Göttsche, Nakajima, and Yoshioka (2011) proved that for a complex projective surface of Seiberg–Witten simple type, the coefficient functions in Mochizuki's residue formula expressing Donaldson invariants through Seiberg–Witten invariants are given by explicit leading terms of Nekrasov's deformed instanton partition function for the N=2 supersymmetric gauge theory with one fundamental matter, and deduced Witten's conjecture and the Mariño–Moore–Peradze superconformal simple type sum rules for such surfaces. They explicitly posed as an open conjecture (Conjecture 4.5) that the same formula, with the same Nekrasov-derived coefficients, holds for every smooth, closed, oriented four-manifold of Seiberg–Witten simple type with first Betti number zero and b+ at least 3 and odd — manifolds for which no complex structure is available. The Feehan–Leness SO(3)-monopole cobordism program yields a proof of Witten's conjecture for all such manifolds that remains conditional on an unproven gluing hypothesis, and it does not identify its universal coefficients with the Nekrasov-derived ones; the directly linked open problems — a rigorous identification of the Feehan–Leness coefficients with the partition-function data, an intrinsic gauge-theoretic definition of the partition function on the Uhlenbeck compactification, extensions beyond simple type, and the deferred proofs of the blow-up formula in the presence of matter — all remain open. The conjecture is open.

Background

For a smooth, closed, oriented four-manifold X, the Donaldson invariants are defined by integrating cohomology classes over the moduli space of irreducible anti-self-dual (ASD) connections — connections whose curvature F_A satisfies F_A = −F_A, the solutions of the four-dimensional Yang–Mills self-duality equations. The µ-map sends a homology class on X to a cohomology class on the moduli space, and the resulting integrals D^(ξ,n)(α^k p^l), for ξ ∈ H²(X,Z), α ∈ H₂(X) and p the point class, are diffeomorphism invariants sensitive to the smooth structure. The Seiberg–Witten (SW) invariants are defined analogously from the monopole equations, a U(1) gauge theory for a spinc structure s (a complex spin structure whose determinant line bundle has first Chern class c₁(s)) coupled to a spinor; they are far easier to compute. X is of SW-simple type when SW(s) vanishes whenever the formal dimension of the monopole moduli space is positive; X is of KM-simple type when its Donaldson series is a finite sum of exponentials. Witten's conjecture (1994) asserts that an X of SW-simple type is of KM-simple type and its Donaldson invariants are given by an explicit universal formula in its SW invariants. Two rigorous programs address it. Feehan and Leness develop a cobordism argument on moduli spaces of SO(3)-monopoles — pairs of an SO(3)-connection and a spinor interpolating between ASD and SW moduli spaces — yielding a formula with universal coefficients that depend only on the characteristic numbers χh(X) = (χ(X)+σ(X))/4 and (K²) := 2χ(X)+3σ(X); those coefficients are not explicitly determined, and the formula is conditional on analytic gluing properties of the monopole moduli spaces. Mochizuki, working algebro-geometrically on complex projective surfaces, proved a residue formula: the Donaldson invariants equal a sum over basic classes of SW invariants times residues at a = ∞ of explicit C-equivariant integrals over products X^[n₁] × X^[n₂] of Hilbert schemes of points (X^[n] parametrizes length-n zero-dimensional subschemes; rank-one sheaves are twisted ideal sheaves, which is why Hilbert schemes appear). His proof joins moduli spaces of stable pairs across a change of stability condition through a 'master space' equipped with an obstruction theory. Nekrasov's deformed partition function is the equivariant integral of a multiplicative class over the moduli space of framed instantons on R⁴ (described by the ADHM construction, i.e., by explicit linear-algebra data), with respect to rotations of R⁴ with equivariant parameters ε₁, ε₂; it has combinatorial expressions as sums over tuples of Young diagrams and is the instanton-counting definition of the partition function of four-dimensional N = 2 supersymmetric SU(r) gauge theory in the Ω-background. Adding N_f fundamental matter hypermultiplets of mass m couples 'matter bundles'. For the rank-2 theory with one fundamental hypermultiplet, the Seiberg–Witten curve of the low-energy theory is y² = 4x²(x+u) + 4mΛ³x + Λ⁶, with u the Coulomb-branch coordinate and Λ the dynamical scale. Göttsche–Nakajima–Yoshioka proved (their Theorem 4.4) that Mochizuki's coefficient functions, specialized at a = m (the locus where the Seiberg–Witten curve becomes singular), are given by the leading terms F₀, H, A, B of this Nekrasov partition function — F₀ by periods of the Seiberg–Witten curve, A and B by explicit complementary expressions — evaluated at (a, m, Λ) = (a, a, Λ^(4/3)a^(−1/3)). Since these expressions involve only intersection pairings and characteristic numbers, they make sense for any smooth four-manifold, and the authors conjectured (Conjecture 4.5) that Mochizuki's formula with these coefficients holds in the smooth category. Their residue analysis shows the differential's contribution at the superconformal point (where the Seiberg–Witten curve acquires a cusp) vanishes exactly when X satisfies the superconformal simple type sum rules of Mariño–Moore–Peradze, so the conjecture would imply Witten's conjecture for all smooth SW-simple-type four-manifolds with b₁(X) = 0 and b⁺(X) ≥ 3 odd (b₁ the first Betti number, b⁺ the maximal dimension of a positive-definite subspace of H²(X,R) for the intersection pairing). A structural obstacle to a gauge-theoretic proof: the partition function is defined on the algebro-geometric framed moduli spaces M(r,n) with universal sheaves, whereas the analytic compactification of the instanton moduli space is the Uhlenbeck compactification M₀(r,n) — which allows instantons to bubble off at points, adding symmetric products of X — and the universal sheaf does not descend to it.

Problem Statement

Primary target — Göttsche–Nakajima–Yoshioka Conjecture 4.5, in the source's precise scope. Let X be a smooth, closed, oriented four-manifold of Seiberg–Witten simple type with b₁(X) = 0 and b⁺(X) ≥ 3 odd. Then Mochizuki's formula (Theorem 4.1 of the source, proved when X is complex projective) continues to hold for X up to sign: the Donaldson invariants of X, assembled as in the source, equal the sum over Seiberg–Witten classes ξ̃₁ = 2ξ₁ − K_X of SW(ξ̃₁) times the residue at a = ∞ of the explicit generating differential B(ξ₁, ξ; a) da of the source's Theorem 4.4, whose coefficients are the leading terms F₀^inst, H, A^inst, B^inst of Nekrasov's deformed partition function for the rank-2 theory with one fundamental hypermultiplet, evaluated at (a, m, Λ) = (a, a, Λ^(4/3)a^(−1/3)); in these expressions the canonical class K_X enters only through the characteristic number (K_X²) — read for smooth X as 2χ(X)+3σ(X), as in the source — and through the combination ξ − K_X, which is replaced by an auxiliary integral class whose only requirement is that it equals (ξ mod 2) + w₂(X) in H²(X,Z/2) (an auxiliary spinc structure), and the stability conditions of the projective case are dropped. The source notes that the conjecture would follow immediately if the universal coefficients f_{k,l} of the Feehan–Leness SO(3)-monopole cobordism formula were identified with these explicit Nekrasov-derived coefficients, but that it does not follow from the Feehan–Leness statement itself, because the superconformal-simple-type sum rules impose nontrivial relations on Seiberg–Witten invariants and make the coefficients non-unique; a direct comparison of the two coefficient systems, without assuming a complex projective model, remains an open problem directly linked to the conjecture. A proof for all such X, or a disproof by an explicit counterexample, resolves the primary target. Two further open problems posed in the same paper concern the gauge-theoretic foundation of the conjecture: (i) give a direct, rigorous identification of the Feehan–Leness universal coefficients with the explicit F₀, H, A, B expressions; (ii) define the partition function and the matter-bundle Euler classes intrinsically on the gauge-theoretic Uhlenbeck compactification M₀(r, n) — the universal sheaf does not descend to it, so this is nontrivial — in a way producing the same series as the algebro-geometric definition through the framed moduli spaces M(r, n). The authors' further directions — extension to four-manifolds not of SW-simple type, to higher-rank gauge groups, and to other numbers of fundamental hypermultiplets, together with completion of the deferred blow-up proofs — are stated follow-up open directions beyond the conjecture itself.

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