Universal error formula relating Donaldson invariants and Pidstrigach-Tyurin spin invariants of smooth four-manifolds for general Dirac index

For a smooth, closed, oriented four-manifold $X$ with $b_1(X)=0$, the spin invariants $P_X^{\Lambda,w}(z)$ of Pidstrigach and Tyurin are defined by integrating $\mu$-classes over the Uhlenbeck closure of the degeneracy locus of the family of Dirac operators coupled to a rank-two bundle over the moduli space $M_\kappa^w$ of projectively anti-self-dual connections, while the Donaldson invariants $D_X^w(z)$ integrate $\mu$-classes over the Uhlenbeck compactification itself. Leness's computation of the Poincaré dual of the degeneracy locus in $\mu$-classes (Theorem 1.1) suggests the equality $P_X^{\Lambda,w}(z)=D_X^w(zT_\kappa^{\Lambda,w})$, but Leung's thesis shows it holds only at Dirac index $n_a=0$: the cobordism between the geometric representative of the computed class and the degeneracy locus need not extend over the Uhlenbeck compactification. The open problem is to compute the error $E_X^{\Lambda,w}(z)=D_X^w(zT_\kappa^{\Lambda,w})-P_X^{\Lambda,w}(z)$ for general Dirac index $n_a\le 0$, to prove the conjecture (Conjecture 4.2 of the source) that it is a polynomial in Donaldson invariants with insertions built from characteristic classes of the index bundles at lower instanton levels, the intersection form $Q_X$, and $\Lambda$, with universal coefficients depending only on topological data, and to determine those coefficients. The central obstruction is the construction of compact replacements (caps) for ends of the geometric representatives that meet multiple overlapping lower Uhlenbeck strata. No resolution is known: the adjacent Donaldson-Seiberg-Witten and Poincaré-Seiberg-Witten legs of the surrounding program advanced substantially, but the direct error formula and its universal coefficients remain open.

Background

Donaldson theory studies a smooth, closed, oriented four-manifold $X$ through moduli spaces of anti-self-dual connections. For a complex rank-two bundle $E_\kappa^w\to X$ with first Chern class $w$ and instanton number $\kappa=-\frac14\langle p_1(\mathfrak{su}(E)),[X]\rangle$, let $M_\kappa^w$ be the moduli space of projectively anti-self-dual (ASD) connections, that is, connections whose self-dual curvature part vanishes up to a central scalar term. The $\mu$-map of Kronheimer and Mrowka assigns to $h\in H_2(X)$ and to the point class $x\in H_0(X)$ cohomology classes $\mu(h)\in H^2(M_\kappa^w)$ and $\mu(x)\in H^4(M_\kappa^w)$ via Chern classes of the universal bundle; the Donaldson invariants $D_X^w(z)$ for $z\in\mathbb A(X)=\mathrm{Sym}(H_2(X;\mathbb R)\oplus H_0(X;\mathbb R))$ are defined by intersecting geometric representatives $\mathcal V(z)$ (submanifolds of the ambient configuration space realizing the $\mu$-classes) with the Uhlenbeck compactification $\bar M_\kappa^w$, which adjoins ideal connections: its lower strata have the form $M_{\kappa-\ell}^w\times\Sigma$ with $\Sigma\subset\mathrm{Sym}^\ell(X)$ recording $\ell$ concentrated instantons.

A spin$^c$ structure $\mathfrak s=(\rho,W)$ on $X$ consists of rank-two Hermitian spinor bundles $W^\pm$ with $\det W^+\cong\det W^-$, and splicing it with $E_\kappa^w$ gives a spin$^u$ structure $\mathfrak t=(\rho\otimes\mathrm{id}_E, W\otimes E)$; write $\Lambda=c_1(W^+)+w$ and let $n_a(\mathfrak t)$ be the index of the coupled Dirac operator $\slashed D_A:\Gamma(W^+\otimes E)\to\Gamma(W^-\otimes E)$. As the connection $A$ ranges over $M_\kappa^w$ these operators form a family of Fredholm operators, and its degeneracy locus $J_\kappa^{\Lambda,w}\subset M_\kappa^w$ (where the kernel is one-dimensional) is, for $n_a(\mathfrak t)\le 0$ and generic perturbations, a smooth submanifold of codimension $2(1-n_a(\mathfrak t))$ by transversality results of Feehan and Teleman. Pidstrigach and Tyurin defined the spin invariants by integrating $\mu$-classes over the Uhlenbeck closure of this locus: $P_X^{\Lambda,w}(z)=\langle\bar\mu(z),[\bar J_\kappa^{\Lambda,w}]\rangle$; they are close enough to the Donaldson invariants that a formula relating the two is expected.

Leness computed, generalizing results of Leung's thesis, the Poincaré dual of $J_\kappa^{\Lambda,w}$ on any compact subset $K_\kappa^w\subset M_\kappa^w$ as an explicit polynomial in the $\mu$-classes $\mu(\mathfrak t)$, $\Omega$, and $\mu(x)$, with coefficients given by a generating function $F(x,y,z)=\exp(\frac12 xJ_1(z)+\frac14 y^2J_2(z)+J_3(z))$ where $J_1(z)=z^{-1}\tan^{-1}(z)$, $J_2(z)=z^{-3}(z-\tan^{-1}(z))$, $J_3(z)=-\frac12 n_a(\mathfrak t)\ln(1+z^2)+\kappa(z^{-1}\tan^{-1}(z)-1)$ (a Koschorke-Porteous/Thom-Porteous type formula for families of Fredholm operators). Writing this class as $\mu(T_\kappa^{\Lambda,w})$, one might expect $P_X^{\Lambda,w}(z)=D_X^w(zT_\kappa^{\Lambda,w})$, but Leung showed this holds only when $n_a(\mathfrak t)=0$: the geometric representative $\mathcal V(T_\kappa^{\Lambda,w})$ and the degeneracy locus are cobordant on compact subsets of $M_\kappa^w$, but the cobordism need not extend to the Uhlenbeck compactification. The discrepancy

$E_X^{\Lambda,w}(z)=D_X^w(zT_\kappa^{\Lambda,w})-P_X^{\Lambda,w}(z)$

is thus localized near the lower Uhlenbeck strata. Taubes' gluing theorems present the ends of $\mathcal V(z)\cap M_\kappa^w$ near a stratum $M_{\kappa-\ell}^w\times\Sigma$ as fiber bundles whose fibers are cones built from moduli spaces of framed, centered ASD connections on $S^4$. For $n_a(\mathfrak t)=-1$, Leung constructed an Ozsvathian cap (a compact replacement for such an end, in the sense of Ozsvath's blow-up formulas) for the single relevant end and used an excision argument to show the resulting intersection correction is universal, obtaining an explicit formula; only partial computations exist for $n_a(\mathfrak t)=-2$. For $n_a(\mathfrak t)<-2$, the closure $\bar{\mathcal V}(z)\cap\bar M_\kappa^w$ meets more than one lower stratum, the ends are covered by several overlapping gluing charts, and constructing caps becomes substantially harder; the same difficulty of computing intersection numbers in unions of overlapping open sets (the overlap problem) appears in the Kotschick-Morgan conjecture on wall crossing for $b_2^+=1$ and throughout the SO(3)-monopole program.

The motivation is the Pidstrigach-Tyurin program: the SO(3)-monopole cobordism of Feehan and Leness relates the Donaldson and Seiberg-Witten invariants, and separately the spin and Seiberg-Witten invariants, through polynomials with universal coefficients depending on topological data; Witten's conjecture (Donaldson series equal Seiberg-Witten series for manifolds of simple type) is the expected explicit form on the Donaldson side. A direct Donaldson-spin relation of the type sought here would provide independent leverage for determining the still-undetermined universal coefficients in those cobordism formulas and could reveal new topological constraints on the invariants of the smooth structure.

Problem Statement

Let $X$ be a smooth, closed, oriented four-manifold with $b_1(X)=0$, let $E_\kappa^w\to X$ be a complex rank-two bundle, and let $\mathfrak t$ be a spin$^u$ structure on $X$ with $\kappa=-\frac{1}{4}p_1(\mathfrak t)$, $c_1(\mathfrak t)=\Lambda$, and Dirac index $n_a(\mathfrak t)\le 0$. Let $J_\kappa^{\Lambda,w}\subset M_\kappa^w$ be the degeneracy locus of the family of coupled Dirac operators, let $\mu(T_\kappa^{\Lambda,w})$ be the rational cohomology class given by Theorem 1.1 of Leness (2012) (the Poincaré dual of $J_\kappa^{\Lambda,w}$ on compact subsets, expressed in $\mu$-classes), and let $P_X^{\Lambda,w}(z)$ and $D_X^w(z)$ be the spin and Donaldson invariants for $z\in\mathbb A(X)=\mathrm{Sym}(H_2(X;\mathbb R)\oplus H_0(X;\mathbb R))$. The problem is to compute, for general Dirac index (beyond the known cases $n_a=0$ and $n_a=-1$ and the partial results for $n_a=-2$), the error

$E_X^{\Lambda,w}(z)=D_X^w(zT_\kappa^{\Lambda,w})-P_X^{\Lambda,w}(z),$

to prove the conjectured structure (Conjecture 4.2 of the source) that $E_X^{\Lambda,w}(z)$ is a polynomial in the Donaldson invariants $D_X^w(zO_\kappa^{\Lambda,w})$, the intersection form $Q_X$, and $\Lambda$, where $O_\kappa^{\Lambda,w}$ is an expression in the characteristic classes of the Dirac index bundles $\mathcal D_{\kappa-i}^{\Lambda,w}$ at lower instanton levels $i>0$, with coefficients depending only on topological invariants of $X$, $\kappa$, and $\Lambda^2$, and to determine these universal coefficients explicitly. The central obstruction identified in the source is that for $n_a(\mathfrak t)<-2$ the closure $\overline{\mathcal V}(z)\cap\bar M_\kappa^w$ meets more than one lower Uhlenbeck stratum, so the ends of $\mathcal V(z)\cap M_\kappa^w$ are covered by multiple overlapping gluing charts modeled on fibered cone-bundles (Taubes' gluing descriptions), and constructing compact replacements (caps) for these ends that allow comparison of the intersection numbers defining the two invariants is the known difficulty.

References

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