A spin structure on a surface means a double covering of its space of non-zero tangent vectors which is non-trivial on each individual tangent space. On an oriented 1-dimensional manifold S it means a double covering of the space of positively-oriented tangent vectors. For purposes of gluing, this is the same thing as a spin structure on a ribbon neighbourhood of S in an orientable surface. Each spin structure has an automorphism which interchanges its sheets, and this will induce an involution T on any vector space which is naturally associated to a 1-manifold with spin structure, giving the vector space a mod 2 grading by its ±1-eigenspaces. A topological-spin theory is a functor from the cobordism category of manifolds with spin structures to the category of super vector spaces with its graded tensor structure. The functor is required to take disjoint unions to super tensor products, and additionally it is required that the automorphism of the spin structure of a 1-manifold induces the grading automorphism T = (−1)^{degree} of the super vector space. This choice of the supersymmetry of the tensor product rather than the naive symmetry which ignores the grading is forced by the geometry of spin structures if the possibility of a semisimple category of boundary conditions is to be allowed. There are two non-isomorphic circles with spin structure: S^{1}_{ns}, with the * Möbius or “Neveu-Schwarz” structure, and S^{1}_{r}, with the trivial or “Ramond” structure*. A topological-spin theory gives us state spaces C

_{ns}and C

_{r}, corresponding respectively to S

^{1}

_{ns}and S

^{1}

_{r}.

There are four cobordisms with spin structures which cover the standard annulus. The double covering can be identified with its incoming end times the interval [0,1], but then one has a binary choice when one identifies the outgoing end of the double covering over the annulus with the chosen structure on the outgoing boundary circle. In other words, alongside the cylinders A^{+}_{ns,r} = S^{1}_{ns,r} × [0,1] which induce the identity maps of C_{ns,r} there are also cylinders A^{−}_{ns,r} which connect S^{1}_{ns,r} to itself while interchanging the sheets. These cylinders A^{−}_{ns,r} induce the grading automorphism on the state spaces. But because A^{−}_{ns} ≅ A^{+}_{ns} by an isomorphism which is the identity on the boundary circles – the * Dehn twist* which “rotates one end of the cylinder by 2π” – the grading on C

_{ns}must be purely even. The space C

_{r}can have both even and odd components. The situation is a little more complicated for “U-shaped” cobordisms, i.e., cylinders with two incoming or two outgoing boundary circles. If the boundaries are S

^{1}

_{ns}there is only one possibility, but if the boundaries are S

^{1}

_{r}there are two, corresponding to A

^{±}

_{r}. The complication is that there seems no special reason to prefer either of the spin structures as “positive”. We shall simply choose one – let us call it P – with incoming boundary S

^{1}

_{r}⊔ S

^{1}

_{r}, and use P to define a pairing C

_{r}⊗ C

_{r}→ C. We then choose a preferred cobordism Q in the other direction so that when we sew its right-hand outgoing S

^{1}

_{r}to the left-hand incoming one of P the resulting S-bend is the “trivial” cylinder A

^{+}

_{r}. We shall need to know, however, that the closed torus formed by the composition P ◦ Q has an even spin structure. The Frobenius structure θ on C restricts to 0 on C

_{r}.

There is a unique spin structure on the pair-of-pants cobordism in the figure below, which restricts to S^{1}_{ns} on each boundary circle, and it makes C_{ns} into a commutative * Frobenius algebra* in the usual way.

If one incoming circle is S^{1}_{ns} and the other is S^{1}_{r} then the outgoing circle is S^{1}_{r}, and there are two possible spin structures, but the one obtained by removing a disc from the cylinder A^{+}_{r} is preferred: it makes C_{r} into a graded module over C_{ns}. The chosen U-shaped cobordism P, with two incoming circles S^{1}_{r}, can be punctured to give us a pair of pants with an outgoing S^{1}_{ns}, and it induces a graded bilinear map C_{r} × C_{r} → C_{ns} which, composing with the trace on C_{ns}, gives a non-degenerate inner product on C_{r}. At this point the choice of symmetry of the tensor product becomes important. Let us consider the diffeomorphism of the pair of pants which shows us in the usual case that the Frobenius algebra is commutative. When we lift it to the spin structure, this diffeomorphism induces the identity on one incoming circle but reverses the sheets over the other incoming circle, and this proves that the cobordism must have the same output when we change the input from S(φ_{1} ⊗ φ_{2}) to T(φ_{1}) ⊗ φ_{2}, where T is the grading involution and S : C_{r} ⊗ C_{r} → C_{r} ⊗ C_{r} is the symmetry of the tensor category. If we take S to be the symmetry of the tensor category of vector spaces which ignores the grading, this shows that the product on the graded vector space C_{r} is graded-symmetric with the usual sign; but if S is the graded symmetry then we see that the product on C_{r} is symmetric in the naive sense.

There is an analogue for spin theories of the theorem which tells us that a two-dimensional topological field theory “is” a commutative Frobenius algebra. It asserts that a spin-topological theory “is” a Frobenius algebra C = (C_{ns} ⊕ C_{r},θ_{C}) with the following property. Let {φ_{k}} be a basis for C_{ns}, with dual basis {φ^{k}} such that θ_{C}(φ_{k}φ^{m}) = δ^{m}_{k}, and let β_{k} and β^{k} be similar dual bases for C_{r}. Then the Euler elements χ_{ns} := ∑ φ_{k}φ^{k} and χ_{r} = ∑ β_{k}β^{k} are independent of the choices of bases, and the condition we need on the algebra C is that χ_{ns} = χ_{r}. In particular, this condition implies that the vector spaces C_{ns} and C_{r} have the same dimension. In fact, the Euler elements can be obtained from cutting a hole out of the torus. There are actually four spin structures on the torus. The output state is necessarily in C_{ns}. The Euler elements for the three even spin structures are equal to χ_{e} = χ_{ns} = χ_{r}. The Euler element χ_{o} corresponding to the odd spin structure, on the other hand, is given by χ_{o} = ∑(−1)^{degβk}β_{k}β^{k}.

A spin theory is very similar to a Z/2-equivariant theory, which is the structure obtained when the surfaces are equipped with principal Z/2-bundles (i.e., double coverings) rather than spin structures.

It seems reasonable to call a spin theory semisimple if the algebra C_{ns} is semisimple, i.e., is the algebra of functions on a finite set X. Then C_{r} is the space of sections of a vector bundle E on X, and it follows from the condition χ_{ns} = χ_{r} that the fibre at each point must have dimension 1. Thus the whole structure is determined by the Frobenius algebra C_{ns} together with a binary choice at each point x ∈ X of the grading of the fibre E_{x} of the line bundle E at x.

We can now see that if we had not used the graded symmetry in defining the tensor category we should have forced the grading of C_{r} to be purely even. For on the odd part the inner product would have had to be skew, and that is impossible on a 1-dimensional space. And if both C_{ns} and C_{r} are purely even then the theory is in fact completely independent of the spin structures on the surfaces.

A concrete example of a two-dimensional topological-spin theory is given by C = C ⊕ C_{η} where η^{2} = 1 and η is odd. The Euler elements are χ_{e} = 1 and χ_{o} = −1. It follows that the partition function of a closed surface with spin structure is ±1 according as the spin structure is even or odd.

The most common theories defined on surfaces with spin structure are not topological: they are 2-dimensional conformal field theories with N = 1 supersymmetry. It should be noticed that if the theory is not topological then one does not expect the grading on C_{ns} to be purely even: states can change sign on rotation by 2π. If a surface Σ has a conformal structure then a double covering of the non-zero tangent vectors is the complement of the zero-section in a two-dimensional real vector bundle L on Σ which is called the spin bundle. The covering map then extends to a symmetric pairing of vector bundles L ⊗ L → TΣ which, if we regard L and TΣ as complex line bundles in the natural way, induces an isomorphism L ⊗_{C} L ≅ TΣ. An N = 1 * superconformal field theory* is a conformal-spin theory which assigns a vector space H

_{S,L}to the 1-manifold S with the spin bundle L, and is equipped with an additional map

Γ(S,L) ⊗ H_{S,L} → H_{S,L}

(σ,ψ) ↦ G_{σ}ψ,

where Γ(S,L) is the space of smooth sections of L, such that G_{σ} is real-linear in the section σ, and satisfies G^{2}_{σ} = D_{σ2}, where D_{σ2} is the * Virasoro action* of the vector field σ

^{2}related to σ ⊗ σ by the isomorphism L ⊗

_{C}L ≅ TΣ. Furthermore, when we have a cobordism (Σ,L) from (S

_{0},L

_{0}) to (S

_{1},L

_{1}) and a holomorphic section σ of L which restricts to σ

_{i}on S

_{i}we have the intertwining property

G_{σ1} ◦ U_{Σ,L} = U_{Σ,L} ◦ G_{σ0}

….