@article{cook_fulp_2011, title={HOLONOMY IN ROGERS SUPERMANIFOLDS WITH APPLICATIONS TO SUPER YANG-MILLS THEORY}, volume={8}, ISSN={["1793-6977"]}, DOI={10.1142/s0219887811005221}, abstractNote={The present paper focuses on a certain class of Banach manifolds we call Rogers supermanifolds since they are indeed supermanifolds modeled on graded Banach spaces. Although the subject of holonomy is well-developed for superanalytic supermanifolds utilizing local ring formulations of supermanifolds this seems not to be the case for supermanifolds modeled on graded Banach manifolds in the sense of Rogers. The proof of our main result requires a partial development of these concepts for such supermanifolds. Our main result determines conditions under which a super connection on a superprincipal bundle [Formula: see text] induces a connection on a quotient superprincipal bundle [Formula: see text] where [Formula: see text] is a foliation of [Formula: see text] and [Formula: see text] is the induced foliation on [Formula: see text]. We also show how such a quotient formulation may be used to describe in a fully geometric fashion the so-called "conventional constraints" of super Yang–Mills theory. One consequence of our development is that instead of requiring two superconnections to describe Yang–Mills theory as is the case in some formulations, we describe the relevant concepts using a single superconnection and moreover we show that the "pregauge transformations" are simply ordinary gauge transformations on the appropriate quotient bundles.}, number={2}, journal={INTERNATIONAL JOURNAL OF GEOMETRIC METHODS IN MODERN PHYSICS}, author={Cook, James S. and Fulp, Ronald}, year={2011}, month={Mar}, pages={429–458} }
@article{cook_fulp_2008, title={Infinite-dimensional super Lie groups}, volume={26}, ISSN={["0926-2245"]}, DOI={10.1016/j.difgeo.2008.04.009}, abstractNote={A super Lie group is a group whose operations are G ∞ mappings in the sense of Rogers. Thus the underlying supermanifold possesses an atlas whose transition functions are G ∞ functions. Moreover the images of our charts are open subsets of a graded infinite-dimensional Banach space since our space of supernumbers is a Banach Grassmann algebra with a countably infinite set of generators. In this context, we prove that if h is a closed, split sub-super Lie algebra of the super Lie algebra of a super Lie group G , then h is the super Lie algebra of a sub-super Lie group of G . Additionally, we show that if g is a Banach super Lie algebra satisfying certain natural conditions, then there is a super Lie group G such that the super Lie algebra g is in fact the super Lie algebra of G . We also show that if H is a closed sub-super Lie group of a super Lie group G , then G → G / H is a principal fiber bundle. We emphasize that some of these theorems are known when one works in the super-analytic category and also when the space of supernumbers is finitely generated in which case, one can use finite-dimensional techniques. The issues dealt with here are that our supermanifolds are modeled on graded Banach spaces and that all mappings must be morphisms in the G ∞ category.}, number={5}, journal={DIFFERENTIAL GEOMETRY AND ITS APPLICATIONS}, author={Cook, James and Fulp, Ronald}, year={2008}, month={Oct}, pages={463–482} }
@article{cook_2006, title={Gauged Wess-Zumino model in noncommutative Minkowski superspace}, volume={47}, ISSN={["1089-7658"]}, DOI={10.1063/1.2162330}, abstractNote={We develop a gauged Wess–Zumino model in noncommutative Minkowski superspace. This is the natural extension of the work of Carlson and Nazaryan, which extended N=1∕2 supersymmetry written over deformed Euclidean superspace to Minkowski superspace. We investigate the coupling of the vector and chiral superfields. Noncommutativity is implemented by replacing products with star products. Although, in general, our star product is nonassociative, we prove that it is associative to the first order in the deformation parameter C. We show that our model reproduces the N=1∕2 theory in the appropriate limit, namely when the deformation parameters C¯α̇β̇=0. Essentially, we find the N=1∕2 theory and a conjugate copy. As in the N=1∕2 theory, a reparametrization of the gauge parameter, vector superfield, and chiral superfield are necessary to write standard C-independent gauge theory. However, our choice of parametrization differs from that used in the N=1∕2 supersymmetry, which leads to some unexpected new terms.}, number={1}, journal={JOURNAL OF MATHEMATICAL PHYSICS}, author={Cook, JS}, year={2006}, month={Jan} }