Advances in High Energy Physics

Volume 2016, Article ID 7350892, 11 pages

http://dx.doi.org/10.1155/2016/7350892

## A -Continuum of Off-Shell Supermultiplets

^{1}Department of Physics & Astronomy, Howard University, Washington, DC 20059, USA^{2}Department of Physics, University of Central Florida, Orlando, FL 32816, USA^{3}Affine Connections, LLC, College Park, MD 20740, USA

Received 4 August 2015; Revised 15 October 2015; Accepted 22 October 2015

Academic Editor: Stefano Moretti

Copyright © 2016 Tristan Hübsch and Gregory A. Katona. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP^{3}.

#### Abstract

Within each supermultiplet in the standard literature, supersymmetry relates its bosonic and fermionic component fields in a fixed way, particularly to the selected supermultiplet. Herein, we describe supermultiplets wherein a* continuously variable* “tuning parameter” modifies the supersymmetry transformations, effectively parametrizing a novel “-continuum” of distinct finite-dimensional off-shell supermultiplets, which may be probed already with bilinear Lagrangians that couple to each other and to external magnetic fields, two or more of these continuously many supermultiplets, each “tuned” differently. The dependence on the tuning parameters cannot be removed by any field redefinition, rendering this “-*moduli space*” observable.

*“Discreteness is the refuge of the clumsy.”*

Jorge Hazzan

#### 1. Introduction, Results, and Synopsis

Supersymmetry has been studied for over forty years [1, 2], has had successful application in nuclear physics [3, 4] and critical phenomena [5, 6], and has recently found applications also in condensed matter physics: see the recent reviews [7, 8], for example. In quantum applications, the supermultiplets must be off-shell, that is, free of any (space) time-differential constraint that could play the role of the Euler-Lagrange (classical) equation of motion. The long-standing challenge of a systematic classification of off-shell supermultiplets [9, 10] has been addressed with significant success in the last decade or so; see [11–19] and references therein. One of the pivotal ideas enabling this recent development was the use of graph-theoretical methods [20–22] in assessing the structure of the supersymmetry transformations within the world-line dimensional reduction of off-shell supermultiplets, which turned out to relate the classification problem to encryption and coding theory [23–25].

Although this research program uncovered trillions of off-shell supermultiplets of world-line -extended supersymmetry, we show herein—corroborated by concurrent research [26]—that they provide merely a discrete subset of a vast *-continuum*, giving rise to a *-moduli space*. Furthermore, we prove herein that this novel -continuum is physically observable.

This may well come as a surprise, since both the continuous Lie algebras and the various discrete symmetry groups familiar from physics applications all have* discrete* sequences of inequivalent unitary and linear and finite-dimensional representations. Reference [27] showed that the infinite sequence of quotient supermultiplets specified in [21] defines a similarly infinite sequence of ever larger unitary, linear, and finite-dimensional off-shell representations of -extended world-line supersymmetry, and [28] finds highly nontrivial and continuously variable dynamics for the simplest of these supermultiplets, even with only bilinear Lagrangians.

Our present results, however, radically extend this line of research. By proving that each of these supermultiplets is merely a special member in a* continuum of distinct supermultiplets*, we prove that already bilinear Lagrangians of [28] can couple* continuously many* distinct supermultiplets. The coupling constants are therefore functions over this novel -moduli space, providing access to physically probe and observe this -continuum.

For simplicity and concreteness, we focus on the -extended world-line supersymmetry algebra without central charges: where is the Hamiltonian (in the familiar units) and are the supercharges, four real generators of supersymmetry. For concreteness, we focus on a particular set of supermultiplets (see (2) below) which were adapted from [28] by replacing one of the component bosons with its -derivative and renaming the component fields. Our present results then apply equally well not only to the supermultiplet of [27, 28] but also to the infinite sequence of ever larger supermultiplets constructed therein. Our present focus on world-line supersymmetry should nevertheless have implications for all supersymmetry, since, (a) by dimensional reduction, (1) is an integral part and common denominator of every supersymmetric theory, (b) it is directly relevant in diverse fields in physics, from candidates for the fundamental description of -theory [29] to the phenomenology of topological insulators and graphene [30], and (c) it shows up in the Hilbert space of every supersymmetric quantum theory. We defer the exploration of these implications to a subsequent effort.

The paper is organized as follows. Section 2 defines an illustrative 1-parameter family of indecomposable off-shell, unitary, and finite-dimensional supermultiplets and identifies the novel -continuum and the corresponding -moduli space, . Section 3 then explicitly constructs Lagrangians that, even though being just bilinear in fields, (1) inextricably depend on the tuning parameter , (2) pairwise couple continuously many inequivalent supermultiplets, and (3) provide for physical probing of this -continuum by coupling to external magnetic fields. Section 4 provides a token example of such nontrivial dynamics which essentially depend on the tuning parameter , and our conclusions are summarized in Section 5.

#### 2. The -Continuum of Off-Shell Supermultiplets

We proceed by way of a concrete example, introducing the following 1-parameter* family of variations* of the off-shell supermultiplet from [28]: Omitting the fourth supersymmetry, , would result in a minimal example of this -continuum; see Main Theorem of [26]. The inclusion of , however, proves that our results are not an artifact of “too few supersymmetries” and also affords possible extensions to higher-dimensional spacetimes to be explored separately.

The supermultiplet (2) may be depicted (graphical depictions of supersymmetry transformation rules are a time-tested intuitive tool [31] but have been rigorously formalized only recently [21], and we adopt those conventions) in the manner of Figure 1. Component fields are depicted as nodes and the -transformations between them are depicted as connecting edges, variously colored to correspond to the four supercharges ; these are drawn solid (dashed) to depict the positive (negative) signs in (2). This graphical rendition of the supermultiplet (2) at once reveals that the supermultiplet (2) consists of two identical submultiplets, and , which supersymmetry connects by the one-way transformations. Such one-way transformations are exemplified by the fact that contains , but does not contain ; this is depicted by the tapering edges crossing the dashed vertical divider in Figure 1.