Comparing Skein and Quantum Group Representations and Their Application to Asymptotic Faithfulness

Abstract

We make two related observations in this paper. First the representations of mapping class groups from the Ising TQFT and its quantum group counterpart SU (2) 2 are neither equivalent as representations nor Galois conjugate to each other. Hence mapping class group representations obtained from quantum skein theory are fundamentally distinct from those obtained from quantum group Reshetikhin-Turaev or geometric quantization constructions. Then we generalize the asymptotic faithfulness of the skein quantum SU (2) representations of mapping class groups of orientable closed surfaces to skein quantum SU (3). We conjecture asymptotic faithfulness holds for skein quantum G representations when G is a simply-connected simple Lie group. The difficulty for such a generalization lies in the lack of an explicit description of the fusion spaces with multiplicities to define an appropriate complexity of state vectors.

Asymptotic Faithfulness of Quantum $\mathrm{Sp}(4)$ Mapping Class Group Representations

“…Asymptotic faithfulness for skein SU (2) k , meaning for Jones-Kauffman TQFTs, was proven in [7], which is independent from the parallel asymptotic faithfulness of the representations from SU (n) k TQFTs [1]. The skein theoretic methods were extended to skein SU (3) k in [5].…”