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This work constructs a projective representation of the dotted Ptolemy groupoid at roots of unity, linking to the SU(2) Wess–Zumino–Witten model.

Cyclic Quantum Teichmüller Theory

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This work constructs a projective representation of the dotted Ptolemy groupoid at roots of unity, linking to the SU(2) Wess–Zumino–Witten model.
Explicit construction of finite-dimensional projective representations of the dotted Ptolemy groupoid.
Reinterpretation of cyclic quantum dilogarithm parameters for ensuring pentagon identity.
Geometric decomposition of quantum states into irreducible modules using the Chekhov–Fock algebra.
Demonstrated that the intertwiner coincides with the transpose of the reduced quantum hyperbolic operator (Baseilhac–Benedetti).
Proved that the mutation relation of coefficients is equivalent to the quantum gluing equation.
Presented partial evidence for the coincidence of the irreducible intertwiner with the Bonahon–Liu intertwiner.

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Randomized trial reveals interactions in quantum cluster algebras and Teichmüller space, indicating new mathematical structures.

Tsukasa Ishibashi (2026) studied this question.

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