No abstract

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 generalize the known method for explicit construction of mirror pairs of (2, 2)-superconformal field theories, using the formalism of Landau-Ginzburg orbifolds. Geometrically, these theories are realized as Calabi-Yau hypersurfaces in weighted projective spaces. This generalization makes it possible to construct the mirror partners of many manifolds for which the mirror was not previously known.