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Yang-Mills Instantons from Gravitational Instantons

We show that every gravitational instantons are SU(2) Yang-Mills instantons on a Ricci-flat four manifold although the reverse is not necessarily true. It is shown that gravitational instantons satisfy exactly the same self-duality equation of SU(2) Yang-Mills instantons on the Ricci-flat manifold determined by the gravitational instantons themselves. We explicitly check the correspondence with several examples and discuss their topological properties.


Introduction
An instanton in gauge theories is a topologically nontrivial solution described by a self-dual or antiself-dual connection with a finite action. Such instantons play an important role in the nonperturbative dynamics of gauge theories, in particular, to understand the vacuum structure of quantum field theories [1]. One of the most powerful uses of instantons in recent years is in the analysis of strongly coupled gauge dynamics where they play a key role in unraveling the plexus of entangled dualities that relates different theories. One of the highlights is the remarkable theory of Seiberg and Witten [2] which determines the low-energy behavior of N = 2 supersymmetric gauge theories exactly. In N = 2 supersymmetric gauge theories, the instantons lead to quantum corrections for the metric on the moduli space of vacua.
A semi-classical evaluation of the path integral requires us to find the complete set of finite-action configurations which minimize the Euclidean action. In pure Yang-Mills theory, the complete set of self-dual gauge fields of arbitrary topological charge k can be obtained by solving some quadratic matrix equations, known as the Atiyah-Drinfeld-Hitchin-Manin (ADHM) equations [3], which are a set of nonlinear algebraic equations constraining a matrix of moduli parameters. It can be shown [4] that the functional integral in the semi-classical approximation reduces to an integral over the instanton moduli space in each instanton sector. In principle the low-energy effective action can also be calculated from first principles via conventional semi-classical methods using instantons.
It is known that the instanton calculus in supersymmetric theories is fully controllable when the theories are weakly coupled. This leads to the idea of testing the Seiberg-Witten theory by calculating the instanton effects and comparing these expressions with those extracted from the Seiberg-Witten curve. For reviews, see, for example, [4,5]. Since the integral over a generic instanton moduli space is too complicated to be done directly, it was fully accomplished only recently by using the localization technique and considering the resolution of the instanton moduli space via the ADHM construction relevant to a noncommutative gauge theory [6]. It has been checked [7] that the results computed using the method of localization perfectly agree with the Seiberg-Witten solution for N = 2 supersymmetric gauge theories.
On the mathematical side, instantons lie at the heart of the recent works on the topology of fourmanifolds [8]. In particular, Donaldson used the moduli space of instantons over a differentiable four-manifold to construct topological invariants of the four-manifold and showed that the moduli spaces of instantons often carry nontrivial and surprising information about the background manifold.
One would like to extend the path integral approach to include gravitation. Although the Euclidean gravitational action is not positive-definite even for real positive-definite metrics, one can evaluate the functional integral by first looking for non-singular stationary points of the action functional and expand about them. Such critical points are finite action solutions to the classical field equations called "gravitational instantons," the gravitational analogue of Yang-Mills instantons [9]. These are defined as complete, non-singular, and positive-definite metrics which are self-dual or anti-self-dual metrics of vacuum Einstein equations [10]. One can show [11] that the self-dual or anti-self-dual metrics are local minima of the action among metrics with zero scalar curvature.
In general relativity, the Lorentz group appears as the structure group acting on orthonormal frames in the tangent space of a Riemannian manifold M [12]. Under a local Lorentz transformation which is the orthogonal rotation group O(4), a matrix-valued spin connection ω (More details will be explained in Sect. 3.) Since the group O(4) is a direct product of normal subgroups SU(2) L and SU(2) R , i.e. O(4) = SU(2) L × SU(2) R , the four-dimensional Euclidean gravity, when formulated as the O(4) gauge theory, will basically be two copies of SU(2) gauge theories.
As we summarized above, Yang-Mills instantons are important to determine the vacuum structure of quantum field theories and the ADHM construction provides a description of all instantons on R 4 in terms of algebraic data. One would expect that gravitational instantons play a similar substantial role in quantum gravity although the quantum aspect of general relativity has encountered long-standing difficulties because there is hardly any common ground between general relativity and quantum mechanics. The well-known divergences in a quantum theory of gravity suggest that a field theory of gravity like as Einstein's general relativity is a purely low-energy or large-distance approximation to some more fundamental theory. Therefore, the gauge theory formulation of gravity may be helpful to glimpse some basic structures of such a fundamental theory because nonperturbative and quantum aspects about gauge theories are relatively well-known.
Whereas gravity is different from gauge theory in several marked ways, underlying mathematical structures are very similar to each other in many ways [10]. See, for example, the Table 1 in [13]. It was shown [14,15,16,17] that certain classes of gravitational instantons such as the asymptotically locally Euclidean (ALE) and the asymptotically locally flat (ALF) hyper-Kähler four-manifolds can be constructed as a hyper-Kähler quotient of a finite-dimensional Euclidean space. 1 This construction is actually akin to the ADHM construction of Yang-Mills instantons on R 4 [3] and has a natural interpretation in terms of D-branes in string theory. Moreover, the hyper-Kähler quotient construction of Yang-Mills instantons on an ALE or ALF space [19,20,21,22] is a natural generalization of the original ADHM construction of instantons on flat space. The study of Yang-Mills theories on a curved manifold has recently received renewed attention because they are involved with effective field theories of D-brane and NS5-brane configurations [23,24]. Now our motivation of this paper has surfaced. In this paper and its sequels, we wish to go beyond a mere formal analogy between gravity and gauge theory and try to answer to the following questions: A. What is the precise relation between gravity and gauge theory variables? B. How much are they parallel? C. How is the topology of a Riemannian manifold M encoded into gauge fields? D. What are crucial differences? E. Can it be applied to examine a quantum nature of gravity? The paper is organized as follows. In Section 2, we will summarize Yang-Mills instantons on a curved four-manifold to set our notation and explain why Yang-Mills instantons on a Ricci-flat manifold is a solution of the coupled equations in Einstein-Yang-Mills theory.
In Section 3, we will employ the decomposition in [25] to explicitly realize that the Lorentz group O(4) is a direct product of normal subgroups SU(2) L and SU(2) R , i.e. O(4) = SU(2) L × SU(2) R . 2 It is then easy to show [27] that the four-dimensional Euclidean gravity, when formulated as the O(4) gauge theory, will basically be two copies of SU(2) gauge theories. In particular, it can be shown that one of SU(2)'s decouples from the theory when considering self-dual or anti-self-dual metrics called gravitational instantons. As a result, one can show that gravitational instantons satisfy exactly the same self-duality equation of SU(2) Yang-Mills instantons on the Ricci-flat manifold determined by the gravitational instantons themselves. Therefore, every gravitational instantons can be interpreted as self-gravitating SU(2) Yang-Mills instantons although the reverse is not necessarily true. This provides a powerful method to find a particular class of Yang-Mills instantons on a general self-dual four manifold.
In Section 4, we will elucidate with explicit examples how it is always possible to find Yang-Mills instantons on a Ricci-flat manifold M using the prescription in Section 3 whenever a gravitational instanton solution is given. Our method vividly realizes the Charap-Duff prescription [27] for SU(2) Yang-Mills instantons on a Ricci-flat manifold (see also [28]). We will easily reproduce already known solutions in literatures [29,30,31,32] in this way and also find new Yang-Mills instantons as a byproduct.
In Section 5, some issues about topological invariants for Riemannian manifolds will be discussed. In the gravity side, there are two topological invariants [10] known as the Euler characteristic χ(M) and the Hirzebruch signature τ (M), while, in the gauge theory side, there is a unique topological invariant up to a boundary term given by the Chern class of gauge bundle. The correspondence between gravitational and Yang-Mills instantons then implies that the two topological invariants for gravitational instantons should be related to each other. We conjecture a possible relation between χ(M) and τ (M) by inspecting several known results in literatures [33,34,35].
In Section 6, we draw our conclusions and discuss open issues for future works. Finally, we set up our index notation which is especially useful for the explicit calculation in Section 4; otherwise diverse spaces we are considering would lead to some confusions.

Yang-Mills Instantons on Riemannian Manifold
Consider a curved four-manifold M whose metric is given by Let π : E → M be an SU(2) bundle over M whose curvature is defined by where we choose an unconventional normalization TrT a T b = −4δ ab for later purpose. Let us introduce at each spacetime point in M a local frame of reference in the form of four linearly independent vectors (vierbeins or tetrads) The above pairing leads to the relation In terms of the non-coordinate (anholonomic) basis in Γ(T M) or Γ(T * M), the metric (2.1) can be written as (2) gauge fields in the non-coordinate basis takes the form where we used the structure equation The frame basis E A = E M A ∂ M ∈ Γ(T M) satisfies the Lie algebra under the Lie bracket are the structure functions in (2.8).
Consider SU(2) Yang-Mills theory defined on the Riemannnian manifold (2.1) whose action is given by The self-duality equation for the action (2.11) can be derived by observing the following identity where ε M N P Q is the metric independent Levi-Civita symbol with ε 1234 = 1. Note that the second term in Eq. (2.12) is a topological term (total derivative) and so does not affect the equations of motion. Because the first term in Eq. (2.12) is positive-definite, the minimum of the action (2.11) can be achieved by the self-dual gauge fields (instantons) satisfying In the non-coordinate basis, the self-duality equation (2.13) can be written as the form . It is easy to check that the SU(2) instantons defined by (2.13) automatically satisfy the equations of motion g M N D M F N P = 0 (2.15) because we have the following relation from the self-duality (2.13) where we used the Bianchi identity for the SU(2) curvature (2.2), i.e.
The covariant derivative in (2.15) is with respect to both the Yang-Mills and gravitational connections, i.e.
where Γ M N P is the Levi-Civita connection.
Now the problem we pose here is how to construct instanton solutions satisfying (2.13). Several questions immediately arise. Is it possible to find an instanton solution satisfying (2.13) on an arbitrary Riemannian manifold ? Or is there any constraint on the background manifold for the existence of Yang-Mills instantons ? What is the moduli space of SU(2) instantons defined on a given fourmanifold M ?
We will show that a large class of Yang-Mills instantons satisfying (2.13) or (2.14) can be solved by gravitational instantons. To be precise, we will show that every gravitational instantons satisfy the self-duality equation (2.13) for SU(2) gauge fields on a Riemannian manifold defined by the gravitational instanton itself. To prepare our setup, let us consider the case when SU(2) Yang-Mills and gravitational fields are both dynamically active. The total action is defined by where the Yang-Mills action S Y M is given by (2.11) and the gravitational action is given by The gravitational field equations read as

Gravitational Instantons
Under local frame rotations in O(4), the vectors transform according to and the covariant derivative is defined by 3) The connection one-forms ω A B = ω M A B dx M satisfy the Cartan's structure equations [12], where T A are the torsion two-forms and R A B are the curvature two-forms. In terms of local coordinates, they are given by Now we impose the torsion free condition, to recover the standard content of general relativity, which eliminates ω M as an independent variable, i.e., where f ABC are the structure functions given by (2.10). The spin connection (3.8) is related to the Levi-Civita connection as follows Since the spin connection ω M AB and the curvature tensor R M N AB are antisymmetric on the AB index pair, one can decompose them into a self-dual part and an anti-self-dual part as follows [25,26] where the 4 × 4 matrices η a AB andη a AB for a =1,2,3 are 't Hooft symbols defined bȳ Note that the 't Hooft matrices intertwine the group structure of the index a with the spacetime structure of the indices A, B. We list some useful identities of the 't Hooft tensors [25,26] η (±)a AB ≡η a AB . Of course all these separations are due to the fact, O(4) = SU(2) L × SU(2) R , stating that any O(4) rotations can be decomposed into self-dual and anti-self-dual rotations. To be explicit, for where λ a (+) (x) and λ a (−) (x) are local gauge parameters in SU(2) L and SU(2) R , respectively. To be specific, let us introduce two families of 4 × 4 matrices defined by According to the definition (3.12), the matrix representation of the generators in (3.21) is given by Then Eqs. (3.17) and (3.18) immediately show that T a ± satisfy SU(2) Lie algebras, i.e., According to the definition (3.21), the self-duality (3.13) leads to the important relation The 't Hooft matrices in (3.21) are two independent spin s = 3 2 representations of SU(2) Lie algebra. A deep geometrical meaning of the 't Hooft symbols is to specify the triple (I, J, K) of complex structures of R 4 ∼ = C 2 as the simplest hyper-Kähler manifold for a given orientation. The triple complex structures (I, J, K) form a quaternion which can be identified with the SU(2) generators T a ± in (3.21) [26].

Now we introduce an O(4)-valued gauge field defined by
M dx M (a = 1, 2, 3) are connection one-forms on M and T a ± are Lie algebra generators of SU(2) L and SU(2) R satisfying (3.24). The identification we want to make is then given by Since the group SO(4) is a direct product of normal subgroups SU(2) L and SU(2) R , i.e. SO(4) = SU(2) L × SU(2) R , we take the 4-dimensional defining representation of the Lorentz generators as follows where T a + and T a − are the SU(2) L and SU(2) R generators given by Eq. (3.21). It is then easy to check using Eqs. (3.24) and (3.19) or Eq. (3.14) that the generators in Eq. (3.27) satisfy the Lorentz algebra.
According to the identification (3.26), SU(2) gauge fields can be defined from the spin connections That is, we get the decomposition (3.10) for spin connections. Using the definition (3.21), the spin connection (3.10) and the curvature tensor (3.11) can be written as follows: Using the Lie algebra (3.24), one can write the field strength (3.31) as the component form which is precisely the same as Eq. (2.2). Therefore, we see that A M T a ± can be identified with SU(2) L,R gauge fields and F ∈ SU(2) L,R are group elements defined by Eq. (3.20). Let us recall the symmetry property of curvature tensors determined by the properties about the torsion and the tangent-space group Also note that the curvature tensors satisfy the first Bianchi identity which is an integrability condition originated by the fact that the spin connections (3.8) are determined by potential fields, i.e., vierbeins. It is easy to see that the following symmetry can be derived by using Eqs. (3.34) and (3.35) The gravitational instantons are defined by the self-dual solution to the Einstein equation Note that a metric satisfying the self-duality equation  A solution of the half-flat equation F (±) = 0 is given by A (±) = Λ ± dΛ −1 ± and then Eq.(3.33) shows that it is always possible to choose a self-dual gauge A (±)a = 0. Therefore, one can see the following important property. If the spin connection is, for example, self-dual, i.e. A (−) M is a pure gauge. In other words, in this self-dual gauge, the problem of finding gravitational instantons is equivalent to one of finding self-dual spin connections [13] which is equivalent to the (anti-)self-dual gauge condition A (±)a M = 0 according to the decomposition (3.29). The gravitational instantons defined by Eq.(3.37) are then obtained by solving the first-order differential equations defined by (3.39).
The self-duality equations (3.37) are imposed on the second group indices [CD] of the curvature tensor R ABCD and they do not touch the first group indices [AB]. But note that the first Bianchi identity (3.35) reshuffles three indices in R ABCD and the symmetry (3.36) is consequently deduced. Thereby the self-duality condition for the second group should necessarily be correlated to the one for the first group [27]. In other words, because the Riemann curvature tensors satisfy the symmetry property (3.36), the gravitational instanton (3.37) is equivalent to the self-duality equation Then, using the decomposition (3.30) again, one can similarly show that the gravitational instanton (3.40) can be understood as an SU (2) Yang-Mills instanton defined by (2.14), i.e.
M N are defined by Eq. (2.7). In a coordinate basis, the self-duality equation (3.41) can be written as the form (2.13) because one can deduce that where √ g = detE A M . Therefore, we see that gravitational instantons defined by Eq. (3.37) are solutions of both (2.13) and (2.23) and so they can be regarded as Yang-Mills instantons in the sense that the self-duality equation of gravitational instantons can always be recast into exactly the same self-duality equation as the SU (2) Yang-Mills instantons on a Ricci-flat manifold. But note that the Yang-Mills instantons as well as the four-dimensional metric used to define Eq. (3.42) are simultaneously determined by gravitational instantons. Therefore, the self-duality in Eq. (3.42) cannot be interpreted as SU (2) instantons in a fixed background. Although every gravitational instantons satisfy the self-duality equation (2.13) for Yang-Mills instantons on a Ricci-flat manifold, the converse is not necessarily true: An SU(2) instanton on a Ricci-flat manifold is not always a gravitational instanton. For example, Yang-Mills instantons on ALE spaces in [19,20] and ALF spaces in [21,22] consist of a more general class of solutions than those obtained from ALE and ALF gravitational instantons.
As was pointed out above, the self-duality in Eq. (3.42) should not be interpreted as SU(2) instantons in a fixed background because we are solving the coupled equations (2.13) and (2.23). We are not solving Eq. (2.13) on a non-dynamical background manifold. Note that the Yang-Mills action (2.11) is invariant under the conformal transformation and, on the right-hand side, it is given by Eq. (2.18). Rewrite the covariant derivative (3.45) as the form Using the decompositions (3.29) and (3.30) and the commutation relations (3.24), we get the covariant derivative is taken on both sides to yield will be satisfied accordingly. So remarkably it turns out that gravitational instantons can actually be identified with Yang-Mills instantons in the sense that the gravitational and Yang-Mills instantons satisfy mathematically the same self-duality equations. But, as we discussed before, the self-duality equation (3.42) must be interpreted as self-gravitating Yang-Mills instantons rather than SU(2) instantons on a rigid background.

Yang-Mills Instantons from Gravitational Instantons
We showed in the previous section that every gravitational instantons satisfy the self-duality equation (2.13) on a Ricci-flat manifold defined by the gravitational instanton itself. We have constructed SU(2) gauge fields as the projection of the spin connection (3.2) onto the self-dual part and the antiself-dual part by using the 't Hooft symbols. The embedding to relate gauge and spin connections was suggested long ago by Charap and Duff [27]. (See also [28].) In this section, we will elucidate with explicit examples how Yang-Mills instantons can be obtained from gravitational instantons.
To be specific, we want to find Yang-Mills instantons satisfying Eq. (2.13) where the background metric g M N is a gravitational instanton obeying Eq. (3.40). First, we will calculate the spin connection (3.8) for a given gravitational instanton metric and then identify SU(2) gauge fields A M according to the identification (3.26). As was shown in (3.42), the corresponding field strength F M N of the SU(2) gauge fields automatically satisfies the self-duality equation (2.13) on a curved manifold M whose metric is given by the gravitational instanton itself.
We will easily reproduce already known solutions in literatures [29,30,31,32] in this way. As a byproduct, we will also find new Yang-Mills instantons on a curved manifold M. It might be emphasized that it is always possible to find Yang-Mills instantons on a Ricci-flat manifold M by the same procedure whenever a gravitational instanton M is given, as will be illustrated with several examples. Here we refer to the index convention in Section 1.

Gibbons-Hawking metric
The Gibbons-Hawking metric [45] is a general class of self-dual, Ricci-flat metrics with the triholomorphic U(1) symmetry which describes a particular class (A-type) of ALE and ALF instantons. The Gibbons-Hawking metric for gravitational multi-instantons is given by where with ǫ = 0 for ALE instantons and ǫ = 1 for ALF instantons. Here we use the world index M = (i, 4 = τ ) with i = 1, 2, 3 and the frame index A = (î,4) withî =1,2,3. Note that ψ = ψ(x), q i = q i (x) and the Killing vector ∂/∂τ generates the triholomorphic U(1) symmetry.
One can easily read off the vierbeins from the metric (4.1) as Using the torsion-free condition, Similarly, the spin connections and the structure functions can be obtained as follows Note that we are explicitly discriminating the three-dimensional world and frame indices as (i, j, k, · · · ) and (î,ĵ,k, · · · ), respectively. It is easy to see that the self-duality equation (3.39) for the spin connection (4.6) is reduced to the equation Using the result (4.8), one can now read off the self-dual SU(2) gauge fields defined by ω AB = A a η a AB : That is, with the notation Eˆiψ = e ψ ∂ i ψ ≡ ∂ˆiψ, It is easy to derive the following relation from Eq. (4.8) Using the above results, one can get the field strengths for SU(2) gauge fields (4.9) Now it is straightforward to check that the above SU(2) field strengths are self-dual, i.e. (4.14) To be specific, one can explicitly see that where the relation (4.11) was used. In terms of the harmonic function in Eq. (4.2), the above field strength can be represented by and Eq. (4.11) can be written as It would be interesting to compare Eq. (4.17) (after the replacement ∂ i → ∂ M since the function V (x) does not depend on τ ) with the 't Hooft ansatz A a µ =η a µν ∂ ν log φ(x) for SU(2) multi-instantons (see Eq. (4.60b) in [1]) satisfying 3 ∂ µ ∂ µ log φ + ∂ µ log φ∂ µ log φ = 0.

Taub-NUT metric
The Taub-NUT metric is the simplest ALF space described by the Gibbons-Hawking metric (4.1) with ǫ = 1 and k = 1. Using the spherical coordinates, it is given by with the coefficients c 1 = c 2 = c 3 given by The Maurer-Cartan one-forms {σ i } satisfy the following exterior algebra [10] dσ i + 1 2 εˆiĵkσ j ∧ σ k = 0. The vierbein bases are given by and where κ i are the basis vectors dual to σ i , i.e. σ i , κ j = δ i j , satisfying Here we indicate no summation convention for the index i with the notation (NS[i]). The spin connections read as Note that the spin connections in Eq. (4.25) are not completely self-dual, but the anti-self-dual part is simply given by ω AB σ a 2 and so their curvature tensors identically vanish thanks to Eq. (4.21). The curvature tensors are so self-dual, i.e. R AB = F a η a AB , which are given by The corresponding SU(2) gauge fields can be identified from (4.25) as Therefore, the field strength of the SU(2) gauge fields (4.27) can be calculated to be (4.29) Note that the SU(2) field strengths in (4.28) are self-dual, i.e. F = * F , which, of course, coincide with the curvature tensor (4.26).
Our result here agrees with the self-dual gauge fields in [31,32].

Eguchi-Hanson metric
The Eguchi-Hanson metric [46] is the simplest ALE space described by the Gibbons-Hawking metric (4.1) with ǫ = 0 and k = 2. Let us consider the metric given by with the function h(r) = 1 − a 4 /r 4 . The Maurer-Cartan one-forms {σ i } satisfy the exterior algebra which are clearly self-dual. The curvature tensors are straightforwardly computed by The self-dual curvature tensors for the Eguchi-Hanson metric (4.30) can be determined by SU(2) gauge fields A a = 1 The corresponding SU(2) field strength coincides with the curvature tensor R AB = F a η a AB in (4.33) where F a = dA a − ε abc A b ∧ A c and they are given by Our result here agrees with the self-dual gauge fields in [29,32].

Atiyah-Hitchin metric
The Atiyah-Hitchin metric [47] describes a four-dimensional hyper-Kähler manifold with SO(3) isometry that was introduced to describe the moduli space of SU(2) BPS monopoles of magnetic charge 2. Let us consider the Bianchi type IX space [11] which is locally described by the metric with an SU(2) or SO(3) isometry group where a τ = a 1 a 2 a 3 and a i 's are functions solely of τ . The self-dual conditions for all Bianchi IX solutions are given by the equations where three constant numbers α i , i = 1, 2, 3, satisfy α i α j = εˆiĵkα k . Choosing (α 1 , α 2 , α 3 ) = (1, 1, 1) will lead to the Atiyah-Hitchin metric [47] while (α 1 , α 2 , α 3 ) = (0, 0, 0) yields the Eguchi-Hanson type I or II metric [46]. Identify the vierbein basis from the metric (4.37) without summation convention for the index i. The left-invariant 1-forms {σ i } on S 3 satisfy the exterior algebra (4.31) and the dual basis vectors {κ i } satisfy the Lie algebra [κ i , κ j ] = −εˆiĵkκ k . Note that the metric (4.37) has the same structure as the Taub-NUT metric (4.19). Therefore, the spin connections also have the same structure as follows where the prime means the derivative with respect to τ . Note that the spin connections in (4.40) are not self-dual in general. One can check using Eq. (4.38) that the spin connections in Eq. (4.40) satisfy the following relation (NS[a]). Let us define SU(2) gauge fields as follows Our previous result (3.41) implies that the field strengths F a = dA a − ε abc A b ∧ A c defined by the SU(2) gauge fields in (4.42) are necessarily self-dual. Now we will show that it is the case. It is straightforward to calculate the SU(2) field strength where a a ≡ a ′ a /a τ . Using the identity [48] derived from Eq. (4.38), 44) we see that the field strength (4.43) has the correct self-dual structure, i.e.

Real heaven
The real heaven metric [49] describes four dimensional hyper-Kähler manifolds with a rotational Killing symmetry which is also completely determined by one real scalar field. The metric is given by where q α = −ε αβ ∂ β ψ, (α = 1, 2) and the function ψ(x) is independent of τ and satisfy the threedimensional continual Toda equation The rotational Killing vector is given by c α ∂ α ψ∂/∂τ with constants c α . We identify the vierbein vectors as From the torsion-free equation It is straightforward to check that the self dual relations, ω31 = ω24 and ω23 = ω14, are satisfied if and only if the continual Toda equation (4.47) is satisfied. However, the relation ω12 = ω34 is not satisfied. In order to cure this mismatch, first note that we can perform the local frame rotation (3.1) as follows The spin connections also transform according to Eq.(3.2) and dτ = −q α e −φα Eα+e φ 4 E4 . Note that the frame rotation (4.51) affects the self-duality condition only for ω34 = ω34 + 1 2 dτ due to the inhomogeneous term (4.53). In other words, ω31 = ω24 and ω23 = ω14 are automatically satisfied thanks to the previous relations. Now it is straightforward to check that ω12 = ω12 = ω34 = ω34 − 1 2 q α e −φα Eα + 1 2 e φ 4 E4 . Therefore, the spin connections in (4.52) become self-dual.
If one introduces SU(2) gauge fields by the corresponding field strengths, F a = dA a −ε abc A b ∧A c , should be self-dual according to the general result (3.41). This can also be proved by using the relation (3.19) which leads to the following result Hence the self-duality of F a results from the self-dual curvature tensors R AB . Or one can check it by a straightforward calculation using Eqs. (4.50) and (4.47) though rather tedious. The self-dual gauge fields in Eq. (4.54) describe a Yang-Mills instanton on the real heaven (4.46), which is a new solution to the extent of our knowledge.

Euclidean Schwarzschild solution
The Euclidean Schwarzschild metric [9] was constructed by the Wick rotation of the Schwarzschild black-hole solution. It is not a gravitational instanton (not a hyper-Kähler manifold) although it is a Ricci-flat manifold. The metric takes the form The radial coordinate is constrained by r ≥ 2m and the time coordinate τ is an angular variable with period 8πm. Hence this solution has the topology R 2 × S 2 . After defining the vierbein basis (E1 = h(r) −1 dr, E2 = rdθ, E3 = r sin θdφ, E4 = h(r)dτ ), it is easy to compute spin connections: where h(r) = 1 − 2m r . The corresponding curvature tensors are given by which are not self-dual anymore although they are Ricci-flat, i.e., R AB ≡ R ACBC = 0.
Because the spin connections in Eq. (4.57) are neither self-dual nor anti-self-dual, we can consider both type of SU(2) gauge fields defined by The field strengths, F (±)a = dA (±)a − ε abc A (±)b ∧ A (±)c , should be either self-dual (for the + sign) or anti-self-dual (for the − sign) because we get the following result which can be derived by using the relation (3.19). According to the general result (3.41), the SU(2) gauge fields in Eq. (4.59) automatically satisfy the self-duality equation (2.13) where the background geometry is given by the metric (4.56). Therefore, the solution (4.59) indeed describes an SU(2) Yang-Mills (anti-)instanton on the space (4.56). The solution (4.59) was originally found by Charap and Duff [27]. The reason for the revival here is that the solution (4.59) exposes an interesting structure for a Ricci-flat manifold. According to the decomposition (3.29) and (3.30), we see that the Euclidean Schwarzschild metric (4.56) describes the sum of an SU(2) L instanton and an SU(2) R anti-instanton. Therefore, an interesting question is whether this kind of feature is generic or not. Remarkably it can be shown [50] that any Einstein manifold satisfying R AB = Λδ AB for either Λ = 0 or Λ = 0 always arises as the sum of SU(2) L instantons and SU(2) R anti-instantons.

Topological Invariants
The correspondence between gravitational and Yang-Mills instantons now raises an intriguing question about topological invariants in gravity and gauge theories. In the gravity side, there are two topological invariants associated with the Atiyah-Patodi-Singer index theorem for an elliptic complex in four dimensions [10], namely the Euler characteristic χ(M) and the Hirzebruch signature τ (M), which can be expressed as integrals of the curvature of a four dimensional metric while, in the gauge theory side, there is a unique topological invariant up to a boundary term given by the Chern class of gauge bundle. Thus a natural question is how the two kinds of topological invariants for self-dual four manifolds can be related to the Chern class of instanton bundle. In particular, the two topological invariants for gravitational instantons should be related to each other, in other words, Hence the homotopy group of O(4) in the gravity theory is isomorphic to two copies of the additive group of integers Consequently, there are two independent gravitational topological charges [10], i.e., the Euler characteristic χ(M) and the Hirzebruch signature τ (M). The Euler number χ(M) for the de Rham complex and the signature τ (M) for the Hirzebruch signature complex are, respectively, defined by 4 where θ AB is the second fundamental form of the boundary ∂M. It is defined by where ω AB are the actual connection 1-forms and ω 0AB are the connection 1-forms if the metric were locally a product form near the boundary [10]. The connection 1-form ω 0AB will have only tangential components on ∂M and so the second fundamental form θ AB will have only normal components on ∂M. And η S (∂M) is the η-function given by the eigenvalues of a signature operator defined over ∂M and depends only on the metric on ∂M [10]. The topological invariants are also related to nuts (isolated points) and bolts (two surfaces), which are the fixed points of the action of one parameter isometry groups of gravitational instantons [33]. We have verified in the previous sections that, for gravitational instantons, one of the SU(2) factors in (5.3) completely decouples from the theory. Therefore, the topological classification of (anti-)self-dual spin connections will essentially be the same as Eq. (5.2) in the gauge theory. That is the reason why we expect the relation (5.1) for the topological invariants in Eqs. (5.5) and (5.6). Now we will confirm the relation (5.1) explicitly determining the coefficients.
Since θ AB in Eq. (5.7) are antisymmetric on the AB index pair, we will decompose them into a self-dual part and an anti-self-dual part according to Eq. (3.10) We take the normal to the boundary to be (A =4)-direction and so we have θˆiĵ = 0. It is then straightforward to express the topological invariants in terms of SU(2) gauge fields using the decompositions (3.10), (3.11) and (5.8): where we defined the volume forms as E1 ∧ E2 ∧ E3 ∧ E4 ≡ √ gd 4 x and E1 ∧ E2 ∧ E3| ∂M ≡ √ hd 3 x. An interesting pattern appears in the topological invariants. First consider a compact Einstein manifold without boundary, i.e. ∂M = 0. It turns out [50] that F (+)a and F (−)a are self-dual and antiself-dual instantons, respectively. Then we see that the Euler number χ(M) = χ + (M) + χ − (M) does not distinguish self-dual and anti-self-dual instantons since both contribute with equal sign while the Hirzebruch signature τ (M) = τ + (M) − τ − (M) distinguishes self-dual and anti-self-dual instantons. Based on the observation, we can draw general properties about 4-dimensional compact Einstein manifolds where all boundary terms vanish. As we mentioned above, the Euler number χ(M) gets equal sign contributions from self-dual and anti-self-dual gauge fields while the Hirzebruch signature τ (M) is not the case. Thus we see that χ(M) ≥ 0 with the equality only if M is flat. This is the Berger's result [10]. We can further refine the Berger's result by looking at the expressions (5.9) and (5.10): because F (−) describes SU(2) anti-instantons. The inequality (5.11) will be saturated if and only if a compact four-manifold is half-flat, i.e. F (−)a = 0. In the result, we get a general relation where the bound is saturated only for T 4 and K3 surface, which are compact self-dual four-manifolds as either trivial or nontrivial gravitational instantons. This result is known as the Hitchin-Thorpe inequality [10]. For noncompact manifolds, there are additional boundary terms as shown in (5.9) and (5.10) which are not separated into the self-dual and anti-self-dual parts unlike as the volume terms. In particular, the eta-invariant η S (∂M) for k self-dual gravitational instantons [35] is given by where ǫ = 0 for ALE boundary conditions and ǫ = 1 for ALF boundary conditions. Because the topological invariants for a noncompact manifold with boundary have nontrivial boundary corrections, it is not easy to demonstrate the relation (5.1) although such a relation should exist for general half-flat manifolds. But, one may infer by investigating known examples so far that the following relation would be satisfied for noncompact gravitational instantons. It turns out [33,34,35] that ALE instantons including all ADE series and ALF instantons of AD series satisfy the relation (5.14). 5 Therefore, the evidence for the relation (5.14) is overwhelming. Since we believe that the relation (5.1) will be generic independently of asymptotic boundary conditions and topology, we conjecture that the relation (5.14) will be true for general noncompact gravitational instantons. It may be proved by showing the following identity for gravitational instantons, e.g., with F (−)a = 0 and so taking the self-dual gauge A (−)a = 0: Indeed, for ALE and ALF spaces, one can derive the relation χ(M) − τ (M) = 1 − 4I 1 2 (S ± , D) using Eqs. (12), (13), (14) and (20) in [34]. If M has a spin structure, the index of the Dirac operator, I 1 2 (S ± , D), must identically vanish [11], and thus we confirm the above identity. For general cases, we do not know how to rigorously prove the above identity and so we leave it as our conjecture.
The topological invariant in SU(2) gauge theory is given by the second Chern number  Table 2 in [51] for the relation (5.14) of all ALE manifolds. See also the where F a Finally we get the topological numbers for the Taub-NUT space We have used the result (5.13) for the η-invariant with k = 1. In this case, the Euler number (5.24) is equal to the instanton number (5.17) because there is no boundary correction [31,32]. And it is straightforward to check the relation (5.15).

Eguchi-Hanson space
For the product metric Note that we have to choose the angular coordinate ranges to remove the apparent singularities in the metric at r = a. Thus the boundary at ∞ becomes RP 3 .
Then we obtain the following result and get the topological numbers for the Eguchi-Hanson space

Euclidean Schwarzschild solution
This solution is interesting because it has a nontrivial Euler number [9] although it is not a gravitational instanton. But it turns out that this solution is actually the sum of SU(2) L instanton and SU(2) R anti-instanton, which explains why it has a nontrivial Euler number. Take the product metric The second fundamental form at the boundary r = r 0 is then given by Using the result (4.58) with the definition F (±)a = 1 4 η (±)a AB R AB , we obtain It is then straightforward to get the topological invariants [9] χ . Hence we confirm that the Euclidean Schwarzschild solution (4.56) is the sum of an SU(2) instanton and an anti-instanton. And the relation (5.14) implies that τ ± (M) = 0 or η(∂M) = − 2 3 . Therefore the SU(2) instanton for the Euclidean Schwarzschild solution (4.56) has the same topological invariants as the Taub-NUT space (4.19) [27]. Note that two instantons belong to different gauge groups, one in SU(2) L and the other in SU(2) R , and so they cannot decay into a vacuum. As a result, the space (4.56) should be stable at least perturbatively. One may ask whether this kind of feature is special or general. Remarkably it can be shown [50] that any Ricci-flat four-manifold always arises as the sum of SU(2) L instantons and SU(2) R anti-instantons. Hence any Ricci-flat manifold should be stable for the same reason.

Topological invariant of Yang-Mills instantons
We have noticed that the instanton number (5.16) for (anti-)self-dual gauge fields satisfying (2.13) is not necessarily integer-valued because it does not take possible boundary corrections into account. But the equivalence of the self-dual systems in (2.13) and (3.37) implies that we need to also consider boundary contributions for the topological charge of Yang-Mills instantons defined on a curved manifold. Thereby we suggest the Chern number for an instanton bundle including boundary corrections which can be identified with the Euler characteristic (5.9) in the self-dual gauge, A (−)a = 0, with the gauge theory normalization A a Y M = −2A a G and F a Y M = −2F a G and is accordingly integer-valued. Note that the boundary term in (5.37) is precisely the Chern-Simons form for the SU(2) vector bundle at an asymptotic infinity. Now we consider the four-manifold M to have two ends, one at an asymptotic infinity ∂M ∞ and the other at an inner boundary ∂M 0 describing nuts and bolts of gravitational instantons [33]. For example, the inner boundary is at r = m for the Taub-NUT space (4.19) and at r = a for the Eguchi-Hanson space (4.30). Using the identity F a ∧ F a = dK where and the boundary operation ∂M = ∂M 0 − ∂M ∞ , 6 one can rewrite the instanton number (5.37) as the Chern-Simons integral on the inner boundary ∂M 0 , i.e., Then we get a very interesting result that the Chern-Simons integral (5.39) on the inner boundary ∂M 0 simply counts the number of nuts plus the twice of the number of bolts in gravitational instantons: It is easy to check the result (5.41) for the Taub-NUT space (♯(nuts) = 1, ♯(bolts) = 0) and for the Eguchi-Hanson space (♯(nuts) = 0, ♯(bolts) = 1), using the previous results with the relation

Discussion
Let us go back to the questions we have raised in Section 1. So far we have focused on the similarity between gauge theory and gravitation. A main source of the similarity is coming from the fact that the O(4)-valued 1-forms ω A B are gauge fields (a connection of the spin bundle SM) with respect to O(4) rotations as shown in Eq. (3.2). Then the Riemann curvature tensors in (3.7) constitute O(4)-valued curvature 2-forms of the spin bundle SM. Therefore, the four-dimensional Euclidean gravity can be formulated as a gauge theory using the language of the O(4) gauge theory. Via the fact that the Lorentz group O(4) is a direct product of normal subgroups SU(2) L and SU(2) R , i.e. O(4) = SU(2) L × SU(2) R , the four-dimensional Euclidean gravity can be decomposed into two copies of SU(2) gauge theories. In particular, the (anti-)self-dual sector satisfying (3.37) can be formulated as an SU(2) gauge theory, as clearly indicated in Eq. (3.33).
Nevertheless, gravity is different from gauge theory in many aspects. A decisive source of the difference is the existence of a Riemannian metric which does not have any counterpart in gauge theory. We highlight some crucial differences between gauge theory and gravitation with the following The metric is constrained to be covariantly constant with respect to the Levi-Civita connection (3.9) or equivalently the vierbeins are constrained to be torsion-free, i.e., T A = dE A + ω A B ∧ E B = 0. This constraint leads to the result that the spin connections ω A B are determined by potential fields, i.e., vierbeins, as Eq. (3.8). As a result, a primary field for gravity is the metric tensor rather than a gauge field (a connection of vector bundle). This extra structure comprises a core origin of the differences in the above table.
Recently one of us showed [25] (see also recent reviews [26] and [52]) that Einstein gravity can be derived from electromagnetism in noncommutative space. In particular, the vierbeins E A in gravity arise from the leading order of noncommutative U(1) gauge fields and higher order terms give rise to derivative corrections to Einstein gravity. Actually the Einstein equations arising from the noncommutative gauge fields and the resulting emergent gravity motivate to newly address the questions in Section 1 in a more broad context to include noncommutative U(1) gauge theories. For example, it was rigorously shown in [53] that noncommutative U(1) instantons are equivalent to gravitational instantons. Therefore, it will be very interesting to find a precise map between noncommutative U(1) instantons and Yang-Mills instantons because a particular class of Yang-Mills instantons can be obtained from gravitational instantons as was shown in this paper. We hope to draw some valuable insights from this line of thought in our future works. Now our method in Section 3 can easily be generalized to get new instanton solutions by the conformal rescaling method [38]. Suppose that (M, g) is a self-dual gravitational instanton and consider a Weyl transformation given by Eq. We can apply the decompositions (3.10) and (3.11) to the transformed spin connection (6.1) and the corresponding curvature tensor R = d ω + ω ∧ ω, respectively. After all, we will get new SU(2) gauge fields defined by A (+)a = A (+)a + A (+)a , A (−)a = A (−)a (6.2) where A (+)a are the self-dual gauge fields determined by the original self-dual spin connection ω AB and A (±)a ≡ 1 2 η (±)a AB (E B log Ω)E A (6.3) and the corresponding SU(2) field strengths will be given by For the first choice (I), we will get a self-dual Yang-Mills instanton while, for the second choice (II), an anti-self-dual Yang-Mills instanton. Then one can show [50] that, for the case (I), the Ricciscalar R = g M N R M N will identically vanish, i.e. R = 0, but the case (II) seems to give rise to an intriguing manifold satisfying R M N − 1 4 g M N R = 0. Because the Ricci scalar transforms under the Weyl transformation (3.43) as Ω 3 R = ΩR − 6✷ g Ω where ✷ g refers to the scalar Laplacian on (M, g), we see that the rescaling function Ω(x) must be harmonic, i.e. Ω −1 ✷ g Ω = 0, for the case (I), taking into account that R = 0. But the harmonic function Ω(x) will allow mild singularities [38] which can be removed by a gauge transformation.
By the same procedure as Eq. (3.42), the self-dualities in Eqs. (6.5) and (6.6) can be written as where g = Ω 4 √ g. However, taking into account the conformal invariance of self-duality, we get the self-duality equation on the original four-manifold (M, g), i.e., Consequently, we get new Yang-Mills instantons on an original Ricci-flat manifold (M, g) after the Weyl transformation (6.1). More details about explicit solutions obtained in this way and their topological properties will be discussed elsewhere.
In this paper we showed that any gravitational instanton is an SU(2) Yang-Mills instanton on the gravitational instanton itself. Regarding to this property, there is an interesting theorem (Example 3 (page 302) in [19] and see also Sect. 7 in [20]) that there always exists an instanton bundle on an ALE manifold M with the instanton number k = 1 − 1 |Γ| (|Γ| denoting the order of Γ in M ∼ = C 2 /Γ) defined by (5.16) such that the moduli space of self-dual connections on the instanton bundle is a four-dimensional hyper-Kähler manifold and coincides with the base manifold M. Inferred from our result, the above property seems to be true for other self-dual manifolds. To be precise, suppose that M(E → M, k) is the moduli space of self-dual connections on a vector bundle E over M with instanton number k where M is a gravitational instanton. Then, each non-empty, non-compact 4-dimensional component of the moduli space M(E → M, k) is isomorphic to the gravitational instanton itself. It will be interesting to clarify this assertion.