Supertranslation algebra

Definition

Let $(V,\eta)$ be a real pseudo-Euclidean vector space, let $S$ be a real spinor module, and choose a $\operatorname{Spin}(V,\eta)$ -equivariant symmetric bilinear map

\Gamma:\operatorname{Sym}^2 S\longrightarrow V.

The corresponding supertranslation algebra is the Lie superalgebra

\mathfrak t=V_{\bar0}\oplus S_{\bar1}

with

[s_1,s_2]=\Gamma(s_1,s_2),\qquad [V,V]=[V,S]=0.

It is two-step nilpotent when $\Gamma\ne0$ .

Because $S$ is odd, graded skew-symmetry makes its odd–odd bracket symmetric, not alternating. Since $V$ is central, the odd–odd–odd Jacobi identity is automatic. Spin-equivariance is the condition needed to let Lorentz transformations act by automorphisms.

Choices and variants

The spinor representation and the admissible bilinear maps depend on dimension, signature, reality condition, and chirality. Extended $\mathcal N$ -supersymmetry replaces $S$ by spinors tensored with a multiplicity space. Central or higher-degree “brane charges” are extensions of the basic supertranslation algebra rather than part of this definition.

The ordinary translation algebra $V$ is the even reduction. Adjoining the Lorentz algebra produces the super-Poincaré algebra, while integrating $\mathfrak t$ produces super-Minkowski space with its supertranslation group structure and supertranslation distribution.

References

P. Deligne and J. W. Morgan, “Notes on supersymmetry (following Joseph Bernstein),” in Quantum Fields and Strings: A Course for Mathematicians, Volume 1, American Mathematical Society, 1999, 41–97. Relevant: spinors and supertranslation algebras.
J. Figueroa-O'Farrill, “Majorana spinors,” lecture notes, 2001. Author manuscript. Relevant: admissible spinor bilinears and supersymmetry algebras.