Holomorphic Factorization for a Quantum Tetrahedron

Abstract

We provide a holomorphic description of the Hilbert space ${\mathcal{H}_{j_1,\ldots,j_n}}$ of SU(2)-invariant tensors (intertwiners) and establish a holomorphically factorized formula for the decomposition of identity in ${\mathcal{H}_{j_1,\ldots,j_n}}$. Interestingly, the integration kernel that appears in the decomposition formula turns out to be the n-point function of bulk/boundary dualities of string theory. Our results provide a new interpretation for this quantity as being, in the limit of large conformal dimensions, the exponential of the Kähler potential of the symplectic manifold whose quantization gives ${\mathcal{H}_{j_1,\ldots,j_n}}$. For the case n = 4, the symplectic manifold in question has the interpretation of the space of “shapes” of a geometric tetrahedron with fixed face areas, and our results provide a description for the quantum tetrahedron in terms of holomorphic coherent states. We describe how the holomorphic intertwiners are related to the usual real ones by computing their overlap. The semi-classical analysis of these overlap coefficients in the case of large spins allows us to obtain an explicit relation between the real and holomorphic description of the space of shapes of the tetrahedron. Our results are of direct relevance for the subjects of loop quantum gravity and spin foams, but also add an interesting new twist to the story of the bulk/boundary correspondence.