Double Field Theory and the Geometry of Duality CMH & Barton - - PowerPoint PPT Presentation

Double Field Theory and the Geometry of Duality

CMH & Barton Zwiebach

arXiv:0904.4664, 0908.1792

CMH & BZ & Olaf Hohm

arXiv:1003.5027, 1006.4664

Closed String Field Theory

• Captures exotic and complicated structure
• f interacting string
• Non-polynomial, algebraic structure not Lie

algebra, cocycles.

• On torus: winding modes, T
• duality.
• Seek subsector capturing exotic structure &

duality, simple enough to analyse explicitly

Strings on d-Torus

• Supergravity limit: symmetry O(d,d)
• String: Perturbative T
• duality O(d,d;Z)
• Kaluza-Klein theory: includes momentum

modes on torus

• Include string winding or brane wrapping

modes? Duality symmetry?

Strings on Td

X = XL(σ + τ) + XR(σ − τ), ˜ X = XL − XR X ˜ X

conjugate to momentum, to winding no.

dX = ∗d ˜ X ∂aX = ab∂b ˜ X

Strings on Td

X = XL(σ + τ) + XR(σ − τ), ˜ X = XL − XR X ˜ X

conjugate to momentum, to winding no.

dX = ∗d ˜ X ∂aX = ab∂b ˜ X

Need “auxiliary” for interacting theory i) Vertex operators ii) String field Kugo & Zwiebach

˜ X eikL·XL, eikR·XR

Φ[x, ˜ x, a, ˜ a]

Strings on Td

X = XL(σ + τ) + XR(σ − τ), ˜ X = XL − XR X ˜ X

conjugate to momentum, to winding no.

dX = ∗d ˜ X ∂aX = ab∂b ˜ X

Need “auxiliary” for interacting theory i) Vertex operators ii) String field Kugo & Zwiebach

˜ X eikL·XL, eikR·XR

Φ[x, ˜ x, a, ˜ a]

Doubled Torus 2d coordinates

Transform linearly under Doubled sigma model CMH 0406102

O(d, d; Z)

X ≡ ˜ xi xi

String Field Theory

• n Minkowski Space

String field

Φ[X(σ), c(σ)]

Expand to get infinite set of fields

Xi(σ) → xi, oscillators gij(x), bij(x), φ(x), . . . , Cijk...l(x), . . .

Integrating out massive fields gives field theory for

gij(x), bij(x), φ(x)

String Field Theory

• n a torus

String field

Φ[X(σ), c(σ)]

Expand to get infinite set of double fields Seek double field theory for

Xi(σ) → xi, ˜ xi, oscillators gij(x, ˜ x), bij(x, ˜ x), φ(x, ˜ x), . . . , Cijk...l(x, ˜ x), . . . gij(x, ˜ x), bij(x, ˜ x), φ(x, ˜ x)

Double Field Theory

• Construct from SFT, DFT of “massless”

fields

• Double field theory on doubled torus
• Novel symmetry, reduces to diffeos + B-field
• trans. in any half-dimensional subtorus
• Backgrounds depending on seen by

particles, on seen by winding modes. Backgrounds with both: unfamiliar.

Earlier versions: Siegel, Tseytlin

{xa} {˜ xa} gij(x, ˜ x), bij(x, ˜ x), φ(x, ˜ x)

• Restriction to “massless” fields NOT a low-

energy limit

• Lowest terms in level expansion
• T
• duality a manifest symmetry
• General solution of SFT: double fields
• DFT needed for non-geometric backgrounds
• Real dependence on full doubled geometry,

dual dimensions not auxiliary or gauge

• artifact. Double geom. physical and dynamical

Strings on a Torus

• Coordinates
• Momentum
• Winding
• Fourier transform
• Doubled Torus
• String Field Theory gives

infinite set of fields

Rn−1,1 × T d

xa ∼ xa + 2π pi = (kµ, pa) wa (pa, wa) ∈ Z2d (kµ, pa, wa) → (yµ, xa, ˜ xa) ˜ xa ∼ ˜ xa + 2π Rn−1,1 × T 2d n + d = D = 26 or 10 φ(yµ, xa, ˜ xa) xi = (yµ, xa)

T

• Duality
• Interchanges momentum and winding
• Equivalence of string theories on dual

backgrounds with very different geometries

• String field theory symmetry, provided fields

depend on both Kugo, Zwiebach

• For fields not Buscher
• Aim: generalise to fields

ψ(y) ψ(y, x, ˜ x) ψ(y, x, ˜ x) x, ˜ x Dabholkar & CMH Generalised T

• duality

Free field equn, M mass in D dimensions M 2 ≡ −(k2 + p2 + w2) = 2 α′ (N + ¯ N − 2) L0 − ¯ L0 = N − ¯ N − pawa = 0 hij → {hµν, hµa, hab} Constraint N = ˜ N = 1 M 2 = 0 pawa = 0 Massless states hij(yµ, xa, ˜ xa), bij(yµ, xa, ˜ xa), d (yµ, xa, ˜ xa) Constrained fields ∆φ = 0 φ(y, x, ˜ x) ∆ ≡ ∂ ∂xa ∂ ∂˜ xa

Constrained fields ∆φ = 0 φ(y, x, ˜ x) ∆ ≡ ∂ ∂xa ∂ ∂˜ xa Momentum space φ(kµ, pa, wa) ∆ = pawa Momentum space: Dimension n+2d Cone: dimension n+2d-1 pawa = 0

Constrained fields ∆φ = 0 φ(y, x, ˜ x) ∆ ≡ ∂ ∂xa ∂ ∂˜ xa Momentum space φ(kµ, pa, wa) ∆ = pawa Momentum space: Dimension n+2d Cone: dimension n+2d-1 pawa = 0 DFT: fields on this cone, with discrete p,w Problem: naive product of fields on cone do not lie

• n cone. Vertices need projectors

Constrained fields ∆φ = 0 φ(y, x, ˜ x) ∆ ≡ ∂ ∂xa ∂ ∂˜ xa Momentum space φ(kµ, pa, wa) ∆ = pawa Momentum space: Dimension n+2d Cone: dimension n+2d-1 pawa = 0 DFT: fields on this cone, with discrete p,w Problem: naive product of fields on cone do not lie

• n cone. Vertices need projectors

Restricted fields: Fields that depend on d of 2d torus momenta, e.g. or Simple subsector, no projectors needed, easier φ(kµ, pa) φ(kµ, wa)

Torus Backgrounds

Gij = ηµν Gab

• ,

Bij = Bab

• ˜

xi = {˜ yµ, ˜ xa} = {0, ˜ xa} xi = {yµ, xa} Eij ≡ Gij + Bij Left and Right Derivatives Di = ∂ ∂xi − Eik ∂ ∂˜ xk , ¯ Di = ∂ ∂xi + Eki ∂ ∂˜ xk α′ = 1 ∆ = 1 2(D2 − ¯ D2) = −2 ∂ ∂˜ xi ∂ ∂xi = 1 2(D2 + ¯ D2) D2 = GijDiDj

Quadratic Action

S(2) =

• [dxd˜

x] 1 2eijeij + 1 4( ¯ Djeij)2 + 1 4(Dieij)2 −2 d Di ¯ Djeij − 4 d d

• δeij =

¯ Djλi + Di¯ λj , δd = −1 4D · λ − 1 4 ¯ D · ¯ λ Invariant under using constraint ∆λ = ∆¯ λ = 0 eij → eji , Di → ¯ Di , ¯ Di → Di , d → d Discrete Symmetry

Comparison with Conventional Actions

Di = ∂i − ˜ ∂i , ¯ Di = ∂i + ˜ ∂i Take Bij = 0 ˜ ∂i ≡ Gik ∂ ∂˜ xk = ∂2 + ˜ ∂2 ∆ = −2 ∂i ˜ ∂i eij = hij + bij Usual quadratic action

• dx L[ h, b, d; ∂ ]

L[ h, b, d; ∂ ] = 1 4 hij∂2hij + 1 2(∂jhij)2 − 2 d ∂i∂j hij −4 d ∂2 d + 1 4 bij∂2bij + 1 2(∂jbij)2

Action + dual action + strange mixing terms

Double Field Theory Action

δhij = ∂iǫj + ∂jǫi + ˜ ∂i˜ ǫj + ˜ ∂j˜ ǫi , δbij = −(˜ ∂iǫj − ˜ ∂jǫi) − (∂i˜ ǫj − ∂j˜ ǫi) , δd = − ∂ · ǫ + ˜ ∂ · ˜ ǫ . Diffeos and B-field transformations mixed. Cubic action found for full DFT S(2) =

• [dxd˜

x]

• L[ h, b, d; ∂ ] + L[ −h, −b, d; ˜

∂ ] + (∂khik)(˜ ∂jbij) + (˜ ∂khik)(∂jbij) − 4 d ∂i ˜ ∂jbij

T

• Duality Transformations of Background

g = a b c d

• ∈ O(d, d; Z)

X ≡ ˜ xi xi

• E′ = (aE + b)(cE + d)−1

X′ = ˜ x′ x′

• = gX =

a b c d ˜ x x

• T
• duality

transforms as a vector

T

• Duality is a Symmetry of the Action

Fields

eij(x, ˜ x), d(x, ˜ x)

Background Eij

E′ = (aE + b)(cE + d)−1 X′ = ˜ x′ x′

• = gX =

a b c d ˜ x x

• M ≡ dt − E ct

¯ M ≡ dt + Etct Action invariant if: eij(X) = Mi

k ¯

Mj

l e′ kl(X′)

d(X) = d′(X′) With general momentum and winding dependence!

Projectors and Cocycles

Naive product of constrained fields does not satisfy constraint String product has explicit projection Double field theory requires projections, novel forms Leads to a symmetry that is not a Lie algebra, but is a homotopy lie algebra SFT has non-local cocycles in vertices, DFT should too Cocycles and projectors not needed in cubic action

L−

0 Ψ1 = 0, L− 0 Ψ2 = 0 but

∆A = 0, ∆B = 0 ∆(AB) = 0

but

L−

0 (Ψ1Ψ2) = 0

General fields

ψ(x)

Fields on Spacetime M

ψ(x, ˜ x)

Restricted Fields on N, T

• dual to M

ψ(x′)

Subsector with fields and parameters all restricted to M or N

• Constraint satisfied on all fields and products of fields
• No projectors or cocycles
• T
• duality covariant: independent of choice of N
• Can find full non-linear form of gauge transformations
• Full gauge algebra, full non-linear action

M,N null wrt O(D,D) metric ds2 = 2dxidxi

Background Independence?

Action S(E, e, d)

δEij = −χij δeij = χij − 1 2

• χi

kekj − χk jeik

• + O(e2)

δχS = 0

is background independent: Constant shift

Background Independence?

Action S(E, e, d)

δEij = −χij δeij = χij − 1 2

• χi

kekj − χk jeik

• + O(e2)

δχS = 0

Background independent field:

δχEij = 0 Eij ≡ Eij + eij + 1 2ei

kekj + O(e3)

is background independent: Constant shift

Background Independence?

Action S(E, e, d)

δEij = −χij δeij = χij − 1 2

• χi

kekj − χk jeik

• + O(e2)

δχS = 0

Background independent field:

δχEij = 0 Eij ≡ Eij + eij + 1 2ei

kekj + O(e3)

E ≡ E +

• 1 − 1

2 e −1 e

is background independent: Constant shift

• Write action and transformations in terms
• f
• and derivatives
• Find manifestly background independent

forms that agree with action and transformations to lowest order

• Unique BI terms that agree with lowest
• rder results. Complete non-linear

structure!

Eij, d

Di = ∂i − Eik ˜ ∂k ¯ Di = ∂i + Eki ˜ ∂k

S =

• dxd˜

x e−2d − 1 4 gikgjl DpEkl DpEij + 1 4gkl DjEikDiEjl + ¯ DjEki ¯ DiElj

• +
• Did ¯

DjEij + ¯ Did DjEji

• + 4Did Did
• BI Action and Transformations

S =

• dxd˜

x e−2d − 1 4 gikgjl DpEkl DpEij + 1 4gkl DjEikDiEjl + ¯ DjEki ¯ DiElj

• +
• Did ¯

DjEij + ¯ Did DjEji

• + 4Did Did
• δEij = ξM∂MEij

+ Di ˜ ξj − ¯ Dj ˜ ξi + DiξkEkj + ¯ DjξkEik δλd = −1 2∂MξM + ξM∂M d

BI Action and Transformations

• Remarkable action, reduces to familiar ones
• uses stringy combination
• checked gauge invariant
• Invariant under O(D,D) if dimensions all

non-compact. Broken to discrete subgroup by boundary conditions.

Eij = gij + bij

2-derivative action

S = S(0)(∂, ∂) + S(1)(∂, ˜ ∂) + S(2)(˜ ∂, ˜ ∂)

Write in terms of usual fields

S(0) Eij = gij + bij e−2d = √−ge−2φ

2-derivative action

S = S(0)(∂, ∂) + S(1)(∂, ˜ ∂) + S(2)(˜ ∂, ˜ ∂)

Write in terms of usual fields

S(0) Eij = gij + bij e−2d = √−ge−2φ

• dx√−ge−2φ

R + 4(∂φ)2 − 1 12H2

Gives usual action (+ surface term)

S(0) = S(E, d, ∂)

2-derivative action

S = S(0)(∂, ∂) + S(1)(∂, ˜ ∂) + S(2)(˜ ∂, ˜ ∂)

Write in terms of usual fields

S(0) Eij = gij + bij e−2d = √−ge−2φ

• dx√−ge−2φ

R + 4(∂φ)2 − 1 12H2

Gives usual action (+ surface term)

S(0) = S(E, d, ∂) S(2) = S(E−1, d, ˜ ∂)

T

• dual!

2-derivative action

S = S(0)(∂, ∂) + S(1)(∂, ˜ ∂) + S(2)(˜ ∂, ˜ ∂)

Write in terms of usual fields

S(0) Eij = gij + bij e−2d = √−ge−2φ

• dx√−ge−2φ

R + 4(∂φ)2 − 1 12H2

Gives usual action (+ surface term)

S(0) = S(E, d, ∂) S(2) = S(E−1, d, ˜ ∂)

T

• dual! strange mixed terms

S(1)

E′(X′) = (aE(X) + b)(cE(X) + d)−1

Generalised T

• duality transformations:

d′(X′) = d(X) h in O(d,d) acts on toroidal coordinates only

X′M ≡

• ˜

x′

i

x′i

• = hXM =

a b c d ˜ xi xi

• Buscher if fields independent of toroidal coordinates

Generalisation to case without isometries

∂M ≡

• ∂i

∂i

• ηMN =

I I

• O(D,D) Covariant Notation

(λ, ¯ λ) → ΣM

M = 1, ..., 2D

XM ≡ ˜ xi xi

• Parameters

Gauge Algebra C-Bracket:

[δΣ1, δΣ2] = δ[Σ1,Σ2]C

[Σ1, Σ2]C ≡ [Σ1, Σ2] − 1 2 ηMNηP Q ΣP

[1 ∂N ΣQ 2]

Lie bracket + metric term

Parameters restricted to N Decompose into vector + 1-form on N C-bracket reduces to Courant bracket on N

ΣM(X)

Same covariant form of gauge algebra found in similar context by Siegel Symmetry is Reducible

ΣM = ηMN∂Nχ

Parameters of the form do not act cf 2-form gauge field Parameters of the form do not act

δB = dα α = dβ

Gauge algebra determined up to such transformations

J(Σ1, Σ2, Σ3) ≡ [ [Σ1, Σ2] , Σ3 ] + cyclic = 0

Jacobi Identities not satisfied! for both C-bracket and Courant-bracket How can bracket be realised as a symmetry algebra?

[ [δΣ1, δΣ2] , δΣ3 ] + cyclic = δJ(Σ1,Σ2,Σ3)

J(Σ1, Σ2, Σ3) ≡ [ [Σ1, Σ2] , Σ3 ] + cyclic = 0

Jacobi Identities not satisfied! for both C-bracket and Courant-bracket How can bracket be realised as a symmetry algebra? Resolution:

J(Σ1, Σ2, Σ3)M = ηMN∂Nχ δJ(Σ1,Σ2,Σ3)

does not act on fields

[ [δΣ1, δΣ2] , δΣ3 ] + cyclic = δJ(Σ1,Σ2,Σ3)

Generalised Metric Formulation

HMN = gij −gikbkj bikgkj gij − bikgklblj

• .

2 Metrics on double space

HMN, ηMN

Generalised Metric Formulation

HMN = gij −gikbkj bikgkj gij − bikgklblj

• .

HMN ≡ ηMP HP QηQN HMP HP N = δM

N

2 Metrics on double space

HMN, ηMN

Constrained metric

Generalised Metric Formulation

HMN = gij −gikbkj bikgkj gij − bikgklblj

• .

hP

MhQ NH′ P Q(X′) = HMN(X)

X′ = hX h ∈ O(D, D) HMN ≡ ηMP HP QηQN HMP HP N = δM

N

2 Metrics on double space

HMN, ηMN

Constrained metric Covariant Transformation

L = 1 8 HMN∂MHKL ∂NHKL − 1 2HMN∂NHKL ∂LHMK − 2 ∂Md ∂NHMN + 4HMN ∂Md ∂Nd

S =

• dxd˜

x e−2d L

O(D,D) covariant action

L = 1 8 HMN∂MHKL ∂NHKL − 1 2HMN∂NHKL ∂LHMK − 2 ∂Md ∂NHMN + 4HMN ∂Md ∂Nd

S =

• dxd˜

x e−2d L

δξHMN = ξP ∂P HMN + (∂MξP − ∂P ξM) HP N + (∂NξP − ∂P ξN) HMP

Gauge Transformation O(D,D) covariant action

L = 1 8 HMN∂MHKL ∂NHKL − 1 2HMN∂NHKL ∂LHMK − 2 ∂Md ∂NHMN + 4HMN ∂Md ∂Nd

S =

• dxd˜

x e−2d L

δξHMN = ξP ∂P HMN + (∂MξP − ∂P ξM) HP N + (∂NξP − ∂P ξN) HMP

δξHMN = LξHMN

Gauge Transformation Rewrite as “Generalised Lie Derivative” O(D,D) covariant action

Generalised Lie Derivative

• LξAM

N ≡ ξP ∂P AM N

+(∂MξP −∂P ξM)AP

N + (∂NξP − ∂P ξN)AM P

A M1...

N1...

Generalised Lie Derivative

• LξAM

N ≡ ξP ∂P AM N

+(∂MξP −∂P ξM)AP

N + (∂NξP − ∂P ξN)AM P

• LξAM

N = LξAM N − ηP QηMR ∂QξR AP N

+ ηP QηNR ∂RξQ AM

P

A M1...

N1...

Generalised Lie Derivative

• LξAM

N ≡ ξP ∂P AM N

+(∂MξP −∂P ξM)AP

N + (∂NξP − ∂P ξN)AM P

• LξAM

N = LξAM N − ηP QηMR ∂QξR AP N

+ ηP QηNR ∂RξQ AM

P

A M1...

N1...

Lξ1 , Lξ2

• = −

L[ξ1,ξ2]C

Algebra given by C-bracket

D-Bracket

• A, B
• D ≡

LAB

• A, B

M

D =

• A, B

M

C + 1

2∂M BNAN

D-Bracket

• A, B
• D ≡

LAB

• A, B

M

D =

• A, B

M

C + 1

2∂M BNAN

• Not skew, but satisfies Jacobi-like identity
• A,
• B, C
• D
• D =
• A, B
• D
• , C
• D +
• B,
• A, C
• D
• D

D-Bracket

• A, B
• D ≡

LAB

On restricting to null subspace N C-bracket ➞ Courant bracket D-bracket ➞ Dorfman bracket Gen Lie Derivative ➞ GLD of Grana, Minasian, Petrini and Waldram

• A, B

M

D =

• A, B

M

C + 1

2∂M BNAN

• Not skew, but satisfies Jacobi-like identity
• A,
• B, C
• D
• D =
• A, B
• D
• , C
• D +
• B,
• A, C
• D
• D

Generalized scalar curvature

R ≡ 4 HMN∂M∂Nd − ∂M∂NHMN − 4 HMN∂Md ∂Nd + 4∂MHMN ∂Nd + 1 8 HMN∂MHKL∂NHKL − 1 2HMN∂MHKL ∂KHNL

Generalized scalar curvature

R ≡ 4 HMN∂M∂Nd − ∂M∂NHMN − 4 HMN∂Md ∂Nd + 4∂MHMN ∂Nd + 1 8 HMN∂MHKL∂NHKL − 1 2HMN∂MHKL ∂KHNL

S =

• dx d˜

x e−2d R

Generalized scalar curvature

R ≡ 4 HMN∂M∂Nd − ∂M∂NHMN − 4 HMN∂Md ∂Nd + 4∂MHMN ∂Nd + 1 8 HMN∂MHKL∂NHKL − 1 2HMN∂MHKL ∂KHNL

S =

• dx d˜

x e−2d R δξR = LξR = ξM∂MR δξ e−2d = ∂M(ξMe−2d)

Gauge Symmetry

Generalized scalar curvature

R ≡ 4 HMN∂M∂Nd − ∂M∂NHMN − 4 HMN∂Md ∂Nd + 4∂MHMN ∂Nd + 1 8 HMN∂MHKL∂NHKL − 1 2HMN∂MHKL ∂KHNL

S =

• dx d˜

x e−2d R δξR = LξR = ξM∂MR δξ e−2d = ∂M(ξMe−2d)

Gauge Symmetry Field equations give gen. Ricci tensor

M-Theory

• Strings: Doubled torus e.g. T7→ T14
• M-Theory: M-torus e.g. T7→ T56
• U-duality E7 acts on T56
• 7 torus coordinates + 49 coords

conjugate brane wrapping modes

Extended Geometry

• Extended geometry: metric and 3-

form gauge field

• Generalised metric
• Rewrite of 11-d sugra action in

terms of this

CMH; Pacheco & Waldram

Hillman; Berman & Perry

Double Field Theory

• Captures some of the magic of string theory
• Constructed cubic action, quartic could have

new stringy features

• T
• duality symmetry, cocycles, homotopy Lie,

constraints

• For fields restricted to null subspace, have full

non-linear action and gauge transformations.

• Background independent, duality covariant
• Courant bracket gauge algebra

• Stringy issues in simpler setting than SFT
• Geometry? Meaning of curvature?
• Use for non-geometric backgrounds?
• Generalised Geometry doubles Tangent

space, DFT doubles coordinates.

• Full theory (no restriction)? Does it close on

a geometric action with just these fields?

• General spaces, not tori?
• Doubled geometry physical and dynamical