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

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 $N\le 10$. 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.

Final version to appear in JHEP. 52 pages. The part with the complete classification of irreps (and the explicit presentation of length-4 irreps of N=9,10,11,12 and N=10 length-5 irreps) is unchanged. An extra section has been added with an entire class of off-shell invariant actions for arbitrary values N of the 1D extended supersymmetry. A non-trivial N=8 off-shell action for the (1,8,7) multiplet has been constructed as an example. It is obtained in terms of the octonionic structure constants

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