Stringy Unification of Type IIA and IIB Supergravities ...ism2012/talks/ISM2012_ImtakJeon.pdf ·...

Stringy Unification ofType IIA and IIB Supergravities under

N=2 D=10 Supersymmetric Double FieldTheory

ISM 2012

Imtak JeonarXiv1210.5078, with Jeong-Hyuck Park, Kanghoon Lee and Yoonji Suh

19 Dec. 2012

Based on works

in collaboration with Jeong-Hyuck Park, Kanghoon Lee

• Differential geometry with a projection: Application to double field theoryJHEP 1104:014 (2011), arXiv:1011.1324

• Double field formulation of Yang-Mills theoryPhys. Lett. B 701:260 (2011), arXiv:1102.0419

• Stringy differential geometry, beyond RiemannPhys. Rev. D 84:044022 (2011), arXiv:1105.6294

• Incorporation of fermions into double field theoryJHEP 1111:025 (2011), arXiv:1109.2035

• Supersymmetric Double Field Theory: Stringy Reformulation ofSupergravity Phys. Rev. D Rapid Comm. (2012), arXiv:1112.0069

• Ramond-Ramond Cohomology and O(D,D) T-dualityJHEP 09 (2012) 079, arXiv:1206.3478

Introduction

• String theory says that gµν , Bµν and φ should be treated on an equal footing ,because they form a multiplet of T-duality.

• This suggests that there should be an unifying description of them, beyond theRiemannian geometry.

• T-duality is an pure stringy effect , and gravity theory is hardly expected tocapture the stringy effect.

• Double Field Theory(DFT) has been suggested as an unifying description ofstring effective theory, which captures the stringy effect by manifesting theT-duality structure [Siegel, Hull, Zwiebach, Hohm] .

Introduction

1. Bosonic Double Field Theory (Review)

2. Stringy differential geometry and Supersymmetric DFT(SDFT)

: unifying covariant description of 10D IIA and IIB SUGRAs

3. Parametrization and gauge fixing

Reduction to generalized geometry, ordinary SUGRAs

how to exchange IIA and IIB under T-duality

4. Summary

1. (Bosonic) Double Field Theory

“Double”: x −→ (x, x)

• This originates from the string field theory.

The fields in SFT on a toroidal background have dependance on pi and wi,

pi ←→ xi wi ←→ xi

so have natural dependance on both x and x.: Φ(xi, xi)

• “String field theory is a double field theory”. [Kugo, Zwiebach]

• In DFT, we focus on "massless sector".

1. Bosonic Double Field Theory

• Dynamical fields in DFT :

dilaton d, ‘generalized metric’HAB (Not metric of DFT) ,

e−2d =√−ge−2φ , HAB =

„g−1 −g−1B

Bg−1 g− Bg−1B

«

• T-duality is realized as an subgroup of O(D,D) rotation, [Giveon, Rabinovici,

Veneziano, Tseytlin, Siegel] :d is scalar andHAB is rank 2 tensor.

• Metric in DFT : O(D,D) metric,

JAB :=

„0 11 0

«

freely raises or lowers the (D + D)-dimensional vector indices, A,B.

1. Bosonic Double Field Theory

• DFT action for NS-NS sector: Hull and Zwiebach , later with Hohm

SDFT =

Zdy2D e−2d LDFT(H, d) ,

where

LDFT(H, d) = HAB `4∂A∂Bd − 4∂Ad∂Bd + 18∂AHCD∂BHCD − 1

2∂AHCD∂CHBD´

+4∂AHAB∂Bd − ∂A∂BHAB .

• O(D,D) structure is manifest and background independent.

• All spacetime dimension is ‘formally doubled’, yA = (xµ, xν),A = 1, 2, · · · ,D+D.

Constraint in Double Field Theory• DFT is a D-dimensional theory written in terms of (D + D)-dimensional

language, i.e. tensors.

• Strong constraint (section condition) from level matching condition:The O(D,D) d’Alembert operator is trivial, acting on arbitrary fields or gaugeparameters as well as their products:

∂A∂A = J AB∂A∂B = 2

∂2

∂xµ∂xµ' 0

• DFT lives on a D-dimensional null hyperplane and O(D,D) rotates the nullhyperplane.

• Up to O(D,D) rotation, we can choose a frame to set

∂

∂xµ' 0 .

• DFT action is (locally) equivalent to the effective action:

SDFT =⇒ Seff. =

ZdxD√−ge−2φ

“Rg + 4(∂φ)2 − 1

12 H2”.

O(D, D) meta-symmetry

• A priori, the O(D,D) structure in DFT is a ‘meta-symmetry’ or ‘hiddensymmetry’ rather than a Noether symmetry,

• Only after dimensional reductions or assuming independence alongcertain n directions, the subgroup O(n, n) can generate a Noethersymmetry,

Gauge symmetry: ‘DFT-diffeomorphism’

Unification of diffeomorphism and B-field gauge symmetry , expressed via

• ‘generalized Lie derivative’ [Siegel, Courant, Grana ...]

LXTA := XB∂BTA + ω∂BXBTA + ∂AXBTB−∂BXATB.

• XA is an unifying gauge parameter (B-field gauge symmetry+diffeomorphism ),

XA = (Λµ, δxν)

• HAB is a rank 2 DFT tensor, and e−2d is a weight 1 DFT scalar,

3. Stringy differential geometry andSupersymmetric DFT

[1011.1324, 1105.6294, 1109.2035, 1112.0069, 1206.3478, 1210.5078]

• connections and curvatures of DFT

• the construction of N=2 SDFT which unifies D = 10 IIA and IIB SUGRAsin a covariant manner

Symmetries of SDFT

• O(D,D) T-duality: Meta-symmetry

• Gauge symmetries

1. DFT-diffeomorphism (generalized diffeomorphism)• Diffeomorphism• B-field gauge symmetry

2. A pair of local Lorentz symmetries, Spin(1,D−1)× Spin(D−1, 1)

3. N = 2 Local SUSY

Use (double) vielbein formulation.

(cf . metric-like formulation for Bosonic DFT [1105.6924 Park, Lee, IJ]

Field contents of type II SDFT

• Bosons

• NS-NS sector

DFT-dilaton: dDouble-vielbeins: VAp , VAp

• R-R potential: Cαα

Fermions•• DFT-dilatinos: ρα , ρ′α

• Gravitinos: ψαp , ψ′αp


• Bosons

• NS-NS sector



• Fermions• DFT-dilatinos: ρα , ρ′α


Index Representation Metric (raising/lowering indices)A, B, · · · O(D,D) vector JABp, q, · · · Spin(1,D−1) vector ηpq = diag(−+ + · · ·+)

α, β, · · · Spin(1,D−1) spinor C+αβ , (γp)T = C+γpC−1

+p, q, · · · Spin(D−1, 1) vector ηpq = diag(+−− · · ·−)

α, β, · · · Spin(D−1, 1) spinor C+αβ , (γ p)T = C+γpC−1

+


• Bosons

• NS-NS sector





All NS-NS fields, d,VAp, VAp, will be equally treated as basic geometric objects.


• Bosons

• NS-NS sector





R-R potential is bi-fundamental spinor representationas a democratic description.

cf . O(D,D) spinor representation Fukuma, Oota Tanaka; Hohm, Kwak, Zwiebach


• Bosons

• NS-NS sector



Fermions•• DFT-dilatinos: ρα , ρ′α



• Bosons

• NS-NS sector





A priori, O(D,D) rotates only the O(D,D) vector indices (capital Roman), andthe R-R sector and all the fermions are O(D,D) T-duality singlet.

The usual IIA⇔ IIB exchange will follow only after fixing a gauge.


• Bosons

• NS-NS sector





• Set the chiralities for IIA and IIB

γ(D+1)Cγ(D+1) = ±C . γ(D+1)ψp = +ψp , γ(D+1)ρ = −ρ ,γ(D+1)ψ′p = ±ψ′p , γ(D+1)ρ′ = ∓ρ′ .

Double-vielbein 1105.6294, 1109.2035

• Double-vielbein simultaneously diagonalizes JAB andHAB,

J =`

V V´ η−1 0

0 η

!`V V

´T, H =

`V V

´ η−1 0

0 −η

!`V V

´T.

• They must satisfy

VApVAq = ηpq , VApVA

p = 0 VApVBp = PAB ,

VApVAq = ηpq , VA

pVBp = PAB ,

PAB, PAB are projection matrices(‘chiral and anti-chiral’ or ‘left and right’),

PABPB

C = PAC , PA

BPBC = PA

C , PABPB

C = 0

which are related toH and J ,

PAB + PAB = JAB , PAB − PAB = HAB

• Projection will be the characteristic property of DFT geometry.

Semi-covariant derivatives• We introduce master ‘semi-covariant’ derivative

DA = ∂A + ΓA + ΦA + ΦA .

• It is also useful to set

∇A = ∂A + ΓA , DA = ∂A + ΦA + ΦA .

• compatibility for the whole NS-NS sector

DAd = 0 , DAVBp = 0 , DAVBp = 0 . (cf . Dµeν a = 0)

It follows that

∇Ad = 0 , ∇APBC = 0 , ∇APBC = 0 , (cf .∇µgνλ = 0)

• Spin connections

ΦApq = VBp∇AVBq , ΦApq = VB

p∇AVBq ,

• Torsion free conection ,Γ0

[ABC] = 0 ,

in determined in terms of basic geometrical variables,

Γ0CAB = 2 (P∂CPP)[AB] + 2

`P[A

DPB]E − P[A

DPB]E´ ∂DPEC

− 4D−1

`PC[APB]

D + PC[APB]D´`∂Dd + (P∂EPP)[ED]

´,

cf . re-derived by Hohm & Zwiebach (1112.5296).

• General torsionful conection ,

ΓCAB = Γ0CAB + ∆CAB ,

As in SUGRA, the torsion can be constructed from the bi-spinorial objects.

Projection-aided covariant derivatives

“semi-covariant derivative” :

combined with the projections , we can generate various covariant quantities:

Examples:

• For O(D,D) tensors:

PCDPA

B∇DTB , PCDPA

B∇DTB ,

PAB∇ATB , PAB∇ATB , Divergences ,

PABPCD∇A∇BTD , PABPC

D∇A∇BTD . Laplacians

• Rule: need opposite chirality or contraction


• For Spin(1,D−1)× Spin(D−1, 1) tensors:

DpTq , DpTq ,

DpTp , DpTp ,

DpDpTq , DpDpTq ,

where we setDp := VA

pDA , Dp := VApDA .

These are the pull-back of the previous results using the double-vielbeins.


• Dirac operators for fermions, ρα, ψαp , ρ′α, ψ′αp :

γpDpρ = γADAρ , γpDpψp = γADAψp ,

Dpρ , Dpψp = DAψ

A ,

γ pDpρ′ = γADAρ

′ , γ pDpψ′p = γADAψ

′p ,

Dpρ′ , Dpψ

′p = DAψ′A ,


• For Spin(1,D−1)× Spin(D−1, 1) bi-fundamental spinors, Cαβ :

γADAC , DACγA .

• Further defineD+C := γADAC + γ(D+1)DACγA ,

D−C := γADAC − γ(D+1)DACγA .

• Especially for the torsionless case, the corresponding operators are nilpotent

(D0+)2C ≡ 0 , (D0

−)2C ≡ 0 ,

• The field strength of the R-R potential, Cαα, is then defined by

F := D0+C .

Curvatures 1105.6294

• We define, as for a key quantity in our formalism, cf . [Siegel; Waldram;Hohm, Zwiebach]

SABCD := 12

`RABCD + RCDAB − ΓE

ABΓECD´.

This is Not covariant tensor, but contracting with projection operators, we canobtain covariant quatities.

• Rank two-tensor:

PIAPJ

BSAB , where SAB := SCACB

• Scalar curvature: defines the Lagrangian for NS-NS sector

(PABPCD − PABPCD)SACBD

• There is no covariant rank 4 tensor.

• Type II (N = 2) D = 10 Supersymmetric Double Field Theory

cf .N = 1 D = 10 SDFT Park, Lee, IJ, 1112.0069

Type II N = 2 D = 10 SDFT

• Lagrangian (full order of fermions ):

LType II = e−2dh

18 (PABPCD − PABPCD)SACBD + 1

2 Tr(FF)− iρFρ′ + iψpγqF γ pψ′q

+ i 12 ργ

pD?p ρ− iψpD?p ρ− i 12 ψ

pγqD?qψp − i 12 ρ′γ pD′?p ρ′ + iψ′pD′?p ρ′ + i 1

2 ψ′pγ qD′?q ψ′p

iwhere F αα denotes the charge conjugation, F := C−1

+ FT C+.

• DA in SACBD, D?A and D′?A are defined by their own torsionful connection ,

• The torsions are determined to satisfy usual 1.5 formalism ,

δLSDFT = δΓABC × 0 .

• The Lagrangian is pseudo : self-duality of the R-R field strength needs to beimposed by hand, just like the ‘democratic’ type II SUGRA Bergshoeff, et al.“

1− γ(D+1)” `F − i 1

2ρρ′ + i 1

2γpψqψ

′pγ

q´ ≡ 0 .

Type II N = 2 D = 10 SDFT• N = 2 Local SUSY (full order of fermions ):

δεd = −i 12 (ερ+ ε′ρ′) ,

δεVAp = iVAq(ε′γqψ

′p − εγpψq) ,

δεVAp = iVAq(εγqψp − ε′γpψ

′q) ,

δεC = i 12 (γpεψ′p − ερ′ − ψpε

′γ p + ρε′) + Cδεd − 12 (VA

q δεVAp)γ(d+1)γpCγ q ,

δερ = −γpDpε+ i 12γ

pε ψ′pρ′ − iγpψqε′γqψ

′p ,

δερ′ = −γ pD′pε′ + i 1

2 γpε′ ψpρ− iγ qψ′pεγ

pψq ,

δεψp = Dpε+ (F − i 12γ

qρ ψ′q + i 12ψ

q ρ′γq)γpε′ + i 1

4εψpρ+ i 12ψpερ ,

δεψ′p = D′pε′ + (F − i 1

2 γqρ′ψq + i 1

2ψ′qργq)γpε+ i 1

4ε′ψ′pρ

′ + i 12ψ′pε′ρ′ .

D is also defined by its own torsionful connection.

• The action is invariant up to the self-duality.

4. Parametrization and gauge fixing

Parametrization

• Assuming that the upper half blocks are non-degenerate, the double-vielbeintakes the most general form,

VAp = 1√2

„(e−1)p

µ

(B + e)νp

«, VAp = 1√

2

„(e−1)p

µ

(B + e)νp

«.

Here eµp and eν p are two copies of the D-dimensional vielbein correspondingto the same spacetime metric,

eµpeν qηpq = −eµ peν qηpq = gµν .

• cf . Another parametrization may provide a natural explanation fornon-geometric flux c.f. Andriot et al. (München)

• From now on, we take the former parametrization and impose ∂∂xµ≡ 0.

Parametrization: Reduction to D dimension

• This reduces (S)DFT to generalized geometryHitchin; Grana, Minasian, Petrini, Waldram

• For example, the O(D,D) covariant Dirac operators become√

2γADAρ ≡ γm `∂mρ+ 14ωmnpγ

npρ+ 124 Hmnpγ

npρ− ∂mφρ´,

√2γADAψp ≡ γm `∂mψp + 1

4ωmnpγnpψp + ωmpqψ

q + 124 Hmnpγ

npψp + 12 Hmpqψ

q − ∂mφψp´,

√2VA

pDAρ ≡ ∂pρ+ 14ωpqrγ

qrρ+ 18 Hpqrγ

qrρ ,

√2DAψ

A ≡ ∂ pψp + 14ωpqrγ

qrψp + ωppqψ

q + 18 Hpqrγ

qrψp − 2∂pφψp .

DFT is locally equivalent to generalized geometry.

Diagonal gauge fixing and Reduction to SUGRA

• Further, we may choose a diagonal gauge:

eµp = eµ p

• This breaks the pair of local Lorentz symmetry,

Spin(1,D−1)× Spin(D−1, 1) =⇒ Spin(1,D−1)D .

• And it reduces SDFT to SUGRA:

N = 2 D = 10 SDFT =⇒ 10D type II democratic SUGRABergshoeff, et al.; Coimbra, Strickland-Constable, Waldram

IIA↔ IIB

• In order to preserve the diagonal gauge, eµp ≡ eµ p, the O(D,D) transformationrule is modified.

• A compensating local Lorentz transformation, Lqp , SL

αβ ∈ Pin(D−1, 1)R ,

must be accompanied:

VAp −→ MA

BVBqLq

p , γ qLqp = S−1

L γ pSL ,

• All the barred indices are now to be rotated. Consistent with Hassan

• If and only if det(L) = −1, the modified O(D,D) rotation flips the chiralityof the theory, since

γ(D+1)SL = det(L) SLγ(D+1) .

Thus, the mechanism above naturally realizes the exchange of type IIA and IIBsupergravities under O(D,D) T-duality.

Summary

• SDFT reformulates and unifies type II supergravities in very simple form indemocratic fashion.

• SDFT manifests simultaneously O(10, 10) T-duality and all gaugesymmetries.

• DFT opens up the new stringy differential geometry beyond Riemann, whereall NS-NS sector are all basic geometric objects.

• Ordinary SUGRA≡ Gauge fixed SDFTSpin(1, 9)L×Spin(9, 1)R → Spin(1, 9)D.

• The gauge fixing modifies the O(D,D) transformation rule and generate theexchange of IIA and IIB theory.

Outlook and extensions• More than reformulation

e.g. Non-geometric background [Andriot, Hohm, Larfors, Lust, Patalon]

Origin of Gauged supergravities [Aldazabal, et. al. Guissbuhler, Berman et. al.]

Noncommutativity and nonassociativity [Lust, Blumenhagen; Zwiebach, Hohm]

• Double Sigma Models [Hull; Berman, Copland, Thompson; Nibbelink, Patalong]

• U-duality, M-theory extension [Berman, Copland, Godazgar, Perry, Thompson; West;

Coimbra, Strickland-Constable, Waldram]

• Double D-branes [Albertsson, Dai, Kao, Lin; Jensen]

• Heterotic formulation [Andriot; Hohm, Kwak]

• Massive Type II [ Hohm, Kwak]

etc...

Related similar works

• N=1 Supersymmetric DFT [ Hohm, Kwak]

• DFT geometry [ Hohm, Zwiebach]

• Generailzed goemetry: type II SUGRA [ Coimbra, Strickland-Constable, Waldram]

Conclusion

DFT may be a proper description for string effective theory including“stringy effect”.

Conclusion

Thank you.

Appendix: Reduction of RR field

• For the R-R sector in type II SDFT, we may parameterize the potential as

C ≡` 1

2

´ D+24P′

p1p! Ca1a2···apγ

a1a2···ap

and obtain the field strength,

F := D0+C ≡

` 12

´ D4P′

p1

(p+1)! Fa1a2···ap+1γa1a2···ap+1

whereP′

p denotes the odd p sum for type IIA and even p sum for type IIB, and

Fa1a2···ap = p`D[a1Ca2···ap] − ∂[a1φ Ca2···ap]

´+ p!

3!(p−3)! H[a1a2a3Ca4···ap]

• In fact, the pair of nilpotent differential operators, D0+ and D0

−, reduce to anexterior derivative and its dual derivative respectively,

D0+ =⇒ d + (H − dφ)∧D0− =⇒ ∗ [ d + (H − dφ)∧ ] ∗

Appendix: Torsions

ΓABC = Γ0ABC + i 1

3 ργABCρ− 2iργBCψA − i 13 ψ

pγABCψp + 4iψBγAψC

+i 13 ρ′γABCρ

′ − 2iρ′γBCψ′A − i 1

3 ψ′pγABCψ

′p + 4iψ′BγAψ

′C .

and D?A and D′?A are defined by their own connection,

Γ?ABC =ΓABC − i 1196 ργABCρ+ i 5

4 ργBCψA + i 524 ψ

pγABCψp − 2iψBγAψC + i 52 ρ′γBCψ

′A ,

Γ′?ABC =ΓABC − i 1196 ρ′γABCρ

′ + i 54 ρ′γBCψ

′A + i 5

24 ψ′pγABCψ

′p − 2iψ′BγAψ

′C + i 5

2 ργBCψA .

Appendix: 10D N = 1 SUGRA 1112.0069

• From D = 11 SUGRA by Cremmer, Julia & Scherk with ansatz,

EMA =

e−

13φeµa 00 e

23φ

!,

Aµνλ = 0 , Aµν11 = 12 Bµν .

Ψa = 16 2

14 e

16φ`5ψa − γabψ

b − γaρ´,

Ψz = − 13 2

14 e

16φ (ρ+ γaψa) ,

γ(10)ψa = ψa , γ(10)ρ = −ρ . γ(10)ε = ε ,

we can derive theN = 1 10D SUGRA. cf. Chamseddine, Bergshoeff et al.Consistent with Coimbra, Strickland-Constable & Waldram.

Appendix: 10D N = 1 SUGRA 1112.0069• Action with full fermion order

L10D = e× e−2φhR + 4∂µφ∂µφ− 1

12 HλµνHλµν

+i2√

2ργm[∂mρ+ 14 (ω + 1

6 H)mnpγnpρ]− i4

√2ψp[∂pρ+ 1

4 (ω + 12 H)pqrγ

qrρ]

−i2√

2ψpγm[∂mψp + 14 (ω + 1

6 H)γnpψp + ωmpqψq − 1

2 Hmpqψq]

+ 124 (ψqγmnpψq)(ψ

rγmnpψr)− 148 (ψqγmnpψq)(ργ

mnpρ)i.

• Supersymmetry

δεφ = i 12 ε(ρ+ γaψa) , δεea

µ = iεγaψµ , δεBµν = −2iεγ[µψν] ,

δερ = − 1√2γa[∂aε+ 1

4 (ω + 16 H)abcγ

bcε− ∂aφε]

+i 148 (ψdγabcψd)γabcε+ i 1

192 (ργabcρ)γabcε+ i 12 (εγ[aψb])γ

abρ ,

δεψa = 1√2[∂aε+ 1

4 (ω + 12 H)abcγ

bcε]

−i 12 (ρε)ψa − i 1

4 (ρψa)ε+ i 18 (ργbcψa)γ

bcε+ i 12 (εγ[bψc])γ

bcψa .

Modified O(D,D) Transformation Rule After The Diagonal Gauge Fixing

d −→ d

VAp −→ MA

B VBp

VAp −→ MA

B VBq Lq

p

Cαα , Fαα −→ Cαβ(S−1L )β α , F αβ(S−1

L )β α

ρα −→ ρα

ρ′α −→ (SL)αβρ′β

ψαp −→ (L−1)pqψαq

ψ′αp −→ (SL)αβψ′βp

• All the barred indices are now to be rotated. Consistent with Hassan

Appendix: Double field Yang-Mills theory 1102.0419

• With the semi-covariant derivative, we may construct DFT for YM:

FAB := ∇AVB −∇BVA − i [VA,VB] , VA =

„φλ

Aµ + Bµνφν

«.

Covariant quantity: PACPB

DFCD ,

Action:

SYM =

ZΣD

e−2d Tr“

PABPCDFACFBD

”≡Z

dxD√−ge−2φTr“

fµν fµν + 2DµφνDµφν + 2DµφνDνφµ + 2i fµν [φµ, φν ]

−[φµ, φν ][φµ, φν ] + 2 (fµν + i[φµ, φν ]) Hµνσφσ + HµνσHµντφ

σφτ”.

• Similar to topologically twisted Yang-Mills, but differs in detail.

• Curved D-branes are known to convert adjoint scalars into one-form,φa → φµ, Bershadsky

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