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In particle physics, a field theory is a specification of how a field (such as functions or vector fields on spacetime) changes with time or with respect to other components of the field. A gauge theory is a field theory in terms of equivalence classes of covector fields.
Supersymmetry is a symmetry that relates bosons and fermions, the two fundamental classes of elementary particles. Together with rotations and translations, they form the super Poincare group. To study this group, we view the translations as generating a polynomial ring and the rotations as a standard (Lie or matrix) group. A representation maps elements of a group or ring to linear transformations or matrices on a vector space. We study a representation of the super Poincare group which maps the polynomial ring and the supersymmetry algebra to linear transformations on a module. Modules are basically vector spaces, except that their scalars come from polynomial rings. We show that in this that the equivalence classes of gauge fields are dual to a special submodule of a free module. This submodule is called the syzygy submodule.
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Tran, Giang, "A Module-Theoretic Approach to Supersymmetric Gauge Theory" (2012). Senior Projects Spring 2012. 214.