BRS Cohomology of the Supertranslations in D=4

TL;DR

This paper investigates the cohomology of supertranslation operators in four-dimensional supersymmetry, revealing a complex structure influenced by spin-orbit coupling, which is fundamental in supersymmetric field theories.

Contribution

It uncovers the detailed cohomology structure of supertranslation operators in D=4, highlighting the role of spin-orbit coupling in supersymmetric theories.

Findings

Cohomology depends on SU(2) spin-orbit coupling.

The structure of the cohomology is complex.

Spin-orbit coupling is fundamental in supersymmetric field theories.

Abstract

Supersymmetry transformations are a kind of square root of spacetime translations. The corresponding Lie superalgebra always contains the supertranslation operator $ \delta = c^{\alpha} \sigma^{\mu}_{\alpha \dot \beta} {\overline c}^{\dot \beta} (\epsilon^{\mu})^{\dag} $. We find that the cohomology of this operator depends on a spin-orbit coupling in an SU(2) group and has a quite complicated structure. This spin-orbit type coupling will turn out to be basic in the cohomology of supersymmetric field theories in general.