Geometric construction of spinors in orthogonal modular categories

Abstract

A geometric construction of ${\mathbb{Z}}_2$–graded odd and even orthogonal modular categories is given. Their 0–graded parts coincide with categories previously obtained by Blanchet and the author from the category of tangles modulo the Kauffman skein relations. Quantum dimensions and twist coefficients of 1–graded simple objects (spinors) are calculated. We show that invariants coming from our odd and even orthogonal modular categories admit spin and ${\mathbb{Z}}_2$–cohomological refinements, respectively. The relation with the quantum group approach is discussed.