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Apparently there is a notion of for example a $G$-connection on a discrete set. I've understood that this is a standard tool in for example lattice gauge theory. I'm looking for references to learn more about this (i.e. discrete 1-forms, connections, etc.) in the discrete setting.

More specifically, suppose I have a set $V$ of vertices and a set $E$ of directed edges. Let $G$ be a finite (perhaps non-abelian) group. What is the right notion of $G$-connection on $(V,E)$? Is there some classification of connections? In particular, I suppose it might matter if the directed graph can be drawn on a torus or on a sphere (or the plane).

Any references to literature where the basic notions are described is greatly appreciated.

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2 Answers

up vote 6 down vote accepted

I'd begin by looking at Oeckl's "Discrete gauge theory": http://www.amazon.com/Discrete-Gauge-Theory-Lattices-Tqft/dp/1860945791

For a finite group $G$, the notion of a $G$-connection is easy to define; it is usually done when you have not just a graph but a 2-complex and add a condition that the monodromy of connection around the boundary of each 2-cell is trivial. If you apply it to a cell decomposition of a closed surface, you get the space of gauge equivalence classes of $G$-bundles on the surface, so teh result does not depend on the choice of cell decomposition. This is a special (and well-known) example of Turaev-Viro theory, which in this case is also known as Chern-Simons theory with a finite gauge group (see paper of Freed and Quinn in CMP vol. 156).

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Thanks! That book looks quite interesting, I'll try to get hold of it. –  Pieter Naaijkens Nov 11 '10 at 15:40
To get things straight, is the monodromy condition a condition on the connection, or does this define a flat connection? –  Pieter Naaijkens Nov 12 '10 at 9:42
For a finite group, every connection in a $G$-bundle over a manifold is automatically flat. –  Sasha Kirillov Nov 13 '10 at 4:30
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You might also want to check out the papers of Tony Phillips and David Stone (MathSciNet link) in the 80s/90s.

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