The most general target category we can consider is a symmetric tensor category: clearly we need a tensor product, and the axiom H_{Y1⊔Y2} ≅ H_{Y1} ⊗ H_{Y2} only makes sense if there is an involutory canonical isomorphism H_{Y1} ⊗ H_{Y2} ≅ H_{Y2} ⊗ H_{Y1} .

A very common choice in physics is the category of super vector spaces, i.e., vector spaces V with a mod 2 grading V = V^{0} ⊕ V^{1}, where the canonical isomorphism V ⊗ W ≅ W ⊗ V is v ⊗ w ↦ (−1)^{deg v deg w}w ⊗ v. One can also consider the category of Z-graded vector spaces, with the same sign convention for the tensor product.

In either case the closed string algebra is a graded-commutative algebra C with a trace θ : C → C. In principle the trace should have degree zero, but in fact the commonly encountered theories have a grading anomaly which makes the trace have degree −n for some integer n.

We define topological-spin^{c} theories, which model 2d theories with N = 2 supersymmetry, by replacing “manifolds” with “manifolds with spin^{c} structure”.

A spin^{c} structure on a surface with a conformal structure is a pair of holomorphic line bundles L_{1}, L_{2} with an isomorphism L_{1} ⊗ L_{2} ≅ TΣ of holomorphic line bundles. A spin structure is the particular case when L_{1} = L_{2}. On a 1-manifold S a spin^{c} structure means a spin^{c} structure on a ribbon neighbourhood of S in a surface with conformal structure. An N = 2 superconformal theory assigns a vector space H_{S;L1,L2} to each 1-manifold S with spin^{c} structure, and an operator

U_{S0;L1,L2}: H_{S0;L1,L2} → H_{S1;L1,L2}

to each spin^{c}-cobordism from S_{0} to S_{1}. To explain the rest of the structure we need to define the N = 2 Lie superalgebra associated to a spin^{c }1-manifold (S;L_{1},L_{2}). Let G = Aut(L_{1}) denote the group of bundle isomorphisms L_{1} → L_{1} which cover diffeomorphisms of S. (We can identify this group with Aut(L_{2}).) It has a homomorphism onto the group Diff^{+}(S) of orientation-preserving diffeomorphisms of S, and the kernel is the group of fibrewise automorphisms of L_{1}, which can be identified with the group of smooth maps from S to C^{×}. The Lie algebra Lie(G) is therefore an extension of the Lie algebra Vect(S) of Diff^{+}(S) by the commutative Lie algebra Ω^{0}(S) of smooth real-valued functions on S. Let Λ^{0}_{S;L1,L2} denote the complex Lie algebra obtained from Lie(G) by complexifying Vect(S). This is the even part of a Lie super algebra whose odd part is Λ^{1}_{S;L1,L2} = Γ(L_{1}) ⊕ Γ(L_{2}). The bracket Λ^{1} ⊗ Λ^{1} → Λ^{0} is completely determined by the property that elements of Γ(L_{1}) and of Γ(L_{2}) anticommute among themselves, while the composite

Γ(L_{1}) ⊗ Γ(L_{2}) → Λ^{0} → Vect_{C}(S)

takes (λ_{1},λ_{2}) to λ_{1}λ_{2} ∈ Γ(TS).

In an N = 2 theory we require the superalgebra Λ(S;L_{1},L_{2}) to act on the vector space H_{S;L1,L2}, compatibly with the action of the group G, and with a similar intertwining property with the cobordism operators to that of the N = 1 case. For an N = 2 theory the state space always has an action of the circle group coming from its embedding in G as the group of fibrewise multiplications on L_{1} and L_{2}. Equivalently, the state space is always Z-graded.

An N = 2 theory always gives rise to two ordinary conformal field theories by equipping a surface Σ with the spin^{c} structures (C,TΣ) and (TΣ,C). These are called the “A-model” and the “B-model” associated to the N = 2 theory. In each case the state spaces are cochain complexes in which the differential is the action of the constant section of the trivial component of the spin^{c}-structure.