Classification of irreps and invariants of theN-extended Supersymmetric Quantum Mechanics

Abstract

We present an algorithmic classification of the irreps of the N -extended onedimensional supersymmetry algebra linearly realized on a finite number of fields. Our work is based on the 1-to-1 [1] 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 ≤ 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 real N = 11 extended supersymmetries is also explicitly presented.Tensoring irreps allows us to systematically construct manifestly (N -extended) supersymmetric multi-linear invariants without introducing a superspace formalism. Multi-linear invariants can be constructed both for unconstrained and multilinearly 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.Tensoring zero-energy irreps leads us to the notion of the fusion algebra of the 1D N -extended supersymmetric vacua.

On dimensional extension of supersymmetry: from worldlines to worldsheets

“…In this spirit, the authors of [9][10][11][12][13][14][15] developed a detailed classification of a huge class (∼ 10 12 for no more than 32 supersymmetries) of worldline supermultiplets wherein each supercharge maps each component field to precisely one other component field or its derivative, and which are faithfully represented by graphs called Adinkras; see also [16][17][18][19][20][21][22]. The subsequently intended dimensional extension has been addressed recently [7,8], and the purpose of the present note is to complement this effort and identify an easily verifiable obstruction to dimensional extension.…”

Gauging N=4 supersymmetric mechanics II: (1,4,3) models from the Delduc1, Ivanov2 2007Nuclear Physics B5031130View full textAdd to dashboardCite

“…In view of the distinguished role of supersymmetric quantum mechanics, both as the appropriate simplified "laboratory" for studying various aspects of supersymmetric quantum field theories and as a theory providing superextensions of some intrinsically one-dimensional systems, it is of importance to fully understand these specific features of the d = 1 supersymmetry and their dynamical manifestations in the corresponding models of supersymmetric mechanics. One of these peculiarities is the so-called "1D automorphic duality" [1]- [3] which relates off-shell d = 1 supermultiplets with the same number of physical fermions and different divisions of the set of bosonic fields into physical and auxiliary components (see also [4,5,6]). The procedure generating the multiplets with a greater number of auxiliary fields from those with the lesser number and the procedure inverse to it can be referred to as the "reduction" and "oxidation", respectively [7] 1 .…”

New Results in Supersymmetric Quantum Mechanics

“…The results here discussed open the way to the construction of off-shell invariant actions for large values of N, when the superfield formalism is not available. Already in [1] we were able to produce an N = 8 supersymmetric quantum mechanical system, not previously identified in the literature. The Cayley-Dickson's connection with the extended supersymmetry can also shed light to a possible off-shell formulation of the 11-dimensional supergravity thought as low-energy limit of the M-theory.…”

“…The problem of classifying the irreducible representations of the N-extended supersymmetry algebra is reduced to the problem of classifying the regular dressed operators. This problem has been solved in [1], by making use of specific properties of the Clifford algebras irreps. For any N, all length-3 multiplets of the type (n − k, n, k) are an irrep of the N-susy.…”

“…The classification of the irreducible representations of the one-dimensional N-extended supersymmetry has been given in [1]. In this talk we briefly summarize the main results contained in that paper and discuss at length one of the most promising frameworks for their application, based on the concept of dimensional reduction (or, conversely, "oxidation" to higher dimensions [2]).…”