Geometric construction of spinors in orthogonal modular categories

A geometric construction of Z_2-graded 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 for 1-graded simple objects (spinors) are calculated. We show that our even orthogonal modular categories admit cohomological refinements and the odd orthogonal ones lead to spin refinements. The relation with the quantum group approach is discussed.