The level-k $U(1)$ Chern-Simons theory is a spin topological quantum field theory for k odd. Its dynamics is captured by the 2d CFT of a compact boson with a certain radius. Recently it was recognized that a dependence on the 2d spin structure can be given to the CFT by modifying it using the so-called Arf invariant. We demonstrate that one can reorganize the torus partition function of the modified CFT into a finite sum involving a finite number of conformal blocks. This allows us to reproduce the modular matrices of the spin theory. We use the modular matrices to calculate the partition function of the spin Chern-Simons theory on the lens space $L(a,\pm 1)$, and demonstrate the expected dependence on the 3d spin structure.

水平-k $ü（\mathrm{1个}）$Chern-Simons理论是k个奇数的自旋拓扑量子场理论。它的动力学被具有一定半径的紧凑玻色子的2 CFT捕获。最近，人们认识到，可以通过使用所谓的Arf不变量修改CFT，从而赋予CFT自旋结构以依赖性。我们证明了可以将修改后的CFT的环面划分函数重组为涉及有限数量的保形区块的有限总和。这使我们能够重现自旋理论的模块化矩阵。我们使用模块化矩阵来计算自旋Chern-Simons理论在透镜空间上的分配函数$\mathrm{大号}（\mathrm{一种}，\pm \mathrm{1个}）$，并证明了对3d自旋结构的预期依赖性。