Strings and (Non)-Geometry Dieter Lst, LMU and MPI Mnchen Meeting - PowerPoint PPT Presentation

Strings and (Non)-Geometry Dieter Lüst, LMU and MPI München Meeting „The particle Physics and Cosmology of Supersymmetry and String Theory“, Spring Meeting at University of Pennsylvania, Philadelphia, March, 16-18, 2012 Donnerstag, 15. März 2012

Outline: I) Introduction II) String scattering at high energies III) Non-geometric flux compactifications and deformed geometries D. Lüst, JEHP 1012 (2011) 063, arXiv:1010.1361 R. Blumenhagen, A. Deser, D.Lüst, E. Plauschinn, F. Rennecke, J. Phys A44 (2011), 385401, arXiv:1106.0316 C. Condeescu, I. Florakis, D. Lüst, arXiv:1202.6366. D. Andriot, M. Larfors, D.L. P. Patalong, arXiv:1106.4015 Talk by Additional work: D. Andriot, O. Hohm, M. Larfors, D.L. P. Patalong, arXiv:1202.3060 D. Andriot IV) Outlook & open problems Donnerstag, 15. März 2012

I) Introduction Question What is gravity? ⇔ What is space-time? Problems: Quantization, Dark Matter & Energy, Hierarchy,.. Donnerstag, 15. März 2012

I) Introduction Question What is gravity? ⇔ What is space-time? Problems: Quantization, Dark Matter & Energy, Hierarchy,.. Geometry in general depends on, with what kind of objects you test it. Point particles in classical Einstein gravity see smooth & continuous manifolds. However Einstein gravity is plagued by singularities ! Donnerstag, 15. März 2012

I) Introduction Question What is gravity? ⇔ What is space-time? Problems: Quantization, Dark Matter & Energy, Hierarchy,.. Geometry in general depends on, with what kind of objects you test it. Point particles in classical Einstein gravity see smooth & continuous manifolds. However Einstein gravity is plagued by singularities ! Space (time?) can be only dissolved up to distances of order . L P is the shortest possible distance! L P Donnerstag, 15. März 2012

String theory: Theory of Quantum Gravity How does a string see space-time? The short distance nature of space can be possibly tested by string scattering at high energies. Shortest possible scale in string theory: L s We expect that geometry is changing at distances of the order of the string length. Donnerstag, 15. März 2012

String theory: Theory of Quantum Gravity How does a string see space-time? The short distance nature of space can be possibly tested by string scattering at high energies. Shortest possible scale in string theory: L s We expect that geometry is changing at distances of the order of the string length. Stringy (non)- geometry: deformed geometry: ● Non-commutative geometry: [ X i , X j ] ≃ O ( L s ) ● Non-associative geometry: [[ X i , X j ] , X k ] ≃ O ( L s ) Donnerstag, 15. März 2012

II) String scattering at high energies String scattering amplitudes exhibit two important properties: (i) The poles of the amplitudes are dominated by the exchange of an infinite number of massive Regge modes. k 1 k 3 ∞ � = n =0 | k ; n � k 2 k 4 ∞ s − π 2 A ( k 1 , k 2 , k 3 , k 4 ; α ′ ) ∼ − Γ ( − α ′ s ) Γ (1 − α ′ u ) γ ( n ) ∼ t 6 tu ( α ′ ) 2 + . . . � = s − M 2 Γ ( − α ′ s − α ′ u ) n n =0 (Veneziano, 1968) Donnerstag, 15. März 2012

Possible discovery of Regge modes in case of low string scale, M s ≃ O ( TeV ) Model-independent dijet- events at LHC due to colored Regge states: (D.L., S. Stieberger, T. Taylor, 2008; L. Anchordoqui, H. Goldberg, S. Nawata, D.L., S. Stieberger, T. Taylor, 2008; W. Feng, D.L., O. Schlotterer, S. Stieberger, T. Taylor, 2010) 2012 ATLAS and CMS data: M s ≥ 4 TeV Donnerstag, 15. März 2012

(ii) The exchanged Regge „particles“ grow in size at higher and higher energies - exchange of extended objects! The strings scale is the shortest possible length L s scale that can be dissolved by string scattering. (Amati, Ciafaloni, Veneziano, 1987) ● Reformulation of Heisenberg uncertainty relation: 1 ∆ p + L 2 ∆ x ≃ s ∆ p ∆ x min = L s ⇒ ● UV/IR mixing in string theory. String scattering amplitudes exhibit non-Wilsonian behavior. Donnerstag, 15. März 2012

At the scale M s ≃ 1 /L s Einstein gravity gets changed. Donnerstag, 15. März 2012

At the scale M s ≃ 1 /L s Einstein gravity gets changed. At the scale M s ≃ 1 /L s the notion of geometry might be changed ! Donnerstag, 15. März 2012

At the scale M s ≃ 1 /L s Einstein gravity gets changed. At the scale M s ≃ 1 /L s the notion of geometry might be changed ! ⇒ Stringy non-Riemannian geometry. Donnerstag, 15. März 2012

III) Non-geometric flux compactifications (Non-commutative/non-associative closed string geometry) Donnerstag, 15. März 2012

III) Non-geometric flux compactifications (Non-commutative/non-associative closed string geometry) Recall standard Riemannian geometry: Donnerstag, 15. März 2012

III) Non-geometric flux compactifications (Non-commutative/non-associative closed string geometry) Recall standard Riemannian geometry: - Flat space: Triangle: α + β + γ = π Triangle: α + β + γ > π ( < π ) - Curved space: Donnerstag, 15. März 2012

III) Non-geometric flux compactifications (Non-commutative/non-associative closed string geometry) Recall standard Riemannian geometry: - Flat space: Triangle: α + β + γ = π Triangle: α + β + γ > π ( < π ) - Curved space: Manifold: need different coordinate charts, which are patched together by coordinates transformations, i.e. group of diffeomorphisms: Di ff ( M ) : f : U → U ′ Donnerstag, 15. März 2012

Properties of Riemannian manifolds: ● distances between two points can be arbitrarily short. ● coordinates commute with each other: [ X i , X j ] = 0 This is the situation, if one is using point particles to probe distance and the geometry of space. Now we want to understand, how extended closed strings may possibly see the (non)-geometry of space. Donnerstag, 15. März 2012

We will encounter two different interesting situations: Donnerstag, 15. März 2012

We will encounter two different interesting situations: - Non-geometric Q-fluxes: spaces that are locally still Riemannian manifolds but not anymore globally. Transition functions between two coordinate patches are not only diffeomorphisms but also T -duality transformations: Di ff ( M ) Di ff ( M ) × SO ( d, d ) → Q-space will become non-commutative: [ X i , X j ] � = 0 Donnerstag, 15. März 2012

We will encounter two different interesting situations: - Non-geometric Q-fluxes: spaces that are locally still Riemannian manifolds but not anymore globally. Transition functions between two coordinate patches are not only diffeomorphisms but also T -duality transformations: Di ff ( M ) Di ff ( M ) × SO ( d, d ) → Q-space will become non-commutative: [ X i , X j ] � = 0 - Non-geometric R-fluxes: spaces that are even locally not anymore manifolds. R-space will become non-associative: [[ X i , X j ] , X k ] + perm . � = 0 Donnerstag, 15. März 2012

We will encounter two different interesting situations: - Non-geometric Q-fluxes: spaces that are locally still Riemannian manifolds but not anymore globally. Transition functions between two coordinate patches are not only diffeomorphisms but also T -duality transformations: Di ff ( M ) Di ff ( M ) × SO ( d, d ) → Q-space will become non-commutative: [ X i , X j ] � = 0 - Non-geometric R-fluxes: spaces that are even locally not anymore manifolds. R-space will become non-associative: [[ X i , X j ] , X k ] + perm . � = 0 Physics is nevertheless smooth and well-defined! Donnerstag, 15. März 2012

T -duality: Consider compactification on a circle with radius R: X ( τ , σ ) = X L ( τ + σ ) + X R ( τ − σ ) � α ′ 1 x n α n e − in ( τ + σ ) , � X L ( τ + σ ) = 2 + p L ( τ + σ ) + i 2 n � =0 � α ′ 1 x � α n e − in ( τ − σ ) X R ( τ − σ ) = 2 + p R ( τ − σ ) + i n ˜ 2 n � =0 (KK momenta ) � M � 1 R + ( α ′ ) − 1 NR = p L , p L + p R = M 2 = p R 1 � M � R − ( α ′ ) − 1 NR = p R p L − p R = ( α ′ ) − 1 NR ˜ = p 2 (dual momenta - winding modes) Donnerstag, 15. März 2012

→ α ′ T -duality: T : R ← R , M ← → N T : p ← p , p L ← → p L , p R ← → − p R . ˜ → ˜ ● Dual space coordinates: X ( τ , σ ) = X L − X R ( X, ˜ X ) : Doubled geometry: (O. Hohm, C. Hull, B. Zwiebach (2009/10)) T -duality is part of stringy diffeomorphism group. ˜ T : X ← X , X L ← → X L , X R ← → − X R → √ ● Shortest possible radius: R ≥ R c = α ′ Donnerstag, 15. März 2012

T -fold: Patching uses T -duality. e.g. torus fibrations E ′ ( U ′ ) E ( U ) E = G + B U U ′ Geometric background: E ′ = aEa t in U ∩ U ′ , a ∈ GL ( d, Z ) Non-geometric background: E ′ = aE + b cE + d in U ∩ U ′ Donnerstag, 15. März 2012

Example: torus bundle over : S 1 1 ds 2 = dx 2 ( dx 2 1 + dx 2 Metric: 3 + 2 ) 1 + x 2 3 x 3 B-field: B x 1 ,x 2 = 1 + x 2 3 Monodromy: is periodic: x 3 E ( x 3 + 2 π ) = aE ( x 3 ) + b cE ( x 3 ) + d ∈ SO (2 , 2; Z ) Donnerstag, 15. März 2012

Mathematical framework: - Doubled field theory: uses completely SO(d,d) invariant formalism. - Generalized complex geometry: uses doubled tangent space . T ⊕ T ∗ Donnerstag, 15. März 2012