Stringy Differential Geometry, beyond Riemann€¦ · Stringy differential geometry We propose a novel differential geometry which • enables us to rewrite the low energy effective

Stringy Differential Geometry, beyond Riemann

Imtak Jeon

Sogang University, Seoul

12 July. 2011

Talk is based on works

in collaboration with Jeong-Hyuck Park and Kanghoon Lee

• Differential geometry with a projection:

Application to double field theory

JHEP 1104:014 (2011), arXiv:1011.1324

• Double field formulation of Yang-Mills theory

Phys.Lett.B701:260-264 (2011) , arXiv:1102.0419

• Stringy differential geometry, beyond Riemann

arXiv: 1105.6294

Introduction

• In Riemannian geometry, the fundamental object is the metric, gµν .

• String theory puts gµν , Bµν and φ on an equal footing.

• This may suggests the existence of a veiled unifying description of them,beyond Riemann.

Introduction

Symmetry

• guides the structure of Lagrangians and organizes the physical laws into simpleforms.

• for example, in Maxwell theory,

• U(1) gauge symmetry forbids m2AµAµ

• Lorentz symmetry unifies the original 4 eqs into 2.

Introduction

• Essence of Riemannian geometry: Diffeomorphism

• ∂µ −→ ∇µ = ∂µ + Γµ

• ∇λgµν = 0 −→ Γλµν = 1

2 gλρ(∂µgνρ + ∂νgµρ − ∂ρgµν)

• Curvature, [∇µ,∇ν ], −→ Rg.

• Main purpose : Generalization to a geometry for stringy theory.

• Key property in string theory : T-duality

Outline

• T-duality in string theory

• String effective action and Double Field Theory

• Stringy differential geometry as underlying geometry of DFT.

• Local inertial frame

• Non-Abelian YM gauge field in DFT.

Closed string on a Toroidal background

• For closed string wraping around a circle, the modes expansion is

X = XL + XR ,

XL(σ+) = 1

2 (x + x) + 12 (p + w)σ+ + · · · ,

XR(σ−) = 1

2 (x− x) + 12 (p− w)σ− + · · · .

where

σ± := τ ± σ, p : momentum mode , w : winding mode ,x : center of mass , x is introduced .

• Physical condition (Virasoro constraint) implies the level matching and theon-shell condition,

L0 − L0 = 0L0 + L0 = 2 ⇐⇒

N − N − p · w = 0M

2 = p2 + w2 + 2(N + N − 2)

T-duality

• T-duality transformation is

XµL + Xµ

R −→ XµL − Xµ

R ,

such that the two pairs, (x, p) and (x, w), are exchanged by each other,

(x, x, p, w) −→ (x, x, w, p) ,

Then, the level matching condition and the mass spectrum is invariant.

• On d-dimensional toroidal background, the T-duality is realized as O(d, d, Z)

transformation preserving J =

„0 11 0

«

2d×2d.

• xi : Conjugate to the momenta pi

xi : Conjugate to the winding number wi

It is natural to introduce the additional coordinate xi

• Closed string field theory treats momenta and winding modes symmetrically.So, target spacetime fields are also depend on both x and x.: Φ(xi, xi)[Kugo,Zwiebach 1992]

Effective theory

• Describes a gravity : gµν , Bµν , φ are on an equal footing completing themassless sector.

• Low energy effective action of them:

Seff. =

ZdxD√

−ge−2φ `Rg + 4∂µφ∂µφ− 1

12 HλµνHλµν´

• Diffeomorphism and one-form gauge symmetry are manifest

xµ→ xµ + δxµ , Bµν → Bµν + ∂µΛν − ∂νΛµ .

Effective theory

• Describes a gravity : gµν , Bµν , φ are on an equal footing completing themassless sector.

• Low energy effective action of them:

Seff. =

ZdxD√

−ge−2φ `Rg + 4∂µφ∂µφ− 1

12 HλµνHλµν´

• Though not manifest, if there is isometry, this enjoys T-duality which mixesgµν , Bµν , φ (Buscher )

• Include the x dependence. −→ double field theory

Double field theory (DFT)• Hull and Zwiebach , later with Hohm constructed action which has explicit

T-duality,

SDFT =

Zdy2D e−2d

hH

AB `4∂A∂Bd − 4∂Ad∂Bd + 1

8 ∂AHCD∂BHCD

−12 ∂AH

CD∂CHBD´

+ 4∂AHAB∂Bd − ∂A∂BH

ABi.

• d is double field theory ’dilaton’ given by

e−2d =√−ge−2φ

and HAB is 2D× 2D matrix, ‘generalized metric’

HAB =

„gµν

−gµκBκσ

Bρκgκν gρσ − BρκgκλBλσ

«.

• Spacetime dimension is ‘formally’ doubled from D to 2D,

xµ→ yA = (xµ, xν) , ∂µ → ∂A = (∂µ, ∂ν) .

Double field theory (DFT)

• Indices, A, B, C, · · · , are 2D-dimensional vector indices, which can be loweredor raised by O(D, D) metric JAB.

JAB :=

„0 11 0

«.

• Under global O(D, D) rotation, L ∈ O(D, D),

HAB(y) −→ LACLB

DHCD(y) , d(y) −→ d(y) ,

manifest that DFT action is invariant.

• Note that O(D, D) T-duality is background independent.

Double field theory (DFT)

• By ’Level matching condition,’

DFT action ≡ D-dimensional closed string effective action

• It possesses a gauge symmetry, say double gauge symmetry, through’generalized Lie derivative’;

double gauge symmetry = diffeomorphism + 1-form gauge symmety

Level matching constraint

• Level matching condition for the massless sector,

p · w ≡ 0 ⇐⇒∂2

∂xµ∂xµ= ∂A∂A

≡ 0 .

• Strong constraint : all the fields and the gauge parameters as well as all of theirproducts should be annihilated by ∂2 = ∂A∂A

∂2Φ ≡ 0 , ∂AΦ1∂AΦ2 ≡ 0 ⇐⇒ ∂ ≡ 0 .

• DFT is restricted in the strong constraint.

• Actually meaning that theory is not truly doubled: we can impose ∂ ≡ 0 byO(D, D) rotation.

• DFT is reorganization of effective action: Upon the level matching constraint,

SDFT =⇒ Seff. =

ZdxD√

−ge−2φ“

Rg + 4(∂φ)2−

112 H2

”.

Double gauge symmetry

• Introducing an unifying doubled gauge parameter,

XA = (Λµ, δxν)

• diffeomorphism and 1-form gauge transformation is expressed in unifiedfashion, upon the level matching constraint,

δXHAB ≡ XC∂CHAB + 2∂[AXC]HC

B + 2∂[BXC]HAC ,

δXè−2d´

≡ ∂A`XAe−2d´

.

• In fact, these coincide with the generalized Lie derivative,

δXHAB = LXHAB , δX(e−2d) = LX(e−2d) = −2(LXd)e−2d.

Generalized Lie derivative

• Definition, Siegel, Courant, Grana ...

LXTA1···An := XB∂BTA1···An + ω∂BXBTA1···An +nX

i=1

2∂[Ai XB]TA1···Ai−1B

Ai+1···An .

• cf. ordinary one,

LXTA1···An := XB∂BTA1···An + ω∂BXBTA1···An +nX

i=1

∂Ai XBTA1···Ai−1B

Ai+1···An .

• Definition of tensor(density) in DFT : ’double gauge covariant’ quantity

δXTA1A2···An = LXTA1A2···An .

• HAB: rank 2 generalized tensor, e−2d: weight 1 generalized scalar,

Algebra of generalized Lie derivative

• Commutator of generalized Lie derivatives is closed, up to the level matchingcondition, by using c-bracket,

[LX, LY ] ≡ L[X,Y]C ,

where [X, Y]C denotes C-bracket

[X, Y]AC := XB∂BYA− YB∂BXA + 1

2 YB∂AXB −12 XB∂AYB ,

known to be euqivalent the Courant bracket if dropping ∂.

Diffeomorphism & one-form gauge symmetry

• Direct computation shows that

LXHAB ≡ XC∂CHAB + 2∂[AXC]HC

B + 2∂[BXC]HAC ,

LXè−2d´

≡ ∂A`XAe−2d´

,

are symmetry of DFT action by Hull, Zwiebach and Hohm,

SDFT =

Zdy2D e−2d

R(H, d) ,

where ’generalized curvature’ is

R(H, d) = HAB `

4∂A∂Bd − 4∂Ad∂Bd + 18 ∂AH

CD∂BHCD −12 ∂AH

CD∂CHBD´

+4∂AHAB∂Bd − ∂A∂BH

AB .

• This expression may be analogous to the case of writing the scalar curvature,Rg, in terms of the metric and its derivative.




B + 2∂[BXC]HAC ,

LXè−2d´

≡ ∂A`XAe−2d´

,

are symmetry of the action by Hull, Zwiebach and Hohm,

SDFT =

Zdy2D e−2d

R(H, d) ,

where

R(H, d) = HAB `

4∂A∂Bd − 4∂Ad∂Bd + 18 ∂AH


CD∂CHBD´


AB .

• What is the underlying geometry?




B + 2∂[BXC]HAC ,

LXè−2d´

≡ ∂A`XAe−2d´

,

are symmetry of the action by Hull, Zwiebach and Hohm,

SDFT =

Zdy2D e−2d

R(H, d) ,

where

R(H, d) = HAB `

4∂A∂Bd − 4∂Ad∂Bd + 18 ∂AH


CD∂CHBD´


AB .

• connection and covariant derivative of the geometry?

Stringy differential geometry

We propose a novel differential geometry which

• enables us to rewrite the low energy effective action as a single term in a

geometrical manner,

Seff. =

Zdx2D e−2d

HABSAB ,

• treats the three objects of the massless sector in a unified way,

• manifests not only diffeomorphism and one-form gauge symmetry but also

O(D, D) T-duality,

Stringy differential geometry• Motivated by the observation that,

H =

„g−1

−g−1BBg−1 g− Bg−1B

«

is of the most general form to satisfy

HACHC

B = δAB , HAB = HBA ,

and the upper left D× D block of H is non-degenerate,

• we focus on a symmetric projection,

PABPB

C = PAC PAB = PBA ,

which is related to H by

PAB = 12 (JAB +HAB) .

• Three basic objects:JAB , PAB , d .


• We postulate a “semi-covariant" derivative,∇A,

∇CTA1A2···An= ∂CTA1A2···An−ωΓBBCTA1A2···An +

nX

i=1

ΓCAiBTA1···Ai−1BAi+1···An .

• In particular,

∇C(e−2d) = ∂Ce−2d− ΓB

BCe−2d = −2(∇Cd)e−2d

=⇒ ∇Cd := ∂Cd + 12 Γ

BBC


• We demand the following compatibility conditions,

∇AJBC = 0 , ∇APBC = 0 , ∇Ad = 0 ,

as for the unifying description of the massless modes(cf. ∇λgµν = 0 in Riemannian geometry).

• Further we require,

ΓCAB + ΓCBA = 0 , ΓABC + ΓCAB + ΓBCA = 0 .


• Then, we may replace ∂A by∇A in LX and also in [X, Y]AC,

LXTA1···An = XB∇BTA1···An + ω∇BXBTA1···An +

Pni=1 2∇[Ai XB]TA1···Ai−1

BAi+1···An ,

[X, Y]AC =XB∇BYA

− YB∇BXA + 1

2 YB∇

AXB −12 XB∇

AYB .

• cf. In Riemannian geometry, torsion free condition implies

LXTµ1···µn = Xν∇νTµ1···µn + ω∇νXνTµ1···µn +

Pni=1∇µi X

νTµ1···µi−1νµi+1···µn ,

[X, Y]µ = Xν∇νYµ

− Yν∇νXµ .


• Explicitly, the connection is

ΓCAB = 2(P∂CPP)[AB] + 2`P[A

DPB]E− P[A

DPB]E´

∂DPEC

−4

D−1

`PC[APB]

D + PC[APB]D´`

∂Dd + (P∂EPP)[ED]

´.

where P is the complementary projection,

P = 12 (1−H) .

and P and P are orthogonal,

PP = PP = 0 .

• Is this derviativ∇A double gauge covariant?


• For usefulness, we set

PCABDEF := PC

DP[A[EPB]

F] + 2D−1 PC[APB]

[EPF]D ,

PCABDEF := PC

DP[A[EPB]

F] + 2D−1 PC[APB]

[EPF]D ,

which satisfy

PCABDEF = PDEFCAB = PC[AB]D[EF] ,

PCABDEFPDEF

GHI = PCABGHI ,

PA

ABDEF = 0 , PABPABCDEF = 0 , etc.

• The connection belongs to the kernel of these rank six-projectors,

PCABDEFΓDEF = 0 , PCAB

DEFΓDEF = 0 .


• Under double-gauge transform, δXPAB = LXPAB and δXd = LXd, thediffeomorphism and the one-form gague tranform,we obtain

(δX−LX)ΓCAB ≡ 2ˆ(P+P)CAB

FDE− δ F

C δ DA δ E

B˜∂F∂[DXE] ,

and

(δX−LX)∇CTA1···An≡

X

i

2(P+P)CAiBFDE∂F∂[DXE]TA1···B···An .

• Hence, these are not double-gauge covariant,

δX = LX .


• However, the characteristic property of our derivative,∇A, is that, combinedwith the projections, it can generate various O(D, D) and double-gaugecovariant quantities:

PCDPA1

B1 PA2B2 · · · PAn

Bn∇DTB1B2···Bn ,

PCDPA1

B1 PA2B2 · · ·PAn

Bn∇DTB1B2···Bn ,

PAB∇ATB , PAB

∇ATB ,

PABPC1D1 · · · PCn

Dn∇A∇BTD1···Dn ,

PABPC1D1 · · ·PCn

Dn∇A∇BTD1···Dn .

• This suggests us to call∇A as semi-covariant derivative .

Curvature

• The usual curvature,

RCDAB = ∂AΓBCD − ∂BΓACD + ΓACEΓBED − ΓBC

EΓAED ,

satisfying

[∇A,∇B]TC1C2···Cn = −ΓDAB∇DTC1C2···Cn +

nX

i=1

RCiDAB TC1···Ci−1D

Ci+1···Cn ,

is NOT double-gauge covariant,

δXRABCD = LXRABCD .

It satisfy RABCD = R[AB][CD], but

RABCD = RCDAB .

Generalized curvature

• Instead, we define, as for a key quantity in our formalism,

SABCD := 12

`RABCD +RCDAB − ΓE

ABΓECD´

.

• This can be read off from the commutaor,

PIAPJ

B[∇A,∇B]TC ≡ 2PIAPJ

BSCDABTD .

• It can be shown, by brute force computation, to satisfy

• just like the Riemann curvature,

SABCD = 12 (S[AB][CD] + S[CD][AB]) ≡ SABCD , SA[BCD] = 0 ,

• and further

P AI P B

J P CK P D

L SABCD ≡ 0 , P AI P B

J P CK P D

L SABCD ≡ 0 , etc.

Generalized curvature

• SABCD is not double-gauge covariant.

• Under the double-gauge transformations, we get

(δX − LX)SABCD ≡ 4∇A

h(P+P)BCD

EFG∂E∂[FXG] .i

.

• Nevertheless, contracting indices we can obtain covariant quantities.

Covariant curvature

By settingSAB= SBA:= SC

ACB ,

which turns out to be traceless,SA

A ≡ 0 ,

and contracting with projection operators, we get

• Double-gauge covariant rank two-tensor,

PIAPJ

BSAB : generalized curvature tensor

• Double-gauge covariant scalar,

HABSAB : generalized curvature scalar

Reproduction of DFT

• Natural DFT action is proposed by

SDFT =

Zdy2D e−2d

HABSAB ,

• In fact, the covariant scalar constitutes the effective action as

HABSAB ≡ Rg + 4φ− 4∂µφ∂µφ− 1

12 HλµνHλµν .

• It also agrees with Hull, Zwiebach and Hohm,

HABSAB ≡ H

AB `4∂A∂Bd − 4∂Ad∂Bd + 1

8 ∂AHCD∂BHCD −

12 ∂AH

CD∂CHBD´


AB .

Deriving Equations of motion

• It is easy to rederive the equation of motion.

• Under arbitrary infinitesimal transformations of the dilaton and the projection,we get

δSeff. ≡

Zdy2D 2e−2d

“δPABSAB − δdHABSAB + δSAB

”.

• The third therm is total derivative as

δSABCD = ∇[AδΓB]CD +∇[CδΓD]AB , ∇Ad = 0 ,

where explicitly

δΓCAB = 2P D[A P E

B]∇CδPDE + 2(P D[A P E

B] − P D[A P E

B])∇DδPEC

−4

D−1 (PC[AP DB] + PC[AP D

B] )(∂Dδd + PE[G∇GδPE

D])

−ΓFDE δ(P + P)CABFDE ,

Deriving Equations of motion

• It is easy to rederive the equation of motion.

• Under arbitrary infinitesimal transformations of the dilaton and the projection,we get

δSeff. ≡

Zdy2D 2e−2d

“δPABSAB − δdHABSAB

”.

• from the relationδP = PδPP + PδPP ,

the equations of motions are easily obtained:

P(IAPJ)

BSAB = 0 , HABSAB = 0 .

Local inertial frame and Double-vielbein

• JAB and HAB can be simultaneously diagonalized,

J =`

V V´ “

η−1 00 −η

” `V V

´t,

H =`

V V´ “

η−1 00 η

” `V V

´t.

Here η and η are two copies of the D-dimensional Minkowskian metric. Both Vand V are 2D×D matrices which we name ‘double-vielbein’.

• They must satisfy

V = PV , Vη−1Vt = P , VtJ V = η , Vt

J V = 0 ,V = PV , V η V t = −P , V t

J V = −η−1 .

• There are two copies of independent vielbein(Siegel, Tseytlin )

Local inertial frame and Double-vielbein

• Our dobule-vielbein is of the general form,

VAm = 1√2

0

@e m

µ

(B + e)νm

1

A, VAn = 1√

2

0

@e nµ

(B− e)νn

1

A ,

where Bνm = Bνλeλm and Bν

n = Bνλe λn.

• Here, eµm and eν

n are two copies of the D-dimensional vielbeincorresponding to the same spacetime metric,

eµmeνm = eµ

neνn = gµν .

• We may identify (B + e)µm and (B− e)ν

n as two copies of the vielbeinfor the winding mode coordinate, xµ, since

(B + e)µm(B + e)νm = (B− e)µ

n(B− e)νn = (g− Bg−1B)µν .

Local inertial frame and Double vielbein

• Internal symmetry group is

SO(1, D−1)× SO(D−1, 1) ,

• Taking single diagonal local Lorentz group SO(1, D−1) or SO(D−1, 1) bygauge fixing corresponds to

VAm = 1√2

„e m

µ

(B + e)νm

«, UA

m := 1√2

„(e−1)mµ

(B− e)νm

«,

or

UAm = 1√2

„e m

µ

(B + e)νm

«, VA

m := 1√2

„(e−1)mµ

(B− e)νm

«.

where we define “twins" of the double-vielbein, by exchanging eµm and eµm inV and V .

• While V and V are O(D, D) covariant, the twins are not O(D, D) covariant.

Double spin connection

• Gauging each diagonal local Lorentz symmetry, we get doubled spinconnection

ΩAmn = PABVCm∇BVC

n − PABUCm∇BUC

n .

orΩAmn = PA

BUCm∇BUCn − PA

BVCm∇BVCn .

• Upon the level matching constraint, they are expressed in terms ofD-dimensional notation,

ΩA ≡

0

@−

12 Hµ

ων −12 BνρHρ

1

A , ΩA ≡

0

@−

12 Hµ

ων −12 BνρHρ

1

A ,

Pull back to D-dimensional theory

• Double-vielbein can pull back the chiral and the anti-chiral 2D indices to themore familiar D-dimensional ones

• We pull back the double-gauge covariant rank two-tensor to obtain,

SABVAmVB

n ≡ Rmn + 2DmDnφ− 14 HmµνHn

µν + (∂λφ)Hλmn −12∇

λHλmn .

• As expected, its symmetric and the anti-symmetric parts correspond to theequations of motion of the effective action for gµν and Bµν respectively.

• Pullback of various covariant quantities

VAlDATk1 k2···kn ≡

1√2

DlTk1 k2···kn ,

VAlDATk1k2···kn ≡

1√2

DlTk1k2···kn ,

PABDATBk1···kn ≡

1√2

DlTlk1···kn −√

2 Dlφ Tlk1···kn ,

PABDATBk1···kn ≡ −

1√2

DlTlk1···kn +√

2 Dlφ Tlk1···kn ,

PABDADBTk1···kn≡

12 DµDµTk1···kn − Dµφ DµTk1···kn ,

PABDADBTk1···kn≡ −

12 DµDµTk1···kn + Dµφ DµTk1···kn ,

where we put, as for D-dimensional tensors,

Tk1k2···kn = TA1A2···An VA1 k1 VA2 k2 · · ·VAn

kn ,Tk1 k2···kn = TA1A2···An VA1

k1VA2

k2· · · VAn

kn ,Tlk1···kn = TBA1···An VB

lVA1 k1 · · ·V

Ankn ,

Tlk1···kn = TBA1···An VBlVA1

k1· · · VAn

kn ,

which are O(D, D) singlets, and we set, for Dm = (e−1)mµDµ and

Dn = (e−1)nµDµ,

Dµ := µ + (ωµ + 12 Hµ) + (ωµ −

12 Hµ) .

Symmetry structure

• Symmetry structure for the double field theory

• O(D, D) ’T-duality’

• Double-gauge symmetry

DiffeomorphismOne-form gauge symmetry for Bµν

• Commutator between so(D, D) and generalized Lie derivative does notgenerate any symmetry of double field theory.

[δh, LX] = LY , ∂CYA∂CTB1B2···Bn = 0 .

• O(D, D) transformation rotates the entire hyperplane on which DFT lives.

Application to Yang-Mills





• two copies of local Lorentz symmetry






• Yang-Mills gauge symmetry

• Apply the formalism to couple the non-Abelian Yang-Mills gauge field to theDFT action.


• We postulate a vector potential, VA, which

• is O(D, D) and double-gauge covariant,• and transforms under non-Abelian gauge group, g ∈ G,

VA −→ gVAg−1− i(∂Ag)g

−1 .


• The usual field strength,

FAB = ∂AVB − ∂BVA − i [VA,VB] ,

is YM gauge covariant, but it is NOT double-gauge covariant,

δXFAB = LXFAB .


• Instead, we consider with the semi-covariant derivative,

FAB := ∇AVB −∇BVA − i [VA,VB] = FAB − ΓCABVC .

• While this is neither YM gauge nor double-gauge covariant,

FAB −→ gFABg−1 + iΓC

AB(∂Cg)g−1 ,

δXFAB = LXFAB ,

• if projected properly, it can be covariant up to level matching condtion,

PACPB

DFCD −→ PA

CPBD

gFCDg−1 ,

δX(PACPB

DFCD) = LX(PA

CPBDFCD) .


PACPB

DFCD is DFT field strength which is fully covariant with respect to

• O(D, D) T-duality

• Gauge symmetry

• Double gauge = Diffeomorphism + one form gauge symmetry• Yang-Mills gauge

Yang-Mills action

• Our double field formulation of Yang-Mills action is

SYM = g−2YM

Zdy2D e−2d Tr

“PABPCD

FACFBD

”,

• Manifestly this action is invariant under O(D, D) T-duality, double-gauge andYang-Mills gauge transformation.

• Corresponding D-dimensional action of the Double field YM action?

Yang-Mills in components

• Decompose the vector potential into chiral and anti-chiral ones,

VA = V+A + V−A ,

V+A = PA

BVB , V−A = PA

BVB .

• Their general forms are

V+A = 1

2

0

@A+λ

(g+B)µνA+ν

1

A , V−A = 12

0

@−A−λ

(g−B)µνA−ν

1

A .


• With the field redefinition,

Aµ :=12(A+

µ + A−µ ) , φµ :=12(A+

µ − A−µ ) ,

we get a general form of double guage field

VA =

„φλ

Aµ + Bµνφν

«.

• Aµ and φν will be YM gauge connection and YM gauge covariant one-formrespectively.


• Turning off the x-dependence reduces the action to

SYM ≡ g−2YM

ZdxD √

−ge−2φ Tr“−

14 f µν fµν

”,

wherefµν := fµν − Dµφν − Dνφµ + i [φµ, φν ] + Hµνλφλ ,

and

Tr“

fµν f µν”

= Tr“

fµν f µν + 2DµφνDµφν + 2DµφνDνφµ− [φµ, φν ][φµ, φν ]

+2i fµν [φµ, φν ] + 2 (f µν + i[φµ, φν ]) Hµνσφσ + HµνσHµντφσφτ

”.

• For T-duality, we need YM gauge covariant 1-form field.

• 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

More on fully covariant quantities

• Even power of the field strength, FAB := PACPB

DFCD,

Tr“F

A1B1 FA2B1 FA2B2 FA3B2 · · · F

AnBnFA1Bn

”.

• For the Abelian group, DBI type action

det“ηAB + κ FACFB

C”

= det“ηAB + κ FCAF

CB

”,

κ is a constant.No square root is necessary since this is a scalar, not a density.

Concluding remarks

• O(D, D) T-duality, diffeomorphism, one-form gauge symmetry fixes the lowenergy effective action,

Seff. =

ZdxD e−2d

HABSAB .

• Non-Abelian Yang-Mills field is incorporated in DFT formulation.

Concluding remarks

• Yet, string theory interpretation of the YM theory is not clear.

• D-brane in DFT Albertsson, Dai, Kao, Lin

• Fermion in DFT Coimbra, Strickland-Constable, Waldram

• Extension to RR fields Hull, Kwak, Zwiebach

• Application to ‘doubled sigma model’ and generalization to M-theory are ofinterest Hull, Berman, Perry, ... ... etc...

• Concluding:Perhaps, our formalism may provide some clue to a new framework for stringtheory, beyond Riemann.

Concluding remarks

• Yet, string theory interpretation of the YM theory is not clear.

• D-brane in DFT Albertsson, Dai, Kao, Lin

• Fermion in DFT Coimbra, Strickland-Constable, Waldram

• Extension to RR fields Hull, Kwak, Zwiebach

• Application to ‘doubled sigma model’ and generalization to M-theory are ofinterest Hull, Berman, Perry, ... ... etc...

• Concluding:Perhaps, our formalism may provide some clue to a new framework for stringtheory, beyond Riemann.

Thank you.

Stringy Differential Geometry, beyond Riemann€¦ · Stringy differential geometry We propose a novel differential geometry which • enables us to rewrite the low energy effective

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