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Research Article | Open Access
Tristan Hübsch, Gregory A. Katona, "A -Continuum of Off-Shell Supermultiplets", Advances in High Energy Physics, vol. 2016, Article ID 7350892, 11 pages, 2016. https://doi.org/10.1155/2016/7350892
A -Continuum of Off-Shell Supermultiplets
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.”
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 —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  showed that the infinite sequence of quotient supermultiplets specified in  defines a similarly infinite sequence of ever larger unitary, linear, and finite-dimensional off-shell representations of -extended world-line supersymmetry, and  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  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  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  to the phenomenology of topological insulators and graphene , 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 : Omitting the fourth supersymmetry, , would result in a minimal example of this -continuum; see Main Theorem of . 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  but have been rigorously formalized only recently , 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.
Surprisingly—and radically extending our previous work on the topic [27, 28]—we find that these one-way -transformations admit a continuous “tuning parameter.” Denoting in the tabulation (2), its value is in no way restricted by the supersymmetry algebra relations (1)! That is, the supersymmetry transformations (2) close the algebraic relations (1) on every given component field with no need of any -differential condition and for each possible value of separately. The tabulation (2) is thus a continuous 1-parameter family of proper off-shell representations of -extended supersymmetry on the world-line. By contrast, the trillions of supermultiplets reported in [22, 25] as well as those of [27, 28] and all known supermultiplets [1, 32] form at most discrete sequences.
The supermultiplet (2) then is one of the simplest examples of the -continuum; see also . In turn the real line is the corresponding coarse -moduli space. Explicit choices of the Lagrangian will determine corresponding actions of a mapping class group, , whereby becomes the true (and model-dependent) -moduli space; see below.
The special value decomposes the supermultiplet (2) into two separate off-shell -dimensional supermultiplets, both of which being the world-line dimensional reduction of the familiar chiral supermultiplet [1, 21]. When , the off-shell supermultiplet (2) cannot be decomposed as a direct sum of two separate supermultiplets. The off-shell supermultiplets of -extended supersymmetry considered in [27, 28] may be similarly extended to depend on a precisely analogous tuning parameter, dialing the “magnitude” of the one-way -transformations connecting the two halves of the supermultiplet; see Figure 1. Those supermultiplets are closely related to the version of (2): except for some renaming of component fields, one merely needs to drop the fourth supersymmetry and replace , effectively lowering the corresponding node (top, left) to the bottom level in the graph in Figure 1. This, however, obstructs dimensional extension even to just world-sheet supersymmetry , providing our main motivation to consider (2) instead of the slightly simpler supermultiplet of [27, 28].
Explicit attempts verify that no local component field redefinition can remove the parameter from the supersymmetry transformations (2). As discussed subsequently, all efforts to eliminate from the -transformations must involve nonlocal transformations; see also (6) below.
The table (2) thus defines a 1-parameter continuum of indecomposable off-shell, unitary, and linear representations of world-line -extended supersymmetry, , parametrizing this -continuum and providing a coarse parametrization for the corresponding -moduli space.
We now turn to show that the supersymmetry tuning parameter does show up in the dynamics, is observable, and makes any two such supermultiplets, each with a different -value, usefully inequivalent in the sense of : using supermultiplets with different tuning parameter values permits constructing Lagrangians which could not be constructed without this variation.
To prove this, we construct sufficiently general Lagrangians for direct use in classical applications and in quantum models using the corresponding Hamiltonian, , or via the partition functional .
3.1. Simple Kinetic Terms
Following the procedure employed in , we use the fact that any Lagrangian of the form is automatically supersymmetric, since its -transformation necessarily produces a total -derivative. This is the direct adaptation of the construction of the so-called -terms in standard treatments of supersymmetry [1, 2].
Dimensional analysis dictates that for kinetic-type Lagrangians we need to be bilinear in the component fields ; this will produce terms of the form , , , and as appropriate for kinetic terms. Table 1 lists the individually supersymmetric Lagrangian summands obtained this way after dropping total -derivatives. As shown, the ten bilinear functions result in six linearly independent terms, so we define and we read off the actual summands from Table 1 to save space. For example, defines the “standard-looking” kinetic terms for this supermultiplet. Herein, the local component field redefinition would completely eliminate the appearance of the continuous tuning parameter from the “standard-looking” Lagrangian (5) and would thus seem to render the supermultiplets (2) with various values of the tuning parameter physically equivalent to each other. We note in passing that field redefinition (6) complicates the transformation table (2), the effect of which is that the partition-crossing edges in the graph in Figure 1 become regular, “two-way” edges, hiding the reducibility of the supermultiplet (2).
|The entry is not simplified further to facilitate comparison with Table 2.|