We explain how the redefinitions of supermultiplet component fields, comprising what we call "frame shifts," can be used in conjuction with the graphical technology of multiplet Adkinras to render manifest the reducibility of off-shell representations of supersymmetry. This technology speaks to possibility of organizing multiplet constraints in a way which complements and extends the possibilities afforded by superspace methods.

We construct a model of an electrically charged magnetic dipole with arbitrary N-extended world-line supersymmetry, which exhibits a supersymmetric Zeeman effect. By including supersymmetric constraint terms, the ambient space of the dipole may be tailored into an algebraic variety, and the supersymmetry broken for almost all parameter values. The so-exhibited obstruction to supersymmetry breaking refines the standard one, based on the Witten index alone.

The recent classification scheme of so-called adinkraic off-shell supermultiplets of N-extended worldline supersymmetry without central charges finds a combinatorial explosion. Completing our earlier efforts, we now complete the constructive proof that all of these trillions or more of supermultiplets have a superfield representation. While different as superfields and supermultiplets, these are still super-differentially related to a much more modest number of minimal supermultiplets, which we construct herein.

We present further progress toward a complete classification scheme for describing supermultiplets of N-extended worldline supersymmetry, which relies on graph-theoretic topological invariants. In particular, we demonstrate a relationship between Adinkra diagrams and quotients of N-dimensional cubes, where the quotient groups are subgroups of $(Z_2)^N$. We explain how these quotient groups correspond precisely to doubly even binary linear error-correcting codes, so that the classification of such codes provides a means for describing equivalence classes of Adinkras and therefore supermultiplets. Using results from coding theory we exhibit the enumeration of these equivalence classes for all cases up to 26 supercharges, as well as the maximal codes, corresponding to minimal supermultiplets, for up to 32 supercharges.

Adinkras are a graphical depiction of representations of the N-extended supersymmetry algebra in one dimension, on the worldline. These diagrams represent the component fields in a supermultiplet as vertices, and the action of the supersymmetry generators as edges. In a previous work, we showed that the chromotopology (topology with colors) of an Adinkra must come from a doubly even binary linear code. Herein, we relate Adinkras to Clifford algebras, and use this to construct, for every such code, a supermultiplet corresponding to that code. In this way, we correlate the well-known classification of representations of Clifford algebras to the classification of Adinkra chromotopologies.

We introduce a variety of four-dimensional N=2 matter multiplets which have not previously appeared explicitly in the literature. Using these, we develop a class of supersymmetric actions supplying a context for a systematic exploration of N=2 matter theories, some of which include Hypermultiplet sectors in novel ways. We construct an N=2 supersymmetric field theory in which the propagating fields are realized off-shell exclusively as Lorentz scalars and Weyl spinors and which involves a sector with precisely the R-charge assignments characteristic of Hypermultiplets.

Previous work has shown that the classification of indecomposable off-shell representations of N-supersymmetry, depicted as Adinkras, may be factored into specifying the topologies available to Adinkras, and then the height-assignments for each topological type. The latter problem being solved by a recursive mechanism that generates all height-assignments within a topology, it remains to classify the former. Herein we show that this problem is equivalent to classifying certain (1) graphs and (2) error-correcting codes.

In this paper, we discuss off-shell representations of N-extended supersymmetry in one dimension, i.e. N-extended supersymmetric quantum mechanics, and following earlier work on the subject, we codify them in terms of graphs called Adinkras. This framework provides a method of generating all Adinkras with the same topology, and so also all the corresponding irreducible supersymmetric multiplets. We develop some graph theoretic techniques to understand these diagrams in terms of a relatively small amount of information, namely, at what heights various vertices of the graph should be "hung." We then show how Adinkras that are the graphs of N-dimensional cubes can be obtained for superfields satisfying constraints that involve superderivatives. This dramatically widens the range of supermultiplets that can be described using the superspace formalism and also organizes them. Other topologies for Adinkras are possible, and we show that it is reasonable that these are also the result of constraining superfields using superderivatives. We arrange the family of Adinkras with an N-cubical topology, and so also the sequence of corresponding irreducible supersymmetric multiplets, in a cyclic sequence, which we call the main sequence. We produce the N = 1 and N = 2 main sequences in detail, and indicate some aspects of the situation for higher N.

We present a counter-example to the recent claim that supermultiplets of N-extended supersymmetry with no central charge and in 1-dimension are specified unambiguously by providing the numbers of component fields in all available engineering dimensions within the supermultiplet.

We demonstrate a method for describing one-dimensional N-extended supermultiplets and building supersymmetric actions in terms of unconstrained prepotential superfields, explicitly working with the Scalar supermultiplet. The method uses intuitive manipulations of Adinkras and GR(d,N) algebras, a variant of Clifford algebras. In the process we clarify the relationship between Adinkras, GR(d,N) algebras, and superspace.

We develop a systematic method for classifying supersymmetric orbifold compactifications of M-theory. By restricting our attention to Abelian orbifolds with low order, in the special cases where elements do not include coordinate shifts, we construct a “periodic table” of such compactifications, organized according to the orbifolding group (order ⩽12) and dimension (up to 7). An intriguing connection between supersymmetric orbifolds and G2 structures is explored.

We explicitly derive a complementary pair of four-dimensional M-theory brane-world models, linked by a five-dimensional bulk, each of which has a unique anomaly-free chiral spectrum. This is done via resolution of local consistency requirements, in the context of the simplest global quotient T7/Γ involving ten-dimensional fixed-planes, for which a chiral four-dimensional spectrum could arise.