The theory of quantum invariants of 3-manifolds was developed by N. Reshetikhin and V. G. Turaev in [Invent. Math. 103, No. 3, 547–597 (1991; Zbl 0725.57007)]. These invariants are known as WRT quantum invariants. This development was inspired by E. Witten’s work in [Comm. Math. Phys. 121, No. 3, 351–399 (1989; Zbl 0667.57005)]. This theory was later formalized by Turaev by introducing the notion of modular categories [V. G. Turaev, Quantum invariants of knots and 3-manifolds. Berlin: Walter de Gruyter & Co. (1994; Zbl 0812.57003)]. His idea was that modular categories give rise to a 3-dimensional Topological Quantum Field Theory (TQFT), and in particular, to quantum 3-manifold invariants.
Modular categories equipped with extra structures allow to define refinements of quantum 3-manifold invariants. The authors construct quantum invariants using modular categories equipped with cohomology classes or generalized spin and complex spin structures. Their construction generalizes all known cohomological or spin type refinements of the WRT 3-manifold quantum invariants. Also, they provide a splitting formula for the refined invariants generalizing the well-known decomposition of quantum invariants into projective ones and those determined by the linking matrix.