A quantization of the SL₂(ℂ) Chern-Simons invariant of tangle exteriors

2025/09/02 by Calvin McPhail-Snyder, McPhail-Snyder, Calvin
Mathematics · #Advanced Operator Algebra Research #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #Quantum Algebra (math.QA)

paper · pdf · doi:10.48550/arxiv.2509.02365

openalex publication_date 2025/09/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We define a sequence of invariants Z_N^ψ of tangles with flat \mathfraksl₂ connections (i.e. hyperbolic structures) on their complements. These can be interpreted as a geometric twist of the Kashaev invariant or as a quantization of the SL₂(ℂ) Chern-Simons invariant. To support the second interpretation we give a new description I^ψ of the Chern-Simons invariant of a tangle exterior. Z_N^ψ directly recovers I^ψ when N = 1. We build Z_N^ψ using modules over unrestricted quantum \mathfraksl₂ at a root of unity and the holonomy R-matrices previously constructed by the author and Reshetikhin (arXiv:2509.02354). Unlike most previous constructions of geometric quantum invariants Z_N^ψ is defined without any phase ambiguity. It is natural to conjecture that Z_N^ψ is related to the quantization of Chern-Simons theory with complex, noncompact gauge group SL₂(ℂ) and we discuss how to interpret our results in this context.

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