Geometric construction of spinors in orthogonal modular categories

A geometric construction of 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 Z(2)-cohomological refinements, respectively. The relation with the quantum group approach is discussed.