QUOTIENT CONSTRUCTIONS
3.6 Potentials
Definition 3.6.1. IfN is a hyperK¨ahler manifold, a function µ: N →R is ahyperK¨ahler potential if for each complex structureAcompatible with the hyperK¨ahler structure of N, µsatisfies
i∂A∂¯Aµ=ωA, where ωA is the K¨ahler form of A.
The result quoted above suggests that the existence of a hyperK¨ahler potential may be linked with the existence of certain group actions.
Proposition 3.6.2. Suppose N is a hyperK¨ahler manifold. If N has a hyperK¨ahler potential thenN admits a local Sp(1)-action which permutes the complex structures and, in the notation of the previous section, is such that the vector field IXI is independent of I.
Conversely, if N admits such an Sp(1)-action, then N has a hy- perK¨ahler potential.
Proof. Let µ be a hyperK¨ahler potential on N. For a complex structure I com- patible with the hyperK¨ahler structure of N, define a vector field XI by dµ = XIyωI =−(IXI)yg, whereg is the metric on N. Thus the vector field X =IXI is independent of the choice of complex structure I. It is now sufficient to show that the bracket [IX, JX] a non-zero multiple of KX. Firstly, if A and B are vector fields then
0 =−d2µ(A, B) =d(Xyg)(A, B) =Ag(X, B)−Bg(X, A)−g(X,[A, B]).
Taking A=IX and B=JX yields g(X,[IX, JX]) = 0, so [IX, JX] is orthogonal to X. If C is also a vector field then
0 =dωI(A, B, C) =AωI(B, C) +BωI(C, A) +CωI(A, B)
−ωI(A,[B, C])−ωI(B,[C, A])−ωI(C,[A, B]).
3.6 Potentials 70 Fix a point n of N and let Y ∈TnN be orthogonal to the quaternionic span of X.
ExtendY locally so that it commutes withIX. TakeA=Y,B =IX andC =JX in the above formula. This gives
0 =g(Y, I[IX, JX]) +g(X,[JX, Y]) +g(JX, I[Y, IX]) =g(Y, I[IX, JX]),
so [IX, JX] is in the span of IX, JX and KX. Now µ is a K¨ahler potential for I, sod(Idµ) =ωI for eachI. ThusLIXωJ =d((KX)yg) =d(K(−Xyg)) =d(Kdµ) = ωK and similarly for cyclic permutations of I, J andK. Also, LIXωI =d(Xyg) = d2µ= 0. Thus
L[IX,JX]ωI = [LIX, LJX]ωI
= (LIXLJX−LJXLIX)ωI
=−LIXωK −0
=ωJ,
showing that [IX, JX] is non-zero and that the Sp(1)-action, resulting from inte- gration, permutes the complex structures.
Conversely, ifN admits a permutingSp(1)-action withIXI fixed, let µI be the moment map associated toI, thendµI =XIyωI =−(IXI)yg. Hitchin et al. (1987) show that µI is a K¨ahler potential for J and K. Now let µJ be the moment map associated to J. Then since JXJ is independent ofJ, we havedµJ =−(JXJ)yg= dµI. NowµJ is a K¨ahler potential forI, so applying∂I to this equation shows that i∂IdµI =ωI and thatµI is a hyperK¨ahler potential.
Note that on the associated bundle of a quaternionic K¨ahler manifold the hy- perK¨ahler potential is the function r2. The hyperK¨ahler potential determines the metric as follows. The first part of the proposition is well-known.
Proposition 3.6.3. 1) If N is a K¨ahler manifold with metric g, Rie- mannian connection∇, complex structure I and K¨ahler formωI, then for a function µ on N
1
2(∇2X,Yµ+∇2IX,IYµ) =g(X, Y) for all X, Y if and only if
i∂I∂¯Iµ=ωI.
2) If N is a hyperK¨ahler manifold, then for a function µ
∇2µ=g if and only if µ is a hyperK¨ahler potential.
Proof. 1) Let X, Y be two vector fields. Now ∇ is torsion-free and I is K¨ahler, so
2(i∂I∂¯Iµ)(X, Y) =−id(dµ+iIdµ)(X, Y)
=d(Idµ)(X, Y)
=X(Idµ)(Y)−Y(Idµ)(X)−Idµ[X, Y]
=X((IY)µ)−Y((IX)µ)−(I[X, Y])µ
=X((IY)µ)−Y((IX)µ)−I(∇XY − ∇YX)µ
=X((IY)µ)− ∇X(IY)µ − Y((IX)µ) +∇Y(IX)µ
=∇2X,IYµ− ∇2Y,IXµ and the result follows.
2) This follows from part (1) by fairly general principles. Leth(X, Y) =∇2X,Yµ;
we need only show that g = h if and only if g(X, Y) = 12(h(X, Y) +h(IX, IY)) for all vector fields X, Y and each compatible complex structure I. If g =h then this is just the fact that I preserves the metric. If the second condition holds then h(IX, IY) = h(JX, JY). Put X′ = IX, Y′ = IY and suppose IJ = K = −JI,
then h(X′, Y′) =h(KX′, KY′), as required.
3.6 Potentials 72 Remark. If N is a hyperK¨ahler manifold with hyperK¨ahler potential µ, then the vector field X dual to dµ is an infinitesimal quaternionic transformation. In the notation of Proposition 3.6.2,X is the vector fieldIXI, so we have a localH∗-action.
The system of equations
∇dµ=λg,
for some constant λ, is over determined and Weitzenb¨ock techniques show that the H∗-orbits are flat and totally geodesic.
Pursuing an idea of Lebrun (1990), suppose M is a quaternionic K¨ahler mani- fold with Levi-Civita connection ∇and thatM admits a hyperK¨ahler metric in the same quaternionic class. The hyperK¨ahler metric trivialises the bundle H and so gives a solutionh of the twistor equation. Quaternionic invariance of this equation, implies that e=∇his a section of E. Frome andhone can construct infinitesimal vector fields for a local H∗-action. We expect such metrics to be locally isometric to an associated bundleU(M′) for some quaternionic K¨ahler manifold M′. If M is four-dimensional, the above discussion shows that it is flat.
The classification of hyperK¨ahler 4-manifolds admitting a permuting Sp(1)- action was carried out by Gibbons & Pope (1979) and completed by Atiyah &
Hitchin (1988). The metrics obtained (upto finite quotients) are the flat metric on H, the taub-nut metric and the hyperK¨ahler metric on the moduli space of charge 2 monopoles. Of these, the only one that can possess a hyperK¨ahler potential is the flat metric.
Maciocia (1989) explicitly constructs the hyperK¨ahler potential for the mod- uli space Mk,r of framed SU(r)-instantons of charge k over R4. If A is such a connection, then the hyperK¨ahler potential is given by the ‘second moment’
m2(a) = 1 16π2
Z
R4kxk2TrFA2, where FA is the curvature of A.
Corollary 3.6.4. The manifold m−12 (x)/SO(3) ⊂ Mk,r/SO(3), for any non-degenerate point x ∈Imm2, is quaternionic K¨ahler.