Moment Maps and Toric Special Holonomy

Transcription

1 Department of Mathematics, University of Aarhus PADGE, August 2017, Leuven DFF

2 Delzant HyperKähler G 2 Outline The Delzant picture HyperKähler manifolds Hypertoric Construction G 2 manifolds Multi-Hamiltonian Toric G 2

3 Delzant HyperKähler G 2 The Delzant picture Compact symplectic toric manifolds Delzant polytopes (M 2n, ω) symplectic with a Hamiltonian action of G = T n : have G-invariant µ : M g R n with d µ, X = X ω X g the moment map µ invariant ω pulls-back to 0 on each T n -orbit b 1 (M) = 0 = each symplectic T n -action is Hamiltonian dim(m/t n ) equals dimension of target space of µ Stabiliser of any point is a (connected) subtorus of dimension n rankdµ

4 Delzant HyperKähler G 2 The Delzant polytope = {a R n a,u k λ k, k = 1,...,m} = µ(m) with faces F k = { µ,u k = λ k } satisfying smoothness F k1 F kr = u k1,...,u kr are part of a Z-basis for Z n R n compactness the intersection of any n faces is a point M is constructed as the symplectic quotient of C m by the Abelian group N given by at level λ = (λ k ) 0 N T m T n 0 0 n R m β R n 0 β(e k ) = u k

5 Delzant HyperKähler G 2 Hypertoric Construction HyperKähler manifolds (M,д, ω I, ω J, ω K ) is hyperkähler if each (д, ω A = д(a, )) is Kähler and I J = K = JI Then dim M = 4n and д is Ricci-flat, holonomy in Sp(n) SU(2n) Ricci-flatness implies: if M is compact, then any Killing vector field is parallel so the holonomy of M reduces So take (M,д) non-compact and complete instead Swann (2016) and Dancer and Swann (2016), following Bielawski (1999), Bielawski and Dancer (2000) and Goto (1994)

6 Delzant HyperKähler G 2 Hypertoric Construction Hypertoric is complete hyperkähler M 4n with tri-hamiltonian G = T n action: have G-invariant map (hyperkähler moment map) µ = (µ I, µ J, µ K ): M R 3 g d µ A, X = X ω A dim(m/t n ) equals dimension 3n of target space of µ Stabiliser of any point is a (connected) subtorus of dimension n 1 3 rankdµ Locally (Lindström and Roček 1983) д = (V 1 ) ij θ i θ j + V ij (dµ i I dµj I + dµi J dµj J + dµi K dµj K ), with V ij positive-definite and harmonic on every affine three-plane X a,v = a + R 3 v

7 Delzant HyperKähler G 2 Hypertoric Construction Hypertoric configuration data For M 4n hypertoric, µ is surjective: µ(m 4n ) = R 3n = R 3 R n = Im H R n All stabilisers are (connected) subtori and µ induces a homeomorphism M/T n R 3n 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 Z n, λ k Im H smoothness H(u k1, λ k1 ) H(u kr, λ kr ) = u k1,...,u kr part of a Z-basis for Z n Get only finitely many distinct vectors u k, but possibly infinitely many λ k s is

8 Delzant HyperKähler G 2 Hypertoric Construction Construction and classification Choose L finite or countably infinite and Λ = (Λ k ) k L H L, so that with λ k = 1 2 Λ kiλ k have k L(1 + λ k ) 1 < Let L 2 (H) = {v H L k L v k 2 < } Hilbert manifold M Λ = Λ + L 2 (H) is hyperkähler with action of Hilbert group T λ = {д (S 1 ) L k L(1 + λ k ) 1 д k 2 < } Suppose u k Z n are such that {H(u k, λ k ) k L} satisfy the smoothness condition and define the Hilbert group N β by 0 N β T λ T n 0 0 n β t λ β R n 0 β(e k ) = u k Then the hyperkähler quotient of M Λ by N β is hypertoric and every simply-connected hypertoric manifold arises in this way up to adding a (positive semi-definite) constant matrix to (V ij )

9 Delzant HyperKähler G 2 Multi-Hamiltonian Toric G 2 G 2 manifolds M 7 with φ Ω 3 (M) pointwise of the form φ = e 123 e 145 e 167 e 246 e 257 e 347 e 356, e ijk = e i e j e k Specifies metric д = e e2 7, orientation vol = e and four-form φ = e 4567 e 2345 e 2367 e 3146 e 3175 e 1256 e 1247 via 6д(X,Y) vol = (X φ) (Y φ) φ Holonomy of д is in G 2 when dφ = 0 = d φ, a parallel G 2 -structure Then д is Ricci-flat

10 Delzant HyperKähler G 2 Multi-Hamiltonian Toric G 2 Multi-Hamiltonian actions Joint work with Thomas Bruun Madsen (M, α) manifold with closed α Ω p (M) preserved by G = T n This is multi-hamiltonian if it there is a G-invariant ν : M Λ p 1 g with d ν, X 1 X p 1 = α(x 1,..., X p 1, ) for all X i g take n > p 2 ν invariant α pulls-back to 0 on each T n -orbit b 1 (M) = 0 = each T n -action preserving α is multi-hamiltonian For (M, φ) a parallel G 2 -structure, can take α = φ and/or α = φ

11 Delzant HyperKähler G 2 Multi-Hamiltonian Toric G 2 Multi-Hamiltonian parallel G 2 -manifolds Proposition Suppose (M, φ) is a parallel G 2 -manifold with T n -symmetry multi-hamiltonian for α = φ and/or α = φ. Then 2 n 4. n = 2, α = φ ν : M Λ 2 R 3 = R 1, 1 < 5 = dim(m/t n ) n = 3, α = φ ν : M Λ 2 R 3 = R 3, 3 < 4 = dim(m/t n ) n = 3, α = φ µ : M Λ 3 R 3 = R, 1 < 4 = dim(m/t n ) n = 3, α = φ and α = φ (ν, µ): M R 3 R = R 4, 4 = 4 = dim(m/t n ) (T) n = 4, α = φ ν : M Λ 2 R 4 = R 6, 6 > 3 = dim(m/t n ) n = 4, α = φ ν : M Λ 3 R 4 = R 4, 4 > 3 = dim(m/t n ) (A) Madsen and Swann (2012) (B) Baraglia (2010) (A) (B)

12 Delzant HyperKähler G 2 Multi-Hamiltonian Toric G 2 Toric G 2 Definition A toric G 2 manifold is a parallel G 2 -structure (M, φ) with an action of T 3 multi-hamiltonian for both φ and φ Let U 1, U 2, U 3 generate the T 3 -action, then φ(u 1,U 2,U 3 ) = 0, multi-moment maps (ν, µ) = (ν 1,ν 2,ν 3, µ): M R 4 dν i = U j U k φ (i j k) = (1 2 3) dµ = U 1 U 2 U 3 φ Example M = S 1 C 3 has standard flat φ = i 2 dx(dz 11 + dz 22 + dz 33 ) + Re(dz 123) preserved by S 1 T 2 S 1 SU(3) 4(ν 1 iµ) = z 1 z 2 z 3, 4ν 2 = z 2 2 z 3 2, 4ν 3 = z 3 2 z 1 2

13 Delzant HyperKähler G 2 Multi-Hamiltonian Toric G 2 Proposition All isotropy groups of the T 3 action are connected and act on the tangent space as maximal tori in 1 SU(3), 1 3 SU(2) or 1 7 At a point p in a principal orbit U 1, U 2, U 3 are contained in a coassociative subspace of T p M and (dν,dµ) has full rank 4 Let M 0 be the points with trivial isotropy Then (ν, µ) induces a local diffeomorphism M 0 /T 3 R 4 Proposition For the flat structure on M = S 1 C 3 the quotient M/T 3 is homeomorphic to R 4 and (ν, µ) induces a homeomorphism Corollary For any toric G 2 -manifold M, the quotient M/T 3 is a topological manifold

14 Delzant HyperKähler G 2 Multi-Hamiltonian Toric G 2 Local form (M, φ) toric G 2 with generating vector fields U i M 0 M 0 /T 3 is a principal torus bundle with connection one-forms θ i Ω 1 (M 0 ) satisfying θ i (U j ) = δ ij, θ i (X ) = 0 X U 1,U 2,U 3 On M 0, put B = ( д(u i,u j ) ) and V = B 1 = 1 det B adj B Theorem On M 0 д = 1 detv θ t adj(v ) θ + dν t adj(v )dν + det(v )dµ 2 φ = det(v )dν dµdν t adj(v ) θ + S i, j,k θ ij dν k φ = θ 123 dµ det(v ) ( dν t adj(v ) θ ) 2 + det(v )dµ S i, j,k θ i dν jk

15 Delzant HyperKähler G 2 Multi-Hamiltonian Toric G 2 Theorem (continued) Such a (д, φ, φ) defines a parallel G 2 -structure if and only if V C (M 0 /T 3, S 2 R 3 ) is a positive-definite solution to and 3 i=1 L(V ) + Q(dV ) = 0 V ij ν i = 0 j = 1, 2, 3 (divergence-free) (elliptic) where L = 2 µ V ij ν i ν j k,l and Q is a quadratic form with constant coefficients

16 Delzant HyperKähler G 2 Multi-Hamiltonian Toric G 2 Proposition Solutions V to the divirgence-free equation are given locally by A C (M 0 /T 3, S 2 R 3 ) via V ii = 2 A j j ν k A kk ν j 2 2 A jk ν j ν k (i j k) = (1 2 3) V ij = 2 A ik + 2 A jk 2 A ij ν j ν k ν i ν k ν 2 k 2 A kk ν i ν j

17 Delzant HyperKähler G 2 Multi-Hamiltonian Toric G 2 E.g. V 3 = µ, V 2 = µ 3 3ν3 2, V 1 = 2µ 5 15µ 2 ν3 2 5ν 2 2 Example solutions Example Bryant-Salamon metrics and their generalisations by Brandhuber et al. (2001) on S 3 R 4 : complete, cohomogeneity one with symmetry group SU(2) SU(2) S 1 Z/2 Example Diagonal V = diag(v 1,V 2,V 3 ). (divergence-free) V i / ν i = 0. Off-diagonal terms in (elliptic) give V i V j ν j ν i = 0. Either V = diag(v 1 (ν 2, µ),v 2 (ν 3, µ),v 3 (ν 1, µ)) linear in each variable Or have an elliptic hierachy V 3 = V 3 (µ), V 2 = V 2 (ν 3, µ), V 1 = V 1 (ν 2,ν 3, µ) 2 V 3 µ 2 = 0 2 V 2 µ 2 + V 2 V 2 3 ν 2 = 0 2 V 1 3 µ 2 + V 2 V 1 2 ν 2 + V 2 V ν 2 = 0 3

18 References References I Baraglia, D. (2010), Moduli of coassociative submanifolds and semi-flat G 2 -manifolds, J. Geom. Phys. 60:12, pp Bielawski, R. (1999), Complete hyper-kähler 4n-manifolds with a local tri-hamiltonian R n -action, Math. Ann. 314:3, pp Bielawski, R. and A. S. Dancer (2000), The geometry and topology of toric hyperkähler manifolds, Comm. Anal. Geom. 8:4, pp Brandhuber, A., J. Gomis, S. S. Gubser and S. Gukov (2001), Gauge Theory at Large N and New G 2 Holonomy Metrics, Nuclear Phys. B 611:1-3, pp Dancer, A. S. and A. F. Swann (2016), Hypertoric manifolds and hyperkähler moment maps, arxiv: [math.dg]. Goto, R. (1994), On hyper-kähler manifolds of type A, Geometric and Funct. Anal. 4, pp

19 References References II Lindström, U. and M. Roček (1983), Scalar tensor duality and N = 1, 2 nonlinear σ-models, Nuclear Phys. B 222:2, pp Madsen, T. B. and A. F. Swann (2012), Multi-moment maps, Adv. Math. 229, pp Swann, A. F. (2016), Twists versus modifications, Adv. Math. 303, pp