Advances in Theoretical and Mathematical Physics

Volume 23 (2019)

Number 8

Homotopy classes of gauge fields and the lattice

Pages: 2207 – 2254

DOI: https://dx.doi.org/10.4310/ATMP.2019.v23.n8.a7

Authors

Claudio Meneses (Centro de Investigación en Matemáticas, Valenciana, Guanajuato, México; and Mathematisches Seminar, Christian-Albrechts Universität zu Kiel, Germany)

José A. Zapata (Centro de Ciencias Matemáticas, Universidad Nacional Autónoma de México (UNAM), Morelia, México; and Department of Applied Mathematics, University of Waterloo, Ontario, Canada)

Abstract

For a smooth manifold $M$, possibly with boundary and corners, and a Lie group $G$, we consider a suitable description of gauge fields in terms of parallel transport, as groupoid homomorphisms from a certain path groupoid in $M$ to $G$. Using a cotriangulation $\mathscr{C}$ of $M$, and collections of finite-dimensional families of paths relative to $\mathscr{C}$, we define a homotopical equivalence relation of parallel transport maps, leading to the concept of an extended lattice gauge (ELG) field. A lattice gauge field, as used in Lattice Gauge Theory, is part of the data contained in an ELG field, but the latter contains further local topological information sufficient to reconstruct a principal $G$-bundle on $M$ up to equivalence. The space of ELG fields of a given pair $(M, \mathscr{C})$ is a covering for the space of fields in Lattice Gauge Theory, whose connected components parametrize equivalence classes of principal $G$-bundles on $M$. We give a criterion to determine when ELG fields over different cotriangulations define equivalent bundles.

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This work was partially supported by grant PAPIITUNAM IN 109415. The first-named author was partially supported by the DFG SPP 2026 priority programme “Geometry at infinity”. The second-named author was supported by a sabbatical grant by PASPA-UNAM.