For each integer N ≥ 2, Donaldson-type polynomial invariants of smooth 4-manifolds can be defined from moduli spaces of anti-self-dual connections on hermitian bundles of rank N; for N ≥ 3 they were predicted by Mariño and Moore from quantum field theory. Daemi and Xie (Geom. Topol. 24, 2020) confirmed the Moore–Mariño prediction for N = 3 for simply connected elliptic surfaces without multiple fibers and certain surfaces of general type, and posed the general problem: Question 7.1, the correct analogue of the simple type condition for U(N)-invariants, and Conjecture 7.2, the closed Moore–Mariño formula expressing the U(3)-series as a universal sum over pairs of Kronheimer–Mrowka/Seiberg–Witten basic classes with explicitly prescribed constants. Daemi–Iida–Scaduto (2024) proved the structure-theorem shape of Conjecture 7.2 for U(3)-simple-type manifolds with b₂⁺ > 1, with undetermined basic classes and coefficients. The surviving open core — universality of U(3)-simple type, identification of the U(3) basic classes and constants with Seiberg–Witten data, and the correct formulation for N ≥ 4 — remains open in the literature cited below.

Background

Instanton invariants of 4-manifolds. Let X be a closed, connected, smooth, oriented 4-manifold with b2+(X)≥2b_2^+(X)\ge 2, where b2+b_2^+ denotes the dimension of a maximal positive-definite subspace of H2(X;R)H^2(X;\mathbb R) for the intersection pairing QQ (the condition b2+≥2b_2^+\ge 2 removes metric/orientation dependence). An anti-self-dual (ASD) connection on a principal PU(N)PU(N)-bundle over X is a connection whose curvature satisfies FA+=(FA+∗FA)/2=0F_A^+=(F_A+*F_A)/2=0. Donaldson's polynomial invariants (the case N = 2) are multilinear functions on homology obtained by integrating cohomology classes over the moduli space of irreducible ASD connections; they distinguish smooth structures on homeomorphic 4-manifolds. Kronheimer (2005) gave a rigorous definition for every rank N≥2N\ge 2, realizing invariants that Mariño and Moore had earlier computed heuristically from the topological u(N)u(N) quantum field theory. Write χ(X)\chi(X) and σ(X)\sigma(X) for the Euler characteristic and signature.

Simple type, basic classes, and Witten's conjecture. For N = 2, X is of simple type (in the sense of Kronheimer–Mrowka) when the invariant scales polynomially in the class x of a point, i.e. DX,w(x2z)=4DX,w(z)D_{X,w}(x^2z)=4D_{X,w}(z) for all z. Their structure theorem then expresses the Donaldson series DX(z)D_X(z) as a finite sum ∑iaieKi⋅z\sum_i a_i e^{K_i\cdot z} of exponentials, where the classes Ki∈H2(X;Z)K_i\in H^2(X;\mathbb Z) (integral lifts of w2(X)w_2(X)) are the basic classes and aia_i are rational. The Seiberg–Witten invariant SWX(s)SW_X(\mathfrak s) of a spinᶜ structure s\mathfrak s (a count of solutions of the monopole equations, for b2+≥2b_2^+\ge 2 independent of the metric) has its own notion of basic classes c1(Ss+)c_1(S_{\mathfrak s}^+) with SWX≠0SW_X\ne 0; X is of Seiberg–Witten simple type when SWX(s)=0SW_X(\mathfrak s)=0 whenever the virtual dimension of the Seiberg–Witten moduli space for s\mathfrak s is nonzero. Witten (1994) conjectured that the instanton basic classes are exactly the Seiberg–Witten basic classes and that the coefficients aia_i are explicit universal multiples of the Seiberg–Witten invariants, so that rank-2 Donaldson theory is determined by Seiberg–Witten theory (established for large classes of manifolds, open in general). It is unknown whether any manifold with b2+≥2b_2^+\ge 2 fails either simple type condition.

Higher rank. Mariño and Moore (Comm. Math. Phys. 199 (1998)) computed the Donaldson–Witten function for gauge groups of rank N > 1 and predicted: a generalized simple-type condition; a closed formula for the U(N)-series of simple-type manifolds in terms of Seiberg–Witten data; and, consequently, that ranks N≥3N\ge 3 carry no new topological information beyond rank 2 and Seiberg–Witten theory. Daemi and Xie (Geom. Topol. 24, 2020) gave the first rigorous confirmation for N = 3 on simply connected elliptic surfaces without multiple fibers and certain simply connected surfaces of general type. To state their condition, the U(N)-polynomial invariants are linear maps DX,wND^N_{X,w} on the tensor power A(X)⊗(N−1)A(X)^{\otimes(N-1)}, where A(X)=Sym∗(H0(X)⊕H2(X))⊗Λ∗H1(X)A(X)=\mathrm{Sym}^*(H_0(X)\oplus H_2(X))\otimes\Lambda^*H_1(X) and, for a class α∈Hi(X)\alpha\in H_i(X), the superscript α(r)\alpha^{(r)} (2≤r≤N2\le r\le N) marks the copy of α\alpha in the (r−1)(r-1)-st tensor factor; the copies of the point class (the generator of H0(X)H_0(X)) are denoted a2,…,aNa_2,\dots,a_N. For b1(X)=0b_1(X)=0, Daemi–Xie define X to have w-simple type with respect to an embedded surface w when DX,w3(a23z)=27 DX,w3(z)D^3_{X,w}(a_2^3 z)=27\,D^3_{X,w}(z) and DX,w3(a3z)=0D^3_{X,w}(a_3 z)=0 for all z, and X to be of U(3)-simple type when this holds for every w in X; the U(3)-series D^X,w3(eΓ(2)+Λ(3)):=DX,w3((1+a2/3+a22/9) eΓ(2)+Λ(3))\hat D^3_{X,w}(e^{\Gamma^{(2)}+\Lambda^{(3)}}):=D^3_{X,w}\big((1+a_2/3+a_2^2/9)\,e^{\Gamma^{(2)}+\Lambda^{(3)}}\big) is the exponential generating function in Γ,Λ∈H2(X)\Gamma,\Lambda\in H_2(X) after this universal normalization involving a2a_2. The 4-dimensional computations also feed a 3-dimensional construction, the rank-3 sutured instanton homology SHI∗3(M,α)SHI^3_*(M,\alpha) for balanced sutured manifolds — a sutured manifold being a compact 3-manifold with boundary together with oriented curves (the sutures) dividing the boundary into positive and negative parts — a Floer group built from projectively flat U(3) connections, the higher-rank analogue of Kronheimer–Mrowka's SU(2) sutured theory, which supplied the paper's motivation for higher rank.

The algebraic model at N = 2. The U(N)-instanton Floer homology of S1×ΣgS^1\times\Sigma_g (Σ_g a genus-g surface) with admissible bundle is identified with H∗(NgN)H^*(\mathcal N_g^N), the cohomology of the moduli space of stable rank-N holomorphic bundles with fixed determinant over Σ_g. For N = 2, Muñoz computed the eigenvalue spectrum of the natural μ-operators (the actions of homology classes on instanton Floer homology induced by cup products on the moduli space, analogous to multiplying by cohomology classes) on this Floer homology (Topology 38, 1999) and identified its product with the quantum cohomology of Ng2\mathcal N_g^2 (Duke Math. J. 98, 1999) — quantum cohomology being a deformation of the cup product governed by genus-zero curve counts. These N = 2 results are the model for Daemi–Xie's Conjectures 7.3–7.5 on the algebra Vg,dN\mathbb V_{g,d}^N; for N = 3 only a complete set of relations for H∗(Ng3)H^*(\mathcal N_g^3) (Earl; Earl–Kirwan) is available, with no recursive presentation, which is the main technical obstruction to extending the N = 2 arguments.

Problem Statement

Determine the relation between higher-rank Donaldson polynomial invariants and Seiberg–Witten invariants of closed smooth 4-manifolds — the rank-N ≥ 3 generalization of Witten's conjecture, as posed by Daemi–Xie (Question 7.1 and Conjecture 7.2). The linked components of one objective are:

(1) Simple type at higher rank (Question 7.1). Formulate the correct analogue of the simple type condition for U(N)-polynomial invariants for each N≥3N\ge 3 — for N = 3 the candidate is the U(3)-simple type condition of Daemi–Xie — and decide whether every closed smooth 4-manifold with b2+≥2b_2^+\ge 2 satisfies it; in particular, clarify the relationship among U(3)-simple type, Kronheimer–Mrowka simple type, and Seiberg–Witten simple type (no manifold with b2+≥2b_2^+\ge 2 is known to fail any of the three).

(2) The Moore–Mariño formula for the U(3)-series (Conjecture 7.2). For X of U(3)-simple type with b2+≥2b_2^+\ge 2, prove that the U(3)-series has the closed form

D^X,w3(eΓ(2)+Λ(3))=eQ(Γ)/2−Q(Λ)∑i,jci,j ζ−w⋅(Ki−Kj)/2 e32(Ki+Kj)⋅Γ+32 i (Ki−Kj)⋅Λ\hat D^3_{X,w}\big(e^{\Gamma^{(2)}+\Lambda^{(3)}}\big)=e^{Q(\Gamma)/2-Q(\Lambda)}\sum_{i,j}c_{i,j}\,\zeta^{-w\cdot(K_i-K_j)/2}\,e^{\frac{\sqrt3}{2}(K_i+K_j)\cdot\Gamma+\frac{\sqrt3}{2}\,i\,(K_i-K_j)\cdot\Lambda}

with ζ=e2πi/3\zeta=e^{2\pi i/3}, where {Ki}\{K_i\} are the Kronheimer–Mrowka basic classes of X and the constants take the universal prescribed values

ci,j=2 χ(X)+32σ(X)+12Ki⋅Kj  3 2+(7χ(X)+11σ(X))/4  SWX(Ki) SWX(Kj),c_{i,j}=2^{\,\chi(X)+\frac{3}{2}\sigma(X)+\frac{1}{2}K_i\cdot K_j}\;3^{\,2+(7\chi(X)+11\sigma(X))/4}\;SW_X(K_i)\,SW_X(K_j),

SWX(K)SW_X(K) denoting the Seiberg–Witten invariant of the spinᶜ structure with basic class K (in the conventions of Daemi–Xie; Daemi–Iida–Scaduto restate the same conjecture at their eq. (9.7) with a minor difference of convention noted in their Remark 2.3). In particular, the basic classes appearing in the U(3) structure theorem coincide with the Kronheimer–Mrowka/Seiberg–Witten basic classes, so U(3)-invariants of simple-type manifolds carry no information beyond the U(2)/Seiberg–Witten package.

Prior partial resolution. Daemi–Iida–Scaduto (arXiv:2402.10448, Theorem 1.10/6.26) already proved that for b2+>1b_2^+>1 and U(3)-simple type the U(3)-series has the displayed finite-sum form for some finite set of classes satisfying a U(3) adjunction inequality, with undetermined ci,j∈Q[3]c_{i,j}\in\mathbb Q[\sqrt3]. The surviving core of (2) is precisely the identification of those classes with the Kronheimer–Mrowka/Seiberg–Witten basic classes and the determination of the constants ci,jc_{i,j}; the surviving core of (1) is the universality of U(3)-simple type and the entire formulation for N ≥ 4. A positive resolution of (1)–(2) confirms the Moore–Mariño prediction for N = 3 and supplies the correct simple-type formulation for N ≥ 4; a single closed 4-manifold with b2+≥2b_2^+\ge 2 failing U(3)-simple type, or of U(3)-simple type but violating the conjectured formula, refutes it and would exhibit smooth-structure information genuinely new to higher rank.

References

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