A K\"ahler Compatible Moyal Deformation of the First Heavenly Equation

We construct a noncommutative K\"ahler manifold based on a non-linear perturbations of Moyal integrable deformations of $D=4$ self-dual gravity. The deformed K\"ahler manifold preserves all the properties of the commutative one, and we obtain the associated noncommutative K\"ahler potential using the Moyal deformed gravity approach. We apply this construction to the Atiyah-Hitchin metric and its K\"ahler potential, which is useful in the description of interactions among magnetic monopoles at low energies.


Introduction
Several applications of hyper-Kähler manifolds in four dimensions involve gravitational instantons, non-linear graviton theory and the heavenly equations [1,2,3,4]; they have also been extensively used in supersymmetric field theories [5,6]. Of particular interest is their appearance in topological field theories and string theory, where in some cases the moduli spaces have a hyper-Kähler structure. The existence of such a structure provides a more profound and alternative understanding of a physical system in general.
Even though hyper-Kähler manifolds have been analysed in great detail, there exist examples where the metric has proved to be difficult or impossible to calculate. Nevertheless, an algebraic description of four-dimensional non-compact hyper-Kähler manifolds possessing one abelian isometry was given in [7]. In that work, a self-duality condition for the Killing vector associated to the isometry plays a fundamental role in the analysis and classification manifolds; the translational or rotational character of the isometry translates into the existence of a translational or rotational Killing vector.
Two particular examples of self-dual manifolds with rotational Killing symmetries are the Eguchi-Hanson and the Taub-NUT metrics; both metrics share SO(3) × SO(2) as a larger group of isometries, but these two group factors act differently on each spacetime. SO(3) is a translation symmetry for the Eguchi-Hanson metric with SO(2) acting as a rotational one; the situation is the opposite for the Taub-NUT spacetime. On the other hand, a relevant example of a spacetime admitting only a rotational isometry is the Atiya-Hitchin (AH) metric that arises on the moduli space M 0 2 of BPS SU(2) monopoles [8,9]. As noted in a series of works [8,9,10], the AH spacetime is a useful tool in the description of interactions among magnetic monopoles. More specifically, the AH metric is the metric on the moduli space of charge-two non-abelian magnetic SU(2) monopole with a fixed centre. Its geodesics describe low-energy monopoles interacting through the exchange of massless photons and scalars [8]; at long distances, it reduces to the Taub-NUT space with negative mass parameter [11]. The structure of the AH metric is a four-dimensional hyper-Kähler manifold with SO(3) isometry; the SO(3) group does not rotate the three Kähler forms, and it is the only specific example of a four-dimensional hyperkähler space without tri-holomorphic isometries. Furthermore, the AH spacetime is a self-dual solution to Einstein field equations [12] and all of its Killing vectors lack a self-dual covariant derivative [13]; the isometry group SU(2) is often identified with a supersymmetry group [14].
In general, the metric related to self-dual vacuum solutions to Einstein's field equations is a solution to the first, or the second, heavenly equations [15]. The original motivation for heavenly equations, heavenly metrics and heavenly spaces was the desire to obtain real solutions to Einstein's equations on real manifolds and spawned several papers on the subject and its generalisations [16,17,18,19,20]. Assuming that the anti-self-dual part of the Weyl tensor was algebraically special, the equivalence between the vacuum Einstein field equations in complex spacetime and the heavenly equation was established in [21]. The heavenly equations are also integrable using the twistor formalism [3,22] and several examples are known [23,24]; they are a constant source of research in mathematics and physics.
It is of interest to consider integrable generalisations of the heavenly equation due to its applications to physical systems; its modifications may allow a description of new phenomena or interactions not present in the standard description of a system. For instance, one of the possible generalisations that may be relevant is the one related to the Moyal ⋆-product. Within the context of particles and fields, the Moyal product provides a straightforward generalisation to noncommutative field theories and introduces, for example, new interactions in the Standard Model that may be probed in the laboratory. Several applications in quantum gravity also exist, in particular regarding the issue of the singularities and the thermodynamical properties of solutions in general relativity.
In the case of the first heavenly equation, a Moyal deformation of this equation has been done independently by Strachan [25] and Takasaki [26]. Furthermore, a suitable deformed differential calculus was introduced in [27] to deal with the deformation. When applied to integrable systems, the equations dΩ = 0, for a 2-form Ω provide a concise writing of the integrability conditions for the deformed system. We want to analyse the consequences of a noncommutative structure on the moduli space of interacting magnetic monopoles; we expect that such a deformation gives rise to interactions that may be identified with some already known or produces new ones. For the classical case, the Kähler potential for the AH metric was obtained by Olivier [28] following an approach based on the existence of a η-self-dual Killing vector in conjunction with previous results obtained by Boyer and Finley [29]. In our approach, we consider a Moyal deformation of this Kähler potential, and we require that the deformed potential Ω must share the same features and properties of the classical one.
The plan of the present paper is as follows. In Sec. 2, we review the noncommutative deformation of the Monge-Ampère or first heavenly equation using the Moyal ⋆-product; in this section, we write the equation satisfied by the noncommutative contributions that preserve the first heavenly equation. In Sec. 3, we also recall the anti-self-dual vacuum Einstein equations that determine the structure of complex four-dimensional metrics of Euclidean signature. How integrability is preserved in the Moyal-deformed case for these spaces and the conditions under which we guarantee that the Moyal-deformed potential is Kähler are presented there.
Afterwards, in Sec. 4, we use the Moyal deformed gravity approach to rewrite the deformed Kähler potential Ω in terms of local frame fieldsê a µ (x, θ); we also obtain general expressions for the corresponding deformed metric elements. We discuss the deformed Kähler potential up to first order on the noncommutative parameter θ in greater detail in Sec. 5; through an integration procedure, we explicitly specify the first order modifications to the deformed metric and vierbien. The results in Secs. 4 and 5 are put together in Sec. 6 to analyse the case of the AH metric. There, we obtain the Moyal deformed AH metric as a function of the original AH metric, the corrections to the Kähler manifold and the complex coordinates used in the original formulation by Olivier. We finally end with our conclusions and some remarks on future work.

Moyal deformation of the first heavenly equation
It is well known that the first and second heavenly equations describe the general metric of a selfdual vacuum space-time [15] and that these equations are integrable by twistor methods [3,22,30]. Plebanski [15] showed that complex metrics with a self-dual Riemann tensor, to be referred to as self-dual metrics, can be described in terms of one function Ω, the Kähler potential, satisfying the first heavenly equation This equation is also called Plebanski' s first equation. The Kähler potential Ω is an unknown function of suitable spacetime coordinates x µ := (p, q,p,q); it gives a local expression of self-dual vacuum Einstein spaces. In this section we follow the procedure in [25] to obtain the deformation of the first heavenly equation using Strachan's idea. The starting point is to replace the Poisson bracket by the Moyal bracket defined as 3) The definition of the Moyal ⋆-product in this expression is where ← → P is the Poisson operator Equivalently, we can also write where ǫ µν = δ μ ı δ ν  ǫī; the non-vanishing elements of the Levi-Civita pseudotensor are then ǫpq = −ǫqp = 1. The Moyal product in Eq. (2.4) has the important property that lim θ→0 f ⋆ g = f g. (2.8) In consequence, the Moyal bracket takes the alternative form Thus the Moyal algebra is a deformation of the Poisson algebra. Using the well-known Taylor expansion of the sine fuction we obtain The above formulation can be also implemented by considering a noncommutative matrix with the ⋆-product defined as Notice that if f and g depend only on the pair of coordinates (p, q), then f ⋆ g = 0.
Following [25] and [31], we present the iterative method for constructing a set of differential equations for the Moyal deformation of the first heavenly equation. The idea in [25] was to construct a perturbative solution in powers of θ for the deformed potential Ω. We consider then a series development in powers of θ for Ω Here Ω (n) are functions to be determined and Ω (0) = Ω is the classical Kähler potential. Plugging this expansion into the integrable deformation of Plebanski's (first heavenly) equation and equating the coefficients of the same powers of θ 2 in both sides of this equation, we get the system of differential equations where the product ⋆ 2s+1 has the form If we define r := 2s + m + n, we have i.e., the first heavenly equation. For any r ≥ 1, the expression in Eq. (2.18) gives To analyse modifications to the first heavenly equation, we set r = 1 to obtain Therefore, once Ω (0) is known, Eq. (2.21) becomes a linear partial differential equation for the first order corrections Ω (1) . The deformed potential Ω defined as before is not necessarily Kähler-like, we need to impose additional conditions to guarantee that it will be; these conditions will be discussed in the next section.

Integrable Systems
Several multidimensional integrable systems were discussed in [27] assuming a symplectic manifold with some associated ⋆-product. All these systems share the characteristic feature that they have associated a 2-form Ω which satisfies the equations dΩ = 0, (3.1) These equations contain the integrability conditions of the systems in a concise geometric way. In the following, we focus on the implications of these relations for the particular case of an integrable deformation of self-dual vacuum Einstein equations.

The self-dual vacuum Einstein equations
Consider a real manifold (M, g) of dimension four and metric We assume that the associated Levi-Civita covariant derivative is torsionless , i.e. D k g ab = 0. Locally it is possible to introduce complex coordinates such that the metric reads The metric takes then the form corresponding to a Kähler manifold. Note that this metric is real since it was so in the original coordinates; as a consequence we have Eq. (3.5) implies that g ij and its conjugate vanish. The indexes i andī are called holomorphic and antiholomorphic respectively; the standard convention is to write the holomorphic index first. Complex 4-metrics of Euclidean signature with vanishing Ricci tensor and anti-self-dual Weyl tensor correspond to anti-self-dual vacuum Einstein solutions. The field equations in this case are invariant under changes of coordinates and may be written in several coordinates; in particular, using the fact that these metrics are automatically Kähler, they may be written in terms of the Kähler potential Ω as g i = ∂ i ∂Ω = ∂ i Ω =: Ω i . It follows that From the curvature conditions, we also obtain the equation governing the Kähler potential; this equation is Plebanski's first heavenly equation [15] Ω pp Ω qq − Ω pq Ω qp = 1. (3.8) The corresponding anti-self-dual Ricci-flat metric is then Eq. (3.8) admits a solution using the Penrose transform [3,32]; it defines a completely multidimensional integrable system with an infinite number of conservation laws, hierarchy and Lax pair formulation as well [33,34,35,36,37]. As mentioned before, a noncommutative deformation of the integrability conditions Eqs. (3.1) and (3.2) was proposed in [27]. More specifically, it was noted that if Ω is the deformed 2-form Ω = dp ∧ dq + λ( Ω pp dp ∧ dp + Ω pq dp ∧ dq + Ω qp dq ∧ dp + Ω qq dq ∧ dq) + λ 2 dp ∧ dq, (3.10) then it clearly satisfies the condition dΩ = 0. Furthermore, it is straightforward to see that The

Deformed properties
As we previously mentioned, we want to construct a Moyal deformed integrable Kähler manifold. For this purpose, we impose that the deformed Kähler potential Ω must share the same features and properties of the undeformed system; they are 1. the 2-form Ω should be closed i.e., d Ω = 0, 2. the metric coefficients Ω i should be real (hermitian property), 3. the determinant of Ω should be equal to one, i.e. det Ω i := Ω pp ⋆ Ω qq − Ω pq ⋆ Ω qp = 1.
The first of the above conditions can be analysed by fixing for the 2-form Ω, the same functional form as that of Eq. (3.10). This fact implies that the following condition must hold (3.14) We now write Eq. (2.20) as The second sum vanishes because of the first property imposed on Ω (n) ; we have thus In terms of the Poisson bracket, we write this equation as where we assume that Ω is given by Eq. (2.14).
The second condition implies that we are constructing a Kähler potential and that the corresponding metric coefficients are hermitian as in the commutative case; therefore, each perturbation Ω (n) should also be hermitian. We have then where the last two conditions hold because we have a deformed Kähler manifold. The third property can be imposed from the curvature condition Eq. (2.1) since the deformed first heavenly equation admits a rewriting as a simple determinant. We know that in the case of the non-deformed Kähler manifold, the determinant of the metric tensor is equal to one. In the case of the deformed case the deformed metric tensor Ω i should then satisfy the same property. We demand that by definition (3.20) The above equation is equivalent to detΩ i = ǫklΩ pkΩql = 1. Using the power series expansion in Eq. (2.14) forΩ, we get the different contributions to detΩ i order by order on θ pk Ω (n) ql . (3.21) The first term in this expression is the Moyal deformation of the first heavenly equation; its value is equal to one. Therefore, we conclude that the second term should vanish, namely In this equation are encoded all the combinations of order θ j such that j > r.

Noncommutative gravity
In the previous section, we studied the noncommutative deformation of the Kähler potential using the Moyal deformation of the first heavenly equation. The deformation functions Ω (n) are unknown; each one of them satisfy their respective differential equations obtained from Eq. (2.20). In this section, we give an Ansatz for the deformed Kähler potentials Ω (n) appearing in Eq. (3.17). As we will see later, the deformed functions Ω (n) can be expressed in terms of a deformed local frame or vierbeinê a µ (x, θ); after an integration procedure, the deformed Kähler potential Ω will be written in terms of the vierbein.

Deformed gauge fields
For some time now, a subject of interest has been the construction of consistent noncommutative deformations of Einstein gravity. Following the standard procedure to construct noncommutative gauge and scalar field theories [38,39], noncommutative versions of the Einstein-Hilbert action have been obtained by replacing the ordinary product by the noncommutative Moyal product where x µ are spacetime coordinates. The noncommutative structure of spacetime is then where the elements θ µν are constant (canonical) parameters and antisymmetric, i.e. θ µν = −θ νµ . In this approach, we can introduce a noncommutative metric aŝ in terms of a vierbeinê a µ (x, θ) and the Minkowski metric η ab ; the vierbeinê a µ (x, θ) reduces to the commutative one when θ = 0. The metricĝ µν is symmetric by construction and real even if the deformed tetrad fieldsê a µ (x, θ) are complex quantities. For θ = 0, we identify this metric with the commutative metric fieldĝ We want now to construct a Moyal deformed spacetime with associated deformed vierbein and metric, sharing the same properties of the undeformed spacetime, and such that the deformed Kähler potential discussed in the subsection 3.2 exists. For this purpose, we first introduce the vector fieldsê b µ asê b µ = e b µ + θ kλ e b µkλ + · · · + θ k 1 λ 1 · · · θ knλn e b µk 1 λ 1 ···knλn + . . . , (4.5) where the elements e b µk 1 λ 1 ···knλn are to be found. With this series expansion in powers of θ, we write then the metric tensor asĝ µν = g µν + iθ kλ g (1) µν kλ + O(θ 2 ), (4.6) up to first order on θ.
Since we are interested in making compatible the non-conmutative deformation of the Kähler metric Eq. (3.13), with the deformation in Eq. (4.6), let us assume the following decomposition for the verbein θ kλ e b µkλ = θ kλ P kλ e (1)b µ , (4.7) to first order on θ; here P kλ and e (1)b µ are unknown quantities. For the n-th order we generalize this Ansatz to θ k 1 λ 1 · · · θ knλn e b µk 1 λ 1 ···knλn = θ kλ P kλ n e (n)b µ , (4.8) implying θ kλ P kλ = θ; it is easy to show now that if we choose P kλ := ∂ k Ω p ∂ λ Ω q where Ω is the undeformed Kähler potential, then Eq. (4.9) is satisfied, where θ kλ is given by Eq. (2.12). In a similar way as for the vierbein, the tensor metric Eq. (4.6) takes the general form where the g (n) µν 's are tensor fields written in terms of the e (n)a µ 's; we fix their form as follows: first, to make compatible this deformation with the structure of a Kähler manifold, we need to impose the constraint ∂ α g (n) µν = 0 with x α = (p, q,p,q). This condition implies the following property for the vierbein ∂ µ e (n)a ν = 0 with x µ = (p, q,p,q); Eq. (4.3) simplifies then tô g µν =ê a µê b ν η ab . (4.11) Using now the expansion ofê a µ in powers of θ into Eq. (4.11), and equating the coefficients of the same power of θ in both sides of the equation, we obtain the n-th tensor field g

The deformed Kähler potential
According with [15], self-dual gravity can be parametrised in terms of the complex coordinates x µ = {p, q,p,q} and the resulting spacetime has the estructure of a Kähler manifold. In our case, the Kähler spacetime given in (3.5) can be written in terms of the classical veirbein e a µ and the local flat spacetime metric η ab defined as As we discussed previously, the vierbein e (n)a µ must have the property ∂ µ e (n)a ν = 0, µ = p, q,p,q. Therefore, we shall assume the following dependence The deformed Kähler potential Ω (n+1) to order n + 1 will depend explicitly on the vierbein to order n and n + 1 i.e., on e (n+1)a µ and e (n)a µ respectively. We begin by analising the deformation to first order: from Eq. (5.2) we obtain Using the undeformed vierbein e a µ given in Eq. To obtain the Kähler potential up to first order, we need to solve the system of equations (5.6)-(5.9). If we take integrate first the above equations with respect to the anti-holomorphyc variables xī = {p,q}, we need to calculate a set of integrals of the form e (1)A α Ω (0) βµ dx ν . After an integration by parts, we see that where we used the properties ∂ µ e (1)a ν = 0, ∂ α Ω (0) µν = 0 of the verbein and the Kähler potential respectively. In consequence, we have where C

Solutions for the Kähler potential to first order
As a particular example of the previous approach, we consider now in detail the deformation of the Kähler potential and the vierbein up to first order on the noncommutative parameter θ. We recall that the curvature condition can be formulated as a simple determinant, that is where Ω = Ω (0) + θΩ (1) up to first order. Substituting this expression in the determinant condition, we obtain the following equations pq Ω for case I, and Ω (1) for case II. We use now Eqs. (5.6)-(5.9) into the previous formulas to obtain, after some simplifications, the following relations among the components of the vierbein to first order where Ω i , with {ij} = {p, q,p,q}, are the metric coefficients of the undeformed metric tensor. For case I, we choose the following Ansatz for the vierbein e (1)4 q = α + β Ω pp + γΩ qq + δΩ pq + σΩ qp , where α, α ′ , β, β ′ , . . . are arbitrary constants. If we substitute the above equations into Eq. (5.27), we obtain the following relationship between the coefficients The solution to this system of coupled linear equations is For case II, we choose the following Ansatz for the vierbein where α, α ′ , β, β ′ , . . . are arbitrary constants. Substitution of these equations into Eq. (5.28) leads to The previous expressions determine the Kähler potential that is compatible with the Moyal deformation of the first heavenly equation. We have thus arrived to a multi-parameter family of solutions for the Moyal-deformed Kähler potential.
On the other hand, let g αβ be a η-self dual Riemman 4-metric with Euclidean signature and let ξ = ξ α ∂ α = ∂/∂φ be a Killing vector of g αβ associated to rotational symmetry. Then, locally we may write Eq. (6.4) implies then the existence of a η-self dual Killing vector field. In [28], a set of complex coordinates for the Atiyah-Hitchin (AH) metric [10] was found as an alternative procedure to the twistor formalism [41,42]. The AH metric is of importance in the analysis of magnetic monopoles interactions; the moduli space admits a metric formulation in the low-energy limit leading to the AH metric, and dispersion processes may be analyised in this way. In the following, we review briefly the main points of this construction; later, we generalise them to the noncommutative framework using the results presented in the previous sections. We recall first that the length element of the AH metric is [10] where and The differential 1-forms (sin ψdθ − sin θ cos ψdφ) , σ 2 y = 1 2 (cos ψdθ + sin θ sin ψdφ) , σ 2 z = 1 2 (dψ + cos θdφ) , (6.10) in the metric are invariant under the SU(2) of supersymmetry; the Killing vector associated to the diagonal U(1) is ξ = ∂/∂φ. By casting the metric in the form Eq. (6.2), we establish the identifications β 2 γ 2 sin 2 θ + δ 2 cos 2 θ β 2 sin 2 ψ + γ 2 cos 2 ψ , γ θψ = − 1 16 δ 2 γ 2 − β 2 cos θ sin θ cos ψ sin ψ , γ ψψ = 1 16 δ 2 sin 2 θ β 2 cos 2 ψ + γ 2 sin 2 ψ . (6.12) The corresponding Kähler potential has a nice simple form as a function of θ, ψ, k 2 , namely [28] where J := 1 8 (βγ + γδ + δβ) − γδ sin 2 θ cos 2 ψ − δβ sin 2 θ sin 2 ψ − βγ cos 2 θ . (6.14) On the other hand, Boyer and Finley [29] studied the Killing vectors in self dual Euclidean Einstein spaces using the formalism of complex H-spaces. They proved that it is always possible to choose complex coordinates on the such that either ξ = ∂ p + ∂p and ξΩ = 0, (6.15) or ξ = i(p∂ p +p∂p) and ξΩ = 0, where Ω satisfies the Monge-Ampère or first heavenly equation where Ω (0) := Ω. Now, we define J := rΩ (0) r (J is conjugated to ln r with respect to Ω (0) ), and use (J, q,q) as a new choice of independent variables to rewrite Eqs. (6.19) and (6.21). For the latter we obtain Ω (0) qq = r J r q rq r (r J ) 2 + 1 , so that the line element ds 2 has the same form as Eq. (6.2) (recall that ξ = ∂/∂θ); this coordinate frame is referred as the Toda frame [41]. It follows that we have the identifications γ ij dx i dx j = dJ 2 + 4rdqdq. (6.24) Eq. (6.14) gives the expression for J. We mentioned in Sec. 2 that a Moyal deformation of the first heavenly equation was possible and that the deformed Kähler potential Ω could be given as a power series expansion on the noncommutative parameter. If we also impose the condition that the moduli space, which is the AH spacetime, preserves its (anti-)self-dual character under the deformation, we must demand that the rotational Killing symmetry be unchanged; this requirement happens if, only if, each Ω (n) in the series expansion of the deformed Kähler potential is a function only of r, q,q. expression for the modified potential as a series expansion on the noncommutative parameter exists, where each term in this expansion satisfies a partial differential equation [25].
In the standard commutative situation, the Kähler potential satisfying the first heavenly equation is integrable. We extended this property to the modified potential by demanding a set of conditions using the Moyal bracket; these conditions also helped to fix the form of the potential in such a way that it becomes Kähler.
We applied these results to the particular case of a Kähler potential associated to self-dual vacuum solutions to Einstein's equations, and we analysed the problem of determining each one of the contributions in the series expansion of the modified Kähler potential. We obtained then explicit expressions for the deformed vierbein up to first order on the noncommutative parameter. With this information, we obtained two multi-parameter solutions for the Kähler potential also to first order; this procedure may be generalised to higher order in a straightforward way.
Finally, we applied this approach to the calculation of the modified Kähler potential associated with the AH spacetime, also up to first order on the noncommutative parameter. By extending the procedure of constructing complex coordinates for the AH metric [28], we obtained thus the modified AH metric in terms of the standard commutative one and the noncommutative contributions to the Kähler potential. Taking into account that the AH metric describes the moduli space of interacting magnetic monopoles at low energies, our results aim to incorporate noncommutative effects on these interactions.