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Clifford Algebras 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 Killing É. Cartan Hurwitz Hopf 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 O ≥ ab = ba ab = ba ! (ab)c = a(bc) (ab)c = a(bc) ! These are all the euclidean normed real division algebras. [Hurwitz] Hopf fibrations S1 S3 S7 S15 S0 S1 S3 S7 S1 S2 S4 S8 " " " " S0 R S1 C S3 H S7 O ⊂ ⊂ ⊂ ⊂ 1 2 3 2 7 2 15 2 S R S C S H S O ⊂ ⊂ ⊂ ⊂ 1 2 4 8 S ∼= RP1 S ∼= CP1 S ∼= HP1 S ∼= OP1 These are the only examples of fibre bundles where all three spaces are spheres. [Adams] Simple Lie algebras (over C) 4 classical series: 5 exceptions: An 1 SU(n + 1) G2 14 ≥ F4 52 Bn 2 SO(2n + 1) ≥ E6 78 Cn 3 Sp(n) ≥ E7 133 Dn 4 SO(2n) E8 248 ≥ [Lie] [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, oneS 15 obtains a triple of exceptional Lie algebras: S7 S8 " E8 “Killing superalgebra” B4 F4 Spinors Clifford Clifford algebras V n , real euclidean vector space !− −# V C!(V ) = filtered associative algebra v v + v 21 ! ⊗ ! | | # C!(V ) ∼= ΛV (as vector spaces) C!(V ) = C!(V ) C!(V ) 0 ⊕ 1 C!(V ) = ΛevenV odd 0 ∼ C!(V )1 ∼= Λ V orthonormal frame e1, . , en n eiej + ejei = 2δij1 C! (R ) =: C!n − Examples: C!0 = 1 = R ! " ∼ 2 C!1 = 1, e1 e = 1 = C 1 − ∼ ! " 2 # 2 C!2 = 1, e1", e2 e = e = 1, e1e2 = e2e1 = H 1 2 − − ∼ ! " # " Classification n C!n Bott periodicity: 0 R 1 C C!n+8 = C!n R(16) 2 H ∼ ⊗ 3 H H ⊕ e.g., 4 H(2) 5 C(4) C!9 ∼= C(16) 6 R(8) C!16 ∼= R(256) 7 R(8) R(8) ⊕ 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 e e e 1 2 · · · n which is central for n odd. Notation : Mn or Mn± Clifford modules n/2 dim Mn = 2! " Spinor representatinos so C! n → n 1 exp Spinn C!n ei ej eiej ⊂ ∧ "→ − 2 n 1 n s Spinn, v R = svs− R ∈ ∈ ⇒ ∈ which defines a 2-to-1 map Spin SO n → n with archetypical example Spin3 = SU2 H ∼ ⊂ 2-1 SO3 = SO (ImH) ∼ " By restriction, every representation of C ! n defines a representation of Sp in n : C! Spin n ⊃ n M = ∆ = ∆+ ∆ ∆ chiral spinors ⊕ − ± M± = ∆ ∆ spinors One can read off the type of representation from Spinn (C!n) = C!n 1 ⊂ 0 ∼ − (n 1)/2 (n 2)/2 dim ∆ = 2 − dim ∆ = 2 − ± Spinor inner product ( , ) − − nondegenerate form on ∆ (ε1, ε2) = (ε2, ε1) (ε , e ε ) = (e ε , ε ) ε ∆ 1 i · 2 − i · 1 2 ∀ i ∈ = (ε , e e ε ) = (e e ε , ε ) ⇒ 1 i j · 2 − i j · 1 2 2 n which allows us to define [ , ] : Λ ∆ R − − → [ε , ε ], e = (ε , e ε ) ! 1 2 i" 1 i · 2 Spin geometry Spin manifolds M n differentiable manifold, orientable, spin g riemannian metric w1 = 0 w2 = 0 GL(M) O(M) SO(M) Spin(M) M M M M " " " " _ _ GLn o ? On o ? SOn o o Spinn 1 Possible Spin(M) are classified by H ( M ; Z / 2) . n n+1 e.g., M = S R ⊂ O(M) = On+1 SO(M) = SOn+1 Spin(M) = Spinn+1 n S ∼= On+1/On ∼= SOn+1/SOn ∼= Spinn+1/Spinn π (M) = 1 = unique spin structure 1 { } ⇒ Spinor bundles C!(T M) Clifford bundle C!(T M) = ΛT M ∼ M " S(M) := Spin(M) ∆ (chiral) ×Spinn spinor S(M) := Spin(M) Spin ∆ bundles ± × n ± 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 (M,g) is Ricci-flat Killing spinors (M,g) is Einstein R = 4λ2n(n 1) − = 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) metric cone + M = R M g = dr2 + r2g Ch. Bär × Killing spinors parallel spinors 1-1 in (M,g) in the cone λ = 1 ± 2 ! " 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. n Holonomy n SOn 2m Um 2m SUm 4m Sp Sp M. Wang m · 1 M. Berger 4m Spm 7 G2 8 Spin7 Or else the cone is flat and M is a sphere. Killing superalgebra Construction of the algebra (M, g) riemannian spin manifold k = k k 0 ⊕ 1 k1 = { Killing spinors } 1 with λ = 2 ! " k0 = [k1, k1] Killing vectors ⊂ { } [ , ] : Λ2k 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! − − ⊗ → ? Kosmann Lichnerowicz X Γ(T M) Killing g = 0 ∈ LX AX := Y Y X ∈ !→ −∇ so(T M) ! : so(T M) EndS(M) spinor representation → := + !(A ) spinorial Lie derivative LX ∇X X cf. LX Y = X Y + AX Y = X Y Y X = [X, Y ] ✓ ∇ ∇ − ∇ Properties X, Y k , Z Γ(T M), ε Γ(S(M)), f C∞(M) ∀ ∈ 0 ∈ ∈ ∈ L (Z ε) = [X, Z] ε + Z L ε X · · · X LX (fε) = X(f)ε + fLX ε ε k1, X k0 [ , ] : k0 k1 k1 ∀ ∈ ∈ − − ⊗ → X ε k1 [L , ]ε = ε L ∈ X ∇Z ∇[X,Z] [LX , LY ]ε = L[X,Y ]ε [X, ε] := X ε ✓ L } The Jacobi identity Jacobi: Λ3k k → (X, Y, Z) [X, [Y, Z]] [[X, Y ], Z] [Y, [X, Z]] !→ − − 4L components(Z ε) = [X, Z: ] ε + Z L ε X · · · X Λ3k k ✓ Jacobi for vector fields 0 → 0 Λ2k k k ✓ 0 ⊗ 1 → 1 2 k0 Λ k1 k0 ✓ [LX⊗, LY ]ε →= L[X,Y ]ε 3 Λ k1 k1 ? but k0 equivariant → − Some examples 7 8 S R k0 = so8 k1 = ∆+ 28 + 8 = 36 so9 ⊂ 8 9 S R k0 = so9 k1 = ∆ 36 +16 = 52 f4 ⊂ 15 16 S R k0 = so16 k1 = ∆+ 120+128 = 248 e8 ⊂ 3 k0 k Λ k∗ = 0 = Jacobi 1 ⊗ 1 ⇒ ! " 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 ε . LX 2) In the cone, X ε = " ( A X ) ε and since X is L linear, the endomorphism A X is constant. It is the natural action on spinors. A (slight) generalisation 9 10 S R C!9 = C(16) M = ∆ C ⊂ ∼ ⊗ irreducible real spinor module Spin(9) SO(16) → 9 9 complex spinor bundle S S dvol(S ) = i → complex symmetric inner product X ψ , ψ = + ψ , X ψ ! · 1 2" ! 1 · 2" 1 Killing spinors K = ψ Γ(S) X ψ = X ψ ± ∈ ∇ ± 2 · k0 = so! (10) " # C " k1 = K+ K ⊕ − Natural brackets well-defined, but Jacobi fails! (Killing-Yano) g0 = so (10) C g1 = k1 C ⊕ [ψ+, ψ ] = + ψ+, ψ − · · · ! −" z C = [z, ψ] = 1 Dψ ∈ ⇒ 3 empirical! Jacobi: [[ψ+, ψ ], χ+] [[χ+, ψ ], ψ+] = 0 ψ , χ K − − − ∀ ± ± ∈ ± is satisfied, even when g Λ3g g 0 = 0 1 → 1 " ! " 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?.
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