ON DONALDSON AND SEIBERG-WITTEN INVARIANTS

PAUL M. N. FEEHAN

Abstract. In these lecture notes, we sketch the proof (due to Thomas Leness and the author) of Witten’s conjecture that the Donaldson and Seiberg-Witten series coincide, modulo powers of degree c + 2, with c = − \frac{1}{4} (7χ + 11σ), for four-manifolds obeying some mild conditions, where χ and σ are their Euler characteristic and signature. We use the moduli space of SO(3) monopoles as a cobordism between a link of the Donaldson moduli space of anti-self-dual SO(3) connections and links of the moduli spaces of Seiberg-Witten monopoles. Gluing techniques allow us to compute contributions from Seiberg-Witten moduli spaces lying in the first (or “one-bubble”) level of the Uhlenbeck compactification of the moduli space of SO(3) monopoles.

1. Introduction

This article consists of a very lightly edited version of the author’s notes for his lecture at the Mathematische Arbeitstagung 2001, hosted by the Max Planck Institut für Mathematik, Bonn. The severe time constraints imposed by the rapid production schedule for the Arbeitstagung proceedings guaranteed that these notes remain rough and informal, glossing over technical details and qualifications appropriate for more complete published accounts elsewhere [7], [8], [11]. Despite all their shortcomings, we hope that they provide a brief introduction to some of the author’s joint work with Thomas Leness and which is reasonably clear, even if not necessarily precise in all respects.

1.1. Witten’s conjecture. Two kinds of invariants can be used to explore the classification problem for compact, smooth 4-manifolds:

• Donaldson invariants, defined using an SO(3) Yang-Mills gauge theory (1986–1994).

• Seiberg-Witten invariants, defined using a U(1) monopole gauge theory (1994). We shall restrict our attention throughout to the case of closed, oriented 4-manifolds with b_{1} = 0 and odd b_{2}^{+} > 1. The conjectured relationship between these gauge theory invariants is described below:

Conjecture 1.1. [25] A 4-manifold X has KM-simple type if and only if it has SW-simple type. If X has simple type, the KM and SW basic classes coincide, and the Donaldson and Seiberg-Witten series obey

D_{X}^{w}(h) = 2^{2−c(X)}e^{\frac{1}{2} Q_{X}(h,h)}SW_{X}^{w}(h), h ∈ H_{2}(X; ℝ). (1)

Here, the 4-manifold X has intersection form

Q_{X} : H_{2}(X; ℤ) × H_{2}(X; ℤ) → ℤ,

Euler characteristic χ, signature σ and

c(X) = −\frac{1}{4}(7χ + 11σ).

Date: This version: June 17, 2001.

The author was supported in part by NSF grant DMS 9704174.

We shall recall the definitions of the Donaldson and Seiberg-Witten series shortly.

1.2. Remarks on the problem. Before proceeding to discuss our work on Witten’s conjecture, it is interesting to compare the mathematician’s and physicists’ approaches to establishing (1).

Witten employs a certain N = 2 supersymmetric quantum Yang-Mills theory. He uses rescaling, g_{t} = t^{2}g, of the Riemannian metric g on X and metric independence of the correlation functions to relate the Donaldson invariants (t → 0) with the Seiberg-Witten invariants (t → ∞).

The mathematical approach to a proof of Witten’s formula proposed by Pidstrigatch and Tyurin [22] instead employs an SO(3) monopole gauge theory which generalizes both the instanton and U(1) monopole gauge theories. All three gauge theories are classical field theories and their solutions are invariant under metric rescaling, whereas Witten’s quantum field theory is sensitive to metric rescaling.

Apparently, the SO(3) monopole gauge theory provides a purely classical field theory alternative to Witten’s quantum field theory method. The problem of determining the relationship between these two approaches is surely an important one worth exploring further.

2. SO(3) monopoles

2.1. Clifford modules and spin structures. Given a Riemannian metric g on X, let V → X be a Hermitian bundle with Clifford map ρ : T^{∗}X → End_{ℂ}(V),

ρ(α)^{†} = −ρ(α), and ρ(α)^{2} = −g(α, α), α ∈ Ω^{1}(X, ℝ).

Then (ρ, V) defines a Clifford or ℂℓ(T^{∗}X) module structure on V. If W → X is a complexrank four Hermitian bundle, then 𝔰 = (ρ, W) is a spin^{c} structure, familiar from Seiberg-Witten theory [23]. If V → X is a complex-rank eight Hermitian bundle, then we call 𝔱 = (ρ, V) a spin^{u} structure.

2.2. From spin^{u} structures to SO(3) bundles. Given 𝔱 = (ρ, V) on X, one obtains

• An SO(3) subbundle,

𝔤_{𝔱} ⊂ 𝔰𝔲(V),

characterized as the span of the sections ξ of 𝔰𝔲(V) such that [ξ, ρ(ω)] = 0, for all ω ∈ Ω^{•}(X, ℝ).

• A complex line bundle,

det^{\frac{1}{2}} (V^{+}),

where V = V^{+} ⊕ V^{−} and V^{±} are the ∓1 eigenspaces of ρ(vol) on V.

• Splittings, if Λ^{2}(T^{∗}X) = Λ^{+} ⊕ Λ^{−} and Λ^{±} are the ±1 eigenspaces of ∗_{g} on Λ^{2}(T^{∗}X), and ρ : Λ^{±} ≅ 𝔰𝔲(W^{±}) are the usual isomorphisms of SO(3) bundles,

𝔰𝔲(V^{±}) ≅ ρ(Λ^{±}) ⊕ ρ(Λ^{±}) ⊗ 𝔤_{𝔱} ⊕ 𝔤_{𝔱}.

Moreover, for any choice of spin^{c} structure 𝔰 = (ρ, W), one further obtains

• A complex-rank two Hermitian bundle,

E = Hom_{ℂℓ(T^{∗}X)}(W, V).

• A Clifford module isomorphism,

V ≅ W ⊗ E.

• An isomorphism of SO(3) bundles

𝔰𝔲(E) ≅ 𝔤_{𝔱}.

• An isomorphism of complex line bundles,

det^{\frac{1}{2}} (V^{+}) ≅ det(W^{+}) ⊗ det(E).

2.3. SO(3)-monopole equations. We call a pair (A, Φ) an SO(3) monopole if

ad^{−1}(F + ; Â) − ρ^{−1}(Φ ⊗ Φ^{∗})_{00} = 0,
D_{A}Φ = 0

where A is a spin connection on V, inducing a fixed connection A|_{det(V^{+})} = 2A_{Λ} on det(V^{+}) and Φ is a section of V^{+}; Â is the induced connection on the SO(3) bundle 𝔤_{𝔱} ⊂ 𝔰𝔲(V); F + ; Â is the self-dual component of the curvature of Â; the term (Φ ⊗ Φ^{∗})_{00} is the component of Φ ⊗ Φ^{∗} lying in ρ(Λ^{+}) ⊗ 𝔤_{𝔱}; D_{A} : C^{∞}(V^{+}) → C^{∞}(V^{−}) is the Dirac operator. We let M_{𝔱} be the space of SO(3) monopoles for 𝔱 = (ρ, V), modulo gauge transformations.

2.4. Singularities in SO(3)-monopole space. We now classify the fixed points of the circle action on M_{𝔱} given by complex multiplication on the spinor components.

An SO(3) monopole (A, Φ) is a Yang-Mills or instanton solution if

Φ ≡ 0 and F + ; Â = 0.

Hence, there is a moduli subspace of SO(3) instantons,

M_{κ}^{w} ↪ M_{𝔱},

where κ = −\frac{1}{4}p_{1}(𝔤_{𝔱}) and w ∈ H^{2}(X; ℤ) lifts w_{2}(𝔤_{𝔱}).

An SO(3) monopole (A, Φ) is a Seiberg-Witten or reducible solution if

A = B ⊕ B ⊗ A_{L} on V = W ⊕ W ⊗ L,

for some line bundle L, and Φ = Ψ ⊕ 0 with Ψ a section of W^{+} obeying

Tr(F_{B}^{+}) − ρ^{−1}(Ψ ⊗ Ψ^{∗})_{0} − F_{A_{Λ}}^{+} = 0,
D_{B}Ψ = 0.

Hence, there are moduli subspaces of Seiberg-Witten monopoles for 𝔰 = (ρ, W), whenever V = W ⊕ W ⊗ L:

M_{𝔰} ↪ M_{𝔱}.

3. Invariants of smooth 4-manifolds

We sketch definitions of the Donaldson series [17] and Seiberg-Witten series [25].

3.1. Donaldson invariants. Set 𝔸(X) = Sym(H_{0}(X; ℝ) ⊕ H_{2}(X; ℝ)), so z ∈ 𝔸(X) is a linear sum of monomials

x^{m}β_{1}β_{2} ⋯ β_{δ−2m},

with x ∈ H_{0}(X; ℤ) positive generator and β_{i} ∈ H_{2}(X; ℝ). Cohomology classes on M_{κ}^{w} can be defined via a map [1], [3],

μ_{p} : H_{i}(X; ℝ) → H^{4−i}(M_{κ}^{w}; ℝ).

The Donaldson invariant is then a linear function

D_{X}^{w} : 𝔸(X) → ℝ,

where, for a monomial z with deg(z) = 2δ,

D_{X}^{w}(z) = ⟨μ_{p}(z), [M_{κ}^{w}]⟩

with μ_{p}(z) = μ_{p}(x)^{m} ⌣ μ_{p}(β_{1}) ⌣ ⋯ ⌣ μ_{p}(β_{δ−2m}).

3.2. Kronheimer-Mrowka structure theorem. One says that a 4-manifold X has KMsimple type if for some w, and all z ∈ 𝔸(X),

D_{X}^{w}(x^{2}z) = 4D_{X}^{w}(z).

One defines the Donaldson series by setting

D_{X}^{w}(h) = D_{X}^{w}((1 + \frac{1}{2}x)e^{h}), h ∈ H_{2}(X; ℝ). (2)

We recall the celebrated

Theorem 3.1. [17] If X has KM-simple type, then there exist a_{r} ∈ ℚ and K_{r} ∈ H^{2}(X; ℤ), the KM-basic classes, such that

D_{X}^{w}(h) = e^{\frac{1}{2} Q_{X}(h,h)} ∑_{r=1}^{s}(−1)^{\frac{1}{2} (w^{2}+w·K_{r})}a_{r}e^{⟨K_{r},h⟩}, h ∈ H_{2}(X; ℝ). (3)

3.3. Seiberg-Witten invariants. The Seiberg-Witten invariants comprise a function,

SW_{X} : Spin^{c}(X) → ℤ,

where

SW_{X}(𝔰) = ⟨μ_{𝔰}(x)^{max}, [M_{𝔰}]⟩,

and μ_{𝔰}(x) ∈ H^{2}(M_{𝔰}; ℤ) is a cohomology class associated to a circle action. One says that a 4-manifold X has SW-simple type if

SW_{X}(𝔰) = 0

when M_{𝔰} has positive dimension and calls c_{1}(𝔰) = c_{1}(W^{+}) an SW-basic class if SW_{X}(𝔰) ≠ 0. We define the Seiberg-Witten series by

SW_{X}^{w}(h) = ∑_{𝔰∈Spin^{c}(X)}(−1)^{\frac{1}{2} (w^{2}+c_{1}(𝔰)·w)}SW_{X}(𝔰)e^{⟨c_{1}(𝔰),h⟩}, h ∈ H_{2}(X; ℝ). (4)

Witten’s prediction then takes the form stated in Conjecture 1.1.

4. SO(3) monopole cobordism

4.1. Bubbling and Uhlenbeck compactness. In order to apply the moduli space M_{𝔱} of SO(3) monopoles as a cobordism, we must use a compactification. If {A_{α}, Φ_{α}}_{α∈ℕ} ⊂ M̃_{𝔱}, then the sequence converges to an ideal SO(3) monopole (A_{∞}, Φ_{∞}, x) in M̃_{𝔱_{ℓ}} × Sym^{ℓ}(X) if

• (A_{α}, Φ_{α}) → (A_{∞}, Φ_{∞}) in C^{∞} on X \ {x}, modulo gauge transformations,

• |F_{A_{α}}|^{2} → |F_{A_{∞}}|^{2} + 8π^{2} ∑_{x∈x} δ(x), as measures. We let M̄_{𝔱} be closure of M_{𝔱} with respect to the Uhlenbeck topology, implicit above, in the space of ideal SO(3) monopoles,

⨆_{ℓ=0}^{N} (M_{𝔱_{ℓ}} × Sym^{ℓ}(X)) ,

where 𝔱_{ℓ} = (ρ, V_{ℓ}) is a spin^{u} structure with characteristic classes

p_{1}(𝔤_{𝔱_{ℓ}}) = p_{1}(𝔤_{𝔱}) + 4ℓ, w_{2}(𝔤_{𝔱_{ℓ}}) = w_{2}(𝔤_{𝔱}), and c_{1}(𝔱_{ℓ}) = c_{1}(𝔱),

and c_{1}(𝔱) := c_{1}(det^{\frac{1}{2}} (V^{+})). The space M̄_{𝔱} is smoothly stratified, with a top or zeroth level M_{𝔱}, and lower levels M_{𝔱_{ℓ}} × Sym^{ℓ}(X), ℓ ≥ 1.

4.2. Strata in space of SO(3) monopoles. For the top level, one has a stratification

M_{𝔱} = M_{κ}^{w} ⊔ M_{𝔱}^{∗,0} ⊔ ⋃_{𝔱=𝔰⊕𝔰′} M_{𝔰}. (5)

The complement M_{𝔱}^{∗,0} in M_{𝔱} of the Yang-Mills and Seiberg-Witten solutions is a smooth manifold, cut out transversely by the SO(3) monopole equations.

A stratification of the form (5) arises in each level, M_{𝔱_{ℓ}} × Sym^{ℓ}(X), of the compactification M̄_{𝔱}. Though dimension-counting arguments rule out contributions from the instanton moduli subspace M_{κ−ℓ}^{w} of M_{𝔱_{ℓ}}, Seiberg-Witten moduli subspaces of M_{𝔱_{ℓ}} can contribute to Donaldson invariants computed using M_{κ}^{w} ⊂ M_{𝔱}. To apply the cobordism, we

• Define a link L̄_{𝔱,κ}^{w} ⊂ M̄_{𝔱}/S^{1} of the instanton moduli subspace,

M̄_{κ}^{w} ⊂ M̄_{𝔱},

by restricting to spinors with L^{2} norm equal to a small positive constant, and

• Use gluing theory [9], [10] to construct links L̄_{𝔱,𝔰} ⊂ M̄_{𝔱}/S^{1} of ideal Seiberg-Witten

moduli subspaces,

M_{𝔰} × Sym^{ℓ}(X) ⊂ M̄_{𝔱}.

These links are considerably more difficult to construct than L̄_{𝔱,κ}^{w}.

4.3. SO(3)-monopole cobordism formula. The cobordism M̄_{𝔱}/S^{1} now yields the raw identity,

⟨μ_{p}(z) ⌣ μ_{c}^{δ_{c}−1}, [L̄_{𝔱,κ}^{w}]⟩ = − ∑_{𝔰∈Spin^{c}(X)}⟨μ_{p}(z) ⌣ μ_{c}^{δ_{c}−1}, [L̄_{𝔱,𝔰}]⟩ , (6)

where μ_{c} ∈ H^{2}(M_{𝔱}^{∗,0}; ℤ) is a class associated to a circle action on M_{𝔱}^{∗,0}. One finds that

⟨μ_{p}(z) ⌣ μ_{c}^{δ_{c}−1}, [L̄_{𝔱,κ}^{w}]⟩

is a multiple of the Donaldson invariant, D_{X}^{w}(z). The sum in (6) is over all 𝔰 ∈ Spin^{c}(X), with L̄_{𝔱,𝔰} empty unless M_{𝔰} × Sym^{ℓ}(X) ⊂ M̄_{𝔱}, for some ℓ(𝔱, 𝔰) ≥ 0.

The difficult aspect of using (6) to derive Witten’s formula (1) is to show that

⟨μ_{p}(z) ⌣ μ_{c}^{δ_{c}−1}, [L̄_{𝔱,𝔰}]⟩

is the correct multiple of the Seiberg-Witten invariant SW_{X}(𝔰); the degree of difficulty grows rapidly with ℓ ≥ 0.

5. Application of the cobordism

We consider following distinct cases, arranged in increasing order of complexity:

• There are no Seiberg-Witten moduli spaces with non-zero invariants in M̄_{𝔱}, so the intersection (M_{𝔰} × Sym^{ℓ}(X)) ∩ M̄_{𝔱} is empty for all ℓ ≥ 0 and SW_{X}(𝔰) ≠ 0. The Donaldson invariants defined by M̄_{κ}^{w} ⊂ M̄_{𝔱} are then zero and, eventually, this observation leads to a vanishing result:

D_{X}^{w}(h) ≡ 0 ≡ SW_{X}^{w}(h) (mod h^{c(X)−2}), h ∈ H_{2}(X; ℝ).

• Contributions from M_{𝔰} ⊂ M_{𝔱} lead to Witten’s formula, mod h^{c(X)}.

• Contributions from M_{𝔰} × X ⊂ M̄_{𝔱} lead to Witten’s formula, mod h^{c(X)+2}.

• Contributions from M_{𝔰} × Sym^{2}(X) ⊂ M̄_{𝔱} lead to Witten’s formula, mod h^{c(X)+4}.

• Calculation of contributions M_{𝔰} × Sym^{ℓ}(X) ⊂ M̄_{𝔱} for ℓ ≥ 3 should lead to a full verification of Witten’s formula.

We have considered the cases ℓ = 0 and ℓ = 1 in detail [7], [8], [11] and we would expect the case ℓ = 2 to follow in a similar manner, by exploiting work of Leness [18] on the wall-crossing formula for Donaldson invariants.

At present, we can compute the general shape of the contributions for ℓ ≥ 3, but complete, direct computations of those contributions appears to be a difficult problem.

5.1. Level-zero Seiberg-Witten contributions. Let B ⊂ H^{2}(X; ℤ) be the set of Seiberg-Witten basic classes and define B^{⊥} ⊂ H^{2}(X; ℤ) as the Q_{X}-orthogonal complement of B. We call a 4-manifold X abundant if Q_{X}|_{B^{⊥}} has a hyperbolic sublattice. Every compact, complex algebraic, simply connected surface with b_{2}^{+} ≥ 3 is abundant [7].

Theorem 5.1. [8] Assume X is abundant, has b_{1} = 0, odd b_{2}^{+} ≥ 3, and SW-simple type. Suppose Λ ∈ B^{⊥} exists with Λ^{2} = 2 − (χ + σ). For such Λ and w ∈ H^{2}(X; ℤ) with w − Λ ≡ w_{2}(X) (mod 2) one has, for all h ∈ H_{2}(X; ℝ),

(7)  D_{X}^{w}(h) ≡ 0 ≡ SW_{X}^{w}(h) (mod h^{c(X)−2}),
(8)  D_{X}^{w}(h) ≡ 2^{2−c(X)}e^{\frac{1}{2} Q_{X}(h,h)}SW_{X}^{w}(h) (mod h^{c(X)}).

The vanishing assertion (7) for the Seiberg-Witten series is a statement that the Moore-Mariño-Peradze conjecture holds for (abundant) 4-manifolds of Seiberg-Witten simple type [6], [19], [20].

5.2. Level-one Seiberg-Witten contributions. With more sophisticated analytical tools, specifically gluing theory, we can compute contributions from M_{𝔰}×X ⊂ M̄_{𝔱} and these computations lead to the

Theorem 5.2. [11] Same hypotheses as Theorem 5.1, but now suppose Λ ∈ B^{⊥} exists with Λ^{2} = 4 − (χ + σ). For such Λ and w ∈ H^{2}(X; ℤ) with w − Λ ≡ w_{2}(X) (mod 2) one has, for all h ∈ H_{2}(X; ℝ),

(9)  D_{X}^{w}(h) ≡ 0 ≡ SW_{X}^{w}(h) (mod h^{c(X)−2}),
(10)  D_{X}^{w}(h) ≡ 2^{2−c(X)}e^{\frac{1}{2} Q_{X}(h,h)}SW_{X}^{w}(h) (mod h^{c(X)+2}).

We expect an identity similar to (10), but mod h^{c(X)+4}, by computing contributions for ℓ = 2, when Λ^{2} = 6 − (χ + σ). The restrictive hypotheses on existence of classes Λ with prescribed even squares can be dropped if one can consider contributions for arbitrary ℓ ≥ 0.

5.3. Seiberg-Witten contributions from arbitrary levels. More generally, we expect the following to hold:

Conjecture 5.3. [11] For Λ, w in H^{2}(X; ℤ) with w − Λ ≡ w_{2}(X) (mod 2), and all d ≥ 0 and h ∈ H^{2}(X; ℝ),

\begin{aligned}D_{X}^{w}(h^{d}) & = 2^{2−c(X)} ∑_{𝔰∈Spin^{c}(X)}(−1)^{\frac{1}{2} (w^{2}+c_{1}(𝔰)·w)}SW_{X}(𝔰) \\ & × ∑_{i=0}^{m} δ_{i} (⟨c_{1}(𝔰), h⟩, ⟨Λ, h⟩) Q_{X}^{ℓ−i}(h, h), \end{aligned} (11)

where ℓ = ℓ(c_{1}(𝔰), Λ) and m(ℓ, d) = min(ℓ, [d/2]) above. The functions δ_{i} are homogeneous polynomials in two variables, whose coefficients are polynomials in 2χ ± 3σ, (c_{1}(𝔰) − Λ)^{2}, Λ^{2}, and (c_{1}(𝔰 − Λ) · c_{1}(𝔰). A similar formula holds for the invariants D_{X}^{w}(x^{m}h^{d−2m}), 0 ≤ m ≤ [d/2].

Conjecture 5.3 is easier to prove than Witten’s, but is still powerful enough to prove that the Seiberg-Witten invariants determine the Donaldson invariants. We have carried out most of the work required to prove this conjecture, though some details remain to be checked.

Witten’s formula (1) should follow from Conjecture 5.3 by indirect calculations of the remaining unknown coefficients, exploiting

• Known Donaldson and Seiberg-Witten invariants for complex surfaces, or

• A possible generalization of the wall-crossing calculations of Göttsche [15] for Donaldson invariants when b_{2}^{+} = 1 (and 1-parameter families) to the case of b_{2}^{+} > 1 (and b_{2}^{+}-parameter families).

6. Outline of proof of Witten’s formula

We shall first sketch how to compute the rough form of the pairings,

⟨μ_{p}(z) ⌣ μ_{c}^{δ_{c}−1}, [L̄_{𝔱,𝔰}]⟩, (12)

or, at least why these pairings have the form

SW_{X}(𝔰) × (Factors depending only on topology).

We use our gluing theory [9], [10] to construct a topological model for a neighborhood in M̄_{𝔱} and hence a link, L̄_{𝔱,𝔰} ⊂ M̄_{𝔱}/S^{1}, of the ‘stratum’

M_{𝔰} × Sym^{ℓ}(X) ⊂ M̄_{𝔱}.

Given this topological model for L̄_{𝔱,𝔰}, we can then apply intersection theory methods to partly compute the pairings (12).

This suffices to prove the ‘rough version’ of Witten’s formula (11), which is enough to show that the Seiberg-Witten invariants determine the Donaldson invariants. We have completed much of this step, though many details remain to be checked. We shall illustrate the method below, often assuming ℓ = 1 for the sake of simplicity [11].

The passage from this stage to Witten’s formula requires us to compute the many universal, but unknown coefficients in the rough version (11) of Witten’s formula. Thus we shall also indicate how one should be able to use existing results [12], [13], [17] for the Donaldson and Seiberg-Witten invariants to determine the remaining unknown coefficients in our expressions for the pairings (12).

6.1. Neighborhood of Seiberg-Witten stratum. Our gluing theory [9], [10] allows us to construct a model for a neighborhood of the level M_{𝔰} × Sym^{ℓ}(X) in a local, ‘virtual’ moduli space,

M̄_{𝔱,𝔰}^{vir} := Ñ_{𝔱_{ℓ},𝔰}(ε) ×_{G_{𝔰}} Ḡl_{𝔱_{ℓ}}(δ). (13)

A neighborhood of M_{𝔰} × Sym^{ℓ}(X) in the true moduli space M̄_{𝔱} then takes the shape

γ (χ^{−1}(0) ∩ M̄_{𝔱,𝔰}^{vir}) = M̄_{𝔱} ∩ γ(M̄_{𝔱,𝔰}^{vir}),

The stratum M_{𝔰} in M_{𝔱_{ℓ}} has a ‘virtual’ normal bundle,

N_{𝔱,𝔰} → M_{𝔰},

and an obstruction bundle with section χ,

Ξ_{𝔱,𝔰} → M_{𝔱_{ℓ}}.

The gluing map,

γ : M̄_{𝔱,𝔰}^{vir} → Configuration space of ideal pairs containing M̄_{𝔱},

gives a homeomorphism from χ^{−1}(0) ∩ M̄_{𝔱,𝔰}^{vir} onto a neighborhood of M_{𝔰} × Sym^{ℓ}(X) in M̄_{𝔱}.

6.2. Instanton component of the topological model. When ℓ = 1, the instanton gluing-data component is given by

(14)  Ḡl_{𝔱_{ℓ}}(δ) = (Fr(𝔤_{𝔱_{1}}) ×_{X} Fr(T^{∗}X) × M̄_{1}^{s,♮}(S^{4}, δ)) /(SO(3) × SO(4)).

Here, M_{k}^{s,♮}(S^{4}, δ) is the moduli space of k-instantons on S^{4}, framed at south pole s, with mass center at the north pole, and scale ≤ δ. For k = 1 there are homeomorphisms,

M_{1}^{s,♮}(S^{4}, δ) ≅ (0, δ] × SO(3),
M̄_{1}^{s,♮}(S^{4}, δ) ≅ c(SO(3)),

where c(SO(3)) is the cone on SO(3).

Our gluing-model (13) admits an Uhlenbeck stratification, given below when ℓ = 1:

M̄_{𝔱,𝔰}^{vir} = M_{𝔱,𝔰}^{vir} ⊔ (N_{𝔱,𝔰}(ε) − M_{𝔰}) × X ⊔ M_{𝔰} × X.

When ℓ ≥ 2, the symmetric product has its usual stratification and one can also construct bundles Gl_{𝔱_{ℓ}}(δ, Σ) → Σ with instanton moduli space fibers, for each stratum Σ ⊂ Sym^{ℓ}(X), following the prescription of Friedman and Morgan [14] and developing the idea of Kotschick and Morgan [16] and Mrowka [21]. The difficult part is to assemble these local gluing data bundles into a space of global gluing data, Gl_{𝔱_{ℓ}}(δ), with a projection map onto Sym^{ℓ}(X). One essentially has

Ḡl_{𝔱_{ℓ}}(δ) = ⋃_{Σ⊂Sym^{ℓ}(X)} Gl_{𝔱_{ℓ}}(δ, Σ),

where we patch together the local bundles Gl_{𝔱_{ℓ}}(δ, Σ) using (non-canonical) transition maps. Of course, the difficulty is to ensure that the transition maps obey the cocycle condition when ℓ ≥ 3.

6.3. Link of the Seiberg-Witten stratum. We define the link in two steps, first considering the Seiberg-Witten component of the link of M_{𝔰} × Sym^{ℓ}(X) ⊂ M̄_{𝔱,𝔰}^{vir}/S^{1},

L̄_{𝔱,𝔰}^{vir,s} := (∂Ñ_{𝔱_{ℓ},𝔰}(ε) ×_{G_{𝔰}} Ḡl_{𝔱_{ℓ}}(δ)) /S^{1},

and, second, defining the instanton component of the link of M_{𝔰} × Sym^{ℓ}(X) ⊂ M̄_{𝔱,𝔰}^{vir}/S^{1},

L̄_{𝔱,𝔰}^{vir,i} := (Ñ_{𝔱_{ℓ},𝔰}(ε) ×_{G_{𝔰}} ∂Ḡl_{𝔱_{ℓ}}(δ)) /S^{1},

a complex disk bundle over M_{𝔰}×∂Ḡl_{𝔱_{ℓ}}(δ)/S^{1}. When ℓ = 1, the boundary ∂Ḡl_{𝔱_{ℓ}}(δ) is defined by restricting to instantons on S^{4} with scale δ. We then define the link of M_{𝔰} × Sym^{ℓ}(X) in the virtual moduli space, M̄_{𝔱,𝔰}^{vir}/S^{1},

L̄_{𝔱,𝔰}^{vir} := L̄_{𝔱,𝔰}^{vir,s} ∪ L̄_{𝔱,𝔰}^{vir,i}.

Finally, we obtain

L̄_{𝔱,𝔰}^{vir} := γ (χ^{−1}(0) ∩ L̄_{𝔱,𝔰}^{vir}) = (M̄_{𝔱}/S^{1}) ∩ γ(L̄_{𝔱,𝔰}),

the link of M_{𝔰} × Sym^{ℓ}(X) in the true moduli space M̄_{𝔱}/S^{1}.

6.4. Factorization in Seiberg-Witten link pairings. The virtual moduli space method reduces the computation of the link pairing (12) to that of the pairing on the right-hand side below,

⟨μ_{p}(z) ⌣ μ_{c}^{δ_{c}}, [L̄_{𝔱,𝔰}]⟩ = ⟨μ_{p}(z) ⌣ μ_{c}^{δ_{c}} ⌣ e, [L̄_{𝔱,𝔰}^{vir}]⟩, (15)

where e is Euler class of the total obstruction bundle over M̄_{𝔱,𝔰}^{vir}, with section χ, and L̄_{𝔱,𝔰} ≅ χ^{−1}(0) ∩ L̄_{𝔱,𝔰}^{vir}. We next show [11] that the pairing (15) with [L̄_{𝔱,𝔰}^{vir}] is expressible in terms of pairings with

[M_{𝔰} × ∂Ḡl_{𝔱_{ℓ}}(δ)/S^{1}].

In particular, we can show that the latter pairings are in turn products of • Pairings with [M_{𝔰}], giving multiples of the Seiberg-Witten invariant, SW_{X}(𝔰), and • Pairings with [∂Ḡl_{𝔱_{ℓ}}(δ)/S^{1}].

The pairings with [∂Ḡl_{𝔱_{ℓ}}(δ)/S^{1}] depend only the topology of the 4-manifold, X, and universal data.

6.5. From the rough version to Witten’s formula. A key remaining point is to compute the pairings with [∂Ḡl_{𝔱_{ℓ}}(δ)/S^{1}], for all ℓ ≥ 2. We note that it is already a hard analytical and topological problem just to construct the spaces Ḡl_{𝔱_{ℓ}}(δ), for ℓ ≥ 1.

We strongly suspect that one should be able to determine the pairings with [∂Ḡl_{𝔱_{ℓ}}(δ)/S^{1}] by exploiting the known Donaldson and Seiberg-Witten invariants for compact, complex algebraic surfaces. For example, when ℓ = 1, only 3 pairings with [∂Ḡl_{𝔱_{ℓ}}(δ)/S^{1}] are required in the proof of Theorem 5.2.

The pairings with [∂Ḡl_{𝔱_{ℓ}}(δ)/S^{1}] also arose in the Kotschick-Morgan wall-crossing formula for Donaldson invariants when b_{2}^{+}(X) = 1, where the Riemannian metric varies in 1-parameter families [15], [16]. Is there an analogue of that wall-crossing formula for b_{2}^{+}- parameter families when b_{2}^{+}(X) > 1? If so, we might expect that Göttsche’s determination of the Donaldson invariant wall-crossing formula may extend in such a way as to allow us to compute the pairings with [∂Ḡl_{𝔱_{ℓ}}(δ)/S^{1}] without relying heavily on examples.

7. Witten’s conjecture and symplectic 4-manifolds

7.1. Gauge theory and Lefschetz fibrations. Away from finitely many critical points, a Lefschetz fibration π : X → S is smooth fiber bundle over a connected base, whose fibers are closed Riemann surfaces, Σ, of given genus. Possibly after blowing up, all symplectic 4-manifolds admit Lefschetz fibrations [2].

One can ask what is the relationship between the gauge theories and the Lefschetz fibration structure? As a first step, one could use the product, X = Σ × S, as a toy model. In this situation, the relationship between the gauge theory moduli spaces and holomorphic maps can be explored via the following techniques:

• Adiabatic limit analysis, by the scaling metric g = g_{Σ} ⊕ g_{S} as g_{ε} = ε^{2}g_{Σ} ⊕ g_{S}, with ε → 0, or

• Restriction of stable, holomorphic bundles over Σ × S to Σ × {z}, as z ∈ S varies. This leads to the following identifications:

1. SO(3)-instantons over Σ × S are identified with holomorphic maps, S → M_{Σ}, where M_{Σ} is the space of flat SO(3) connections over Σ. This identification goes back to Dostoglou and Salamon [5].

2. Seiberg-Witten U(1)-monopoles over Σ × S are identified with holomorphic maps, S → M_{Σ,d}, where M_{Σ,d} is the space of vortices on the line bundle L → Σ with d = ⟨c_{1}(L), [Σ]⟩. This identification has been outlined by Salamon [24].

3. SO(3)-monopoles over Σ × S are identified with holomorphic maps from S to a space of non-abelian vortices over Σ. Non-abelian vortices of this kind have been studied by Bradlow, Garcia-Prada, and others.

Of course, there are many unanswered questions:

• What is the relationship between the gauge theory compactifications and the compactifications of spaces of holomorphic maps?

• What is the relationship between the gauge theory invariants and Gromov-Witten invariants of spaces of flat connections or vortices over Riemann surfaces?

• Can one extend the analysis for the toy model, Σ×S, to the case of non-trivial surface bundles or Lefschetz fibrations?

• Can symplectic 4-manifolds provide additional data we can use to determine of Witten’s formula?

We note that the relationship between Seiberg-Witten invariants and Lefschetz fibration data bears on a closely related program due to Simon Donaldson and Ivan Smith to give Lefschetz-fibration of proofs of results for symplectic 4-manifolds derived via Seiberg-Witten theory [4].

References

[1] S. K. Donaldson, Polynomial invariants for smooth four-manifolds, Topology 29 (1990), 257–315.

[2] , Lefschetz fibrations in symplectic geometry, Proceedings of the International Congress of Mathematicians, Vol. II (Berlin, 1998), 1998, pp. 309–314 (electronic).

[3] S. K. Donaldson and P. B. Kronheimer, The geometry of four-manifolds, Oxford Univ. Press, Oxford, 1990.

[4] S. K. Donaldson and I. Smith, Lefschetz pencils and the canonical class for symplectic 4-manifolds, math.SG/0012067.

[5] S. Dostoglou and D. A. Salamon, Self-dual instantons and holomorphic curves, Ann. of Math. (2) 139 (1994), 581–640.

[6] P. M. N. Feehan, P. B. Kronheimer, T. G. Leness, and T. S. Mrowka, PU(2) monopoles and a conjecture of Mariño, Moore, and Peradze, Math. Res. Lett. 6 (1999), 169–182, math.DG/9812125.

[7] P. M. N. Feehan and T. G. Leness, PU(2) monopoles and links of top-level Seiberg-Witten moduli spaces, J. Reine Angew. Math., to appear, math.DG/0007190.

[8] , PU(2) monopoles. II: Top-level Seiberg-Witten moduli spaces and Witten’s conjecture in low degrees, J. Reine Angew. Math., to appear, dg-ga/9712005.

[9] , PU(2) monopoles. III: Existence of gluing and obstruction maps, submitted to a print journal; math.DG/9907107.

[10] , PU(2) monopoles. IV: Surjectivity of gluing maps, in preparation.

[11] , PU(2) monopoles, level-one Seiberg-Witten moduli spaces, and Witten’s conjecture in low degrees, Topology Appl., to appear.

[12] R. Fintushel and R. Stern, Donaldson invariants of 4-manifolds with simple type, J. Differential Geom. 42 (1995), 577–633.

[13] , Rational blowdowns of smooth 4-manifolds, J. Differential Geom. 46 (1997), 181–235, alggeom/9505018.

[14] R. Friedman and J. W. Morgan, Smooth four-manifolds and complex surfaces, Springer, Berlin, 1994.

[15] L. Göttsche, Modular forms and Donaldson invariants for 4-manifolds with b^{+} = 1, J. Amer. Math. Soc. 9 (1996), 827–843, alg-geom/9506018.

[16] D. Kotschick and J. W. Morgan, SO(3) invariants for four-manifolds with b^{+} = 1, II, J. Differential Geom. 39 (1994), 433–456.

[17] P. B. Kronheimer and T. S. Mrowka, Embedded surfaces and the structure of Donaldson’s polynomial invariants, J. Differential Geom. 43 (1995), 573–734.

[18] T. G. Leness, Donaldson wall-crossing formulas via topology, Forum Math. 11 (1999), 417–457, dgga/9603016.

[19] M. Mariño, G. Moore, and G. Peradze, Four-manifold geography and superconformal symmetry, Math. Res. Lett. 6 (1999), 429–437, math.DG/9812042.

[20] , Superconformal invariance and the geography of four-manifolds, Comm. Math. Phys. 205 (1999), 691–735, hep-th/9812055.

[21] T. S. Mrowka, private communication.

[22] V. Y. Pidstrigatch and A. N. Tyurin, Localisation of Donaldson invariants along the Seiberg-Witten classes, dg-ga/9507004.

[23] D. Salamon, Spin geometry and Seiberg-Witten invariants, Birkhäuser, Boston, to appear.

[24] D. A. Salamon, Seiberg-Witten invariants of mapping tori, symplectic fixed points, and Lefschetz numbers, Proceedings of 6th Gökova Geometry-Topology Conference, vol. 23, 1999, pp. 117–143.

[25] E. Witten, Monopoles and four-manifolds, Math. Res. Lett. 1 (1994), 769–796, hep-th/9411102.

Department of Mathematics, Rutgers University, Piscataway, NJ 08854-8019

   E-mail address: feehan@math.rutgers.edu

Current address: Department of Mathematics, Trinity College, University of Dublin, Dublin 2, Ireland

   E-mail address: feehan@maths.tcd.ie
   URL: http://www.maths.tcd.ie/∼feehan