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Irreducible self-adjoint representations of quantum Teichm\'uller space and the phase constants
- Publication Year :
- 2020
-
Abstract
- Quantization of the Teichm\"uller space of a non-compact Riemann surface has emerged in 1980's as an approach to three dimensional quantum gravity. For any choice of an ideal triangulation of the surface, Thurston's shear coordinate functions on the edges form a coordinate system for the Teichm\"uller space, and they should be replaced by suitable self-adjoint operators on a Hilbert space. Upon a change of triangulations, one must construct a unitary operator between the Hilbert spaces intertwining the quantum coordinate operators and satisfying the composition identities up to multiplicative phase constants. In the well-known construction by Chekhov, Fock and Goncharov, the quantum coordinate operators form a family of reducible representations, and the phase constants are all trivial. In the present paper, we employ the harmonic-analytic theory of the Shale-Weil intertwiners for the Schr\"odinger representations, as well as Faddeev-Kashaev's quantum dilogarithm function, to construct a family of irreducible representations of the quantum shear coordinate functions and the corresponding intertwiners for the changes of triangulations. The phase constants are explicitly computed and described by the Maslov indices of the Lagrangian subspaces of a symplectic vector space, and by the pentagon relation of the flips of triangulations. The present work may generalize to the cluster $\mathscr{X}$-varieties.
- Subjects :
- 20G05, 43A25, 47A67, 81R60, 22D10, 16G99
Pure mathematics
010102 general mathematics
Hilbert space
General Physics and Astronomy
01 natural sciences
Linear subspace
Fock space
Quantization (physics)
Symplectic vector space
symbols.namesake
0103 physical sciences
symbols
Quantum gravity
010307 mathematical physics
Geometry and Topology
Unitary operator
0101 mathematics
Self-adjoint operator
Mathematics - Representation Theory
Mathematical Physics
Mathematics
Subjects
Details
- Language :
- English
- Database :
- OpenAIRE
- Accession number :
- edsair.doi.dedup.....a724b1599fceadca428badbf1071dfdc