Existence and moduli of irreducible self-dual connections on the four-sphere for compact simple gauge groups at low instanton number

Atiyah, Hitchin and Singer's 1978 twistor-space analysis of the self-duality equations on the four-sphere established that the moduli space $\mu(G,k)$ of irreducible self-dual connections with compact simple structure group $G$ and instanton number $k$ is, at irreducible points, a smooth manifold of dimension $4h^\vee(G)k-\dim G$ ($h^\vee$ the dual Coxeter number), that it is empty when this dimension is negative, and — by an inductive construction along chains of group embeddings combined with a holonomy semi-continuity argument — that irreducible solutions exist for $SU(n)$ with $k\ge n/2$, $Sp(n)$ with $k\ge n$, $Spin(n)$ with $k\ge n/4$ ($n\ge 7$), $G_2$ with $k\ge 2$, and $F_4$, $E_6$, $E_7$, $E_8$ with $k\ge 3$. The construction is one-sided and leaves open a precisely delimited gap region where the dimension count permits solutions but the construction does not reach: $SU(3)$ and $SU(4)$ at $k=1$, $Sp(2)$ at $k=1$, $G_2$ at $k=1$, $Sp(3)$ and $Sp(4)$ at $k=2$, $Spin(n)$ at $k=2$ for $9\le n\le 14$, and $F_4$, $E_6$, $E_7$ at $k=2$, among other cases. The subsequent ADHM construction solved the classification for the unitary series at every charge — though it does not by itself settle the low-charge irreducibility question — but has no known analogue for the exceptional groups. The 1978 paper leaves open the following problem: complete the existence determination for every pair $(G,k)$ in the gap region — construction or proof of non-existence — and describe the structure (presentation, orbit-type stratification, connected components) of the resulting moduli spaces.

Background

A connection $A$ on a principal bundle over a compact Riemannian 4-manifold has curvature $F_A$, and its Yang–Mills energy $\int\mathrm{tr}(F_A\wedge *F_A)$ ($*$ the Hodge star) is minimized exactly by the anti-self-dual connections, those with $*F_A=-F_A$; which sign is called "self-dual" is a convention of orientation, and the 1978 paper uses the opposite sign to this one. Finite-action solutions of these first-order equations on Euclidean 4-space, called instantons, were introduced by Belavin–Polyakov–Schwartz–Tyupkin, with the general conformal family written down by Jackiw–Nohl–Rebbi; finite-action solutions on Euclidean 4-space correspond to smooth solutions on the round four-sphere $S^4$ with the point at infinity removed. Isomorphism classes of principal bundles with compact connected simple structure group $G$ over $S^4$ are classified by $\pi_3(G)\cong\mathbb{Z}$, so each bundle carries an integer $k$, the instanton number (the second Chern class when $G=SU(n)$, the group of unitary matrices of determinant one).

Write $\mu(G,k)$ for the set of gauge-equivalence classes of irreducible anti-self-dual connections of instanton number $k$, where irreducible means that the connection is not obtained from one for a proper subgroup of $G$, equivalently that the stabilizer of the gauge action is the center of $G$. The infinitesimal deformations of such a connection are counted by the index of an elliptic complex (kernel minus cokernel: deformations minus obstructions), and Atiyah–Hitchin–Singer proved that $\mu(G,k)$ is a smooth manifold at irreducible points of dimension

$\dim\mu(G,k)=4h^\vee(G)\,k-\dim G,$

where $h^\vee(G)$ is the dual Coxeter number, a standard invariant of the Lie algebra ($n$ for $SU(n)$, $n+1$ for $Sp(n)$ the compact symplectic group, $n-2$ for $Spin(n)$ the spin double cover of the rotation group $SO(n)$, and $4,9,12,18,30$ for the exceptional groups $G_2,F_4,E_6,E_7,E_8$). The parameter count itself is due to Bernard–Christ–Guth–Weinberg. Two consequences are immediate: $\mu(G,k)$ is empty when the dimension would be negative, and for $SU(2)$ at $k=1$ the formula gives $5$, the position (four parameters) and scale (one parameter) of the basic instanton.

Atiyah–Hitchin–Singer analyzed these moduli spaces by the twistor method: to an oriented conformal 4-manifold whose Weyl tensor is self-dual (automatic for $S^4$) one associates its twistor space — for $S^4$ the complex projective 3-space $\mathbb{CP}^3$ fibred over $S^4$ with fibre $\mathbb{CP}^1$ — and the Ward correspondence identifies anti-self-dual $G$-connections with holomorphic bundles on the twistor space (principal $G$-bundles, or vector bundles after choosing a faithful unitary representation of $G$) that are trivial on each fibre $\mathbb{CP}^1$ and satisfy a reality condition. Building on this correspondence and on an induction along chains of group embeddings ($SU(n)\subset SU(n+1)$, $Sp(n)\times Sp(1)\subset Sp(n+1)$, $SU(3)\subset G_2$, $Spin(9)\subset F_4$, $F_4\subset E_6$, up to $E_7\times SU(2)\subset E_8$) combined with a semi-continuity argument for holonomy groups, they proved that irreducible instantons exist whenever $k\ge n/2$ for $SU(n)$, $k\ge n$ for $Sp(n)$, $k\ge n/4$ for $Spin(n)$ with $n\ge 7$, $k\ge 2$ for $G_2$, and $k\ge 3$ for $F_4$, $E_6$, $E_7$, $E_8$. Their construction is one-sided: it starts from an irreducible instanton of sufficiently large charge inside an embedded subgroup, so it says nothing about the pairs $(G,k)$ below these thresholds for which the dimension count is still permissive.

That gap region — $4h^\vee(G)k-\dim G\ge 0$ but $k$ below the construction threshold — is nonempty exactly for $SU(n)$ with $n\ge 3$ (for example $SU(3)$ and $SU(4)$ at $k=1$, where $\dim\mu$ equals $4$ and $1$), $Sp(n)$ with $n\ge 2$ (roughly $n/2\le k<n$; for example $Sp(2)$ at $k=1$ with $\dim\mu=2$), $Spin(n)$ with $n\ge 9$ (for example $k=2$ for $9\le n\le 14$), and $F_4$, $E_6$, $E_7$ at $k=2$ (with $\dim\mu=20$, $18$ and $11$) and $G_2$ at $k=1$ (with $\dim\mu=2$); for $SU(2)$, $E_8$, $Spin(7)$ and $Spin(8)$ the dimension bound and the construction threshold coincide, so there is no gap.

Soon after, Atiyah–Hitchin–Drinfeld–Manin gave the ADHM construction, under which all instantons on $S^4$ with structure group $U(n)$ — equivalently $SU(n)$, since $H^2(S^4)=0$ makes the central $U(1)$ part topologically trivial — of every charge $k\ge 1$ arise from finite-dimensional linear-algebra data (a pair of matrices and two maps satisfying quadratic moment-map equations), the framed moduli spaces — those fixing a trivialization of the bundle at a chosen base point, which removes the residual quotient action — being hyperkähler quotients (quotients by a compact group cut out by moment-map equations, carrying a richer quaternionic geometry) of dimension $4nk$. The same formalism extends to the other classical groups, and any compact group embedded in a classical group can in principle be treated by imposing the extra structure on the data; but this gives a constrained description of classical-group instantons rather than an intrinsic construction for the embedded group, and it does not by itself determine which low charges support irreducible connections. This settles construction and classification for the unitary series, but no intrinsic analogue is known for the exceptional groups: Benvenuti–Hanany–Mekareeya state that an ADHM construction is not known for exceptional gauge groups. The geometry of instanton moduli spaces meanwhile became a field in itself — Uhlenbeck's removable-singularity theorem and the compactification of the moduli spaces by ideal connections (unions of lower-charge instantons concentrated at points), the hyperkähler nature of the natural $L^2$ metric (Hitchin–Karlhede–Lindström–Roček), the Atiyah–Jones conjecture on the stable topology of the based $SU(2)$ moduli spaces (posed by Atiyah–Jones, proved by Boyer–Hurtubise–Mann–Milgram), and Donaldson's application of the $SU(2)$ moduli spaces to invariants of smooth 4-manifolds — but essentially all of this development concerns the classical groups. What remains from the 1978 paper's own limits is the existence determination in the gap region and a structural description of the nonempty moduli spaces, particularly for the exceptional groups.

Problem Statement

Let $S^4$ carry its standard round metric and let $G$ be a compact connected simple Lie group. For an integer $k$ (the instanton number, normalized so that the dimension formula below holds; $k=c_2$ for $G=SU(n)$), let $\mu(G,k)$ denote the moduli space of gauge-equivalence classes of irreducible anti-self-dual connections on principal $G$-bundles over $S^4$ of instanton number $k$ (anti- versus self-duality is an orientation convention). At irreducible points $\mu(G,k)$ is a smooth manifold of dimension

$\dim\mu(G,k)=4h^\vee(G)\,k-\dim G,$

and $\mu(G,k)$ is empty whenever this number is negative.

Part 1 (existence in the gap region). For every pair $(G,k)$ with $4h^\vee(G)k-\dim G\ge 0$ and $k$ below the Atiyah-Hitchin-Singer construction threshold — $k<n/2$ for $SU(n)$, $k<n$ for $Sp(n)$, $k<n/4$ for $Spin(n)$ with $n\ge 7$, $k<2$ for $G_2$, $k<3$ for $F_4$, $E_6$, $E_7$ (for $SU(2)$, $E_8$, $Spin(7)$ and $Spin(8)$ there is no such $k$) — determine whether $\mu(G,k)$ is nonempty (existence or non-existence of an irreducible anti-self-dual connection of instanton number $k$). The first cases are $SU(3)$ and $SU(4)$ at $k=1$, $Sp(2)$ at $k=1$, $G_2$ at $k=1$, and, at $k=2$, $SU(5)$ through $SU(8)$, $Sp(3)$, $Sp(4)$, $Spin(n)$ for $9\le n\le 14$, and $F_4$, $E_6$, $E_7$; the families continue for all larger ranks (for example $SU(7)$, $SU(8)$, $Sp(4)$ and $Spin(n)$ for $13\le n\le 22$ also have gap cases at $k=3$).

Part 2 (structure of the nonempty moduli spaces). For each $(G,k)$ in the gap region with $\mu(G,k)$ nonempty, give a presentation of the moduli space, determine its stratification by orbit type and its number of connected components, consistent with the dimension formula above and with the known charge-one $SU(2)$ picture (five parameters: position and scale).

References

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