Chris Hull Imperial College London

Generalized Geometry and String Theory Chris Hull Imperial College London

Tuesday, 14 July 15 Generalised Geometry

• Much recent work in string theory, supergravity... GG and its generalisations • Geometric framework for metric + p-form gauge ﬁelds • Uniﬁcation of various geometric structures • Organizing principle : Duality symmetries • New mathematics for old geometries • New ‘non-geometries’ in string theory

Tuesday, 14 July 15 Generalised Geometry

Studies structures on a d-dimensional manifold M on which there is a natural action of O(d,d) Hitchin

Manifold with metric + B-ﬁeld, natural action of O(d,d), tangent space “doubled” to T ⊕ T ∗

Extended/Exceptional Geometry: O(d,d) replaced by Ed+1 acting on extended tangent bundle

Organizing principle : Duality symmetries e.g. O(d,d), Ed

Tuesday, 14 July 15 Geometry of String Theory Spacetime: supergravity background. 10-dimensional manifold M with metric + p-form gauge ﬁelds

Type II strings

Metric, signature 9+1 gµ⌫ Dilaton: scalar ﬁeld 2-form Gauge ﬁeld Bµ⌫ H = dB

p-form gauge ﬁelds Cp

Cp = dp 1 + ..., Gp+1 = Gp + ..., Gp+1 = G9 p ⇤ Type IIA: p odd. Type IIB: p even

Tuesday, 14 July 15 3,1 M = R K6 Calabi-Yau ⇥ Supersymmetric background: admits Killing spinors Spinor ﬁelds generating supersymmetries preserving soln • 6-fold K: Calabi-Yau if Kahler Ricci-ﬂat complex structure J: integrable map J : TK TK • ! J 2 = 1 • Hermitian metric g Kahler 2-form ! ! = g J k = ! • ij ik j ji • Holomorphic (3,0) form ⌦ d! =0, ! =0 Forms from Killing r d⌦ =0, ⌦ =0 spinor bilinears r Generalisation to FLUX BACKGROUNDS? B2,Cp =0 Tuesday, 14 July 15 6 Duality Symmetries fromTorus Reductions

• Compactify gravity or supergravity on Td • Scalars in coset space G/H, • Truncated theory: non-comapact G symmetry • Full theory: discrete subgroup G(Z)

Tuesday, 14 July 15 • General Relativity in 3+1 dimensions on S1 EHLERS SYMMETRY SL (2 , R ) Scalars in coset space SL (2 , R ) U(1) • GR in D dimensions, reduced on torus Td, Symmetry GL ( d, R ) GL(d, ) Scalars g ij in coset space R O(d) • Full theory: Kaluza-Klein spectrum, discrete momentum p, breaking GL ( d, R ) to GL ( d, Z ) Group of large diffeomorphisms of d-torus

Tuesday, 14 July 15 • Metric, 2-form gauge ﬁeld B, reduced on Td Symmetry O ( d, d ) O(d, d) Scalars g ij B ij in target space O(d) O(d) ⇥ • Type II Supergravity, reduced on Td Symmetry E d +1 (Cremmer-Julia) Ed+1 Scalars g ij ,B ij ,C i 1 ...i p , in target space Hd+1

Full theory: GL ( d, R ) G breaks to GL(d, Z) ✓ String Theory: Discrete Symmetries of full theory O(d, d) O(d, d; Z) T-duality, on p + string winding ! Ed+1 Ed+1(Z) ! U-duality, on p + brane wrapping Tuesday, 14 July 15 Generalised Geometry

Conventional manifold M, doubled tangent space

∗ GL(d, R) acting on TM O ( d, d ) acting on T ⊕ T Natural structure (M,g,B) instead of (M,g) Doubled Geometry

GL(d, Z) acting on Td O ( d, d, Z ) acting on T2d Doubled manifold

Tuesday, 14 July 15 Generalised Geometry Hitchin, Gualtieri Generalised Vector =Vector + 1-form i ∗ I v V = v + ξ ∈ T ⊕ T V = ! ξi " O(d,d) V → gV 0 O(d,d) Metric η = gtηg = η ! 0 "

t i ⌘(v + ⇠,v+ ⇠)=V ⌘V =2v ⇠i Lie Bracket Courant Bracket

Tuesday, 14 July 15 Generalised Metric HIJ Positive deﬁnite metric on T ⊕ T ∗ compatible with η η−1Hη−1 = H−1

S = η−1H satisﬁes S2 = Real structure Parameterised by G = Gt,B= −Bt E = G + B Metric G and B-ﬁeld B G − BG−1BBG−1 H = −1 −1 ! −G BG" O(d,d): t −1 H→g Hg E → (aE + b)(cE + d) Parameterise coset O(d, d) G,B combined in O(d) × O(d) gen metric

Tuesday, 14 July 15 Forms on M Λ•T ∗ O(d,d) spinors

i i j i j k φ = α + βidX + γijdX ∧ dX + δijkdX ∧ dX ∧ dX + ... 2d components Clifford action on forms V = v + ξ ∈ T ⊕ T ∗ • ∗ • ∗ ΓV :Λ T → Λ T

ΓV : φ !→ ιvφ + ξ ∧ φ

′ ΓV ΓV ′ +ΓV ′ ΓV = −2η(V,V ) Even forms Chiral spinor C0 + C2 + C4 + ... C+ Odd forms Anti-Chiral spinor − C1 + C3 + C5 + ... C

Tuesday, 14 July 15 Generalised Complex Structure J Hitchin

Endomorphism of T ⊕ T ∗ J 2 = − Courant-integrable η hermitian

Interpolates between symplectic & complex structures, generalising both.

Generalised Kahler Gualtieri H, J S = η−1H [S, J ]=0 J ′ = J S

Tuesday, 14 July 15 Generalised Bihermitian Kahler Gualtieri Geometry

′ (M; H, J , J ) (M; G, B, J+,J−)

HIJ Gij,Bij I i J J (J±) j Two Complex structures on M, G Hermitian ± ± −1 ∇ J± =0 ∇ = ∇±G H

(2,2) SUSY Sigma Model with torsion Gates,CMH,Rocek SUSY analysis gives general structure, potential Lindstrom, Rocek, von Unge, Zabzine

Tuesday, 14 July 15 Generalised Calabi-Yau • Gen Complex geometry can be encoded by pair of (pure) O(d,d) spinors i.e. differential forms ⌦, eB+i! ⇠ ⇠ • Generalised CY: these forms closed, and the spinors non-degenerate Hitchin, Gualtieri • RR ﬁelds are also an O(d,d) spinor • Gen CY with ﬂux: Conditions for SUSY have elegant formulation in terms of these spinors Grana, Minasian, Petrini, Tomasiello

Tuesday, 14 July 15 Strings on Circle M = S1 X ⇥ Discrete momentum p=n/R If it winds m times round S1, winding energy w=mRT Energy = p2+w2+....

T-duality: Symmetry of string theory p w m $ n $ R 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)

Tuesday, 14 July 15 T-Duality G(Y ),B(Y ) Xi • Space has d-torus ﬁbration • G,B on ﬁbres Y m • T-Duality O(d,d;Z), mixes G,B • Mixes Momentum and Winding • Changes geometry and topology −1 E → (aE + b)(cE + d) ab h = ∈ O(d, d; Z) Eij = Gij + Bij ! cd" 1 On circle, radius R: O(1, 1; Z)=Z2 : R !→ R

Tuesday, 14 July 15 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 Hull • Patching with T-duality: T-FOLDS • Patching with U-duality: U-FOLDS

Tuesday, 14 July 15 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

Tuesday, 14 July 15 ′ ′ E(Y ) E (Y ) Torus ﬁbration

Y Y ′

U U ′ Geometric background: G, H=dB tensorial T-fold:Transition functions involve T-dualities (as well as diffeomorphisms and 2-form gauge transformations) E=G+B Non-tensorial ′ −1 ′ O(d, d; Z) E =(aE + b)(cE + d) in U ∩ U Glue using T-dualities also ➞ T-fold Physics smooth, as T-duality a symmetry Fixes Not conventional smooth geometry moduli!

Tuesday, 14 July 15 Double Field Theory Hull & Zwiebach

• From sector of String 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

Tuesday, 14 July 15 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. gij(x, x˜),bij(x, x˜), (x, x˜)

Tuesday, 14 July 15 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, GG.

Tuesday, 14 July 15 O(D,D) covariant action using generalised metric, dilaton d

2d S = dxdxe˜ L 1 1 L = MN KL MN KL 8 H M H N HKL 2H N H LHMK 2 d MN +4 MN d d M N H H M N Hohm, Hull & Zwiebach L cubic! Indices raised and lowered with ⌘

If independent of dual coord, X˜ Gives usual action (+ surface term)

2 2 1 2 dx⇥ ge R + 4(⇥) H 12 ⇥ ⇤

Tuesday, 14 July 15 Extended Geometry, M-Theory

T ⊕ Λ2T ∗⊕Λ5T ∗ ⊕ Λ6T Hull; Pacheco & Waldram • Extended geometry: extends tangent space • Metric and 3-form gauge ﬁeld, action of exceptional U-duality group • 10-d type II and 11-d sugra in terms of extended geometry Coimbra, Strickland-Constable & Waldram

Double ﬁeld theory generalises Berman, Perry et al, • Hohm & Sambtleben, ...

Tuesday, 14 July 15 Generalised Parallelalisability

Lee, Strickland Constable, Waldram

• Generalised (or extended) tangent space is generalised parallelalisable if it admits a global frame. Important examples include S3,S4,S7 • Explains consistent truncations of supergravity on S3,S4,S7 • Gives insights into exotic supergravity gaugings of Dell’Agata, Inverso,Trigiante

Tuesday, 14 July 15 Conclusions • Generalised geometry gives elegant formulation for gravity + p-form gauge ﬁelds • Uniﬁed treatment of various structures, allows general results • Generalised complex geometry, generalised CY, superymmetric spaces. Gen parallelisability. • String theory sees a doubled or extended spacetime, physical extra dimensions • String theory has non-geometric solns, which look geometric in doubled formulation

Tuesday, 14 July 15 Tuesday, 14 July 15 Tuesday, 14 July 15 Tuesday, 14 July 15 Tuesday, 14 July 15 Tuesday, 14 July 15 Tuesday, 14 July 15 Tuesday, 14 July 15 Doubled Geometry for T-fold d Xi Hull T torus ﬁbres have XI = I =1, ..., 2d doubled coords ! Xi # " Transforms linearly under O(d, d; Z) T-fold transition: mixes X, X˜ No global way of separating “real” space coordinate X from “auxiliary”X! Duality covariant formulation in terms of X Transition functions O(d, d; Z) ⊂ GL(2d; Z) can be used to construct bundle with ﬁbres T2d Doubled space is smooth manifold! Sigma Model on doubled space Tuesday, 14 July 15 Tuesday, 14 July 15 Type I Extended Geometry

Action of O(d,d)

T ⊕ T ∗ cf Brane charges Generalised metric G, B2 O(d, d) HIJ ∈ O(d) × O(d)

Tuesday, 14 July 15 Type I Extended Geometry

Action of O(d,d)

T ⊕ T ∗ cf Brane charges Generalised metric G, B2 O(d, d) HIJ ∈ O(d) × O(d)

Type II: O(d,d) Ed+1, add RR ﬁelds

Tuesday, 14 July 15 Type I Extended Geometry

Action of O(d,d)

T ⊕ T ∗ cf Brane charges Generalised metric G, B2 O(d, d) HIJ ∈ O(d) × O(d)

Type M: O(d,d) Ed, B 2 C3 T ⊕ T ∗ T ⊕ Λ2T ∗

Tuesday, 14 July 15 Type II Extended Geometry d ≤ 4

Action of Ed+1

T ⊕ T ∗ ⊕S± cf Brane charges Generalised metric ∓ G, B2, Φ,C Ed+1 HIJ ∈ Hd+1

+ IIB C C0 + C2 + C4 + ... − RR ﬁelds IIA C C1 + C3 + C5 + ...

Tuesday, 14 July 15 Type II Extended Geometry d ≤ 6

Action of Ed+1

T ⊕ T ∗ ⊕S± ⊕Λ5T ⊕ Λ5T ∗ cf Brane charges Generalised metric ∓ ˜ G, B2, Φ,C , B6 Ed+1 HIJ ∈ Hd+1

+ IIB C C0 + C2 + C4 + ... − RR ﬁelds IIA C C1 + C3 + C5 + ...

Tuesday, 14 July 15 Type M Extended Geometry d ≤ 4

Action of Ed

T ⊕ Λ2T ∗ cf Brane charges Generalised metric G, C3 Ed HIJ ∈ Hd

Tuesday, 14 July 15 Type M Extended Geometry d ≤ 7

Action of Ed

T ⊕ Λ2T ∗ ⊕Λ5T ∗ ⊕ Λ6T cf Brane charges

˜ Generalised metric G, C3 , C6 Ed HIJ ∈ Hd

Tuesday, 14 July 15 Tuesday, 14 July 15 • More general non-geometric backgrounds. Gives uplift of GENERIC gauged Sugras Shelton, Taylor & Wecht 2005 Dabholkar & Hull 2005 • Explicit doubled geometries constructed for T-folds and “spaces with R-ﬂux” Hull & Reid-Edwards 2008-9 • Sigma models on doubled spaces; quantisation Hull 2004-6 • Other approaches to quantisation Tseytlin; Berman, Thompson, Copland; Hackett-Jones & Motsopoulos

Tuesday, 14 July 15 Tuesday, 14 July 15 Strings on Td

X = XL(σ + τ)+XR(σ − τ), X˜ = XL − XR

X conjugate to momentum, X˜ to winding no. ˜ b ˜ dX = ∗dX ∂aX = ϵab∂ X

Tuesday, 14 July 15 Strings on Td

X = XL(σ + τ)+XR(σ − τ), X˜ = XL − XR

X conjugate to momentum, X˜ to winding no. ˜ b ˜ dX = ∗dX ∂aX = ϵab∂ X Need “auxiliary” X ˜ for interacting theory i) Vertex operators eikL·XL ,eikR·XR ii) String ﬁeld Kugo & Zwiebach [x, x,˜ a, a˜]

Tuesday, 14 July 15 Strings on Td

X = XL(σ + τ)+XR(σ − τ), X˜ = XL − XR

X conjugate to momentum, X˜ to winding no. ˜ b ˜ dX = ∗dX ∂aX = ϵab∂ X

Strings on torus see DOUBLED GEOMETRY! T-duality group O(d, d; Z)

Doubled Torus 2d coordinates x˜ X i Transform linearly under O(d, d; Z) xi ⇥ Sigma model on doubled torus Tseytlin; Hull

Tuesday, 14 July 15