The Homotopy Type of Seiberg-Witten Configuration Space

Let X be a closed smooth 4-manifold. In the Theory of the SeibergWitten Equations, the configuration space is Aα ×Gα Γ (S α ), where Aα is defined as the space of u1-connections on a complex line bundle over X, Γ (S α ) is the space of sections of the positive complex spinor bundle over X and Gα is the gauge group. It is shown that Aα×Gα Γ (S α ) has the same homotopic type of the Jacobian Torus T b1(X) = H1(X,R) H1(X,Z) , where b1(X) = dimRH(X,R).


Introduction
Although the physical meaning of the Seiberg-Witten equations (SW α -eq.) is yet to be discovered, the mathematical meaning is rather deep and efficient to understand one of the most basic phenomenon of differential topology in four dimension, namely, the existence of non-equivalent differential smooth structures on the same underlying topological manifold.The Seiberg-Witten equations arose through the ideas of duality described in Witten [12].It is conjectured that the Seiberg-Witten equations are dual to Yang-Mills equations; the duality being at the quantum level.A necessary condition is the equality of the expectation values for the dual theories.In topology, this means that fixed a 4-manifold its Seiberg-Witten invariants are equal to Donaldson invariants.A basic reference for SW α -eq. is [2].
Let (X, g) represent a fixed riemannian structure on X. Originally, the SW αequations were 1 st -order differential equations and their solutions (A, φ) satisfying 2000 Mathematics Subject Classification: 58J05, 58E50 C. M. Doria φ = 0 were called SW α -monopoles.These equations were not obtained by a variational principle.In [12], Witten used some special identities to obtain an integral useful to prove that the moduli spaces of sw-monopoles were empty, but a finite number of them.This integral defines the SW α -functional on the configuration space A α × Gα Γ (S + α ) and satisfies the Palais-Smale condition, as proved by Jost-Peng-Wang in [7].
The nature of the SW α -monopoles is rather subtle than the anti-self-dual connections considered in the Donaldson theory.It is known that the scalar curvature k g plays a important role to the non-existence of SW α -monopoles on X, e.g.: on S 4 endowed with the round metric, the only solution is the trivial one (0, 0).It is a open question to find necessary and sufficient conditions for the existence of a SW α -monopoles in A α × G α Γ (S + α ).In [11], Taubes shows that if X admits a symplectic structure then the spin c -structure α defined by the canonical class admits a SW α -monopole.In [5], Fintushel-Stern proved that whenever there is a class α ∈ Spin c (X) which SW α -invariant is non-zero, then we can construct many closed 4-manifolds Y , all non diffeomorphic to X, admitting a SW β -monopole for some β ∈ Spin c (Y ).
Once the Seiberg-Witen theory can be formulated in a variational framework, and the functional satisfies the Palais-Smale condition, it is natural to search for a Morse Theory framework.As a first step, our attempt is to prove that the homotopy type of the configuration space is completely determined by the classical Hodge theory .This fact contrast with Donaldson theory, where there are an abundace of instantons and the Atiyah-Jones conjecture shows the interplay among the homotopy type of the moduli space of connections and the homotopy type of the moduli space of instantons.
By considering the embedding of the Jacobian Torus in the configuration space, the variational formulation of the SW α -equations give us a interpretation to the topology of A α × G α Γ (S + α ).Theorem 1.1.Let X be a closed smooth 4-manifold endowed with a riemannian metric g which scalar curvature is k g .Let 1.If k g ≥ 0, then the gradient flow of the SW α -functional defines an homotopy equivalence among α ) has the same homotopy type of T b 1 (X) .

Basic Set Up
From a duality principle applicable to SUSY theories in Quantum Field Theory, Seiberg-Witten discovered a nice coupling of the self-dual(SD) equation, of a U 1 Yang-Mills Theory, to the Dirac c equation.The coupling is performed by a particular isomorphism relating the space Ω 2 + (X), of self-dual 2-forms, and the bundle End 0 (S α + ).
By considering the projection p 1 : H 2 (X, Z) ⊕ H1 (X; Z 2 ) → H 2 (X, Z), the space of Spin c -strutures on X is given by For each α ∈ Spin c (X), there is a representation ρ α : SO 4 → Cl 4 , induced by a Spin c representation, and consequently, a pair of vector bundles (S α + , L α ) over X (see [8]), where It is called the determinant line bundle associated to the Spin c -struture α.
Thus, for a given α ∈ Spin c (X) we associate a pair of bundles: From now on, we consider • a Riemannian metric g over X, • a Hermitian structure h on S α .
Remark 2.1.Let E → X be a vector bundle over X; 1.The space of sections of E (usually denoted by Γ (E)) is denoted by Ω 0 (E).
2. The space of p-forms (1 ≤ p ≤ 4) with values in E is denoted by Ω p (E).
3. For each fixed covariant derivative 1 on E, there is a 1 st -order differential operator d : For each class α ∈ Spin c (X) corresponds a U 1 -principal bundle over X, denoted P α , with c 1 (P α ) = α.Also, we consider the adjoint bundles Ad(U 1 ) is a fiber bundle with fiber U 1 , and ad(u 1 ) is a vector bundle with fiber isomorphic to the Lie Algebra u 1 .Once a covariant derivative is considered on ad(u 1 ), it induces the sequence The 2-form of curvature F , induced by the connection , is the operator Since Ad(U 1 ) ∼ X ×U 1 and ad(u 1 ) ∼ X ×u 1 , the spaces Ω 0 (ad(u 1 )) and Γ (Ad(U 1 )) are identified, respectively, to the spaces Ω 0 (X, iR) and M ap(X, U 1 ).It is well known from the theory (see in [3]) that a u 1 -connection defined on L α can be identified with a section of the vector bundle Ω 1 (ad(u 1 )), and a Gauge transformation with a section of the bundle Ad(U 1 ).
On a complex vector bundle E over (X,g), endowed with a hermitian metric and a covariant derivative , we consider the Sobolev Norm of a section φ ∈ Ω 0 (E) as and the Sobolev Spaces of sections of E as The space G α is the Gauge Group acting on C α by the action Since we are in dimension 4, the vector bundle Ω 2 (ad(u 1 )) splits as where (+) is the seld-dual component and (-) the anti-self-dual.The 1 st -order (original) Seiberg-Witten equations are defined over the configuration space where • The quadratic form σ : Γ (S + α ) → End 0 (S + α ) given by performs the coupling of the ASD-equation with the Dirac c operator.Locally, for φ = (φ 1 , φ 2 ) the quadratic form takes the value 2  2 .
The SW α -monopoles form the set of solutions of equations ( 2.4), this space can be described as the inverse image F −1 (0) by a map F α :

The W-Homotopy Type of
The space A α × Gα Γ (S + α ) isn't a manifold since the action isn't free.We observe that the isotropic group Therefore, G (A,0) iso U 1 and we consider the Gauge Group From now on, instead of the G α -action, we consider on C α the G α -action; consequently, the quotient space A α × G α Γ (S + α ) is a manifold.Nevertheless, the spaces A α /G α and A α / G α are diffeomorphic because all elements A ∈ A α have the same isotropic group.
In this section, the hypothesis of the theorem A.4 are checked to the space A α × Gα Γ (S + α ), and the study of its weak homotopy type is performed.We begin with the following remarks;

The quotiente spaces B
2. the G α -action on A α is not free since the action of the subgroup of constant maps g : M → U 1 , g(x) = g, ∀x ∈ M , acts trivially on A α .
As mentioned before, instead of the G α -action, we consider on C α the G α -action.On A α , the G α -action is free, and so, the space B α = A α / G α is a manifold.
The G α -action on Γ (S + α ) is free except on the 0-section, where the isotropic group is the full group G α .The action also preserves the spheres in Γ (S + α ), consequently, the quotient space is a cone over the quotient of a sphere by the G α -action.Therefore, the quotient space is contractible.
It follows from the Corollary of A.3 that there exists the fibration By the contractibility of Γ (S + α ), it follows that In [1], they studied the homotopy type of the space B * = A G * , where A is the space of connections defined on a G-Principal Bundle P and is a subgroup of the Gauge Group G = Γ (Ad(P )).They observed that G * acts freely on A, and so, the quotiente space B * is a manifold.We need to compare the G α and G * α actions on A α , neverthless, they turn out to be equal.Proposition 3.1.The gauge groups G α and G * α are diffeomorphic and perform the same action on A α .
Proof.The projection ρ in the exact sequence The same computation implies that In this way, the results of [1] can be applied to the understanding of the topology of the space A α / G α .
The weak homotopy type of B * α has been studied in [1] and [3] where they proved the following; Theorem 3.2.Let L α be a complex line with c 1 (L α ) = α, EU 1 be the Universal bundle associated to U 1 and From Algebraic Topology, we know that 1.There is a 1-1 correspondence

The space of isomorphic classes of complex line bundles is
In other words, Consequently, we can perform the computation of π n (M ap 0 α (X, CP ∞ )) by fixing a class of [X, CP ∞ ].
For a class α ∈ H 2 (X, Z), we fix a map f : X → CP ∞ representing α, x 0 ∈ X and a ∈ S n .Thus, However, Hence, Proposition 4.1.For each α ∈ Spin c (X), let L α be the determinant line bundle associated to α and (A,φ) ∈ C α .Also, assume that k g =scalar curvature of (X,g).Then, The identities above are applied to the functional ( 4.1) and, as consequence, a new functional turns up into the scenario.The new functional is defined as follows; Definition 4.1.For each α ∈ Spin c (X), the Seiberg-Witten Functional is the functional SW α : C α → R given by where k g = scalar curvature of (X,g).
Let k g,X = min x∈X k g and 1. Since X is compact and 3. The SW α -functional is bounded below by 0, and it is equal to 0 if and only if either there exists a SW α -monopole or an self-dual U 1 -connection.
Proposition 4.2.The Euler-Lagrange equations of the SW α -functional (4.2) are ) where Φ : Remark 4.3.Locally, in a orthonormal basis {η i } 1≤i≤4 of T * X, the operator Φ * can be written as The regularity of the solutions of ( 4.4) and ( 4.5) was studied by Jost-Peng-Wang in [7].They observed that the L ∞ estimate of φ, already known to be satisfied by the SW α -monopoles, is also obeyed by the solutions of ( 4.4) e ( 4.5).The estimate is the following; where k − g,X = max x∈X {0, −k 1 2 g (X)}.
In [7], Jost-Peng-Wang studied the analytical properties of the SW α -functional.They proved that the Palais-Smale Condition, up to gauge equivalence, is satisfied.Therefore, by the Minimax Principle the SW α -functional always attains its minimum in A α × Gα Γ (S + α ) and, consequently, on A α × G α Γ (S + α ).In this way, it is left the following question: Under which conditions the minimum in As a consequence of the estimate ( 4.6), if the Riemannian metric g on X has non-negative scalar curvature then the only solutions are (A, 0), where The following result is well known [3]; Proposition 4.4.Let X be a closed, smooth 4-manifold.The solutions of d * F A = 0, module the G α -action, define the Jacobian Torus Final Remark: The main result implies that the nature of the SW α -monopoles are rather subtle than we may expect.The following questions were not reached by the methods applied to prove the main result; 1. Is SW α a Morse function for a generic subset of metrics on X ?If the answer is positive, is it possible to transform SW α into a perfect Morse function ?
2. Under which condition there exists a SW α -monopole in A α × G α Γ (S + α )? 3. Are there unstable critical points of the SW α -functional?
A. The Diagonal Action and its Quotient Space Let M, N be smooth manifolds endowed with G-actions α M , α N (respec.).About the G-actions, we will assume that; 1.There exists a subgroup H of G such that for all m ∈ M G m is conjugate to H.

2.
The quotient spaces M G and N G are Hausdorff spaces.
The product action of G × G on the manifold M × N , is defined by or equivalently,