A geometric construction of exceptional Lie algebras José Figueroa-O’Farrill Maxwell Institute & School of Mathematics

Leeds, 13 February 2008

2007 will be known as the year where E8 (and Lie groups) went mainstream...

Introduction

Hamilton

Cayley

Lie

É. Cartan

Hurwitz

Hopf

Killing

J.F. Adams

This talk is about a relation between exceptional objects:

• Hopf bundles • exceptional Lie algebras using a geometric construction familiar from supergravity: the Killing (super)algebra.

Real division algebras R

C

H

ab = ba

ab = ba

ab != ba

(ab)c = a(bc)

(ab)c = a(bc)

(ab)c = a(bc)

O

≥ (ab)c != a(bc)

These are all the euclidean normed real division algebras. [Hurwitz]

Hopf fibrations S1  0 S "

S3  1 S "

S7  3 S "

S 15  7  S "

S0 ⊂ R

S1 ⊂ C

S3 ⊂ H

S7 ⊂ O

S1

1

S ⊂R

S2

2

S1 ∼ = RP1

3

S ⊂C

S4

2

S2 ∼ = CP1

7

S8

2

15

2

⊂O

S ⊂H

S

S4 ∼ = HP1

S8 ∼ = OP1

These are the only examples of fibre bundles where all three spaces are spheres. [Adams]

Simple Lie algebras (over C)

4 classical series: An≥1 Bn≥2

SU (n + 1)

G2

14

SO(2n + 1)

F4

52

E6

78

E7

133

E8

248

Cn≥3

Sp(n)

Dn≥4

SO(2n)

[Lie]

5 exceptions:

[Killing, Cartan]

Supergravity Supergravity is a nontrivial generalisation of Einstein’s theory of General Relativity. The supergravity universe consists of a lorentzian spin manifold with additional geometric data, together with a notion of Killing spinor. These spinors generate the Killing superalgebra. This is a useful invariant of the universe.

Applying the Killing superalgebra construction to the exceptional Hopf fibration, one obtains a triple of exceptional Lie algebras:

S 15  7  S " S8

E8 B4

“Killing superalgebra”

F4

Spinors

Clifford

Clifford algebras V

n

real euclidean vector space

!−, −# !

V C!(V ) = filtered associative algebra 2 !v ⊗ v + |v| 1# C!(V ) ∼ = ΛV

(as vector spaces)

C!(V ) = C!(V )0 ⊕ C!(V )1 even ∼ C!(V )0 = Λ V

odd ∼ C!(V )1 = Λ V

orthonormal frame ei ej + ej ei = −2δij 1

e1 , . . . , en C! (R ) =: C!n n

Examples: C!0 = !1" ∼ =R " 2 ! # C!1 = 1, e1 "e1 = −1 ∼ =C " 2 ! # 2 C!2 = 1, e1 , e2 "e1 = e2 = −1, e1 e2 = −e2 e1 ∼ =H

Classification n 0 1 2 3 4 5 6 7

C!n R C H H⊕H H(2) C(4) R(8) R(8) ⊕ R(8)

Bott periodicity: C!n+8 ∼ = C!n ⊗ R(16)

e.g., C!9 ∼ = C(16) C!16 ∼ = R(256)

From this table one can read the type and dimension of the irreducible representations.

C!n has a unique irreducible representation if

n is even and two if n is odd. They are distinguished by the action of e1 e2 · · · en

which is central for n odd. Notation :

Mn or M± n dim Mn = 2!n/2"

Clifford modules

Spinor representatinos son → C!n

ei ∧ ej "→

s ∈ Spinn ,

exp

1 − 2 ei ej

v∈R

n

=⇒

Spinn ⊂ C!n −1

svs

∈R

n

which defines a 2-to-1 map Spinn → SOn with archetypical example

Spin3 ∼ = SU2 ⊂ H   " 2-1 SO3 ∼ = SO(ImH)

By restriction, every representation of C!n defines a representation of Spinn : C!n ⊃ Spinn M = ∆ = ∆ + ⊕ ∆−

M =∆ ±

∆± ∆

chiral spinors spinors

One can read off the type of representation from Spinn ⊂ (C!n )0 ∼ = C!n−1 dim ∆ = 2(n−1)/2

dim ∆± = 2(n−2)/2

Spinor inner product (−, −) nondegenerate form on ∆ (ε1 , ε2 ) = (ε2 , ε1 ) (ε1 , ei · ε2 ) = − (ei · ε1 , ε2 )

∀εi ∈ ∆

=⇒ (ε1 , ei ej · ε2 ) = − (ei ej · ε1 , ε2 )

which allows us to define

[−, −] : Λ ∆ → R 2

![ε1 , ε2 ], ei " = (ε1 , ei · ε2 )

n

Spin geometry

Spin manifolds M

n

g

differentiable manifold , orientable, spin riemannian metric

GL(M )   "

O(M )   "

M

GLn

w1 = 0

SO(M )   "

!

M

!"

On

w2 = 0

!

M

!"

SOn ! !

Spin(M )   " M

Spinn

Possible Spin(M) are classified by H (M ; Z/2) . 1

e.g.,

M =S ⊂R n

n+1

O(M ) = On+1 SO(M ) = SOn+1 Spin(M ) = Spinn+1 S ∼ = Spinn+1 /Spinn = SOn+1 /SOn ∼ = On+1 /On ∼ n

π1 (M ) = {1} =⇒

unique spin structure

Spinor bundles C!(T M )   " M

Clifford bundle C!(T M ) ∼ = ΛT M

S(M ) := Spin(M ) ×Spinn ∆

(chiral)

S(M )± := Spin(M ) ×Spinn ∆±

bundles

spinor

We will assume that C!(T M ) acts on S(M )

The Levi-Cività connection allows us to differentiate spinors ∇ : S(M ) → T M ⊗ S(M ) ∗

which in turn allows us to define parallel spinor

∇ε = 0

Killing spinor

∇X ε = λX · ε

Killing constant

If (M,g) admits parallel spinors Killing spinors

(M,g) is Ricci-flat

(M,g) is Einstein R = 4λ n(n − 1) 2

=⇒

λ ∈ R ∪ iR

Today we only consider real λ.

Killing spinors have their origin in supergravity. The name stems from the fact that they are “square roots” of Killing vectors. ε1 , ε2 Killing

[ε1 , ε2 ] Killing

Which manifolds admit real Killing spinors? (M, g) (M , g) Ch. Bär

M = R+ × M

Killing spinors in (M,g) !

λ=

metric cone

1 ±2

"

g = dr2 + r2 g

1-1

parallel spinors in the cone

More precisely... If n is odd, Killing spinors are in one-to-one correspondence with chiral parallel spinors in the cone: the chirality is the sign of λ. If n is even, Killing spinors with both signs of λ are in one-to-one correspondence with the parallel spinors in the cone, and the sign of λ enters in the relation between the Clifford bundles.

This reduces the problem to one (already solved) about the holonomy group of the cone.

M. Berger

n n 2m 2m 4m 4m 7 8

Holonomy SOn Um SUm Spm · Sp1 Spm G2 Spin7

M. Wang

Or else the cone is flat and M is a sphere.

Killing superalgebra

Construction of the algebra riemannian spin manifold

(M, g) k = k0 ⊕ k1 k1 =

{ Killing! spinors } " with λ =

k0 = [k1 , k1 ] ⊂

1 2

{ Killing vectors }

[−, −] : Λ k → k 2

?

[−, −] : Λ k0 → k0

✓ [—,—] of vector fields

2

[−, −] : Λ k1 → k0

✓ g([ε1 , ε2 ], X) = (ε1 , X · ε2 )

[−, −] : k0 ⊗ k1 → k1

? spinorial Lie derivative!

2

Kosmann

Lichnerowicz

X ∈ Γ(T M )

Killing

LX g = 0

∈

AX := Y !→ −∇Y X

so(T M ) ! : so(T M ) → EndS(M ) LX := ∇X + !(AX )

spinor representation spinorial Lie derivative

cf. LX Y = ∇X Y + AX Y = ∇X Y − ∇Y X = [X, Y ]

✓

Properties ∀X, Y ∈ k0 ,

}

Z ∈ Γ(T M ),

ε ∈ Γ(S(M )),

f ∈ C (M ) ∞

LX (Z · ε) = [X, Z] · ε + Z · LX ε LX (f ε) = X(f )ε + f LX ε [LX , ∇Z ]ε = ∇[X,Z] ε [LX , LY ]ε = L[X,Y ] ε

∀ε ∈ k1 , X ∈ k0 LX ε ∈ k1

[−, −] : k0 ⊗ k1 → k1 [X, ε] := LX ε

✓

The Jacobi identity Jacobi: Λ k → k 3

(X, Y, Z) !→ [X, [Y, Z]] − [[X, Y ], Z] − [Y, [X, Z]]

4 components : Λ k0 → k0

✓

Jacobi for vector fields

Λ k0 ⊗ k1 → k1

✓

[LX , LY ]ε = L[X,Y ] ε

k0 ⊗ Λ k1 → k0

✓

LX (Z · ε) = [X, Z] · ε + Z · LX ε

Λ3 k1 → k1

?

but k0 − equivariant

3

2

2

Some examples 8

k0 = so8

k1 = ∆ +

28 + 8 = 36

so9

S 8 ⊂ R9

k0 = so9

k1 = ∆

36 +16 = 52

f4

k0 = so16

k1 = ∆ +

120+128 = 248

e8

7

S ⊂R

S

15

!

⊂R

k1 ⊗

16

" 3 ∗ k0 Λ k1

=0

=⇒

Jacobi

Resulting Lie algebras are simple.

A sketch of the proof Two observations: 1) The bijection between Killing spinors and parallel spinors in the cone is equivariant under the action of isometries. Use the cone to calculate LX ε . 2) In the cone, LX ε = "(AX )ε and since X is

linear, the endomorphism AX is constant. It is the natural action on spinors.

A (slight) generalisation 9

S ⊂R

10

C!9 ∼ = C(16)

M=∆⊗C

irreducible real spinor module Spin(9) → SO(16)

complex spinor bundle

S→S

9

complex symmetric inner product !X · ψ1 , ψ2 " = + !ψ1 , X · ψ2 "

dvol(S ) = i 9

Killing spinors

" # K± = ψ ∈ Γ(S)"∇X ψ = ± 12 X · ψ !

k0 = soC (10)

k 1 = K+ ⊕ K−

Natural brackets well-defined, but Jacobi fails! (Killing-Yano)

g0 = soC (10) ⊕ C

[ψ+ , ψ− ] = · · · + !ψ+ , ψ− " z ∈ C =⇒ [z, ψ] = 13 Dψ

empirical!

g1 = k1

Jacobi: [[ψ+ , ψ− ], χ+ ] − [[χ+ , ψ− ], ψ+ ] = 0

∀ψ± , χ± ∈ K±

is satisfied, even when !

Λ g1 → g1 3

" g0

"= 0

The resulting Lie algebra is E6 (complexified)

Open questions

• Other exceptional Lie algebras? E7 should follow from the 11-sphere, but this is still work in progress. G2?

• Are the Killing superalgebras of the Hopf spheres related?

• What structure in the 15-sphere has E8 as automorphisms?