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Quaternionic Kähler and Hyperkähler Manifolds

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Quaternionic Kähler and Hyperkähler Manifolds CHAPTER 12 Quaternionic K¨ahler and Hyperk¨ahler Manifolds In this chapter we will recall some basic results concerning various quaternionic geometries. Our main focus will be on the quaternionic K¨ahler (QK) and the hyeprk¨ahler (HK) manifolds as these two geometries are of special importance in the description and understanding of 3-Sasakian structures, the main topic of our next chapter. It would be impossible here, in a single chapter, to give a complete account of what is currently known about QK and HK spaces. Each case would require a separate monograph. Our goal is to describe some of the properties of such manifolds relevant to the Sasakian Geometry. Quaternionic K¨ahler geometry is traditionally defined by the reduction of the holonomy group Hol(M,g)toa subgroup of Sp(n)Sp(1) ⊂ SO(4n, R). Observe that Sp(1) · Sp(1) SO(4) so any oriented Riemannian 4-manifold has this property. It is generally accepted and, as we shall see later, quite natural, to extend this definition in dimension 4 via an additional curvature condition: an oriented Riemannian manifold (M 4,g) is said to be QK if the metric g is self-dual and Einstein. Interest in QK manifolds and this holonomy definition dates back to the celebrated Berger Theorem [Ber55]. The Lie group Sp(n)Sp(1) appears on the list of possible restricted holonomy groups of an oriented Riemannian manifold (M,g) which is neither locally product nor locally symmetric [Ber55]. In particular, the holonomy reduction implies that QK manifolds are always Einstein, though their geometric nature very much depends on the sign of the scalar curvature. The model example of a QK manifold with positive scalar curvature (positive QK manifold) is that of the quaternionic projective space HPn. The model example of a QK manifold with negative scalar curvature (negative QK manifold) is that of the quaternionic hyperbolic ball HHn. When the scalar curvature vanishes a QK manifold is necessarily locally hyperk¨ahler. In the language of holonomy the hyperk¨ahler manifolds are characterized by the reduction of the holonomy group Hol(M,g) to a subgroup of Sp(n) ⊂ SO(4n, R). Hence, the HK Geometry is a special case of the QK Geometry and, just as in the previous case, we find the Lie group Sp(n) on the Berger’s list. The model example of the HK geometry is that of the quaternionic vector space Hn with the flat metric. The first non-trivial complete examples of HK metrics, as well as the terminology, is due to Calabi [Cal79] who constructed such metrics on M = T ∗CPn. 12.1. Quaternionic Geometry of Hn, HPn and HHn The purpose of this section is to describe quaternionic geometries of some model examples of quaternionic manifolds. We will do it in considerable detail using terms which, in greater generality, will only be defined later. The quaternions H are the associative, non-commutative real algebra 0 1 2 3 a 4 H = {u | u = u + u i1 + u i2 + u i3,u∈ R} R . 307 308 12. QUATERNIONIC KAHLER¨ AND HYPERKAHLER¨ MANIFOLDS The imaginary units are often denoted by {i1,i2,i3} = {i, j, k}. The imaginary 3 quaternions Im(H)=span(i1,i2,i3) R and the multiplication rules are given by the formula 3 (12.1.1) iaib = −δab + abcic, c=1 We define the quaternionic conjugateq ¯ and the norm |u| by 3 3 0 a 2 a 2 u¯ = u − u ia, and |u| = (u ) . a=1 a=0 The non-zero quaternions H \{0} = H∗ = GL(1, H) from a group isomorphic to R+ × Sp(1), where Sp(1) is the subgroup of unit quaternions and the isomorphism is given explicitly by the map u → (|u|,u/|u|). The group of unit quaternions (12.1.2) Sp(1) = {σ ∈ H∗ GL(1, H) | σσ¯ =1}, as a manifold, is just the unit 3-sphere in R4. Furthermore, we have the group isomorphism f : Sp(1) → SU(2) explicitly given by 0 1 2 3 σ + σ i1 σ + σ i1 (12.1.3) f(σ)= 2 3 0 1 . −σ + σ i1 σ − σ i1 It is known that Spin(4) = Sp(1) × Sp(1) and SO(4) Sp(1)Sp(1), where custom- arily Sp(1)Sp(1) denotes the quotient by the diagonal Z2. This is yet another group isomorphism between classical groups which can be explained using the quaternionic 4 geometry of H R . Consider the action of G = Sp(1)+ × Sp(1)− on H given by ¯ (12.1.4) ϕ(σ,λ)(u)=σuλ. We assume the convention that the Sp(1)+ factor acts by the left quaternionic multiplication while the Sp(1)− factor acts from the right. Clearly, the two actions commute and the Z2 subgroup generated by (−1, −1) acts trivially. The quotient acts on R4 preserving the Euclidean metric and orientation. This is the special 4 orthogonal group SO(4). It is worthwhile to write this action on R .TheSp(1)+ part is given by the following group homomorphism A+ : Sp(1) → SO(4): ⎛ ⎞ σ0 −σ1 −σ2 −σ3 ⎜σ1 σ0 −σ3 σ2 ⎟ (12.1.5) A (σ)=⎜ ⎟ = σ01l + σ1I+ + σ2I+ + σ3I+, + ⎝σ2 σ3 σ0 −σ1⎠ 4 1 2 3 σ3 −σ2 σ1 σ0 + + where the matrices Ii = A (ei) (12.1.6)⎛ ⎞ ⎛ ⎞ ⎛ ⎞ 0 −10 0 00−10 00 0 −1 ⎜1000⎟ ⎜00 01⎟ ⎜00−10⎟ I+ = ⎜ ⎟ ,I+ = ⎜ ⎟ ,I+ = ⎜ ⎟ , 1 ⎝000−1⎠ 2 ⎝10 00⎠ 3 ⎝01 0 0⎠ 0010 0 −100 10 0 0 { + + +} R4 give a globally defined hypercomplex structure I1 ,I2 ,I3 on . For a purely + 2 imaginary τ = −τ¯ in Sp(1) one sets I (τ)=A+(τ) and gets the whole S -family of complex structures. We obtain the right hyperk¨ahler structure on H by further ω+ + + + ⊗ setting g0 + := g0 + i1ω1 + i2ω2 + i3ω3 = du du¯, where the multiplication 12.1. QUATERNIONIC GEOMETRY OF Hn, HPn AND HHn 309 in H is used to interpret the left hand side as an H-valued tensor. This gives the standard Euclidean metric g0 and the three symplectic forms + b ∧ c 0 ∧ a (12.1.7) ωa = du du + du du where (a, b, c) is any cyclic permutation of (1, 2, 3). We can also introduce an H- valued differential 2-form 1 (12.1.8) ω+ = i ω+ + i ω+ + i ω+ =Im(du ⊗ du¯)= du ∧ du.¯ 1 1 2 2 3 3 2 The 2-from du ∧ du¯ is purely imaginary as α ∧ β =(−1)pqβ¯ ∧ α¯, where p, q are the respective degrees. The Sp(1)− part is given by A− : Sp(1) → SO(4) with ⎛ ⎞ λ0 λ1 λ2 λ3 ⎜− 1 0 − 3 2 ⎟ ⎜ λ λ λ λ ⎟ 0 1 − 2 − 3 − (12.1.9) A−(λ)= = λ 1l + λ I + λ I + λ I . ⎝−λ2 λ3 λ0 −λ1⎠ 4 1 2 3 −λ3 −λ2 λ1 λ0 − − The matrices Ii = A (ei) (12.1.10)⎛ ⎞ ⎛ ⎞ ⎛ ⎞ 0100 0010 0001 ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ − −100 0 − 0001 − 00−10 I = ⎜ ⎟ ,I= ⎜ ⎟ ,I= ⎜ ⎟ , 1 ⎝ 000−1⎠ 2 ⎝−1000⎠ 3 ⎝ 0100⎠ 0010 0 −100 −1000 { − − −} give a globally defined hypercomplex structure I1 ,I2 ,I3 . Furthermore, with the − Euclidean metric one gets the left hyperk¨ahler structure on H by setting g0 −ω := − − − − − − ⊗ g0 iω1 i2ω2 i3ω3 = du¯ du. This gives 1 (12.1.11) ω− = i ω− + i ω− + i ω− = −Im(du ⊗ du¯)=− du ∧ du,¯ 1 1 2 2 3 3 2 where, as before, we get the three symplectic forms − b ∧ c − 0 ∧ a (12.1.12) ωa = du du du du , for each cyclic permutation of (1, 2, 3). These are clearly fundamental 2-forms { −} associated to the complex structures Ia . Note that by construction, for any (σ, λ) one has [A+(σ),A−(λ)] = 0 and the product A+(σ)A−(λ) ∈ SO(4). In particular, { + + +} { − − −} the two hypercomplex structures I1 ,I2 ,I3 and I1 ,I2 ,I3 commute. The − − hyperk¨ahler structure (g0,Ia ,ωa ) is preserved by Sp(1)+ (hypek¨ahler isometry) 2 while Sp(1)− acts by rotating the complex structures on S . The role of Sp(1)+ and + + Sp(1)− reverses for (g0,Ia ,ωa ). With only little extra effort one can “compactify” this example to see that another Lie group U(2) is a compact manifold with two commuting hypercomplex structures, though U(2) admits no hyperk¨ahler metric. Remark 12.1.1: Consider the group of integers Z acting on H by translations of the real axis. The action preserves both hypercomplex structures and the metric, hence, the quotient H/Z S1 × R3 is also a flat hyperk¨ahler manifold with infinite fundamental group π1 = Z. To indicate the difference, we will write the flat metric 2 0 in this case as g0 = dθ + dx · dx replacing x with the angle coordinate θ. Example 12.1.1: Quaternionic vector spaces. Much of the above discussion Hn { | 0 1 2 3 ∈ H } extends to = u =(u1,...,un) uj = uj +uj i1+uj i2+uj i3 ,j=1,...,n . 310 12. QUATERNIONIC KAHLER¨ AND HYPERKAHLER¨ MANIFOLDS Here and from now on we will choose to work with the left hyperk¨ahler structure on Hn, i.e., with the symplectic 2-forms are given by n (12.1.13) g0 − ω = du¯j ⊗ duj j=1 so that n 3 n a 2 b ∧ c − 0 ∧ a (12.1.14) g0 = (dxj ) ,ωa = dxj dxj dxj dxj j=1 a=0 j=1 for any cyclic permutation (a, b, c)of(1, 2, 3). The corresponding hypercomplex structures is then given by the left multiplication by {¯i1,¯i2,¯i3} = {−i, −j, −k} with the standard basis as in 12.1.10, where 0, 1 are now matrices of size n × n.
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