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

2009

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

Freidel, Laurent, Livine, Etera R.

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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.

Built on

J. Hoppe, Diffeomorphism Groups, Quantization, and SU ⁡ ( ∞ ) \mathrm{SU}(\infty)
1989
Earlier work this paper cites.
C. Quesne, q q -bosonic operators as double irreducible tensors for u q ( n ) + u q ( m ) u_{q}(n)+u_{q}(m)
1993
Earlier work this paper cites.
A. Barbieri, “Quantum tetrahedra and simplicial spin networks,” Nucl. Phys. B 518
1998
Earlier work this paper cites.
K. Krasnov, The Area Spectrum in Quantum Gravity
1998
Earlier work this paper cites.
A. Ashtekar, J. Baez and K. Krasnov, Quantum Geometry of Isolated Horizons and Black Hole Entropy
2000
Earlier work this paper cites.

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Then

E.R. Livine and S. Speziale, A new spinfoam vertex for quantum gravity
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Later among the works it cites.
E.R. Livine and D. Terno, Bulk Entropy in Loop Quantum Gravity
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