# Torus actions and Ricci-flat metrics

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1 Department of Mathematics, University of Aarhus November 2016 / Trondheim To Eldar Straume on his 70th birthday DFF

2 Delzant HyperKähler G2

3 Outline Delzant HyperKähler G2 1 The Delzant picture 2 HyperKähler manifolds Dimension four Toric hyperkähler 3 G 2 manifolds

4 Delzant HyperKähler G2 The Delzant picture (M 2n, ω) symplectic with a Hamiltonian action of T n = The moment map µ : M R n = t identifies the orbit space M/T n with a convex polytope in R n. Each such (M 2n, ω) may be constructed as a symplectic quotient of R 2m by an Abelian subgroup of T m.

5 Delzant HyperKähler G2 Symplectic moment maps Let (M, ω) be symplectic: dω = 0. If X X(M) preserves ω, then Cartan s formula gives 0 = L X ω = d(x ω), so (locally) X ω = dµ X for some µ X : M R. For G Abelian acting on M preserving ω, the action is Hamiltonian if there is a G-invariant map µ : M g such that d µ, X = X ω for each X g. M simply-connected and G connected Abelian = Hamiltonian all orbits are isotropic ω(x, Y) = 0 for all X, Y g.

6 Delzant HyperKähler G2 The Delzant polytope Faces of = µ(m) are of the form F k = { µ, u k = λ k } and the Delzant polytope is = {a R n a, u k λ k k}. u k R n with points over (F k ) having stabiliser the subtorus with Lie algebra {v R n = t v, u k = 0}, so u k Q n. Smoothness of M is equivalent to: F k1 F kr = = the corresponding u k1,..., u kr are part of a Z-basis. This restricts the possible u i s locally.

7 Delzant HyperKähler G2 4d toric HyperKähler manifolds (M, ω I, ω J, ω K ) is hyperkähler if: 1 each ω A is a symplectic two-form, 2 the tangent bundle endomorphisms I = ω 1 K ω J, etc., satisfy I 2 = 1 = J 2 = K 2, IJ = K = JI, etc., and g = ω A (A, ) is independent of A and positive definite. Consequences dim M = 4n, g is Ricci-flat, with holonomy contained in Sp(n) SU(2n).

8 Delzant HyperKähler G2 4d toric Symmetry considerations Ricci-flatness implies: if M is compact, then any Killing vector field is parallel, so the holonomy of M reduces and M splits as a product, if M is homogeneous then g is flat, so M is a quotient of flat R 4n by a discrete group (Alekseevskiĭ and Kimel fel d 1975). Take (M, g) complete and G Abelian group of tri-holomorphic isometries. Assume the action is tri-hamiltonian, so there is a hyperkähler moment map: a G-invariant map with d µ A, X = X ω A. This forces 4 dim G dim M. µ = (µ I, µ J, µ K ) : M R 3 g

9 Delzant HyperKähler G2 4d toric Gibbons-Hawking Ansatz in 4d X a tri-hamiltonian vector field on hyperkähler M 4 Away from M X, locally g = 1 V (dt + ω)2 + V(dx 2 + dy 2 + dz 2 ) where V = 1/g(X, X), dx = X ω I = dµ I, etc., and on R 3. In particular, dω = 3 dv µ = (µ I, µ J, µ K ) is locally a conformal submersion to (R 3, dx 2 + dy 2 + dz 2 ), V is locally a harmonic function on R 3.

10 Examples Delzant HyperKähler G2 4d toric V(p) = c i Z 1 p p i, c 0, p i R 3 distinct c = 0, Z < : multi-euguchi Hanson metrics Z space flat R 4 T CP(1)... c > 0, Z < : multi-taub-nut metrics Z space flat S 1 R 3 Taub-NUT R 4 T CP(1)... Z countably infinite: require V(p) to converge at some p R 3, get A metrics (Anderson et al. 1989; Goto 1994), e.g. Z = N >0, p n = (n 2, 0, 0), and their Taub-NUT deformations.

11 Delzant HyperKähler G2 4d toric Classification Theorem (Swann 2016) The above potentials V(p) = c i Z 1 p p i, c 0, p i R 3 distinct, with 0 < V(p) < for some p R 3, classify all complete hyperkähler four-manifolds with tri-hamiltonian circle action. When Z <, this is due to Bielawski (1999), and the first parts of the proof are essentially the same.

12 Delzant HyperKähler G2 4d toric Proof structure The only special orbits are fixed points µ : M/S 1 R 3 is a local homeomorphism near a fixed point x, V(µ(y)) = 1 2 µ(y) µ(x) 1 + φ(µ(y)) with φ positive harmonic (Bôcher s Theorem; a Chern class) µ : N 3 = (M \ M X )/S 1 R 3 is conformal: conformal factor V, positive harmonic can adjust to V superharmonic, so that N becomes complete and use Schoen and Yau (1994) to show µ : N R 3 is injective with image Ω having boundary that is polar V is then given by a Martin integral representation supported on Ω; completeness of M forces Ω to be discrete.

13 Delzant HyperKähler G2 4d toric Toric hyperkähler (Dancer and Swann 2016) M 4n complete hyperkähler with tri-hamiltonian action of T n. Is given locally by the Pedersen-Poon Ansatz: g = (V 1 ) ij (dt + ω i )(dt + ω j ) + V ij (dx i dx j + dy i dy j + dz i dz j ), with V ij = 2 F x i x j with F a positive function on R 3 R n harmonic on every affine three-plane X a,v = a + R 3 v. For generic X a,v, then Y = µ 1 (X a,v ) is smooth with free T n 1 action, Y/T n 1 is complete hyperkähler of dimension 4 with S 1 -action. Above analysis then fixes V on X a,v, and F, providing a classification. All examples may be constructed as hyperkähler quotients of flat affine subspaces of Hilbert spaces, cf. Goto 1994; Hattori 2011.

14 Delzant HyperKähler G2 4d toric Hypertoric configuration data In the hypertoric situation µ is surjective: µ(m 4n ) = R 3n = R 3 R n = Im H R n. Polytope faces are replaced by affine flats of codimension 3: H k = H(u k, λ k ) = {a Im H R n a, u k = λ k }, u k R n, λ k Im H. Again stabilisers of points mapping to H k are contained in the subtorus with Lie algebra orthogonal to u k, forcing u k Q n. This time H(u k1, λ k ) H(u kn, λ kr ) = whenever u k1,..., u kr are linearly independent. Smoothness implies each such set u k1,..., u kr is part of a Z-basis for Z n, giving a global restriction on the u k s. Get only finitely many distinct vectors u k, but possibly infinitely many λ k s.

15 Delzant HyperKähler G2 G 2 manifolds M 7 with ϕ Ω 3 (M) pointwise of the form ϕ = e e e e 246 e 257 e 356 e 347, e ijk = e i e j e k. ϕ specifies a metric g and an orientation The holonomy of g lies in G 2 when dϕ = 0 = d ϕ g is then Ricci-flat

16 Delzant HyperKähler G2 Multi-moment maps: T 2 symmetry (Madsen and Swann 2012) Suppose T 2 acts preserving (M, ϕ), holonomy in G 2, with generating vector fields U, V. The Cartan formula implies U V ϕ is closed. A function ν : M R with dν = U V ϕ is called a multi-moment map. At regular values X 4 = ν 1 (x)/t 2 is a four manifold carrying three symplectic forms of the same orientation induced by U ϕ, V ϕ, U V ϕ. These do not form a hyperkähler structure in general. (M, ϕ) may be recovered from the four-manifold X 4 by building a T 2 -bundle Y 6, constructing an SU(3) geometry (σ, ψ + ) on this bundle and then using an adaptation of the Hitchin flow for time derivatives of these forms.

17 Delzant HyperKähler G2 Multi-moment maps: T 3 symmetry Suppose T 3 acts preserving (M, ϕ), holonomy in G 2, with generating vector fields U, V, W. Multi-moment maps ν U, ν V, ν W given by dν U = V W ϕ, etc. Hamiltonian condition is that this should be T 3 -invariant, is equivalent to ϕ(u, V, W) = 0. There is a fourth multi-moment µ associated to ϕ via dµ = U V W ϕ. (ν U, ν V, ν W, µ) : M 7 R 4 has generic fibre T 3. What is the analogue of the Gibbons-Hawking Ansatz?

18 References I References Alekseevskiĭ, D. V. and B. N. Kimel fel d (1975), Structure of homogeneous Riemannian spaces with zero Ricci curvature, Funktsional. Anal. i Prilozhen. 9:2, pp. 5 11, trans. as Structure of homogeneous Riemannian spaces with zero Ricci curvature, Functional Anal. Appl. 9:2, pp Anderson, M. T., P. B. Kronheimer and C. LeBrun (1989), Complete Ricci-flat Kähler manifolds of infinite topological type, Comm. Math. Phys. 125:4, pp Bielawski, R. (1999), Complete hyper-kähler 4n-manifolds with a local tri-hamiltonian R n -action, Math. Ann. 314:3, pp Dancer, A. S. and A. F. Swann (2016), Hypertoric manifolds and hyperkähler moment maps, arxiv: [math.dg].

19 References II References Goto, R. (1994), On hyper-kähler manifolds of type A, Geometric and Funct. Anal. 4, pp Hattori, K. (2011), The volume growth of hyper-kähler manifolds of type A, Journal of Geometric Analysis 21:4, pp Madsen, T. B. and A. F. Swann (2012), Multi-moment maps, Adv. Math. 229, pp Schoen, R. and S.-T. Yau (1994), Lectures on differential geometry, Conference Proceedings and Lecture Notes in Geometry and Topology, I, Lecture notes prepared by Wei Yue Ding, Kung Ching Chang [Gong Qing Zhang], Jia Qing Zhong and Yi Chao Xu, Translated from the Chinese by Ding and S. Y. Cheng, Preface translated from the Chinese by Kaising Tso, International Press, Cambridge, MA, pp. v+235.

20 References III References Swann, A. F. (2016), Twists versus modifications, Adv. Math. 303, pp

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