Invariants of Spin Networks with Boundary in Quantum Gravity and TQFTs
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.
KeywordsSpin Variable Spin Network Topological Field Theory Inverse Move Empty Boundary
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- 1.Ponzano G., Regge T. (1968): Semiclassical Limit of Racah Coefficients, in Spectroscopic and Group Theoretical Methods, ed. by F. Bloch, F. in Physics, pp. 1–58. North-Holland, Amsterdam, pp. 1—58Google Scholar
- 4.Carter J.S., Kauffman L.H., Saito M. (1998): Structure and diagrammatics of four dimensional topological lattice field theories. Preprint, math. GT/9806 023Google Scholar
- 6.Carbone G., Carfora M., Marzuoli A.: Wigner symbols and combinatorial invariants of three-manifolds with boundary. Preprint DFNT-T 14/98 and SISSA 118/98/FMGoogle Scholar
- 7.Yutsis A.P., Levinson LB., Vanagas V.V. (1962): The Mathematical Apparatus of the Theory of Angular Momentum. Israel Program for Sci. Transi. Ltd. JerusalemGoogle Scholar
- 10.Carter J.S., Flath D.E., Saito M. (1995): The Classical and Quantum 6j-Symbols. Math. Notes 43. Princeton University Press, PrincetonGoogle Scholar
- 12.Varshalovich D.A., Moskalev A.N., Khersonski V.K. (1988): Quantum Theory of Angular Momentum. World Scientific, SingaporeGoogle Scholar