HyperK¨ ahler Quotients

base spaces may be recovered from the hyperK¨ahler structures. The construction is a moment map construction based on the observation that the total spaces admit a special type of Sp(1)-action. Finally we return to the Sp(1)-actions and show that these hyperK¨ahler metrics are determined by special K¨ahler potentials.

3.1 HyperK¨ ahler Quotients

LetM be a 4n-dimensional hyperK¨ahler manifold, with Riemannian metricg, com- plex structures I, J and K, and symplectic form η = η_Ii+η_Jj +η_Kk. If X is a vector field onM, letL_X denote the Lie derivative with respect toX. We say that X isKillingifL_Xg= 0,triholomorphicifL_XI =L_XJ =L_XK = 0, orHamiltonian if L_Xη = 0. Note that any two of these conditions imply the third one.

Now suppose that G is a compact Lie group acting freely and smoothly on M preserving gand η. For any vector fieldX generated by this action we haveLXg= 0 =LXη. Now η is closed, so

0 =L_Xη =Xydη+d(Xyη) =d(Xyη).

If the first cohomology group H¹(M,R) vanishes then there is a function µ_X ∈ Ω⁰(M,ImH) such that

Xyη=dµ_X.

These functions can be combined into a single map µ: M →g^∗⊗ImH defined by

hµ(m), Xi=µ_X(m), (3.1.1)

for m ∈ M. Note that µ_X is only defined upto an additive constant, but if G is compact, then (Guillemin & Sternberg, 1984) there is no obstruction to choosing

these constants so that µ is equivariant with respect to the actions of G on M and g^∗. In this case, µ (or µ_X) is called amoment map for the action of G on M.

Given a moment mapµ, supposea∈g^∗⊗ImHis invariant under the (coadjoint) action of G. Then, if we set M_a =µ⁻¹(a), M_a is a G-invariant submanifold of M of dimension 4n−3k, wherek is the rank ofG. NowG acts freely and isometrically on Ma, so ˆMa = Ma/G is a Riemannian manifold of dimension 4n−4k. Since Galso preserves η, it can be shown that ηand the almost complex structures on M define a hyperK¨ahler structure on ˆM_a. Details of these assertions may be found in Hitchin, Karlhede, Lindstr¨om & Roˇcek (1987).

This construction may be generalised to pseudo-hyperK¨ahler manifolds. We require that if we consider the distribution A of vectors tangent to the action of G on M, then the restriction of the pseudo-metric g to A is non-degenerate. This ensures that g gives a G-invariant distribution of horizontals in T M_a for M_a → M_a/G (see Lang, 1984). With this additional hypothesis, the proofs cited above show that M_a/G is pseudo-hyperK¨ahler. Hitchin (1988) has also studied the case where we have no metric, just a hypersymplectic structure, and obtains a further generalisation of the quotient construction.

The first example of a hyperK¨ahler quotient is that of U(1) acting on flat space Hⁿ via

e^iθ·(a+jb) =e^iθa+je^−iθb.

The value of the moment map for this action at q ∈ Hⁿ is ¯q^tiq and we obtain a hyperK¨ahler quotient for eacha ∈ImH. Ifais non-zero, there is anα∈Hsuch that

¯

αiα=a and theU(1)-equivariant mapq 7→qα⁻¹ takes the level set associated toa to that associated toi. Thus the hyperK¨ahler quotient associated toais homothetic

3.1 HyperK¨ahler Quotients 48 that associated toi. Writingq =a+jb, the moment map equation ¯q^tiq =ibecomes

|a|²− |b|² = 1, b^ta = 0.

The hyperK¨ahler quotient is topologically T^∗CP(n −1), since a defines a point of CP(n−1) and b⊗a¯ defines an element of T_[a]^∗ CP(n−1) ∼= (Hom(hai,hai^⊥))^∗. Hitchin (1986) shows that the metric obtained on T^∗CP(n−1) coincides with the one constructed by Calabi (1979, 1980).

The 0-level set of the moment map is

|a|² =|b|² and b^ta = 0.

The map (a, b)7→a⊗¯bis well-defined on the quotient space. The image inMn(C)∼= u(n,C) is the highest root orbit, together with 0, and the action of Sp(1) on Hⁿ on the right descends to the action described in Chapter 2. So the hyperK¨ahler quotient is the associated bundle U(Gr₂(Cⁿ)) with its hyperK¨ahler metric and we may view this metric as a singular limit of the Calabi metric on T^∗CP(n−1). Note that Galicki (1986) obtains this Calabi metric as a limit of the quaternionic K¨ahler metrics on Gr₂(Cⁿ⁺¹).

The twistor space of Gr2(Cⁿ) is the flag manifold

F_1,2,n = U(n)

U(n−2)×U(1)×U(1).

This clearly fibres overCP(n−1), and, away from the zero sections,T^∗CP(n−1) may be regarded asO(−2). Letting =⇒denote a degenerate limit of a family of metrics, and using the subscripts _qK and _hK to indicate whether the space in question is

to be regarded as a quaternionic K¨ahler or hyperK¨ahler manifold respectively, we have the following diagram (away from zero sections).

U(Gr₂(Cⁿ))_qK =⇒ U(Gr₂(Cⁿ))_hK ⇐= T^∗CP(n−1) ⇐= Gr₂(Cⁿ⁺¹)

ց ւ

F_1,2,n

ւ ց

Gr₂(Cⁿ) CP(n−1)

More generally, we can consider the bundleU(M) associated to a quaternionic K¨ahler manifold M with non-zero scalar curvature.

Lemma 3.1.1. If X is a Killing vector field on M such that L_XΩ = 0, where Ω is the fundamental 4-form of M, then X may be lifted to a Hamiltonian Killing vector fieldX˜ on U(M).

Proof. Let φ be an isometry of M such that φ^∗Ω = Ω. Then φ_∗ defines a map φ_∗: Fx →F_φ(x) which commutes with the Sp(n)Sp(1)-action on the reduced frame bundle F, since

(φ_∗u(ξ))·(A, q) =φ_∗u(Aξq) =¯ φ_∗(u(ξ)·(A, q)),

for u ∈ F, ξ ∈ H and (A, q) ∈ Sp(n) × Sp(1). Similarly, the pull-back of φ_∗ toF×(H^∗/Z₂) commutes with the Sp(n)Sp(1)-action, so we have an induced map φ_∗: U(M)→ U(M).

Now φ^∗ preserves θ, for ifu ∈F, v∈TuF then

φ^∗θu(v) =θφ∗u(φ_∗∗v) = (φ_∗u)⁻¹(π_∗φ_∗∗v)

and we have that πφ_∗ = φπ, so φ^∗θ_u(v) = u⁻¹φ_∗⁻¹(φ_∗π_∗v) = θ_u(v). Also, φ^∗ commutes with d and the connection forms ω_± are uniquely determined by

3.1 HyperK¨ahler Quotients 50 dθ = −ω+ ∧θ−θ∧ω₋, so φ^∗ω₋ = ω₋. The induced map φ_∗ acts trivially on H, and so preserves x and dx. Thus φ^∗ is an isometry on U(M) which preserves the 2-form υ.

Let φ_t ba a local 1-parameter group of local isometries generating X. Since L_XΩ = 0, we may assume that φ^∗_tΩ = Ω. As above, we can lift φ_t to a local 1-parameter group φ_t∗ of local isometries on U(M). The vector field ˜X generated by φt∗ is thus a Hamiltonian Killing field on U(M). Note that π_∗^HX˜ =X.

Suppose a compact Lie group G acts on M freely and isometrically and that this action preserves Ω. The lemma above then tells us that the action ofG may be lifted toU(M) and that this action is isometric and triholomorphic. Sinceπ_∗^HX˜ =X for each vector field generated by the action, the induced action preserves the fibres of π^H: U(M)→M, and is thus free.

LetY be a Killing field generated by the action ofGonM and letX denote the lift of Y to U(M). The proof of the lemma yields an Sp(n)Sp(1)-invariant vector field, which we again denote byX, on F and̟_∗X =X. If we define µX on F×H^∗ by

µ_X =−Xy(xω₋x),¯ (3.1.2) then µ_X is Sp(n)×Sp(1)×Z₂-invariant and so is well-defined on U(M).

Proposition 3.1.2. Letµ: U(M)→g^∗⊗ImHbe defined from(3.1.2)as in(3.1.1). Thenµis a moment map for the induced action ofG onU(M).

Proof. First we check that dµ_X =Xyυ. Now, since M is Einstein we have

dµX =−(LX−Xyd)(xω₋x)¯

=−L_X(xω₋x) +¯ Xy(dx∧ω₋x¯+xcθ¯^t∧θx¯−xω₋∧ω₋x¯−xω₋∧d¯x).

But Xydx= 0 and LXx= 0, so

dµ_X =−x(L_Xω₋)¯x+Xy(α∧α¯+cxθ¯^t∧θx).¯

Let φ_t be a 1-parameter group of isometries preserving Ω and generating Y. Then φ_t∗ generates X and it preserves ω₋, so L_Xω₋ = 0. Hence, dµ_X = Xyυ. The equivariance of µ follows directly from the definition ofµ_X.

Define αY to be Yyg, where g is the metric on M. Then ∇αY is an element of Λ²T^∗M, since Y is a Killing field (see Kobayashi & Nomizu, 1963). We have a subbundle sp(1) of Λ²T^∗M whose fibre is the Lie algebra sp(1) ∼= ImH; this bundle is just the isometric embedding of G in Λ²T^∗M. Let (∇α_Y)^{sp (1)} denote the orthogonal projection of ∇α_Y onto sp(1). If we choose a frame u ∈ F, then (∇α_Y)^sp_π(u)⁽¹⁾ =ω₋(X)_u, whereX is the lift ofY toT F, as above. Thus,µ_X vanishes at a point a ∈ U(M) if and only if (∇α_Y)^sp_πH(a)⁽¹⁾ = 0. So, if µ_X vanishes ata, then µX vanishes on the whole fibre through a and if X is a lifted Killing vector field, then X is horizontal over each point of the fibre through a.