DIFFERENTIAL GEOMETRY AND THE QUATERNIONS. Nigel Hitchin (Oxford) The Chern Lectures Berkeley April 9th-18th 2013

DIFFERENTIAL GEOMETRY AND THE QUATERNIONS Nigel Hitchin (Oxford) The Chern Lectures Berkeley April 9th-18th 2013

16th October 1843

ON RIEMANNIAN MANIFOLDS OF FOUR DIMENSIONS 1 SHIING-SHEN CHERN Introduction. It is well known that in three-dimensional elliptic or spherical geometry the so-called Clifford's parallelism or parataxy has many interesting properties. A group-theoretic reason for the most Let #o, #1» ^2, #3 be the coordinates of a point with respect to a frame Co, ei, e2, e 3, as defined by (2). To these coordinates we associate a unit quaternion (4) X = XQ + xii + X2J + xzk, N(X) = 1, where N(X) denotes the norm of X. Let (4a) X* = x* + x*i+ x*j + x?k. S-S.Chern, On Riemannian manifolds of four dimensions, Bull. Amer. Math. Soc. 51 (1945) 964 971.

Quaternions came from Hamilton after his best work had been done, and though beautifully ingenious, they have been an unmixed evil to those who have touched them in any way Lord Kelvin 1890

GEOMETRY OVER THE QUATERNIONS

q H quaternions q = x 0 + ix 1 + jx 2 + kx 3 algebraic variety? f(q 1,..., q n ) = 0 q 2 + 1 = 0: 2-sphere q = ix 1 + jx 2 + kx 3, x 2 1 + x2 2 + x2 3 = 1

submanifold M H n T x M H n T x M quaternionic for all x M M = H m H n

INTRINSIC DIFFERENTIAL GEOMETRY

quaternionic structure on the tangent bundle T affine connection X Y zero torsion X Y Y X = [X, Y ]

H n n-dimensional quaternionic vector space left action by GL(n, H) commutes with right action of H GL(n, H) H

metric maximal compact subgroup Sp(n) Sp(1) GL(n, H) H Levi-Civita connection : preserving metric unique torsion-free connection Quaternionic Kähler preserves quaternionic structure

GL(n, H) preserves action of H on tangent bundle T I,J,K End(T ) such that I 2 = J 2 = K 2 = IJK = 1 metric Sp(n) GL(n, H) Levi-Civita connection : preserving metric unique torsion-free connection Hyperkähler preserves I,J,K

SL(n, H) U(1) preserves action of C on tangent bundle T if a torsion-free connection preserves this structure, it is unique complex quaternionic complex manifold volume form U(1) connection on K

SL(n, H) U(1) SL(1, H) U(1) = Sp(1) U(1) = SU(2) U(1) = U(2) for n = 1 complex quaternionic = Kähler complex surface with zero scalar curvature n > 1 complex quaternionic is non-metric

Lecture 1 Quaternionic manifolds Lecture 2 Hyperkähler moduli spaces Lecture 3 Twistors and holomorphic geometry Lecture 4 Correspondences and circle actions

THE HYPERKÄHLER QUOTIENT

hyperkähler manifold M 4k complex structures I, J, K + metric g Kähler forms ω 1, ω 2, ω 3

hyperkähler manifold M 4k complex structures I, J, K + metric g Kähler forms ω 1, ω 2, ω 3 ω i : T T, K = ω 1 1 ω 2 etc.

Lie group G acting on M, fixing ω 1, ω 2, ω 3 a g vector field X a d(i Xa ω i ) + i Xa dω i = L Xa ω i = 0

Lie group G acting on M, fixing ω 1, ω 2, ω 3 a g vector field X a d(i Xa ω i ) + i Xa dω i = L Xa ω i = 0 moment map i Xa ω i = dµ a i

µ : M g R 3 If G acts properly and freely on µ 1 (0) then...... the quotient metric on µ 1 (0)/G is hyperkähler...... of dimension dim M 4 dim G

EXAMPLE M = H n = C n + jc n flat hyperkähler manifold ω 1 = i 2 (dz k d z k + dw k d w k ) ω 2 + iω 3 = dz k dw k G = U(1) action u (z, w) = (uz, u 1 w)

EXAMPLE M = H n = C n + jc n flat hyperkähler manifold ω 1 = i 2 (dz k d z k + dw k d w k ) ω 2 + iω 3 = dz k dw k G = U(1) action u (z, w) = (uz, u 1 w) µ(z, w) = (z k z k w k w k, z k w k ) + const. R C = R 3 choice

µ(z, w) = (z k z k w k w k, z k w k ) + (1, 0) R C = R 3 µ 1 (0) : z 2 w 2 + 1 = 0 and z k w k = 0

µ(z, w) = (z k z k w k w k, z k w k ) + (1, 0) R C = R 3 µ 1 (0) : z 2 w 2 + 1 = 0 and z k w k = 0 w = 0 projection µ 1 (0) CP n 1 µ 1 (0)/U(1) = T CP n 1 Calabi metric, Eguchi-Hanson (n=2)

HERMITIAN SYMMETRIC SPACES O. Biquard, P. Gauduchon, Hyperkähler metrics on cotangent bundles of Hermitian symmetric spaces, in Lecture Notes in Pure and Appl. Math 184, 287 298, Dekker (1996) p : T (G/H) G/H ω 1 = p ω + dd c h h =(f(ir(ix,x))x, X), R curvature tensor, X T 1+u 1 log 1+ 1+u f(u) = 1 u 2

EXAMPLE M = H + H and G = R action t (q 1, q 2 ) = (e it q 1, q 2 + t)

EXAMPLE M = H + H and G = R action t (q 1, q 2 ) = (e it q 1, q 2 + t) µ 1 (0) : z 1 2 w 1 2 = im z 2 and z 1 w 1 = w 2 µ 1 (0)/R = C 2, coordinates (z 1, w 1 ) Taub-NUT metric

V harmonic function on R 3 dv = dα g = V (dx 2 1 + dx2 2 + dx2 3 )+V 1 (dθ + α) 2. ω 1 = Vdx 2 dx 3 + dx 1 (dθ + α) V = 1 2r + c

NJH, A. Karlhede, U. Lindström & M. Roček, Hyperkähler metrics and supersymmetry, Comm. Math. Phys. 108 (1987), 535 589. K.Galicki & H.B Lawson Jr. Quaternionic reduction and quaternionic orbifolds, Math. Ann. 282 (1988) 121.

QUATERNIONIC KÄHLER AND HYPERKÄHLER

Sp(n) Sp(1) GL(n, H) H Levi-Civita connection : preserving metric unique torsion-free connection Quaternionic Kähler preserves quaternionic structure

Sp(n) Sp(1) GL(n, H) H Levi-Civita connection : preserving metric unique torsion-free connection Quaternionic Kähler preserves quaternionic structure principal Sp(1) bundle with connection

T is a module over a bundle of quaternions (e.g. HP n ) equivalently a rank 3 bundle of 2-forms ω 1, ω 2, ω 3

T is a module over a bundle of quaternions (e.g. HP n ) equivalently a rank 3 bundle of 2-forms ω 1, ω 2, ω 3 ω 1 = θ 2 ω 3 θ 3 ω 2 curvature K 23 = dθ 1 θ 2 θ 3 etc. in fact K 23 = cω 1, c constant scalar curvature

P = SO(3) frame bundle θ i well-defined 1-forms on P dim P R + = 4n + 4

P = SO(3) frame bundle θ i well-defined 1-forms on P dim P R + = 4n + 4 define ϕ i = d(tθ i ) (t = R + coordinate) three closed 2-forms ϕ 1, ϕ 2, ϕ 3

T (P R + )=H V on H, θ i = 0 and dt = 0, ϕ i = tcω i on V, ϕ 1 = dt θ 1 + t 2 θ 2 θ 3 etc. algebraic relations for hyperkähler if c>0 Lorentzian version Sp(1,n)if c<0

EXAMPLE M = HP n quaternionic projective space P = S 4n+3 H n+1 P R + = H n+1 \{0}

P R + = Swann bundle or hyperkähler cone G preserves quaternionic Kähler structure induced action on P preserves ϕ 1, ϕ 2, ϕ 3 Quaternionic Kähler quotient hyperkähler quotient on Swann bundle

P R + = Swann bundle or hyperkähler cone G preserves quaternionic Kähler structure induced action on P preserves ϕ 1, ϕ 2, ϕ 3 Quaternionic Kähler quotient hyperkähler quotient on Swann bundle... at zero value of the moment map

EXAMPLE M = Sp(2, 1)/Sp(2) Sp(1) and G = R R = SO(1, 1) Sp(1, 1) Sp(2, 1) Quotient = deformation of hyperbolic metric on B 4 self-dual Einstein

Math. Ann. 290, 323-340 (1991) Anm 9 Springer-Verlag 1991 The hypercomplex quotient and the quaternionic quotient Dominic Joyce Merton College, Oxford, OX1 4JD, UK Received November 30, 1990 1 Introduction When a symplectic manifold M is acted on by a compact Lie group of isometries F, then a new symplectic manifold of dimension dimm-2dimf can be defined, called the Marsden-Weinstein reduction of M by F [MW]. Kfihler manifolds are

QUATERNIONIC KÄHLER AND COMPLEX QUATERNIONIC

M quaternionic Kähler locally defined 2-forms ω 1, ω 2, ω 3 span a subbundle E Λ 2 T invariant closed 4-form Ω = ω 2 1 + ω2 2 + ω2 3 stabilizer Sp(n) Sp(1)

action of G preserving Ω (and therefore the metric) i Xa Ω = dµ a 2-form µ a moment form µ Λ 2 T g

vector field X 1-form X (dx ) + = component in E Λ 2 T = µ up to a constant multiple

locally µ = µ 1 ω 1 + µ 2 ω 2 + µ 3 ω 3 if µ = 0 distinguished almost complex structure 1 µ (µ 1I + µ 2 J + µ 3 K)

locally µ = µ 1 ω 1 + µ 2 ω 2 + µ 3 ω 3 if µ = 0 distinguished almost complex structure 1 µ (µ 1I + µ 2 J + µ 3 K) PROP: This is integrable. F.Battaglia, Circle actions and Morse theory on quaternion- Kähler manifolds, J.Lond.Math.Soc 59 (1999) 345 358.

locally µ = µ 1 ω 1 + µ 2 ω 2 + µ 3 ω 3 if µ = 0 distinguished almost complex structure 1 µ (µ 1I + µ 2 J + µ 3 K) PROP: This is integrable. F.Battaglia, Circle actions and Morse theory on quaternion- Kähler manifolds, J.Lond.Math.Soc 59 (1999) 345 358. PROP: M carries a canonical complex quaternionic structure.

THE CONNECTION Levi-Civita connection holonomy Sp(k) Sp(1) torsion-free, holonomy in GL(k, H) H : 1-form α Z Y = Z Y + α(z)y + α(y )Z α(iy )IZ α(iz)iy α(jy )JZ α(jz)jy α(ky )KZ α(kz)ky

I = 0? torsion-free, holonomy in GL(k, H) H : choose local gauge µ = µω 1 I = θ 2 K θ 3 J

µ = i i X ω i ω i dµ = i X ω 1 µθ 2 = i X ω 3 µθ 3 = i X ω 2 I = θ 2 K θ 3 J I =0 α = Jθ 2 /2=Kθ 3 /2 α = d log µ/2.

Riemannian volume form v g v g = (2k + 2)(d log µ)v g µ (2k+2) v g invariant volume form holonomy SL(k, H) U(1)

DIMENSION 4 SL(1, H) U(1) = U(2) = Levi-Civita connection of µ 2 g self-dual Einstein scalar-flat Kähler K.P.Tod, The SU( )-Toda field equation and special fourdimensional metrics, in Geometry and physics (Aarhus, 1995 Dekker, 1997, 317 312 A.Derdzinski, Self-dual Kähler manifolds and Einstein man folds of dimension four, Comp. Math. 49 (1983) 405-433

EXAMPLE HP 1 = S 4 g S 4 = 1 (1 + ρ 2 + σ 2 ) 2(dρ2 + ρ 2 dϕ 2 + dσ 2 + σ 2 dθ 2 ) X = / θ X = σ 2 dθ/(1 + ρ 2 + σ 2 ) 2 = u 2 dθ u = σ/(1 + ρ 2 + σ 2 ),v =(ρ 2 + σ 2 1)/ρ g S 4 = 1 1 4u 2du2 + u 2 dθ 2 + 1 4u2 (v 2 + 4) 2dv2 + 1 4u2 v 2 +4 dϕ2

µ =(1 4u 2 ) 1/2 g = 1 1 (1 4u 2 ) 2du2 + (1 4u 2 ) u2 dθ 2 1 + (v 2 + 4) 2dv2 + 1 v 2 +4 dϕ2 H 2 S 2 scalar curvature 4+4=0 (u = (tanh 2x)/2 and v =2tany)... on S 4 minus the circle ρ =0, σ =1

COMPACT QUATERNION KÄHLER MANIFOLDS Wolf spaces G/K symmetric Sp(n + 1)/Sp(n) Sp(1), SU(n + 2)/S(U(n) U(2)), SO(n + 4)/SO(n) SO(4) E 6 /SU(6) SU(2), E 7 /Spin(12) Sp(1), E 8 /E 7 Sp(1) F 4 /Sp(3) Sp(1), G 2 /SO(4)

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