By Michael Artin
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Additional info for Grothendieck topologies: Notes on a seminar. Spring, 1962
A Here γi is a real-analytic path in S, T e is the holonomy of A along this path, and f is a continuous complex-valued function of finitely many such holonomies. Then we define an inner product on Fun(A) and complete it to obtain the Hilbert space L2 (A). To define this inner product, we need to think about graphs embedded in space: Definition 1. A finite collection of real-analytic paths γi : [0, 1] → S form a graph in S if they are embedded and intersect, it at all, only at their endpoints. We then call them edges and call their endpoints vertices.
We conclude with a word about double covers. We can also express general relativity in 3 dimensions as a BF theory by taking the double cover Spin(2, 1) ∼ = SL(2, R) or Spin(3) ∼ = SU(2) as gauge group and letting P be the spin bundle. This does not affect the classical theory. As we shall see, it does affect the quantum theory. Nonetheless, it is very popular to take these groups as gauge groups in 3-dimensional quantum gravity. The question whether it is ‘correct’ to use these double covers as gauge groups seems to have no answer — until we couple quantum gravity to spinors, at which point the double cover is necessary.
This is not true in the smooth context: for example, two smoothly embedded paths can intersect in a Cantor set. One can generalize the construction of L2 (A) and L2 (A/G) to the smooth context, but one needs a generalization of graphs known as ‘webs’. The smooth and real-analytic categories are related as nicely as one could hope: a paracompact smooth manifold of any dimension admits a real-analytic structure, and this structure is unique up to a smooth diffeomorphism. 2. There are various ways to modify a spin network in S without changing the state it defines: – We can reparametrize an edge by any orientation-preserving diffeomorphism of the unit interval.