \{0,1\}$. The bracket is complex-bilinear, preserves parity, and obeys $[x,y]=-(-1{)}^{\mathrm{\mid }x\mathrm{\mid }\mathrm{\mid }y\mathrm{\mid }}[y,x]$. Its complete nonzero table, apart from the graded-skew consequences, is

$[(A,B),(A^{\mathrm{\prime }},B^{\mathrm{\prime }})]=([A,A^{\mathrm{\prime }}],[B,B^{\mathrm{\prime }}]),[(A,B),T]=(A\otimes I+I\otimes B)T,$

$[(A,B),Q_{\alpha }^I]=A^{\gamma }{}_{\alpha }{Q}_{\gamma }^I,[(A,B),{\bar{Q}}_{\dot{\alpha }I}]=B^{\dot{\gamma }}{}_{\dot{\alpha }}{\bar{Q}}_{\dot{\gamma }I},$

Here $A^{\gamma }{}_{\alpha }$ is the entry in row $\gamma$, column $\alpha$, so $A{s}_{\alpha }={\sum }_{\gamma }{A}^{\gamma }{}_{\alpha }{s}_{\gamma }$; dotted entries use the same convention. Thus on the translation basis the tensor action is $[(A,B),T_{\alpha \dot{\beta }}]={\sum }_{\gamma }{A}^{\gamma }{}_{\alpha }{T}_{\gamma \dot{\beta }}+{\sum }_{\dot{\gamma }}{B}^{\dot{\gamma }}{}_{\dot{\beta }}{T}_{\alpha \dot{\gamma }}$.

$\{Q_{\alpha }^I,{\bar{Q}}_{\dot{\beta }J}\}=-2{\delta }^I{}_J{T}_{\alpha \dot{\beta }},\{Q_{\alpha }^I,Q_{\beta }^J\}=2{ϵ}_{\alpha \beta }{Z}^{IJ},\{{\bar{Q}}_{\dot{\alpha }I},{\bar{Q}}_{\dot{\beta }J}\}=2{ϵ}_{\dot{\alpha }\dot{\beta }}{\bar{Z}}_{IJ}\mathrm{.}$

All other brackets vanish: $V$ and $C_N$ are abelian, $C_N$ is central, and $V$ brackets trivially with the odd space. The two antisymmetric factors in each equal-chirality odd bracket make that bracket symmetric in the combined spinor-flavor labels. The Lie-superalgebra and real-form properties are established by the local Jacobi result.

Define a conjugate-linear involution $\ast$ on the generators by

$(A,B{)}^{\ast }=(-\bar{B},-\bar{A}),Q_{\alpha }^I{}^{\ast }={\bar{Q}}_{\dot{\alpha }I},{\bar{Q}}_{\dot{\alpha }I}{}^{\ast }=Q_{\alpha }^I,T_{\alpha \dot{\beta }}^{\ast }=T_{\beta \dot{\alpha }},(Z^{IJ}{)}^{\ast }={\bar{Z}}_{IJ},({\bar{Z}}_{IJ}{)}^{\ast }=Z^{IJ}\mathrm{.}$

It uses the operator-adjoint convention $[x,y{]}^{\ast }=-(-1{)}^{\mathrm{\mid }x\mathrm{\mid }\mathrm{\mid }y\mathrm{\mid }}[x^{\ast },y^{\ast }]$. The associated real Lie superalgebra is the fixed space of the conjugate-linear map $\tau (x)=-x^{\ast }$ on even elements and $\tau (x)=i{x}^{\ast }$ on odd elements. Its real Lorentz algebra is the fixed diagonal copy of $\mathfrak{s}\mathfrak{l}(2,\mathbb{C})$; its real translation generators are the $\tau$-fixed skew-Hermitian bispinors. The star-fixed translation coordinates defined below are abstract elements of the complex algebra; their multiples by $i$ form a real basis of the tau-fixed translation space.

For the metric convention $\eta =\mathrm{diag}(-1,1,1,1)$, put ${\sigma }^0=I_2$ and

${\sigma }^1=\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right),{\sigma }^2=\left(\begin{array}{cc}0& -i\\ i& 0\end{array}\right),{\sigma }^3=\left(\begin{array}{cc}1& 0\\ 0& -1\end{array}\right),{\bar{\sigma }}^0=I_2,{\bar{\sigma }}^j=-{\sigma }^j\mathrm{.}$

They obey

${\sigma }^{\mu }{\bar{\sigma }}^{\nu }+{\sigma }^{\nu }{\bar{\sigma }}^{\mu }=-2{\eta }^{\mu \nu }{I}_2\mathrm{.}$

Define abstract translation coordinates $p_0=(T_{1\dot{1}}+T_{2\dot{2}})\mathrm{/}2$, $p_1=(T_{1\dot{2}}+T_{2\dot{1}})\mathrm{/}2$, $p_2=(T_{2\dot{1}}-T_{1\dot{2}})\mathrm{/}(2i)$, and $p_3=(T_{1\dot{1}}-T_{2\dot{2}})\mathrm{/}2$. Then $T_{\alpha \dot{\beta }}={\sigma }_{\alpha \dot{\beta }}^{\mu }{p}_{\mu }$, $p_{\mu }^{\ast }=p_{\mu }$, and $\tau (i{p}_{\mu })=i{p}_{\mu }$. A representation of the complex supertranslation algebra $\mathfrak{t}=V\oplus C_N\oplus {\mathfrak{g}}_{\bar{1}}$ on a graded complex vector space $\mathcal{D}$ means a parity-preserving complex-linear map $\rho :\mathfrak{t}\to {\mathrm{End}}_{\mathbb{C}}(\mathcal{D})$ preserving the bracket, where the target bracket on homogeneous endomorphisms is $[X,Y]=XY-(-1{)}^{\mathrm{\mid }X\mathrm{\mid }\mathrm{\mid }Y\mathrm{\mid }}YX$. Write $P_{\mu }=\rho (p_{\mu })$ and use the charge symbols for their images. Thus $\rho (T_{\alpha \dot{\beta }})={\sigma }_{\alpha \dot{\beta }}^{\mu }{P}_{\mu }$ and the represented mixed relation is $\{Q_{\alpha }^I,{\bar{Q}}_{\dot{\beta }J}\}=-2{\sigma }_{\alpha \dot{\beta }}^{\mu }{P}_{\mu }{\delta }^I{}_J$, with $P_{\mu }={\eta }_{\mu \nu }{P}^{\nu }$. Complex linearity gives $\rho (i{p}_{\mu })=i{P}_{\mu }$, hence $-i\rho (i{p}_{\mu })=P_{\mu }$. For a real number $M>0$, a massive rest module here means such a supertranslation representation with $P^0=M{I}_{\mathcal{D}}$ and $P^j=0$ for $j=1,2,3$. Consequently $P_0=-M{I}_{\mathcal{D}}$ and its mixed anticommutator is $2M{\delta }_{\alpha \dot{\beta }}{\delta }^I{}_J{I}_{\mathcal{D}}$. This is an algebraic specialization, with no assertion of a Hilbert adjoint or a spectral fiber. Here $N$ counts left Weyl spinors; the scalar central-charge matrix is antisymmetric and therefore vanishes for $N=1$. Tensorial or topological extensions are outside this scalar-central-charge algebra. The convention uses natural units $\mathrm{\hslash }=c=1$; translation and central-charge generators have mass dimension one and supercharges have mass dimension one-half.

Martin, §§1–3.1.

Gates et al., §3.2.

The complex numbers as $\mathbb{R}[x]\mathrm{/}(x^2+1)$, with the real embedding and imaginary unit $i$: The complex numbers as $\mathbb{R}[x]\mathrm{/}(x^2+1)$, with the real embedding and imaginary unit $i$.

$\mathbb{C}=\mathbb{R}[x]\mathrm{/}(x^2+1)$ is a field, every element is uniquely $a+bi$, and every nonzero element has inverse $(a-bi)\mathrm{/}(a^2+b^2)$: $\mathbb{C}=\mathbb{R}[x]\mathrm{/}(x^2+1)$ is a field, every element is uniquely $a+bi$, and every nonzero element has inverse $(a-bi)\mathrm{/}(a^2+b^2)$.

Real and imaginary parts, complex conjugation, and modulus: Real and imaginary parts, complex conjugation, and modulus.

The separate positive unitary representation definition supplies a nonzero graded complex Hilbert space $\mathcal{H}$, a joint momentum projection-valued measure $E$, self-adjoint coordinate integrals $p_{\mu }(E)$ on their square-integrability domains, and a common dense invariant domain $\mathcal{D}$. In that setting the algebraic images above are $P_{\mu }=p_{\mu }(E){\mathrm{\mid }}_{\mathcal{D}}$, and $-i\rho (i{p}_{\mu })=P_{\mu }$ on $\mathcal{D}$. Self-adjointness and spectral assertions require those additional hypotheses; they do not follow from a representation on a graded complex vector space. The rest-module relation above applies on any separately supplied invariant domain on which the momenta have the stated scalar values; no existence of a nonzero spectral fiber is asserted here.

positive unitary representation definition