The Fine Structure of SU(2) Intertwiners from U(N) Representations

TL;DR

This work reveals that the space of N-valent SU(2) intertwiners at fixed total area forms an irreducible representation of ${\mathrm{U}}(N)$, realized through a Schwinger-oscillator construction that yields a closed ${\mathfrak{u}}(N)$ algebra commuting with ${\mathfrak{su}}(2)$. The authors identify the ${\mathrm{U}}(N)$ highest weight as $[l,l,0,\dots]$, interpret the total area as the ${\mathrm{U}}(1)$ Casimir, and show the action corresponds to area-preserving diffeomorphisms of the boundary surface, linking it to polyhedral geometry and spin-network dynamics. They develop generating functionals to count intertwiners, relate intertwiner counts to ${\mathrm{U}}(N)$ representation dimensions, and address black hole entropy via a binomial-transformed, no-trivial-puncture count, obtaining holographic leading behavior and a universal $-\tfrac{3}{2}\ln l$ correction. The formalism extends naturally to spin networks on graphs, with a per-vertex ${\mathrm{Gr}}_{2,N}$ description, and offers generalizations to ${\mathrm{SU}}(d)$, supersymmetric, and quantum-deformed cases, suggesting deep connections to diffeomorphism symmetry, matrix models, and possible integrable structures in loop quantum gravity.

Abstract

In this work we study the Hilbert space space of N-valent SU(2) intertwiners with fixed total spin, which can be identified, at the classical level, with a space of convex polyhedra with N face and fixed total boundary area. We show that this Hilbert space provides, quite remarkably, an irreducible representation of the U(N) group. This gives us therefore a precise identification of U(N) as a group of area preserving diffeomorphism of polyhedral spheres. We use this results to get new closed formulae for the black hole entropy in loop quantum gravity.