Skein Module Deformations and the Asymptotics of Quantum Invariants in Witten-Reshetikhin-Turaev Theory

Mathematical Applications of Science Fiction We present a rigorous derivation of the asymptotic expansionof the Witten-Reshetikhin-Turaev (WRT) invariants Zk(M )for closed 3-manifolds as the level k → ∞. By constructinga deformation quantization of the SL(2, C) character varietyvia the Kauffman bracket skein module SA(M ), we establisha correspondence between the semi-classical limit of the skeinalgebra and the symplectic geometry of the moduli space offlat connections. We introduce the Transfinite Skein Sheaf,a cohomological apparatus that resolves the singularities ofthe character variety, allowing for a precise WKB analysisof the partition function. We prove that the WRT invariantsatisfies a holonomic difference equation (the AJ-conjecture)and that its asymptotic phase is determined by the Chern-Simons functional evaluated on the geometric component ofthe character variety.