About a Family of ALF Instantons with Conical Singularities

We apply the techniques developed in our previous article to describe some interesting families of ALF gravitational instantons with conical singularities. In particular, we completely understand the 5-dimensional family of Chen-Teo metrics and prove that only 4-dimensional subfamilies can be smoothly compactified so that the metric has conical singularities.


Introduction
In a previous paper [3], the authors of the present paper have provided a complete classification, as well as an effective mode of construction, of so-called toric Hermitian ALF gravitational instantons.These are four-dimensional, complete, non-compact oriented Ricci-flat Riemannian (positive definite, smooth) manifolds, which are toric, i.e., admits an effective metric action of the torus T 2 = S 1 × S 1 , are conformally Kähler -but non-Kähler -and, at infinity, are diffeomorphic to the product R × L, where L is locally a S 1 -bundle over the sphere S 2 ; the AF case is when L = S 2 × S 1 .
This class of gravitational instantons includes the Riemannian versions, obtained by Wick rotations, of well-known Lorentzian space-times, namely (i) the Schwarzschild space, (ii) the family of Kerr spaces, (iii) the self-dual Taub-NUT space, equipped with the opposite orientation to the one induced by its hyperkähler structure, and (iv) the Taub-bolt space, discovered in 1978 by Don Page [15].Apart from the Taub-NUT space, these spaces share the feature of being of type D + D − , meaning that their self-dual and anti-self-dual Weyl tensors, W + and W − respectively, are both degenerate and non-vanishing, hence giving rise to an ambikähler structure, as defined in [2], in their conformal cases, cf.Section 2.2 below, see also [9] for the Kerr spaces (in this terminology, the self-dual Taub-NUT space is of type D + O − , meaning that W + is degenerate and non-vanishing, while W − ≡ 0).
It has been conjectured for a long time that no other 4-manifold could be admitted in the family of toric Hermitian ALF gravitational instantons.In 2011 however, a new 1-parameter family of toric ALF -in fact AF -gravitational instantons was discovered by Yu Chen and Edward Teo, [6], and these instantons were shown to be Hermitian by Steffen Aksteiner and Lars Andersson [1].In contrast with the former examples, the Chen-Teo instantons are not derived from a Lorentzian space by a Wick rotation, and don't give rise anymore to an ambitoric structure, but nevertheless still admit a toric Kähler structure in their conformal class.Moreover, in view of the ALF condition, this toric Kähler structure, as well as the toric Kähler structure associated to any toric Hermitian ALF gravitational instanton, can be chosen to be locally close at infinity to the product of the standard sphere S 2 , of curvature 1, with a hyperbolic cusp, of curvature −1.
In [3], it has been shown that, together with the above mentioned gravitational instantons, issuing from classical Lorentzian spaces, the Chen-Teo instanton provides the missing puzzlepiece needed for a full classification of the toric Hermitian ALF gravitational instantons.In this approach, the conformal Kähler structure plays a crucial role and, together with a recent ansatz due to Paul Tod, allows for a simple construction of these instantons, as explained in the next section.In particular, it provides a new, simple definition of the Chen-Teo instantons.As a matter of fact, this approach first provides a general description of these metrics on the open set where the toric action is free.The actual construction of instantons then amounts to smoothly compactifying the manifold together with the metric along the invariant divisors, encoded by the edges of the moment polytope.This phase of the construction actually requires strong restrictions, and eventually ends up with the above mentioned complete classification.
A larger class is obtained by smoothly compactifying the manifold as above but by allowing conical singularities of the metric along some of them.Metrics of this kind will be called regular.Examples of such metrics can be found inside the well-known family of so-called Kerr-Taub-NUT metrics introduced by Gibbons-Perry in [12], cf.[3, Section 7.2.1].
In 2015, Chen-Teo introduced a new 5-parameter family of toric ALF Ricci-flat metrics -in fact, a 4-parameter family, if we ignore the scaling, as we shall do in the sequel -including the above mentioned 1-parameter sub-family of Chen-Teo instantons.
Apart from the latter, none of them is smooth -a consequence of the classification in [3], see also Section 3.2 below -but some of them are regular, as defined above.The main goal of this paper is to provide an alternative description of the Chen-Teo family, to detect inside this family the regular elements and the topology of their boundary at infinity, cf.Theorem 3.3 below, and to describe some distinguished regular sub-families and their limits.

Toric Hermitian ALF gravitational instantons: a quick review
In this section, we recall some main features of toric Hermitian ALF gravitational instantons, taken from [3].

General presentation
It turns out, cf.[3,Corollary 5.2], that each of them is completely encoded by a positive, continuous, convex, piecewise affine function f on R of the form for some positive integer r, where z denotes the standard parameter of R, A is a positive real number, the coefficients a i are positive real numbers, with r i=1 a i = 1, and the z i , i = 1, . . ., r, called angular or turning points, denote the points of discontinuity on R of the slope of f .For convenience, we put: z 0 := −∞ and z r+1 := +∞, cf. Figure 1 for the piecewise affine function of the Chen-Teo instanton.On each open interval (z i , z i+1 ), i = 0, . . ., r, the slope of f is constant, denoted by f ′ i .It is required that The coefficients a i are related to the slopes f ′ i by According to the Tod ansatz, cf.[17] and also [3,Section 3], the geometry of a toric Hermitian Ricci-flat metric is determined by a harmonic, axisymmetric (real) function U = U (ρ, z), defined on the Euclidean space R 3 , with the following notation: if u 1 , u 2 , u 3 denote the standard coordinates of R 3 , the pair (ρ, z) -the so-called Weyl-Papapetrou coordinates -are defined by 2 , z = u 3 ; U being axisymmetric means that it is invariant by the S 1 -action: e iθ • u = (cos θu 1 + sin θu 2 , − sin θu 1 + cos θu 2 , u 3 ), hence is a function of ρ, z, and the condition of being harmonic is then expressed by U zz + U ρρ + 1 ρ U ρ = 0, where, as usual, U ρ , U z , U ρ,z etc. denote the partial derivatives with respect to ρ and z.For any such generating function U , the corresponding metric is then given, in the Harmark form [5,13], by on the open set, where ρ ̸ = 0, where t, x 3 are angular coordinates, and V , F , e 2ν are functions of ρ, z, defined by where H, the conjugate function of U , is defined, up to an additive constant, by cf. Section 2.2 for the significance of the constant k.
The functions H and F are both defined up to an additional constant.Indeed, in the expression (2.2) of the metric, the 1-form η = dt − F dx 3 is well defined, but the pair (t, F ) is subject to the transform (t, F ) → (t + cx 3 , F + c), for any constant c, by which η, the vector field ∂ t and x 3 remain unchanged, while the vector field ∂ x 3 becomes ∂ x 3 − c∂ t .In particular, the vector field ∂ x 3 + F ∂ t remains unchanged.
In the current ALF case, it was shown in [3, Section 5] that the generating function U of any toric Hermitian ALF gravitational instanton is defined on the whole space R 3 , except on the z-axis ρ = 0, that, near the z-axis, U is close to f (z) log ρ 2 , while, at infinity, it is asymptotic to the harmonic axisymmetric function U 0 defined by It follows that the generating function U of any toric Hermitian ALF gravitational instanton is actually entirely determined by the above piecewise affine function f (z), via the formula By setting and by noticing that the constant k in (2.3)-(2.4) is equal to 2A, cf.below, we get the following expressions of U , its first and second derivatives, and H: up to constant.We then get and It then readily follows that and that V tends to 1 at infinity.On the z-axis ρ = 0, for any z where f ′ (z) ̸ = 0, we infer From (2.13), we infer that on any interval (z i , z i+1 ), i = 0, . . ., r, F (0, z) is constant, say equal to , since H is continuous on the z-axis, we then have Propsition 7].From (2.13) again, we get up to an additional constant, hence It follows that In particular, the metric is AF, i.e., satisfies F r − F 0 = 0, if and only if Remark 2.1.As explained above, to any convex, piecewise affine function f (z) as defined in (2.1) is associated a generating function U (ρ, z), defined by (2.6), hence by (2.7); conversely, it follows from (2.8) that f (z) is determined by U (ρ, z) via the formula f (z) = 1 2 (ρU ρ )(0, z).The corresponding Ricci-flat metric g is then expressed by (2.2), where the functions V (ρ, z), F (ρ, z), e 2ν (ρ, z) are given by (2.3)-(2.4),hence by (2.9)- (2.11).For any real constants α > 0, β, f (z) may be replaced by the function f (z , and the corresponding generating function is then replaced by The corresponding Ricci-flat metric is then g(ρ, z, t, x 3 ) := 1 α 2 g(αρ, αz+β, αt, x 3 ), meaning that g is homothetic to g by a factor 1/α 2 , via the change of variables (ρ, z, t, x 3 ) → (αρ, αz +β, αt, x 3 ).Also notice that, by denoting fi := f (z i ) and

The Kähler environment
By definition, a toric Hermitian ALF gravitational instanton, say (M, g), admits a Kähler metric, g K , in the conformal class of g, which is actually toric as well, meaning that the torus action is Hamiltonian, i.e., admits a moment map The fact that g is conformally Kähler implies that the self-dual Weyl tensor W + of g, regarded as a (symmetric, trace-less) operator on the self-dual part of Λ 2 M , is degenerate, meaning that W + has a simple, non-vanishing, simple eigenvalue, say λ, and a repeated eigenvalue − λ 2 .It is also required that λ is not constant and everywhere non-vanishing.According to [8], the Kähler metric g K is then equal to λ 2/3 g or a constant multiple.In view of the ALF condition, it is convenient to set: where the constant k is chosen in such a way that g K is asymptotic at infinity to the product of the standard sphere of radius 1 and the Poincaré cusp of sectional curvature −1 (more detail in [3, Section 2]).
The conformal factor k −1 λ 2 3 is then equal to x 2 1 , where x 1 denotes the moment of the Hamiltonian Killing vector field ∂ t , and the scalar curvature, Scal g K , of the Kähler metric g K is then equal to 6k 2 3 λ 1 3 = 6kx 1 , and tends to 0 at infinity.In particular, g K = x 2 1 g is extremal, even Bach-flat, since it is conformal to an Einstein metric.The constant k is actually the same as the constant k appearing in (2.3)-(2.4),and turns out to be equal to 2A [3, equation (89)].
In terms of the generating function U , the Kähler form, ω K , and the volume form, v g K , of g K have the following expression: From (2.15), we can infer that the volume of (M, g K ) is finite, and the image of the moment map is a convex, pre-compact polytope in the Lie algebra t of the torus T 2 , cf. Figure 2, which is the picture, taken from [3, Section 8], of the moment polytope of the Chen-Teo instanton.Notice that, apart from the dashed edge E ∞ , representing the boundary at infinity, each edge E i , i = 0, 1, . . ., r is associated to the interval (z i , z i+1 ) on the z-axis ρ = 0.The moment with respect of ω K of the Killing vector fields ∂ t and ∂ x 3 -which, in general, don't form a basis of Λ -are denoted by x 1 and µ respectively, with where, we recall, H is defined by (2.5) [3, Proposition 6.1].Notice however that in general ∂ t and ∂ x 3 , regarded as elements of the Lie algebra t of the torus T 2 don't form a basis of the lattice Λ in t induced by T 2 .In restriction to the boundary ρ = 0, the moments are functions of z, with the following expressions on each interval (z i , z i+1 ), where f ′ i ̸ = 0, The expression (2.2) of the metric g holds on the open set, M 0 , where ρ ̸ = 0, i.e., where the torus action is free.In the toric Kähler setting, the boundary ρ = 0 of this open set is formed of (r − 1) compact invariant divisors, isomorphic to 2-spheres, and of two divisors isomorphic to punctured spheres, corresponding to a point at infinity for each of them, encoded by the r + 1 edges of the moment polytope.To each edge of the moment polytope, itself encoded by an interval (z i , z i=1 ), i = 0, . . ., r, is associated a Killing vector field, v i , regarded as an element of t, actually a primitive element of Λ: v i is then the generator of a S 1 -action of period 2π, and vanishes on the corresponding invariant divisor.It follows from (2.12) that v i has the following form: More generally, if the metric admits a conical singularity along the invariant divisor E i , of angle 2πα i , then (2.17) The conditions that (M 0 , g) will smoothly extend to the boundary, possibly with conical singularities of g on the invariant divisors, is that each pair v i , v i+1 be a basis of the lattice Λ, i.e., that each pair be related to the next one by an element of the group GL(2, Z) of 2 × 2 matrices with integer coefficients and determinant equal to ±1, i.e., where the ℓ i are integers and ϵ = ±1, cf.[3, Section 8].
As already mentioned, these conditions turn out to be quite restrictive, in particular impose that r cannot exceed 3.For each value 1, 2 or 3 of r, the only toric Hermitian ALF gravitational instantons are then as follows, cf.Theorems A and 8.
(iii) The Euclidean Schwarzschild metric, discovered by K. Schwarzschild in 1918, with f (z) = m + 1 2 |z + m| + 1 2 |z − m|, which can be viewed as a particular case of Euclidean Kerr metric, with a = 0 and b = m.r = 3: The 1-parameter family of Chen-Teo instantons, discovered by Yu Chen and Edward Teo in 2011, cf.[6], with ) (2.20) In contrast with the previous cases, the Chen-Teo instantons are not the Euclidean form of Lorentzian spaces, and their anti-self-dual Weyl tensor W − is not degenerate, as shown in [1].(0, 0) 2 ), −1 2 .The slope of the edge between V 2 and V 3 is −1 for any value of the parameter p.
More generally, we shall consider smooth completions of the metric g given by (2.2) admitting conical singularities along the invariant divisors, as described by Theorem 7.5 in [3, Section 7].As shown in [3,Section 7], this can be done for the whole family of Kerr-Taub-NUT family, introduced by G.W. Gibbons and M.J. Perry in 1980, which includes the instantons mentioned above when r = 2.
This also concerns the Chen-Teo 4-parameter family introduced in [7], which we shall explore in the next section.

The self-dual Eguchi-Hanson metric
The Eguchi-Hanson metric was first discovered by Tohru Eguchi and Andrew J. Hanson in [10], and by Eugenio Calabi in [4]; it is also a member of the Gibbons-Hawking family of hyperkähler metrics [11].Like the Taub-NUT metric quoted above, also a member of the Gibbons-Hawking family, the Eguchi-Hanson metric is of type O + D − with respect to the orientation determined by the hyperkähler structure, meaning that W + ≡ 0, while W − is degenerate, but non-zero.With respect to the opposite orientation it is then of type D + O − and will then be called the self-dual Eguchi-Hanson metric.This can be written in Harmark form (2.2), with It can be shown that the simple eigenvalue of W + is λ + = 2b2 r 3 and the conformal Kähler metric is then conveniently chosen to be g K =1 r 2 g, whose Kähler class is then ω K = − dr r 2 ∧ (dt + cos θdx 3 ) − 1 r sin θdθ ∧ dx 3 , so that the moment x 1 of ∂ t be equal to 1 r and k = 2b 2 .Unlike the self-dual Taub-NUT metric, the self-dual Eguchi-Hanson metric is ALE, not ALF, but its generating function, U EH , is nevertheless of the same type (2.6) as the generating functions of the gravitational instantons considered in this note, namely , and the corresponding piecewise affine function is then It may be observed that the positive constant A appearing in the general expression (2.1) is here equal to 0 and that the identity k = 2A is here no longer valid, showing again that the self-dual Eguchi-Hanson metric does not belong to the family of gravitational instantons considered in this paper.It may however be viewed as a limit, as already observed by Don Page in [15], cf. also [3, Section 7.2].In the current setting, this can be viewed by considering the following one-parameter family of metrics encoded by their piecewise affine functions of the form normalized by the condition A + b = 1, cf.Remark 2.1, with A, b ≥ 0; notice that f 1 = f 2 = 1, and that most metrics in this family have conical singularities along invariant divisors, of angles 2πα i , i = 1, 2, 3.The vector fields attached to the corresponding polytopes, cf.(2.16)-(2.17),are then v 0 = −α 0 (∂ , and the regularity condition is then: , for some integer ℓ and ϵ = ±1; we then have α 0 = ϵα 2 , hence ϵ = 1, α 0 = α 2 , and we can actually assume α 0 = α 2 = 1, and ℓα = A(F 2 − F 0 ), by setting α 1 = α, hence, by (2.14) , where we can assume that ℓ is equal to 1, 0 or −1.When A runs in the open interval (0, 1), the corresponding metric is smooth in the following three cases: , corresponding to the "positive" Taub-bolt metric, the Schwarzschild metric and the "negative" Taub-bolt metric respectively. 1 When A ∈ 0, 1 2 we can take ℓ = −1, the topology is that of the negative Taub-bolt metric (the total space of O(1)), the angle 4π(1 − 2A) goes from 0 when A → 1 2 to 4π when A → 0, hence the metric tends to the pull-back, from O(2) to O(1), of the self-dual Eguchi-Hanson metric, with a conical singularity of angle 4π.When A ∈ 1 2 , 1 , we have ℓ = 1, the topology is that of the positive Taub-bolt metric (the total space of O(−1)), again the angle 4π(2A − 1) goes from 0 when A → 1 2 to 4π when A → 1.The limit for A = 1 is the Taub-Nut metric on R 4 which is generated by the function f 1 (z) = 1 + |z|.
There is a symmetry around A = 1 2 : the metrics for A = 1 2 ± a are the same with the orientation reversed, up to scale.So it may seem curious that the limits for A = 0 and A = 1 are the selfdual Eguchi-Hanson metric and the Taub-NUT metric.This contradiction is solved by understanding that these are limits at different scales: the Taub-NUT metric is obtained when A → 1 by shrinking the 2-sphere to a point, and by rescaling there is a bubble which is O(−1) with the 2-cover of the Eguchi-Hanson metric.This is precisely what we see on the other side A → 0, with the opposite orientation.
Finally notice the change of topology and of orientation at A = 1 2 , ℓ = 0, b = 1 2 , encoding the (Riemannian) Schwarzschild metric, which lives on the product S 2 × R 2 , with its natural two orientations.

The Chen-Teo family
The Chen-Teo 4-parameter family is actually relevant to the general treatment of the preceding section, i.e., is included and probably coincides with the family of toric Hermitian ALF gravitational instantons, with r = 3, when the z-axis admits 3 angular points, z 1 < z 2 < z 3 .
The convex piecewise affine function f as then the following general form: where are the slopes of f , on the open intervals (−∞, z 1 ), (z 1 , z 2 ), (z 2 , z 3 ), (z 3 , ∞) respectively.The pair (p, q) then belongs to the open domain of R 2 defined by We denote f 1 := f (z 1 ), f 2 := f (z 2 ), f 3 := f (z 3 ), and, in addition to p, q, we introduce two positive parameters a, b by Alternatively, Notice that the parameters a, b, as well as the parameters p, q, are insensitive to the transform described in Remark 2.1.From (3.1), we get For further use, it will be convenient to "normalize" the convex piecewise affine function f (z), via the transform described in Remark 2.1, in order that z 2 = 0 and The convex piecewise affine function f (z) is then given by (3.1), with

Regularity
As in the introduction, we call a metric regular if on some smooth compactification it has at worst conical singularities.In order to test the regularity of these metrics, we introduce the angles 2πα 0 , 2πα 1 , 2πα 2 , 2πα 3 , attached to each divisor, where the α i are all positive, and we consider the corresponding Killing vector fields, when p ̸ = 0, q ̸ = 0, where, for i = 0, 1, 2, 3, F i denotes the (constant) value of F in the interval (z i , z i+1 ) on the axis ρ = 0, cf.[3, Lemma 7.1].The regularity conditions are then where ℓ 1 , ℓ 2 are integers, and ϵ 1 , ϵ 2 are equal to ±1, hence or else, in view of (3.4), ) ) In view of (3.5), in (3.7) the F i may be replaced by F i +c for any constant c, and likewise in (3.8) in view of (3.6).Also recall, cf.[3, Proposition 7.3], that the F i are related by In particular, As observed above, in view of (3.5)-(3.6),(3.7)-(3.8)can be rewritten as Since α 1 and α 2 are both positive, it follows that or, equivalently, In view of (3.9)-(3.10),n, defined by (3.13), has the following expression: and will be called the normalized total NUT-charge, cf.[7, Section III.B].By (3.13), n is then an integer, whenever the metric is regular.Notice that n = 0 if and only if F 3 − F 0 = 0, i.e., if and only if the metric is AF.
Remark 3.2.Notice that (3.14) can be rewritten as It follows from (3.14) and (3.2) that n is well defined at p = 0 or q = 0 and that the quantity q has the sign of n.In particular, a regular metric is AF if and only if the parameters p, q, a, b are related by a + b − (a−1) p − (b−1) q = 0. From (3.5)-(3.8),we infer ) In view of (3.9), we then get from which we infer since the pair (v 0 , v 1 ) is a basis of the lattice Λ.From (3.15)-(3.17)and (3.9), we easily infer that the integers ℓ 1 , ℓ 2 can be rewritten as From (3.18) and (3.19), the conical parameters α 1 , α 2 are given by while the relations (3.5), (3.6), (3.11) and (3.12) are expressed by ) By using the expressions of α 1 , α 2 given by (3.23), it is easily checked that these relations are all satisfied.So far, we assumed that pq ̸ = 0.In view of (3.3), the cases when p = 0, q > 0 or q = 0, p > 0 are then obtained by continuity.When p tends to 0, then q > 0, since p + q > 0, and and Similarly, when q tends to 0, then p > 0 and Recall that a metric of the Chen-Teo 4-parameter family is said to be regular if it can be smoothly compactified along the invariant divisors, D i , encoded by the edges E i of the momentum polytope, i = 0, . . ., r, with a suitable choice of conical singularities of angles 2πα i along each D i .In view of the above, this happens if and only if we can choose α 0 , α 1 , α 2 , α 3 , all positive, satisfying the conditions (3.24)-(3.27),hence, equivalently, the conditions (3.21)-(3.23), in fact (3.21)-(3.22)only, since α 1 and α 2 are then be defined by (3.23).We first observe that, in this case, it follows from (3.23) that α 1 and α 2 are completely determined by α 0 and α 3 , as p + q > 0; moreover, only the quotient α 0 /α 3 is relevant, so that we can arrange that, say, α 0 = 1.This being understood, we can formulate the following statement: Theorem 3.3.Let (M, g) be an element of the 4-parameter Chen-Teo family, of parameter p, q, a, b; let n be the total NUT-charge of g: (1) (M, g) is regular (that is has a smooth compactification on which it has at worst conical singularities) if and only if n is an integer.
(2) If (M, g) is regular, then the boundary at infinity is diffeomorphic to L, where L is (i) a lens space of type ℓ/n, where ℓ is a factor Proof .(i) We already know that n is an integer whenever g is regular.For the converse, in view of the above, we simply have to show that if n is an integer there always exist α 0 , α 3 positive, in fact only α 3 > 0 if we choose α 0 = 1, satisfying the conditions (3.21)- (3.22),where ℓ 1 , ℓ 2 is some pair of integers such that n = ℓ 1 ℓ 2 − 1.From (3.14), we infer that If n ≥ 0, hence ℓ 1 ℓ 2 > 0, it follows from (3.28) that either 1+ aq(1−p) p+q n and 1+ bp(1−q) p+q n are both positive or both negative; the second case is in fact excluded, due to the constraints on the parameters p, q, a, b: indeed, since n ≥ 0, if p > 0, q > 0, then 1 + aq (1−p)  p+q n and 1 + bp(1−q) p+q n are clearly both positive, and this is still the case if p ≥ 0, q < 0, because of (3.29), or if p < 0, q ≥ 0, because of (3.30).Thus, ℓ 1 , ℓ 2 are both positive, and α 3 is then defined by (3.21)- (3.22).
The limit for the angle 4π is again obtained by blowing down the S 2 to a point and is a Kerr metric.The limit for the angle 2π(1 − q) is a Kerr-Taub-bolt metric with a conical singularity: it changes the topology at infinity because there is a bubble at infinity.The special case q = 0 was already studied in Section 3.3, case (iv).

Figure 1 .
Figure 1.The piecewise affine function of the Chen-Teo metric.