Введение
1. Introduction 2
2. Preliminary material 8
2.1 Quaternion Kahler and hyperkahler spaces 8
2.1.1 Quaternionic Kahler spaces in dimension higher than four 9
2.1.2 Quaternion Kahler manifolds in dimension four 12
2.2 Hypercomplex structures 14
2.2.1 Basic concepts 14
2.2.2 Some explicit examples 16
2.3 Hyperkahler spaces in four dimensions 18
2.4 Quaternion Kahler and hyperkahler metrics in d = 4 with at least one isometry 21
2.4.1 Hyperkahler metrics with Killing vectors that are not self-dual 21
2.4.2 Integrability of the axial continuum Toda equation 23
2.4.3 Einstein-Weyl structures and hyperkahler metrics 25
2.4.4 Self-dual structures with one Killing vector 27
2.4.5 The Joyce spaces 29
2.4.6 Identification of the quaternionic-Kahler metrics with at least one isometry 31
2.4.7 Examples of toric quaternion Kahler spaces 33
2.5 Higher dimensional hypergeometry 39
2.5.1 Construction of in hyperkahler manifolds with T" tri-holomorphic isometry 39
2.5.2 Quaternion Kahler spaces in quaternion notation 41
2.5.3 The Swann extension 43
2.6 Spaces with C?2 holonomy 45
2.6.1 The group C?2 and the octonions 45
2.6.2 G% holonomy and self-duality 51
2.6.3 The Bryant-Salamon construction 53
2.6.4 Weak G2.7 f?2 holonomy spaces and M-theory compactifications 59
3. Heterotic geometry without isometries 62
3.1 Introduction 62
3.2 Hyperkahler torsion manifolds 65
3.2.1 Main properties 65
3.2.2 Relation with the Plebanski-Finley conformal structures 67
3.3 The general solution 70
3.4 Discussion 75
4. D-instanton sums for matter hypermultiplets 77
4.1 Ooguri-Vafa solution 77
4.2 Pedersen-Poon Ansatz 79
4.3 D-instanton sums 80
4.4 Discussion 82
5. Hyperkahler spaces with local triholomorphic U(l) x [/(1) isometry and su pergravity solutions 83
5.1 Toric hyperkahler metrics of the Swaim type 83
5.2 Supergravity solutions related to hyperkahler manifolds 86
6. G2 toric metrics and supergravity backgrounds 89
7. Conclusion 93
7 LINK1 Hypercomplex structures LINK1
As we have seen, the problem of classifying the possible quaternion Kahler spaces appearing in d = 4 is equivalent to find all the Einstein spaces with self-dual Weyl tensor. In this section we focus in four dimensional spaces for which the second property hold namely, self-dual spaces. Let us consider a metric g and denote with [g] the family of metrics obtained from g by arbitrary conformal transformation g -» Q2g. Because the Weyl tensor is conformally invariant it follows that the self-duality of g implies the self-duality of [g], the converse is also true. An important example of self-dual families are the hypercomplex structures [60], [61]. To consider them has several advantages for our purposes. It is easier to understand the Ashtekar-Jacobson-Smolin [149] and the Plebanski [76] description of hyperkahler spaces once the properties of hypercomplex structures are understood. Moreover we will show that hypercomplex structures and weak torsion hyperkahler structures are the same concept in four dimensions. As it follows from equation (2.7), for hyperkahler spaces there exist a frame for which that is, for which the Kahler triplet is closed. Conversely, it was shown in [76] that any space in d = 4 satisfying (2.34) is automatically hyperkahler. As we will see below this implies the curvature tensor Щы of an hyperkahler space is automatically self-dual. A simple calculation show that if we make the transformation g = Sl2g then the Kahler triplet is transformed as J = fi27 and (2.34) became Relation (2.35) is not the same as (2.34) and therefore (2.34) is not invariant under conformal transformations. A conformally invariant generalization of (2.34) is being a an arbitrary 1-form. Under conformal transformation we have a-f 2dlog(Q) and (2.36) is unaltered [61]. This property define a conformal family [g] with all the element sharing the property (2.36). If a is a gradient (a — "Чф, being ф certain function depending on all the coordinates), then it will exist a representative g of [g] with self-dual curvature and therefore hyperkahler. From the expression (2.32) of the Weyl tensor it is possible to check that if R is self-dual, then W is also self-dual. Thus the family [g] corresponding to an hyperkahler metric will have self-dual Weyl tensor (2.32). But the explicit calculation of the Wabcd for a metric satisfying (2.36) shows that a disappear in its calculation.
Therefore the condition to be gradient is not necessary to ensure W- = 0 and condition (2.36) defines a self-dual structure in general [61]. One family of metrics satisfying (2.36) are the so called hypercomplex structures [60]. To define them, let us consider as before Riemanian space M of real dimension 4n endowed with a metric g and a set of three almost complex structures J (i = 1,2,3) satisfying the quaternionic algebra and for which the metric g satisfies for any X,Y in TXM. The complex structure is said to be integrable if the Niejenhuis tensor vanish for every pair of vector fields A , Y. An hypercomplex structure is a conformal structure [g] for which the triplet J1, J2 and J3 are integrable. The integrability condition is conformally invariant. In four dimensions we can select the self-dual complex structures up to an SU(2) transformation. The action of (2.40) over the tangent space TMX is defined by The annulation of the tensor N (X, Y) will be equivalent to the conditions for some set of functions Лі,Л2,Л3, Л4 [59]. This is a direct consequence of (2.41) and (2.39). We immediately recognize that in the case Aa — 0 we recover the Ashtekar-Jacobson-Smolin quadratic equations for self-dual spaces [149]. If we consider the conformal transformation g — a2g, then it is seen that (2.42) is also satisfied with Aa — Aa + 2ealog(fi). This mean that condition (2.42) define a conformal family of metrics [g] and therefore the definition of hypercomplex structures is consistent. Hypercomplex condition implies, but is not implied by, that [g] is self-dual [59]-[60]. This is because (2.42) implies the self-duality condition (2.36). To see the last statement clearly consider the connection us given by It is well known that the antisymmetric part wp is related to the structure functions defined by the Lie Consider now the form This is equivalent to (2.36) with a — A — x- The same formula holds for J and J . Therefore we reach to the conclusion that hypercomplex structures satisfies (2.36) and therefore are self-dual [59]. It is straightforward to prove that under the conformal change g —У Q2g the forms a, A and \ transform Hypercomplex structures arc not the only self-dual structures in four dimensions. Counterexamples are for instance the Joyce spaces that will be introduced in another section. It will be instructive to present some examples of hypercomplex spaces. Let us look now for self-dual structures [g] two commuting [/(1) Killing vectors satisfying (2.42). The representatives [g] of such structures take the Gowdy form The latin and greek indices takes values 1 and 2. Both gab and ga$ are supposed to be independent of the coordinates xa = (0, ip). Then the Killing vectors are д/дв and д/дц and are commuting, so there By Gauss theorem there exists a local scale transformation g - Q2g which reduce (2.46) to being gQp function of the new coordinates (p,rj). We can express where Ax and B, are unknown functions of (p, rj) and the factor p was introduced by convenience. But instead to work with (2.48) we will work with the following equivalent expression Anzatz (2.49) looks more complicated than (2.48), but has the advantage that the inverse einbeins ЄІ takes the simple form Inserting (2.50) and (2.51) into the quadratic system (2.42) gives Q = 1 and the Cauchy Riernman equations and the same equations for J% and By.
This means that the metric (2.49) is described in terms of two holomorphic functions We can find the same result in another way. Let us consider four functions fi,..., f and .9ь ---,94. depending on the coordinates x1 = p and x2 — 77 and let us define the vector fields Clearly ea are the most general vector fields for a metric with two commuting isometries up to a conformal scaling. Introducing this expressions into (2.57) gives the system of equations /i)„ - ШР = 0, Ы„ - {92), = 0 and from the first we see that /3 = {H)p and /4 = (H)4 for some function H(p 77) and therefore we can make the coordinate change (p, 77, в, tp) —у (p — H, 77, в, p) and eliminate /3 and /4. The same holds for g$ and 54 and therefore we are dealing with the case described in (2.51). The corresponding hypcrkahler metric is then conformal to which is the same metric as above with fl = 1. We have redefined e\ = A0, e2 = BQ, f\ = Ai, /2 = B2 in (2.55). There exists another family of hypercomplex structures that can be constructed in terms of holomorphic functions. It is direct to check that Ashtekar equations (2.57) can be cast in the following complex form Let М be a complex surface with holomorphic coordinates (z , z2) and let us define four vector fields ЄІ as being fj a complex function on M. Then (2.56) implies that dfj/dzk = 0 and therefore we can construct an hypercomplex structure using four arbitrary holomorphic functions or two holomorphic vector In a well known work [149] Ashtekar, Jacobson and Smolin introduced a formulation for self-dual manifolds in which they reduced the problem to solve certain quadratic equations for the dual vector fields and a volume form preserving condition. Here we review their construction in the context of hypercomplex structures. Proposition 3 Consider an oriented manifold M and four vector fields в\, e2, єз ап& e4 forming an oriented basis for TM at each point. Let us suppose that the fields satisfies the quadratic equations and the volume preserving condition ЄцО = 0 for some -form 0. Then the vectors ea are conformal to an orthonormal frame of a self-dual Ricci-flat metric. Before to explain Proposition 3 let us note that is (2.57) with the J40 S equal to zero. Then (3.328) implies that x = a- Then if x were a gradient, then the form a also would be a gradient and there will exist a conformal change taking the condition to d T = 0. Therefore the corresponding metric will be conformal to an hyperkahler one, which are automatically self-dual Ricci-flat. As we will see, this is essentially the content of the volume preserving condition of proposition 3, In general the covariant divergence of an arbitrary vector field V is obtained by the formula The divergence of a tetrad e a is given by being c6 defined by Let us define a factor Q through and take the derivative of (2.60) along eQ = 2-1ea. Using the condition CaQ = 0 together with (2.58) The last implication is just the definition (3.328). The transformation rule (2.45) implies that under д and therefore a is a gradient. Then g is conformal to an hyperkahler metric.
