Classification of irreps and invariants of the N-extended Supersymmetric Quantum Mechanics

TL;DR

This paper classifies the irreducible representations of N-extended supersymmetry in one dimension, providing a systematic way to construct supersymmetric invariants and actions without superspace formalism, up to N=10.

Contribution

It introduces an algorithmic classification of irreps based on Clifford algebras, explicitly classifies irreps for N up to 10, and constructs supersymmetric invariants systematically.

Findings

Complete classification of irreps up to N=10.

Construction of supersymmetric invariants without superspace.

Explicit example of N=8 off-shell action for (1,8,7) multiplet.

Abstract

We present an algorithmic classification of the irreps of the $N$-extended one-dimensional supersymmetry algebra linearly realized on a finite number of fields. Our work is based on the 1-to-1 \cite{pt} correspondence between Weyl-type Clifford algebras (whose irreps are fully classified) and classes of irreps of the $N$-extended 1D supersymmetry. The complete classification of irreps is presented up to $N\leq 10$. The fields of an irrep are accommodated in $l$ different spin states. N=10 is the minimal value admitting length $l>4$ irreps. The classification of length-4 irreps of the N=12 and {\em real} N=11 extended supersymmetries is also explicitly presented.\par Tensoring irreps allows us to systematically construct manifestly ($N$-extended) supersymmetric multi-linear invariants {\em without} introducing a superspace formalism. Multi-linear invariants can be constructed both for {\em unconstrained} and {\em multi-linearly constrained} fields. A whole class of off-shell invariant actions are produced in association with each irreducible representation. The explicit example of the N=8 off-shell action of the $(1,8,7)$ multiplet is presented.\par Tensoring zero-energy irreps leads us to the notion of the {\em fusion algebra} of the 1D $N$-extended supersymmetric vacua.