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 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 as the Adinkra 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 symbolic method for organizing the representation theory of one-dimensional superalgebras. This relies on special objects, which we have called adinkra symbols, which supply tangible geometric forms to the still-emerging mathematical basis underlying supersymmetry.
Adinkras are diagrams that describe many useful supermultiplets in D = 1 dimensions. We show that the topology of the Adinkra is uniquely determined by a doubly even code. Conversely, every doubly even code produces a possible topology of an Adinkra. A computation of doubly even codes results in an enumeration of these Adinkra topologies up to N = 28, and for minimal supermultiplets, up to N = 32.
The vector-tensor multiplet is coupled off-shell to an N = 2 vector multiplet such that its central charge transformations are realized locally. A gauged central charge is a necessary prerequisite for a coupling to supergravity and the strategy underlying our construction uses the potential for such a coupling as a guiding principle. The results for the action and transformation rules take a nonlinear form and necessarily include a Chern-Simons term. After a duality transformation the action is encoded in a homogeneous holomorphic function consistent with special geometry.December 1995 † Wetenschappelijk Medewerker, NFWO, Belgium 1 For a recent discussion of N = 2 dilaton assignments, see [6]
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