Infinite discrete symmetries near singularities and modular forms - PowerPoint PPT Presentation

Infinite discrete symmetries near singularities and modular forms Axel Kleinschmidt (Albert Einstein Institute, Potsdam) IHES, January 26, 2012 Based on work with: Philipp Fleig, Michael Koehn, Hermann Nicolai and Jakob Palmkvist D 80 (2009) 061701(R), arXiv:0907.3048] [KKN = Phys. Rev. [KNP = arXiv:1010.2212] [FK, to be published] Symmetries and modular forms – p.1

Context and Plan Hidden symmetries and cosmological billiards in super- gravity [Damour, Henneaux 2000; Damour, Henneaux, Nicolai 2002] Minisuperspace models for quantum gravity and quantum cosmology [DeWitt 1967; Misner 1969] U-dualities constraining string scattering amplitudes [Green, Gutperle 1997; Green, Miller, Russo, Vanhove 2010; Pioline 2010] Symmetries and modular forms – p.2

Context and Plan Hidden symmetries and cosmological billiards in super- gravity [Damour, Henneaux 2000; Damour, Henneaux, Nicolai 2002] Minisuperspace models for quantum gravity and quantum cosmology [DeWitt 1967; Misner 1969] U-dualities constraining string scattering amplitudes [Green, Gutperle 1997; Green, Miller, Russo, Vanhove 2010; Pioline 2010] Plan Cosmological billiards and their symmetries Quantum cosmological billiards: arithmetic structure Modular forms for hyperbolic Weyl groups and infinite Chevalley groups Generalization and outlook Symmetries and modular forms – p.2

Cosmological billards: BKL Supergravity dynamics near a space-like singularity simplify. [Belinskii, Khalatnikov, Lifshitz 1970; Misner 1969; Chitre 1972] T = T 1 T = T 2 < T 1 T = 0 x 2 x 1 (conj.) Spatial points decouple ⇒ dynamics becomes ultra-local. Reduction of degress of freedom to spatial scale factors β a d ds 2 = − N 2 dt 2 + e − 2 β a dx 2 � ( t ∼ − log T ) a a =1 Symmetries and modular forms – p.3

Cosmological billiards: Dynamics Effective Lagrangian for β a ( t ) ( a = 1 , . . . , d ) d L = 1 � � β a ˙ β b − V eff ( β ) G ab : DeWitt metric � n − 1 G ab ˙ (Lorentzian signature) 2 a,b =1 M β Close to the singularity V eff con- sists of infinite potentials walls, obstructing free null motion of β a . Symmetries and modular forms – p.4

Cosmological billiards: Dynamics Effective Lagrangian for β a ( t ) ( a = 1 , . . . , d ) d L = 1 � � β a ˙ β b − V eff ( β ) G ab : DeWitt metric � n − 1 G ab ˙ (Lorentzian signature) 2 a,b =1 M β Close to the singularity V eff con- sists of infinite potentials walls, obstructing free null motion of β a . Billiard table Resulting billiard geometry that = E 10 Weyl chamber of E 10 Weyl chamber ( D = 11 , type (m)IIA and IIB). [Damour, Henneaux 2000] Symmetries and modular forms – p.4

Cosmological billiards: Geometry The sharp billiard walls come from � c A e − 2 w A ( β ) V eff ( β ) = A with w A ( β ) a set of linear forms on β -space. For G ab β a β b → −∞ (towards the singularity) the potential term becomes 0 or ∞ , defining two sides of a wall. Symmetries and modular forms – p.5

Cosmological billiards: Geometry The sharp billiard walls come from � c A e − 2 w A ( β ) V eff ( β ) = A with w A ( β ) a set of linear forms on β -space. For G ab β a β b → −∞ (towards the singularity) the potential term becomes 0 or ∞ , defining two sides of a wall. For the dominant terms c A ≥ 0 [Damour, Henneaux, Nicolai 2002] . Furthermore, the scalar product between the normals to those faces coincides with E 10 Cartan matrix. Associated E 10 Weyl group W ( E 10 ) are the symmetries of the unique even self-dual lattice II 9 , 1 = Λ E 8 ⊕ II 1 , 1 . Finite (hyperbolic) volume ⇒ Chaos! [Damour, Henneaux 2000] Symmetries and modular forms – p.5

Quantum cosmological billiards Setting n = 1 one has to quantize � d � 2   d d L = 1 β b = 1 β a ) 2 − � β a G ab ˙ ˙ � ( ˙ � ˙ β a   2 2 a =1 a =1 a,b =1 β b = 0 on billiard domain. with null constraint ˙ β a G ab ˙ π a = G ab ˙ H = 1 β b 2 π a G ab π b . Canonical momenta: ⇒ Symmetries and modular forms – p.6

Quantum cosmological billiards Setting n = 1 one has to quantize � d � 2   d d L = 1 β b = 1 β a ) 2 − � β a G ab ˙ ˙ � ( ˙ � ˙ β a   2 2 a =1 a =1 a,b =1 β b = 0 on billiard domain. with null constraint ˙ β a G ab ˙ π a = G ab ˙ H = 1 β b 2 π a G ab π b . Canonical momenta: ⇒ Wheeler–DeWitt (WDW) equation in canonical quantization H Ψ( β ) = − 1 2 G ab ∂ a ∂ b Ψ( β ) = 0 Klein–Gordon ‘inner product’. Symmetries and modular forms – p.6

Quantum cosmological billiards (II) Introduce new coordinates ρ and ω a ( z ) from ‘radius’ and co- ordinates z on unit hyperboloid β a = ρω a , ω a G ab ω b = − 1 ρ 2 = − β a G ab β b Symmetries and modular forms – p.7

Quantum cosmological billiards (II) Singularity: ρ → ∞ Introduce new coordinates ρ and ω a ( z ) from ‘radius’ and co- ρ ordinates z on unit hyperboloid β a = ρω a , ω a G ab ω b = − 1 ωa ( z ) ρ 2 = − β a G ab β b Timeless WDW equation in these variables � � � � − ρ 1 − d ∂ ρ d − 1 ∂ + ρ − 2 ∆ LB Ψ( ρ, z ) = 0 ∂ρ ∂ρ ✻ Laplace–Beltrami operator on unit hyperboloid Symmetries and modular forms – p.7

Solving the WDW equation � � � � − ρ 1 − d ∂ ρ d − 1 ∂ + ρ − 2 ∆ LB Ψ( ρ, z ) = 0 ∂ρ ∂ρ Separation of variables: Ψ( ρ, z ) = R ( ρ ) F ( z ) For − ∆ LB F ( z ) = EF ( z ) get � 2 R ± ( ρ ) = ρ − d − 2 E − ( d − 2 2 ) 2 ± i [Positive frequency coming out of singularity is R − ( ρ ) .] Left with spectral problem on hyperbolic space. Symmetries and modular forms – p.8

∆ LB and boundary conditions The classical billiard ball is constrained to Weyl chamber with infinite potentials ⇒ Dirichlet boundary conditions v Use upper half plane model u ∈ R d − 2 , v ∈ R > 0 z = ( � u, v ) , � ∆ LB = v d − 1 ∂ v ( v 3 − d ∂ v ) + v 2 ∂ 2 ⇒ � u � u Symmetries and modular forms – p.9

∆ LB and boundary conditions The classical billiard ball is constrained to Weyl chamber with infinite potentials ⇒ Dirichlet boundary conditions v Use upper half plane model u ∈ R d − 2 , v ∈ R > 0 z = ( � u, v ) , � ∆ LB = v d − 1 ∂ v ( v 3 − d ∂ v ) + v 2 ∂ 2 ⇒ u � � u With Dirichlet boundary conditions ( d = 3 in [Iwaniec] ) � 2 � d − 2 − ∆ LB F ( z ) = EF ( z ) ⇒ E ≥ 2 Symmetries and modular forms – p.9

Arithmetic structure (I) Beyond general inequality details of spectrum depend on shape of domain. (‘Shape of the drum’ problem) Focus on maximal supergravity ( d = 10 ). Domain is ② 8 determined by E 10 Weyl group. ② ② ② ② ② ② ② ② ② -1 0 1 2 3 4 5 6 7 Symmetries and modular forms – p.10

Arithmetic structure (I) Beyond general inequality details of spectrum depend on shape of domain. (‘Shape of the drum’ problem) Focus on maximal supergravity ( d = 10 ). Domain is ② 8 determined by E 10 Weyl group. ② ② ② ② ② ② ② ② ② -1 0 1 2 3 4 5 6 7 u ∈ O 9 -dimensional upper half plane with octonions: u ≡ � On z = u + i v the ten fundamental Weyl reflections act by w − 1 ( z ) = 1 z , w 0 ( z ) = − ¯ z + 1 , w j ( z ) = − ε j ¯ zε j ¯ ε j simple E 8 rts. [Feingold, AK, Nicolai 2008] Symmetries and modular forms – p.10

Arithmetic structure (II) Iterated action of w − 1 ( z ) = 1 z , w 0 ( z ) = − ¯ z + 1 , w j ( z ) = − ε j ¯ zε j ¯ generates whole Weyl group W ( E 10 ) . Even Weyl group W + ( E 10 ) gives ‘holomorphic’ maps W + ( E 10 ) = PSL 2 ( O ) . Modular group over the integer ‘octavians’ O . [Example of family of isomorphisms between hyperbolic Weyl groups and modular groups over division algebras [Feingold, AK, Nicolai 2008] .] Symmetries and modular forms – p.11

Modular wavefunctions (I) Weyl reflections on wavefunction Ψ( ρ, z ) � +Ψ( ρ, z ) Neumann b.c. Ψ( ρ, w I · z ) = − Ψ( ρ, z ) Dirichlet b.c. Use Weyl symmetry to define Ψ( ρ, z ) on the whole upper half plane, with Dirichlet boundary conditions ⇒ Ψ( ρ, z ) is Symmetries and modular forms – p.12

Modular wavefunctions (I) Weyl reflections on wavefunction Ψ( ρ, z ) � +Ψ( ρ, z ) Neumann b.c. Ψ( ρ, w I · z ) = − Ψ( ρ, z ) Dirichlet b.c. Use Weyl symmetry to define Ψ( ρ, z ) on the whole upper half plane, with Dirichlet boundary conditions ⇒ Ψ( ρ, z ) is Sum of eigenfunctions of ∆ LB on UHP Invariant under action of W + ( E 10 ) = PSL 2 ( O ) . Anti-invariant under extension to W ( E 10 ) . Symmetries and modular forms – p.12

Modular wavefunctions (I) Weyl reflections on wavefunction Ψ( ρ, z ) � +Ψ( ρ, z ) Neumann b.c. Ψ( ρ, w I · z ) = − Ψ( ρ, z ) Dirichlet b.c. Use Weyl symmetry to define Ψ( ρ, z ) on the whole upper half plane, with Dirichlet boundary conditions ⇒ Ψ( ρ, z ) is Sum of eigenfunctions of ∆ LB on UHP Invariant under action of W + ( E 10 ) = PSL 2 ( O ) . Anti-invariant under extension to W ( E 10 ) . ⇒ Wavefunction is an odd Maass wave form of PSL 2 ( O ) [cf. [Forte 2008] for related ideas for Neumann conditions] Symmetries and modular forms – p.12