By A. I. Fetísov
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Baez Again both sides describe the projection on the subspace of vectors that transform in the trivial representation, and again we can write the formula as a generalized skein relation: ρ2 ❉ ρ1 ▲ ❉ ▲ ❉ ▲ ❉ ▲ ❉ ▲❉ ▲❉ ▲❉ ❉ ι ▲t ρ3 ▲ ✬✩ g dg = ✫✪ ... ...... ρ4 ι∗ t ☞ ☞ ☞ ☞ ι ☞ ☞ . ρ1........ ☞.. ☞ ☞ ρ4 ρ2 ρ3 ρ4 ρ3 ρ2 We use this formula once for each dual edge — or equivalently, once for each tetrahedron in the triangulation — to do the integral over all group elements in the partition function.
In this case of BF theory this halfway house works as follows. As before, let us start with an (n − 1)-dimensional real-analytic manifold S representing space. Given any triangulation of S we can choose a graph in S called the ‘dual 1-skeleton’, having one vertex at the center of each (n − 1)-simplex and one edge intersecting each (n − 2)-simplex. Using homotopies and skein relations, we can express any state in Fun(A0 /G) as a linear combination of states coming from spin networks whose underlying graph is this dual 1skeleton.
Xn , yn satisfying the relation −1 −1 −1 R(xi , yi ) := (x1 y1 x−1 1 y1 ) · · · (xn yn xn yn ) = 1. A point in hom(π1 (S), G) may thus be identified with a collection g1 , h1 , . . , gn , hn of elements of G satisfying R(gi , hi ) = 1, and a point in hom(π1 (S), G)/G is an equivalence class [gi , hi ] of such collections. The cases G = SU(2) and G = SO(3) are particularly interesting for their applications to 3-dimensional Riemannian general relativity. When G = SU(2), all G-bundles over a compact oriented surface S are isomorphic, and A0 /G = hom(π1 (S), G)/G.
Acerca de la Demostración en Geometría by A. I. Fetísov