DIFFERENTIAL GEOMETRY AND THE QUATERNIONS. Nigel Hitchin (Oxford) The Chern Lectures Berkeley April 9th18th 2013


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1 DIFFERENTIAL GEOMETRY AND THE QUATERNIONS Nigel Hitchin (Oxford) The Chern Lectures Berkeley April 9th18th 2013
2 16th October 1843
3
4 ON RIEMANNIAN MANIFOLDS OF FOUR DIMENSIONS 1 SHIINGSHEN CHERN Introduction. It is well known that in threedimensional elliptic or spherical geometry the socalled Clifford's parallelism or parataxy has many interesting properties. A grouptheoretic 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. SS.Chern, On Riemannian manifolds of four dimensions, Bull. Amer. Math. Soc. 51 (1945)
5 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
6 GEOMETRY OVER THE QUATERNIONS
7 q H quaternions q = x 0 + ix 1 + jx 2 + kx 3 algebraic variety? f(q 1,..., q n ) = 0 q = 0: 2sphere q = ix 1 + jx 2 + kx 3, x x2 2 + x2 3 = 1
8 submanifold M H n T x M H n T x M quaternionic for all x M M = H m H n
9 INTRINSIC DIFFERENTIAL GEOMETRY
10 quaternionic structure on the tangent bundle T affine connection X Y zero torsion X Y Y X = [X, Y ]
11 H n ndimensional quaternionic vector space left action by GL(n, H) commutes with right action of H GL(n, H) H
12 metric maximal compact subgroup Sp(n) Sp(1) GL(n, H) H LeviCivita connection : preserving metric unique torsionfree connection Quaternionic Kähler preserves quaternionic structure
13 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) LeviCivita connection : preserving metric unique torsionfree connection Hyperkähler preserves I,J,K
14 SL(n, H) U(1) preserves action of C on tangent bundle T if a torsionfree connection preserves this structure, it is unique complex quaternionic complex manifold volume form U(1) connection on K
15 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 nonmetric
16 Lecture 1 Quaternionic manifolds Lecture 2 Hyperkähler moduli spaces Lecture 3 Twistors and holomorphic geometry Lecture 4 Correspondences and circle actions
17 THE HYPERKÄHLER QUOTIENT
18 hyperkähler manifold M 4k complex structures I, J, K + metric g Kähler forms ω 1, ω 2, ω 3
19 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.
20 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
21 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
22 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
23 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)
24 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
25 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 = 0 and z k w k = 0
26 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 = 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, EguchiHanson (n=2)
27 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, , 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
28 EXAMPLE M = H + H and G = R action t (q 1, q 2 ) = (e it q 1, q 2 + t)
29 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 ) TaubNUT metric
30 V harmonic function on R 3 dv = dα g = V (dx dx2 2 + dx2 3 )+V 1 (dθ + α) 2. ω 1 = Vdx 2 dx 3 + dx 1 (dθ + α) V = 1 2r + c
31 NJH, A. Karlhede, U. Lindström & M. Roček, Hyperkähler metrics and supersymmetry, Comm. Math. Phys. 108 (1987), K.Galicki & H.B Lawson Jr. Quaternionic reduction and quaternionic orbifolds, Math. Ann. 282 (1988) 121.
32 QUATERNIONIC KÄHLER AND HYPERKÄHLER
33 Sp(n) Sp(1) GL(n, H) H LeviCivita connection : preserving metric unique torsionfree connection Quaternionic Kähler preserves quaternionic structure
34 Sp(n) Sp(1) GL(n, H) H LeviCivita connection : preserving metric unique torsionfree connection Quaternionic Kähler preserves quaternionic structure principal Sp(1) bundle with connection
35 T is a module over a bundle of quaternions (e.g. HP n ) equivalently a rank 3 bundle of 2forms ω 1, ω 2, ω 3
36 T is a module over a bundle of quaternions (e.g. HP n ) equivalently a rank 3 bundle of 2forms 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
37 P = SO(3) frame bundle θ i welldefined 1forms on P dim P R + = 4n + 4
38 P = SO(3) frame bundle θ i welldefined 1forms on P dim P R + = 4n + 4 define ϕ i = d(tθ i ) (t = R + coordinate) three closed 2forms ϕ 1, ϕ 2, ϕ 3
39 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
40 EXAMPLE M = HP n quaternionic projective space P = S 4n+3 H n+1 P R + = H n+1 \{0}
41 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
42 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
43 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 selfdual Einstein
44 Math. Ann. 290, (1991) Anm 9 SpringerVerlag 1991 The hypercomplex quotient and the quaternionic quotient Dominic Joyce Merton College, Oxford, OX1 4JD, UK Received November 30, Introduction When a symplectic manifold M is acted on by a compact Lie group of isometries F, then a new symplectic manifold of dimension dimm2dimf can be defined, called the MarsdenWeinstein reduction of M by F [MW]. Kfihler manifolds are
45 QUATERNIONIC KÄHLER AND COMPLEX QUATERNIONIC
46 M quaternionic Kähler locally defined 2forms ω 1, ω 2, ω 3 span a subbundle E Λ 2 T invariant closed 4form Ω = ω ω2 2 + ω2 3 stabilizer Sp(n) Sp(1)
47 action of G preserving Ω (and therefore the metric) i Xa Ω = dµ a 2form µ a moment form µ Λ 2 T g
48 vector field X 1form X (dx ) + = component in E Λ 2 T = µ up to a constant multiple
49 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)
50 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)
51 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) PROP: M carries a canonical complex quaternionic structure.
52 THE CONNECTION LeviCivita connection holonomy Sp(k) Sp(1) torsionfree, holonomy in GL(k, H) H : 1form 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
53 I = 0? torsionfree, holonomy in GL(k, H) H : choose local gauge µ = µω 1 I = θ 2 K θ 3 J
54 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.
55 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)
56 DIMENSION 4 SL(1, H) U(1) = U(2) = LeviCivita connection of µ 2 g selfdual Einstein scalarflat Kähler K.P.Tod, The SU( )Toda field equation and special fourdimensional metrics, in Geometry and physics (Aarhus, 1995 Dekker, 1997, A.Derdzinski, Selfdual Kähler manifolds and Einstein man folds of dimension four, Comp. Math. 49 (1983)
57 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 u2 (v 2 + 4) 2dv u2 v 2 +4 dϕ2
58 µ =(1 4u 2 ) 1/2 g = 1 1 (1 4u 2 ) 2du2 + (1 4u 2 ) u2 dθ (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
59 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)
60 NEXT LECTURE... Hyperkähler manifolds Hyperholomorphic bundles Moduli space examples
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