Complex Quantum Chern-Simons

2014/09/03 by Jørgen Ellegaard Andersen, Andersen, Jørgen Ellegaard, Rinat Kashaev +1 · 2 citations
Mathematics · Physics and Astronomy · #Algebraic structures and combinatorial models #Black Holes and Theoretical Physics #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Quantum Algebra (math.QA)

paper · pdf · doi:10.48550/arxiv.1409.1208

openalex publication_date 2014/09/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We lay down a general framework for how to construct a Topological Quantum Field Theory Z_A defined on shaped triangulations of orientable 3-manifolds from any Pontryagin self-dual locally compact abelian group A. The partition function for a triangulated manifold is given by a state integral over the LCA A of a certain combinations of functions which satisfy Faddeev's operator five term relation. In the cases where all elements of the LCA A are divisible by 2 and it has a subgroup B whose Pontryagin dual is isomorphic to A/B, this TQFT has an alternative formulation in terms of the space of sections of a line bundle over (A/B)². We apply this to the LCA ℝ× ℤ/Nℤ and obtain a TQFT, which we show is Quantum Chern-Simons theory at level N for the complex gauge group SL(2,ℂ) by the use of geometric quantization.

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• Quantum Dilogarithms and New Integrable Lattice Models in Three Dimensions
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