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 m R $ $ $ w n 1/RT Fourier transf of discrete p,w gives periodic coordinates X, X Circle + dual circle Stringy 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 = XL X R X conjugate to momentum, X to winding no. dx = d X a X = ϵ ab b X Need auxiliary X for interacting theory 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 = XL X R X conjugate to momentum, X to winding no. dx = d X a X = ϵ ab b X Strings on torus see DOUBLED TORUS T-duality group O(d, d; Z) Doubled Torus 2d coordinates Transform linearly under O(d, d; Z) X x i 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 O(d, d; Z) E=G+B Non-tensorial E =(ae + b)(ce + d) 1 in U U Glue using T-dualities also T-fold Physics smooth, as T-duality a symmetry T-fold transition: mixes X, X

Extra Dimensions Torus compactified theory has charges arising in SUSY algebra, carried by BPS states PM : Momentum in extra dimensions ZA: wrapped brane & wound string charges But PM,ZA related by dualities. Can ZA 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,YM.

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 (p, w) Fourier transform to doubled space: (x, x) Double Field Theory from closed string field theory. Some non-locality in doubled space Subsector? e.g. T-duality is a manifest symmetry g ij (x, x), b ij (x, x), (x, x)

Double Field Theory Double field theory on doubled torus General solution of string theory: involves doubled fields (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 YMN 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 dualitysymmetric 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 M4xM6 Can be written as Extended Field Theory in space with 6 +16 coordinates with SO(5,5) Symmetry Berman, Godzagar, Perry; Hohm, Sambtleben Is SO(5,5) a real symmetry for generic solutions M4xM6? Similar story for MdxM10-d ; MdxM11-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 U-Duality symmetry SO(4,20;Z) 6+24 dims? Duality symmetry seems to be background dependent; makes background independent formulation problematic

DFT gives O(D,D) covariant formulation O(D,D) Covariant Notation x i x i MN = 0 I I 0 X M @ M M =1,..., 2D @ i @ i! @2 @x i @ x i = 1 2 @M @ M Constraint M M A =0 on all fields and parameters Weak Constraint or weak section condition Arises from string theory constraint (L 0 L0 ) =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 M (AB) =0 ( M A)( M B)=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 = xm x m M = m m Linearised Gauge Transformations h ij = i j + j i + i j + 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 parameter for diffeomorphisms m x m g ij (x),b ij (x),d(x) m parameter for B-field gauge transformations

Generalised Metric Formulation Hohm, H &Z H MN = g ij g ik b kj b ik g kj g ij b ik g kl b lj. 2 Metrics on double space H MN, MN H MN MP H PQ QN Constrained metric H MP H PN = M N

Generalised Metric Formulation Hohm, H &Z H MN = g ij g ik b kj b ik g kj g ij b ik g kl b lj. 2 Metrics on double space H MN, MN H MN MP H PQ QN Constrained metric H MP H PN = M N Covariant O(D,D) Transformation h P M h Q NH PQ (X ) = H MN (X) X = hx h O(D, D)

O(D,D) covariant action S = dxd xe 2d L L = 1 8 HMN M H KL NH KL 1 2 M d N H MN +4H MN M d N d 2 HMN NH KL LH MK Lagrangian L CUBIC in fields! Indices raised and lowered with O(D,D) covariant (in ) R 2D MN

2-derivative action S = S (0) (, )+S (1) (, )+S (2) (, ) Write S (0) in terms of usual fields Gives usual action (+ surface term) dx ge 2 R + 4( ) 2 1 12 H2 S (0) = S(E,d, )

2-derivative action S = S (0) (, )+S (1) (, )+S (2) (, ) Write S (0) in terms of usual fields Gives usual action (+ surface term) dx ge 2 R + 4( ) 2 1 12 H2 S (0) = S(E,d, ) S (2) = S(E 1,d, ) T-dual! S (1) strange mixed terms

O(D,D) covariant action S = dxd xe 2d L L = 1 8 HMN M H KL NH KL 1 2 M d N H MN +4H MN M d N d 2 HMN NH KL LH MK Gauge Transformation H MN = P P H MN +( M P P M ) H PN +( N P P N ) H MP Write as Generalised Lie Derivative H MN = L H MN

Conclusions Duality symmetries lead to extension of geometry to allow non-geometric solutions String theory on torus: T-duality symmetry. Winding modes: doubled geometry, infinite number of doubled fields DFT: with strong constraint, get conventional sugra in duality symmetric formulation More generally, this applies locally in patches. Use DFT gauge and O(D,D) symmetries in transition functions. Get T-folds etc.

Full theory with weak constraint: non-local, stringy How much of this is special to tori? Other topologies may not have windings, or have different numbers of momenta and windings. No T-duality? No doubling? Duality symmetry gives deep insight into stringy geometry. But seems to be very different on different backgrounds, e.g. T 4, K3 Much remains to be understood about string/ M theory