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

U(1) spin Chern-Simons theory and Arf invariants in two dimensions