Kevin Iga

Codes and supersymmetry in one dimension

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.

Dimensional Enhancement via Supersymmetry

We explain how the representation theory associated with supersymmetry in diverse dimensions is encoded within the representation theory of supersymmetry in one time-like dimension. This is enabled by algebraic criteria, derived, exhibited, and utilized in this paper, which indicate which subset of one-dimensional supersymmetric models describes “shadows” of higher-dimensional models. This formalism delineates that minority of one-dimensional supersymmetric models which can “enhance” to accommodate extra dimensions. As a consistency test, we use our formalism to reproduce well-known conclusions about supersymmetric field theories using one-dimensional reasoning exclusively. And we introduce the notion of “phantoms” which usefully accommodate higher-dimensional gauge invariance in the context of shadow multiplets in supersymmetric quantum mechanics.

On the matter of N=2 matter

Abstract 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.

Super-Zeeman embedding models on N-supersymmetric world-lines

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.

A SUPERFIELD FOR EVERY DASH-CHROMOTOPOLOGY

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.

A Dynamical Systems Proof of Fermat's Little Theorem

This theorem, not to be confused with its more famous brother, Fermats Last Theorem, is useful not only when you want to compute what 69397 is modulo 13 (though it is useful for that kind of problem); it is also foundational to the RSA code in moder cryptology (Churchhouses book on codes [2] gives a readable account of this). Fermats little theorem and its generalizations make frequent appearances in any course in number theory and abstract algebra. There are many elementary proofs of this result: Almost every number theory textbook (like Niven and Zuckermans book An Introduction to the Theory of Numbers [5]) and abstract algebra textbook (like Artins Algebra [1] and Hersteins Topics in Algebra [4]) states, proves, and applies this result. Usually, the theorem is proved using algebraic ideas-some of them very abstract. In this note we will prove it with a simple argument from dynamical systems on the unit interval [0, 1]. The basic idea is to count points of minimum period p of a particular dynamical system, and note that this number must be divisible by p. A referee pointed out another dynamical-systems proof of this result, by Hausner [3]. Although that proof yields other impressive results beyond Fermats little theorem, it is much less visual.

Structural Theory and Classification of 2D Adinkras

Adinkras are combinatorial objects developed to study (1-dimensional) supersymmetry representations. Recently, 2D Adinkras have been developed to study -dimensional supersymmetry. In this paper, we classify all D Adinkras, confirming a conjecture of T. Hubsch. Along the way, we obtain other structural results, including a simple characterization of Hubsch’s even-split doubly even codes.

Hardness of learning problems over Burnside groups of exponent 3

In this work, we investigate the hardness of learning Burnside homomorphisms with noise (

The Truck Driver's Straw Problem and Cantor Sets

B_{n} \hbox {-}\mathsf {LHN}