Low Dimensional Topology by Samuel J. Lomonaco (ed.)

By Samuel J. Lomonaco (ed.)

Derived from a unique consultation on Low Dimensional Topology geared up and carried out by means of Dr Lomonaco on the American Mathematical Society assembly held in San Francisco, California, January 7-11, 1981

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Low Dimensional Topology

Derived from a distinct consultation on Low Dimensional Topology equipped and performed by way of Dr Lomonaco on the American Mathematical Society assembly held in San Francisco, California, January 7-11, 1981

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The area operators for surfaces do not commute when the surfaces intersect. There are three ways to chop a tetrahedron in half using a parallelogram, but we cannot simultaneously diagonalize the areas of these parallelograms, since they intersect. We can describe a basis of states for the quantum tetrahedron using 5 numbers: the areas of its 4 faces and any one of these parallelograms. Different ways of chopping tetrahedron in half gives us different bases of this sort, and the matrix relating these bases goes by the name of the ‘6j symbols’: ..

These reduce to Penrose’s original spin networks when the group is SU(2) and the graph is trivalent. Similarly, a spin foam is a 2-dimensional complex built from vertices, edges and polygonal faces, with the faces labeled by group representations and the edges labeled by intertwining operators. When the group is SU(2) and three faces meet at each edge, this looks exactly like a bunch of soap suds with all the faces of the bubbles labeled by spins — hence the name ‘spin foam’. If we take a generic slice of a spin foam, we get a spin network.

These are called ‘reducible’ connections. A more careful definition of the physical phase space would have to take these points into account. 2. The space A0 /G is called the ‘moduli space of flat connections on P |S ’. We can understand it better as follows. Since the holonomy of a flat connection around a loop does not change when we apply a homotopy to the loop, a connection A ∈ A0 determines a homomorphism from the fundamental group π1 (S) to G after we trivialize P at the basepoint p ∈ S that we use to define the fundamental group.

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