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Invariants of Spin Networks with Boundary in Quantum Gravity and TQFTs

  • G. Carbone
  • M. Carfora
  • A. Marzuoli

Abstract

The search for classical or quantum combinatorial invariants of compact n-dimensional manifolds (n = 3, 4) plays a key role both in topological field theories and in lattice quantum gravity (see, e.g., [1, 2, 3, 4]). We present here a generalization of the partition function proposed by Ponzano and Regge to the case of a compact 3-dimensional simplicial pair (M 3, ∂M 3). The resulting state sum Z[(M 3, ∂M 3)] contains both Racah-Wigner 6j symbols associated with tetrahedra and Wigner 3 jm symbols associated with triangular faces lying in ∂M 3. The analysis of the algebraic identities associated with the combinatorial transformations involved in the proof of the topological invariance makes manifest a common structure underlying the 3-dimensional models with empty and non-empty boundaries respectively. The techniques developed in the 3-dimensional case can be further extended in order to deal with combinatorial models for n = 2,4 and possibly to establish a hierarchy among such models. As an example we derive here a 2-dimensional closed state sum model including suitable sums of products of double 3 jm symbols, each of them being associated with a triangle in the surface.

Keywords

Spin Variable Spin Network Topological Field Theory Inverse Move Empty Boundary 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Italia 2000

Authors and Affiliations

  • G. Carbone
  • M. Carfora
  • A. Marzuoli

There are no affiliations available

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