Symmetry and Geometry in String Theory
Symmetry and Extra Dimensions
String/M Theory • Rich theory, novel duality symmetries and exotic structures • Supergravity limit - misses stringy features • Infinite set of fields: misses dualities • Perturbative string - world-sheet theory • String field theory: captures interactions, T - duality, some algebraic structure • Fundamental formulation?
Strings in Geometric Background Manifold, background tensor fields G ij , H ijk , Φ Fluctuations: modes of string Treat background and fluctuations the same? Stringy geometry? Singularity resolution? Dualities: mix geometric and stringy modes Non-Geometric Background? String theory: solutions that are not “geometric” Moduli stabilisation. Richer landscape?
Duality Symmetries • Supergravities: continuous classical symmetry, broken in quantum theory, and by gauging • String theory: discrete quantum duality symmetries; not field theory symms • T -duality: perturbative symmetry on torus, mixes momentum modes and winding states • U-duality: non-perturbative symmetry of type II on torus, mixes momentum modes and wrapped brane states
Symmetry & Geometry • Spacetime constructed from local patches • All symmetries of physics used in patching • Patching with diffeomorphisms, gives manifold • Patching with gauge symmetries: bundles • String theory has new symmetries, not present in field theory. New non-geometric string backgrounds • Patching with T -duality: T-FOLDS • Patching with U-duality: U-FOLDS
Extra Dimensions • Kaluza-Klein theory: extra dimensions to spacetime, geometric origin for “photons” • String theory in 10-d, M-theory in 11-d • Duality symmetries suggest strings see further dimensions. Strings on torus see doubled spacetime: double field theory • Extended spacetime as arena for M-theory?
Strings on Circle M = S 1 × X Discrete momentum p=n/R If it winds m times round S 1 , winding energy w=mRT Energy = p 2 +w 2 +.... T -duality : Symmetry of string theory p w ↔ m n ↔ R 1/RT ↔ • Fourier transf of discrete p,w gives periodic coordinates Circle + dual circle X, ˜ X • S tringy symmetry, not in field theory • On d torus, T -duality group O ( d, d ; Z )
Strings on T d ˜ X = X L ( σ + τ ) + X R ( σ − τ ) , X = X L − X R conjugate to momentum, to winding no. ˜ X X ∂ a X = � ab ∂ b ˜ dX = ∗ d ˜ X X Need “auxiliary” for interacting theory ˜ X Vertex operators e ik L · X L , e ik R · X R
Strings on T d ˜ X = X L ( σ + τ ) + X R ( σ − τ ) , X = X L − X R conjugate to momentum, to winding no. ˜ X X ∂ a X = � ab ∂ b ˜ dX = ∗ d ˜ X X Strings on torus see DOUBLED TORUS T-duality group O ( d, d ; Z ) Doubled Torus 2d coordinates � ⇥ x i ˜ X ≡ Transform linearly under O ( d, d ; Z ) x i DOUBLED GEOMETRY Duff;Tseytlin; Siegel;Hull;…
T -fold patching R 1/R Glue big circle (R) to small (1/R) Glue momentum modes to winding modes (or linear combination of momentum and winding) Not conventional smooth geometry
E ′ ( Y ′ ) E ( Y ) Torus fibration Y Y ′ U ′ U T-fold :Transition functions involve T -dualities E=G+B Non-tensorial E ′ = ( aE + b )( cE + d ) − 1 in U ∩ U ′ O ( d, d ; Z ) -dualities also ➞ T-fold Glue using T Physics smooth, as T -duality a symmetry T -fold transition: mixes X, ˜ X
E ′ ( Y ′ ) E ( Y ) Torus fibration Y Y ′ U ′ U T-fold :Transition functions involve T -dualities E=G+B Non-tensorial E ′ = ( aE + b )( cE + d ) − 1 in U ∩ U ′ O ( d, d ; Z ) -dualities also ➞ T-fold Glue using T Physics smooth, as T -duality a symmetry T -fold transition: mixes X, ˜ X But doubled space is smooth manifold! Double torus fibres, T -duality then acts geometrically
Extra Dimensions • Torus compactified theory has charges arising in SUSY algebra, carried by BPS states • P M: Momentum in extra dimensions • Z A : wrapped brane & wound string charges • But P M ,Z A related by dualities. Can Z A be thought of as momenta for extra dimensions? • Space with coordinates X M ,Y A ?
Extended Spacetime • Supergravity can be rewritten in extended space with coordinates X M ,Y A . Duality symmetry manifest. • But fields depend only on X M (or coords related to these by duality). • Gives a geometry for non-geometry: T -folds • Actual stringy symmetry of theory can be quite different from this sugra duality: Background dependence? • In string theory, can do better... DOUBLE FIELD THEORY, fields depending on X M ,Y M.
Double Field Theory Hull & Zwiebach • From sector of String Field Theory. Features some stringy physics, including T -duality, in simpler setting • Strings see a doubled space-time • Necessary consequence of string theory • Needed for non-geometric backgrounds • What is geometry and physics of doubled space?
Strings on a Torus • States: momentum p, winding w • String: Infinite set of fields ψ ( p, w ) • Fourier transform to doubled space: ψ ( x, ˜ x ) • “Double Field Theory” from closed string field theory. Some non-locality in doubled space • Subsector? e.g. g ij ( x, ˜ x ) , b ij ( x, ˜ x ) , φ ( x, ˜ x ) • T -duality is a manifest symmetry
Double Field Theory • Double field theory on doubled torus • General solution of string theory: involves doubled fields ψ ( x, ˜ x ) • Real dependence on full doubled geometry, dual dimensions not auxiliary or gauge artifact. Double geom. physical and dynamical • Strong constraint restricts to subsector in which extra coordinates auxiliary: get conventional field theory locally. Recover Siegel’s duality covariant formulation of (super)gravity
M-Theory • 11-d sugra can be written in extended space. • Extension to full M-theory? • If M-theory were a perturbative theory of membranes, would have extended fields depending on X M and 2-brane coordinates Y MN • But it doesn’t seem to be such a theory • Don’t have e.g. formulation as infinite no. of fields. Only implicit construction as a limit. • Extended field theory gives a duality- symmetric reformulation of supergravity
Type IIA Supergravity Compactified on T 4 Duality symmetry SO(5,5) BPS charges in 16-dim rep Type IIA Supergravity on M 4 xM 6 Can be written as “Extended Field Theory” in space with 6 +16 coordinates Berman, Godzagar, Perry; with SO(5,5) Symmetry Hohm, Sambtleben Is SO(5,5) a “real” symmetry for generic solutions M 4 xM 6 ? Similar story for M d xM 10-d ; M d xM 11-d
Extension to String Theory? Type IIA Superstring Compactified on T 4 U-Duality symmetry SO(5,5;Z) 6+16 dims? But for other backgrounds, symmetry different Type IIA Superstring Compactified on K3 6+24 dims? U-Duality symmetry SO(4,20;Z) Duality symmetry seems to be background dependent; makes background independent formulation problematic
DFT gives O(D,D) covariant formulation O(D,D) Covariant Notation � ˜ ˜ ! ∂ i ⇥ x i X M ≡ ∂ M ≡ x i ∂ i � ⇥ I 0 η MN = M = 1 , ..., 2 D I 0 ∂ 2 = 1 2 ∂ M ∂ M ∆ ≡ ∂ x i ∂ ˜ x i Constraint ∂ M ∂ M A = 0 Weak Constraint or on all fields and parameters weak section condition Arises from string theory constraint ( L 0 − ¯ L 0 ) Ψ = 0
• Weakly constrained DFT non-local. Constructed to cubic order Hull & Zwiebach • ALL doubled geometry dynamical, evolution in all doubled dimensions • Restrict to simpler theory: STRONG CONSTRAINT • Fields then depend on only half the doubled coordinates • Locally, just conventional SUGRA written in duality symmetric form
Strong Constraint for DFT Hohm, H &Z ( ∂ M A ) ( ∂ M B ) = 0 ∂ M ∂ M ( AB ) = 0 on all fields and parameters If impose this, then it implies weak form, but product of constrained fields satisfies constraint. This gives Restricted DFT, a subtheory of DFT Locally, it implies fields only depend on at most half of the coordinates, fields are restricted to null subspace N. Looks like conventional field theory on subspace N • Siegel’s duality covariant form of (super)gravity
✓ ˜ ✓ ˜ ◆ ◆ x m ✏ m X M = ⇠ M = x m ✏ m Linearised Gauge Transformations ⇧ i ⇥ j + ⇧ j ⇥ i + ˜ ⇥ j + ˜ � h ij = ⇧ i ˜ ⇧ j ˜ ⇥ i , � b ij = − (˜ ⇧ i ⇥ j − ˜ ⇧ j ⇥ i ) − ( ⇧ i ˜ ⇥ j − ⇧ j ˜ ⇥ i ) , � d = − ⇧ · ⇥ + ˜ ⇧ · ˜ Invariance needs constraint ⇥ . Diffeos and B-field transformations mixed. If fields indep of , conventional theory g ij ( x ) , b ij ( x ) , d ( x ) ˜ x m parameter for diffeomorphisms ✏ m parameter for B-field gauge transformations ˜ ✏ m
Generalised Metric Formulation Hohm, H &Z � g ij − g ik b kj ⇥ H MN . = b ik g kj g ij − b ik g kl b lj 2 Metrics on double space H MN , η MN H MN ≡ η MP H P Q η QN H MP H P N = δ M Constrained metric N
Generalised Metric Formulation Hohm, H &Z � g ij − g ik b kj ⇥ H MN . = b ik g kj g ij − b ik g kl b lj 2 Metrics on double space H MN , η MN H MN ≡ η MP H P Q η QN H MP H P N = δ M Constrained metric N Covariant O(D,D) Transformation h P M h Q N H � P Q ( X � ) = H MN ( X ) h ∈ O ( D, D ) X � = hX
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