Explicit Spin Structures on the Torus

Question

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 T_pE 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 g_αβg_βγ=g_αγ on $U_\alpha \cap U_\beta \cap U_\gamma$. Perhaps there are other definitions better for explicit computations.

Answer 1

On a torus spin structures have a particularly simple representation, because you have an essentially standard trivialization of the tangent bundle given by your euclidean metric.

From your definition of spin structure, there is the associated $spin(2)$-bundle which double covers the unit tangent bundle of the surface. Let's call it $spin(SE) \to SE$. Both of these are circle bundles over the surface $E$.

Identify $\pi_1 E$ with your lattice $\mathbb Z \oplus \lambda \mathbb Z$ with generators $1$ and $\lambda$. Given either generator, lift it to a loop in $SE$ the unit circle bundle -- since we're in such a simple situation choose it to be a loop coming from parallel transport. That loop lifts to a path in $spin(SE)$, and it may or may not be a closed path. This is a $\mathbb Z_2$-valued invariant of your spin structure (closed produces $0$, not closed produces $1$), and you have two of them, one for each generator. That's how you identify your spin structures with $\mathbb Z_2^2 = H^1(E;\mathbb Z_2)$.

If you didn't have the standard parallel-transport lift to work with you'd have to refine this construction. In general $H^1(E;\mathbb Z_2)$ acts freely and transitively on the spin structures, there is no "base-point". The action is basically given by the argument above by removing the parallel transport construction and simply considering a lift of a loop in $E$ to $SE$, and then to two distinct spin bundles covering $SE$ -- if the loops either both lift to closed paths (or both don't lift to close paths), give that a $0$, if one lifts to a closed path and the other doesn't, give that a $1$.

Answer 2

Other posters explain some of the topology of spin structures. Here's a differential-geometric answer relevant to Dirac operators. The exercise you have set yourself, of understanding Dirac operators on the 2-torus, is a good one. Rather than trying to do it for you, I'll instead discuss the 3-torus (cf. Kronheimer-Mrowka, "Monopoles and 3-manifolds").

So: a spin-structure on a Riemannian 3-manifold $Y$ can be understood in the following workmanlike way: we give a rank 2 hermitian vector bundle $S \to Y$ (the spinor bundle); a unitary trivialization of $\Lambda^2 S$; and a Clifford multiplication map $\rho \colon TY\to \mathfrak{su}(S)$, such that at each $y\in Y$ there is some oriented orthonormal basis $(e_1,e_2,e_3)$ for $T_y Y$ such that $\rho(e_i)$ is the $i$th Pauli matrix $\sigma_i$. More invariantly, one can instead say that $\rho$ is an isometry (with respect to the inner product $(a,b)=tr(a^\ast b)/2$) and satisfies the orientation condition $\rho(e_1)\rho(e_2)\rho(e_3)=1$.

If we have two spin-structures, with spinor bundles $S$ and $S'$, we can look at the sub-bundle of $\mathrm{SU}(S,S')$ consisting of those fibrewise special isometries that intertwine the Clifford multiplication maps. This bundle has fibre $\{ \pm 1 \}$: it is a 2-fold covering of $Y$. As such it is classified by a class in $H^1(Y;\mathbb{Z}/2)$, whose non-vanishing is clearly the only obstruction to isomorphism of the two spin-structures. Conversely, by tensoring everything by real orthogonal line bundles (work out what this means concretely!), you can construct all spin structures, up to isomorphism, from a chosen one.

On flat $T^3$, all the data can be taken translation invariant. The Dirac operator is then $D = \sum_i{\sigma_i\partial_i}$. Tensoring with an orthogonal line bundle $\lambda$ (constructed, if you will, from a character $\pi_1(T^3)\to O(1)$) the formula becomes $D_\lambda =D \otimes 1_\lambda$.

In 2 dimensions, the story will be similar; the new feature is that the spinor bundle splits into two line bundles. The translation-invariant Dirac operator is nothing but the Cauchy-Riemann operator $\partial/\partial x + i \partial/\partial y$.

Answer 3

As an algebraic geometer, here is how I would describe the correspondence between spin structures on E and H₁(E,Z₂). While this might not be what you're looking for, maybe it still helps to see another perspective.

1. We can define a spin structure on a curve X to be the choice of a line bundle L such that $L^{\otimes 2} \cong \Omega_X$. Note that by "line bundle", I really mean "invertible sheaf". (More generally, the same definition with 2 replaced by d defines the notion of a d-spin structure.)
2. On an elliptic curve, $\Omega_X \cong \mathcal{O}_X$.
3. The fact that $L^{\otimes d} \cong \mathcal{O}_X$ can be rephrased exactly as saying that L is a d-torsion element in the Jacobian of X. (Jacobian = group of line bundles of degree zero under tensor product.)
4. For any curve X, there is a canonical isomorphism between H¹(X,Z_d) and the group of d-torsion points in Jac X.

Note in particular the role of point 2 above. Point 1,3 and 4 say that for a general curve X the space of d-spin structures is a torsor for H¹(X,Z_d), since if L and M are line bundles such that $L^{\otimes d} \cong M^{\otimes d} \cong \Omega_X$, then $L \otimes M^{-1}$ is a d-torsion point on Jac X. Then the second point says that for elliptic curves, the isomorphism between d-spin structures and H¹(X,Z_d) is actually canonical: since the trivial line bundle $\mathcal{O}_X$ is a d-spin structure, there is a distinguished base point in the space of d-spin structures.

Answer 4

Why are you after representations of $\mathrm{SU}(2)$? Since you are looking at a two-dimensional spin manifold, the spinor bundles have structure group $\mathrm{Spin}(2)$, which is the double cover of the $\mathrm{SO(2)}$ structure group of the tangent bundle.

Also, there is only one action on the tangent bundle $TE$, since that is uniquely determined by the reduction of the structure of the frame bundle to $\mathrm{SO}(2)$, which is the result of introducing the riemannian structure. The different spin structures correspond to different ways of lifting the oriented orthonormal frame bundle of the tangent bundle to a principal $\mathrm{Spin}(2)$ bundle.

A good place to learn about spin structures on Riemann surfaces is the 1971 paper Riemann surfaces and spin structures by Atiyah, published in the Annales Scientifiques de l'École Normale Supérieure. A more physicsy paper is Theta functions. modular invariance and strings by Alvarez-Gaumé, Moore and Vafa. IIRC, this paper goes into some detail about the different spin structures on the torus and how the modular group transforms them.

Answer 5

There is a straightforward Cech covering that gives you the spin structures you want. View the torus as an identification space on a parallelogram, so there is a 0-cell, 2 1-cells, and a 2-cell. Take one of the open sets to be a small ball around the 0-cell, take another to be the interior of the 2-cell or something slightly smaller, and take the last two to be small open neighborhoods of the 1-cells. If you chose the sets well, the intersections have a small number of contractible components. Then for each $\pm 1$-valued 1-cocycle, you can choose locally constant transition functions on these intersections with values in $\pm 1 = \operatorname{Ker}(\operatorname{Spin}\_2(\mathbb{R}) \to SO\_2(\mathbb{R}))$ that realize the cocycle.

Answer 6

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.