Kauffman skein algebras and quantum Teichmüller spaces via factorization homology

Abstract

We compute the factorization homology of the four-punctured sphere and punctured torus over the quantum group [Formula: see text] explicitly as categories of equivariant modules using the framework developed by Ben-Zvi et al. We identify the algebra of [Formula: see text]-invariants (quantum global sections) with the spherical double affine Hecke algebra of type [Formula: see text], in the four-punctured sphere case, and with the “cyclic deformation” of [Formula: see text] in the punctured torus case. In both cases, we give an identification with the corresponding quantum Teichmüller space as proposed by Teschner and Vartanov as a quantization of the moduli space of flat connections.