Toric gravitational instantons with four fixed points

Introduction

A gravitational instanton is a complete, non-compact Ricci-flat Riemannian 4-manifold whose curvature has finite energy. Gravitational instantons were introduced by Hawking in Euclidean quantum gravity [19]. The classical examples, among them the Euclidean Schwarzschild and Kerr metrics, the Eguchi–Hanson and Taub–NUT metrics, and the multi-centre metrics of Gibbons and Hawking with collinear centres, are toric: the torus $T^2$ acts on them effectively by isometries. A natural measure of the complexity of a toric instanton is the number of fixed points of the torus action. This paper classifies the simply connected toric gravitational instantons with four fixed points, the smallest number of fixed points for which such an instanton can be Hermitian for neither orientation. Here a metric is called Hermitian for an orientation if it admits a complex structure that is compatible with the metric and induces this orientation.

The geometry of a toric instanton $(M,g)$ is encoded in its orbit space. If $M$ is simply connected and the action has $n \ge1$ fixed points, the orbit space $M/T^2$ is a closed half-plane, and the fixed points map to $n$ points of its boundary line, the axis. We call them the turning points. They cut the axis into $n-1$ bounded intervals, the finite rods, whose lengths $L_1,\ldots,L_{n-1}$ are the rod lengths, and two semi-infinite rods. Over each rod the stabiliser of the action is a circle subgroup of $T^2$. It is generated by a primitive vector of the integer lattice in the Lie algebra of $T^2$, the rod vector of the rod, which is determined up to sign. The turning points, the lines spanned by the rod vectors, and the type of the end, described below, form the rod structure of the instanton [25]. Li and Sun draw it like the moment polygon of a toric symplectic 4-manifold, with each rod as an edge orthogonal to its rod vector (Figure 1). Number the rods from left to right, with rod vectors $v_0,\ldots,v_n$. Since the action is smooth, the rod vectors of the two rods that meet at a turning point form a basis of the lattice [30]. Consequently, for suitable signs, the sum $v_{j-1}+v_{j+1}$ of the neighbours of the rod vector $v_j$ of each finite rod is an integer multiple $\kappa_jv_j$ of it. The integers $\kappa_1,\ldots,\kappa_{n-1}$ form the chain, which determines the topology. The preimage of the closure of each finite rod is an embedded 2-sphere whose self-intersection number is the corresponding entry of the chain, and $M$ is the interior of a linear plumbing of disc bundles over these spheres. For example, the multi-Eguchi–Hanson metrics with four collinear centres have, in their hyperkähler orientation, the chain $(-2,-2,-2)$. The spheres are the $(-2)$-curves of the minimal resolution of $\mathbb{C}^2/\mathbb{Z}_4$.

At infinity a toric instanton is modelled on one of three types of ends [25]. An ALE end is asymptotic to the end of a quotient $\mathbb{R}^4/\Gamma$ of flat space by a finite group. An ALF end is asymptotic, up to a finite cover, to the end of the Taub–NUT metric, a circle bundle over the complement of a ball in $\mathbb{R}^3$ whose fibres have asymptotically constant length. It is ALF$^+$ or ALF$^-$ according as the orientation of $M$ corresponds to the hyperkähler orientation of the Taub–NUT metric or to the opposite one. An AF end is asymptotic to the end of the quotient of $\mathbb{R}^3 \times\mathbb{R}$ by an isometry that translates $\mathbb{R}$ and rotates $\mathbb{R}^3$ about an axis through the angle $2\pi\beta$. The number $\beta$, defined modulo $\mathbb{Z}$, is the twist of the end, and its sign depends on an orientation of the torus. Geodesic balls of radius $r$ have volume of order $r^4$ for ALE ends and of order $r^3$ for ALF and AF ends. The end is AF if and only if the two semi-infinite rods have the same stabiliser.

The classification rests on the soliton representation of toric instantons proved in [3]. With two commuting Killing fields, the Ricci-flat equations reduce to an equation for the Gram matrix of the Killing fields which is the integrability condition of a linear system with a spectral parameter. The soliton transformations of Belinski and Zakharov [7, 8] add finitely many poles in the spectral parameter to a known solution and produce the new metric by linear algebra. Geroch conjectured that an infinite-dimensional group of such transformations generates all these metrics from flat space [13, 14]. In [3] a solitonic form of this conjecture is proved for toric instantons: each of them is obtained from a flat metric by finitely many soliton transformations with poles on the axis. Each such instanton is given by a closed formula in finitely many soliton data: a positive weight at each turning point, a weight at infinity, which is zero for ALE ends and positive otherwise, and an integer, the degree index. Up to a common factor, the components of the rod vectors are Hankel determinants of measures built from the soliton data, hence polynomials in them. Smoothness, which is the relation between consecutive rod vectors expressed by the chain, becomes a system of polynomial equations with integer coefficients.

Conversely, every solution with positive weights, the weight at infinity included, and a degree index between $1$ and $n-1$ defines a smooth complete instanton, whose end is ALF or AF. The classification of toric instantons with a given number of turning points is thus a question about the positive real solutions of polynomial systems, one for each chain, type of end, and degree index.

We say that two metrics agree up to scale if one is isometric to a constant positive multiple of the other. The main result is the following.

Theorem 2.7 in § 2.5 is the detailed version of Theorem A: it gives the chains, the soliton data of $g_{\mathrm{ALF}}$, which lie in $\mathbb{Q}(\sqrt{2},\sqrt{5})$, the types of the ends, and the conventions for the twist. Parts (a) and (b) are known: the toric hyperkähler instantons are the Gibbons–Hawking metrics with collinear centres, by the classification of hyperkähler instantons (§ 2.2) or directly by Li and Sun [25]. The new content concerns the instantons that are hyperkähler for neither orientation: none of them has an ALE end, only one has an ALF end, and those with an AF end live on $X_3$, one up to scale for each twist. The instanton $g_{\mathrm{ALF}}$ was found in [3 Rem. 11.5(a)] and independently by Jones [20], and by Theorem A it is, up to scale, the only simply connected toric instanton with four fixed points and an ALF end that is hyperkähler for neither orientation. Part (d) proves Li and Sun’s conjecture [25 Conj. 8.5] for $X_3$: for each twist the rod lengths, and with them the instanton up to scale, are unique. The AF instantons of (d) are, as outlined in § 5.5, the metrics found by Teo [31], which are therefore smooth complete instantons for every twist in $(0,1)$.

Li and Sun’s degree (§ 2.1) links the rod structure to the Hermitian geometry. For a given orientation, a toric instanton has degree $0$ if and only if it is hyperkähler, and degree $1$ if and only if it is Hermitian non-Kähler, that is, Hermitian with nonvanishing self-dual Weyl curvature for this orientation [25 Thm. 7.4, 24 §2]. The degrees for the two orientations add up to $n-1$ for ALE ends and to $n$ for ALF and AF ends, where $n$ is the number of fixed points [3 Rem. 2.8]. An instanton that is Hermitian for neither orientation has degree at least $2$ for both, and the first case in which this is possible is that of four fixed points and an ALF or AF end. Li and Sun proved that such instantons exist. For every $m\geq1$ and every twist $\beta\in(0,1)$ their manifold $X_m$ (§ 2.1) carries a toric AF instanton with the twist $\beta$ [25 Thm. 1.1], which is Hermitian for neither orientation when $m\geq3$, and they conjecture that it is unique for each twist.

Overview of this paper. Section 2 gives background and preliminaries and ends with the detailed main theorem and an outline of its proof (§ 2.5). Section 3 carries out the reduction, and Sections 4 and 5 treat the ALF and the AF chain. Section 6 gives the metrics explicitly, and Section 7 describes what is known beyond four turning points. The appendices contain the fold argument (Appendix A) and the convexity argument (Appendix B). The role of computation and of an AI assistant is described in a note at the end of the main text.

Background

A toric instanton has two descriptions that the proof combines: its rod structure, which records the stabilisers of the torus action along the boundary of the orbit space, and its soliton data, which give the metric in closed form [3].

Rod structures, chains, and ends

A gravitational instanton is a complete, non-compact, oriented Ricci-flat Riemannian 4-manifold $(M,g)$ whose curvature has finite energy, $\int_M \lvert\operatorname{Rm}_g\rvert^2 \, \mathrm{dvol}_g < \infty$. Following Li and Sun [25 Def. 2.1] and the companion paper [3 Defs. 2.1, 2.2], we call it toric if it carries an effective isometric action of the torus $T^2$ in which every point has a connected stabiliser. By Lemma 2.1 below, every effective isometric action of $T^2$ on a simply connected gravitational instanton has connected stabilisers, so the instanton with this action is toric.

We write $\mathfrak{t}$ for the Lie algebra of $T^2$ and $\Lambda\subset\mathfrak{t}$ for the lattice of elements $X$ with $\exp(2\pi X)=1$, so that $T^2=\mathfrak{t}/\Lambda$. Orlik and Raymond list the possible stabilisers of an effective action of $T^2$ on an orientable 4-manifold together with their slice representations [30 §1.4]. For closed manifolds $M$ with $H_1(M;\mathbb{Z})=0$, Orlik and Raymond also exclude finite non-trivial stabilisers [30 Lemma 5.2]. The following lemma treats non-compact manifolds.

Proof. Let $x \in M$, let $H \subset T^2$ be its stabiliser and $O=T^2\cdot x$ its orbit, and let $V \subset T_xM$ be the orthogonal complement of $T_xO$. For small $\epsilon>0$ and the open ball $V_\epsilon\subset V$ of radius $\epsilon$ about $0$, the map

is an equivariant diffeomorphism onto an open neighbourhood $U$ of $O$ [9 Ch. VI, Thm. 2.2, Cor. 2.4]. Here $H$ acts on $V$ by the differentials of its elements at $x$. Since $T^2$ is abelian, these differentials are the identity on $T_xO$. An isometry of the connected manifold $M$ that fixes $x$ and whose differential at $x$ is the identity is the identity map. Since the action is effective, $H$ therefore acts faithfully on $V$. Every element of $H$ is joined to the identity by a path in $T^2$ and therefore preserves the orientation of $M$, which is orientable since it is simply connected. Its differential at $x$ thus has positive determinant, and $H$ is a closed subgroup of $SO(V)$ with $\dim V=2+\dim H$.

If $\dim H=2$, then $H=T^2$. If $\dim H=1$, then the identity component $H_0$ of $H$ is the group of rotations of the 3-dimensional space $V$ about a line $V_0$. Since $H$ is abelian, each element of $H$ maps $V_0$, the fixed line of $H_0$, onto itself. A rotation of $V$ with this property is a rotation about $V_0$ or a half-turn about a line orthogonal to $V_0$. A half-turn of this kind conjugates each rotation about $V_0$ to its inverse, so it does not commute with $H_0$. Hence $H=H_0$ is a circle. If $\dim H=0$, then $H$ is a finite subgroup of $SO(2)$, hence cyclic of some order $k$, and it remains to show that $k=1$.

Suppose that $k\geq2$. Then $H$ acts on $V\cong\mathbb{C}$ by multiplication with the $k$-th roots of unity. Let $h\in H$ be the element that acts by $e^{2\pi i/k}$, and choose $\zeta\in\mathfrak{t}$ with $\exp(2\pi\zeta)=h$. The lattice $\Lambda_x=\Lambda+\mathbb{Z}\zeta$ contains $\Lambda$ with $\Lambda_x/\Lambda\cong H$, and $X\mapsto\exp(2\pi X)\cdot x$ identifies $\mathfrak{t}/\Lambda_x$ with $O$. Let $N=M/T^2$ be the orbit space, $\pi:M\to N$ the orbit map, and $y=\pi(x)$. By (2.1), $D = \pi(U)$ is homeomorphic to $V_{\epsilon}/H$, hence to an open disc by $v \mapsto v^{k}$, with $y$ as its centre. By the slice theorem at all points, $N$ is a surface with boundary [30 §1.4]. It is connected, it is non-compact since $\pi$ is proper, and it is simply connected since $\pi_{*}: \pi_{1}(M) \to\pi_{1}(N)$ is onto [9 Ch. II, Cor. 6.3]. Gluing $\partial N \times[0,1)$ to $N$ along $\partial N$ gives a connected non-compact surface without boundary which deformation retracts onto $N$, so $H_{2}(N;\mathbb{Z}) = 0$ [17 Prop. 3.29]. Since also $H_{1}(N;\mathbb{Z}) = 0$ and $H_{1}(D;\mathbb{Z}) = 0$, the Mayer–Vietoris sequence of the open cover of $N$ by $N \setminus\{y\}$ and $D$ shows that the inclusion induces an isomorphism

where the first isomorphism is the winding number about the centre in the coordinate $v^{k}$ of $D$.