Quantum Supersymmetric Bianchi IX Cosmology and its Hidden Kac-Moody Structure. Thibault DAMOUR

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1 Quantum Supersymmetric Bianchi IX Cosmology and its Hidden Kac-Moody Structure Thibault DAMOUR Institut des Hautes Études Scientifiques Gravitation, Solitons and Symmetries, May 20 23, 2014 Thibault Damour (IHES) Quantum Susy Bianchi IX Cosmology Gibbons, Studium, May / 33

2 Hidden Kac-Moody symmetries (E 10 or AE n ) in (super)gravity Damour, Henneaux 01; Damour, Henneaux, Julia, Nicolai 01; Damour, Henneaux, Nicolai 02; Damour, Kleinschmidt, Nicolai 06; de Buyl, Henneaux, Paulot 06; Kleinschmidt, Nicolai 06;... [first conjectured by Julia 82; related conjectures Ganor 99, 04; West (E 11 ) 01; DHN work stemmed from a study of Belinski-Khalatnikov- Lifshitz chaotic billiard dynamics] Gravity/Coset conjecture (DHN02) Duality between D = 11 supergravity (or, hopefully, M-theory) and the (quantum) dynamics of a massless spinning particle on E 10 /K (E 10 ) Thibault Damour (IHES) Quantum Susy Bianchi IX Cosmology Gibbons, Studium, May / 33

3 Gravity/Coset Correspondence E 10 : Damour, Henneaux, Nicolai 02; related: Ganor 99 04; E 11 : West 01 Thibault Damour (IHES) Quantum Susy Bianchi IX Cosmology Gibbons, Studium, May / 33

4 A concrete case study (Damour, Spindel 2013, 2014) Quantum supersymmetric Bianchi IX, i.e. quantum dynamics of a supersymmetric triaxially squashed three-sphere time Susy Quantum Cosmology: Obregon et al 1990; D Eath, Hawking, Obregon, 1993, D Eath 1993, Csordas, Graham 1995, Moniz 94,... Thibault Damour (IHES) Quantum Susy Bianchi IX Cosmology Gibbons, Studium, May / 33

5 A concrete case study (Damour, Spindel 2013, 2014) Technically: Reduction to one, time-like, dimension of the action of D = 4 simple supergravity for an SU(2)-homogeneous (Bianchi IX) cosmological model g µν dx µ dx ν = N 2 (t)dt 2 + g ab (t)(τ a (x) + N a (t)dt)(τ b (x) + N b (t)dt), τ a : left-invariant one-forms on SU(2) S 3 : dτ a = 1 2 ε abc τ b τ c Thibault Damour (IHES) Quantum Susy Bianchi IX Cosmology Gibbons, Studium, May / 33

6 Dynamical degrees of freedom Gauss-decomposition of the metric: 3 g bc = e 2βa Sâ b (ϕ 1, ϕ 2, ϕ 3 ) Sâ c(ϕ 1, ϕ 2, ϕ 3 ) â =1 six metric dof: β a = (β 1 (t), β 2 (t), β 3 (t)) = cologarithms of the squashing parameters a, b, c of 3-sphere a = e β1, b = e β2, c = e β3 and three Euler angles: ϕ a = (ϕ 1 (t), ϕ 2 (t), ϕ 3 (t)) parametrizing the orthogonal matrix S^a b Thibault Damour (IHES) Quantum Susy Bianchi IX Cosmology Gibbons, Studium, May / 33

7 Dynamical degrees of freedom Gravitino components in specific gauge-fixed orthonormal frame θ α canonically associated to the Gauss-decomposition θ 0 = N(t)dt, θâ = b e βa (t) Sâ b (ϕ c(t))(τ b (x) + N b (t)dt) redefinitions of the gravitino field: Ψ A^α (t) := g1/4 ψ A^α and Φ a A := Σ B γ^a AB ΨB^a (no summation on ^a) 3 4 gravitino components Φ a A, a = 1, 2, 3; A = 1, 2, 3, 4. Thibault Damour (IHES) Quantum Susy Bianchi IX Cosmology Gibbons, Studium, May / 33

8 Supersymmetric action (first order form) S = dt [ π a β a + p ϕ a ϕ a + i 2 G ab Φ a A Φ b A + Ψ A 0 S A ÑH Na H a ] G ab : Lorentzian-signature quadratic form: G ab dβ a dβ b a ( ) 2 (dβ a ) 2 dβ a a G ab defines the kinetic terms of the gravitino, as well as those of the β a s: 1 2 G ab β a β b Lagrange multipliers Constraints S A 0, H 0, H a 0 Thibault Damour (IHES) Quantum Susy Bianchi IX Cosmology Gibbons, Studium, May / 33

9 Quantization Bosonic dof: π a = i β a ; p ϕa = i ϕ a Fermionic dof: Φ a A Φ b B + Φ b B Φ a A = Gab δ AB This is the Clifford algebra Spin (8 +, 4 ) The wave function of the universe Ψ σ (β a, ϕ a ) is a 64-dimensional spinor of Spin (8, 4) and the gravitino operators Φ a A are gamma matrices acting on Ψ σ, σ = 1,..., 64 Similar to Ramond string ψ µ 0 γµ spinorial amplitude Thibault Damour (IHES) Quantum Susy Bianchi IX Cosmology Gibbons, Studium, May / 33

10 Dirac Quantization of the Constraints Ŝ A Ψ = 0, Ĥ Ψ = 0, Ĥ a Ψ = 0 Diffeomorphism constraint p ϕa Ψ = i ϕ a Ψ = 0 : s wave w.r.t. the Euler angles Wave function Ψ(β a ) submitted to constraints Ŝ A ( π, β, Φ) Ψ(β) = 0, Ĥ( π, β, Φ) Ψ(β) = 0 π a = i β a PDE s for the 64 functions Ψ σ (β 1, β 2, β 3 ) Heavily overdetermined system of PDE s Thibault Damour (IHES) Quantum Susy Bianchi IX Cosmology Gibbons, Studium, May / 33

11 Explicit form of the SUSY constraints (γ 5 γ^0^1^2^3, β 12 β 1 β 2, Φ 12 Φ 1 Φ 2 ) Ŝ A = 1 π a Φ a A e 2βa (γ 5 Φ a ) A 2 a a 1 8 coth β 12(Ŝ12(γ 12 Φ12 ) A + (γ 12 Φ12 ) A Ŝ 12 ) + cyclic (123) (Ŝcubic A + Ŝcubic A ) Ŝ 12 ( Φ) = 1 2 [( Φ3 γ^0^1^2 ( Φ 1 + Φ 2 )) + ( Φ1 γ^0^1^2 Φ1 ) + ( Φ2 γ^0^1^2 Φ2 ) ( Φ1 γ^0^1^2 Φ2 )], ŜA cubic = 1 ( Ψ0 γ 0 Ψâ) γ 0 ΨAâ 1 ( Ψâ γ 0 γâ 4 8 Ψ b) ΨA b a a,b + 1 ( Ψ0 γâ Ψ 8 Ψ b)(γâ A b + γ b Ψ A â ), a,b Thibault Damour (IHES) Quantum Susy Bianchi IX Cosmology Gibbons, Studium, May / 33

12 (Open) Superalgebra satisfied by the ŜA s and Ĥ Ŝ A Ŝ B + ŜB ŜA = 4 i C L C AB (β) ŜC Ĥδ AB [ŜA, Ĥ] = M B A ŜB + N A Ĥ Thibault Damour (IHES) Quantum Susy Bianchi IX Cosmology Gibbons, Studium, May / 33

13 Dynkin Diagrams (= Cartan Matrix) of E 10 and AE Figure: Dynkin diagram of E 10 with numbering of nodes Cartan matrix of AE 3 : (A ij ) = α 7 α 1 7 α 2 Dynkin diagram AE 3 Thibault Damour (IHES) Quantum Susy Bianchi IX Cosmology Gibbons, Studium, May / 33

14 Root diagram of AE 3 = A dimensional Lorentzian-signature space: metric in Cartan sub-algebra G ab dβ a dβ b = a ( ) 2 (dβ a ) 2 dβ a a (directly linked to Einstein action: K 2 ij (K i i ) 2 ) Thibault Damour (IHES) Quantum Susy Bianchi IX Cosmology Gibbons, Studium, May / 33

15 Kac-Moody Structures Hidden in the Quantum Hamiltonian 2 Ĥ = Gab ( π a + i A a )( π b + i A b ) + µ 2 + Ŵ (β), G ab metric in Cartan subalgebra of AE 3 Ŵ (β) = W bos g (β) + Ŵ g spin (β) + Ŵ sym spin (β). W bos g (β) = 1 2 e 4β1 e 2(β2 +β 3) + cyclic 123 Linear forms α g ab (β) = βa + β b six level-1 roots of AE 3 Thibault Damour (IHES) Quantum Susy Bianchi IX Cosmology Gibbons, Studium, May / 33

16 Kac-Moody Structures Hidden in the Quantum Hamiltonian Ŵ spin g (β, Φ) = e αg 11 (β) Ĵ 11 ( Φ) + e αg 22 (β) Ĵ 22 ( Φ) + e αg 33 (β) Ĵ 33 ( Φ). Ŵ spin sym (β) = 1 2 (Ŝ12( Φ)) 2 1 sinh 2 α sym 12 (β) + cyclic 123, Linear forms α sym 12 (β) = β1 β 2, α sym 23 (β) = β2 β 3, α sym 31 (β) = β3 β 1 three level-0 roots of AE 3 Thibault Damour (IHES) Quantum Susy Bianchi IX Cosmology Gibbons, Studium, May / 33

17 Spin dependent (Clifford) Operators coupled to AE 3 roots Ŝ 12 ( Φ) = 1 2 [( Φ 3 γ^0^1^2 ( Φ 1 + Φ 2 )) + ( Φ 1 γ^0^1^2 Φ1 ) + ( Φ 2 γ^0^1^2 Φ2 ) ( Φ 1 γ^0^1^2 Φ2 )], Ĵ 11 ( Φ) = 1 2 [ ^Φ1 γ^1^2^3 (4 Φ 1 + Φ 2 + Φ 3 ) + ^Φ2 γ^1^2^3 Φ3 ]. Ŝ12, Ŝ 23, Ŝ 31, Ĵ 11, Ĵ 22, Ĵ 33 generate (via commutators) a 64- dimensional representation of the (infinite-dimensional) maximally compact sub-algebra K (AE 3 ) AE 3. [The fixed set of the (linear) Chevalley involution, ω(e i ) = f i, ω(f i ) = e i, ω(h i ) = h i, which is generated by x i = e i f i.] Thibault Damour (IHES) Quantum Susy Bianchi IX Cosmology Gibbons, Studium, May / 33

18 Maximally compact sub-algebra of a Kac-Moody alg. Generalization of so(n) gl(n) : gl(n): arbitrary matrices: m ij so(n): antisymmetric matrices: a ij = a ij projection: gl(n) so(n): m ij 1 2 (m ij m ji ) Kac-Moody alg. generated by h i, e i, f i : Maximally compact subalgebra generated by x i = e i f i : i.e. fixed set of the (linear) Chevalley involution ω(e i ) = f i, ω(f i ) = e i, ω(h i ) = h i, which corresponds to a a T Thibault Damour (IHES) Quantum Susy Bianchi IX Cosmology Gibbons, Studium, May / 33

19 The squared-mass Quartic Operator µ 2 in Ĥ In the middle of the Weyl chamber (far from all the hyperplanes α i (β) = 0): 2 Ĥ π2 + µ 2 where µ 2 Φ4 gathers many complicated quartic-in-fermions terms (including Ŝ 2 ab and the infamous ψ4 terms of supergravity). Remarkable Kac-Moody-related facts: µ 2 Center of the algebra generated by the K (AE 3 ) generators Ŝab, Ĵ ab µ 2 is the square of a very simple operator Center µ 2 = Ĉ2 F where ĈF := 1 2 G ab Φ a γ^1^2^3 Φb. Thibault Damour (IHES) Quantum Susy Bianchi IX Cosmology Gibbons, Studium, May / 33

20 Solutions of SUSY constraints Overdetermined system of 4 64 Dirac-like equations ( ) i Ŝ A Ψ = 2 Φa A β a +... Ψ = 0 Space of solutions is a mixture of discrete states and continuous states, depending on fermion number N F = C F 3. solutions for both even and odd N F. continuous states (parametrized by initial data comprising arbitrary functions) at C F = 1, 0, +1. Our results complete inconclusive studies started long ago: D Eath 93, D Eath-Hawking-Obregon 93, Csordas-Graham 95, Obregon 98,... Thibault Damour (IHES) Quantum Susy Bianchi IX Cosmology Gibbons, Studium, May / 33

21 Fermionic number operator Fermionic number operator ^N F = ^C F + 3, with ĈF := 1 2 G ab Φ a γ^1^2^3 Φb. Eigenvalues C F = 3, 2, 1, 0, 1, 2, 3 or 0 N F 6 Fermionic annihilation operators b± a and creation operators b b with ɛ, σ = ± b+ a = Φ a 1 + i Φ a 2, ba = Φ a 3 i Φ a 4 b + a = Φ a 1 i Φ a 2, ba = Φ a 3 + i Φ a 4 { bɛ, a b } σ b = 2 G ab δ ɛ σ, { } { bɛ, a bσ b = ba ɛ, b } σ b = 0. Vacuum state (empty) b a ɛ 0 = 0 : N F = 0 [Filled state: b a ɛ 0 + = 0 : ^N F = 6] ^N F = G ab ba + b b + + G ab ba bb Thibault Damour (IHES) Quantum Susy Bianchi IX Cosmology Gibbons, Studium, May / 33

22 Spin diamond Spin (8, 4) = 64-dim space spanned by the spinor wavefunction Ψ σ can be sliced by a fermionic number operator ^N F = ^C F + 3. Among these, the solutions (discrete = crosses or continuous = circle) of the susy constraints are marked in red. Thibault Damour (IHES) Quantum Susy Bianchi IX Cosmology Gibbons, Studium, May / 33

23 Solution space of quantum susy Bianchi IX: N F = 0 Level N F = 0 : unique ground state f = f (β) 0 with ( ) f = f 0 [(y x)(z x)(z y)] 3/8 e x + 1 y + 1 z (x y z) 5/4 where x := e 2β 1 = 1 a 2, y := e 2β 2 = 1 b 2, z := e 2β 3 = 1 c 2 f differs from previously discussed ground state exp ( 1 2 (a2 + b 2 + c 2 ) ) (Moncrief-Ryan, D Eath,...) by factors (x y) 3/8... vanishing on the symmetry walls. [ centrifugal barriers ] Thibault Damour (IHES) Quantum Susy Bianchi IX Cosmology Gibbons, Studium, May / 33

24 Solution space of quantum susy Bianchi IX: N F = 1 Previous approaches (D Eath, Graham,...) concluded to the absence of states at odd fermionic levels. Level N F = 1 : two dimensional solution space: c + f a ba c f a ba 0 f a = {x(y z), y(z x), z(x y)} (x y z) 5/4 ((x y)(y z)(z x)) 3/8 e 1 2 ( 1 x + 1 y + 1 z ) Thibault Damour (IHES) Quantum Susy Bianchi IX Cosmology Gibbons, Studium, May / 33

25 Solution space: N F = 2 σ,σ =± k,k =1,...,3 f σσ kk bk σ bk σ 0, f σσ kk = f σ σ k k mixture of discrete (finite-dimensional) and continuous (infinitedimensional; free functions) space of solutions Example of two-dimensional subspace of discrete solutions { {f12 ɛɛ, f23 ɛɛ, f31 ɛɛ } = f ɛɛ x(y z) yz+ xyz } 2, y(z x) zx+xyz 2, z(x y) xy+xyz 2 with the prefactor functions f ɛɛ given by ( ) f ɛɛ = e x + 1 y + 1 z (xyz) 3/4 [(x y)(y z)(x z)] 1/8 [C 1 (x z) 1/2 + ɛ C 2 (y z) 1/2] Thibault Damour (IHES) Quantum Susy Bianchi IX Cosmology Gibbons, Studium, May / 33

26 Solution space: N F = 2 (continued) Example of infinite-dimensional space of solutions k (ab) ba + bb 0 with a symmetric k ab = f + (ab), with 6 components, that satisfies Maxwelltype equations i 2 k k kp + ϕ k k kp 2 ρ kl pk kl = 0 i 2 [p k q]r ϕ [p k q]r + µ pq k k kr + 2 ρ [p r k k q]k = 0 similar to δk 0, dk 0. The compatibility equations (linked to d 2 = 0 = δ 2 ) are satisfied because of Bianchi-like identities. Then one can separate the problem into: (i) five constraint equations (containing only spatial β x (ii) six evolution equations (containing time derivative β 0 with β 0 β 1 + β 2 + β 3 ) derivatives) Thibault Damour (IHES) Quantum Susy Bianchi IX Cosmology Gibbons, Studium, May / 33

27 N F = 2, 3, 4, 5, 6 Finally, the general solution is parametrized by giving, as free data, two arbitrary functions of two variables (β x, β y ). These free data allow one to compute the Cauchy data for the six k ab, which can then be evolved by the six evolution equations: 0 k ab =... At level N F = 3, there is a similar mixture between discrete solutions and continuous ones. Moreover, when 4 N F 6, there is a duality mapping solutions for N F into solutions for N F = N F 6. E.g. unique state at N F = 6: ( ) [(y x)(z x)(z y)] 3/8 e x + 1 y + 1 z with b a ɛ 0 + = 0 (filled state) (x y z) 5/4 0 + Thibault Damour (IHES) Quantum Susy Bianchi IX Cosmology Gibbons, Studium, May / 33

28 Quantum Supersymmetric Billiard The spinorial wave function of the universe Ψ(β a ) propagates within the (various) Weyl chamber(s) and reflects on the walls (= simple roots of AE 3 ). In the small-wavelength limit, the reflection operators define a spinorial extension of the Weyl group of AE 3 (Damour Hillmann 09) defined within some subspaces of Spin(8, 4) R αi with Ĵα i = {Ŝ23, Ŝ31,Ĵ11} and ε 2 α i = Id = exp ( i π ) 2 ε α i Ĵ αi Thibault Damour (IHES) Quantum Susy Bianchi IX Cosmology Gibbons, Studium, May / 33

29 Fermions and their dominance near the singularity Crucial issue of boundary condition near a big bang or big crunch or black hole singularity: DeWitt 67, Vilenkin 82..., Hartle-Hawking 83...,..., Horowitz-Maldacena 03 Finding in Bianchi IX SUGRA: the WDW square-mass term µ 2 is negative (i.e. tachyonic) in most of the Hilbert space (44 among 64). µ 2 = , 3, , , , 3, (1) 0 This is a quantum effect quartic in fermions: 1 2 ρ 4 ψ 4 µ 2 (V 3 ) 2 = µ 2 (a b c) 2 = µ 2 ā 6 which dominates the other contributions near a small volume singularity abc 0. Thibault Damour (IHES) Quantum Susy Bianchi IX Cosmology Gibbons, Studium, May /

30 Bouncing Universes and Quantum Boundary Conditions at a Spacelike Singularity When µ 2 is negative, such a stiff, negative ρ 4 = p 4 classically leads to a quantum avoidance of a singularity, i.e. a bounce of the universe. Quantum mechanically, the general solution of the WDW equation (in hyperbolic polar coordinates β a = ργ a ) ( 1 ρ 2 ρ ρ 1 ) ρ 2 γ + ^µ 2 Ψ (ρ, γ a ) = 0 behaves, after a quantum-billiard mode-expansion Ψ (ρ, γ a ) = n R n(ρ) Y n (γ a ), as ρ R n (ρ) u n (ρ) a n e µ ρ + b n e + µ ρ, as ρ + This suggests to impose the boundary condition Ψ e µ ρ 0 at the singularity, which is, for a black hole crunch, a type of final-state boundary condition. Thibault Damour (IHES) Quantum Susy Bianchi IX Cosmology Gibbons, Studium, May / 33

31 Classical Bottle Effect Classical confinement between µ 2 < 0 for small volumes, and the usual closed-universe recollapse (Lin-Wald) for large volumes periodic, cyclically bouncing, solutions (Christiansen-Rugh-Rugh 95). Thibault Damour (IHES) Quantum Susy Bianchi IX Cosmology Gibbons, Studium, May / 33

32 Quantum Bottle Effect?? a set of discrete quantum states, corresponding ( à la Selberg- Gutwiller) to the classical periodic solutions? These would be excited avatars of the N F = 0 ground state [ ] 3/8 Ψ 0 = (abc) (b 2 a 2 )(c 2 b 2 )(c 2 a 2 ) e 2(a 1 2 +b 2 +c 2 ) 0 and define a kind of quantum storage ring of near-singularity states (ready for tunnelling, via inflation, toward large universes). Thibault Damour (IHES) Quantum Susy Bianchi IX Cosmology Gibbons, Studium, May / 33

33 Conclusions The case study of the quantum dynamics of a triaxially squashed 3- sphere (Bianchi IX model) in (simple, D = 4) supergravity confirms the hidden presence of hyperbolic Kac-Moody structures in supergravity. [Here, AE 3 and K (AE 3 )] The wave function of the universe Ψ(β 1, β 2, β 3 ) is a 64-dimensional spinor of Spin(8, 4) which satisfies Dirac-like, and Klein-Gordon-like, wave equations describing the propagation of a quantum spinning particle reflecting off spin-dependent potential walls which are built from quantum operators Ŝ12, Ŝ23, Ŝ31, Ĵ11, Ĵ22, Ĵ33 that generate a 64-dim representation of K (AE 3 ). The squared-mass term µ 2 in the KG equation belongs to the center of this algebra. The space of solutions is a mixture of discrete states and continuous states (depending on fermion number N F = C F + 3). µ 2 is mostly negative and can lead to a quantum avoidance of the singularity, and to a bottle effect near the singularity. Thibault Damour (IHES) Quantum Susy Bianchi IX Cosmology Gibbons, Studium, May / 33