DIFFERENTIAL GEOMETRY AND THE QUATERNIONS

Nigel Hitchin (Oxford)

The Chern Lectures Berkeley April 9th-18th 2013

3 26th December 1843 • 16th October 1843 2 3 RIEMANNIAN MANIFOLDS OF FOUR DIMENSIONS 965

We call a frame an ordered set of four mutually perpendicular unit vectors eo, Ci, C2, 63. There exists one and only one rotation carrying 3 one frame to another. The coordinates Xof X\, &2, Xz of a point ï&S with respect to the frame eo, ei, C2, e3 are defined by the equation

(2) % = XQCO + Xtfi + X2t2 + Xst3.

Let eo*, ei*, e2*, e3* be a frame related to eo, ei, e2, e3 by means of the rela- tions

3 (3) e«* = ]F) daftp, a = 0, 1, 2, 3,

OwherN RIEMANNIAe (aap) is a propeN MANIFOLDr orthogonaSl OmatrixF FOU, anR d DIMENSIONSlet #0*, #1*, #2*1 , #3* be the coordinates of the same point x with respect to the frame SHIING-SHEN CHERN eo*, ei*, e2*, e3*. Then we have Introduction. It is well known that3 in three-dimensional elliptic or spherica(3a) l geometry the so-calle x*d Clifford'= X) a<*p%p,s parallelis m or paratax ot = 0,y 1ha, 2s, 3. locallymany interestinµ = µ ωg properties+ µ ω .+ Aµ group-theoretiω c reason for the most • 1 1 2 2 3 3 importanThe propertiet of thess oef propertiespherical sgeometr is the facy art etha thost the ewhich universa, whel ncoverin expresseg d grouin pterm of ths eo fprope coordinater orthogonas withl grourespecp itn tfouo ar framevariable, remais is thn e invariandirect t ifproducµunde= 0tr chango distinguishedf thee universaof the framel almostcoverin. g complexgroups of structuretwo proper orthogonal • group￿ s in three variables. This last-mentioned property has no ana- Let 1 logue for #oorthogona, #1» ^2, #l3 groupbe th(esµ icoordinate1nI n+ (>4µ2)J svariables+ ofµ a3 Kpoin). Ot n£ thwite hothe respecr handt to, a a framknowledge Co, eie , oe2f , ethree-dimensiona, as defineµ d by (2)l ellipti. To thesc ore coordinatespherical geometrs we associaty is e 3 ￿ ￿ usefua unil fot rquaternio the studny of orientable Riemannian manifolds of four di- mensions, because their tangent spaces possess a geometry of this PROP:(4) This is integrable.X = XQ + xii + X2J + xzk, N(X) = 1, • kind. It is the purpose of this note to give a study of a compact orient- ablwhere Riemanniae N(X) denoten manifols thde onorf foum ro fdimension X. Let s at each point of which F.Battaglia, Circle actions and Morse theory on quaternion- is (4aattache) d a three-dimensiona X* = x* +l spherica x*i+ x*jl space + . x?k. Thi s necessitates a K¨ahlermorThee carefun manifolds,the l followinstudy ogJ.Lond.Math.Socf theoresphericam l isgeometr well knowy tha59n n[ (1999)l hithert] : o 345given – in 358. the literature, except, so far as the writer is aware, in a paper by S-S.Chern,E. StudTHEOREy [2].MOn2 0u1. r The Riemannianmai n proper resul t orthogonalconsist manifoldss o f grouptwo formulas of(3a) four can, whic be dimensions, expressedh expres s in Bull.twtheo topologica Amer.quaternion Math.l notationinvariant Soc. ins theo51f forma (1945)compac t 964–971.orientable differentiable manifol(5) d of four dimensions as X*integral = AXB,s ove r the manifold of differ- ential invariants constructed from a Riemannian metric previously PROP:givewheren on AMth, eB carriesmanifold are unit a. quaternions.Thes canonicale two topologica It complex containsl invariant the quaternionic two s subgroups have a linea r • combinatio(6a) n which is the Euler-PoincarX* = AX,é characteristic. 8 1(6b. Three-dimensiona) l sphericaX*l =geometry XB, . We consider an ori- ented Euclidean space of four dimensions E4 with the coordinates called the subgroupsA of left and right translations respectively. A left xo, translationXi, x2, X3. I isn Ea rightlet S* translation be the oriente whend anduni t only hyperspher when it e is defineX* = d± b X.y the equation It is important to 2giv e a2 distinctio2 2n between the left and right (1) Xo + Xl + X2 + #3 = I- Three-dimensional spherical geometry is concerned with properties on 5s which remain invariant under the rotation group (that is, the proper orthogonal group) of E4 leaving the origin fixed. Received by the editors June 22, 1945. 1 The content of this paper was originally intended to be an illustration in the author's article, Some new viewpoints in differential geometry in the large, which is due to appear in this Bulletin. Later it appeared more advisable to publish these results separately, but a comparison with the above-mentioned article, in particular §7, is recommended. 2 Numbers in brackets refer to the references cited at the end of the paper. 964 “Quaternions came from Hamilton after his best work had been done, and though beautifully ingenious, they have been an un- mixed evil to those who have touched them in any way”

Lord Kelvin 1890 GEOMETRY OVER THE QUATERNIONS

2 q H quaternions q = x + ix + jx + kx • ∈ 0 1 2 3

algebraic variety? f(q , . . . , qn) = 0 • 1

q2 + 1 = 0: 2-sphere q = ix + jx + kx , x2 + x2 + x2 = 1 • 1 2 3 1 2 3

5 submanifold M Hn • ⊂

n TxM H • ⊂

m n TxM quaternionic for all x M M = H H • ∈ ⇒ ⊂

6 INTRINSIC DIFFERENTIAL GEOMETRY

3 quaternionic structure on the tangent bundle T •

affine connection Y • ∇X

zero torsion Y X = [X, Y ] • ∇X − ∇Y

2 Hn n-dimensional quaternionic vector space •

left action by GL(n, H) •

commutes with right action of H •

GL(n, H) H • · ∗

3 metric maximal compact subgroup • ⇔

Sp(n) Sp(1) GL(n, H) H • · ⊂ · ∗

Levi-Civita connection : unique torsion-free connection • ∇ preserving metric

Quaternionic K¨ahler preserves quaternionic structure • ⇔ ∇

4 GL(n, H) preserves action of H on tangent bundle T •

I,J,K End(T ) such that I2 = J2 = K2 = IJK = 1 • ∈ −

metric Sp(n) GL(n, H) • ⊂

Levi-Civita connection : unique torsion-free connection • ∇ preserving metric

Hyperk¨ahler preserves I,J,K • ⇔∇

3 K¨ahler form ω Ω1,1 • ∈ SL(n, H) U(1) preserves action of C on tangent bundle T • · dω =0 • if a torsion-free connection preserves this structure, it is • ∇ locallyunique ω = ddcf = dIdf •

fcomplexK¨ahler potential quaternionic – complex manifold •

volume form U(1) connection on K • ⇒

2 3 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 = K¨ahler complex surface • with zero scalar curvature

n > 1 complex quaternionic is non-metric •

7 Lecture 1 Quaternionic manifolds •

Lecture 2 Hyperk¨ahler moduli spaces •

Lecture 3 Twistors and holomorphic geometry •

Lecture 4 Correspondences and circle actions •

6 THE HYPERKAHLER¨ QUOTIENT

2 hyperk¨ahler manifold M4k •

complex structures I, J, K + metric g •

K¨ahler forms ω , ω , ω • ⇒ 1 2 3

1 ω : T T , K = ω− ω etc. • i → ∗ 1 2

3 hyphyperkerk¨ahler¨ahler manifoldmanifold MM44kk ••

complexcomplex strstructuresuctures II,,JJ,,KK ++ metricmetric gg ••

KK¨ahler¨ahler foformsrms ωω1,,ωω2,,ωω3 •• ⇒⇒ 1 2 3

11 ωωi ::TT TT ,, KK ==ωω−− ωω2 etc.etc. •• i →→ ∗∗ 11 2

33 Lie group G acting on M, fixing ω , ω , ω • 1 2 3

a g vector field Xa • ∈

d(i ω ) + i dω = ω = 0 • Xa i Xa i LXa i

moment map i ω = dµa • Xa i i

13 Lie group G acting on M, fixing ω1, ω2, ω3 • Lie group G acting on M, fixing ω1, ω2, ω3 •

a g vector field Xa • a ∈ g vector field Xa • ∈

d(i ω ) + i dω = ω = 0 • d(iXaωi) + iXadωi = LXaωi = 0 • Xa i Xa i LXa i

moment map i ω = dµa • moment map iXaωi = dµai • Xa i i

13 13 µ : M g R3 • → ∗ ⊗

If G acts properly and freely on µ 1(0) then... • −

... the quotient metric on µ 1(0)/G is hyperk¨ahler... • −

... of dimension dim M 4 dim G • −

5 EXAMPLE EXAMPLE M = Hn = Cn + jCn flat hyperk¨ahler manifold • M EXAMPLE= Hn = Cn + jCn flat hyperk¨ahler manifold • EXAMPLE • M = Hn = Cn + jCn flat hyperk¨ahler manifold i • = n = n + n flat hypω1 =erk¨ahler(dzk manifolddz¯k + dwk dw¯k) •M H C jC i 2 ∧ ∧ • ω = (dz dz¯ + dw dw¯ ) 1 2 k ∧ k k ∧ k • i ω •= (dz dz¯ + dw diw¯ ) ω2 + iω3 = dz dw 1 k k k k k ∧ k 2 ∧ ω1 =∧ ω(dz+kiω dz=¯k +dzdwkdwdw¯k) 22 ∧3 k ∧ ∧ k 1 G = U(1) action u (z, w) = (uz, u− w) • · 1 ωG2 =+ Uiω(1)3 =actiondzk dwu k(z, w) = (uz, u− w) • ∧ ·ω2 + iω3 = dzk dwk ∧ 3 µ(z, w) = (zkz¯k wkw¯k + c, zkwk) R C = R • 1 − ∈ 3× G = U(1) action u (µz(,zw,)w=) =(uz(z,kuz¯−k ww) kw¯k + c, zkwk) R C = R • •·G = U(1) action−u (z, w) = (uz, u∈1w)× • · − 3 3 3 µ(z, w) = (zkz¯k wkw¯k + c, zkwk) R C = R • − µ(z, w) = (z z¯∈ w× w¯ + c, z w ) R C = R3 • k k − k k k k ∈ × 3 3 EXAMPLE EXAMPLE M = Hn = CnEXAMPLE+ jCn flat hyperk¨ahler manifold • M EXAMPLE= Hn = Cn + jCn flat hyperk¨ahler manifold • EXAMPLE M = Hn = Cn + jCn flat hyperk¨ahler manifold • • M = Hn = Cn + jCn flat hyperk¨ahler manifold i • •M = Hn = Cn + jCn flat hypω1 =erk¨ahler(dzk manifolddz¯k + dwk dw¯k) • i 2 ∧ ∧ ω1 = (dzk dz¯k + dwk dw¯k) • 2i ∧ ∧ ω1 = (dzk dz¯k + dwk dw¯k) • i 2 ∧ ∧ ω •= (dz dz¯ + dw diw¯ ) ω2 + iω3 = dz dw 1 k k k k k ∧ k 2 ∧ ω1 =∧ ω(dz+kiω dz=¯k +dzdwkdwdw¯k) 22 ∧3 k ∧ ∧ k ω2 + iω3 = dz dw 1 G = U(1) action u (z, wk)∧= (uzk , u− w) • · 1 ωG2 =+ Uiω(1)3 =actiondzk dwu k(z, w) = (uz, u− w) • ∧ ·ω2 + iω3 = dzk dwk G = U(1) action u (z, w) = (uz, u∧1w) 3 • µ(z, w) = ·(zkz¯k wkw¯k +−c, zkwk) R C = R = (1) action ( ( ) =) =•( ( ¯ 1 ) ¯ +− ) =∈ 3× G U u µz,zw, w uzz,kuz−k wwkwk c, zkwk 1R C R • •·G = U(1) action−u (z, w) = (uz, u∈− w)× • µ(z, w) = (z z¯ w· w¯ , z w ) + const. R C = R3 3 k k k k k k 3 • − 3 ∈ × µ(z, w) = (zkz¯k wkw¯k + c, zkwk) R C = R • − µ(z, w) = (z z¯∈ w× w¯ + c, z w ) R C = R3 3 • k k − k k k k ∈ × 3 3 choice

NJH, A. Karlhede, U. Lindstr¨om & M. Roˇcek, Hyperk¨ahler met- rics and supersymmetry, Comm. Math. Phys. 108 (1987), 535–589.

K.Galicki & H.B Lawson Jr. Quaternionic reduction and quater- nionic orbifolds, Math. Ann. 282 (1988) 121. µ(z, w) = (z z¯ w w¯ , z w ) + (1, 0) R C = R3 • k k − k k k k ∈ ×

µ 1(0) : z 2 w 2 + 1 = 0 and z w = 0 • − ￿ ￿ − ￿ ￿ k k

w = 0 projection µ 1(0) CP n 1 • ￿ ⇒ − → −

µ 1(0)/U(1) = T CP n 1 • − ∼ ∗ −

Calabi metric, Eguchi-Hanson (n=2)

6 3 µ((z, w))==((zk¯z¯k wk¯w¯k, zkwk))++(1(1,0)0) R C== R3 • µ z, w µ(z,zwkz)k=−(wzkz¯wk, zkwwkw¯ , z w, ) +∈(1R, 0)×C R R C = R3 • • − k k − k k k k ∈ × ∈ ×

µ 11(0) : z 22 w 22 + 1 = 0 and z w = 0 • µ−− (0)µ: ￿1z(0)￿ −: ￿wz ￿2 + 1w=2 0+and1 =z0kkwandkk=z0w = 0 • • −￿ ￿ − ￿￿ ￿￿ − ￿ ￿ k k

w = 0 projection µ 11(0) CPnn 11 • w =￿ 0 ⇒w =projection0 projectionµ−− (0)µ→1C(0)P −− CP n 1 • ￿ • ⇒ ￿ ⇒ →− → −

µ 11(0)/U(1) = T CPnn 11 • µ−− (0)/Uµ (1)1(0)=∼/U∼T(1)∗∗CP= −T− CP n 1 • • − ∼ ∗ −

Calabi metric, Eguchi-Hanson (n=2) Calabi Calabimetric,metric,Eguchi-HEguchi-Hanson (n=2)anson (n=2)

6 6 6 HERMITIAN SYMMETRIC SPACES

O. Biquard, P. Gauduchon, Hyperk¨ahler 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 = p ω + ddch • 1 ∗

h =(f(IR(IX,X))X, X), R curvature tensor, X T • ∈ ∗ • 1 1+√1+u 8 f(u)= 1+u 1 log • u ￿ − − 2 ￿ ￿ 1 1+√1+u f(u)= 1+u 1 log u ￿ − − 2 ￿ ￿ EXAMPLE

M = H + H and G = R •

action t (q , q ) = (eitq , q + t) • · 1 2 1 2

µ 1(0) : z 2 w 2 = im z and z w = w • − | 1| − | 1| 2 1 1 2

µ 1(0)/R = C2, coordinates (z , w ) • − ∼ 1 1

Taub-NUT metric

4 EXAMPLE EXAMPLE EXAMPLE M = H + H and G = R • M = H + H and G = R • M = H + H and G = R • it action t (q1, q2) = (e q1, q2 + t) it • action t· (q1, q2) = (e q1, q2 + t) • · action t (q , q ) = (eitq , q + t) • · 1 2 1 2 µ 1(0) : z 2 w 2 = im z and z w = w • − 1 | 1| 2− | 1| 2 2 1 1 2 µ− (0) : z1 w1 = im z2 and z1w1 = w2 • | | − | | µ 1(0) : z 2 w 2 = im z and z w = w • − | 1| − | 1| 2 1 1 2 µ 1(0)/R = C2, coordinates (z , w ) • − 1 ∼ 2 1 1 µ− (0)/R =∼ C , coordinates (z1, w1) • µ 1(0)/R = C2, coordinates (z , w ) • − ∼ 1 1 Taub-NUT metric Taub-NUT metric Taub-NUT metric 4 4 4 3 V harmonic function on R 3 • V harmonic function on R • V harmonic function on R3 • V harmonic function on R3 • dV = dα • ∗ dV = dα • ∗dV = dα • ∗dV = dα • ∗ • • 2 2 2 1 2 • g = V (dx1 + dx2 +2 dx3)+2 V − 2(dθ + α)1 . 2 • g = V (dx21 + dx22 + dx23)+V −1 (dθ + α)2 . g = V (dx12 + dx22 + dx32)+V − 1(dθ + α) 2. g = V (dx + dx + dx )+V − (dθ + α) . TAUB-NUT1 2 3 ω1 = Vdx2 dx3 + dx1 (dθ + α) • ω1∧= Vdx2 dx∧3 + dx1 (dθ + α) • ω1 = Vdx2 ∧dx3 + dx1 ∧(dθ + α) • ω = Vdx ∧ dx + dx ∧ (dθ + α) • 1 2 ∧ 3 1 ∧ • 1 V = + c 3 2 3 r 3 3

2 choice

NJH, A. Karlhede, U. Lindstr¨om & M. Roˇcek, Hyperk¨ahler met- rics and supersymmetry, Comm. Math. Phys. 108 (1987), 535–589.

K.Galicki & H.B Lawson Jr. Quaternionic reduction and quater- nionic orbifolds, Math. Ann. 282 (1988) 121. QUATERNIONIC KAHLER¨ AND HYPERKAHLER¨

6 metric maximal compact subgroup • ⇔

Sp(n) Sp(1) GL(n, H) H • · ⊂ · ∗

Levi-Civita connection : unique torsion-free connection • ∇ preserving metric

Quaternionic K¨ahler preserves quaternionic structure • ⇔ ∇

11 metric maximal compact subgroup • ⇔

Sp(n) Sp(1) GL(n, H) H • · ⊂ · ∗

Levi-Civita connection : unique torsion-free connection • ∇ preserving metric

Quaternionic K¨ahler preserves quaternionic structure • ⇔ ∇

principal Sp(1) bundle with connection • 11

16 T is a module over a bundle of quaternions (e.g. HP n) •

equivalently a rank 3 bundle of 2-forms ω , ω , ω • 1 2 3

ω = θ ω θ ω • ∇ 1 2 ⊗ 3 − 3 ⊗ 2

curvature K = dθ θ θ etc. • 23 1 − 2 ∧ 3

in fact K = cω , c constant scalar curvature • 23 1 ∼

4 T is a module over a bundle of quaternions (e.g. HP n) • T is a module over a bundle of quaternions (e.g. HP n) •

equivalently a rank 3 bundle of 2-forms ω1, ω2, ω3 • equivalently a rank 3 bundle of 2-forms ω1, ω2, ω3 •

ω1 = θ2 ω3 θ3 ω2 • ∇ω1 = θ2 ⊗ ω3 − θ3 ⊗ ω2 • ∇ ⊗ − ⊗

curvature K23 = dθ1 θ2 θ3 etc. • curvature K23 = dθ1 − θ2 ∧ θ3 etc. • − ∧

in fact K23 = cω1, c constant scalar curvature • in fact K23 = cω1, c constant ∼ scalar curvature • ∼

4 4 P = SO(3) frame bundle •

θ well-defined 1-forms on P • i

dim P R+ = 4n + 4 • ×

define ϕ = d(tθ ) (t = R+ coordinate) • i i

three closed 2-forms ϕ , ϕ , ϕ • 1 2 3

5 PP == SSOO(3)(3) frameframe bundlebundle ••

θθi wwell-definedell-defined 1-fo1-formsrms onon PP •• i

+ dimdimPP RR+ ==44nn++44 •• ××

+ definedefine ϕϕi ==dd((ttθθi)) ((tt==RR+ cocooordinate)rdinate) •• i i

threethree clclosedosed 2-fo2-formsrms ϕϕ1,,ϕϕ2,,ϕϕ3 •• 1 2 3

55 T (P R+)=H V • × ⊕

on H, θ = 0 and dt = 0, ϕ = tcω • i i i

on V , ϕ = dt θ + t2θ θ etc. • 1 ∧ 1 2 ∧ 3

algebraic relations for hyperk¨ahler if c>0 • Lorentzian version Sp(1,n)ifc<0

23 EXAMPLE

M = HP n quaternionic projective space •

P = S4n+3 Hn+1 • ⊂

P R+ = Hn+1 0 • × \{ }

2 P R+ = Swann bundle or hyperk¨ahler cone • ×

G preserves quaternionic K¨ahler structure induced action • ⇒ on P preserves ϕ1, ϕ2, ϕ3

Quaternionic K¨ahler quotient hyperk¨ahler quotient on Swann • ⇔ bundle

7 K¨ahler form ω Ω1,1 • ∈

dω =0 • P R+ = Swann bundle or hyperk¨ahler cone • × locally ω = ddcf = dIdf • G preserves quaternionic K¨ahler structure induced action • ⇒ on P preserves ϕ , ϕ , ϕ f K¨ahler potential1 2 3 •

Quaternionic K¨ahler quotient hyperk¨ahler quotient on Swann • volume form U(1) connection⇔ on K • bundle ⇒

..... at zero value of the moment map •

4 7 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 B4 •

self-dual Einstein •

8 Math. Ann. 290, 323-340 (1991) Anm 9 Springer-Verlag1991

Math. Ann. 290, 323-340 (1991) Anm 9 Springer-Verlag1991

The hypercomplex quotient and the quaternionic quotient Dominic Joyce Merton College, Oxford, OX1 4JD, UK ReceivedThe hypercomplex November 30, 1990 quotient and the quaternionic quotient 1 Introduction Dominic Joyce WhenMerton a College, symplectic Oxford, manifold OX1 4JD, M isUK acted on by a compact Lie group of isometries F, then a new symplectic manifold of dimension dimM-2dimF can be defined, calledReceived the November Marsden-Weinstein 30, 1990 reduction of M by F [MW]. Kfihler manifolds are important examples of symplectic manifolds, and in this case the Marsden- Weinstein reduction yields a new K/ihler manifold, which as a complex manifold is the1 Introduction quotient of the set of stable points of M by the complexified action of F. This is called the K/ihler quotient. WhenRecently, a symplectic these manifoldconstructions M is actedhave on been by a extendedcompact Lieto grouptwo other of isometries classes ofF, manifolds.then a new In symplectic the classification manifold of of Riemannian dimension dimM-2dimFmanifolds by holonomy can be defined, [S 2], K/ihlercalled the manifolds Marsden-Weinstein are manifolds reduction with holonomy of M by FU(n), [MW]. and Kfihler related manifolds to these areare hyperk/ihlerimportant examples manifolds of with symplectic holonomy manifolds, Sp(n), and and quaternionic in this case Kfihler the manifoldsMarsden- withWeinstein holonomy reduction Sp(n)Sp(1). yields Aa newquotient K/ihler process manifold, for hyperk/ihler which as a complex manifolds manifold has been is describedthe quotient by ofHitchin the set etof al. stable [HKLR] points thatof M reduces by the complexifieddimension by action 4dimH, of F.and This this is wascalled generalised the K/ihler by quotient. Galicki and Lawson [-GL] to a quotient for quaternionic K/thlerRecently, manifolds. these constructions have been extended to two other classes of manifolds.Now in parallelIn the classificationwith the classification of Riemannian of Riemannian manifolds manifolds by holonomy by holonomy [S 2], thereK/ihler is manifoldsa theory [B] are thatmanifolds classifies with manifolds holonomy with U(n), torsion-free and related connections to these areby holonomy.hyperk/ihler K/ihler, manifolds hyperk/ihler, with holonomy and quaternionic Sp(n), and quaternionic K/ihler manifolds Kfihler havemanifolds ana- logueswith holonomy in this theory: Sp(n)Sp(1). the analogue A quotient of process a Kfihler for manifold hyperk/ihler is a manifoldscomplex manifold has been [withdescribed holonomy by Hitchin GL(n, et C)], al. the[HKLR] analogue that ofa reduces hyperkfihler dimension manifold by 4dimH, is a hypercom- and this plexwas generalisedmanifold [with by Galickiholonomy and GL(n, Lawson ~-I)], [-GL] and theto aanalogue quotient of for a quaternionicquaternionic K/ihlerK/thler manifoldmanifolds. is a quaternionic manifold [with holonomy GL(n, I-I)GL(I, ~)]. TheNow purpose in parallel of thiswith paper the classification is to present of quotient Riemannian constructions manifolds for by hypercom- holonomy plexthere and is aquaternionic theory [B] thatmanifolds classifies that manifolds are analogous with totorsion-free those already connections known forby hyperk/~hlerholonomy. K/ihler, and quaternionic hyperk/ihler, K/ihler and quaternionicmanifolds. There K/ihler is an manifolds essential havedifference ana- betweenlogues in the this new theory: constructions the analogue and the of aknown Kfihler ones, manifold which iswill a complexnow be explained. manifold [withThe holonomy Marsden-Weinstein GL(n, C)], the reduction analogue and ofa the hyperkfihler other reductions manifold above is a hypercom- are two- stageplex manifoldprocesses. [withFirst, holonomya moment map GL(n, is defined,~-I)], and which the analogue is a map fromof a thequaternionic manifold K/ihler manifold is a quaternionic manifold [with holonomy GL(n, I-I)GL(I, ~)]. The purpose of this paper is to present quotient constructions for hypercom- plex and quaternionic manifolds that are analogous to those already known for hyperk/~hler and quaternionic K/ihler manifolds. There is an essential difference between the new constructions and the known ones, which will now be explained. The Marsden-Weinstein reduction and the other reductions above are two- stage processes. First, a moment map is defined, which is a map from the manifold QUATERNIONIC KAHLER¨ AND COMPLEX QUATERNIONIC

2 M quaternionic K¨ahler •

locally defined 2-forms ω , ω , ω span a subbundle E Λ2T • 1 2 3 ⊂ ∗

invariant closed 4-form Ω = ω2 + ω2 + ω2 • 1 2 3

stabilizer Sp(n) Sp(1) • ·

4 action of G preserving Ω (and therefore the metric) •

i Ω = dµa • Xa

2-form µa •

moment form µ Λ2T g • ∈ ∗ ⊗ ∗

5 vector field X 1-form X￿ • ⇒

(dX￿)+ = component in E Λ2T • ⊂ ∗

= µ up to a constant multiple •

6 locally µ = µ ω + µ ω + µ ω • 1 1 2 2 3 3

if µ = 0 distinguished almost complex structure • ￿ 1 (µ1I + µ2J + µ3K) µ ￿ ￿

PROP: This is integrable. • F.Battaglia, Circle actions and Morse theory on quaternion- K¨ahler manifolds, J.Lond.Math.Soc 59 (1999) 345 – 358.

7 locallylocally µµ==µµ1ωω1 ++µµ2ωω2 ++µµ3ωω3 •• 1 1 2 2 3 3

ifif µµ== 0 0 distinguished distinguished almost almost complex complex structure structure •• ￿ ￿ 1 1 (µ I + µ J + µ K) (µ11I + µ22J + µ33K) µµ ￿￿ ￿￿

PROP:PROP: ThisThis is is integrable. integrable. •• F.Battaglia,F.Battaglia, CircleCircle actions actions and and Morse Morse theory theory on on quaternion- quaternion- K¨ahlerK¨ahler manifolds, manifolds, J.Lond.Math.SocJ.Lond.Math.Soc 5959 (1999)(1999) 345 345 – – 358. 358.

77 locally µ = µ1ω1 + µ2ω2 + µ3ω3 locally µ = µ ω + µ ω + µ ω • • locally µ = µ11ω11 + µ22ω22 + µ33ω33 • locally µ = µ ω + µ ω + µ ω • 1 1 2 2 3 3 if µ = 0 distinguished almost complex structure if µ = 0 distinguished almost complex structure • ￿ • if µ￿ = 0 distinguished almost complex structure 1 • if µ =￿ 0 distinguished almost complex structure (µ I + µ J + µ K) • ￿ 11 1 2 3 (µ1I + µ2J + µ3K) µ µ1 (µ1I + µ2J + µ3K) ￿ ￿ µ (µ1I + µ2J + µ3K) ￿￿µ￿￿ ￿ ￿ PROP: This is integrable. • PROP:PROP: ThisThis is is integrable. integrable. •• PROP: This is integrable. F.Battaglia, Circle actions and Morse theory on quaternion- • F.Battaglia,F.Battaglia, CircleCircle actions actions and and Morse Morse theory theory on on quaternion- quaternion-K¨ahler manifolds, J.Lond.Math.Soc 59 (1999) 345 – 358. F.Battaglia, Circle actions and Morse theory on quaternion- K¨ahlerK¨ahler manifolds, manifolds, J.Lond.Math.SocJ.Lond.Math.Soc 5959 (1999)(1999) 345 345 – – 358. 358. K¨ahler manifolds, J.Lond.Math.Soc 59 (1999) 345 – 358. PROP: M carries a canonical complex quaternionic • PROP: M carries a canonical complex quaternionic structure. • 77 structure. 7

7 THE CONNECTION

Levi-Civita connection holonomy Sp(k) Sp(1) • ∇ ·

torsion-free, holonomy in GL(k, H) H : • · ∗

1-form α •

˜ Y = Y + α(Z)Y + α(Y )Z α(IY )IZ α(IZ)IY • ∇Z ∇Z − − α(JY )JZ α(JZ)JY α(KY )KZ α(KZ)KY − − − −

10 ˜ I = 0? • ∇

torsion-free,˜ I = 0? holonomy in GL(k, H) H : •• ∇ · ∗

torsion-free, holonomy in GL(k, ) : ˜ Iµ== 0? i iXωi ωi H H∗ ••• ∇∇ ⊗ · ￿ = torsion-free,chooseµ locali iXω holonomyi gaugeωi µ = inµGLω1 (k, H) H : ••• ∇ ⊗ · ∗ ￿ choose local gauge = dµµ==iXωi1 ω µωθ2 =µiXωµ3ω1 µθ3 = iXω2 ••• ∇ i X i ⊗ i − ￿

choosedµI==iθX2 localω1 K gaugeµθ3θ2 =µJ i=Xωµ3ω µθ3 = iXω2 ••• ∇ ⊗ − ⊗ 1 − ˜ I =0 α = Jθ2/2=Kθ3/2 ∇ = ⇔ dµI= iθ2ω K µθ3θ =J i ω µθ = i ω 12 •• ∇ X ⊗1 − 2⊗ X 3 3 − X 2 ˜ I =0 α = Jθ /2=Kθ /2 ∇ ⇔ 2 3 I = θ K θ J 12 • ∇ 2 ⊗ − 3 ⊗ ˜ I =0 α = Jθ /2=Kθ /2 ∇ ⇔ 2 3 12 ˜˜II == 0? 0? •• ∇∇ ˜ I = 0? • ∇ torsion-free,torsion-free, holonomy holonomy in in GLGL((k,k,HH)) HH∗:: •• ·· ∗ torsion-free, holonomy in GL(k, H) H : • · ∗ µµ == i iiXωωi ωωi •• ∇∇ i X i ⊗⊗ i ￿￿ µ = i ω ω • ∇ i X i ⊗ i choosechoose local local gauge gauge µµ == µµωω1 •• ￿ 1 choose local gauge µ = µω • 1 dµdµ == iiXωω1 µµθθ2 == iiXωω3 µµθθ3 == iiXωω2 •• X 1 2 X 3 3 −− X 2 dµ = i ω µθ = i ω µθ = i ω • X 1 2 X 3 3 − X 2 II == θθ2 KK θθ3 JJ •• ∇∇ 2 ⊗⊗ −− 3 ⊗⊗ ˜ ˜III==0=0θ Kαα ==θJJθθ22//J2=2=KKθθ33//22 • ∇∇∇ 2 ⇔⊗⇔ − 3 ⊗ 1212 ˜ I =0 α = Jθ2/2=Kθ3/2 ∇ α =⇔d log µ/2. • ⇔ − 12 THE CONNECTION

Levi-Civita connection holonomy Sp(k) Sp(1) • ∇ ·

torsion-free, holonomy in GL(k, H) H : • · ∗ Riemannian volume form vg • 1-form α • ˜ vg = (2k + 2)(d log µ)vg • ∇ − ˜ Y = Y + α(Z)Y + α(Y )Z α(IY )IZ α(IZ)IY • ∇Z ∇Z − − (2k+2) µ−α(JY )JZvg invariantα(JZ)JY volumeα(KY form)KZ α(KZ)KY • − − − −

holonomy SL(k, H) U(1) • ·

11

11 locally µ = µ1ω1 + µDIMENSION2ω2 + µ3ω3 4 •

if µ = 0 distinguished almost complex structure • SL(1￿, H) U(1) = U(2) • · 1 (µ1I + µ2J + µ3K) µ ˜ = Levi-Civita connection￿ ￿ of µ 2g • ∇ − PROP: This is integrable. • self-dual Einstein scalar-flat K¨ahler • F.Battaglia, Circle∼ actions and Morse theory on quaternion- K¨ahler manifolds, J.Lond.Math.Soc 59 (1999) 345 – 358. K.P.Tod, The SU( )-Toda field equation and special four- ∞ dimensionalS-S.Chern, metrics,On Riemannianin “Geometry manifolds and physics of four (Aarhus, dimensions, 1995)”, Dekker,Bull. Amer. 1997, Math. 317–312 Soc. 51 (1945) 964–971.

A.Derdzinski, Self-dual K¨ahler manifolds and Einstein mani-

folds of dimension four, Comp. Math. 49 (1983) 405-433.12 11 EXAMPLE HP1 = S4

1 2 2 2 2 2 2 g 4 = (dρ + ρ dϕ + dσ + σ dθ ) S (1 + ρ2 + σ2)2

X = ∂/∂θ X￿ = σ2dθ/(1 + ρ2 + σ2)2 = u2dθ •

u = σ/(1 + ρ2 + σ2),v =(ρ2 + σ2 1)/ρ • −

2 2 1 2 2 2 1 4u 2 1 4u 2 g 4 = du + u dθ + − dv + − dϕ S 1 4u2 (v2 + 4)2 v2 +4 −

13 µ =(1 4u2)1/2 • µ =(1− 4u2)1/2 • −

1 2 1 2 2 1 2 1 2 g = 1 du + 1 u dθ + 1 dv + 1 dϕ g =(1 4u2)2 du2+(1 4u2) u2dθ2+(v2 + 4)2 dv2+v2 +4 dϕ2 (1− 4u2)2 (1− 4u2) (v2 + 4)2 v2 +4 − − H2 S2 scalar curvature 4+4=0 • H2× S2 scalar curvature− 4+4=0 • × − (u = (tanh 2x)/2 and v =2tany) (u = (tanh 2x)/2 and v =2tany)

..... on S4 minus the circle ρ =0, σ =1 • 14

14 COMPACT QUATERNION KAHLER¨ 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 /SU(6) SU(2),E/Spin(12) Sp(1),E/E Sp(1) • 6 · 7 · 8 7 ·

F /Sp(3) Sp(1),G/SO(4) • 4 · 2

15 NEXT LECTURE...

Hyperk¨ahler manifolds •

Hyperholomorphic bundles •

Moduli space examples •

2