Super Wilson Loops and Holonomy on Supermanifolds

Josua Groeger

Communications in Mathematics (2014)

  • Volume: 22, Issue: 2, page 185-211
  • ISSN: 1804-1388

Abstract

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The classical Wilson loop is the gauge-invariant trace of the parallel transport around a closed path with respect to a connection on a vector bundle over a smooth manifold. We build a precise mathematical model of the super Wilson loop, an extension introduced by Mason-Skinner and Caron-Huot, by endowing the objects occurring with auxiliary Graßmann generators coming from S -points. A key feature of our model is a supergeometric parallel transport, which allows for a natural notion of holonomy on a supermanifold as a Lie group valued functor. Our main results for that theory comprise an Ambrose-Singer theorem as well as a natural analogon of the holonomy principle. Finally, we compare our holonomy functor with the holonomy supergroup introduced by Galaev in the common situation of a topological point. It turns out that both theories are different, yet related in a sense made precise.

How to cite

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Groeger, Josua. "Super Wilson Loops and Holonomy on Supermanifolds." Communications in Mathematics 22.2 (2014): 185-211. <http://eudml.org/doc/269843>.

@article{Groeger2014,
abstract = {The classical Wilson loop is the gauge-invariant trace of the parallel transport around a closed path with respect to a connection on a vector bundle over a smooth manifold. We build a precise mathematical model of the super Wilson loop, an extension introduced by Mason-Skinner and Caron-Huot, by endowing the objects occurring with auxiliary Graßmann generators coming from $S$-points. A key feature of our model is a supergeometric parallel transport, which allows for a natural notion of holonomy on a supermanifold as a Lie group valued functor. Our main results for that theory comprise an Ambrose-Singer theorem as well as a natural analogon of the holonomy principle. Finally, we compare our holonomy functor with the holonomy supergroup introduced by Galaev in the common situation of a topological point. It turns out that both theories are different, yet related in a sense made precise.},
author = {Groeger, Josua},
journal = {Communications in Mathematics},
keywords = {supermanifolds; holonomy; group functor; supermanifolds; holonomy; group functor},
language = {eng},
number = {2},
pages = {185-211},
publisher = {University of Ostrava},
title = {Super Wilson Loops and Holonomy on Supermanifolds},
url = {http://eudml.org/doc/269843},
volume = {22},
year = {2014},
}

TY - JOUR
AU - Groeger, Josua
TI - Super Wilson Loops and Holonomy on Supermanifolds
JO - Communications in Mathematics
PY - 2014
PB - University of Ostrava
VL - 22
IS - 2
SP - 185
EP - 211
AB - The classical Wilson loop is the gauge-invariant trace of the parallel transport around a closed path with respect to a connection on a vector bundle over a smooth manifold. We build a precise mathematical model of the super Wilson loop, an extension introduced by Mason-Skinner and Caron-Huot, by endowing the objects occurring with auxiliary Graßmann generators coming from $S$-points. A key feature of our model is a supergeometric parallel transport, which allows for a natural notion of holonomy on a supermanifold as a Lie group valued functor. Our main results for that theory comprise an Ambrose-Singer theorem as well as a natural analogon of the holonomy principle. Finally, we compare our holonomy functor with the holonomy supergroup introduced by Galaev in the common situation of a topological point. It turns out that both theories are different, yet related in a sense made precise.
LA - eng
KW - supermanifolds; holonomy; group functor; supermanifolds; holonomy; group functor
UR - http://eudml.org/doc/269843
ER -

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