THE SEIBERG–WITTEN MODULI SPACE

The Seiberg–Witten equations.

Let X be a smooth oriented Riemannian 4-manifold. Let P → X be the principal SO(4)-bundle associated to the tangent bundle T X → X, and let P̃ → X be a choice of Spin^{c}-structure on X, i.e. a principal Spin^{c}(4)-bundle with a Spin^{c}(4) → SO(4)-equivariant bundle map to P → X. Let ℂ^{2} → S^{±}(P̃) → X be the plus and minus complex spin bundles associated to the Spin^{c}-structure, and let ℂ^{1} → L → X be their associated determinant bundle.

Let

C = A(L)_{L_{2}^{2}} × Γ(S^{+}(P̃))_{L_{2}^{2}}

be the configuration space, where A(L) denotes the affine space of U(1)-connections A on L, and Γ(S^{+}(P̃)) denotes the space of (smooth) sections ψ in the plus complex spin bundle S^{+}(P̃) → X, i.e. a positive spinor field. The subscript L_{2}^{2} denotes completion with respect to the L_{2}^{2}-Sobolev norm. The tangent space of C at (A, ψ) is

T_{(A,ψ)}C = Ω^{1}(X, iℝ)_{L_{2}^{2}} × Γ(S^{+}(P̃))_{L_{2}^{2}}.

Let

D = Ω_{+}^{2}(X; iℝ)_{L_{1}^{2}} × Γ(S^{−}(P̃))_{L_{1}^{2}},

where Ω_{+}^{2}(X; iℝ) denotes the self-dual 2-forms on X with coefficients in iℝ = u(1), i.e. those in the +1-eigenspace for the involution induced by the Hodge ∗-operation, or equivalently for the multiplication by the complex unit ω_{ℂ}. As above Γ(S^{−}(P̃)) denotes the space of sections in the minus complex spinor bundle S^{−}(P̃), i.e. a negative spinor field. The subscript L_{1}^{2} denotes completion with respect to the L_{1}^{2}-Sobolev norm. The tangent space of D at (ω, μ) is

T_{(ω,μ)}D = Ω^{2}(X, iℝ))_{L_{1}^{2}} × Γ(S^{−}(P̃))_{L_{1}^{2}}.

Let F: C → D be the Seiberg–Witten functional given by

F (A, ψ) = (F_{A}^{+} − q(ψ), / ∂_{A}ψ).

Here F_{A}^{+} is the self-dual (positive) part of the curvature 2-form of the connection A, and q(ψ) = ψ ⊗ ψ^{∗} − \frac{1}{2}|ψ|^{2} · id is the trace 0 part of the rank 1 endomorphism ψ ⊗ ψ^{∗}: μ ↦ ⟨μ, ψ⟩ψ, viewed as a self-dual 2-form through the identification

Ω_{+}^{2}(X; ℂ) ≅ End_{ℂ}(S^{+}(P̃))_{0},

and / ∂_{A} is the Dirac operator for the Levi–Civita connection on P → X and the connection A on L. Then F is smooth. The derivative of F at (A, ψ) has matrix

DF_{(A,ψ)} = [ \begin{matrix}P_{+}d & Dq_{w} \\ ·\frac{1}{2}ψ & / ∂_{A} \end{matrix} ]

Typeset by AMS-TEX

where P_{+} : Ω^{2} → Ω_{+}^{2} denotes projection to the plus part.

Let the gauge group

G = Ω^{0}(X, S^{1})_{L_{3}^{2}}

be the Sobolev L_{3}^{2}-completion of the space of (smooth) maps σ: X → S^{1}. We identify σ with the gauge equivalence P̃ → P̃ over P given by right scalar multiplication through S^{1} ⊂ Spin^{c}(4). Then G acts smoothly on C and D by the formulas

(A, ψ) · σ = (det(σ)^{∗}A, S^{+}(σ^{−1})(ψ) = (A + 2σ^{−1}dσ, σ^{−1} · ψ)
(ω, μ) · σ = (ω, S^{−}(σ^{−1})(μ)) = (ω, σ^{−1} · μ),

and the map F is G-equivariant.

Local structure of B.

The action of G on C has slices. Let (A, ψ) ∈ C be a configuration. Then the action map m: G → C has derivative

Dm = Dm_{id} : T_{id}G → T_{(A,ψ)}C

at the identity given by Dm(f) = (2df, −f · ψ). Let K ⊂ T_{(A,ψ)}C be the orthogonal complement of the image of Dm, with respect to the L^{2}-inner product on T_{(A,ψ)}C. Equivalently K is the kernel of the adjoint (Dm)^{∗} to Dm. Then for U ⊂ K a (small) Stab_{(A,ψ)}-invariant neighbourhood of 0 ∈ K the subspace

S = (A, ψ) + U ⊂ C

is a slice for the action of G on C.

Let B = C/G be the orbit space for the gauge action. Let N = F^{−1}(0) ⊂ ℂ be the solution space of configurations satisfying the Seiberg–Witten equations F (A, ψ) = 0. This subspace is G-invariant. Let M = N /G ⊂ B be the Seiberg– Witten moduli space of gauge equivalence classes of solutions to the Seiberg–Witten equations. When it makes sense, the Seiberg–Witten invariant of X (with the given Spin^{c}-structure) will be defined as the bordism class of M → B.

[figure: G; G; N; π; C; F; π; D; M; B]

A configuration (A, ψ) is reducible if ψ ≡ 0 and irreducible otherwise. Let C^{∗} ⊂ C be the open subset of irreducible configurations, This subspace is G-invariant. Similarly let N^{∗} ⊂ N, B^{∗} ⊂ B and M^{∗} ⊂ M be the open subsets of (gauge equivalence classes of) irreducible configurations in the given spaces. Write [A, ψ] for the gauge equivalence class of (A, ψ).

The action of G is free on the irreducible configurations C^{∗}, and has stabilizer S^{1} ⊂ G on the reducible configurations. Hence B^{∗} can be given the structure of a (Hausdorff) Banach manifold with local parametrizations S ⊂ C^{∗} → B^{∗} near [A, ψ], when S is a slice for the action at (A, ψ). Its tangent space there is then

T_{[A,ψ]}B^{∗} = coker(Dm) = T_{(A,ψ)}C/Dm(T_{id}G) ≅ K ⊂ T_{(A,ψ)}C.

Since G acts freely with slices on C^{∗} there is a fiber bundle G → C^{∗} → B^{∗}. This bundle is universal because C^{∗} is contractible, so B^{∗} has the homotopy type of

B Map(X, S^{1}) ≃ ℂP^{∞} × BH^{1}(X; ℤ),

suitably completed.

Local structure of the moduli space.

Let (A, ψ) ∈ N^{∗} be an irreducible solution. We have an elliptic complex

0 ⟶ T_{id}G ⟶^{Dm_{id}} T_{(A,ψ)}C ⟶^{DF_{(A,ψ)}} T_{0}D ⟶ 0

denoted E = E(A, ψ). Here Dm is injective because the action of G on (A, ψ) is free, so H^{0}(E) = 0.

We say that M is smooth at [A, ψ] if F: C → D is a submersion at (A, ψ), i.e. if DF_{(A,ψ)} is surjective. Then H^{2}(E) = 0.

Now suppose that (A, ψ) is a smooth irreducible solution. Then by the implicit function theorem N ⊂ C is a Banach submanifold locally near (A, ψ), with tangent space

T_{(A,ψ)}N = ker(DF) ⊂ T_{(A,ψ)}C.

Then the orbit of G acting on (A, ψ) is contained in N and has tangent space im(Dm). Then K ∩ T_{(A,ψ)}N is its orthogonal complement in T_{(A,ψ)}N, and gives a slice for the G-action on N at (A, ψ). For the action map of G on the slice has derivative

(K ∩ T_{(A,ψ)}N) × T_{id}G ⟶ T_{(A,ψ)}N ⊂ T_{(A,ψ)}C

at ((A, ψ), id), which is the inclusion on the first factor and Dm on the second. We claim that this map is an isomorphism. It is injective because K ∩ im(Dm) = 0. Given n ∈ T_{(A,ψ)}N write n = k+g with k ∈ K and g ∈ im(Dm). Then g ∈ T_{(A,ψ)}N so k = n − g ∈ K ∩ T_{(A,ψ)}N, so the map is also surjective.

Hence N → M is locally a fiber bundle with structure group G, near [A, ψ], and the smooth irreducible points of M form a (Hausdorff) Banach manifold locally modelled on K ∩ T_{(A,ψ)}N. It has tangent space

T_{[A,ψ]}M = ker(DF)/ im(Dm) ≅ K ∩ T_{(A,ψ)}N

equal to the Zariski tangent space H^{1}(E) at [A, ψ].

An application of the Atiyah–Singer index theorem shows that H^{1}(E) has finite dimension

d = \frac{1}{4}(c_{1}^{2}(L) − 2χ(X) − 3 sign(X))

and so in fact M is locally a d-dimensional (ordinary) Hausdorff manifold near its smooth irreducible points.