Modular categories are a well-known source of quantum 3-manifold invariants. In this paper we study structures on modular categories which allow to define refinements of quantum 3-manifold invariants involving cohomology classes or generalized spin and complex spin structures. A crucial role in our construction is played by objects which are invertible under tensor product. All known examples of cohomological or spin type refinements of the Witten-Reshetikhin-Turaev 3-manifold invariants are special cases of our construction. In addition, we establish a splitting formula for the refined invariants, generalizing the well-known product decomposition of quantum invariants into projective ones and those determined by the linking matrix.

中文翻译：

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自旋模块化类别

模范畴是量子 3-流形不变量的著名来源。在本文中，我们研究了模块化类别的结构，这些结构允许定义涉及上同调类或广义自旋和复杂自旋结构的量子 3 流形不变量的改进。在我们的构造中，在张量积下可逆的对象发挥了关键作用。所有已知的 Witten-Reshetikhin-Turaev 3-流形不变量的上同调或自旋类型改进的例子都是我们构造的特例。此外，我们建立了一个细化不变量的分裂公式，将众所周知的量子不变量的乘积分解推广为射影不变量和由链接矩阵确定的那些。