4 research outputs found
Generating Functions for Coherent Intertwiners
We study generating functions for the scalar products of SU(2) coherent
intertwiners, which can be interpreted as coherent spin network evaluations on
a 2-vertex graph. We show that these generating functions are exactly summable
for different choices of combinatorial weights. Moreover, we identify one
choice of weight distinguished thanks to its geometric interpretation. As an
example of dynamics, we consider the simple case of SU(2) flatness and describe
the corresponding Hamiltonian constraint whose quantization on coherent
intertwiners leads to partial differential equations that we solve.
Furthermore, we generalize explicitly these Wheeler-DeWitt equations for SU(2)
flatness on coherent spin networks for arbitrary graphs.Comment: 31 page
On the volume conjecture for polyhedra
22 pages, 2 figuresInternational audienceWe formulate a generalization of the volume conjecture for planar graphs. Denoting by the Kauffman bracket of the graph G whose edges are decorated by real "colors" c, the conjecture states that, under suitable conditions, certain evaluations of grow exponentially as k goes to infinity and the growth rate is the volume of a truncated hyperbolic hyperideal polyhedron whose one-skeleton is G (up to a local modification around all the vertices) and with dihedral angles given by c. We provide evidence for it, by deriving a system of recursions for the Kauffman brackets of planar graphs, generalizing the Gordon-Schulten recursion for the quantum 6j-symbols. Assuming that does grow exponentially these recursions provide differential equations for the growth rate, which are indeed satisfied by the volume (the Schlafli equation); moreover, any small perturbation of the volume function that is still a solution to these equations, is a perturbation by an additive constant. In the appendix we also provide a proof outlined elsewhere of the conjecture for an infinite family of planar graphs including the tetrahedra