TWISTORS AND THE OCTONIONS Penrose 80. Nigel Hitchin. Oxford July 21st 2011

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1 TWISTORS AND THE OCTONIONS Penrose 80 Nigel Hitchin Oxford July 21st 2011

2 8th August 1931

3 8th August an oblong arrangement of terms consisting, suppose, of lines and columns. This will not in itself represent a determinant, but is, as it were, a Matrix out of which we may form various systems of determinants.. JJ Sylvester, An Essay on Canonical Forms, Supplement to a Sketch of a Memoir on Elimination, Transformation and Canonical Forms

4 26th December 1843

5 26th December 1843 John T Graves

6 16th October 1843

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8 Hamilton to Graves October 1843

9 Graves to Hamilton December 1843 (a 2 +b 2 +c 2 +d 2 +e 2 +f 2 +g 2 +h 2 )(m 2 +n 2 +o 2 +p 2 +q 2 +r 2 +s 2 +t 2 ) = v v2 2 + v2 3 + v2 4 + v2 5 + v2 6 + v2 7 + v2 8 where, v 1 = am bn co dp eq fr gs ht v 2 = bm + an + do cp + fq er hs + gt v 3 = cm dn + ao + bp + gq + hr es ft etc.

10

11 CAYLEY NUMBERS e 2 i = e2 j = e2 k = e ie j e k = 1 (ijk) = (123), (145), (624), (653), (725), (734), (176)

12 CAYLEY NUMBERS = OCTONIONS e 2 i = e2 j = e2 k = e ie j e k = 1 (ijk) = (123), (145), (624), (653), (725), (734), (176)

13 AUTOMORPHISMS quaternions x 0 + x 1 i + x 2 j + x 3 k automorphism group SO(3)

14 AUTOMORPHISMS quaternions x 0 + x 1 i + x 2 j + x 3 k automorphism group SO(3) octonions x 0 + x 1 i 1 + x 2 i 2 + x 3 i 3 + x 4 i 4 + x 5 i 5 + x 6 i 6 + x 7 i 7 automorphism group G 2 SO(7) 14-dimensional compact simple Lie group

15 the crazy old uncle nobody lets out of the attic J C Baez, The Octonions, BAMS (2002)

16 Matrices Octonions O

17 Matrices Octonions O What is SL(2, O)?

18 Lorentz group and the conformal group of Minkowski space to be discussed in terms of groups of complex matrices: SO(3, 1 ) ~SL(2, C), SO(4, 2) ~ SU(2, 2) ~Spt(4, C ), (1) (2)

19 Lorentz group and the conformal group of Minkowski space to be discussed in terms of groups of complex matrices: SO(3, 1 ) ~SL(2, C), SO(4, 2) ~ SU(2, 2) ~Spt(4, C ), (1) (2) in terms of 2 X 2 blocks. In the critical dimension of the fermionic string, namely (9 + 1 )-dimensional Minkowski space-time M to, there are similar isomorphisms involving groups of octonionic matrices [ 1 ]: S0(9, 1) ~SL(2, O ), (5) SO(10, 2) --- Sp*(4, O ). (6) KW Chung & A Sudbery, Octonions and the Lorentz and conformal groups of ten-dimensional space-time, Phys Lett B 198 (1987)

20 Spin(9, 1) spinors S, S 16-dimensional real spaces

21 Spin(9, 1) spinors S, S 16-dimensional real spaces Claim: GL(2, O) is an open set in S R 2

22 TWISTORS

23 conformal transformations of S 4 : SO(5, 1) spin representations S, S complex 4-dimensional 2-dimensional quaternionic spaces Spin(5, 1) = SL(2, H)

24 a complex vector space S is quaternionic if it has an antilinear automorphism J such that J 2 = 1 q = (a 0 + ia 1 ) + (a 2 + ia 3 )J A : S S complex linear is quaternionic if AJ = JA (left action of a quaternionic matrix commutes with right multiplication by q)

25 eigenvalues: Av = λv AJv = JAv = J(λv) = λjv complex determinant of A is real and 0 quaternionic n n matrix A det A is a real polynomial of degree 2n

26 REALIZATION IN CONFORMAL GEOMETRY spinor bundles S +, S Dirac operator Dψ = i e i i ψ Twistor operator Dψ = i e i i ψ + 1 n e i e i Dψ

27 REALIZATION IN CONFORMAL GEOMETRY spinor bundles S +, S Dirac operator Dψ = i e i i ψ conformal weight n 1 2 Twistor operator Dψ = i e i i ψ + 1 n e i e i Dψ conformal weight 1 2

28 on S 4, Dψ = 0 with ψ S + has a 4-dimensional space of solutions the space of twistors T for ψ S the solutions are the dual twistor space T

29 on S 4, Dψ = 0 with ψ S + has a 4-dimensional space of solutions the space of twistors T for ψ S the solutions are the dual twistor space T stereographic projection ψ = x ϕ + ϕ + where ϕ,ϕ + are constant spinors on R 4

30 on S 4, Dψ = 0 with ψ S + has a 4-dimensional space of solutions the space of twistors T for ψ S the solutions are the dual twistor space T stereographic projection ψ = x ϕ + ϕ + where ϕ,ϕ + are constant spinors on R 4 conformal transformations act on T, T as the representations S, S of Spin(5, 1).

31 on R 4 f = ( 2 f) ij 1 n δ ij f conformal weight 1 f = 0 has a 6-dimensional space of solutions f = ar 2 + b i x i + c: f 2 δ ij is an Einstein metric

32 on R 4 f = ( 2 f) ij 1 n δ ij f conformal weight 1 f = 0 has a 6-dimensional space of solutions f = ar 2 + b i x i + c: f 2 δ ij is an Einstein metric conformal transformations act via the vector representation V V

33 on R 4 f = ( 2 f) ij 1 n δ ij f conformal weight 1 f = 0 has a 6-dimensional space of solutions f = ar 2 + b i x i + c: f 2 δ ij is an Einstein metric conformal transformations act via the vector representation V V Lorentzian inner product (f, f) = b i b i 4ac = - scalar curvature of the metric

34 solutions ψ 1, ψ 2 of twistor equation hermitian inner product on S + : f = ψ 1, ψ 2 scalar of weight 1

35 solutions ψ 1, ψ 2 of twistor equation hermitian inner product on S + : f = ψ 1, ψ 2 scalar of weight 1 f = 0 f = ψ, ψ real Einstein metric (f, f) = 0 flat metric on S 4 \pt.

36 ψ T f = ψ, ψ vanishes at a point in S 4 f = x ϕ vanishes at x = 0: one-dimensional quaternionic subspace = S + 0 of T S 4 = HP 1 = P(T)

37 GL(2, H)

38 T R 2 left action of End H (T) right action of H + right action of End(R 2 ) T R 2 = Hom H (H 2, T)

39 DETERMINANT ρ = (ψ 1, ψ 2 ) T R 2 ψ a, ψ b = f ab (f 11, f 22 ) (f 12, f 21 ) (real) quartic function µ(ρ) µ(ρ) = 3 det A, A : H 2 T

40 {ρ T R 2 : µ(ρ) 0} = Iso H (H 2, T) quaternionic bases of twistor space any two differ by an action of GL(2, H)

41 GL(2, O)

42 conformal transformations of S 8 : SO(9, 1) spin representations S, S real 16-dimensional twistor space T = S ψ = x ϕ + ϕ + vanishes at a point: S 8 = OP 1

43 ρ = (ψ 1, ψ 2 ) T R 2 ψ a, ψ b = f ab (f 11, f 22 ) (f 12, f 21 ) real quartic µ(ρ)

44 PROP: {ρ T R 2 : µ(ρ) 0} is an open orbit of the group Spin(9, 1) GL(2, R) with stabilizer G 2 SL(2, R). T.Kimura, Introduction to prehomogeneous vector spaces, AMS (2003)

45 PROP: {ρ T R 2 : µ(ρ) 0} is an open orbit of the group Spin(9, 1) GL(2, R) with stabilizer G 2 SL(2, R). T.Kimura, Introduction to prehomogeneous vector spaces, AMS (2003) G 2 SL(2, R) G 2 SO(2, 1) SO(7) SO(2, 1) SO(9, 1) T = O O T R 2 = 2 2 octonionic matrices

46 PROP: {ρ T R 2 : µ(ρ) 0} is an open orbit of the group Spin(9, 1) GL(2, R) with stabilizer G 2 SL(2, R). T.Kimura, Introduction to prehomogeneous vector spaces, AMS (2003) G 2 SL(2, R) G 2 SO(2, 1) SO(7) SO(2, 1) SO(9, 1) T = O O T R 2 = 2 2 octonionic matrices {ρ T R 2 : µ(ρ) 0} = octonionic bases in T

47 IN TWISTOR TERMS twistors ψ = x ϕ + ϕ + ϕ, ϕ + constant spinors in 8 dimensions e unit vector ϕ e ϕ isomorphism as representations of Spin(7)

48 IN TWISTOR TERMS twistors ψ = x ϕ + ϕ + ϕ, ϕ + constant spinors in 8 dimensions e unit vector ϕ e ϕ isomorphism as representations of Spin(7) G 2 Spin(7) stabilizer of a spinor ϕ ρ = (ψ 1, ψ 2 ) = (x ϕ, e ϕ)

49 Spin(9, 1) spinors S, S 16-dimensional real spaces Claim: GL(2, O) is an open set in S R 2

50 so(9, 1) + gl(2) = g 2 + sl(2) + gl(2) O µ=const.

51 so(9, 1) + gl(2) = g 2 + sl(2) + gl(2) O µ=const. µ=const. A gl(2) O tangent space if tr A is imaginary Definition: {ρ T R 2 : µ(ρ) = 3} = SL(2, O)

52 THE INVARIANT METRIC

53 SL(2, O) = hypersurface µ = 3 in R 32 Hessian metric g = 2 µ invariant under Spin(9, 1) SL(2, R) tangent space sl(2) + gl(2) im O

54 SL(2, O) = hypersurface µ = 3 in R 32 Hessian metric g = 2 µ invariant under Spin(9, 1) SL(2, R) tangent space sl(2) + gl(2) im O (A, A) = 3(tr A tr A 2 i )

55 (A, A) = 3(tr A tr A 2 i ) signature ( , ) = (9, 22)

56 (A, A) = 3(tr A tr A 2 i ) signature ( , ) = (9, 22) Replace O by H metric = Killing form on SL(2, H) = Spin(5, 1) signature ( , ) = (5, 10)

57 (A, A) = 3(tr A tr A 2 i ) signature ( , ) = (9, 22) Replace O by H metric = Killing form on SL(2, H) = Spin(5, 1) signature ( , ) = (5, 10) maximal compact Spin(5)

58 (A, A) = 3(tr A tr A 2 i ) signature ( , ) = (9, 22) Replace O by H?? metric = Killing form on SL(2, H) = Spin(5, 1) signature ( , ) = (5, 10) maximal compact Spin(5)

59 STIEFEL MANIFOLDS

60 V 2 (F n ) = orthonormal pairs of vectors V 2 (C 2 ) = U(2) V 2 (H 2 ) = Sp(2)

61 V 2 (F n ) = orthonormal pairs of vectors V 2 (C 2 ) = U(2) V 2 (H 2 ) = Sp(2) V 2 (O 2 ) = Spin(9)/G 2 dim = = 22

62 M = Spin(9)/G 2 T x M = 2 2 octonionic matrices A such that... Ā T = A

63 H (U(2)) = H (S 1 S 3 ) H (Sp(2)) = H (S 3 S 7 ) H (Spin(9)/G 2 ) = H (S 7 S 15 )

64 H (U(2)) = H (S 1 S 3 ) H (Sp(2)) = H (S 3 S 7 ) H (Spin(9)/G 2 ) = H (S 7 S 15 ) Spin(7)/G 2 Spin(9)/G 2 Spin(9)/Spin(7) S 7 S 15

65 Spin(9)/G 2 has trivial tangent bundle WA Sutherland, A note on the parallelizability of spherebundles over spheres, J. London Math. Soc (1964)

66 Spin(9)/G 2 has trivial tangent bundle WA Sutherland, A note on the parallelizability of spherebundles over spheres, J. London Math. Soc (1964) The product of two harmonic forms is harmonic D Kotschick, D & S Terzic, Geometric formality of homogeneous spaces and of biquotients, Pacific J. Math (2011)

67 REAL FORMS

68 µ : S R 2 R dµ(ρ) = ˆρ S R 2

69 µ : S R 2 R dµ(ρ) = ˆρ S R 2 Spin(5, 1), ρ quaternionic matrix A Â = (Ā T ) 1

70 v V, (v, v) 0 ψ v ψ defines S = S real form ˆρ = v ρ (v, v) < 0 Sp(2), (v, v) > 0 Sp(1, 1)

71 µ : S R 2 R Spin(9, 1) (v, v) < 0 Spin(9)/G 2 = SU(2; O) (v, v) > 0 Spin(8, 1)/G 2 = SU(1, 1; O)

72 WHAT NEXT?

73 8-DIMENSIONAL RIEMANNIAN GEOMETRY? M 8 Riemannian manifold principal Spin(8)-bundle P P/G 2 modelled on SU(2, O)

74 D = 10, N = (2, 0) SUPERGRAVITY? M 9,1 space time supermanifold S R 2 M principal SL(2, O) bundle?

75 ... Of course, mathematical beauty is a worthy end in itself, but it would be even more delightful if the octonions turned out to be built into the fabric of nature. As the story of the complex numbers and countless other mathematical developments demonstrates, it would hardly be the first time that purely mathematical inventions later provided precisely the tools that physicists need. J C Baez & J Huerta, The Strangest Numbers in String Theory, Scientific American, May (2011)