TITLE: Explicit Spin Structures on the Torus
QUESTION [10 upvotes]: Basically, I am trying to build explicit examples of Dirac operators.  To this end, I'm looking at the surface E = C/(Z + λZ) - for some λ in H \ SL(2,Z) - with the Euclidean metric and the flat connection.  The, since the torus has genus 1, there are 22=4 spin structures on the tangent bundle of this elliptic curve.  
What are the four ways do define representations of SU(2) Spin(2)=U(1) on TpE for each $p \in$ E?  There is probably one spin structure for each element of the homology ring with coefficients in Z2.
For reference: A spin structure on E is an open covering $\{ U_\alpha : \alpha \in A\}$ and transition functions $g_{\alpha\beta}: U_\alpha \cap U_\beta \to Spin(2) = U(1)$ satisfying a cocycle condition gαβgβγ=gαγ on $U_\alpha \cap U_\beta \cap U_\gamma$.  Perhaps there are other definitions better for explicit computations.

REPLY [3 votes]: On oriented surfaces $F$, spin structures are in one-to-one correspondence with quadratic forms on the group $H_1(F;\mathbb{Z}/2\mathbb{Z})$ refining the intersection form. This correspondence is explicit and constructive -- see Atiyah, Riemannian surfaces and spin structures, Ann. Sci. Ec. Norm. Sup. (4) 4 (1971) and Johnson, Spin structures and quadratic forms on surfaces, Proc. London Math. Soc. 22 (1980). If you pick a basis of $H_1$, specifying your quadratic form on the basis elements defines it uniquely, so there are $2^{2g}$ spin structures. "One for each element of $H_1$" gives you this same number, but it's not really the natural way of counting them.