# Explicit Spin Structures on the Torus

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.

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There aren't any representations of $SU(2)$ on $T_pE$, as $SU(2)$ is a 3-dimensional compact Lie group, in particular it doesn't have any connected normal subgroups. Spin structures have many formulations, what flavour are you looking for? Usually you start by taking the direct sum of the tangent bundle with a trivial line bundle, when you're dealing with 2-dimensional vector bundles. – Ryan Budney Jan 30 '10 at 21:20
Probably a very naive question in the context of this thread but still probably not fully out of context to request for detailed expositions on spin structures. Can you give references which will teach in details the whole idea of spin bundles and spin connections and spinors? – Anirbit Jan 31 '10 at 8:47
Start with getting a feel for all the various equivalent defininitions of orientation here: mathoverflow.net/questions/10966/… then you'll have a launching pad for thinking about spin structures. For example, one way to say a tangent bundle admits a spin structure is it is orientable and every map $S^2 \to M$ admits a lift $S^2 \to O(TM)$. See item (2) in my post to the linked thread. Also see the references below, and the text by Milnor and Stasheff. – Ryan Budney Jan 31 '10 at 17:26
In the back of my head I was thinking the dimensions weren't right $Spin(2) = SO(2)$. Also, let me put in the specific definition of Spin structure from "Lectures on the Seiberg-Witten Invariants" by John Moore. – john mangual Jan 31 '10 at 20:52

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 $\partial/\partial x + i \partial/\partial y$.

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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 $1$.

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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,Zd) 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,Zd), 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,Zd) 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.

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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.

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One common way to represent spin structures on $2$-dimensional vector bundles is to consider the $SO(2)$ as the subgroup of $SO(3)$ that fixes an axis, so the double cover of $SO(2)$ is naturally a subgroup of $Spin(2) \equiv SU(2)$. So I suspect John is using a formulation of spin structures that factors through this construction. – Ryan Budney Jan 30 '10 at 21:35
Another standard spin structure reference would be Milnor's l'Enseignement paper. – Ryan Budney Jan 30 '10 at 21:52
Thanks for this last reference. I knew there was a paper on l'Enseignement, but I had forgotten the author and title! It's in my office, but I'm at home now. As to your first comment, $\mathrm{SU}(2)$ is not isomorphic to $\mathrm{Spin}(2)$, but to $\mathrm{Spin}(3)$. – José Figueroa-O'Farrill Jan 30 '10 at 22:10
Ah, yes, typo. I meant to say $Spin(3)$. – Ryan Budney Jan 30 '10 at 23:28
I'm sorry for causing so much confusion. These references look helpful, as does "Spin Geometry" by Lawson and Michelson. – john mangual Jan 31 '10 at 21:19

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.

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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.

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