What remains of a geometrical notion like that of a principal bundle when the base space is not a manifold but a coarse graining of it, like the poset formed by a base for the topology ordered under inclusion? Motivated by the search for a geometrical framework for developing gauge theories in algebraic quantum field theory, we give, in the present paper, a first answer to this question. The notions of transition function, connection form and curvature form find a nice description in terms of cohomology, in general non-Abelian, of a poset with values in a group G. Interpreting a 1-cocycle as a principal bundle, a connection turns out to be a 1-cochain associated in a suitable way with this 1-cocycle; the curvature of a connection turns out to be its 2-coboundary. We show the existence of nonflat connections, and relate flat connections to homomorphisms of the fundamental group of the poset into G. We discuss holonomy and prove an analogue of the Ambrose-Singer theorem.

Roberts, J.E., & Ruzzi, G. (2006). A cohomological description of connections and curvature over posets. THEORY AND APPLICATIONS OF CATEGORIES, 16, 855-895.

A cohomological description of connections and curvature over posets

ROBERTS, JOHN ELIAS;RUZZI, GIUSEPPE
2006

Abstract

What remains of a geometrical notion like that of a principal bundle when the base space is not a manifold but a coarse graining of it, like the poset formed by a base for the topology ordered under inclusion? Motivated by the search for a geometrical framework for developing gauge theories in algebraic quantum field theory, we give, in the present paper, a first answer to this question. The notions of transition function, connection form and curvature form find a nice description in terms of cohomology, in general non-Abelian, of a poset with values in a group G. Interpreting a 1-cocycle as a principal bundle, a connection turns out to be a 1-cochain associated in a suitable way with this 1-cocycle; the curvature of a connection turns out to be its 2-coboundary. We show the existence of nonflat connections, and relate flat connections to homomorphisms of the fundamental group of the poset into G. We discuss holonomy and prove an analogue of the Ambrose-Singer theorem.
Pubblicato
Rilevanza internazionale
Articolo
Sì, ma tipo non specificato
Settore MAT/05 - Analisi Matematica
English
Con Impact Factor ISI
POSET;CONNECTIONS;CATEGORIES;GAUGE
Roberts, J.E., & Ruzzi, G. (2006). A cohomological description of connections and curvature over posets. THEORY AND APPLICATIONS OF CATEGORIES, 16, 855-895.
Roberts, Je; Ruzzi, G
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/2108/27716
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