Witten's conjecture: Donaldson invariants of four-manifolds with $b_1=0$ and odd $b^+>1$ are determined by Seiberg–Witten invariants

Witten's conjecture predicts that for every closed, smooth, connected, oriented four-manifold $X$ with first Betti number $b_1(X)=0$ and positive-definite part of the intersection form of odd rank $b^+(X)\ge 3$, Seiberg–Witten simple type forces Donaldson (Kronheimer–Mrowka) simple type, that the two theories have the same basic classes, and that the full Donaldson series of $X$ is an explicit universal expression in its Seiberg–Witten invariants, intersection form, Euler characteristic, and signature. The conjecture is known unconditionally only for restricted classes — elliptic surfaces and their blow-ups and rational blow-downs by direct computation of both sides, and complex projective surfaces of simple type via the Göttsche–Nakajima–Yoshioka algebraic-geometric route — and conditionally for all four-manifolds with $b_1=0$ and odd $b^+>1$ of Seiberg–Witten simple type via the Feehan–Leness $SO(3)$-monopole cobordism formula, which is itself established modulo an unresolved gluing-map hypothesis (their Hypothesis 3.1). The problem is to prove (or refute) the conjecture in full generality, with no restriction to symplectic, complex-projective, or geometrically constrained four-manifolds and with no unproved gluing hypothesis assumed.

Background

Donaldson theory studies smooth four-manifolds $X$ through moduli spaces of anti-self-dual connections — Yang–Mills connections on principal $SU(2)$ or $SO(3)$ bundles whose curvature $F_A$ satisfies $F_A = -*F_A$ for a generic Riemannian metric, where $*$ is the Hodge star. Intersection pairings of certain cohomology classes (the $\mu$-classes) on these moduli spaces define the Donaldson invariants, a collection of differential-topological polynomial invariants $D^w_X$ depending on a class $w\in H^2(X;\mathbb{Z})$. Write $b_1$ for the rank of $H^1(X;\mathbb{Z})$ and $b^+$ for the maximal dimension of a positive-definite subspace of $H^2(X;\mathbb{R})$ under the intersection form $Q$; when $b_1=0$ and $b^+>1$ the invariants are metric-independent. A four-manifold $X$ has Kronheimer–Mrowka (Donaldson) simple type when $D^w_X(x^2 z)=4D^w_X(z)$ for all classes $z$, where $x\in H_0(X;\mathbb{Z})$ is the point class; in that case the invariants of high degree are governed by the Donaldson series $\mathcal{D}^w_X(h)=D^w_X((1+\tfrac12 x)e^h)=\sum_n D^w_X(h^n)/n!+\tfrac12\sum_n D^w_X(xh^n)/n!$ (the standard normalization used in item (3) of the Problem Statement), and the Kronheimer–Mrowka structure theorem proves that if such an $X$ has $b_1=0$ and odd $b^+>1$, then $\mathcal{D}^w_X(h)=e^{Q(h)/2}\sum_r c_r e^{K_r\cdot h}$ with finitely many rational coefficients $c_r\neq 0$. The classes $K_r\in H^2(X;\mathbb{Z})/\text{torsion}$ — characteristic elements of the intersection form, meaning $K_r\cdot c\equiv c^2$ mod 2 for every $c\in H^2(X;\mathbb{Z})$, equivalently $K_r\equiv w_2(X)$ mod 2, where $w_2(X)$ is the second Stiefel–Whitney class — are the Donaldson basic classes.

Seiberg–Witten theory associates to each $\text{spin}^c$ structure $\mathfrak{s}$ on $X$ (a rank-2 complex spinor bundle whose determinant line has first Chern class $c_1(\mathfrak{s})$) a count $SW_X(\mathfrak{s})$ of solutions of the Seiberg–Witten monopole equations; for $b^+>1$ it is a diffeomorphism invariant. The classes $c_1(\mathfrak{s})$ with $SW_X(\mathfrak{s})\neq 0$ are the Seiberg–Witten basic classes; for such a class $K\in H^2(X;\mathbb{Z})$, $SW_X(K)$ denotes the sum of $SW_X(\mathfrak{s})$ over all $\text{spin}^c$ structures with $c_1(\mathfrak{s})=K$ (the convention used in item (3) of the Problem Statement). The four-manifold $X$ has Seiberg–Witten simple type if $K^2=c_1^2(X):=2e(X)+3\sigma(X)$ for every such class, where $e$ and $\sigma$ are the Euler characteristic and signature.

In 1994 Witten proposed, on the basis of supersymmetric quantum field theory, a direct universal identity between the two theories: for $b_1=0$ and odd $b^+>1$, Seiberg–Witten simple type should force Donaldson simple type, the basic classes should coincide, and the Donaldson series should be computed by an explicit formula from the Seiberg–Witten invariants (the precise statement is Conjecture 1.1 of Feehan–Leness, Adv. Math. 2019, and Conjecture 2.4 of Sivek, Int. Math. Res. Not. 2015). A tight surface is an embedded surface $S\subset X$ of genus $\ge 2$ with self-intersection $[S]^2=2g(S)-2>0$ (also called $c$-embedded); Kronheimer and Mrowka proved that a four-manifold with $b_1=0$ and odd $b^+>1$ containing a tight surface has Kronheimer–Mrowka simple type, and in their 2004 “Property P” paper they established Witten's conjecture for further classes — e.g. manifolds with a tight surface and a $(-1)$-sphere whose Euler characteristic and signature match those of a smooth even hypersurface in $\mathbb{CP}^3$ of degree $\ge 6$ — but that derivation passes through the Feehan–Leness $SO(3)$-monopole cobordism formula, whose published proof is complete only modulo the gluing hypothesis discussed next. The remaining fully unconditional cases in the published record are direct computations of both sides, notably Fintushel–Stern's for elliptic surfaces and their blow-ups and rational blow-downs, and complex projective surfaces of simple type via the Göttsche–Nakajima–Yoshioka algebraic-geometric route. Feehan and Leness pursued the Pidstrigach–Tyurin program — analyzing moduli spaces of $SO(3)$ monopoles (pairs of an $SO(3)$ connection and a spinor, interpolating between the two theories) — culminating in an $SO(3)$-monopole cobordism formula (Mem. Amer. Math. Soc. 2018) from which they derive, in two 2019 Advances in Mathematics papers, that every closed four-manifold with $b_1=0$ and odd $b^+\ge 3$ of Seiberg–Witten simple type has superconformal simple type (a condition on the Seiberg–Witten series, proposed by Mariño–Moore–Peradze, stronger than the conclusions of Witten's conjecture) and hence satisfies Witten's conjecture — but the cobordism formula itself is stated by its authors only under an unproved “Hypothesis 3.1” about local gluing maps for $SO(3)$ monopoles, whose proof they describe as a book in preparation. Along a different route, Göttsche–Nakajima–Yoshioka proposed an explicit universal formula relating the two theories — derived from Mochizuki's formula (an algebraic-geometric formula for the Donaldson invariants of a complex projective surface, in terms of integrals over Hilbert schemes of points on the surface) and Nekrasov's deformed instanton-counting partition function — and deduced Witten's conjecture in the complex projective setting; for general smooth four-manifolds their formula remains conjectural.

Sivek, Int. Math. Res. Not. 2015 applies the Kronheimer–Mrowka special cases of the conjecture — which, per the final published Feehan–Leness account, are themselves applications of the conditional $SO(3)$-monopole cobordism formula — together with Muñoz's gluing formulae for Donaldson invariants of connected sums along surfaces, to prove that symplectic four-manifolds with $b_1=0$ and $b^+>1$ have nonzero Donaldson invariants, that the canonical class $K_X$ is always a Donaldson basic class, and to characterize basic classes of Lefschetz fibrations (fibrations over $S^2$ with finitely many nodal singular fibers) that evaluate maximally on the fiber. This builds on Taubes's theorems that symplectic four-manifolds have Seiberg–Witten simple type and $K_X$ among their Seiberg–Witten basic classes. The full Witten conjecture would collapse the entire Donaldson theory of such manifolds into Seiberg–Witten theory, and Sivek's paper states it explicitly as the natural open problem that would remove all technical hypotheses from its arguments.

Problem Statement

Prove or refute Witten's conjecture in its full generality (Conjecture 2.4 of Sivek, Int. Math. Res. Not. 2015; Conjecture 1.1 of Feehan–Leness, Adv. Math. 356 (2019), article 106821): let $X$ be a closed, smooth, connected, oriented four-manifold with $b_1(X)=0$ and $b^+(X)\ge 3$ odd, where $b_1$ is the first Betti number and $b^+$ the positive rank of the intersection form $Q$ on $H_2(X;\mathbb{Z})$. If $X$ has Seiberg–Witten simple type, then:

(1) $X$ has Kronheimer–Mrowka (Donaldson) simple type;

(2) the Donaldson basic classes of $X$ coincide with the Seiberg–Witten basic classes;

(3) for every $w\in H^2(X;\mathbb{Z})$ and $h\in H_2(X;\mathbb{R})$, the Donaldson series, in the standard normalization, satisfies

$$\mathcal{D}^w_X(h) \;=\; 2^{\,2-(\chi_h-c_1^2)}\, e^{Q(h)/2} \sum_{K\in\mathcal{B}(X)} (-1)^{(Q(w)+K\cdot w)/2}\, SW_X(K)\, e^{\langle K,h\rangle},$$

where $\mathcal{B}(X)$ is the set of Seiberg–Witten basic classes, $SW_X$ is the Seiberg–Witten invariant, $\chi_h=(e(X)+\sigma(X))/4$ and $c_1^2=2e(X)+3\sigma(X)$ are built from the Euler characteristic and signature, $Q(w)$ denotes the square of $w$ under the intersection pairing, and $\langle K,h\rangle$ is the pairing between cohomology and homology classes. The claim must be established for all such four-manifolds — no restriction to symplectic or complex-projective manifolds, no hypothesis of the existence of tight ($c$-embedded) surfaces, and no unverified analytic hypothesis (such as Feehan–Leness's Hypothesis 3.1 on gluing maps for $SO(3)$ monopoles) may be assumed. A disproof consists of a single explicit four-manifold meeting the hypotheses for which one of the three conclusions fails.

References

1. Steven Sivek, Donaldson invariants of symplectic manifolds, International Mathematics Research Notices 2015, no. 6, 1688–1716. DOI: 10.1093/imrn/rnt345; arXiv:1301.0377; https://arxiv.org/abs/1301.0377
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