Seiberg-Witten monopoles with multiple spinors on a surface times a circle
Aleksander Doan


Seiberg–Witten monopoles with multiple spinors on S^1×Σ]Seiberg–Witten monopoles with multiple spinors on a surface times a circle






The Seiberg–Witten equation with multiple spinors generalises the classical Seiberg–Witten equation in dimension three. In contrast to the classical case, the moduli space of solutions  can be non-compact due to the appearance of so-called Fueter sections. In the absence of Fueter sections we define a signed count of points in  and show its invariance under small perturbations. We then study the equation on the product of a Riemann surface and a circle, describing  in terms of holomorphic data over the surface. We define analytic and algebro-geometric compactifications of , and construct a homeomorphism between them.  For a generic choice of circle-invariant parameters of the equation, Fueter sections do not appear and  is a compact Kähler manifold. After a perturbation it splits into isolated points which can be counted with signs, yielding a number independent of the initial choice of the parameters. We compute this number for surfaces of low genus.


[
    Aleksander Doan
    December 30, 2023
=====================





§ INTRODUCTION


This article addresses the questions of transversality and compactness for the moduli spaces of Seiberg–Witten multi-monopoles: solutions of the Seiberg–Witten equation with multiple spinors introduced in <cit.>. We begin by considering the equation on an arbitrary closed Riemannian spin three-manifold Y but the majority of the paper concerns three-manifolds of the form Y = S^1 ×Σ for a closed Riemann surface Σ.



 §.§ Instantons in higher dimensions

The motivation for studying generalised Seiberg–Witten equations comes from higher-dimensional gauge theory. We only briefly outline this relationship as it will not be essential for understanding the results presented here. Donaldson, Thomas, and Segal initiated a programme of defining an enumerative invariant of Riemannian 7–manifolds with holonomy contained in the exceptional Lie group G_2 <cit.>. The putative invariant counts, in the sense of Fredholm differential topology, G_2–instantons: a class of Yang–Mills connections solving a 7–dimensional analogue of the anti-self-duality equation. A sequence of G_2–instantons can concentrate along a three-dimensional associative submanifold Y, developing a bubble singularity in a way familiar from the four-dimensional theory <cit.>. This phenomenon causes the moduli space of G_2–instantons to be non-compact. In particular, for a family (g_t)_t∈[0,1] of G_2–metrics, the cobordism between the moduli spaces of G_2–instantons with respect to g_0 and g_1, given by the one-parameter moduli space, may fail to be compact. If this is the case, the count of G_2–instantons with respect to g_t jumps as t varies, and is not a deformation invariant. Haydys and Walpuski <cit.> proposed to compensate for such jumps by adding the number of Seiberg–Witten multi-monopoles on Y, counted with signs. This number itself is not a topological invariant of Y; it can jump when the Riemannian metric on Y and other parameters of the equation are deformed. The point is that such jumps should occur exactly when a G_2–instanton is created or destroyed along Y. It is conjectured that a combination of the two numbers: the count of G_2–instantons and that of multi-monopoles summed over all associative submanifolds is invariant under deformations.

The case Y = S^1 ×Σ discussed in this article is particularly relevant to G_2–manifolds of the form S^1 × X for a Calabi–Yau three-fold X containing Σ as a holomorphic curve. G_2–instantons over S^1 × X correspond to Hermitian–Yang–Mills connections on X and so one expects that there is a relationship between multi-monopoles on S^1 ×Σ and Hermitian–Yang–Mills connections on X whose energy is highly concentrated around Σ. From this perspective, our work can be seen as the first step towards a gauge-theoretic interpretation of local Donaldson–Thomas invariants in algebraic geometry <cit.>. 




 §.§ Multi-monopoles on three-manifolds
 
Let Y be a closed three-manifold and let E → Y and L → Y be vector bundles with the structure group  (n) and (1) respectively. Fix an (n)–connection B on E. A spin structure on Y and a choice of a Riemannian metric endow Y with the spinor bundle S: a rank two Hermitian bundle with trivial determinant line bundle and a compatible connection. The Seiberg–Witten equation with n spinors is the following differential equation for a pair (A,Ψ) of a connection on L and a section of Hom(E, S ⊗ L):

    {[                D_ABΨ = 0,; F_A = ΨΨ^* - 1/2 | Ψ |^2. ].
 
Here, _AB is the Dirac operator on (E, S ⊗ L) twisted by the connections A and B, F_A is the curvature two-form, and in the second equation we identify imaginary-valued two-forms with traceless skew-Hermitian endomorphisms of S using the Clifford multiplication.

For n=1 this is the standard Seiberg–Witten equation in dimension three. As in the classical setting, one introduces the moduli space of gauge-equivalence classes of solutions which depends on the choice of the parameters of the equation: the Riemannian metric on Y, the connection B, and a closed two-form used to perturb the equation to guarantee transversality. 

Denote by  the space of all parameters of the equation as described above and for ∈ let () be the corresponding moduli space of solutions to the Seiberg–Witten equation with n spinors; see section <ref>, in particular Definition <ref>, for more details.

The dependence on  does not play an important role for n=1.
In that case, for a generic[By generic we mean: from a residual subset of the space of objects in question. A subset of a topological space is residual if it contains a countable intersection of open and dense subsets. Baire’s theorem asserts that a residual subset of a complete metric space is dense. Unless said otherwise, we use the C^∞–topology. ] choice of , the moduli space of irreducible solutions is an oriented, compact, zero-dimensional manifold, that is: a finite collection of points equipped with signs. If b_1(Y) > 1, there are no reducible solutions and the moduli spaces for two different choices of  are connected through an oriented, compact, one-dimensional cobordism. As a consequence, the signed count of points in (), for any generic , is a topological invariant of Y <cit.>. 

A new feature of the equation for n ≥ 2 is that () may be non-compact for some choices of . Building on work of Taubes <cit.>, Haydys and Walpuski <cit.> showed that a sequence of points in () which does not have any  convergent subsequence can be rescaled so that it converges in an appropriate sense to a Fueter section, a section of a fibre bundle over Y satisfying a non-linear analogue of the Dirac equation. We review Haydys and Walpuski's compactness theorem in subsection <ref>; Fueter sections are introduced in Definition <ref>. An important point is that the differential equation obeyed by a Fueter section, just like the Seiberg–Witten equation itself, depends on the parameter .
As a result, Fueter sections may or may not exist depending on the choice of ∈. 


Denote by  ⊂ the set of all parameters ∈ for which the moduli space () is compact. Haydys and Walpuski's theorem and Corollary <ref> imply that  contains an open neighbourhood of the set of all ∈ for which no Fueter sections exist.


Our first result, proved in section <ref>, shows that the existence of Fueter sections is the only obstruction to defining a signed count of multi-monopoles.


 
Let Y be a closed oriented spin three-manifold with b_1(Y) > 1. Fix vector bundles E→ Y and L→ Y with the structure group (n) and (1) respectively.


  * There exists a locally constant function → such that if () is Zariski smooth with the obstruction bundle →(), then () equals the integral of the Euler class of  over (); see Definitions <ref>, <ref>, and <ref>.

  * For a generic ∈ the moduli space () is a compact, zero-dimensional manifold equipped with a natural orientation described in subsection <ref>. In particular, in this case () equals the signed count of points in ().



At present, little is known about the set  and the existence of Fueter sections. The main difficulty in understanding Fueter sections lies in the fact that they are defined only on the complement of a singular set—which is known to be closed and of Hausdorff dimension at most one <cit.>—and thus not amenable to standard elliptic theory. Conjecturally,  is open and dense in  and its complement has codimension one <cit.>, a prediction strongly supported by recent work of Takahashi <cit.>. If this is the case,  has multiple connected components on which the function → is constant. These components are separated by codimension one walls on which Fueter sections appear and the value of  jumps as we cross one of them. It is exactly this jumping phenomenon that indicates a relationship between the enumerative theories for multi-monopoles and G_2–instantons. 





In view of Theorem <ref>, the central problem in the study of multi-monopoles on three-manifolds—one of importance to the applications in higher-dimensional gauge theory—is that of the existence and properties of Fueter sections. We make some progress on this problem by describing the moduli spaces of multi-monopoles and Fueter sections for a particular class of parameters .  In particular, we exhibit the first examples of Y and  such that

  
  * ∈ and () is non-empty, compact, and consists of irreducible solutions,
  
  * ∉ and () contains sequences of solutions converging to a Fueter section,
  
  * there exist Fueter sections whose singular sets are non-empty [In fact, more is true: these Fueter sections do not arise from everywhere defined harmonic spinors and so are examples of singular harmonic _2 spinors as in <cit.>.],
  
  * there exist Fueter sections that do not appear in the compactification of ().
 
  

Since the first version of this paper appeared, Theorem <ref> has been used to study the wall-crossing phenomenon for multi-monopoles and to prove the existence of Fueter sections, for some choice of , on all closed spin three-manifolds with b_1 > 1 <cit.>.




 §.§ Gauge theory on Riemann surfaces

The examples of Y and  mentioned in the four bullet points above are constructed by means of dimensional reduction. We consider three-manifolds of the form Y=S^1×Σ equipped with a spin structure induced from a spin structure on Σ. The bundles E and L are assumed to be pulled-back from bundles on Σ, and in the space of all parameters of the equation  we distinguish the subspace _Σ consisting of the parameters pulled-back from Σ; see Definition <ref>.

For n=1 the classical Seiberg–Witten equation on S^1 ×Σ reduces to the vortex equation on Σ and the moduli space of solutions is homeomorphic to the symmetric product of Σ <cit.>. Similarly, in section <ref> we prove that for ∈_Σ all irreducible solutions of the Seiberg–Witten equation with multiple spinors are pulled back from solutions of a generalised vortex equation on Σ. In fact, we prove this for a much broader class of three-dimensional Seiberg–Witten equations associated with representations; see Theorem <ref>.

In section <ref> we show that solutions of the dimensionally reduced equation correspond to triples of the form (ℒ, α, β) where ℒ→Σ is a holomorphic line bundle and α, β are holomorphic sections of certain holomorphic vector bundles over Σ twisted by ℒ. This is a version of the Hitchin–Kobayashi correspondence, following earlier work of Bryan and Wentworth <cit.>.  We construct the moduli space _() of isomorphism classes of such triples. In section <ref> we introduce also its natural complex-geometric compactification _() which is a compact complex analytic space containing _ as a Zariski open dense subset. In parallel, guided by an enhanced version of the Haydys and Walpuski's compactness theorem <cit.>, we define also a gauge-theoretic compactification () of the Seiberg–Witten moduli space. In what follows we assume for simplicity that E = 2.

 
For every ∈_Σ there is a homeomorphism () →_() restricting to an isomorphism of real analytic spaces  () →_().










The gauge-theoretic compactification () is a compact real analytic space containing () as a Zariski open dense subset.



It is desirable to define () and prove Corollary <ref> without the assumptions Y = S^1 ×Σ and ∈_Σ. Even though () is expected to be compact for a generic choice of ∈, one needs a good notion of a compactification in order to study generic one-parameter families of moduli spaces and the wall-crossing phenomenon <cit.>.
We hope that the construction of () and the analysis constituting the proof of Theorem <ref> offer some guidance in the study of compactness for arbitrary three-manifolds.


The holomorphic description of Fueter sections allows us to prove that a generic point of _Σ belongs to the set  introduced earlier—this is the main result of sections <ref> and <ref>.





Let Σ be a closed spin surface of genus g(Σ)≥ 1. Suppose that Y= S^1×Σ is equipped with a spin structure induced from the spin structure on Σ,  and that the bundles E and L are pulled-back from bundles over Σ. Let d = ⟨ c_1(L), [Σ] ⟩ be the degree of L.

For a generic choice of ∈_Σ there exist no Fueter sections with respect to  and the moduli space () is a compact Kähler manifold of dimension 

    _() = g(Σ)-1 ± 2d,

where the sign depends on .
In this case, the signed count of Seiberg–Witten multi-monopoles is, up to a sign, the Euler characteristic of the moduli space:

    () = (-1)^g(Σ)-1χ(()).

Moreover, () does not depend on the choice of a generic ∈_Σ.



We should stress that here  is generic in _Σ (parameters pulled-back from Σ) but not in  (all parameters on Y). In particular, () may have positive dimension even though the expected dimension is zero. In other words, the moduli space is Zariski smooth and obstructed as in Theorem <ref>. This is familiar from the study of gauge-theoretic invariants of complex surfaces <cit.>, <cit.>, <cit.>.


The main idea behind the proof of Theorem <ref> is to interpret the existence of Fueter sections as a Fredholm problem. One difficulty in doing so stems from the possible non-smoothness of the singular sets  of Fueter sections. However, an argument using a Weitzenböck formula shows that in fact such a  set is a finite collection of circles of the form S^1 ×{point}. We then show that these singularities can be removed—in an appropriate sense—so that every Fueter section gives rise to a globally defined holomorphic object. As a result, the deformation theory of Fueter section is described by a complex-linear Fredholm operator of non-positive index. Using this, we prove that Fueter sections appear in real codimension at least two and can be avoided in a generic one-dimensional family of parameters in _Σ.










Finally, in section <ref> we use complex-geometric methods and classical results on stable vector bundles on Riemann surfaces <cit.> to describe () and compute () in some cases. The generic properties of the moduli spaces are summarised in Theorem <ref>. We study also some non-generic cases which provide us with examples of Fueter sections and non-compact moduli spaces; see Examples <ref>, <ref>, Proposition <ref>, and Remark <ref>


  For a genus two surface Σ the space of parameters _Σ contains a copy of ^3, thought of as the moduli space of semi-stable (2,)–bundles on Σ. A point  in ^3 corresponds to a plane H_ in the dual projective space (^3)^*. Let T^4 / _2 be the singular Kummer surface in (^3)^*. For a generic choice of ∈^3, the plane H_ intersects T^4/_2 transversely and misses all of the 16 singular points of T^4/_2, so that (T^4/_2) ∩ H_ is a smooth genus three surface. The moduli space () is then the preimage of this intersection under the double covering T^4 → T^4/_2. It is a genus five Riemann surface and so () = 8. When  is chosen so that H() passes through one of the singular points of T^4/_2, the moduli space is necessarily singular and non-compact. 





















 §.§ Acknowledgements
 
 The work presented here is part of my PhD thesis at Stony Brook University. I am most grateful to my advisor Simon Donaldson for his guidance, support, and encouragement. Discussions with Thomas Walpuski and Andriy Haydys have contributed greatly to this paper.
I thank also Aliakbar Daemi, Cristian Minoccheri, Vicente Muñoz, Benjamin Sibley, David Stapleton, Ryosuke Takahashi, Alex Waldron, Malik Younsi, and two anonymous referees for sharing their insights and helping me improve this article.

My research is supported by the https://sites.duke.edu/scshgap/Simons Collaboration on Special Holonomy in Geometry, Analysis, and Physics.



 §.§ Notation and conventions

 

 

Λ^p Y = Λ^p T^*Y         exterior product of the cotangent bundle 

Γ(Y,V) or Γ(V)         sections of a vector bundle V → Y 

Ω^p(Y,V) or Ω^p(V)         differential forms with values in V 

(Y,V) or (V)         connections on V compatible with the structure group 

Cl(α) or α ·         Clifford multiplication by a form α∈Λ^*Y 

⟨·, ·⟩         Euclidean inner product 

(·, · )         Hermitian inner product

g(Σ)         genus of a surface Σ

J^d         Jacobian of degree d holomorphic line bundles


 


When it is not likely to cause confusion, we will use the same notation for a principal (n) or (n) bundle and the associated rank n Hermitian vector bundle. We use the following sign convention for the Clifford multiplication: under the identification of the spinor space of ^3 with the quaternions ℍ, with the complex structure given by right-multiplication by i, the action of e^1, e^2, e^3 is given by left-multiplication by i, j, k respectively.



§ SEIBERG–WITTEN MONOPOLES WITH MULTIPLE SPINORS


In this section we introduce the Seiberg–Witten equation with multiple spinors, discuss transversality and orientations, and review the compactness theorem of Haydys and Walpuski.
The main results are Propositions <ref> and <ref> which lead to the proof of Theorem <ref>.




 §.§ The Seiberg–Witten equation with multiple spinors
 
Let (Y,g) be a compact, connected, oriented Riemannian three-manifold equipped with a spin structure. Let S → Y be the spinor bundle, E → Y an (n)–bundle and L → Y a (1)–bundle. Fix a connection B ∈(E) and a closed two-form η∈Ω^2(Y, i).


The Seiberg–Witten equation with multiple spinors with parameters (g,B,η)[We do not need to assume d η = 0 in order to write down equation (<ref>) but Lemma <ref> below and the Bianchi identity imply that there are no solutions unless dη = 0.] is the following differential equation for a pair (A,Ψ) ∈(L) ×Γ((E, S ⊗ L))

    {[       _ABΨ = 0,; F_A = μ(Ψ) + η. ].

Here, F_A ∈Ω^2(i) is the curvature and  μ is the quadratic map

    μ(Ψ) := ( ΨΨ^* - 1/2 | Ψ |^2   id),

with Ψ^* ∈Γ((S⊗ L, E)) being the adjoint of Ψ.
Using the natural isomorphism L ⊗ L^* ≅, we consider the composition ΨΨ^* as a section of End(S). 
It actually takes values in the subspace i 𝔰𝔲(S) of trace-free self-adjoint endomorphisms which in the equation we implicitly identify with i Λ^2 Y using the Clifford multiplication. 

We will refer to solutions of the equation as Seiberg–Witten monopoles with multiple spinors or, following  <cit.>, Seiberg–Witten multi-monopoles[The name multi-monopoles is often used in reference to solutions of the Bogomolny equation.].


The Seiberg–Witten equation with multiple spinors was introduced in <cit.> and <cit.>. If we trivialise E and represent Ψ by an n–tuple (Ψ_1, …, Ψ_n) with Ψ_i ∈Γ(S ⊗ L), then

    μ(Ψ) =  ∑_i=1^n ( Ψ_i Ψ_i^* - 1/2 | Ψ_i^2  |  id).

In particular, for n=1 one recovers the classical three-dimensional Seiberg–Witten equation. 


Let = C^∞(Y, S^1) be the group of unitary automorphisms of L. An element u ∈ acts on a pair (A,Ψ) by

    u(A, Ψ) := (A - u^-1du, u Ψ).

(A,Ψ) is called irreducible if its –stabiliser is trivial, i.e. Ψ≠ 0, and reducible otherwise. 


To make equation (<ref>) elliptic modulo the action of  we introduce an additional unknown f ∈ C^∞(Y,i). The modified equation for a triple (A,Ψ,f) reads

    {[             _ABΨ + f Ψ = 0,; F_A - ∗ d f - μ(Ψ) - η = 0. ].

In the next proposition we show that this modification does not produce new solutions. 
For Ψ, Φ∈(E, S ⊗ L) let

    μ(Ψ, Φ) := 1/2{ΨΦ^* + ΦΨ^* - ⟨Ψ, Φ⟩ id}.


 
Considering μ(Ψ, Φ) as an imaginary-valued two-form, we have

    ∗ d ( μ(Ψ, Φ) ) = i  ( _ABΨ, Φ ) - i  ( Ψ,  _ABΦ).



This is a straightforward computation; see <cit.>. 






















 If (A, Ψ, f) is a solution of (<ref>), then f is constant and (A,Ψ) is a solution of the original equation (<ref>). Moreover, if Ψ≠ 0, then f = 0.


Applying ∗ d to both sides of the equation F_A - ∗ df - μ(Ψ) - η = 0 results in 

    0    = i  ( _ABΨ, Ψ ) - i  ( Ψ, _ABΨ ) - Δ f 
       = -i  ( f Ψ, Ψ ) + i  ( Ψ, f Ψ )- Δ f = -( 2| Ψ|^2 + Δ)f,

where we have used that _ABΨ = - f Ψ and f is purely imaginary. If Ψ = 0, then Δ f = 0 and f is constant. If Ψ≠ 0, then Δ +2 | Ψ |^2 is invertible and f=0.




 §.§ Deformation theory
 
We want to study the space of solutions of  (<ref>). Although it is naturally equipped with the C^∞–topology, it is convenient to use the Sobolev topology instead.


For k≥ 2 we consider the following Sobolev spaces:



_k(V)         W^k,2 connections on a bundle V → Y, 

Γ_k(V)         W^k,2 sections of V, 

Ω^p_k(V)         W^k,2 differential forms with values in V, 

_k = _k(L) ×Γ_k((E,S⊗ L))         W^k,2 configurations, 

^*_k = _k(L) × (Γ_k((E,S⊗ L)) ∖{ 0 })         W^k,2 irreducible configurations, 

_k(U)         W^k,2_ configurations on an open subset U ⊂ Y, 

_k+1         W^k+1,2 gauge transformations of L.



_k and _k^* are Hilbert manifolds modelled on the Hilbert space Ω_k^1(i) ⊕Γ_k((E, S ⊗ L)) and _k+1 is a Hilbert Lie group modelled on Ω^0_k+1(i). 


The action of  on (L) ×Γ(Y, (E, S ⊗ L)) extends to a smooth, proper action of _k+1 on _k with a metrisable, second countable quotient ℬ_k := _k / _k+1.  Moreover, at every point of _k there exists a local slice for the action and, as a result, ℬ_k^* := _k^* / _k+1 has the structure of a Hilbert manifold.

For the definition of local slices and the proof see <cit.>. We only mention that a local slice passing through (A, Ψ) ∈_k can be chosen to have the form

    (A,Ψ) + K_A,Ψ,

 where K_A,Ψ⊂ T_(A,Ψ)_k is a neighbourhood of zero in G^*, where G is the linearised action of the gauge group at (A,Ψ) and G^* is the formal adjoint of G restricted to _k. 
 

The moduli space of solutions (g,B, η) is the subspace of ℬ_k consisting of gauge equivalence classes of pairs (A, Ψ) satisfying  (<ref>) with parameters (g,B,η). We define also the moduli space of irreducible solutions ^*(g,B,η)  = (g, B,η) ∩ℬ_k^*. 



We will often write σ = (B,η) and denote the corresponding moduli spaces by (g,σ) and ^*(g,σ). When (g,σ) is clear from the context we write simply  and ^*. 


 and ^* are endowed with the subspace topology induced from ℬ_k; hence they are metrisable and second countable. The particular choice of k ≥ 2 is immaterial: if the perturbation σ is of class W^l,2, then the moduli spaces for different choices of k ≤ l+1 are naturally homeomorphic. If moreover l = ∞, then they are all homeomorphic to the space of smooth solutions modulo smooth gauge transformations.

 
Suppose that σ = (B,η) is of class W^l,2 for some l ≥ 1. 


  * For every (A,Ψ) ∈_2 satisfying (<ref>) there exists  u ∈_3 such that u (A, Ψ) ∈_l+1. Two such gauge transformations differ by an element of _l+2. 

  * If (A_i, Ψ_i) is a convergent sequence of solutions in _2, then there is a sequence u_i ∈_3 such that u_i(A_i, Ψ_i) are in _l+1 and converge in _l+1. 



This is a direct consequence of Hodge theory and elliptic regularity for  and d + d^*. Without loss of generality we fix k=2 and drop the subscript k, writing , ℬ, , and so on. The local structure of  at a solution (A,Ψ) is encoded in the deformation complex 

    Γ_3(i) rG_A, Ψ   Ω^1_2(i) ⊕Γ_2(E^* ⊗ S ⊗ L) ⊕Ω^0_2( i) rS_A, Ψ   Γ_1(E^* ⊗ S ⊗ L) ⊕Ω^2_1( i ).

Here, G_A, Ψ is the linearisation of the gauge group action

    G_A,Ψ(h) := (-dh, hΨ, 0),

and S_A, Ψ is the linearisation of the modified equation (<ref>) at  (A, Ψ, 0) 

    S_A,Ψ(a, ϕ, v) := (_ABϕ + a·Ψ + vΨ, da - 2μ(Ψ, ϕ) - ∗ d v).

The complex is well-defined by the Sobolev multiplication theorem. It is elliptic and has index zero. One way to see this is to introduce the formal adjoint

    G_A,Ψ^*(a, ϕ, v) := -d^*a + i  ( Ψ, ϕ )

and consider the extended Hessian L_A,Ψ = S_A,Ψ⊕ G_A,Ψ^*. After rearranging the direct sums and identifying the spaces of two-forms and functions via the Hodge star, L_A,Ψ is given by

    Ω^1_2(i) ⊕Ω^0_2(i ) ⊕Γ_2(E^* ⊗ S ⊗ L) rL_A,Ψ   Ω^1_1(i) ⊕Ω^0_1(i ) ⊕Γ_1(E^* ⊗ S ⊗ L),
  

    L_A,Ψ( [ a; v; ϕ ]) =
    (
    [ ∗ da - dv - 2∗μ(ϕ, Ψ);  - d^*a + i  ( Ψ, ϕ );     a ·Ψ + v Ψ + _ABϕ ]).

L_A,Ψ is elliptic because up to terms of order zero it agrees with the direct sum of _AB and the total operator of the de Rham complex. Representing L_A,Ψ by the matrix

    L_A,Ψ = (
    [         ∗ d         - d - 2∗μ(Ψ, ·);      - d^*a           0 i  ( Ψ, · );    Cl( ·) Ψ       (·) Ψ         _AB ]).

we also see that it is self-adjoint, and so the deformation complex is elliptic and has index zero. 

Let H^0_A,Ψ, H^1_A,Ψ, and H^2_A,Ψ be the homology groups of the deformation complex at (A,Ψ). They are finite dimensional vector spaces. If the solution is irreducible, then H^0_A,Ψ = 0 and, since L_A,Ψ is self-adjoint, H^1_A,Ψ =  L_A,Ψ and H^2_A,Ψ = coker L_A,Ψ are naturally isomorphic. We will later need an explicit description of these groups. 

  H^1_A,Ψ consists of pairs (a,ϕ) solving

    {[ - d^*a + i  ( Ψ, ϕ ) = 0,;          _ABϕ + a ·Ψ = 0,;       da - 2 μ(Ψ, ϕ) = 0. ].


The proof is similar to that of Proposition <ref>.















 
We call an irreducible solution (A,Ψ) regular if H^1_A,Ψ = H^2_A,Ψ = 0. 


 
By elliptic regularity for L_A,Ψ, the property of being a regular solution does not depend on the chosen Sobolev setup (compare with <cit.>). 


The elliptic complex gives rise to a Kuranishi model for the moduli space.


For an irreducible solution (A, Ψ) there is a smooth map 

    κ H^1_A, Ψ→ H^2_A, Ψ

 such that κ(0) = 0, dκ(0) = 0, and a neighbourhood of (A,Ψ) in ^* is homeomorphic to a neighbourhood of 0 in κ^-1(0). 



The proof is standard but we need it for future reference. 
Let U be a local slice passing through (A,Ψ) as in (<ref>). Let V = ×Ω^0_2(i), W = Γ_1(E^* × S × L) ×Ω^2_1(i ), and ℱ V → W be given by the left-hand side of (<ref>). The restriction of ℱ to U ×Ω^0_2(i), which we denote by f, is a smooth Fredholm map. By the implicit function theorem, there is a neighbourhood of (A,Ψ) in U ×Ω^0_2(i) and local charts in which f takes the form

    f  E × d f_A,Ψ,0→ E ×coker d f_A,Ψ,0,


    f(e, x) = (e, K(e,x))

for a Banach space E and a smooth map K  d f_A,Ψ,0→coker d f_A,Ψ,0. The point (A,Ψ) corresponds to (0,0) in E × d f_A,Ψ,0, and we have K(0,0) = (0,0) and dK(0,0) = 0. Let κ(x) = K(0,x). Then the zero set f^-1(0) is locally homeomorphic to κ^-1(0). This proves the proposition because H^1_A,Ψ≅ d f_A,Ψ,0 and H^2_A,Ψ≅coker df_A,Ψ,0.


 
The existence of local Kuranishi models allows us to equip ^* with the structure of a real analytic space; since ℱ is real analytic, we can choose the Kuranishi map κ to be analytic as well. Thus, ^* is locally homeomorphic to the real analytic set V = κ^-1(0). As we vary κ and V, the rings of analytic functions / I(V) glue to a globally defined structure sheaf making ^* into a real analytic space; see <cit.>. 
 
 
  In section <ref> we will face a situation in which the moduli space ^* is not regular, but it still has the structure of a smooth manifold. 
  
 
The moduli space  is said to be Zariski smooth if = ^* and for every [A,Ψ] ∈ there is a local Kuranishi model with the map κ H^1_A,Ψ→ H^2_A,Ψ being zero. 


For a Zariski smooth moduli space the homology group H^1_A,Ψ is the tangent space to  at [A,Ψ]. The groups H^2_A,Ψ form a vector bundle over the moduli space. To see this, consider the trivial bundles over ^*

    𝒱_0     := ^* × (Γ_2(E^* ⊗ S ⊗ L) ⊕Ω^2_2(i)) 
    𝒲_0     := ^* × (Ω^1_1(i ) ⊕Γ_1(E^* ⊗ S ⊗ L) ⊕Ω^0_1(i ))

and the section s_0 ∈Γ((𝒱_0, 𝒲_0))  which at a point (A,Ψ) is given by S^*_A,Ψ. The triple (𝒱_0, 𝒲_0, s_0) is -equivariant and descends to a triple (𝒱, 𝒲, s) of vector bundles 𝒱 and 𝒲 over ℬ^* and a section s ∈Γ( ( 𝒱, 𝒲)). By construction, s([A,Ψ]) ≅ H^2_A,Ψ. 
 
 
The obstruction bundle is the subspace of 𝒱 given by

    := { ([A,Ψ], v)  |  [A,Ψ] ∈^*,  v ∈ s([A,Ψ])}

together with the map →^* induced from the bundle projection 𝒱→ℬ^*. 


 
If  is Zariski smooth, then it is a submanifold of ℬ^* and → is a vector bundle isomorphic to the tangent bundle T. 


More precisely,  is a disjoint union of smooth manifolds of possibly different dimensions. If C is a connected component containing a point [A,Ψ], then C is a smooth submanifold of ℬ^* of dimension H^1_A,Ψ and the restriction of →  to C is a smooth vector bundle isomorphic to TC as unoriented real vector bundles. The proof is standard and we omit it.













 §.§ Perturbations and transversality

The moduli space (g, σ) depends on the choice of a Riemannian metric g and a parameter σ = (B,η). In this subsection we fix g and study the properties of (g,σ) for a generic choice of σ.



We introduce the following Fréchet manifolds of C^∞ parameters 



         the space of Riemannian metrics on Y, 

          the space of closed imaginary-valued two-forms, 

:= (E) ×         the space of perturbations,



and their Sobolev completions _k and _k with respect to the W^k,2-topology.
The space of parameters of the equation, appearing in the introduction and Theorem <ref>, is 

    := ×.
 However, typically we will fix a Riemannian metric and perturb only σ∈. 


Recall that  a subset of a topological space is called residual if it contains a countable intersection of open and dense subsets. Baire’s theorem asserts that a residual subset of a complete metric space is dense.

 
For every g ∈, the set

    {σ∈ |  all solutions in ^*(g, σ) are regular}

 is residual in .



We first prove the corresponding statement for _k in place of , with solutions in  the configuration space _k+1^*. Consider the smooth map

    𝒮_k ×_k+1^* ×Ω^0_k+1(i)    ⟶Γ_k(E^* ⊗ S ⊗ L) ×Ω^2_k(i ) 
    𝒮(B, η, A, Ψ, f)    = ( _A BΨ + f Ψ, F_A - ∗ d f- μ(Ψ) - η).

We will show that 𝒮 is a submersion at all irreducible solutions.  
Let (A,Ψ) ∈_k+1^* be such that 𝒮(B,η, A, Ψ, 0) = 0. The derivative at (B, η, A, Ψ, 0) is 

    d 𝒮(b, ξ, a, ϕ, v) = S_A,Ψ (a, ϕ, v) + (b ·Ψ, -ξ).

Recall that S_A,Ψ is the linearisation operator (<ref>) introduced in the previous subsection. Since  S_A,Ψ⊕ G^*_A,Ψ is Fredholm, so is d 𝒮⊕ G^*_A,Ψ, and the image of d 𝒮 is closed and of finite codimension. Suppose that (Ξ, ω) is L^2–orthogonal to the image of d 𝒮. We will prove ω = 0. 
First, observe that (Ξ,ω) satisfies 

    - i ( Ψ, Ξ )+ ∗ d ω   = 0, 
    _ABΞ - 2 ω·Ψ   = 0, 
    
    d^* ω + μ(Ψ,Ξ)    = 0.

The first equation is a consequence of varying a alone, the second of varying ϕ alone, and the third of varying v alone.
On the other hand, ω is L^2–orthogonal to _k, as a result of varying ξ∈_k alone,  so in particular d^* ω = 0. Applying ∗ d^* to μ(Ψ,Ξ) = 0 and using <cit.>, we obtain

    0    = ∗ d^*μ(Ψ,Ξ) = μ(Ψ, _ABΞ) - ∗i/2(∇_ABΨ, Ξ) - ∗i/2 i  (Ψ, ∇_ABΞ) 
       = μ(Ψ, 2ω·Ψ) + d^* d ω.

Taking the L^2-inner product with ω gives us

    2 ω·Ψ_L^2^2 +  d ω_L^2^2 = 0,

so d ω = 0. We conclude that ω = 0, _ABΞ = 0, and (Ψ, Ξ )= 0.  Next, we prove Ξ = 0. Suppose by contradiction that Ξ is not identically zero. By the unique continuation theorem for harmonic spinors <cit.>, the set { | Ξ | > 0  } is open and dense in Y. The same is true for Ψ and so there exists x ∈ Y such that | Ψ(x) | > 0 and | Ξ(x) | > 0. On the other hand, for all a ∈Ω^1_k+1(i ) and b ∈Ω^1_k( 𝔰𝔲(E) ) 

    0 = ⟨ d 𝒮( b, 0, a, 0 , 0 ) , Ξ⟩_L^2 =  ((a+b) ·Ψ , Ξ )_L^2.

However, we can find a(x) ∈Λ^1_x ⊗ i and b(x) ∈Λ^1_x ⊗𝔰𝔲(E_x) such that

    ( (a(x) + b(x)) ·Ψ(x), Ξ(x))> 0.

This is an elementary fact of linear algebra; it is obvious when E =1 and for E ≥ 2 see Lemma <ref> below. We extend a(x), b(x)  to differential forms a and b satisfying

    ( (a+b) ·Ψ , Ξ )_L^2 > 0.

This is a contradiction; we conclude that Ξ = 0.

We have shown that 𝒮 is a submersion at all irreducible solutions. By the implicit function theorem, 𝒮^-1(0) is a submanifold of _k ×_k+1^* ×Ω^0_k+1(i). The gauge group _k+2 acts freely on 𝒮^-1(0). Denote the quotient space by 𝒳. The projection on _k induces a smooth map π𝒳→_k whose fibre over σ∈_k is ^*(g,σ). The derivative dπ is Fredholm and at a point (A, Ψ) ∈^*(g,σ) there is a natural identification coker dπ_A,Ψ = H^2_A,Ψ. By the Sard-Smale theorem, the set of regular values of π is residual in _k. For every such regular value σ all solutions in ^*(g,σ) are regular. This proves the statement for _k. 

The final part of the proof is to replace _k by . We follow an argument due to Taubes used in the context of pseudo-holomorphic curves <cit.>. For N > 0 define _N ⊂ as the set of all σ such that all solutions (A,Ψ) in ^*(g,σ) satisfying 

    1/N≤Ψ_L^∞≤ N

are regular. We need to show that the intersection ⋂_N ≥ 1_N is residual. First, we show that _N is open. Let σ_i be a sequence in ∖_N converging to σ in C^∞. By the definition of _N, there is a sequence of solutions (A_i, Ψ_i) solving equations (<ref>) with parameters σ_i, satisfying (<ref>) and H^2_A_i, Ψ_i≠ 0. By the first part of Theorem <ref> from the next subsection, after passing to a subsequence (A_i,Ψ_i) converges smoothly modulo gauge. The limit (A,Ψ) satisfies condition (<ref>) and the equations with parameter σ. Since L_A_i, Ψ_i converges to L_A,Ψ,  the latter operator is not surjective and σ∉_N. We conclude that ∖_N is closed.
The last step is to show that _N is dense in . We use the statement for _k proved earlier. Let _k,N be the subspace of _k defined analogously to _N. Since _k,N contains the set of regular values of π discussed earlier, it is dense in _k. It is also open in _k by the same argument as before. By Proposition <ref> and Remark <ref>, _N = _k,N∩ for all k ≥ 2. Every element σ of  can be approximated in the W^k,2–topology by elements of _k,N. In fact, it can be approximated by elements of the intersection _k,N∩ because _k,N is open in _k and  is dense in _k. Therefore, we have shown that σ can be approximated by elements in _N in any Sobolev norm. By the usual diagonal argument and the Sobolev embedding theorem, we can find a sequence in _N converging to σ in C^∞. This shows that _N is dense in .



  
  Let n ≥ 2 and V_n = ^n be the standard representation of (n).
  For every pair of non-zero vectors v, w ∈ V_2 ⊗ V_n there exists b ∈(2) ⊗(n) such that 
  
    ( bv, w )> 0.
 
 


  The proof is similar to that of <cit.>. 
  Let e_1, …, e_n be an orthonormal basis of V_n. 
  Write v and w as
  
    v = ∑_i=1^n v_i ⊗ e_i   and  
        w = ∑_i=1^n w_i ⊗ e_i,

  for v_i, w_i ∈ V_2. 
  Likewise, denoting by σ_1, σ_2, σ_3 the standard basis of (2), we can write b as
  
    b = ∑_k=1^3 σ_k ⊗ b_k

  for some b_k ∈(n), so that
  
    ( bv, w )    = ∑_k=1^3 ∑_i,j ( σ_k v_i, w_j ) ( e_i, b_k e_j).

  Suppose by contradiction that ( bv, w ) = 0 for all b ∈(2) ⊗(n). 
  In particular, setting b_k to be elementary off-diagonal anti-Hermitian matrices, we see that for k=1,2,3 and i ≠ j
  
    ( σ_k v_i, w_j ) -  ( σ_k v_j, w_i )    = 0, 
    
        i ( σ_k v_i, w_j ) + i ( σ_k v_j, w_i )    = 0.
 
  Hence,
  
    ( σ_k v_i, w_j ) = 0

  for k=1,2,3 and i≠ j. 
  Suppose without loss of generality that v_1 ≠ 0. 
  Then σ_1 v_1, σ_2 v_1, σ_3 v_1 are linearly independent over  and thus span V_2 over . 
  It follows that w_j = 0 for j = 2, 3, … n. 
  On the other hand, setting b_k = diag(1,-1,0,…, 0) ∈(n), we obtain that for k=1,2,3 
  
    ( σ_k v_1,  w_1 ) = 0,

  which shows that w_1 = 0 and so w = 0, yielding a contradiction.













































 §.§ Reducible solutions
 
The moduli space  might contain reducible solutions at which it develops singularities. In this paper we focus on the favourable case when reducibles can be avoided. As in the classical setting <cit.>, this is guaranteed by b_1(Y) > 1. 

  For a fixed g ∈ the subset of parameters σ∈ for which (g, σ) contains a reducible solution is contained in a closed affine subspace of  of codimension b_1(Y).  



If (A, 0) is a reducible solution of equations (<ref>) with a parameter σ = (B, η), then F_A = η and passing to the de Rham cohomology we obtain

    [ η ] = - 2π i c_1(L) ∈ H^2(Y, i).

Consider the affine subspace of  given by

    V = {η∈ |  [ η] = - 2π i c_1(L) }.

In other words, V is the preimage of -2π i c_1(L) under the projection π→ H^2(Y, ) associating to each closed form its de Rham class. The map π is linear and surjective. It is also continuous because by the Hodge decomposition theorem it is continuous with respect to the W^k,2–topology for all k. Therefore, V is a closed affine subspace of codimension H^2(Y, ) =  H^1(Y, ) = b_1. 




 §.§ Compactness and Fueter sections

The moduli space of multi-monopoles needs not be compact. 



Haydys and Walpuski <cit.> studied sequences of solutions whose L^2–norms diverge to infinity. They proved that in the limit we obtain a harmonic spinor taking values in the zero set μ^-1(0). In general, it is defined only on the complement of a closed, nowhere dense subset of Y. 

 
Let Z ⊊ Y be a closed, proper subset and (A, Ψ) ∈_k(Y ∖ Z) (see Definition <ref>). A triple (A, Ψ, Z) is called a Fueter section[For the explanation of this name see <cit.>, <cit.>, <cit.>.] with singular set Z if it satisfies 

  
  * _ABΨ = 0 and μ(Ψ) = 0,
  
  * ∫_Y ∖ Z |Ψ|^2 =1 and ∫_Y∖ Z | ∇_A Ψ |^2 < ∞,
  
  * | Ψ | extends to a Hölder continuous function on Y such that Z = | Ψ |^-1(0).




The definition of a Fueter section depends, through the Dirac equation, on the choice of the metric g and the background connection B. We will call (A,Ψ, Z) a Fueter section with respect to (g,B) when we want to stress this dependence. 


There is elliptic regularity theory for Fueter sections <cit.>, which implies that if B is of class W^l,2, then after a gauge transformation (A,Ψ) is in _l+1(Y ∖ Z). Taubes <cit.> proved that the singular set of a Fueter section has Hausdorff dimension at most one. 

 
Let (g_i, σ_i) ∈× be a sequence converging to (g, σ) and [A_i, Ψ_i] a sequence in ℬ such that [A_i, Ψ_i] ∈(g_i, σ_i). 


  * If lim sup_i →∞Ψ_i _L^2 < ∞, then after passing to a subsequence and applying gauge transformations  the sequence (A_i, Ψ_i) converges smoothly to a solution (A,Ψ) representing a point in (g, σ). 

  * If lim sup_i →∞Ψ_i _L^2 = ∞, then there exists a Fueter section (A, Ψ, Z) with respect to (g,B) such that after passing to a subsequence and applying gauge transformations A_i → A weakly in W^1,2_ and Ψ_i / Ψ_i _L^2→Ψ weakly in W^2,2_ over Y ∖ Z. 



  
Let g ∈ and σ = (B, η) ∈.  If there are no Fueter sections with respect to (g,B), then there is an open neighbourhood of (g,σ) in × such that for all (g',σ') from this neighbourhood the moduli space (g', σ') is compact.



Let K = sup{Ψ_L^2 |  [A,Ψ] ∈(g, σ)}. We have K < ∞ by Theorem <ref> and the assumption that there are no (g, B)-Fueter sections. We claim that there is an open neighbourhood of (g, σ) such that for all (g',σ') from this neighbourhood 

    sup_[A,Ψ] ∈(g',σ')Ψ_L^2 < K + 1,

and the proposition follows by Theorem <ref>.  Assume by contradiction that there is a sequence (g_i, σ_i) converging to (g,σ) and violating (<ref>). There is a corresponding sequence (A_i, Ψ_i) representing points in (g_i, σ_i) and satisfying Ψ_i _L^2≥ K+1. The first alternative of Theorem <ref> cannot hold because otherwise we would extract a subsequence converging modulo gauge to a solution (A,Ψ) with [A,Ψ] ∈(g,σ) and Ψ_L^2≥ K +1 > K. On the other hand, the second alternative would imply the existence of a (g, B)-Fueter section. 





 §.§ Orientations
 

If ^* is regular, one prescribes an orientation to every point in ^*. We briefly outline this construction as we will need it later. For (A,Ψ, f) ∈×Ω^0_2(i) let S_A,Ψ,f be the linearisation of (<ref>) at (A,Ψ, f)[Previously we have only dealt with the linearisation at irreducible solutions for which f = 0.] and let L_A,Ψ, f = G_A,Ψ^* ⊕ S_A,Ψ,f. Let Det→^* ×Ω^0_2(i ) be the determinant line bundle of the family L_A,Ψ,f; see <cit.>. As explained in <cit.>, Det descends to a real line bundle Λ→ℬ^* ×Ω^0_2(i ) which is shown to be trivial by considering the –equivariant homotopy { L_A, tΨ, tf}_t ∈ [0,1] connecting L_A,Ψ,f with L_A,0,0 = (d + d^*) ⊕_AB. The kernel and cokernel of d + d^* are naturally identified with H^0(Y,)⊕ H^1(Y,), while _AB is complex-linear and has trivial determinant. It follows that the determinant bundle of the family of operators L_A,0,0 is trivial, and therefore so is Λ. Denote the global orientation on Λ given by this trivialisation by or_Λ. 

On the other hand, L_A,Ψ, f is formally self-adjoint, so coker L_A,Ψ, f and L_A,Ψ,f are naturally isomorphic. More precisely, for a self-adjoint Fredholm operator T we define the Knudsen–Mumford isomorphism  ( T) ⊗ (coker T)^* → by

    ω⊗η↦ (-1)^ T  (  T + 1) / 2η ( ω),

where coker  T =  T^* =  T. These isomorphisms do not yield a global trivialisation of Det because the dimension of the kernel jumps. However, they induce a natural trivialisation of Det over every stratum of ℬ^* ×Ω^0_2(i) over which L_A,Ψ, f has constant rank <cit.>. We denote the induced orientation on Λ, restricted to every stratum, by or_0.


If ^* is regular, we define for every x ∈^*

    (x) := {[                   +1 if or_Λ(x) = or_0(x);                   -1  if or_Λ(x) ≠or_0(x) ].
 



The definition can be extended to the case when   is Zariski smooth since for every connected component C of  the dimension of L_A,Ψ,f is constant along C. 



If  is Zariski smooth and C is a connected component of , set

    (C) := {[                   +1 if or_Λ(x) = or_0(x);                   -1  if or_Λ(x) ≠or_0(x) ].
 
where x is any point in C.



Alternatively, we have Λ =  T⊗ (  )^*, so the trivialisation or_Λ induces a relative orientation on → which orients  the zero set of every transverse section of .



 §.§ Counting solutions
  
We come to the main point of this section, that is defining the signed count of Seiberg–Witten multi-monopoles. The next two propositions are deduced in the standard way from Proposition <ref>, Proposition <ref>, and the discussion of orientations.


For every g∈ there is a residual subset ^reg(g) ⊂ such that for every σ∈^reg(g) the moduli space ^*(g, σ) is an oriented zero-dimensional manifold. If b_1(Y) > 0, we may additionally assume that there are no reducible solutions and ^*(g, σ) = (g, σ).







 Let σ_0 and σ_1 be perturbations in . Denote by (σ_0, σ_1) the space of smooth paths {σ_t }_t ∈ [0,1] of elements in  connecting σ_0 and σ_1. The subspace topology induced from the space C^∞([0,1], ) makes (σ_0, σ_1) into a Fréchet manifold.  



For every smooth path { g_t }_t ∈ [0,1] in  and σ_0 ∈^reg(g_0), σ_1 ∈^reg(g_1) there is a residual subset of  (σ_0, σ_1) such that for all smooth paths  {σ_t }_t∈[0,1] from this subset the one-parameter moduli space

    ⋃_t ∈ [0,1]^*(g_t, σ_t)

is an oriented one-dimensional cobordism between ^*(g_0, σ_0) and ^*(g_1, σ_1). If b_1(Y) > 1, we may additionally assume that ^*(g_t, σ_t) = (g_t, σ_t) for all t ∈ [0,1]. 






 
If (g,σ) is compact and consists of irreducible and regular solutions, set

    (g,σ) :=  ∑_x ∈(g,σ)(x).



The definition of  can be extended to the case when  is compact and Zariski smooth.
 
If (g,σ) is compact and Zariski smooth, set

    (g,σ) := ∑_C(C) χ(C),

where the sum is taken over all connected components of  and χ is the Euler characteristic.


The Gauss–Bonnet theorem and the Poincaré–Hopf index theorem give us two equivalent descriptions of .  


Assume that (g,σ) is compact and Zariski smooth.


  * If s is a transverse section of →(g,σ), then 
 
    (g,σ) = ∑_x ∈ s^-1(0)(x),

 where (x) is obtained by comparing the orientation on s^-1(0) induced fom the relative orientation on → with the natural orientation of a point.

  * Suppose that all components of  are orientable and orient them arbitrarily. The relative orientation on → makes  into an oriented real vector bundle. If e() is the Euler class of , then

    (g,σ) =  ∫_ e( ).













For (g,σ) ∈ such that (g,σ) consists of irreducible and regular solutions (g,σ) is well-defined as in Definition <ref>. If (g,σ) ∈ is an arbitrary point, define (g,σ) to be equal to (g',σ') for any (g',σ') in the same connected component of  and such that (g',σ') consists of irreducible and regular solutions. For two such pairs (g_0, σ_0) and (g_1, σ_1), by Proposition <ref> there is a path (g_t, σ_t) in  such that the union

    𝒲 := ⋃_t ∈ [0,1](g_t, σ_t)

is an oriented compact cobordism from (g_0, σ_0) to (g_1, σ_1). It follows that  (g_0, σ_0) = (g_1, σ_1) and so (g,σ) is well-defined for any (g,σ) ∈. It remains to show that if (g,σ) is Zariski smooth, then the integer obtained in this way is equal to that from Definition <ref>. This is a general fact; see <cit.>. 







§ A DIMENSIONAL REDUCTION
 
We focus on the case Y = S^1 ×Σ. Assuming that the parameters of the equation are invariant in the circle direction, we  show that irreducible Seiberg–Witten multi-monopoles on Y are pulled back from configurations on Σ obeying a generalised vortex equation. This is analogous to the correspondence between classical Seiberg–Witten monopoles and vortices, which in turn correspond to effective divisors <cit.>, <cit.>, <cit.>. 





 §.§ Seiberg–Witten equations and quaternionic representations
 
Equation (<ref>) is an example of the Seiberg–Witten equation associated with a quaternionic representation introduced in <cit.> and <cit.>; see also  <cit.>, <cit.>, <cit.>, <cit.>. The language of quaternionic representations is well-suited for proving the circle-invariance of solutions.

Let M be a quaternionic vector space equipped with a Euclidean inner product ⟨· , ·⟩ compatible with the complex structures i, j, k. Let (M) be the group of quaternionic automorphisms of M preserving ⟨· , ·⟩. Suppose that G and H are compact connected Lie groups together with a representation G × H →(M). There is an associated hyperkähler moment map μ M →𝔰𝔲(2) ⊗𝔤, where 𝔤 is the Lie algebra of G, determined by the identity

    ⟨ a, μ(x) ⟩ = ⟨ a x, x ⟩

for all x ∈ M and a ∈𝔰𝔲(2) ⊗𝔤. Here, we think of 𝔰𝔲(2) as the space of imaginary quaternions.






 
Let Y be a Riemannian spin  three-manifold as before.  Let P → Y and Q → Y be principal bundles with structure groups G and H, respectively. Define

    𝕄 = (P ×_Y Q ×_Y S) ×_G × H × SU(2) M,

𝕄 has the structure of a left module over the Clifford bundle of Y. The moment map descends to a fibre-preserving map μ𝕄→Λ^2 Y⊗𝔤_P, where 𝔤_P is the adjoint bundle of P.
Fix a connection B on Q. For any connection A on P, denote by ∇_AB the covariant derivative on 𝕄 induced from A, B, and the Levi–Civita connection. The pair (𝕄, ∇_AB) is a Dirac bundle in the sense of <cit.>, and as such is equipped with the Dirac operator _ABΓ(Y,𝕄) →Γ(Y,𝕄). The Seiberg–Witten equation associated with the quaternionic representation (G× H,M) is the following differential equation for a pair (A,Ψ) ∈(P) ×Γ(𝕄): 

    {[   _ABΨ = 0,; F_A = μ(Ψ). ].


 
A pair (A,Ψ) ∈(P) ×Γ(Y, 𝕄) is called irreducible if there exists x ∈ Y such that the G–stabiliser of Ψ(x) in M is trivial. 



The Seiberg–Witten equation with multiple spinors (<ref>) corresponds to 

    M = ⊗̋^n,      H = (n),      G = (1)

with the standard representations.




 §.§ Circle-invariance of solutions
 
The goal of the next few paragraphs, which will be achieved in Proposition <ref>, is to describe solutions of (<ref>) under the following assumptions:


  * Y = S^1 ×Σ for a closed Riemann surface Σ; we endow Y with the product metric.

  * The spin structure on Y is induced from a spin structure K^1/2 on Σ [Recall that a spin structure on Σ is equivalent to a complex line bundle K^1/2→Σ together with an isomorphism of K^1/2⊗ K^1/2 with the canonical bundle K = Λ^1,0Σ. As a principal bundle, S → Y is pulled back from the (2)–bundle associated to K^-1/2, the dual of K^1/2, via the inclusion (1) ↪(2). As a vector bundle, S = K^1/2⊕ K^-1/2.].

  * Q and B are pulled back from a bundle and a connection on Σ.

  * P is pulled back from a bundle over Σ [This will eventually follow from the existence of an irreducible solution of (<ref>).].


To keep the notation simple we use the same symbols K^1/2, P, Q, B, and so forth for the corresponding objects over Σ and its pull-back to Y. Observe that 𝕄→ Y is pulled back from 

    𝕄 = ( P ×_Σ Q ×_Σ K^-1/2 ) ×_G × H ×(1)  M.

The action of unit quaternions (2) on M rotates the sphere of complex structures with (1) ⊂(2) being the stabiliser of i; thus, 𝕄→Σ is a complex vector bundle. Consider the quaternionic vector bundle V = (P ×_Σ Q) ×_G × H M; then 𝕄 = V ⊗_ K^-1/2, where the complex structure on V is given by i. The remaining part of the quaternionic structure is encoded in an anti-linear involution j  V → V. Taking the tensor product of j with the anti-linear map K^1/2→ K^-1/2 given by the metric, we obtain an anti-linear isomorphism

    σ V ⊗ K^1/2→ V ⊗ K^-1/2.

We define similarly a map in the opposite direction, also denoted by σ, so that σ^2 = -1. Equivalently, σ can be seen as a map σ𝕄⊗ K →𝕄,  which is a two-dimensional manifestation of the Clifford multiplication.  

Next, we relate sections and connections on Y to those on Σ. Write (Σ, P) and (Y, P) for the spaces of connections on P →Σ and its pull-back to Y. Let t ∈ [0,1] be the coordinate on S^1 in S^1 ×Σ. Any connection A_Y ∈(Y,P) can be uniquely written in the form

    A_Y = A(t) + b(t) dt

for one-periodic families A(t) of connections in (Σ,P) and sections b(t) of 𝔤_P.  (Σ, P) embeds in (Y,P) by pulling-back connections. Its image consists of those A_Y = A(t) + bdt  for which b(t) = 0 and A(t) is independent of t. Likewise, any Ψ∈Γ(Y, 𝕄) can be identified with a one-periodic family Ψ(t) ∈Γ(Σ, 𝕄) and Γ(Σ, 𝕄) embeds into Γ(Y, 𝕄) as sections independent of t. The gauge group (Σ, P) is naturally a subgroup of (Y, P).
  

A circle-invariant configuration is an element of the image of 

    (Σ, P) ×Γ(Σ, 𝕄) ↪(Y,P) ×Γ(Y, 𝕄).

We will identify a circle-invariant configuration on Y with the corresponding pair on Σ.



If two circle-invariant configurations differ by g ∈(Y,P), then g ∈(Σ, P). In particular, (Σ, P) ×Γ(Σ, 𝕄) / (Σ, P) is a submanifold of (Y,P) ×Γ(Y, 𝕄) / (Y,P). 








The Dirac operator on Y can be expressed in terms of the Dolbeault operator on Σ. In the simplest case Y = ^3 = × and M = ℍ, denoting coordinates on × by t and z = x + iy, we have for a map Ψℝ^3 →ℍ 

    Ψ =  i ∂Ψ/∂ t + j ∂Ψ/∂ x + k ∂Ψ/∂ y =   ∂Ψ/∂ t + j ( ∂Ψ/∂ x - i ∂Ψ/∂ y) = i ∂Ψ/∂ t + 2 j ∂Ψ/∂ z.

 In general, ∂ / ∂ z is replaced by the Dolbeault operator 
 
    ∂_ABΓ(Σ, 𝕄) →Γ(Σ, 𝕄⊗ K)

 or equivalently
  
    ∂_ABΓ(Σ, V ⊗ K^-1/2) →Γ(Σ, V ⊗ K^1/2),

which is defined as the (1,0)–part of the covariant derivative on 𝕄. The proof of the next lemma is a simple calculation in conformal coordinates, almost the same as (<ref>) [The difference between the constants 2 in (<ref>) and  √(2) in Lemma <ref> comes from the fact that |dz| = √(2) with respect to the Euclidean metric on .].
 
    

Let A_Y = A(t) + b(t)dt be a connection in (Y, P) and Ψ = Ψ(t) a section in Γ(Y, 𝕄). The Dirac operator _A_Y, B acting on Γ(Y, 𝕄) is given by

    _A_Y BΨ = i ( ∂Ψ/∂ t  + b(t) Ψ)+ √(2)σ( ∂_A(t)BΨ).

































In the dimensionally-reduced setting, we use the splitting of the hyperkähler moment map μ M →𝔤⊗^3 into the real and complex parts: μ_ M →𝔤 and μ_ M →𝔤⊗. If μ = (μ_i,μ_j,μ_k) are the three components of μ, then μ_ = μ_i and μ_ = μ_j + i μ_k. The following identity will be useful later:

    ⟨μ_(x) j x, x ⟩ = | μ_(x) |^2

Under the reduction of the structure group of Y from (3) to (1), the splitting ^3 = ⊕ gives us 𝔰𝔲(S) = ⊕ K^-1. Accordingly, μ𝕄→𝔰𝔲(S) ⊗𝔤_P  splits into the direct sum of
 
    μ_𝕄→𝔤_P     and    μ_𝕄→ K^-1⊗𝔤_P.

μ_ is holomorphic when restricted to fibres. Similarly, we have the conjugate maps

    μ_𝕄→𝔤_P     and    μ_𝕄→ K ⊗𝔤_P,

which satisfy 

    μ_∘σ = - μ_    and    μ_∘σ = μ_.





Let A_Y = A(t) + b(t)dt a connection in (Y,P) and Ψ = Ψ(t) a section in Γ(Y, 𝕄).  The generalised Seiberg–Witten equation (<ref>) for (A_Y, Ψ) is equivalent to 

    {[ i (∂Ψ/∂ t + b Ψ) + √(2)σ( ∂_ABΨ) = 0,; ( ∂ A/∂ t + d_A b )^0,1 = - i/2μ_(Ψ),;                       ∗ F_A  = μ_(Ψ). ].

In particular, for a circle-invariant configuration (A,Ψ) the equation simplifies to

    {[      ∂_ABΨ = 0,;     μ_ (Ψ) = 0,; ∗ F_A = μ_ (Ψ). ].




By Lemma <ref>, the first equation in (<ref>) is equivalent to _A_Y BΨ = 0. The remaining two equations are obtained from the identifications

    𝔰𝔲(S) ≅Λ^0 Σ⊕ K^-1  and  μ≅μ_⊕μ_

discussed earlier. Under the decomposition

    Λ^2  Y =  ( Λ^2  Σ) ⊕( Λ^1 S^1 ⊗Λ^1 Σ)

the curvature F_A_Y decomposes into

    F_A_Y = F_A + dt ∧( ∂ A/∂ t + d_A b ).

We need to identify the splittings (<ref>) and (<ref>)  under the isomorphism  Λ^2 Y ≅𝔰𝔲(S). For simplicity, consider the flat case Y = ×, with coordinates t and z = x + i y—the general case differs from it by a conformal factor. The isomorphism Λ^2 ^3 ≅𝔰𝔲(2) is given by

    dx ∧ dy ↦ i ,    dy ∧ dt ↦ j,    dt ∧ dx ↦ k.

On the other hand, 𝔰𝔲(2) is identified with ⊕ via the map

    ai  + bj + ck ↦ (a, b+ic).

Let α + dt ∧β be a two-form on ^3, where 

    α = a dx ∧ dy,   β = b_1 dx + b_2 dy.

Under the identifications Λ^2 ^3 = 𝔰𝔲(2) = ⊕,

    α + dt ∧β↦ a i - b_2 j + b_1 k ↦ (a, -b_2 + i b_1).

Observe that a = ∗α and (- b_2 + ib_1) dz̅ = 2i β^0,1, where β^0,1 is (0,1)-part of β. It follows that under the splittings (<ref>) and (<ref>) the isomorphism

    ( Λ^2  Σ) ⊕( Λ^1 S^1 ⊗Λ^1 Σ) ≅Λ^0 Σ⊕ K^-1

is the direct sum of the Hodge star Λ^2 Σ→Λ^0 Σ and the map Λ^1 Σ→Λ^0,1Σ taking a one-form β to 2i β^0,1. Thus, F_A_Y = μ(Ψ) is equivalent to the last two equations in (<ref>).


 
Since it is more common to consider holomorphic rather than aholomorphic sections, we can complete the picture by considering the conjugate bundle

    𝕄 = (Q × P × K^1/2) ×_G × H × U(1) M = V ⊗ K^1/2 = 𝕄⊗ K.

We have the Dolbeault operators

    ∂_ABΓ(Σ, 𝕄) = Γ(Σ, V ⊗ K^-1/2) ⟶Γ(Σ, V ⊗ K^1/2 ) = Γ(Σ, 𝕄),


    _ABΓ(Σ, 𝕄) = Γ(Σ, V ⊗ K^1/2) ⟶Γ(Σ, V ⊗ K^-1/2 ) = Γ(Σ, 𝕄),

and the maps σ𝕄→𝕄 and σ𝕄→𝕄 that intertwine them:

    σ∂_AB = _ABσ .

Thus, σ maps aholomorphic sections of 𝕄 to holomorphic sections of 𝕄 and vice versa. It follows from the Kähler identities that

    _AB = - ∂_AB^*

where ∂_AB^* is the formal adjoint of ∂_AB.
















Using σ, we can rewrite (<ref>) as a system of equations for Ψ := σ(Ψ) ∈Γ(Σ,𝕄):

    {[          _ABΨ = 0,;         μ_(Ψ) = 0,; ∗ F_A + μ_(Ψ) = 0. ].

This is an example of a symplectic vortex equation discussed in <cit.>. The target singular symplectic space[For example, for the classical Seiberg–Witten equation μ^-1_(0) = { (x,y) ∈^2  |  xy = 0 }.] is the zero locus μ_^-1(0) ⊂ M.


We now proceed to the main result of this section.


Suppose that conditions (A), (B), (C) listed at the beginning of this subsection are satisfied. If (A_Y, Ψ) is an irreducible solution of the generalised Seiberg–Witten equation (<ref>), then  P → Y is pulled-back from a bundle over Σ (that is: condition (D) is satisfied) and (A_Y, Ψ) is gauge-equivalent to a circle-invariant configuration obeying equation (<ref>).



Theorem <ref> asserts that there is a natural one-to-one correspondence between gauge equivalence classes of solutions of the Seiberg–Witten equation on Y = S^1 ×Σ with the hyperkähler target M and gauge equivalence classes of solutions of the symplectic vortex equation on Σ with the Kähler target μ_^-1(0).



Assume for simplicity that the flavour symmetry H is trivial—the general proof is the same after adjusting the notation. Identify S^1 with [0,1] with the endpoints glued together. Pull-back the data on Y=S^1 ×Σ to one-periodic data on [0,1] ×Σ. Since [0,1] ×Σ is homotopy equivalent to Σ, there is a principal G-bundle P_Σ→Σ and a gauge transformation g ∈(Σ, P_Σ)  such that P is the quotient of [0,1] × P_Σ by the relation (0,p) ∼ (1, g(p)). The isomorphism class of P depends only on the homotopy class of g. Similarly,  𝕄 = (P × S) ×_G × SU(2) M over Y is obtained from pulling-back  𝕄_Σ =  (P_Σ× K^-1/2)_G × U(1) M to [0,1] ×Σ and identifying the fibres over 0 and 1 using g. As before there is an anti-linear map σ𝕄_Σ⊗ K →𝕄_Σ.  
 
For A_Y ∈(Y,P) and Ψ∈Γ(Y, 𝕄) we have
 
    A_Y = A(t) + b(t) dt,   Ψ = Ψ(t),

 where A(t), b(t), and Ψ(t) are families of connections and sections on Σ, as discussed earlier. The only difference now is that the families are periodic with respect to the action of g:
 
    A(1) = g ( A(0) ),    b(1) = g (b(0) ),   Ψ(1) = g( Ψ(0) ).

Define a gauge transformation h over [0,1] ×Σ by

    h(t) = exp( ∫_0^t b(s) ds ).

h does not necessarily descend to an automorphism of P → Y; this happens if and only if h(1) = Ad_g(h_0) = id.  In any case, h is well-defined over [0,1]×Σ and the new connection

    C  h(A_Y) = A_Y - h^-1 d_A_Y h = h(t)( A(t) )

does not have a dt part—it is in a temporal gauge. Thus, it is identified with a path of connections { C(t) }_t ∈ [0,1] on P_Σ satisfying C(1) = h(1)g(C(0)). Likewise, we identify the section h(Ψ) with a path {Φ(t) }_t∈[0,1] of sections of 𝕄_Σ→Σ satisfying Φ(1) = h(1)g(Φ(0)). 

By Proposition <ref>, the Seiberg–Witten equation for (C,Φ) is equivalent to

    {[  i ∂Φ/∂ t + √(2)σ( ∂_C Φ) = 0,; ( ∂ C/∂ t)^0,1 = - i/2μ_( Φ ),;                 ∗ F_C = μ_(Φ). ].

Differentiating the first equation with respect to t and using (<ref>), we obtain

    0     = i ∂^2 Φ/∂ t^2 + √(2)σ{( ∂ C/∂ t)^1,0Φ + ∂_C ( ∂Φ/∂ t)   }
        = i ∂^2 Φ/∂ t^2 + √(2)/2σ iμ_ (Φ) Φ + 2  σ  i∂_C σ∂_C Φ
        = i ∂^2 Φ/∂ t^2 -  √(2)/2 i μ_(Φ) σΦ - 2 i  σ∂_C σ∂_C Φ.

We have used the anti-linearity of σ and the fact that  ∂ C / ∂ t is a real 𝔤–valued one-form, so its (1,0) part is conjugate to the (0,1) part. Multiplying the obtained identity by i and taking the pointwise inner product with Φ yields

    0 = ⟨ - ∂^2 Φ/∂ t^2, Φ⟩ +  √(2)/2⟨μ_(Φ) σΦ, Φ⟩ + 2 ⟨σ∂_C σ∂_C Φ , Φ⟩.

 By formula (<ref>) the second term simplifies to √(2)/2 | μ_(Φ)|^2.  Remark <ref> implies that

    σ∂_C σ∂_C = _C ( σ^2 ) ∂_C = - _C ∂_C = ∂_C^* ∂_C.
 
We conclude that 

    ⟨σ∂_C σ∂_C Φ , Φ⟩_L^2(Σ) =  ⟨∂_C^* ∂_C Φ, Φ⟩_L^2(Σ) = ∂_C Φ_L^2(Σ)^2.

For a fixed value of t integration of (<ref>) over Σ yields

    0 = ∫_Σ⟨ - ∂^2 Φ/∂ t^2 , Φ⟩vol_Σ +   √(2)/2μ_( Φ) _L^2(Σ)^2 + 2 ∂_C Φ_L^2(Σ)^2.

Integrate the last equality by parts with respect to t ∈ [0,1]. The boundary terms vanish because  Φ is periodic up to the action of h(1)g which preserves the inner product. We obtain

    0 = ∂Φ/∂ t^2_L^2 + √(2)/2μ_(Φ) ^2_L^2 + 2 ∂_C Φ^2_L^2,

which shows that 

    ∂Φ/∂ t = 0,   ∂ C/∂ t = 0.

Thus, the families C(t) = C and Φ(t) = Φ are constant and

    C = C(1) = k ( C(0) ) = k(C),   Φ = Φ(1) = k( Φ(0) ) = k(Φ)

for the gauge transformation k = h(1)g over Σ. The first equality implies d_C k = 0, so k is covariantly constant. On the other hand, by irreducibility, there exists a point x ∈Σ such that the G–stabiliser of Φ(x) is trivial. Hence, k(x) = id, so k = id everywhere and g = h(1)^-1. The path h(t)^-1 is a homotopy of gauge transformations connecting g with h(0)^-1 = id and so P → Y is pulled-back from P_Σ→Σ. In particular, we could have chosen g=id, then h(1) = id and h descends to a gauge transformation of P mapping (A_Y, Ψ) to the circle-invariant solution (C,Φ). By Proposition <ref>, (C,Φ) satisfies equation (<ref>).  



Much of this discussion can be extended to the setting when M is a hyperkähler manifold with an isometric (2)–action rotating the sphere of complex structures. The Dirac operator _AB and equation (<ref>) have natural generalisations <cit.>. For Y = S^1 ×Σ one introduces the non-linear Dolbeault operator ∂_AB as in <cit.> so that Lemma <ref> and Proposition <ref> hold. However, our proof of Theorem <ref> makes use of the vector space structure on M and does not immediately generalise to the non-linear setting. We expect the result to be true but in the proof one should use the Weitzenböck formulae for non-linear Dirac operators <cit.>.




 §.§ An abelian vortex equation

We apply Theorem <ref> to the Seiberg–Witten equation with multiple spinors. In this case, M = ⊗̋^n, H = (n), and G = (1). The action of (n) on M is the transpose of the standard representation. We identify $̋ with⊕via

    a + b i + cj + dk = (a + b i) + (c - d i) j ↦ ( a + bi, c + di) ,

so that the action of(1)from the left preserves the complex structure given by the left multiplication byi. ThenMis identified with^n ⊕^nand the moment maps are

    μ_(x,y) = i ( |x|^2 - |y|^2),     μ_(x,y) = y^* x.
 
Suppose thatE →YandL →Yare pulled back from bundles overΣ, and so is the background connectionB. We form the associated bundle

    𝕄 = (E × L × K^1/2) ×_SU(n) × U(1) M = (E ⊗ L^* ⊗ K^1/2) ⊕ (E^* ⊗ L ⊗ K^1/2).

Every circle-invariant sectionΨof𝕄can be written in the formΨ= (β, α)for 

    α∈Γ(Σ, E^* ⊗ L ⊗ K^1/2)    and  β∈Γ(Σ, E ⊗ L^* ⊗ K^1/2) .

The real moment map isμ_(β, α) = i( |β|^2 - |α|^2)and the complex moment map is

    Γ(Σ, E^* ⊗ L ⊗ K^1/2 ) ×Γ(Σ, E ⊗ L^* ⊗ K^1/2) →Γ(Σ, K),

    (α, β) ↦αβ.

Suppose that the closed one-formηused to perturb (<ref>) is pulled back fromΣ. By Remark <ref>, for circle-invariant configurations equation (<ref>) reduces to

    {[                              _ABα = 0,;                              _ABβ = 0,;                                αβ = 0,; i ∗  F_A + | α |^2 - |β|^2 - i ∗η = 0. ].


Now is a good point to introduce the space_Σappearing in Theorem <ref>.


  Let Y = S^1 ×Σ be equipped with a spin structure pulled-back from Σ. Let E → Y be an (n)–bundle pulled-back from Σ.
  
  We introduce the following Fréchet manifolds equipped with the C^∞–topology 
_Σ       the space of Riemannian metrics on Σ, 
_Σ       the space of closed imaginary-valued two-forms on Σ
_Σ := (Σ, E) ×_Σ       the space of perturbations pulled-back fromΣ,



The space of parameters of the equation pulled-back from Σ is 

    _Σ := _Σ×_Σ.
 
We consider it as a subspace of the full parameter space  introduced in <ref>.

Assume the situation described in Definition <ref>.
If (A, Ψ) is an irreducible solution of the Seiberg–Witten equation with multiple spinors (<ref>) with respect to a Riemannian metric g ∈_Σ and a parameter σ = (B,η) ∈_Σ, then 

  * L is pulled back from a bundle over Σ, and 

  * (A,Ψ) is gauge-equivalent to a circle-invariant configuration satisfying equation (<ref>).


§ A HOLOMORPHIC DESCRIPTION OF THE MODULI SPACE


The main result of this section, Theorem <ref> below, identifies the moduli space of Seiberg–Witten monopoles onY = S^1 ×Σwith a certain moduli space of holomorphic data onΣ. 



 §.§ Bryan–Wentworth moduli spaces

In the situation of Proposition <ref>, the moduli space^*=^*(g,σ)has the following description. Let

    _Σ = (Σ, L) ×Γ(Σ, E^* ⊗ L ⊗ K^1/2) ×Γ(Σ, E ⊗ L^* ⊗ K^1/2).

Consider the subspace of_Σconsisting triples(A,α,β)satisfying equations (<ref>) and condition(α,β) ≠(0,0). The gauge group(Σ) = C^∞(Σ, S^1)acts freely on this subspace. By Proposition <ref> and Lemma <ref>, the quotient is homeomorphic to^*. 

The next result extends the work of Bryan and Wentworth <cit.> who described Seiberg–Witten multi-monopoles on Kähler surfaces under the assumption that the background bundleEis trivial andBis the product connection.  Before stating the theorem, we introduce

    d = (L) := ⟨ c_1(L), [Σ] ⟩    and    τ := ∫_Σi η/2π.

Ifd - τ< 0, then the last equation of (<ref>) forcesαto be non-zero for

    0 = ∫_Σ{ i F_A - i η + (| α |^2 - | β |^2) vol_Σ} = 2π( d - τ ) + α_L^2^2 - β_L^2^2.
 
Likewise, ifd - τ> 0, thenβmust be non-zero. In both cases there are no reducible solutions and= ^*. Whend - τ= 0, either bothαandβare non-zero or both of them vanish yielding a reducible solution. 

Recall that in dimension two every unitary connection equips the underlying vector bundle with a holomorphic structure. In particular,K^1/2andKare holomorphic line bundles.


 Denote by _B →Σ the holomorphic (n,)–bundle obtained from E^* with the dual connection B^*. We will write simply  in a context in which B is fixed. 

If d - τ< 0, then ^* is isomorphic as real analytic spaces to the moduli space _ of triples (ℒ, α, β) consisting of

  *  a degree d holomorphic line bundle ℒ→Σ, 

  *  holomorphic sections

    α∈ H^0(Σ, _B ⊗ℒ⊗ K^1/2 )   and  β∈ H^0(Σ, _B^* ⊗ℒ^* ⊗ K^1/2)

satisfying α≠ 0 and αβ = 0 ∈ H^0(Σ, K).

Two such triples (ℒ,α,β) and (ℒ', α', β') correspond to the same point in the _ if there is a holomorphic isomorphism ℒ→ℒ' mapping α to α' and β to β'. 

The statement still holds when d - τ≥ 0 with the difference that for d - τ  > 0 it is β instead of α that is required to be non-zero and for d - τ = 0 both α and β are required to be non-zero.


The next few paragraphs are occupied with a construction of_. A related construction was considered in <cit.>. The first three equations in  (<ref>),

    {[ _ABα = 0,; _ABβ = 0,;   αβ = 0, ].

are invariant under the action of the complexified gauge group^c(Σ) := C^∞(Σ, ^*)of complex automorphisms ofL. The action ofg Σ→^*on(A,α,β) ∈_Σis given by

    g(A, α, β) = (A + g^-1∂g - g^-1 g, g α, g^-1β).
 
In terms of the associated Dolbeault operators we have

    [     _g(A)B = g  _B A g^-1 on Γ(Σ, E^* ⊗ L ⊗ K^1/2),;        _g(A)B = g^-1_BA g on Γ(Σ, E ⊗ L^* ⊗ K^1/2). ]

Consider the subspace of _Σ consisting of triples (A,α,β) satisfying equations (<ref>) and subject to the condition

    {[           α≠ 0 if  d - τ < 0,;           β≠ 0 if  d - τ > 0,; α≠ 0  and β≠ 0 if  d - τ = 0. ].

We define _ to be the quotient of this subspace by the action of ^c(Σ). The points of _ parametrise the isomorphism classes of triples ( ℒ, α, β ) considered in Theorem <ref>.

The moduli space _ depends on the conformal class of the metric g on Σ, the holomorphic bundle _B, the degree d of L, and the sign of d - τ.


Employing the methods of subsection <ref>, one shows that_is metrisable, second countable, and has a natural complex analytic structure given by local Kuranishi models as in Remark <ref>. The discussion is almost the same as that for the Seiberg–Witten equation, so we only outline the details. To set up the Fredholm theory, consider the modified equation

    {[ _ABα + i f β̅ = 0,; _ABβ - i f α̅ = 0,;      αβ + ∂ f = 0. ].

A solution of (<ref>) is a quadruple(A, α, β, f)whereA,α, andβare as before andf ∈C^∞(Σ, ). The equation is elliptic modulo the action of^c(Σ). An analogue of Proposition <ref> is


If (A,α,β,f) is a solution of (<ref>) with (α,β) ≠ 0, then f = 0.


Using the linearisation of (<ref>) together with the complex Coulomb gauge fixing we represent_as the zero set of a Fredholm section. The local structure of the moduli space is encoded in the elliptic complex at a solution(A, α, β, 0):

    Ω^0() rG^c_A,α,β   Ω^0,1⊕Γ(E^* ⊗ L ⊗ K^1/2) ⊕Γ(E ⊗ L^* ⊗ K^1/2) ⊕Ω^0() r   
    rF_A,α,β   Γ(E^* ⊗ L ⊗ K^-1/2) ⊕Γ(E ⊗ L^* ⊗ K^-1/2) ⊕Ω^1,0

whereG^c_A,α,βis the linearised action of the complexified gauge group

    G^c_A,α,β(h) = (-  h, hα, -hβ, 0),

andF_A,α,βis the linearisation of equations (<ref>)
    F_A,α,β(a^0,1, u, v, t) = 
    (
    [ _AB u + a^0,1α + i t β̅; _AB v - a^0,1β - i t α̅;         u β + α v + ∂ t ]).

Even though the map given by the left-hand side of (<ref>) is not holomorphic, its derivativeF_A,α,βat a solution(A,α,β,0)is complex linear and so the cohomology groupsH^0_A,α,β,H^1_A,α,β,H^2_A,α,βare complex vector spaces. If the solution is irreducible, thenH^0_A,α,β = 0. We are left with complex vector spacesH^1_A,α,βandH^2_A,α,βof the same dimension. They have the following description—the proof is yet another variation of that of Proposition <ref>.


Let (A,α, β, f) be a solution of (<ref>) with (α, β) ≠ 0 and f = 0. Then the deformation space H^1_A,α,β is the quotient of the space of solutions 

    (a^0,1, u, v) ∈Ω^0,1() ⊕Γ(E^* ⊗ L ⊗ K^1/2 ) ⊕Γ(E ⊗ L^* ⊗ K^1/2),

    {[ _AB u + a^0,1α = 0,; _AB v - a^0,1β = 0,;      u β + α v = 0. ].
 
by the subspace generated by ( -  h, hα, - hβ) for h ∈Ω^0(). The obstruction space H^2_A, α, β is canonically isomorphic to the dual space  (H^1_A, α, β)^* as complex vector spaces. 






























The analytic structure on a neighbourhood of[A,α,β]in_is induced from a Kuranishi mapκH^1_A,α,β →H^2_A,α,β. Since the derivative ofF_A,α,βis complex linear at a solution,κcan be taken to be complex analytic which shows that_is a complex analytic space.



 §.§ A homeomorphism between the moduli spaces


Since equation (<ref>) is part of (<ref>) and(Σ)is a subgroup of^c(Σ), every point of^*gives rise to a point in_.


The natural map ^* →_ is a homeomorphism.


The proof relies on a generalisation of a classical theorem of Kazdan and Warner.


Let X be a compact Riemannian manifold and let P, Q, and w be smooth functions on X with P and Q non-negative, and

    ∫_X P - Q > 0,     ∫_X w > 0.

Then the equation 

    Δ u + P e^u - Q e^-u = w

has a unique solution u ∈ C^∞(X).

We refer to <cit.> for the proof of Lemma <ref>, but let us remark that it can be easily extended to the case when ∫ w = 0 and both P and Q are not identically zero (without the assumption on the sign of ∫ P - Q). First, observe that if we replace P and Q by P' = e^C P and Q' = e^-C Q respectively, then solving the corresponding equation

    Δ u + P' e^u - Q' e^-u = w

is equivalent to solving the original equation  with P and Q. Indeed,  if u solves the former, then the function u + C is a solution the latter. Since both P and Q are not identically zero, their integrals are positive and by choosing the constant

    C = 1/2log( ∫_X Q /∫_X P),

we can guarantee that ∫ P' - Q' = 0. Thus, we may as well assume that this holds for the original functions P and Q. After this adjustment, we simply repeat the proof from <cit.>. The only difference is the construction of sub- and super-solutions, that is functions u_- and u_+ satisfying u_- ≤ u_+ and

    Δ u_- + Pe^u_- - Q e^-u_- - w ≤ 0,

    Δ u_+ P e^u_+ - Q e^-u_+ - w ≥ 0.

Let v_1 and v_2 be solutions of Δ v_1 = w and Δ v_2 = - P + Q. Choose a constant M such that M ≥sup | v_1 + v_2|  and set 

    u_+ = v_1 + v_2 + M     and     u_- = v_1 + v_2 - M.

Then clearly u_- ≤ u_+ and

    Δ u_- + Pe^u_- - Q e^-u_- - w = P ( e^v_1 + v_2 - M -1 ) + Q (1- e^-v_1 - v_2 + M) ≤ 0,

    Δ u_+ + Pe^u_+ - Q e^-u_+ - w = P( e^v_1 + v_2 + M - 1) + Q ( 1 - e^-v_1 - v_2 -M) ≥ 0.
 
The remaining part of the proof from <cit.> goes through unchanged.

It is clear that the map ^* →_ is continuous, so it remains to construct a continuous inverse _→^*.
Let (A, α, β) be a solution of (<ref>). As in <cit.>, we seek h ∈^c(Σ) such that h (A, α, β) = (A', α', β') satisfies also the third equation of (<ref>). We can assume h = e^f for f Σ→. We have 

    (A', α', β') = (A -  f + ∂ f, e^f α, e^-fβ)

so the curvature of A' is 

    F_A' = F_A - 2 ∂ f = F_A - i ∗Δ f,

where Δ is the positive definite Hodge Laplacian. Thus, (<ref>) for (A', α', β') is equivalent to

    0    = i ∗ F_A' + |α'|^2 - |β'|^2 - iη
       =  Δ f + e^2f | α |^2 - e^-2f | β |^2 + i (∗ F_A - η).

Assume d - τ< 0 and set P = | α |^2, Q = | β|^2 and w = -i ∗ (F_A - η). We need to solve

    Δ f + P e^2f - Q e^-2f  = w.

If d - τ < 0, then α is assumed to be non-zero. After applying a gauge transformation of the form h = e^C for C constant, we may assume that

    ∫_Σ (P-Q)  = ∫_Σ | α |^2 - | β|^2 > 0.

Moreover, we have

    ∫_Σ w =  - ∫_Σ( i F_A - η) = -2π (d - τ) > 0.

The hypotheses of Lemma (<ref>) are satisfied and there is a unique solution f ∈ C^∞(Σ) to (<ref>). This shows that there exists h ∈^c(Σ), unique up to an element of (Σ), mapping (A,α, β) to a solution of (<ref>). If d - τ> 0 the proof is the same with P = | β |^2, B = |α |^2, w = i ∗( F_A - η) and f replaced in the equation by -f. The case d = 0 follows from Remark <ref>.

This gives us an inverse to ^* →_; it remains to show that it is continuous.  Let [A_i, α_i, β_i] be a convergent sequence of points in _. Let (A_i', α_i', β_i') be the corresponding solutions of (<ref>). There is a sequence h_i = u_i e^f_i such that h_i (A_i', α_i', β_i') converges in _Σ. The functions f_i satisfy (<ref>) with coefficients P_i, Q_i, w_i converging in C^∞(Σ). It follows from the proof of <cit.> that for every k there is a C^k bound for f_i, independent of i. By the Arzelà–Ascoli theorem, after passing to a subsequence, f_i converges in C^∞(Σ). It follows that [A_i',α_i',β_i'] converges in ^*, which proves the continuity of _→^*.


 §.§ Proof of Theorem <ref>

It remains to compare the deformation theories of the two moduli spaces to show that the homeomorphism^* →_is an isomorphism of real analytic spaces.

^* is isomorphic to the moduli space _Σ of solutions of (<ref>).

Let_Σbe the space of(Σ)–orbits of triples

    (A, α, β) ∈(Σ,L) ×Γ(Σ, E^* ⊗ L ⊗ K^1/2) ×Γ(Σ, E ⊗ L^* ⊗ K^1/2)

 satisfying(α,β) ≠0and

    {[                             _ABα = 0,;                             _ABβ = 0,;                               αβ = 0,; i ∗  F_A + | α |^2 - |β|^2 - i ∗η = 0 ].

Endow_Σwith a real analytic structure using local Kuranishi models[As in the constructions of  and _ one introduces an extra term f ∈ C^∞(Σ, i) to make the equation elliptic modulo gauge.We can ignore this because analogues of Proposition <ref> and Lemma <ref> hold.]. Let^*_Σand^*be the  spaces of irreducible configurations(A,Ψ)overΣandYrespectively. It follows from Proposition <ref> and Lemma <ref> that the inclusion

    ℬ^*_Σ = ^*_Σ / (Σ) ↪^* / (Σ)  = ℬ^*

induces a homeomorphism_Σ →^*. The Seiberg–Witten moduli space^*is, at least locally, given as the zero set of a Fredholm sectionsof a bundle overℬ^*. On the other hand, the restriction ofstoℬ^*_Σgives a Fredholm section defining_Σ. In order to show that the induced real analytic structures agree we need to prove

    H^1_A,Ψ =  ds_(A,Ψ) =  d (sℬ^*_Σ)_(A,Ψ)

 for every[A, Ψ] ∈_Σ = ^*. The corresponding equality of cokernels follows then from the natural isomorphism betweenH^1_A,ΨandH^2_A,Ψ(and likewise for the equations overΣ). 

Equality (<ref>) is the linearised version of Theorem <ref>.  Let(A,Ψ)be a circle-invariant solution.  According to Lemma <ref>,H^1_A,Ψ = S_A,Ψ ⊕G_A,Ψ^*, whereS_A,Ψis the linearisation of the equation without the extra termfandG_A,Ψthe infinitesimal gauge group action. Using Proposition <ref>, we identifyH^1_A,Ψwith the space of pairs

    (a(t) + b(t) dt, ϕ(t)) ∈Γ(S^1 ×Σ, Λ^1(i ) ⊕Λ^0( i ) ⊕ (E^* ⊗ S ⊗ L ) )

satisfying

    {[ i ( ∂ϕ/∂ t + b Ψ) + √(2)σ( ∂_ABϕ + a^1,0Ψ) = 0,;             ∂ a^1,0/∂ t + ∂ b - i μ_(Ψ, ϕ) = 0,;                         ∗ da + 2μ_ ( Ψ, ϕ) = 0,;               - d^* a - ∂ b/∂ t + i ⟨Ψ, ϕ⟩ = 0. ].

Equality (<ref>) will be established by showing that any solution(a + bdt, ϕ)satisfies

    ∂ a/∂ t = 0,   ∂ϕ/∂ t = 0,    b = 0.

This is done in the same way as in the proof of Theorem <ref>. First,  apply∂/ ∂tto the first two equations, then get rid of the terms∂ϕ/ ∂t,∂b / ∂t, and∂a^1,0 / ∂t.  This results in 

    - ∂^2 ϕ/∂ t^2 + 2 ∂_AB^* ∂_ABϕ - i ⟨Ψ, ϕ⟩Ψ + d^* a ·Ψ + 2 (∂^* a^1,0 ) Ψ + √(2)σμ_(Ψ, ϕ) Ψ = 0,

    - ∂^2 a^1,0/∂ t^2 +  ∂∂^* a^1,0 +  i ∂⟨Ψ, ϕ⟩ + √(2)μ_ ( Ψ, σ∂ϕ) + √(2)μ_ ( Ψ, σ a^1,0Ψ ) = 0.

  Take the realL^2–product of the first equation withϕand the second equation witha^1,0. Integrating by parts as in the proof of Theorem <ref>, we obtain

    ∂ϕ/∂ t^2_L^2 + ∂ a/∂ t^2_L^2 + 2 ∂_ABϕ + a^1,0Ψ_L^2^2 +  - d^*a  + i ⟨Ψ, ϕ⟩^2_L^2 + √(2)μ_(Ψ, ϕ) _L^2^2 = 0.

We have used identity  (<ref>) to relateμ_to the inner product. Thus, we have proved thatb = 0,ϕandaare pulled-back fromΣand satisfy

    {[     ∂_ABϕ + a^1,0Ψ  = 0,;            μ_(Ψ, ϕ) = 0,;  ∗ da + 2μ_ ( Ψ, ϕ) = 0,; - d^* a -  i ⟨Ψ, ϕ⟩ = 0. ].

Recall that with the conventions of subsection <ref> we identifyΨwith a pair(α, β). After a conjugation equation (<ref>) translates to the following equation foraandϕ = (u,v){[                   _AB u + a^0,1α = 0,;                   _AB v - a^0,1β = 0,;                        α v + u β = 0,;   ∗ i da + 2 ⟨α, u ⟩ - 2 ⟨β, v ⟩ = 0,; - d^* a -  i ⟨α, u ⟩ - i ⟨β, v ⟩ = 0. ].

This is the linearisation of (<ref>) together with the Coulomb gauge fixing condition. We conclude that (<ref>) holds and^*is isomorphic to_Σas real analytic spaces.

_Σ is isomorphic to _.

The proof is similar to that of <cit.>, so we only outline the argument. As before, the main point is to show an isomorphism of the deformation spaces for_Σand_. The former is given by (<ref>) and the latter consists of solutions of the first three equations together with a choice of a local slice for the action of^c(Σ). The Lie algebra of^c(Σ)splits is the direct sum ofC^∞(Σ, )and the Lie algebra of(Σ). Under this splitting, we can choose a slice of^c(Σ)–action imposing the standard Coulomb gauge condition for(Σ), which is the last equation of (<ref>), together with a choice of a slice for the action ofC^∞(Σ,):

    e^f (A, α, β) = (A + ∂ f -  f, e^f α, e^-fβ).

The linearisation of this action at(A,α,β)is 

    f ↦ (-  f + ∂ f, f α, -f β).

A local slice for the action ofC^∞(Σ,)can be obtained from any subspace of 

    Ω^1(i) ⊕Γ(E^* ⊗ L ⊗ K^1/2) ⊕Γ(E ⊗ L^* ⊗ K^1/2)

 which is complementary to the image of (<ref>). Hence, to show that the deformation spaces of_Σand_are  isomorphic it is enough to prove that the subspace given by 

    i ∗ da + 2 ⟨α, u ⟩ - 2 ⟨β, v ⟩ = 0

is complementary to the image of (<ref>). In other words, we need to know that for any triple(a,u,v)there is a unique functionf ∈C^∞(Σ,)such that

    0    = i ∗ d(a -  f + ∂ f) + 2 ⟨α, u + fα⟩ - 2 ⟨β, v - f β⟩
       = {Δ + 2( | α |^2 + | β |^2) } f + i ∗ da + 2 ⟨α, u ⟩ - 2 ⟨β, v ⟩.
 
This is true because(α, β) ≠0and so the operatorΔ+ 2 ( |α|^2 + | β|^2)is invertible onC^∞(Σ,). In the same way as in <cit.> we conclude that (<ref>) provides a local Fredholm model for both_Σand_and so the two spaces have isomorphic analytic structures.


§ A TALE OF TWO COMPACTIFICATIONS

The goal of this section is to define natural compactifications of^*and_and to extend the isomorphism^* ≅_to a homeomorphism between these compactifications, thus completing the proof of Theorem <ref>. We assumed - τ< 0so in particular= ^*. The discussion can be easily adapted to the casesd - τ= 0andd - τ> 0.



 §.§ A complex-geometric compactification_has a natural compactification analogous to the one described in <cit.>. Consider the subspaceS ⊂_Σ ×given by

    S := { (A, α, β, t)  |   (A,α,β)  satisfies equations (<ref>), α≠ 0,  and  (β, t) ≠ (0,0) }.

The group^c(Σ) ×^*acts freely onSby the standard action of the first factor on_Σand

    λ(A, α, β, t) = (A, α, λβ, λ t)   for λ∈ℂ^×.

We define _ to be the quotient of S by ^c(Σ) ×^*. 


This is analogous to compactifying^Nbyℂℙ^Nwhich is the quotient of(^N ×) ∖{ (0,0) }by the free action of^*; in fact,_is obtained by applying this construction fibrewise.


Let  be the subspace of _ consisting of triples of the form (A, α, 0). Equivalently,  is the space of ^c(Σ)–orbits of pairs (A,α) satisfying _ABα = 0 and α≠ 0. 


We will see momentarily thatis compact. The natural projection(A, α, β) ↦(A, α)induces a surjective mapπ_ →. Let[A, α] ∈and denote byℒ_Athe holomorphic structure onLinduced byA. The fibreπ^-1([A, α])is the kernel of the homomorphism

    H^0(Σ, _B^* ⊗ℒ_A ⊗ K^1/2) rα    H^0(Σ, K)

given by pairing withα. The compactification_is obtained by replacing each fibreαwith the projective spaceℙ( α⊕)containing it.


The space _ is metrisable, compact, and contains _ as an open dense subset. Moreover, the complex analytic structure on _ extends to a complex analytic structure on _ with respect to which _ is Zariski open.

It is clear that _ is metrisable and _⊂_ is open and dense. In order to show that _ is compact, consider a sequence [A_i, α_i, β_i, t_i] ∈_; we need to argue that there are sequences h_i ∈^c(Σ) and λ_i ∈^* such that after passing to a subsequence h_i λ_i (A_i, α_i, β_i, t_i) converges smoothly in S. This is the content of Step 1 in <cit.>.


Let _Σ^* ⊂_Σ be the subset of configurations (A, α, β) with α≠ 0. The group ^c(Σ) acts freely on _Σ^* with quotient ℬ_Σ^*, a complex Banach manifold. There is a holomorphic vector bundle 𝒲→ℬ_Σ^* such that _ is the zero set of a holomorphic Fredholm section 𝒮ℬ_Σ^* →𝒲. The zero set of such a section carries a natural complex analytic structure  <cit.>. The complex analytic structure on _ is extended to _ by extending 𝒮 to a Fredholm section whose zero set is _. Replace _Σ^* by the subspace of _Σ× consisting of quadruples (A, α, β, t) for which α≠ 0 and (β, t) ≠ (0, 0). Let ℬ_Σ^* be the quotient of this space by the action of ^c(Σ) ×^*: it contains ℬ_Σ^* as an open subset. Let 𝒲→ℬ^*_Σ be the vector bundle obtained as the quotient of 𝒲×^* by the lifted action of ^c(Σ) ×^*. There is a holomorphic Fredholm section 𝒮 extending 𝒮 so that _ = 𝒮^-1(0). When restricted to the open subset ℬ_Σ^*, this reduces to the construction of _ described above, so the inclusion _⊂_ is compatible with the induced analytic structures. Moreover, _∖_ is the intersection of 𝒮^-1(0) with the analytic subset ℬ_Σ^*∖ℬ_Σ^* given by the equation t = 0. We conclude that  _∖_ is an analytic subset of _, and so _ is Zariski open.
⊂_ is compact. Furthermore, _ is compact if and only if = _.
 consists of equivalence classes [A, α, β] for which β = 0; it is compat by Step 1 in <cit.>. If = _, then _ is compact. To prove the converse, observe that if _ is non-compact, then _∖_ is non-empty by Proposition <ref>. On the other hand, _∖_ consists of ^c(Σ) ×^*–orbits of the form [A, α, β, 0] with β≠ 0 so every element of _∖ gives rise to an element of _∖_.


 §.§ A gauge-theoretic compactification

For an arbitrary three-manifold a good compactification ofis yet to be constructed—see <cit.> for a discussion of analytical difficulties involved in such a construction. 

ForY = S^1 ×Σwe can overcome these obstacles thanks to a refined compactness theorem <cit.>.


For the remaining part of the paper we make the assumption that E is an (2)–bundle. In this case the description of Fueter section simplifies <cit.>, <cit.>. The discussion below should easily generalise to the higher rank case.
 
If (A_i,Ψ_i = (α_i, β_i)) is a sequence of solutions of (<ref>) with Ψ_i _L^2→∞, then after passing to a subsequence and applying gauge transformations (A_i, Ψ_i / Ψ_i _L^2) converges in C^∞_ on the complement of a finite set D = { x_1, …, x_N }. The limiting configuration (A, Ψ = (α, β)) is defined on Σ∖ D and satisfies

  * Ψ_L^2 = 1 and | Ψ | > 0 on Σ∖ D,

  * _ABα = 0, _ABβ = 0, αβ = 0, and | α | = | β| on Σ∖ D.

  * A is flat on Σ∖ D and has holonomy contained in _2. 

  *  There are non-zero integers a_1, …, a_N such that ∑_k=1^N a_k = 2d and

    ∗i/2π F_A_i⟶1/2∑_k=1^N a_k δ_x_k

in the sense of measures.

  *  For each k=1, … , N we have  

    | Ψ (x) | = O( dist(x_k,x)^|a_k|/2 ).

Let D ⊂Σ be a finite set. We say that a gauge transformation in (Σ∖ D) or ^c(Σ∖ D) is simple if it has degree zero around each point of D. Denote by _0(Σ∖ D) and ^c_0(Σ∖ D) the subgroups of simple gauge transformations.

Let D ⊂Σ be a finite subset. With any flat connection A ∈(Σ∖ D, L) we associate a measure i ∗ F_A on Σ as follows. For x ∈ D let B ⊂Σ be a small disc centred at x and not containing other points of D. In a unitary trivialisation of LB we have A = d + a for a one-form a ∈Ω^1( B ∖{ x }, i ). Denote

    q_x = ∫_∂ B ia.

The measure i ∗ F_A is defined by

    i ∗ F_A := ∑_x ∈ D q_x δ_x.

One easily checks that i∗ F_A is well-defined and invariant under simple gauge equivalences.















A limiting configuration is a triple (A, Ψ, D) comprising of a finite subset D = { x_1, … , x_N }⊂Σ, a connection A ∈(Σ∖ D,L), and a pair Ψ = (α, β) of nowhere-vanishing sections α∈Γ( Σ∖ D, E^* ⊗ L ⊗ K^1/2) and β∈Γ(Σ∖ D, E ⊗ L^* ⊗ K^1/2) satisfying

  * Ψ_L^2 = 1 and |Ψ| > 0 on Σ∖ D
  * _ABα = 0, _ABβ = 0, αβ = 0, and | α | = | β| on Σ∖ D.

  * A is flat on Σ∖ D and has holonomy contained in _2.

  *  There are non-zero integers a_1, …, a_N such that ∑_k=1^N a_k = 2d and

    ∗i/2π F_A = 1/2∑_k=1^N a_k δ_x_k  as measures.

  *  For each k=1, … , N we have  

    | Ψ (x) | = O( dist(x_k,x)^|a_k|/2 ).
(A,Ψ,D) and (A',Ψ',D') are simple gauge equivalent if D = D' and they differ by an element u ∈_0(Σ∖ D). Let ℱ be the set of simple gauge equivalence classes of limiting configurations.


The spaceℱcan be equipped with a natural topology in which a sequence[A_i, Ψ_i, D_i]converges to[A,Ψ,D]if and only ifi ∗F_A_i' →i ∗F_Aweakly as measures and after applying a sequence in_0(Σ∖D)we haveA_i →AandΨ_i →ΨinC^∞_onΣ∖D. 


Let [A,Ψ,D] be an equivalence class in ℱ. For ϵ > 0, δ > 0, an integer k ≥ 0, and a continuous function f Σ→ we define V_ϵ,δ,k,f(A,Ψ,D) ⊂ℱ to be the set of the elements of ℱ which have a representative (A',Ψ',D') satisfying

  * D' ⊂ D_ϵ where D_ϵ :=   { x ∈Σ |  dist(x, D) < ϵ},

  * A' - A _C^k(Σ∖ D_ϵ) < δ,

  * Ψ' - Ψ_C^k(Σ∖ D_ϵ) < δ. 

  * | ∫_Σ (i ∗ F_A') f - ∫_Σ ( i ∗ F_A) f | < δ.

The family of subsets

    { V_ϵ,δ,k,f(A,Ψ,D) }

forms a base of a Hausdorff topology on ℱ. 

The proof is a simple application of <cit.>. 











The next step is to combineandℱinto one topological space. For this purpose it is convenient to identify points ofwith gauge equivalence classes of triples(A, Ψ, t), wheret ∈(0, ∞), Ψ_L^2 = 1, and

    {[       _ABΨ = 0,; t^2 F_A = μ(Ψ). ].

The map(A, Ψ, t) ↦(A, t^-1 Ψ)gives a homeomorphism between the space of such classes and the moduli space^*. Recall that in our setting there are no reducibles so there is no boundary att →∞. The boundary att →0is obtained by gluing in the space of limiting configurations.


Let [A,Ψ,D] be an equivalence class in ℱ. For ϵ > 0, δ > 0, an integer k ≥ 0, and a continuous function f Σ→ we define

    W_ϵ, δ, k, f(A,Ψ,D) ⊂

to be the set of the elements of  that have a representative (A', Ψ', t) satisfying

  * t < δ,

  * A' - A _C^k(Σ∖ D_ϵ) < δ,

  * Ψ' - Ψ_C^k(Σ∖ D_ϵ) < δ,

  * | ∫_Σ ( i ∗ F_A')f - ∫_Σ (i ∗ F_A)f | < δ.

The compactified moduli space is

    ∪ℱ

equipped with the topology whose basis are the subsets

    W_ϵ,δ,k,f(A,Ψ,D)  W_ϵ, δ, k,f(A,Ψ,D)  ∪ V_ϵ, δ, k,f(A,Ψ,D).


Let  be a base of the topology on . The family of subsets 

    {W_ϵ,δ,k(A,Ψ,D) }∪

 forms a base of a Hausdorff topology on . 












 §.§ A homeomorphism at infinity

The main ingredient in the proof of Theorem <ref> is


The spaces _∖_ and ℱ are homeomorphic. 


The proof of Proposition is preceded by auxiliary results about limiting configurations. The first of them is a complex-geometric analogue of the statement that a limiting configuration induces a flat connection with_2holonomy.


Let (A, α, β) be a solution of (<ref>) with α≠ 0 and β≠ 0. Denote by ℒ the holomorphic line bundle (L, _A). Let D_1 and D_2 be the zero divisors of α and β respectively, and (D_1 - D_2) the holomorphic line bundle associated to D_1 - D_2. There is a canonical holomorphic isomorphism  

    φ_αβ(D_1 - D_2) ⟶ℒ^2.

Recall that α is a holomorphic section of ⊗ℒ⊗ K^1/2 and β is a holomorphic section of ^* ⊗ℒ^* ⊗ K^1/2. Since αβ = 0 and the rank of  is two, we have the short exact sequence

    0 r   ℒ^-1⊗ K^-1/2⊗(D_1) rα   rβ   ℒ^-1⊗ K^1/2⊗(-D_2) r    0.

The associated isomorphism of the determinant line bundles is

    ℒ^-2⊗(D_1 - D_2) ≅≅,

where the last isomorphism follows from the fact that  is a holomorphic SL(2,)-bundle. Tensoring both sides with ℒ^2, we obtain the desired isomorphism φ_αβ. It is canonically determined by α and β.


The lemma below provides an upper bound on the number of components, counted with multiplicities, of the singular set of a limiting configuration. 


There exists M ≥ 0, depending only on the holomorphic bundle , with the following significance. If (ℒ, α, β) is a triple as in Lemma <ref>, then 

    D_1 +  D_2 ≤ M.

Tensoring exact sequence (<ref>) with K^1/2 we see that ℒ^-1⊗(D_1) is a holomorphic subbundle of ⊗ K^1/2. It is an elementary fact that the degrees of line subbundles of a given holomorphic bundle are bounded above <cit.>.  In fact, if ℱ is a holomorphic vector bundle and A ⊂ℱ a line subbundle, then

    A ≤ h^0(Σ, ℱ) + 2g(Σ)-2

where g(Σ) is the genus of Σ. (We will use this bound later.) Thus, we have an upper bound on the degree of ℒ^-1⊗(D_1). On the other hand, ℒ^2 is isomorphic to (D_1 - D_2), so

    ( ℒ^-1⊗(D_1) ) = - ℒ +  D_1 = 1/2(  D_1 +  D_2 ),

which proves the lemma.


The next result will be useful in proving the convergence of measures.


Let f Σ→ be a continuous function, γ > 0, and D ⊂Σ a finite subset. Then there exist ϵ> 0 and δ > 0 with the following property. Suppose that D' ⊂Σ is another finite subset, and A and A' are two flat connections over Σ∖ D and Σ∖ D' respectively, satisfying

  * D' ⊂ D_ϵ,

  * A' - A _C^0(Σ∖ D_ϵ) < δ,

  *  the measures ∗ i F_A and ∗ i F_A' have integer weights.

Then we have

    | ∫_Σ ( i∗ F_A' )f - ∫_Σ  (i ∗ F_A)f | ≤γ i ∗ F_A'.

where i ∗ F_A' is the total variation of the measure i ∗ F_A' given by

    i ∗ F_A'  = ∑_x ∈ D' | q_x |

 for   i ∗ F_A' = ∑_x ∈ D' q_x δ_x . 
 
Let D = { x_1, …, x_N } and a_1, …, a_N be the integer weights of the measure i ∗ F_A as in Definition <ref>. Suppose that ϵ is small enough so that the discs B_i of radius ϵ centred x_i are pairwise disjoint.  Partition the set D' into disjoint subsets D_1', …, D_N' consisting of points within ϵ–distance from x_1, …, x_N respectively. For each i choose small disjoint discs E_i1, E_i2, … centred at points of D_i' and contained in B_i. Let b_i1, b_i2… be the weights of the points in D_i' in the measure i ∗ F_A'. 

In a unitary trivialisation of L over each B_i we have

    A = d + a     and     A' = d + a',

where one-form a is defined on B_i ∖{ x_i } and a' is defined on B_i ∖ D_i'. By the hypothesis of the lemma a - a' _C^0(∂ B_i) < δ. Thus, for sufficiently small δ,

    | a_i - ∑_j b_ij| =| ∫_∂ B_i i a - ∑_j ∫_∂ E_ij ia' | = 
      | ∫_∂ B_i ia - ∫_∂ B_i ia' | < 1.

 Since all numbers a_i, b_ij are integers, so we conclude that

    a_i = ∑_j b_ij.

For each i denote the points of D_i' by { x_ij}. Then

    | ∫_Σ ( i∗ F_A' )f - ∫_Σ  (i ∗ F_A)f |    =  | ∑_i a_i f(x_i) - ∑_i j b_ij f(x_ij) | 
       ≤∑_ij |b_ij| | f(x_i) - f(x_ij) |.

By the continuity of f we can choose ϵ sufficiently small so that 

    sup_x ∈ B_i | f(x_i) - f(x) | ≤γ

for all i = 1, …, N. Then
 
    | ∫_Σ ( i∗ F_A' )f - ∫_Σ  (i ∗ F_A)f |  ≤γ i ∗ F_A'.


The last lemma allows us to extend a limiting configuration to a holomorphic section. The proof is a minor variation of <cit.>.


Let L be a Hermitian line bundle over the unit disc B ⊂. Suppose that A is a unitary connection on LB ∖{ 0 } and φ a section of L over B ∖{ 0 } satisfying _Aφ = 0 and | φ | = 1. Denote by φ the degree of φ∂ B. Then

  * F_A = 0 on B ∖{ 0 } and i ∗ F_A =( 2πφ )δ_0 as measures.

  *  There exists a complex gauge transformation h  B ∖{ 0 }→^* such that h has degree zero around zero and in some trivialisation of L around zero h(A) is the trivial connection and hφ = z^k.

















Set 𝒳 = _∖_. We will construct a continuous bijection 𝒳→ℱ. Since the domain is compact by Proposition <ref> and the target space is Hausdorff by Lemma <ref>, such a map is necessarily a homeomorphism.

 
The construction of 𝒳→ℱ.

Let [A, α, β, t] ∈𝒳. By definition, t=0, α≠ 0, β≠ 0, and (<ref>) is satisfied. Let D_1, D_2 be the zero divisors of α, β, respectively. We will interchangeably consider them as divisors or as subsets of Σ. Set D = D_1 ∪ D_2. We claim that there is a simple complex gauge transformation h ∈^c_0(Σ∖ D) such that (hA, hα, h^-1β) is a limiting configuration. A necessary condition is

    | h α | = | h^-1β |   on Σ∖ D.

 A transformation satisfying (<ref>) exists since α and β are both non-zero on Σ∖ D and we can set h = √( | β| / | α| ); any other choice of h will differ from that one by an element of _0(Σ∖ D). 

The map  𝒳→ℱ is defined by [A,α,β,0] ↦ (A', α', β') := (h(A), hα, h^-1β). We need to show that (A', α', β') represents a class in ℱ. We clearly have 

    {[      _A'Bα' = 0,;      _A'Bβ' = 0,; | α' | = | β' |. ].

Moreover, for Ψ' := (α', β')
    | Ψ' | = √( | α' |^2 + | β' |^2 ) = √(  | h|^2 |α|^2 + |h|^-2 | β |^2 ) = √( 2 | α | | β |).

Integrating over Σ yields

    Ψ' _L^2 = √(∫_Σ 2 | α |  | β | ).

After rescaling β, which does not change the class of [A,α,β,0]∈𝒳, we can assume Ψ_L^2 = 1. We also see that  | Ψ | > 0 on Σ∖ D and in a neighbourhood of every x ∈ D
    | Ψ(y) | = O( dist(y,x)^ord_x(α) + ord_x(β)/2),

where ord_x(α) and ord_x(β) denote the order of vanishing of α and β at x. 

It remains to prove that A' is flat and the measure i ∗ F_A' is as in Definition <ref>. Let φ_αβ(D_1 - D_2) → L^2 be the A–holomorphic isomorphism from Lemma <ref>. The construction of Lemma <ref> is local, so we can also define an analogous map φ_α' β' corresponding to sections α' and β'. Since they are both nowhere vanishing, φ_α' β' is an A'–holomorphic isomorphism of the trivial bundle over Σ∖ D and L^2Σ∖ D; thus, φ_α' β' is a nowhere vanishing A'–holomorphic section of L^2 over Σ∖ D. Moreover, on Σ∖ D
    | φ_α' β' | =  | α' | / | β' | = 1   and  φ_α' β' = h^2 φ_αβ

By Lemma <ref>, the tensor product connection A ⊗ A on L^2 is flat; so A itself is flat and for every x ∈ D the weight of the measure i/2π∗ F_A at x is equal to half of the degree of φ_α' β' around x.  Since h has degree  zero around each point x ∈ D, the degrees of φ_αβ and φ_α' β' around x agree. Denote this degree by q_x ∈. Since the zero divisor of φ_αβ is D_1 - D_2, 

    ∑_x ∈ D q_x =  D_1 -  D_2 =  ( L^2) = 2d.


Finally, observe that for every x ∈ D we have

    q_x  = ord_x( α )- ord_x (β) ≤ord_x (α) + ord_x (β).

Together with (<ref>) this shows that (A', α', β', D) is a limiting configuration. It is easy to check that if we replace (A,α,β,0) by a different quadruple in the same orbit of the ^c(Σ) ×^*–action, then the resulting limiting configurations are simple gauge-equivalent.

𝒳→ℱ is injective.

Suppose that (A_1, α_1, β_1, 0) and (A_2, α_2, β_2, 0) give rise to limiting configurations that are simple gauge equivalent. In particular, they have the same singular set, D say. Suppose that β_1 and β_2 are scaled so that

    ∫_Σ 2 | α_1 | | β_1 | = ∫_Σ 2 | α_2 | | β_2 | = 1.

Composing the simple gauge equivalence of the limiting configurations with complex gauge transformations satisfying (<ref>), we obtain t ∈^c_0(Σ∖ D) such that on Σ∖ D
    A_2 = t(A_1),   α_2 = t α_1,   β_2 = t^-1β_1,   φ_α_2β_2 =  t^2 φ_α_1β_1.

Even though t is not defined at the points of D, the holomorphic data is. In particular,  φ_α_1 β_1 and φ_α_2 β_2 have zeroes at every point of D. Moreover, the zeroes are of the same order—this is equivalent to  the measures of the corresponding limiting configurations being equal. We conclude that t is bounded around D. Since it is also (A_1,A_2)–holomorphic it extends to a holomorphic isomorphism between (L, _ A_1) and (L, _A_2) and so (A_1, α_1, β_1, 0) and (A_2, α_2, β_2, 0) give rise to the same point in 𝒳 and the map 𝒳→ℱ is injective.

𝒳→ℱ is surjective.

 Let (A', Ψ' = (α', β'), D) ∈ℱ. We need to find h ∈^c_0(Σ∖ D) such that (A, α, β)  (h(A'), hα', h^-1β') extends smoothly to the whole of Σ. Furthermore, we should have D = D_1 ∪ D_2 where D_1 and D_2 are the zero divisors of α and β respectively. 

Let φ_α' β'∈Γ(Σ∖ D, L^2) be as before. Then _A'φ_α' β' = 0 and | φ_α' β'| = 1 on Σ∖ D. By Lemma <ref>, applied to the connection A' ⊗ A' and section φ_α' β', there exists k ∈^c_0(Σ∖ D, L) such that C  k(A' ⊗ A') extends to a connection on a line bundle T →Σ and k φ_α' β' extends to a meromorphic section of (T, _C). We claim that T =L^2 as unitary bundles, that k = h^2 for some h ∈^c_0(Σ∖ D), and that C = A ⊗ A for A = h(A'). This follows from the assumption on the measure i ∗ F_A' induced by the limiting configuration (A', α', β', D); by Lemma <ref> for every point x ∈ D the meromorphic section h^2 φ_α' β' vanishes to the order q_x defined by

    i/2π F_A' = 1/2∑_x ∈ D q_x δ_x.

(x is a pole of order | q_x| if q_x < 0.) The claim is then a consequence of the assumption ∑_x ∈ D q_x = 2d = ( L^2) and the fact that k has degree zero around the points of D. Thus, A = h(A') extends. We need to show that α = h α' and β = h^-1β' extend. Observe that 

    | α | | β | = | α' | | β' | = 1/2 | Ψ' |^2,

where we have used | α' | = | β'| and Ψ' = (α', β'). As a consequence, for every x ∈ D
    |α(y) | | β(y) | = O( dist(x,y)^|q_x|).

On the other hand, we have

    | α | / | β |  = | h^2 |  | α' | / | β' |  = | h^2 φ_α' β'|,

so around x ∈ D
    | α (y) | / | β(y) |  = O ( dist(x,y)^q_x).

We conclude that

    | α(y) | = O( dist(x,y)^ |q_x| + q_x/2)    and      | β(y) | = O( dist(x,y)^ |q_x| - q_x/2),

which shows that both α and β are bounded over Σ∖ D. Since they are also holomorphic, they extend to globally defined sections. Hence, (A, α, β, 0) represents a point in 𝒳 corresponding to (A', α', β', D) under 𝒳→ℱ.

𝒳→ℱ is continuous.

Suppose that [A_i, α_i, β_i] → [A, α, β] in 𝒳. Let (A_i', α_i', β_i', D_i) and (A', α', β', D) be the corresponding points in ℱ. We will prove that (A_i', α_i', β_i', D_i) converges to (A', α', β', D) as limiting configurations. We easily check that the points of D_i concentrate around D and modulo simple gauge transformations

    A_i' → A',     α_i' →α',     β_i' →β'

in C^∞_ on  Σ∖ D. By Lemma <ref>, i ∗ F_A_i'→ i ∗ F_A' as measures provided that the sequence of total variations i ∗ F_A_i' is bounded. i ∗ F_A_i' is up to a constant equal to the degree of D_1^i + D_2^i where D_1^i and D_2^i are the zero divisors of α_i and β_i. By Lemma <ref> this degree is bounded above. We conclude that [A_i', α_i', β_i', D_i] → [A', α', β', D] in ℱ.


 §.§ Proof of Theorem <ref>

We construct a bijective map_ →from the homeomorphisms_ →from Theorem <ref> and_ ∖_ →ℱfrom Proposition <ref>. This map is continuous when restricted toand its complement. It remains to show that it is continuous; indeed,_is compact by Proposition <ref> andis Hausdorff by Lemma <ref>, so a continous bijection_ →is a homeomorphism.

Let(A_i, α_i, β_i, t_i)be a sequence representing points in_and(A_i', Ψ_i', t_i')the corresponding sequence of solutions of (<ref>). Suppose thatt_i →0and(A_i, α_i, β_i)converges inC^∞to(A, α, β)withα≠0andβ≠0. This limit represents an element of _ ∖_and thus corresponds to a limiting configuration(A',Ψ', D). We need to show that after applying gauge transformations the sequence of Seiberg–Witten solutions(A_i', Ψ_i', t_i')converges in the sense of Definition <ref> to a limiting configuration which is simple gauge-equivalent to(A', Ψ', D). By Theorem <ref>, the sequence converges and by Proposition <ref>, the limiting configuration is simple gauge-equivalent to(A',Ψ',D). This shows that the map_ →is continuous.


§ FUETER SECTIONS AND COMPLEX GEOMETRY

The main result of this section will ensure the compactness of(g,σ)for any product metricgonY=S^1×Σandσgeneric among the parameters pulled-back fromΣ.

  *  For every product Riemannian metric g on Y = S^1 ×Σ there is a residual subset (g) ⊂(Σ, E) with the property that if B ∈(g), then there exist no Fueter sections with respect to (g,B).
 
  *  Let (g_t)_t∈[0,1] be a path of product metrics and B_0 ∈(g_0), B_1 ∈(g_1). For a generic path (B_t)_t∈[0,1] in (Σ,E) connecting B_0 and B_1 there exist no Fueter sections with respect to (g_t,B_t) for all t ∈ [0,1].


Recall that_contains a compact subspaceconsisting of holomorphic triples(ℒ, α, β)withβ= 0. As a result of Theorems <ref>, <ref>, and Corollary <ref>, we obtain

 For a generic choice of B ∈(Σ,E) we have 

    _ = _ = .

If d - τ = 0 then _ is empty for a generic choice of B ∈(Σ, E). 


Here is an outline of the proof of Theorem <ref>: first, we describe Fueter sections in terms of complex-geometric data onΣ. Next, we show that this data is described by a Fredholm problem of non-positive index given by the Riemann–Roch theorem. As a result, we can apply the Sard–Smale theorem to exclude the existence of such data for a generic choice ofB∈(Σ,E).



 §.§ Circle-invariance of Fueter sections


For the remaining part of the section we continue to assume thatgis a product metric onY = S^1×ΣandB ∈(Σ,E).

Let(A,Ψ,Z)is a Fueter section as in Definition <ref>. Suppose thatZ =S^1 ×DforD ⊂Σ, and that(A,Ψ)is pulled back fromΣ. Then, as in subsection <ref>, we haveΨ= (α, β)where

    α∈Γ(Σ∖ D, E^* ⊗ L ⊗ K^1/2),

    β∈Γ(Σ∖ D, E ⊗ L^* ⊗ K^1/2).

The Fueter equations_ABΨ=0andμ(Ψ)=0are equivalent to

    {[ _ABα = 0,   _ABβ = 0,;               αβ = 0,;        | α | = | β |.;                       ].

Let (A,Ψ,Z) be a Fueter section as in Definition <ref>. Then Z = S^1 × D for a finite subset D ⊂Σ. Moreover, there is a gauge transformation u ∈(Y ∖ Z) such that u has degree zero around each component of Z and u(A, Ψ) is pulled-back from Σ∖ D.

The proof is similar to that of Theorem <ref>. We use the notation from subsection <ref> and ignore the background connection B; the general proof is the same. 


A Weitzenböck formula.


Let t be the coordinate on the S^1 factor of S^1 ×Σ. Unlike in the proof of Theorem <ref>, we cannot put A in a temporal gauge, even after pulling-back to ×Σ, because a priori the singular set Z could intersect the t–axis in a complicated way. However, we still have

    0 = _AΨ = - σ∇_t Ψ +  _AΨ,

where ∇_t = ∇_A( ∂ / ∂ t) and _A is the Dolbeault operator induced by A on the { t }×Σ slice. Let ∇_Σ be the part of the covariant derivative ∇_A in the Σ-direction. Since A is flat, we have 

    0 = ∇_A^2 = ∇_t ∇_Σ + ∇_Σ∇_t.

Applying σ and ∇_t^* = - ∇_t to (<ref>) and using the above commutation relation, we obtain

    0 =  ∇_t^* ∇_t Ψ + σ_A σ_A Ψ =  ∇_t^* ∇_t Ψ + _A^* _A Ψ.

Integration by parts; the circle-invariance of Z.

We want to integrate (<ref>) by parts to conclude ∇_t Ψ = 0 and _A Ψ = 0. The equality holds only on  Y ∖ Z, so we need to use a cut-off function. Let f → [0,1] be smooth and such that

    {[        f = 0 on  (-∞, 0],;          f=1   on  [1, ∞) ].

For every ϵ > 0 we define the cut-off function χ_ϵ Y→ [0,1] by

    χ_ϵ(x) = f( | Ψ(x) | - ϵ/ϵ).
 
Let Z_ϵ be the subset of points in Y satisfying | Ψ(x) | < ϵ. We have 

    {[      χ_ϵ = 0    on   Z_ϵ,;       χ_ϵ =1 on  Y ∖ Z_2ϵ ].

and χ_ϵ is smooth on Y. Take the inner product of (<ref>) with χ_ϵ^2 Ψ and integrate by parts:

    0    = ∫_Y | ∇_t( χ_ϵΨ ) |^2 + ∫_Y | _A ( χ_ϵΨ ) |^2 - ∫_Y ( | ∂_t χ_ϵ |^2 + | χ_ϵ |^2  )| Ψ |^2  
       ≥∫_Y | ∇_t( χ_ϵΨ ) |^2 + ∫_Y | _A ( χ_ϵΨ ) |^2 - 2 ∫_Y | d χ_ϵ |^2 | Ψ |^2.

We need to show that the last term becomes arbitrarily small as ϵ tends to zero. By definition, | Ψ | ≤ 2ϵ on Z_2 ϵ. Let P_ϵ = Z_2ϵ∖ Z_ϵ. By Kato's inequality

    ∫_Y | Ψ |^2 | d χ_ϵ |^2 
        ≤∫_P_ϵ | Ψ |^2 | f' (  | Ψ(x) | - ϵ/ϵ) |^2/ϵ^2 | ∇_A Ψ |^2 
       ≤ 4  f ^2_C^1∫_P_ϵ | ∇_A Ψ |^2 
       ≤ C (P_ϵ) ∇_A Ψ_L^2(M∖ Z)^2.

The right-hand side converges to zero as ϵ→ 0 since | ∇_A Ψ |^2 is integrable, Z = ∩_ϵ >0 Z_ϵ, and (Z) = 0 by Taubes <cit.>. Taking ϵ→ 0 in  (<ref>), we conclude that on Y ∖ Z
    ∇_t Ψ = 0     and    _A Ψ = 0.

In particular, 

    ∂_t | Ψ |^2 = 2 ⟨∇_t Ψ, Ψ⟩ = 0

so |Ψ| is invariant under the circle action on Y ∖ Z. It is also continuous on the whole of Y and | Ψ |^-1(0) = Z, so  Z is necessarily of the form S^1 × D for a proper subset D ⊂Σ. 

(A,Ψ) is pulled-back from Σ∖ D.

We put A in a temporal gauge over S^1 × (Σ∖ D)  as in the proof of Theorem <ref>. The gauge transformation (<ref>) used to do that is the exponential of a smooth function Σ∖ D → i when restricted to each slice { t }× (Σ∖ D); thus, it has degree zero around the components of Z. The same argument as in the proof of Theorem <ref> shows that LY ∖ Z is pulled back from a bundle on Σ∖ D and (A,Ψ) is pulled-back from a configuration on Σ∖ D satisfying (<ref>).

D is a finite set of points.

It is enough to show that D is locally finite. Suppose that Σ is a unit disc and that L and E are trivial. The complement Σ∖ D is a non-compact Riemann surface and 
 (L, _A) defines a holomorphic line bundle over Σ∖ D which is necessarily trivial <cit.>. Thus, there is h ∈^c(B ∖ D) such that h(A) agrees with the product connection on the trivial bundle, and h α and h^-1β correspond to holomorphic maps Σ∖ D →^2. Let γ = (h α) ⊗ (h^-1β); it is a holomorphic map Σ∖ D →^2 ⊗^2 = ^4 satisfying
 
    | γ | = | α | | β | = 1/2 | Ψ |^2,

 so D is the zero set of |γ|. Thus, γ is continuous on Σ and holomorphic on Σ∖ D. By a theorem of Radó<cit.>, γ is holomorphic on Σ and so D=γ^-1(0) is locally finite.


 §.§ A holomorphic description of Fueter sections

If (A,Ψ,Z) is a Fueter section, with Ψ = (α,β) and Z = S^1× D as in Proposition <ref>, then there exist h ∈^c_0(Σ∖ D) and divisors D_1, D_2 such that

  * D = D_1 ∪ D_2 as sets and the divisor D_1 + D_2 is effective,

  * Ã := h(A) extends to a unitary connection on a line bundle over Σ, not necessarily isomorphic to L, defining a holomorphic line bundle ℒ→Σ,

  *  sections α̃ = hα and β̃ = h^-1β extend to holomorphic sections that fit into the short exact sequence

    0 r   ℒ^-1⊗ K^-1/2⊗(D_1) rα̃   rβ̃   ℒ^-1⊗ K^1/2⊗(-D_2) r    0.

Conversely, every set of holomorphic data (Ã, α̃, β̃, D_1, D_2) satisfying conditions (1), (2), (3) can be obtained from a Fueter section (A,Ψ,Z) in this way.

This is similar to Step 3 in the proof of Proposition <ref>. Using Lemma <ref>, we find h ∈^c_0(Σ∖ D, L) such that Ã = h(A) extends yielding a holomorphic line bundle ℒ, say, and h^2 φ_αβ extends to a meromorphic section of ℒ^2. Let α̃ = h α and β̃ = h^-1β. Then 

    | α̃ |/ | β̃ |  = | h^2 φ_αβ |     and      |α̃| | β̃ | = |α | | β | = 1/2 | Ψ |^2.
 
Since h^2 φ_αβ is meromorphic and |Ψ| extends to a continuous function on Σ, it follows that α̃ and β̃ extend to meromorphic sections. Let D_1 and D_2 be the associated divisors of zeroes and poles. We have D = D_1 ∪ D_2 as sets and the condition D = | Ψ |^-1(0) implies that D_1 + D_2 ≥ 0.  The existence of the short exact sequence involving α̃ and β̃ was established in (<ref>).


The next lemma provides a restriction on the possible holomorphic bundlesfitting into the short exact sequence (<ref>).


Under the assumptions of Proposition <ref> there exists a holomorphic line bundle M satisfying  h^0( M^2 ) > 0 and h^0( ⊗ K^1/2⊗ M^-1 ) > 0.  

Recall that by Lemma <ref> we have ℒ^2 = (D_1 - D_2). Set M = ℒ^-1⊗(D_1). Then 

    M^2 = ℒ^-2⊗(2D_1) = (D_2 - D_1 + 2D_1) = (D_1 + D_2).

We have h^0(M^2) > 0  because the divisor D_1 + D_2 is effective. On the other hand, multiplying exact sequence (<ref>) by M^-1⊗ K^1/2, we obtain an injective map →⊗ M^-1⊗ K^1/2, which is the same as a nowhere vanishing section of  ⊗ K^1/2⊗ M^-1.


 §.§ Proof of Theorem <ref>

Fix k ≥ 0. Let ⊂(Σ,E) be the subset consisting of those connections B for which there exists a degree k holomorphic line bundle M →Σ satisfying

    h^0(M^2) > 0   and   h^0( _B ⊗ K^1/2⊗ M^-1) > 0.

The complement (Σ,E) ∖ is residual. Furthermore, for all B_0,B_1 ∈(Σ,E) ∖, a generic path in (Σ,E) connecting B_0 and B_1 is disjoint from .

We use a transversality argument similar to the one used to show Proposition <ref>. As in that case we pass to suitable Sobolev completions of the spaces of connections and sections (for simplicity we keep the same notation). The statement for C^∞ topology will follow from Taubes' trick discussed in the proof of Proposition <ref>.

Let T →Σ be a unitary line bundle of degree k. Denote F = E^* ⊗ K^1/2⊗ T^-1 and consider 

    (Σ,E) ×(Σ,T) ×Γ(F) ×Γ(T^2) →Ω^0,1 (F) ×Ω^0,1( T^2 ),

    (B, A, ψ, α) ↦ (_ABψ, _Aα).
 
This map is ^c(Σ)–equivariant. Let  𝒳 be the open subset of (Σ,T)×Γ(F)×Γ(T^2)/ ^c(Σ) given by { [B,A,ψ, α]  |  ψ≠ 0,  α≠ 0 }. Let 𝒱→(Σ, E) ×𝒳 be the Banach vector bundle obtained from taking the ^c(Σ)–quotient of the trivial bundle with fibre Ω^0,1 (F) ×Ω^0,1( T ). Then the map introduced above descends to a smooth section s  (Σ, E) ×𝒳→𝒱. For every B ∈(Σ,E) the restriction s_B = s(B, ·) is a Fredholm section whose index is the Euler characteristic of the elliptic complex

    Ω^0() r   Ω^0,1() ⊕Γ(F) ⊕Γ(T^2) r   Ω^0,1(F) ⊕Ω^0,1(T^2).

The first arrow in the complex is the linearised action of ^c(Σ), whereas the second is the linearisation of the map (A, ψ, α) ↦ (_ABψ, _A α). This elliptic complex agrees up to terms of order zero with the direct sum of the complexes

    Ω^0() r   Ω^0,1() r    0 
        and

    0 r   Γ(F) ⊕Γ(T^2) rr_AB⊕_A      Ω^0,1(F) ⊕Ω^0,1(T^2).

By the Riemann–Roch theorem, the Euler characteristic of this complex is

    χ(  ) - χ(F) - χ(T^2) = (1-g) - ((F)+2-2g) - (2 (T) + 1-g) = 0

because (F) = 2g-2 - 2(T).  Thus, s_B is a Fredholm section of index zero.

The proof will be completed if we can show that s is transverse to the zero section at all points [B,A,ψ,α] ∈ s^-1(0) ⊂𝒳. Indeed, if this is the case, then by the Sard–Smale theorem, the same is true for s_B for B from a residual subset of (Σ,E). For every such B the set 

    { [A, ψ, α]  |  _ABψ = 0,  _A α = 0,  ψ≠ 0  α≠ 0 }

is a zero-dimensional submanifold of 𝒳. This submanifold must be empty as otherwise it would contain a subset homeomorphic to ^* given by [A, t ψ, α] for t ∈^*. This proves that for a generic B there is no holomorphic line bundle M = (T, _A) together with non-zero α∈ H^0(M^2) and ψ∈ H^0( _B ⊗ K^1/2⊗ M^-1). The statement for paths is proved in the same way.


It remains to show that s is transverse to the zero section. At a point [B,A,ψ,α] ∈ s^-1(0) the first map in  (<ref>) is injective. Thus, it is enough to prove the surjectivity of the operator combining the second map of (<ref>) and the linearisation of _AB with respect to B: 

    Ω^0,1(End(F)) ⊕Ω^0,1() ⊕Γ(F) ⊕Γ(T^2) ⟶Ω^0,1(F) ⊕Ω^0,1(T^2)

    (b,a,u,v) ↦ ( (b+a)ψ + _AB u, a α + _A v).

If the map were not surjective, there would exist a non-zero (p,q) ∈Ω^0,1(F) ⊕Ω^0,1(T^2)L^2–orthogonal to the image; which in turn would imply _AB^* p = 0, _A^* q = 0, and

    ⟨ b ψ , p ⟩_L^2 = 0,     ⟨ a α, q ⟩_L^2 = 0

for all b ∈Ω^0,1(End(F)) and a ∈Ω^0,1(). Note that ψ and α are both non-zero and holomorphic;  p and q are anti-holomorphic and at least one of them is non-zero. Using a bump function as in the proof of Proposition <ref> it is easy to construct b and a such that

    ⟨ b ψ , p ⟩_L^2 + ⟨ a α, q ⟩_L^2 > 0.


Theorem <ref> follows immediately from the previous results.
LetB∈(Σ,E)and denote by_Bthe corresponding holomorphic bundle. Propositions <ref> and <ref> show that a Fueter sections with respect to(g,B)corresponds to a holomorphic triple(ℒ,α, β)fitting into the short exact sequence (<ref>). On the other hand, by Lemmas <ref> and <ref>, for a generic choice ofBthe holomorphic bundle_Bdoes not fit into any such sequence. The same is true whenBvaries in a generic one-parameter family by the second part of Lemma <ref>. 

 §.§ Fueter sections and limiting configurations



Every limiting configurations, as in Definition <ref>, is an example of a Fueter section onY=S^1×Σ. The results established in this section allow us to construct a counterexample to the converse statement.


Suppose that the genus of Σ is positive so that the canonical divisor K is effective. Let C_1 and C_2 be two divisors satisfying C_1 + C_2 = 1/2K ≥ 0. Set 

    ℒ := (C_1 - C_2),      D_1 := 2C_1,      D_2 := 2C_2,

Then 

    ℒ^-1⊗ K^-1/2⊗(D_1) = (C_1 + C_2 - 1/2K) = ,

    ℒ^-1⊗ K^1/2⊗(-D_2) = (-C_1 - C_2 + 1/2K) = .

Set  := ⊕. A Fueter section is given by maps α̃ and β̃ making the sequence (<ref>) exact. In the present setting,   (<ref>) is equivalent to

    0 r   rα̃   ⊕rβ̃   r    0,

so there is an obvious choice of α̃ and β̃ making the sequence exact. According to Proposition <ref>, this gives rise to a Fueter section with singular set D = D_1 ∪ D_2. However, such a Fueter section is not a limiting configuration unless both divisors C_1 and C_2 are effective. 


§ MODULI SPACES OF FRAMED VORTICES

By Theorem <ref>, for a generic choice of a circle-invariant parameter,is homeomorphic to the compact spaceintroduced in Definition <ref>. In this section we prove thatis a Kähler manifold and that the signed count of Seiberg–Witten multi-monopoles onY=S^1×Σis the signed Euler characteristic of, which proves Theorem <ref>. We then establish some general properties ofusing methods of complex geometry.



 §.§ Framed vortices

We continue to assume throughout this section thatEis an(2)–bundle and thatd - τ< 0whered = Landτ= ∫_Σ iη/ 2π[Most of the results generalise easily to the other cases. We will later discuss the role of the sign of d - τ.].depends on the conformal structure onΣ, holomorphic structure_B = (E^*, _B), andd. Its points can be interpreted in three ways. 


  *  As isomorphism classes of pairs (ℒ, α), where ℒ→Σ is a degree d holomorphic line bundle and α is a non-zero holomorphic section of _B ⊗ℒ⊗ K^1/2. 

  *  As ^c(Σ)–equivalence classes of pairs 

    (A, α)  ∈(Σ,L) ×Γ(Σ, E^* ⊗ L ⊗ K^1/2)

satisfying _ABα = 0 and α≠ 0.

  *  As (Σ)–equivalence classes of pairs (A, α) as above satisfying

    {[          _ABα = 0   and α≠ 0,; i ∗ F_A + | α |^2 - i ∗η = 0, ].

Following <cit.>, we will refer toas the moduli space of framed vortices. 


 §.§ Deformation theory

Here we relate the deformation theories ofand_. 

For every conformal class of a metric g on Σ there exists a residual subset ^reg(g) ⊂(Σ, E) such that for every B ∈^reg(g)
  * = (g,_B) is a compact Kähler manifold of complex dimension g(Σ)-1+2d,

  * _ = _(g,_B) is Zariski smooth and the inclusion ↪_ is a homeomorphism inducing an isomorphism of Zariski tangent spaces at every point,

  *  the relative orientation on the obstruction bundle →_ is compatible with the orientation of the cotangent bundle T^* → induced from the complex structure; equivalently, for every connected component C of _ we have (C) = (-1)^g(Σ)-1. 

The following conditions are equivalent:

  *  is regular as the moduli space of framed vortices.

  * _ is compact and equal to .

  *  There exist no triple (ℒ, α, β) consisting of a degree d holomorphic line bundle ℒ→Σ and non-zero holomorphic sections α∈ H^0( _B ⊗ℒ⊗ K^1/2) and β∈ H^0( _B^* ⊗ℒ^* ⊗ K^1/2) satisfying αβ = 0 ∈ H^0(K).

The equivalence of (1) and (2)  follows from Corollary <ref> and the identification of the obstruction bundle ⋃_A,α H^2_A,α with _∖_, shown in the proof of Theorem <ref>. The equivalence of (2) and (3) is obvious from the definition of _. 

 The diffeomorphism type of (g, _B) does not depend on the metric on Σ and the connection B ∈(Σ,E), as long as B is generic. 

Let g_0, g_1 be metrics on Σ and B_0 ∈(g_0), B_1 ∈(g_1) as in Theorem <ref>. For a generich path (B_t) in (Σ,E) connecting B_0 and B_1 there exist no Fueter sections with respect to (g_t,B_t) and so (g_t,B_t) is compact for all t∈[0,1]. By Corollary <ref>, (g_t,_B_t) = (g_t,B_t) is compact and regular as the moduli space of framed vortices. Thus, ⋃_t∈[0,1](g_t,_B_t) → [0,1] is a smooth fibre bundle and every fibre (g_t,_B_t) is diffeomorphic to (g_0,_B_0).



The construction of an analytic structure onfollows the general scheme that by now is familiar to the reader. 
Consider the elliptic complex associated to a solution(A,α)of_AB α= 0:

    Ω^0() rG^c_A,α   Ω^0,1() ⊕Γ(E^* ⊗ L ⊗ K^1/2) rT_A, α   Ω^0,1(E^* ⊗ L ⊗ K^1/2).

whereG^c_A,αis the linearised action of^c(Σ)G^c_A,α(f) = (-  f, f α)     for  f ∈Ω^0(),

andT_A,αis the linearisation of the Dolbeault operator

    T_A,α( a^0,1, ϕ) = ( _ABϕ + a^0,1α )     for  (a^0,1, ϕ) ∈Ω^0,1() ⊕Γ(E^* ⊗ L ⊗ K^1/2).

Denote byH^0_A, α,H^1_A,α, andH^2_A, αthe homology groups of this complex. By definitionconsists of solutions withα≠0, soH^0_A,α = 0. On the other hand, the deformation complex is isomorphic modulo lower order term to the sum of the Dolbeault complexes foronΩ^0()and_ABonE^* ⊗L ⊗K^1/2(with a shift). By the Riemann–Roch theorem the expected complex dimension ofis

    _ H^1_A, α -  _ H^2_A,α =  χ(Σ, ) - χ(Σ, ⊗ℒ⊗ K^1/2 ) = g(Σ) -1 + 2d.

The proof proceeds in three steps.
 is a compact Kähler manifold.

We already know that  is compact. By Corollary <ref>, _  = for a generic B. 
One can show that  is generically smooth in the same way as in Proposition <ref>. 

Alternatively, we can interpret the elements of H^2_A,α as Fueter sections:

    H^2_A,α =  T_A,α^* = { q ∈Γ(E^* ⊗ L ⊗ K^-1/2)  |  α q = 0,  _AB^* q = 0 }.

Every non-zero element of H^2_A,α gives rise to a non-zero β = q∈Γ(E ⊗ L^* ⊗ K^1/2) satisfying _ABβ = 0 and αβ = 0. Thus, the triple (A,α,β) is an element of _∖_ corresponding to a Fueter section as in Proposition <ref>.  By Theorem <ref>, for a generic B there are no Fueter sections so H^2_A,α = 0 for all [A,α] ∈. This implies that  is a complex manifold of dimension g(Σ) - 1 + 2d whose holomorphic tangent space at [A,α] is H^1_A,α. It admits a natural Hermitian metric induced from the L^2–inner product on the space of connections and sections. This metric is Kähler because  is the moduli space of solutions of the framed vortex equations (<ref>), which is an infinite-dimensional Kähler quotient. For details, see <cit.>.

H^1_A,α is naturally isomorphic to the Zariski tangent space to _ at [A,α,0].
H^1_A,α consists of pairs (a^0,1,u) ∈Ω^0,1() ⊕Γ(E^* ⊗ L ⊗ K^1/2) satisfying the linearised equation

    _AB u + a^0,1α = 0

together with the complex Coulomb gauge  (G^c_A,α)^*(a^0,1,u) = 0. By (<ref>), the tangent space to _ at consists of triples (a^0,1, u, v) where a^0,1, u are as above, v ∈Γ(E ⊗ L^* ⊗ K^1/2), and  

    {[ _AB u + a^0,1α = 0,;          _AB v = 0,;             α v = 0 ].

together with the complex Coulomb gauge for (a^0,1, u,v). Any non-zero v satisfying the conditions above would give an element (A, α, v) of _∖_. Since B has been chosen so that  _∖_ is empty,  v = 0 and the equations obeyed by (a^0,1, u, 0) are identical to the ones defining H^1_A, α. We conclude that the Zariski tangent spaces to  and _ are equal.  


Comparing the orientations.

Let (A, Ψ) be an irreducible solution of the Seiberg–Witten equations. We have Ψ = (α, 0) where (A,α) is a solution of the framed vortex equations. Consider the extended Hessian operator introduced in subsection <ref>:

    L_A,ΨΩ^1(i ) ⊕Ω^0(i ) ⊕Γ(E^* ⊗ S ⊗ L ) ⟶Ω^1(i ) ⊕Ω^0(i ) ⊕Γ(E^* ⊗ S ⊗ L )

Write L_A,Ψ = L_A,0 + P, where 

    L_A,0 = 
    (
    [   ∗ d    -d     0; - d^*     0     0;     0     0   _AB ])

and 

    P(a,v, ϕ) = ( i ⟨Ψ, ϕ⟩ , - 2 ∗μ(ϕ, Ψ) , - a ·Ψ + v Ψ ).

The kernel and cokernel of L_A,0 are naturally identified with 

    H^1( Y, i ) ⊕ H^0 ( Y, i ) ⊕_AB.

The isomorphism between L_A,0 and  L_A,Ψ, defining the relative orientation on the obstruction bundle, factors through the determinant space P of the finite dimensional map

    P  H^1( Y, i ) ⊕ H^0 ( Y, i ) ⊕_AB→ H^1( Y, i ) ⊕ H^0 ( Y, i ) ⊕_AB

induced from the zeroth order operator P defined above (for simplicity we use the same letter to denote the induced finite dimensional map).  As in the proof of Proposition <ref>, we have  H^1(Y, i ) = H^1( S^1 , i  ) ⊕ H^0,1(Σ). Consider the complex structure on H^1(S^1, i) ⊕ H^0(Y, i) coming from the identification

    H^1(S^1, i) ⊕ H^0(Y, i) = i⊕ i  = .

Let dt be the one-form spanning H^1(S^1, ). Under the Clifford multiplication, dt acts as the multiplication by i on S, and so i dt acts as the multiplication by -1. Hene, under the isomorphism H^1(S^1, i) ⊕ H^0(Y, i) =, the map (a,v) ↦ (- a ·Ψ + v Ψ) is given by (x+iy) ↦ (x+iy) Ψ  and so, in particular, it is complex linear. Next, consider the first two components of P, that is the map ϕ↦ (i (Ψ, ϕ), -2∗μ(ϕ, Ψ)).  Decompose the moment map into μ = μ_⊕μ_ as in the proof of Proposition <ref>. The map ϕ↦μ_(ϕ, Ψ) is complex linear from _AB to H^0,1(Σ). We are left with the map from _AB to H^1(S^1, i ) ⊕ H^0(Y, i) given by

    ϕ↦ (i (Ψ, ϕ) - 2 ∗μ_(ϕ, Ψ)).

We have ϕ = (u,v) under the splitting S = K^1/2⊕ K^-1/2. Following the identifications from the proof of Proposition <ref> we find that ∗μ_(ϕ, Ψ) = - 2 i  ( α, u) and so our map is
 
    ϕ = (u,v) ↦ (i   ( α, u) , - 4 i  (α, u ).

Up to a constant, it coincides with the complex linear map 

    u ↦ - (α,u) + i(α, u) = - (α, u) = - (u, α).

We conclude that the isomorphism P ≅ L_A,0 agrees with the orientations induced from the complex structures on the cohomology groups. The same is true for P ≅ L_A,Ψ where the complex structures on H^1_A,Ψ =  L_A,Ψ and H^2_A,Ψ = coker L_A,Ψ come from the isomorphism of analytic spaces ≅_ given by Theorem <ref>. The tangent and obstruction spaces to _ are canonically identified with the tangent space to . Therefore, the relative orientation on the obstruction bundle agrees on the complex orientation on T^* →. 





















 §.§ Dependence on the perturbing two-form

We have so far ignored the fact that for fixedd = Lthere are two definitions of_depending on the sign ofd - τ[The case d- τ =0 is uninteresting as the moduli space is generically empty, see Corollary <ref>.], see Definition <ref>. Recall thatτ:= ∫_Σ i η/2πdepends on the choice of the perturbing two-formη. In classical Seiberg–Witten theory, the moduli space of solutions onS^1 ×Σis either^d+g-1 Σor^-d+g-1 Σ, depending on the sign ofd - τ. On the other hand, the Seiberg–Witten invariant does not depend on the choice of the perturbing two-form, which is reflected by the identity

    χ(^d+g-1Σ) = χ(^-d+g-1Σ).

 
Theorem <ref> gives us an analogous identity for the moduli spaces of Seiberg–Witten multi-monopoles. Fix(g,B), withBgeneric, and denote by_^+(d)and_^-(d)the moduli spaces corresponding to a given degreedand two choices of the sign ofd - τ. (In the previous subsections we have always assumed_ = _^+.) The map(ℒ, α, β) ↦(ℒ^*, β, α)induces an isomorphism_^+(d) ⟶_^-(-d). This is in agreement with the standard involution in Seiberg–Witten theory <cit.>. 

Ifg(Σ) ≥1, the count of Seiberg–Witten multi-monopoles(-1)^g(Σ)-1χ(_^+(d))does not depend on the choice of a generic choice of(B,η)as long asd - τ< 0. The same is true for(-1)^g(Σ)-1χ(_^-(d))whend - τ> 0. On the other hand, the choice ofηis immaterial from the viewpoint of the three-dimensional theory—by Theorem <ref>,(g,B,η)does not depend on the choice ofηas long as the moduli space is compact and Zariski smooth. We conclude that for a generic choice ofB ∈(Σ,E)we haveχ( _^+(d) ) = χ( _^-(d)). Combining this with the isomorphism_^+(d) ⟶_^-(-d), we obtain


If g(Σ) ≥ 1, then for a generic choice of B ∈(Σ,E) we have 

    χ( _^+(d) ) = χ( _^+(-d)).


Although the moduli spaces can be defined in terms of the complex geometry ofΣ, it is far from obvious how to prove the above equality without a reference to the three-dimensional theory. We will see in the next section that_^+(d)and_^-(d)can be non-homeomorphic.



§ EXAMPLES AND COMPUTATIONS


In this section we study_using methods of complex geometry. We prove some general properties of the moduli spaces and give their complete description whenΣis a  Riemann surface of genus zero,  one, or two.  


Let g be a product metric on Y = S^1 ×Σ,  B a generic connection pulled-back from Σ, and η∈Ω^2(Σ, i) a two-form satisfying τ(η) > 0. 

Set =(g,B,η) and = (g,B, η). 

  *  If d < (1-g(Σ))/2, then  is empty and = 0.

  *  If d ≥ 0, then  admits a holomorphic map to the Jacobian torus of Σ. Its fibres are projective spaces. If d > 0, this map is surjective. If d = 0, its image is a divisor in the linear system |2 Θ| where Θ is the theta divisor.

  *  If d ≥ g(Σ)-1, then  is biholomorphic to the projectivisation of a rank 2d holomorphic vector bundle over the Jacobian of Σ and = 0.

  *  If d = 0 and g(Σ) = 1, then  consists of two points and = 2.

  *  If d = 0 and g(Σ) = 2, then  is biholomorphic to a closed Riemann surface of genus five and = 8.

We will denote by _(d, ) the moduli space defined using a degree d line bundle L,   holomorphic structure  on E^*, and any perturbing two-form η satisfying d - η < 0; similarly we define (d, ). A property will be said to hold for a generic  if it holds for all  of the form _B = (E, _B) for B from a residual subset of (Σ,E).


 §.§ Generalised theta divisors


Ford=0the moduli spaces of framed vortices are related to generalised theta divisors<cit.>. Let→Σbe a rank two holomorphic stable bundle with trivial determinant. For any line bundleA ∈J^g(Σ)-1the Riemann–Roch theorem gives us

    χ( ⊗ A) = 2 (A) + 2(1-g(Σ)) = 0,

so we expectH^0(⊗A)= H^1( ⊗A) = 0ifAis generic. 

The generalised theta divisor of is 

    θ() := { A ∈ J^g(Σ)-1 |  h^0(⊗ A) > 0 };

One can show thatθ()is a divisor[This is no longer true if  is of higher rank as it can happen that θ()=J^g(Σ)-1.] inJ^g(Σ)-1in the linear system| 2 Θ| = ℂℙ^2g(Σ)-1, whereΘis the classical theta divisor

    Θ := { A ∈ J^g(Σ)-1 |  h^0(A) > 0 }.

If  is a rank two holomorphic stable bundle with trivial determinant, then there is a surjective morphism (0, ) →θ() whose fibres are projective spaces.

A point in (0, ) is an equivalence class [ℒ, α] where ℒ∈ J^0 and α∈ H^0( ⊗ℒ⊗ K^1/2), with α≠ 0. Since (ℒ⊗ K^1/2) = g(Σ)-1, we have ℒ⊗ K^1/2∈θ(). The morphism (0, ) →θ() is given by [ ℒ, α ] ↦ℒ⊗ K^1/2. The preimage of ℒ⊗ K^1/2 is ℙ H^0( ⊗ℒ⊗ K^1/2).

The divisor θ() can be described geometrically as follows.  By a theorem of Lefschetz, the linear system | 2 Θ | is base-point free and gives rise to a holomorphic map J^g(Σ)-1→ | 2 Θ |^*. It follows that θ(), as a subset of J^g(Σ)-1, is the preimage of a hyperplane in | 2 Θ |^* under the map J^g(Σ)-1→ |2 Θ|^*. This hyperplane is easy to identify—it is exactly θ() thought of as a point in | 2 Θ | or, equivalently, as a hyperplane in | 2 Θ |^*. Varying the background bundle , we vary the corresponding hyperplane and therefore the divisor θ() ⊂ J^g(Σ)-1. 


 §.§ General properties of the moduli spaces

For a generic choice of  the following holds. 

  *  If d < 1-g(Σ)/2, then _(d, ) is empty.

  *  If d ≥ 0, then _(d, ) is non-empty. 

  *  If d ≥ g(Σ)-1, then _(d, ) is Zariski smooth with the underying complex manifold biholomorphic to the projectivisation of a rank 2d vector bundle over J^d.

Proposition <ref> shows that the most interesting case is (1-g(Σ))/2 ≤ d < 0. It is an interesting question whether _(d, ) is generically non-empty for d in this range.


The proof of Proposition <ref> relies on the following general result about holomorphic vector bundles on Riemann surfaces. Recall thatstable if for any holomorphic line bundleAthe existence of a non-zero holomorphic mapA →implies(A) < 0. 

If g(Σ) ≥ 2, then there is an open dense subset of (Σ, E) such that for every connection B from this subset the corresponding holomorphic bundle _B is stable. 
 fails to be stable if and only if there is a holomorphic line bundle ℒ with ( ℒ) = d ≥ 0 and a non-zero map θℒ→_B. In other words, if L is a unitary bundle underlying ℒ and A is a connection inducing ℒ, then θ∈Γ(L^* ⊗ E) satisfies _ABθ = 0.
Consider 

    _d := { B∈(Σ,E)  |  _AB = { 0 } for all  A ∈(Σ,L) }

for a fixed degree d unitary bundle L.

_d is open.

Let B∈_d. For every A∈(Σ,L) there is a neighbourhood of (A,B) in (Σ,L) ×(Σ, E) such that for all (A',B') from this neighbourhood _A'B' = 0. Since this condition is invariant under the action of ^c(Σ), and A(Σ, L) / ^c(Σ) is homeomorphic to a torus—in particular, compact—there is a neighbourhood of B in (Σ, E) such that for all B' from this neighbourhood _AB' = 0 for all A. All such B' belong to _d which proves that _d is open.

_d is dense.

We only outline the proof as it is similar to that of Proposition <ref>. In what follows we replace the spaces of connections and sections by their Sobolev completions. Cover (Σ, L) / ^c(Σ) by finitely many charts that can be lifted to ^c(Σ)–slices in (Σ,L). Let V be such a chart; it is an open subset of ^2g parametrising a smooth family of connections { A_x }_x ∈ U. The claim will follow if can show that 

    _V := { B∈(Σ,E)  |  _A_x B = { 0 } for all  x∈ V }

is dense in (Σ,E). To prove this, consider 

    S := {θ∈Γ(Σ, L^* ⊗ E)  |  θ_L^2 = 1 }

and the map

    f (Σ, E) × V × S ⟶Ω^0,1(Σ, L^* ⊗ E)

    f(B,x,θ) := _A_x Bθ.

For every B ∈(Σ,E) the restriction f_B  f(B, ·, ·) is a Fredholm map (between suitable Sobolev spaces) because its derivative is the sum of _A_x B and the derivative with respect to x, which is a finite-dimensional operator. By the Riemann–Roch theorem,

    ind_ df_B    =  V + 2 ind _A_x B - 1
    
       = 2g + 4(-d+1-g(Σ)) -1 
       = 2(-2d + 2 - g(Σ)) - 1 ≤ 0,

where we subtract 1 because _A_x B is restricted to the tangent space to S. A computation similar to that in the proof of Proposition <ref> shows that the derivative of the full map f is surjective at every point of f^-1(0). By the Sard–Smale theorem, the set f_B^-1(0) is empty for B from a dense subset of (Σ,E); all such B belong to _V.

:=⋂_d≥ 0_d is open and dense.
 is dense by Baire's theorem; it is open by the following argument. By (<ref>), the existence of a destabilising map θℒ→_B implies 

    0 ≤ d ≤ h^0(Σ, _B) + 2g(Σ)-2.

The right-hand side can only decrease when B is replaced by a sufficiently close B'. (Indeed, if we split Γ(Σ,E) into _B and its L^2–orthogonal complement Q, then by the elliptic estimate  _B' is non-degenerate when restricted to Q for all nearby connections B'; it follows that the projection _B'→_B is injective. See also 
 <cit.> for an algebro-geometric proof.) Therefore, to guarantee that a nearby connection B' belongs to  it is enough to check that it belongs to _d for finitely many values of d. Thus, for every B ∈ there are finitely many open neighbourhoods of B whose intersection lies entirely in .

By Theorem <ref>, for a generic choice of  the moduli space _ = is a compact complex manifold of dimension g(Σ) - 1 + 2d. If d < (1-g(Σ))/2, then this dimension is negative and _ must be empty.

 The case d = 0 was discussed in the previous subsection. Let (ℒ) = d > 0; then 

    h^0( ⊗ℒ⊗ K^1/2 ) - h^1( ⊗ℒ⊗ K^1/2) = 2d > 0;

thus,  ℒ is in the image of the projection π_→ J^d given by [ℒ, α, β]  ↦ℒ. For a generic , by Theorem <ref>,  _ = and so π^-1(ℒ) =  H^0( ⊗ℒ).

For the proof of the third item, assume g(Σ) ≥ 2; the cases g(Σ) = 0, 1 will be considered separately in the next section. By Lemma <ref>, a generic  is stable and by Serre duality,

    h^1( ( ⊗ℒ⊗ K^1/2  ) = h^0 (( ^* ⊗ℒ^* ⊗ K^1/2 ).

Any element of H^0 (( ^* ⊗ℒ^* ⊗ K^1/2 ) gives a holomorphic map ℒ⊗ K^-1/2→^*. We have

    (ℒ⊗ K^-1/2) = d - g + 1 ≥ 0.

Since ^* is stable, it follows that any holomorphic map ℒ⊗ K^-1/2→^* is trivial. Thus, h^1( ( ⊗ℒ⊗ K^1/2  ) = 0 and by the Riemann–Roch theorem h^0( ⊗ℒ⊗ K^1/2  ) = 2d for every ℒ∈ J^d. We conclude that π_→ J^d is the projectivisation of the rank 2d vector bundle whose fibre over ℒ is the cohomology group H^0( ⊗ℒ⊗ K^1/2  ) = ^2d. 


 §.§ Genus zero
 
LetΣ= ℂℙ^1with its unique complex structure. Fork∈ℤdenote by(k)the unique holomorphic line bundle of degreek;K^1/2 = (-1)is the unique spin structure.  By a theorem of Grothendieck, every holomorphic bundle overℂℙ^1is the direct sum of line bundles. In particular, every holomorphic(2,)–bundle is of the form= (k) ⊕(-k)for somek ≥0, withk = 0being the generic case.

Let Σ = ℂℙ^1 and = (k) ⊕(-k) for k ≥ 0. 

  *  If d ≤ 0 and k ≤ |d|, then _(d, ) is empty.

  *  If d > 0 and k ≤ d, then _(d, ) is Zariski smooth with the underlying complex manifold biholomorphic to ℂℙ^2d-1.

  *  If k > |d|, then  _(d, ) is non-compact and its compactification _(d, ) is homeomorphic to a locally trivial ℂℙ^k-d–fibration over ℂℙ^k+d. 

The fact that the Euler characteristic of the moduli space depends on the sign of d is consistent with the three-dimensional theory. Since b_1( S^1 ×ℂℙ^1) = 1  we do not expect the count of Seiberg–Witten multi-monopoles to be invariant, even when there are no Fueter sections; by Proposition <ref>, there are two chambers in the set of parameters which are separated by a codimension one wall where reducibles appear. As discussed in subsection <ref>, replacing d by -d can be  seen as varying τ so that the sign of d - τ changes; any path of parameters joining such two choices of τ will pass through the wall of reducibles.
_(d, ) consists of the equivalences classes of pairs (α, β) such that

    α∈ H^0( (k+d-1) ) ⊕ H^0((-k + d-1)),

    β∈ H^0( (-k-d-1) ) ⊕ H^0( (k-d-1) ),
α≠ 0, and αβ = 0 ∈ H^0( (-2))—this is automatically satisfied since h^0((-2)) = 0.
If  is generic, so that k = 0, then d ≤ 0 which implies that α = 0 and _(d,) is empty. If d > 0, then α∈ℂ^2d and β = 0; it follows that _(d, ) = (d, ) = ℂℙ^2d-1.  

The same description of _(d, ) is valid in the non-generic case k ≠ 0 as long as k ≤ |d|. When k > |d| the moduli space is no longer compact and Fueter sections appear. If k > d > 0, then α∈^k+d, β∈^k-d and _(d, ) is the total space of the vector bundle (-1)^⊕ (k-d) over ℂℙ^k+d-1. The compactification  _(d, ) is the ℂℙ^k-d-bundle over ℂℙ^k+d-1 obtained from the projectivisation of the vector bundle (-1)^⊕ (k-d)⊕. 


 §.§ Genus one
 LetΣ= S^1 ×S^1equipped with a complex structure making it into an elliptic curve. Isomorphism classes of line bundles of a given degreedonΣform the JacobianJ^dwhich is isomorphic to the dual torusΣ^*. The canonical bundle ofΣis trivial and without loss of generality we can takeK^1/2also to be trivial. Holomorphic vector bundles over elliptic curves were classified by Atiyah <cit.>. A generic holomorphicSL(2,)–bundleis of the form= A ⊕A^-1for a degree zero line bundleA →Σ. We may moreover assume thatA^2 ≠since there are only four line bundles satisfyingA^2 = . 


Let Σ be an elliptic curve. Suppose that  is generic, that is of the form = A ⊕ A^-1 for A ∈ J^0 satisfying A^2 ≠. 

  *  If d < 0, then _(d, ) is empty.
 
  *  If d > 0, then _(d, ) is Zariski smooth with the underlying complex manifold biholomorphic to the projectivisation of a rank 2d vector bundle over J^d.
  
  *  If d = 0, then _(d, ) is regular and consists of two points. 
  
If d >0, then the cohomology ring H^*( _(d, ), ) is isomorphic as H^*( J^d, )–modules to H^*( J^d, ) [ H ] / (H^2d) where (H) =2. In particular, 

    χ( _(d, ) ) = χ( J^d ) χ( ℂℙ^2d-1) = 0,

which is consistent with the fact that χ( _(d, ) ) is invariant under the change d ↦ -d. 
_(d, ) consists of equivalence classes of triples (ℒ, α, β) where ℒ∈ J^d and

    α∈ H^0( A ⊗ℒ) ⊕ H^0(A^-1⊗ℒ),

    β∈ H^0(A^-1⊗ℒ^-1) ⊕ H^0( A ⊗ℒ^-1),

satisfying α≠ 0 and αβ = 0 in H^0( ) =. 

For d < 0 we must have α = 0 and so the moduli space is empty. For d = 0 the only choices of ℒ for which α is possibly non-zero are ℒ = A^-1 and ℒ = A. If ℒ = A^-1, then

    α∈ H^0( ) ⊕ H^0( A^-2),

    β∈ H^0( ) ⊕ H^0( A^2).

Since A^2 and A^-2 are non-trivial, so they have no non-zero sections. The only possibly choice for α, up to scaling, is therefore α = (1,0) and the condition αβ = 0 forces β to be zero since the pairing H^0(  ) × H^0( ) → H^0( ) is simply the multiplication ×→. We repeat the same argument for ℒ = A and conlude that _(d, ) consists of two isolated points. In particular it is compact and so Zariski smooth thanks to Corollary <ref>. Since it is also has the correct dimension zero, we conclude that each of the points is regular.

Consider now the case d > 0. For every ℒ∈ J^d the Riemann–Roch theorem gives us

    h^0( ℒ⊗ A ) - h^1( ℒ⊗ A) = d

and by Serre duality, h^1( ℒ⊗ A ) = h^0( ℒ^-1⊗ A) = 0 because (ℒ^-1)<0, Thus H^0 (ℒ⊗ A) = ^d and the same is true if A is replaced by A^-1. Therefore, α is identified with a non-zero vector in ^2d. On the other hand, β = 0 since A^± 1⊗ℒ^-1 has no non-trivial sections. Since the above discussion is valid for any ℒ∈ J^d, it follows that  _(d, ) is a locally trivial ℂℙ^2d-1–fibration over J^d: the projectivisation of a  rank 2d holomorphic vector bundle over J^d given by the push-forward of the Poincaré line bundle  → J^d ×Σ to the first factor.


It is worthwhile discussing some non-generic examples. The cases when= A ⊕A^-1and either(A) ≠0orA^2 = are similar to the ones already considered. Another possibility is thatis indecomposable, in which case it is of the form= _0 ⊗AwhereA ∈J^0satisfiesA^2 = and_0is the unique non-trivial bundle obtained as an extension

    0 r   r   _0 r   r    0.
 
The line bundleAis uniquely determined by. 


Suppose without loss of generality that  = _0. It is shown in <cit.> that if h^0( ⊗ℒ ) ≠ 0, then either (ℒ)>0 or ℒ = in which case we have h^0( ) = 1. We conclude that when d < 0 or d > 0 the moduli space _(d, ) is, respectively, empty or the projectivisation of a vector bundle over J^d. On the other hand, for d = 0 the only choice of ℒ for which h^0(⊗ℒ) > 0 is ℒ = and we look for holomorphic sections

    α∈ H^0(  ) = ,

    β∈ H^0( ^* ) =

such that α≠ 0 and αβ = 0. Up to scaling,  α = 1. We will show now that  the pairing H^0(  ) × H^0( ^* ) → is trivial and, as a consequence, β can be chosen arbitrarily.  Let Ω∈ H^0( Λ^2 ) be a nowhere vanishing holomorphic volume form. It induces an isomorphism →^* given by v ↦Ω(v, ·). If α is a generator of H^0( ), then γ = Ω(α, ·) is a non-zero holomorphic section of H^0( ^*) and so it must be a generator. On the other hand, γ(α) = Ω(α, α) = 0 since Ω is skew-symmetric—this shows that αβ = 0 for every β∈ H^0( ^*). Therefore, _(0, ) is homeomorphic to . Its compactification _(0, ) is homeomorphic to ℂℙ^1.


 §.§ Genus two
 LetΣbe a genus two Riemann surface. By Lemma <ref>, a generic holomorphic bundle onΣis stable. The proof of the next lemma can be found in <cit.>. 


Let W be a stable rank two vector bundle with trivial determinant over a genus two closed Riemann surface Σ. If ξ→Σ is a degree 1 line bundle, then

  * h^0(W ⊗ξ) ≤ 1.

  *  Any non-zero homomorphism ξ^* → W is everywhere injective.

Let Σ be a closed Riemann surface of genus two. For a generic holomorphic (2,)–bundle →Σ we have the following description of _(d, ).

  *  If d < 0, then _(d, ) is empty.

  *  If d = 0, then _(d, ) is Zariski smooth with the underlying complex manifold biholomorphic to a closed Riemann surface of genus five.

  *  If d > 0, then _(d, ) is Zariski smooth with the underlying complex manifold biholomorphic to the projectivisation of a rank 2d vector bundle over J^d.

Items (2) and (3) follow from Proposition <ref> For d = 0 we use the relation between the moduli space of framed vortices and theta divisors described in subsection <ref>. Let →Σ be a stable (2,)–bundle. Let 𝒮𝒰(2) be the compactification of the moduli space of such bundles obtained by adding the S–equivalence classes of semi-stable bundles. As explained in <cit.>, we have | 2 Θ | = ℂℙ^3 and the map introduced in subsection <ref>
    θ𝒮𝒰(2) →ℂℙ^3

    ↦θ()

 is an isomorphism. Recall that θ() can be seen either as a subset of the Jacobian J^1
    θ() = { A ∈ J^1  |  h^0( ⊗ A) > 0 }

 or as the corresponding point in | 2 Θ |. 
 
 J^1 is a two-dimensional complex torus and the map J^1 → |2 Θ|^* = (ℂℙ^3 )^* induces a degree four embedding of the Kummer surface J^1 / _2. Thus, as a subset θ() ⊂ J^1 is the preimage of the intersection of the Kummer surface in (ℂℙ^3 )^* with the hyperplane θ() ∈ℂℙ^3 under the quotient map J^1 → J^1 / ^2. Since θ𝒮𝒰(2) →ℂℙ^3 is an isomorphism, by changing the background bundle  we can obtain all hyperplanes in (ℂℙ^3)^*. In particular, for a generic choice of , the hyperplane θ() will avoid  all the 16 singular points of J^1 / _2 and the intersection J^1/ _2 ∩θ() will be a smooth complex curve of degree four and genus three. Its preimage under J^1 → J^1 / _2 is a smooth curve C⊂ J^1 of genus five, by the Hurwitz formula.
 
Let π_( 0 , ) → C be the composition of _(0, ) →(0, ) with the projection (0, ) →θ() = C from subsection <ref>.  We claim that this map is an isomorphism for a generic choice of . In order to prove that, it is enough to check that the fibre over any line bundle ℒ⊗ K^1/2 in C consists of one point. This is equivalent to showing that  H^0( ⊗ℒ⊗ K^1/2) is spanned by a single non-zero section α and if β∈ H^0( ^* ⊗ℒ^* ⊗ K^1/2) is any section satisfying αβ = 0, then β = 0.  The first claim follows immediately from Lemma <ref>. As regards the second claim, assume that α and β are as above and  β≠ 0. By Lemma <ref>, the homomorphisms

    αℒ^* ⊗ K^-1/2⟶  and  βℒ⊗ K^-1/2⟶^*

are everywhere injective. Since =2, αβ = 0 implies the exactness of the sequence

    0 r   ℒ^* ⊗ K^-1/2rα   rβ^t   ℒ^* ⊗ K^1/2r    0.

Since =, we conclude ℒ^2 =. There are  16 line bundles satisfying this condition: order two elements of J^0. For each of them, all possible non-trivial extensions  as above are classified by the corresponding extension class in ℙH^1(K^-1) = ℂℙ^2. (Note that the extension is non-trivial because  is stable.) Thus, all stable bundles  that can be represented as such an extension form a proper subvariety of 𝒮𝒰(2) = ℂℙ^3 consisting of the images of 16 maps ℂℙ^2 →𝒮𝒰(2). A generic stable bundle   will not belong to this subvariety. In this case, we conclude that each fibre of the map π consists of a single point and π is an isomorphism. In particular, _(0, ) is compact and therefore Zariski smooth by Corollary <ref>.

Note that the last part of the proof was unnecessary; we already know that generically _ = is compact and Zariski smooth which is enough to conclude = C. On the other hand, the argument presented above identifies the locus of those semi-stable bundles ∈𝒮𝒰(2) for which Fueter sections appear. It consists of strictly semi-stable bundles A ⊕ A^-1, for some A ∈ J^0, which form the Kummer surface J^1 / _2 in 𝒮𝒰(2) = ℂℙ^3, and stable bundles  that arise from an extension of the form (<ref>) for some element ℒ∈ J^0 of order two. The latter form a subvariety covered by the images of 16 maps ℂℙ^2 →ℂℙ^3. 
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