Manifolds with exceptional holonomy
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1 Manifolds with exceptional holonomy Simon Salamon Giornata INdAM, 10 June 2015, Bologna
2 What is holonomy? 1.1 Holonomy is the subgroup generated by parallel transport of tangent vectors along all possible loops on a smooth manifold M. This notion has sense whenever a connection is assigned on TM. This applies to any Riemannian manifold with metric g ij dx i dx j i,j and Christoffel symbols Γ k ij = 1 2 m g km( g im x j + g jm x i g ij x m ), which define the Levi-Civita covariant derivative operator. But often, g and will not be known explicitly!
3 The 2-sphere 1.2 With longitude x 1 and latitude x 2, the metric on S 2 is g = (cos x 2 ) 2 (dx 1 ) 2 + (dx 2 ) 2, giving orthonormal tangent vectors e 1 =(sec x 2 ) x 1, e 2 = x 2. The vector field X shown (along x 2 = π/4) satisfies e1 X = 0.
4 The 2-sphere 1.2 With longitude x 1 and latitude x 2, the metric on S 2 is g = (cos x 2 ) 2 (dx 1 ) 2 + (dx 2 ) 2, giving orthonormal tangent vectors e 1 =(sec x 2 ) x 1, e 2 = x 2. The vector field X shown (along x 2 =21π/90) satisfies e1 X = 0. The holonomy of S 2 = CP 1 lies in SO(2) = U(1). The complex structure J is defined infinitesimally by Je 1 = e 2 and Je 2 = e 1.
5 Complex projective space 1.3 The prototypical compact complex manifold is CP n. The Fubini-Study distance between two points represented by unit vectors w, z C n+1 equals 2 arccos ρ where ρ = w, z z, w w, z 2 = w, w z, z w 2 z 2 is a cross ratio and transition probability. The holonomy group of the associated metric g is the subgroup U(n) of SO(2n). By definition, g is a Kähler metric. Using this, one can show that the isometry group of CP n is generated by SU(n+1)/Z n+1 and complex conjugation [Wigner].
6 Kähler manifolds 1.4 Any algebraic submanifold M of CP N has an induced metric g and orthogonal complex structure J such that J 2 = 1, J = 0 making M a Kähler manifold with holonomy U(n) (2n =dim R M ). There is an associated symplectic form ω given by ω(x, Y ) = g(x, JY ), dω = 0. Example. { 5 K = The induced metric on the smooth intersection i=0 } { 5 } { 5 } zi 2 = 0 λ i zi 2 = 0 µ i zi 2 = 0 i=0 i=0 of 3 quadrics in CP 5 has holonomy U(2). Any K3 surface actually has a metric with holonomy SU(2) [Yau], making it hyperkähler.
7 Timeline (act 1) : É. Cartan classifies symmetric spaces G/H (e.g. E 6 /F 4 ). 1952: De Rham shows that reducible holonomy gives products. 1955: Berger lists potential holonomy representations remaining : Sp(n)U(1) and Spin 9 eliminated from his list, leaving the following (for oriented irreducible manifolds with R 0): real dim holonomy geometric type n SO(n) generic 2n U(n) Kähler 2n SU(n) Ricci-flat Kähler 4n Sp(n) hyperkähler 4n Sp(n)Sp(1) quaternion-kähler 7 G 2 exceptional 8 Spin 7 exceptional Metrics with exceptional holonomy are Ricci-flat.
8 Georges de Rham and Marcel Berger 2.2
9 Timeline (act 2) : It is claimed that there do not exist compact manifolds with holonomy group Sp(n) and n > 1. Pessimism about the existence of metrics with holonomy G 2 or Spin 7 even locally. 1978: Yau proves the Calabi conjecture, and so the existence of compact manifolds with holonomy equal to SU(n). 1983: Beauville discovers two families of compact hyperkähler manifolds (holonomy Sp(n)) on Hilbert schemes of points. 1987: Bryant establishes local existence of metrics with exceptional holonomy G 2 and Spin : First explicit examples are cones. Bryant-S go on to describe families of complete examples.
10 Timeline (act 3) : Joyce proves the existence of metrics with holonomy G 2 and Spin 7 on compact manifolds (resolutions of T 7 /Γ and T 8 /Γ) : Joyce develops lots of new families of examples by resolving Calabi-Yau orbifolds. 2003: Kovalev glues asymptotically cylindrical spaces from two Fano 3-folds to obtain new compact G 2 manifolds. 2012: Corti-Haskins-Nordström-Pacini introduce new rigour and find millions of new compact G 2 manifolds via semi-fano 3-folds. 2013: Kovalev extends his work to Spin : Foscolo-Haskins find new compact nearly-kähler (Einstein) metrics on S 6 and S 3 S 3 (invariant by actions of SU(2) SU(2) or cohomogeneity one, input from Conti-S and Podestà-Spiro).
11 Spin 7 and G 2 holonomy: triality 3.1 Spin 8 has three inequivalent representations on R 8. One remains irreducible when restricted to Spin 7 Spin 8 Aut(R 8 ). The subgroup of Spin 7 fixing v R 8 is the 14-dimensional exceptional Lie group G 2. Moreover, G 2 SO(7) Aut(R 7 ), and S 7 = Spin 7/G 2. The subgroup of G 2 fixing w R 7 is SU(3) and S 6 = G 2 /SU(3). The possible (non-symmetric, irreducible) holonomy groups all act transitively on the unit sphere in T x M [Simons].
12 James Simons s contribution 3.2
13 Bianchi and Einstein 3.3 The Riemann curvature tensor R ijkl of an n-dimensional manifold is skew in i, j and k, l but symmetric in (i, j) (k, l), and ( ) R ker S 2 (Λ 2 (R n )) Λ 4 (R n ). The Ricci tensor R jl = i,k g ik R ijkl defines an element of S 2 (R n ). If the holonomy group is H then h so(n) = Λ 2, and R S 2 (h). Example. S 2 (g 2 ) = V 77 S 2 0 (R 7 ) R. Corollary. A metric with holonomy G 2 has R V 77 and R jl = 0. A metric with holonomy Spin 7 has R W 168 and is also Ricci-flat.
14 G 2 as the stablizer of a 3-form 3.4 On a Kähler manifold (holonomy U(n)), J = 0 and dω = 0. The action of G 2 on R 7 = ImO fixes (in suitable coordinates) ϕ = e 125 e e 136 e e 147 e e 567 Λ 3 (R 7 ). In fact G 2 = {g GL(7, R) : g ϕ = ϕ}, and GL(7, R)/G 2 is an open orbit of Λ 3 (R 7 ). Proof: = ( 7 3). The 3-form ϕ determines the metric and the 4-form ϕ. Lemma [Fernández-Gray]. ϕ=0 dϕ=0 and d ϕ=0.
15 Conical G 2 holonomy 3.5 In polar coordinates, the metric R 7 with holonomy group {e} is g = dr 2 + r 2 h where h is the standard metric on S 6 = G 2 /SU(3). The latter is nearly-kähler and admits a 2-form ω and a 3-form ψ such that dω = ψ and d ψ = ω ω [Reyes-Carrión]. Then defines a G 2 structure with ϕ = ω dr + r ψ, dϕ = 0, ϕ = ψ dr + r ω ω, d ϕ = 0. If we replace S 6 with one of the other nearly-kähler spaces N = S 3 S 3, CP 3, F = SU(3)/T 2, the conical metric on R + N has holonomy group equal to G 2. End of first half
16 Complete SU(2) holonomy: Eguchi-Hanson 4.1 The singular space X = C 2 /±1 has functions u = z 2 1, v = z2 2, w = z 1z 2, embedding it in C 3 as the cone uv = w 2. With the origin removed, it is identified with {α T CP 1 : α 0} since Y = T CP 1 = O( 2) = L L. There is a crepant resolution ρ: Y X. The cotangent bundle admits a symplectic from ω 2 +iω 3 and a metric g of the form (r 2 + 1) 1/2 (e 1 e 1 +e 2 e 2 ) + (r 2 + 1) 1/2 (e 3 e 4 +e 4 e 4 ) }{{}}{{} horizontal vertical defining ω 1 with dω 1 = 0 also, so g has holonomy SU(2). The space Y asymptotically locally Euclidean (ALE).
17 Compact SU(2) holonomy: K3 4.2 An algebraic curvature tensor R ijkl S 2 (Λ 2 (C 4 )) defines a quartic Kummer surface { K R = [x i ] CP 3 : } R iapq R bjkr R cdsl ε abcd ε pqrs x i x j x k x l = 0 with 16 singular points. In fact, K R = T 4 /Z 2 ; each singularity is modelled on C 2 /±1. There is a global resolution K K R where (as before) K = Q 1 Q 2 Q 3 CP 5 admits a metric with holonomy SU(2), by Yau s theorem. This is a model for producing compact 7-manifolds with holonomy G 2.
18 Complete G 2 holonomy 4.3 Theorem [Bryant-S]. There exist complete metrics with holonomy G 2 on the rank 3 vector bundles Λ 2 T M for M = S 4 or CP 2. The metrics arise from the subgroup SO(4) of G 2 for which R 7 = R 4 R 3 = e 1, e 2, e 3, e 4 e 5, e 6, e 7 enabling one to identify R 3 = Λ 2 (R 4 ) and define a 3-form ϕ = (e 12 e 34 ) e 5 + (e 13 e 42 ) e 6 + (e 14 e 23 ) e 7 + e 567. With more careful scaling, the associated metric (r 2 + 1) 1/2 4 e i e i +(r 2 + 1) 1/2 7 e i e i 1 }{{} horizontal 5 } {{ } vertical has holonomy equal to G 2, and is asymptotic to the conical metric on R + CP 3 or R + F as r.
19 Compact G 2 holonomy: Joyce 4.4 Idea. Seek a finite subgroup Γ acting on T 7 = R 7 /Z 7 preserving the 3-form ϕ, so T 7 /Γ is a G 2 orbifold with amenable singularities. We shall use inclusions SU(2) + SO(4) G 2 Example. Let Γ be the abelian group (Z 2 ) 3 generated by α(x) = (x 1, x 2, x 3, x 4, x 5, x 6, x 7, ) β(x) = (x 1, x 2, x 3, x 4, x 5, 1 2 x 6, x 7 ) γ(x) = ( x 1, x 2, x 3, x 4, 1 2 x 5, x 6, 1 2 x 7) While α, β, γ each fix 16 tori T 3, βγ, γα, αβ, αβγ have no fixed points. The singular set of T 7 /Γ consists of 12 disjoint 3-tori T 3 each with normal space C 2 /±1. Each Z 2 acts within SU(2) {e} on R 4 R 3 = R 7 for different R 4 s, so Γ G 2.
20 Compact G 2 holonomy: perturbation 4.5 In the example T 7 /Γ, we resolve each singular point by replacing its transverse neighbourhood by an open subset of the ALE space. Each singular T 3 is then surrounded by a tube RP 3 T 3 outside of which the ALE metric merges with the flat one. In this way, we get a smooth 7-manifold with a 1-parameter family of closed 3-forms ϕ t of G 2 -type for t > 0, and d ϕ L 14 t 16/7. Theorem [Joyce]. In this and similar situations, for sufficiently small t, there exists a G 2 3-form ϕ = ϕ t + dη such that ϕ = 0. Proof. Involves setting up a non-linear PDE for the 2-form η, and sophisticated estimates involving curvature and injectivity radius.
21 G 2 geometry in the 21st century 5.1 Donaldson Kovalev: arxiv:math/ Kovalev-Lee: arxiv: Kovalev-Nordström: arxiv: Corti-Haskins-Nordström-Pacini: arxiv: Corti-Haskins-Nordström-Pacini: arxiv: Crowley-Nordström: arxiv:
22 Semi-Fano 3-folds 5.2 A Fano 3-fold is a projective variety with ample anticanonical bundle K. There are 105 deformation types, of which 17 have b 2 =1 (including a quadric, cubic, quartic in CP 4 ). A weak Fano 3-fold Y is a generalization (requiring K to be nef and K 3 = 2g 2 > 0) that comes equipped with a resolution ρ: Y X where X is a (possibly) singular Fano 3-fold. If ρ is semi-small (no divisor maps to a point) Y is semi-fano. Example. An important case is that in which X is nodal, with a finite number of ordinary double points x. Then ρ replaces x by CP 1 with ν = O( 1) O( 1). A general quartic X containing a plane Π in CP 4 has 9 nodes on Π, and two small projective resolutions Y X with Y semi-fano.
23 Asymptotically cylindrical Calabi-Yau spaces 5.3 By blowing up a semi-fano 3-fold Y along a curve (the base locus of a pencil of anticanonical divisors), one obtains a smooth 3-fold Z with a morphism f : Z CP 1 such that f 1 ( ) = K K is a K3 surface that generates H 2 (Z, Z). Topologically, V = Z \ K K S 1 R +. Theorem [à la Tian-Yau]. V = Z \ K is ACCY: it possesses a Ricci-flat metric with holonomy SU(3), approximating the product metric with holonomy SU(2) {e} on the cylinder, with all derivatives bounded by e λt (some λ > 0) as t.
24 Twisted connected sums 5.4 Take two ACCY 3-folds V ±. Trick is to identify the 7-manifolds V ± S 1 by gluing along the neck as follows K + S 1 S 1 (T, T +1) r K S 1 S 1 (T, T +1) using an isometry r that performs a hyperkähler rotation to mimic the switch of S 1 factors. By the Torelli theorem, r is determined by r : H 2 (K, Z) H 2 (K +, Z). The gluing ensures that the underlying G 2 structures on each R 7 = R 4 R 3 are identified. The resulting compact simply connected 7-manifold M has a closed G 2 form ϕ with k (d ϕ) = O(e λt ), k. For sufficiently large T, ϕ can be perturbed (in its cohomology class, as before) so that ϕ = 0.
25 Enumeration 5.5 We need building block 3-folds Z ± for ACCY s to glue together. There is at least one for almost all Fano or semi-fano 3-folds. H 2 (Z ±, Z) defines a lattice N ± in Pic(K ) H 2 (K ) = ( E 8 ) ( E 8 ) H H H. Typically, we need to arrange N +, N to be orthogonal, which is easy if rkn + + rkn 11. There is no problem matching the 17 Fano 3-folds with b 2 = 1. The 153 = ( ) 18 2 choices give at least 82 topologically distinct smooth manifolds M with holonomy G 2, plotted by b 3 (M)
26 Mass production 5.6 There are 105 deformation classes of Fano 3-folds of the 5564= ( ) pairs can be matched and have b2 = 0. But at most 560 diffeomorphism classes (determined by b 3 and p 1 ). Toric Fano 3-folds correspond to 3-dimensional reflexive polytopes. There are 4319 of these, including 18 from smooth Fano 3-folds and 82 terminal (with nodal singularities). Theorem. There are 526,130 isomorphism classes of smooth toric semi-fano 3-folds (1009 with nodal base), and 435,459 are rigid. Of these, 39,584 pairs satisfy n + + n 11 but give no new b 3 s. Restricting one summand to semi-fano s of rank 1 and 2, one can generate at the very least 246, 446.( ) = 50, 027, 726 types of metrics with holonomy G 2 on 7-manifolds with b 2 = 0 and 55 b 3 287, suggesting the moduli of G 2 metrics is very rich.
27 Related topics 6.1 Interpretations of holonomy via parallel and Killing spinors [Bär]. This is exploited in M-theory, combining N =1 SUSY with 11D SUGRA [Witten, Acharya]. Calibrated submanifolds (associative or coassociative in M 7, Cayley in M 8, fibrations by K3 surfaces, mirror symmetry). Instantons over manifolds M 7 and M 8 with exceptional holonomy. Constructions of conical/complete/compact 8-manifolds with holonomy Spin 7. Understanding (i) the topology of manifolds admitting a metric of exceptional holonomy, and (ii) the moduli spaces of such metrics.
28 Spin 7 by evolution 6.2 Any hypersurface of R 8 inherits a G 2 structure ϕ with d ϕ = 0 from a standard Spin 7 invariant 4-form. In fact, any 7-manifold M (with w 1 = w 2 = 0) admits such a co-calibrated G 2 structure. If this evolves in time t, the 4-form Φ = dt ϕ(t) + ϕ(t), defines a Spin 7 structure on R M. It is closed provided ( ϕ) = dϕ t a flow which has short-time existence. Therefore, (0, ε) M will always admit a metric with holonomy contained in Spin 7. Example. Berger s homogeneous space M 7 = SO(5)/SO(3) has a G 2 structure with dϕ = ϕ. Therefore, R + M 7 has an explicit conical metric with holonomy Spin 7.
29 Spin 7 to the aid of G Lemma. If a closed 8-manifold W has a Spin 7 or Sp(2)Sp(1) structure then 48Â = 3σ χ, where 5760Â = 7p2 1 4p 2. Examples. HP 2 and G 2 /SO(4) satisfy the equality with Â=0 and σ =1, but cannot admit a metric with holonomy Spin 7. Any closed G 2 manifold M 7 can be represented as the boundary of a Spin 7 manifold W, so we can define a diffeomorphism invariant ν(m) = χ(w ) 3σ(W ) mod 48. If M is a twisted connected sum ν(m)=24, but a refinement of the Eells-Kuiper invariant can distinguish matchings. The existence is known of pairs of metrics with holonomy G 2 that are: (i) disconnected within the same homotopy class of G 2 structures, (ii) on homeomorphic non-diffeomorphic manifolds [Crowley et al.]
30 From dimension 5 to 6 to A metric h on a 6-manifold M is nearly-kähler iff the cone dr 2 + r 2 h has holonomy contained in G 2 [Bär]. Such a manifold has an SU(3) structure such that ( X J)(X ) = 0, X. The nearly-kähler spaces S 6, S 3 S 3, CP 3 all admit an action by SU(2) SU(2) with generic orbit E = S 2 S 3. A metric k on a 5-manifold is Einstein-Sasaki iff the cone dy 2 + y 2 k has holonomy contained in SU(3). This is true of E : its conifold has two Calabi-Yau desingularizations in which the vertex is replaced by S 2 and S 3 respectively. Corollary. The sine cone dθ 2 + (sin θ) 2 k makes E = (0, π) E nearly-kähler. Proof. dr 2 + r 2( dθ 2 }{{} +(sin θ)2 k ) = dx 2 + ( dy 2 }{{} +y 2 k ).
31 Two new nearly-kähler metrics 6.5 We consider finally the action of SU(2) SU(2) of cohomogeneity one on compactifications of E = (0, π) S 2 S 3. The sine cone can also be desingularized in two ways. Deforming the metrics one recovers the homogeneous nearly-kähler spaces and more beside: S 6 : S 2 E S 3 S 3 S 3 : S 3 E S 3 CP 3 : S 2 E S 2 Theorem [Foscolo-Haskins 2015]. S 6 and S 3 S 3 each admit a non-homogeneous nearly-kähler metric (and a new almost complex structure such that their cone has a metric with holonomy G 2 ).
32 Updated references 6.6 Kovalev: arxiv:math/ (35 pp) Kovalev-Lee: arxiv: (30 pp) Kovalev-Nordström: arxiv: (36 pp) Corti-Haskins-Nordström-Pacini: arxiv: (83 pp) Corti-Haskins-Nordström-Pacini: arxiv: (107 pp) Crowley-Nordström: arxiv: (26 pp) Crowley-Nordström: arxiv: (24 pp) Foscolo-Haskins, arxiv: (53 pp) Crowley-Goette-Nordström, arxiv: (26 pp) Coates-Haskins-Kasprzyk-Nordström salamon salamon
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