Stringy Gravity: O D completion of General Relativity

Stringy Gravity: O(D,D) completion of General Relativity

Einstein Double Field Equations

GAB = 8πGTAB

Hereafter A,B are O(D,D) indices

박정혁 (朴廷爀) Jeong-Hyuck Park Sogang University

25th International Summer Institute on Phenomenology of Elementary Particle Physics and Cosmology

19th August 2019

O(D,D) completion of General Relativity: GAB = 8πGTAB 1804.00964 (long), 1904.04705 (short), 1905.03620 (COSMO)

• General Relativity is a beautiful theory of gravity,

while modified gravities involve arbitrariness and are thus often ugly.

• Today I am going to talk about a certain modified gravity which

– has no arbitrariness;

– completes GR with Symmetry Principle, O(D,D);

– and is perhaps twice (doubly) more beautiful than GR.1

1One may quantify ‘beauty’ counting the number of symmetries.


























Introduction

– General Relativity is based on Riemannian geometry, where the only geometric andgravitational field is the Riemannian metric, gµν . Other fields are meant to be extra matter.

– On the other hand, string theory suggests to put a two-form gauge potential, Bµν , and ascalar dilaton, φ, on an equal footing along with the metric:

• They form the closed string massless (NS-NS) sector, being ubiquitous in all string theories,

∫dDx

√−ge−2φ

(Rg + 4∂µφ∂

µφ− 1

12 HλµνHλµν)

where H = dB .

This action hides O(D,D) symmetry of T-duality which transforms g,B, φ into one another. Buscher 1987

– T-duality hints at a natural extension of GR, or the O(D,D) completion of GR, in which theabove closed string massless sector constitutes the fundamental gravitational multiplet.

Double Field Theory (DFT), initiated by Siegel 1993 & Hull, Zwiebach 2009-2010, turns out toprovide a concrete realization for this idea of Stringy Gravity by manifesting O(D,D) T-duality.


Introduction

– General Relativity is based on Riemannian geometry, where the only geometric andgravitational field is the Riemannian metric, gµν . Other fields are meant to be extra matter.

– On the other hand, string theory suggests to put a two-form gauge potential, Bµν , and ascalar dilaton, φ, on an equal footing along with the metric:

• They form the closed string massless (NS-NS) sector, being ubiquitous in all string theories,

∫dDx

√−ge−2φ

(Rg + 4∂µφ∂

µφ− 1

12 HλµνHλµν)

where H = dB .

This action hides O(D,D) symmetry of T-duality which transforms g,B, φ into one another. Buscher 1987

– T-duality hints at a natural extension of GR, or the O(D,D) completion of GR, in which theabove closed string massless sector constitutes the fundamental gravitational multiplet.

Double Field Theory (DFT), initiated by Siegel 1993 & Hull, Zwiebach 2009-2010, turns out toprovide a concrete realization for this idea of Stringy Gravity by manifesting O(D,D) T-duality.


Philosophy

– To view DFT as Stringy Gravity, or the O(D,D) completion of GR, where the wholeclosed-string massless (NS-NS) sector is to be the fundamental gravitational multiplet.

The previous Lagrangian itself should be identified as a generalized scalar curvature,

Rg + 4∂µφ∂µφ− 112 HλµνHλµν ≡ S(0) : Pure Gravity

– To employ novel stringy differential geometry, beyond Riemann.

– To formulate the theory in terms of O(D,D) covariant fields only, rather than conventionalones in SUGRAs such as gµν , Bµν , φ, p-forms,...

– Consequently, DFT not only reformulates SUGRA but also unifies non-Riemannian or chiralgravities (Newton-Cartan, Carroll, Gomis-Ooguri,...).

Further, it implies, rather inevitably, modifications to General Relativity.


Plan

I. Review of covariant derivatives, ∇A, and curvatures, SAB , S(0) in DFT.

: classification of the most general DFT backgrounds.

II. Derivation of the Einstein Double Field Equations, GAB = 8πGTAB , as the unifying singleexpression for the closed-string massless NS-NS sector:

GAB = 4P[AC PB]

DSCD − 12JABS(0) , ∇AGAB = 0 ,

TAB = e2d(

8PC[APB]DδLmatterδHCD

− 12J

AB δLmatterδd

), ∇AT AB = 0 .

III. i) Applicatioin to Cosmology, O(D,D) completion of the Friedmann equations;

ii) Spherical (regular) solution to GAB = 8πGTAB for a ‘stringy star’

Collaborators:

Stephen Angus, Kyungho Cho, Guilherme Franzmann, Shinji Mukohyama, Kevin Morand (recent)

as well as Imtak Jeon and Kanghoon Lee (earlier)


Notation

Index Representation Metric (raising/lowering indices)

A,B, · · · ,M,N, · · · O(D,D) vector JAB =

0 1

1 0

p, q, · · · Spin(1,D−1

)L vector ηpq = diag(−+ + · · ·+)

α, β, · · · Spin(1,D−1

)L spinor Cαβ , (γp)T = CγpC−1

p, q, · · · Spin(D−1, 1

)R vector ηpq = diag(+−− · · ·−)

α, β, · · · Spin(D−1, 1

)R spinor Cαβ , (γp)T = CγpC−1

– The twofold local Lorentz symmetries indicate two distinct locally inertial frames for theleft-moving and the right-moving closed string sectors separately ⇒ Unification of IIA and IIB.


Closed-string massless sector as ‘Stringy Graviton Fields’

The stringy graviton fields consist of the DFT dilaton, d , and DFT metric, HMN :

HMN = HNM , HKLHM

NJLN = JKM .

Combining JMN and HMN , we get a pair of symmetric projection matrices,

PMN = PNM = 12 (JMN +HMN ) , PL

M PMN = PL

N ,

PMN = PNM = 12 (JMN −HMN ) , PL

M PMN = PL

N ,

which are orthogonal and complete,

PLM PM

N = 0 , PMN + PM

N = δMN .

Further, taking the “square roots" of the projectors,

PMN = VMpVN

qηpq , PMN = VMpVN

q ηpq ,

we get a pair of DFT vielbeins satisfying their own defining properties,

VMpV Mq = ηpq , VMpV M

q = ηpq , VMpV Mq = 0 ,

or equivalentlyVM

pVNp + VMpVNp = JMN .


Solution to the defining relation, HMN = HNM , HKLHM

NJLN = JKM ?

HMN =

g−1 −g−1B

Bg−1 g − Bg−1B

or HMN = JMN =

0 1

1 0

The left one is well-known: it contains a Riemannian metric and reduces DFT to SUGRA.

The right one admits no Riemannian nor SUGRA interpretation.

Thus, DFT describes not only Riemannian SUGRA but also non-Riemannian novel geometries.


Classification Kevin Morand & JHP 1707.03713

The most general form of the DFT metric, HMN = HNM , HKLHM

NJLN = JKM , is

characterized by two non-negative integers, (n, n), 0 ≤ n + n ≤ D:

HAB =

Hµν −HµσBσλ + Yµi X iλ − Yµı X ıλ

BκρHρν + X iκYνi − X ıκYνı Kκλ − BκρHρσBσλ + 2X i

(κBλ)ρYρi − 2X ı(κBλ)ρYρı

=

1 0

B 1

H Yi (X i )T − Yı(X ı)T

X i (Yi )T − X ı(Yı)T K

1 −B

0 1

i) Symmetric and skew-symmetric fields : Hµν = Hνµ, Kµν = Kνµ, Bµν = −Bνµ ;

ii) Two kinds of zero eigenvectors: with i, j = 1, 2, · · · , n & ı, = 1, 2, · · · , n,

HµνX iν = 0 , Hµν X ıν = 0 , KµνYνj = 0 , Kµν Yν = 0 ;

iii) Completeness relation: HµρKρν + Yµi X iν + Yµı X ıν = δµν .

• Note HAA = 2(n − n) and O(D,D)

O(t+n,s+n)×O(s+n,t+n)having dimension D2 − (n − n)2 .



The most general form of the DFT metric, HMN = HNM , HKLHM

NJLN = JKM , is

characterized by two non-negative integers, (n, n), 0 ≤ n + n ≤ D:

HAB =

Hµν −HµσBσλ + Yµi X iλ − Yµı X ıλ

BκρHρν + X iκYνi − X ıκYνı Kκλ − BκρHρσBσλ + 2X i

(κBλ)ρYρi − 2X ı(κBλ)ρYρı

=

1 0

B 1

H Yi (X i )T − Yı(X ı)T

X i (Yi )T − X ı(Yı)T K

1 −B

0 1

i) Symmetric and skew-symmetric fields : Hµν = Hνµ, Kµν = Kνµ, Bµν = −Bνµ ;

ii) Two kinds of zero eigenvectors: with i, j = 1, 2, · · · , n & ı, = 1, 2, · · · , n,

HµνX iν = 0 , Hµν X ıν = 0 , KµνYνj = 0 , Kµν Yν = 0 ;

iii) Completeness relation: HµρKρν + Yµi X iν + Yµı X ıν = δµν .

• Note HAA = 2(n − n) and O(D,D)

O(t+n,s+n)×O(s+n,t+n)having dimension D2 − (n − n)2 .



I. (n, n) = (0, 0) corresponds to the Riemannian case or Generalized Geometry à la Hitchin :

HMN ≡

g−1 −g−1B


, e−2d ≡√|g|e−2φ Giveon, Rabinovici, Veneziano ’89, Duff ’90

II. Generically, on worldsheet, string becomes chiral and anti-chiral over the n and n dimensions:

X iµ ∂+xµ(τ, σ) ≡ 0 , X ıµ ∂−xµ(τ, σ) ≡ 0 ,

as can be shown using ‘doubled-yet-gauged’ string action.

– Non-Riemannian examples include

• (1, 0) Newton-Cartan gravity (ds2 = −c2dt2 + dx2, limc→∞

g−1 is finite & degenerate)

• (1, 1) Gomis-Ooguri non-relativistic string Melby-Thompson, Meyer, Ko, JHP 2015

• (D−1, 0) ultra-relativistic Carroll gravity

• (D, 0) is uniquely given byH = J : maximally non-Riemannian.This is the completely O(D,D)-symmetric vacuum of DFT. It naturally realizes Siegel’s chiral string.

“Spacetime emerges after SSB of O(D,D), identifying gµν ,Bρσ as Goldstone bosons. "Berman, Blair, and Otsuki 2019

Further, taken as an internal space, it gives a ‘moduli-free’ (Scherk-Schwarz twistable) Kaluza-Kleinreduction of DFT, in fact, to heterotic supergravity/string. Cho, Morand, JHP 2018



I. (n, n) = (0, 0) corresponds to the Riemannian case or Generalized Geometry à la Hitchin :

HMN ≡

g−1 −g−1B


, e−2d ≡√|g|e−2φ Giveon, Rabinovici, Veneziano ’89, Duff ’90

II. Generically, on worldsheet, string becomes chiral and anti-chiral over the n and n dimensions:

X iµ ∂+xµ(τ, σ) ≡ 0 , X ıµ ∂−xµ(τ, σ) ≡ 0 ,

as can be shown using ‘doubled-yet-gauged’ string action.

– Non-Riemannian examples include

• (1, 0) Newton-Cartan gravity (ds2 = −c2dt2 + dx2, limc→∞

g−1 is finite & degenerate)

• (1, 1) Gomis-Ooguri non-relativistic string Melby-Thompson, Meyer, Ko, JHP 2015

• (D−1, 0) ultra-relativistic Carroll gravity

• (D, 0) is uniquely given byH = J : maximally non-Riemannian.This is the completely O(D,D)-symmetric vacuum of DFT. It naturally realizes Siegel’s chiral string.

“Spacetime emerges after SSB of O(D,D), identifying gµν ,Bρσ as Goldstone bosons. "Berman, Blair, and Otsuki 2019

Further, taken as an internal space, it gives a ‘moduli-free’ (Scherk-Schwarz twistable) Kaluza-Kleinreduction of DFT, in fact, to heterotic supergravity/string. Cho, Morand, JHP 2018


Semi-covariant formalism Imtak Jeon, Kanghoon Lee, & JHP 2010, 2011

• Semi-covariant derivative :

∇CTA1A2···An := ∂CTA1A2···An − ωT ΓBBCTA1A2···An +

n∑i=1

ΓCAiBTA1···Ai−1BAi+1···An ,

for which the ‘DFT-Christoffel’ connection can be uniquely fixed,

ΓCAB =2(P∂C PP)[AB]+2(

P[AD PB]

E−P[AD PB]

E)∂D PEC−

4D−1

(PC[APB]

D +PC[APB]D)(∂D d+(P∂E PP)[ED]

)

by demanding compatibility with JAB ,HAB , d, torsionless condition, and projection property,

∇APBC = ∇APBC = ∇Ad = 0 , L∂ξ = L∇ξ ⇔ Γ[ABC] = 0 , (P + P)ABCDEF ΓDEF = 0 ,

where multi-indexed projectors are

PABCDEF := PA

DP[B[E PC]

F ] + 2PM

M−1PA[BPC]

[E PF ]D , same for PABCDEF with PAB ↔ PAB .

The “generalized Lie derivative" (DFT-diffeomorphism generator) is

LξTA1···An := ξC∂CTA1···An + ωT ∂Cξ

CTA1···An +n∑

i=1

(∂Ai ξC − ∂C

ξAi )TA1···C···An ,

and in particular,LξHAB = 8P(A

[CPB)D]∇CξD , Lξd = − 1

2∇AξA.

∗ There are no normal coordinates where ΓCAB would vanish point-wise: Equivalence Principle isbroken for string (i.e. extended object), but recoverable coupled to point particle (scalar field).



• Semi-covariant derivative :

∇CTA1A2···An := ∂CTA1A2···An − ωT ΓBBCTA1A2···An +

n∑i=1

ΓCAiBTA1···Ai−1BAi+1···An ,

for which the ‘DFT-Christoffel’ connection can be uniquely fixed,

ΓCAB =2(P∂C PP)[AB]+2(

P[AD PB]

E−P[AD PB]

E)∂D PEC−

4D−1

(PC[APB]

D +PC[APB]D)(∂D d+(P∂E PP)[ED]

)

by demanding compatibility with JAB ,HAB , d, torsionless condition, and projection property,

∇APBC = ∇APBC = ∇Ad = 0 , L∂ξ = L∇ξ ⇔ Γ[ABC] = 0 , (P + P)ABCDEF ΓDEF = 0 ,

where multi-indexed projectors are

PABCDEF := PA

DP[B[E PC]

F ] + 2PM

M−1PA[BPC]

[E PF ]D , same for PABCDEF with PAB ↔ PAB .

The “generalized Lie derivative" (DFT-diffeomorphism generator) is

LξTA1···An := ξC∂CTA1···An + ωT ∂Cξ

CTA1···An +n∑

i=1

(∂Ai ξC − ∂C

ξAi )TA1···C···An ,

and in particular,LξHAB = 8P(A

[CPB)D]∇CξD , Lξd = − 1

2∇AξA.

∗ There are no normal coordinates where ΓCAB would vanish point-wise: Equivalence Principle isbroken for string (i.e. extended object), but recoverable coupled to point particle (scalar field).



• Semi-covariant Riemann curvature :

SABCD = S[AB][CD] = SCDAB := 12

(RABCD + RCDAB − ΓE

ABΓECD), S[ABC]D = 0 ,

where RABCD denotes the ordinary “field strength”, RCDAB=∂AΓBCD−∂BΓACD+ΓACE ΓBED−ΓBC

E ΓAED .

By construction, it varies as ‘total derivative’,

δSABCD = ∇[AδΓB]CD +∇[CδΓD]AB ,

which is useful for Lagrangian variation, i.e. action principle.

• Semi-covariant ‘Master’ derivative :

DA := ∂A + ΓA + ΦA + ΦA = ∇A + ΦA + ΦA .

The two spin connections are determined in terms of the DFT-Christoffel connection,

ΦApq = V Bp∇AVBq , ΦApq = V B

p∇AVBq ,

by requiring the compatibility with the vielbeins,

DAVBp = ∇AVBp + ΦApqVBq = 0 , DAVBp = ∇AVBp + ΦAp

qVBq = 0 .



• Semi-covariant Riemann curvature :

SABCD = S[AB][CD] = SCDAB := 12

(RABCD + RCDAB − ΓE

ABΓECD), S[ABC]D = 0 ,

where RABCD denotes the ordinary “field strength”, RCDAB=∂AΓBCD−∂BΓACD+ΓACE ΓBED−ΓBC

E ΓAED .

By construction, it varies as ‘total derivative’,

δSABCD = ∇[AδΓB]CD +∇[CδΓD]AB ,

which is useful for Lagrangian variation, i.e. action principle.

• Semi-covariant ‘Master’ derivative :

DA := ∂A + ΓA + ΦA + ΦA = ∇A + ΦA + ΦA .

The two spin connections are determined in terms of the DFT-Christoffel connection,

ΦApq = V Bp∇AVBq , ΦApq = V B

p∇AVBq ,

by requiring the compatibility with the vielbeins,

DAVBp = ∇AVBp + ΦApqVBq = 0 , DAVBp = ∇AVBp + ΦAp

qVBq = 0 .


Anomaly is under control owing to the six-indexed projectors

• Semi-covariance:

δξ(∇CTA1···An

)= Lξ

(∇CTA1···An

)+

n∑i=1

2(P+P)CAiBDEF∂D∂EξF TA1···Ai−1BAi+1···An ,

δξSABCD = LξSABCD + 2∇[A

((P+P)B][CD]

EFG∂E∂F ξG

)+ 2∇[C

((P+P)D][AB]

EFG∂E∂F ξG

).

• This is due to

δξΓCAB = LξΓCAB + 2[(P + P)CAB

FDE − δ FC δ D

A δ EB

]∂F∂[DξE ] .

Ideally one might desire to cancel these red-colored anomalies by adding extra terms to ΓCAB .

But, since

δHAB = (PδHP)AB + (PδHP)AB , δξ(∂CHAB) = Lξ(∂CHAB) + 8P(ADPB)

E∂C∂[DξE ] ,

it is impossible to construct such compensating terms out of the derivatives of HAB .

• However, we can easily project out the anomalies.





)= Lξ

(∇CTA1···An

)+

n∑i=1



((P+P)B][CD]

EFG∂E∂F ξG

)+ 2∇[C

((P+P)D][AB]

EFG∂E∂F ξG

).

• This is due to


FDE − δ FC δ D

A δ EB

]∂F∂[DξE ] .


But, since


E∂C∂[DξE ] ,







)= Lξ

(∇CTA1···An

)+

n∑i=1



((P+P)B][CD]

EFG∂E∂F ξG

)+ 2∇[C

((P+P)D][AB]

EFG∂E∂F ξG

).

• This is due to


FDE − δ FC δ D

A δ EB

]∂F∂[DξE ] .


But, since


E∂C∂[DξE ] ,




Complete covariantization

– Tensors:

PCD PA1

B1 · · · PAnBn∇DTB1···Bn =⇒ DpTq1 q2···qn ,

PCDPA1

B1 · · ·PAnBn∇DTB1···Bn =⇒ DpTq1q2···qn ,

DpTpq1 q2···qn , DpTpq1q2···qn ; DpDpTq1 q2···qn , DpDpTq1q2···qn .

– Yang-Mills:

Fpq := FABV ApV B

q where FAB := ∇AWB −∇BWA − i [WA,WB ] .

– Spinors, ρα, ψαp :

γpDpρ , Dpρ , γ

pDpψq , Dpψp,

– RR sector, Cαα :

D±C := γpDpC ± γ(D+1)DpCγ p

, (D±)2 = 0 =⇒ F := D+C ( RR flux ) .

– Curvatures:

Spq := SABV ApV B

q ( Ricci ) , S(0) := (PACPBD − PAC PBD)SABCD ( scalar ⇒ ‘pure’ DFT ) .



– Tensors:

PCD PA1


PCDPA1



– Yang-Mills:

Fpq := FABV ApV B



γpDpρ , Dpρ , γ

pDpψq , Dpψp,



, (D±)2 = 0 =⇒ F := D+C ( RR flux ) .

– Curvatures:

Spq := SABV ApV B




– Tensors:

PCD PA1


PCDPA1



– Yang-Mills:

Fpq := FABV ApV B



γpDpρ , Dpρ , γ

pDpψq , Dpψp,



, (D±)2 = 0 =⇒ F := D+C ( RR flux ) .

– Curvatures:

Spq := SABV ApV B



O(D,D) symmetric coupling to matter

• D = 10 Maximally Supersymmetric DFT Jeon-Lee-JHP-Suh 2012 [Full order construction]

Ltype II = e−2d[

18 S(0) + 1

2 Tr(FF) + i ρFρ′ + iψpγqF γpψ′q + i 12 ργ

pDpρ− i 12 ρ′γpDpρ

′

−iψpDpρ− i 12 ψ

pγqDqψp + iψ′pDpρ′ + i 12 ψ′p γqDqψ

′p

]which unifies IIA & IIB SUGRAs, and Gomis-Ooguri gravity as different solution sectors.

• Minimal coupling to the D = 4 Standard Model, Kangsin Choi & JHP 2015 [PRL]

LSM = e−2d

116πGN

S(0)

+∑V Tr(FpqFpq) +

∑ψ ψγ

aDaψ +∑ψ′ ψ

′γaDaψ′

−HAB(DAφ)†DBφ − V (φ) + yd q·φ d + yu q·φ u + ye l ′·φ e′

? Every single term above is completely covariant, w.r.t. O(D,D), DFT-diffeomorphisms, and

twofold local Lorentz symmetries.


Derivation of EDFE from General Covariance of Stringy Gravity 1/2

• Henceforth, we consider a general action for Stringy Gravity coupled to matter fields, Υa,

Action =

∫Σ

e−2d[

116πG S(0) + Lmatter

(Υa,DAΥb

) ],

and seek the variation of the action induced by all the fields, d ,VAp, VAp,Υa .

Note δVAp = (P + P)ABδVBp = VAq V BqδVBp + (δVB[pV B

q])VAq . The 2nd term is a local Lorentz rotation

and can be absorbed into δΥa. Thus, only the projected variation, V BqδVBp = −V B

pδVBq , appears.

– Firstly, the ‘pure’ Stringy Gravity term transforms, up to total derivatives ('), as

δ(

e−2d S(0)

)' 4e−2d

(V Bq

δVBpSpq − 1

2 δd S(0)

).

– Secondly, the matter Lagrangian transforms as

δ(

e−2d Lmatter

)' e−2d

(−2V Aq

δVApKpq + δd T(0) + δΥa

δLmatter

δΥa

)where we have been naturally led to define

Kpq :=12

(VAp

δLmatter

δVAq− VAq

δLmatter

δVAp

), T(0) := e2d ×

δ(

e−2d Lmatter

)δd

.




Action =

∫Σ

e−2d[


(Υa,DAΥb

) ],





pδVBq , appears.


δ(

e−2d S(0)

)' 4e−2d

(V Bq

δVBpSpq − 1

2 δd S(0)

).


δ(

e−2d Lmatter

)' e−2d

(−2V Aq


δLmatter

δΥa


Kpq :=12

(VAp

δLmatter

δVAq− VAq

δLmatter

δVAp

), T(0) := e2d ×

δ(

e−2d Lmatter

)δd

.




Action =

∫Σ

e−2d[


(Υa,DAΥb

) ],





pδVBq , appears.


δ(

e−2d S(0)

)' 4e−2d

(V Bq

δVBpSpq − 1

2 δd S(0)

).


δ(

e−2d Lmatter

)' e−2d

(−2V Aq


δLmatter

δΥa


Kpq :=12

(VAp

δLmatter

δVAq− VAq

δLmatter

δVAp

), T(0) := e2d ×

δ(

e−2d Lmatter

)δd

.




Action =

∫Σ

e−2d[


(Υa,DAΥb

) ],





pδVBq , appears.


δ(

e−2d S(0)

)' 4e−2d

(V Bq

δVBpSpq − 1

2 δd S(0)

).


δ(

e−2d Lmatter

)' e−2d

(−2V Aq


δLmatter

δΥa


Kpq :=12

(VAp

δLmatter

δVAq− VAq

δLmatter

δVAp

), T(0) := e2d ×

δ(

e−2d Lmatter

)δd

.



• Combining the two results, the variation of the action reads

δAction =

∫Σ

e−2d[

14πG V AqδVA

p(Spq − 8πGKpq)− 18πG δd(S(0) − 8πGT(0)) + δΥa

δLmatter

δΥa

].

• Specifically when the variation is generated by diffeomorphisms, we have δξΥa = LξΥa and

V AqδξVAp = V AqLξVA

p = 2D[AξB]V AqV Bp , δξd = − 12 e2d Lξ

(e−2d) = − 1

2DAξA .

• The Diffeomorphic General Covariance of the Action then implies

0 =

∫Σ

e−2d[

18πG ξ

BDA

4V[ApVB]

q(Spq − 8πGKpq)− 12JAB(S(0) − 8πGT(0))

+ δξΥa

δLmatter

δΥa

].

This gives the definitions of the off-shell conserved stringy Einstein curvature,

GAB := 4V[ApVB]

qSpq − 12JABS(0) , DAGAB = 0 (off-shell) ,

JHP-Rey-Rim-Sakatani 2015

and the on-shell conserved stringy Energy-Momentum tensor,

TAB := 4V[ApVB]

qKpq − 12JABT(0) , DAT AB = 0 (on-shell) .




δAction =

∫Σ

e−2d[

14πG V AqδVA


δLmatter

δΥa

].




(e−2d) = − 1

2DAξA .


0 =

∫Σ

e−2d[

18πG ξ

BDA

4V[ApVB]


+ δξΥa

δLmatter

δΥa

].


GAB := 4V[ApVB]




TAB := 4V[ApVB]





δAction =

∫Σ

e−2d[

14πG V AqδVA


δLmatter

δΥa

].




(e−2d) = − 1

2DAξA .


0 =

∫Σ

e−2d[

18πG ξ

BDA

4V[ApVB]


+ δξΥa

δLmatter

δΥa

].


GAB := 4V[ApVB]




TAB := 4V[ApVB]



• All the EOMs of the vielbeins and dilaton can be unified into a single expression,

GAB = 8πGTAB : Einstein Double Field Equations

which is naturally consistent with our central idea that Stringy Gravity treats the entireclosed-string massless sector as the geometrical stringy graviton multiplet.

I Restricting to the Riemannian (0, 0) parametrization,EDFE reduce to

Rµν + 25µ(∂νφ)− 14 HµρσHν

ρσ = 8πGK(µν) ,

e2φ5ρ(

e−2φHρµν)

= 16πGK[µν] ,

R + 42φ− 4∂µφ∂µφ− 1

12 HλµνHλµν = 8πGT(0) .

I For other non-Riemannian cases, (n, n) 6= (0, 0),

EDFE govern the dynamics of ‘chiral’ gravities,

e.g. Newton-Cartan, Carroll, Gomis-Ooguri, etc.


• All the EOMs of the vielbeins and dilaton can be unified into a single expression,

GAB = 8πGTAB : Einstein Double Field Equations

which is naturally consistent with our central idea that Stringy Gravity treats the entireclosed-string massless sector as the geometrical stringy graviton multiplet.

I Restricting to the Riemannian (0, 0) parametrization,EDFE reduce to

Rµν + 25µ(∂νφ)− 14 HµρσHν

ρσ = 8πGK(µν) ,

e2φ5ρ(

e−2φHρµν)

= 16πGK[µν] ,

R + 42φ− 4∂µφ∂µφ− 1

12 HλµνHλµν = 8πGT(0) .

I For other non-Riemannian cases, (n, n) 6= (0, 0),

EDFE govern the dynamics of ‘chiral’ gravities,

e.g. Newton-Cartan, Carroll, Gomis-Ooguri, etc.


Examples: TAB := 4V[ApVB]

qKpq − 12JABT(0)

• Scalar field,

Lϕ = − 12H

MN∂Mϕ∂Nϕ− V (ϕ) , Kpq = ∂pϕ∂qϕ , T(0) = −2Lϕ .

• Spinor field,

Lψ = ψγpDpψ + mψψψ , Kpq = − 14 (ψγpDqψ −Dqψγpψ) , T(0) ≡ 0 .

• RR sector,

LRR = 12 Tr(FF) , Kpq = − 1

4 Tr(γpF γqF) , T(0) = 0 .

• Fundamental string: with Di yM = ∂i yM −AMi (doubled-yet-gauged),

e−2d Lstring = 14πα′

∫d2σ

[− 1

2

√−hhij Di yM Dj yNHMN (y)− εij Di yMAjM

]δD(x − y(σ)

),

Kpq = 14πα′

∫d2σ√−hhij Di yM Dj yN VMpVNq e2d(x)δD(x − y(σ)

), T(0) = 0 .

– More examples include Yang-Mills, point particle, Green-Schwarz superstring, etc. 1804.00964


Cosmology Stephen Angus, Kyungho Cho, G. Franzmann, S. Mukohyama & JHP 1905.03620

• O(D,D) completion of the Friendamann equations:

8πG3ρe2φ +

h2

12a6= H2 − 2

(φ′

N

)H +

23

(φ′

N

)2

+ka2

4πG3

(ρ + 3p)e2φ +h2

6a6= −H2 −

H′

N+

(φ′

N

)H −

23

(φ′

N

)2

+1N

(φ′

N

)′8πG

3

(ρe2φ −

12

T(0)

)= −H2 −

H′

N+

23N

(φ′

N

)′which imply the conservation equation,

ρ′ + 3NH(ρ + p) + φ

′T(0)e−2φ = 0 .

Here most general homogeneous and isotropic cosmological ansatzes have been adopted:

ρ :=(−K t

t + 12 T(0)

)e−2φ , p :=

(K r

r − 12 T(0)

)e−2φ , H(3) = hr2√

1−kr2sinϑ dr ∧ dϑ ∧ dϕ .

I This gives an enriched and novel framework beyond typicalstring cosmology, enjoying two equation-of-state parameters,w = p/ρ (conventional) and λ = T(0)e

−2φ/ρ (new).

Most previous literatures assumed no dilaton coupling, λ = 0.


In remaining time,

I wish to discuss briefly the

most general spherically symmetric, asymptotically flat, regular solution to D= 4 EDFE,

namely

stringy ‘star’ of radius rc :

GAB =

8πGTAB for r ≤ rc (spherical)

0 for r > rc

This will contrast Stringy Gravity with GR as

D2+1 components in TAB vs. D(D+1)2 components in Tµν = Tνµ

Stringy Gravity = enriched and novel framework beyond GR


Spherical Symmetry in DFT

• In GR, isometries are addressed in terms of ordinary Lie derivative,

Lξgµν = 0 , LξHλµν = 0 ⇐⇒ LξBµν = ∂ν ξµ − ∂µξν .

• In DFT, the spherical symmetry should be analyzed by generalized Lie derivative,

with triple Killing vectors, ξMa = (ξaµ, ξνa ), a = 1, 2, 3,

LξHAB = 8P(A[CPB)

D]∇CξD = 0 , Lξd = − 12∇Aξ

A = 0 , [ξa, ξb]C =∑

c εabcξc .

I However, while generalized Lie derivative is diffeomorphism covariant,

δξ(LζTM1···Mn ) = Lζ(δξTM1···Mn ) + LδξζTM1···Mn = LζLξTM1···Mn + LLξζTM1···Mn

= LξLζTM1···Mn + L[ζ,ξ]C+LξζTM1···Mn = Lξ(LζTM1···Mn )

it is anomalous under local Lorentz rotations. Hence, potentially problematic with vielbeins:

LξVAp = PABV C

p

(LξPBC

)−(ξBΦBpq + 2D[pξq]

)VA

q ,

where ξq = ξAVAq . This result can be rewritten as

ξBDBVAp + 2D[AξB]VB

p + 2D[pξq]VAq = PA

B(LξPBC

)V C

p .







LξHAB = 8P(A[CPB)

D]∇CξD = 0 , Lξd = − 12∇Aξ

A = 0 , [ξa, ξb]C =∑

c εabcξc .





LξVAp = PABV C

p

(LξPBC


)VA

q ,




B(LξPBC

)V C

p .







LξHAB = 8P(A[CPB)

D]∇CξD = 0 , Lξd = − 12∇Aξ

A = 0 , [ξa, ξb]C =∑

c εabcξc .





LξVAp = PABV C

p

(LξPBC


)VA

q ,




B(LξPBC

)V C

p .


Further-generalized Lie derivative, Lξ

• We generalize the generalized Lie derivative one step further by including additional localLorentz rotations:

LξTMppαα := ξNDNTMppαα + ωT DNξNTMppαα + 2D[MξN]T N

ppαα

+2D[pξq]TMq

pαα − 12D[r ξs](γ

rs)βαTMppβα

+2D[pξq]TMpqαα − 1

2D[r ξs](γr s)β αTMppαβ ,[

Lζ , Lξ]

= L[ζ,ξ]C+ ωpq(ζ, ξ) + ωpq(ζ, ξ) .

• This is covariant under both diffeomorphisms and local Lorentz symmetries, and

allows to analyze the spherical SO(3) symmetry:

Lξa VAp =(Lξa PAC

)V C

p ≡ 0 , Lξa VAp =(Lξa PAC

)V C

p ≡ 0 , Lξa Kpq ≡ 0 , · · ·

which is, after all, consistent with the Riemannian geometry,



Further-generalized Lie derivative, Lξ

• We generalize the generalized Lie derivative one step further by including additional localLorentz rotations:

LξTMppαα := ξNDNTMppαα + ωT DNξNTMppαα + 2D[MξN]T N

ppαα

+2D[pξq]TMq

pαα − 12D[r ξs](γ

rs)βαTMppβα

+2D[pξq]TMpqαα − 1

2D[r ξs](γr s)β αTMppαβ ,[

Lζ , Lξ]

= L[ζ,ξ]C+ ωpq(ζ, ξ) + ωpq(ζ, ξ) .

• This is covariant under both diffeomorphisms and local Lorentz symmetries, and

allows to analyze the spherical SO(3) symmetry:

Lξa VAp =(Lξa PAC

)V C

p ≡ 0 , Lξa VAp =(Lξa PAC

)V C

p ≡ 0 , Lξa Kpq ≡ 0 , · · ·

which is, after all, consistent with the Riemannian geometry,



Solution: stringy ‘star’ of radius rc

• Outside the star, r ≥ rc, the vacuum geometry is given by [Burgess-Myers-Quevedo ’94]

e2φ = γ+

(r−αr+β

) b√a2+b2 + γ−

(r+βr−α

) b√a2+b2 , H(3) = h sinϑ dt ∧ dϑ ∧ dϕ ,

ds2 = e2φ

[−(

r−αr+β

) a√a2+b2 dt2 +

(r+βr−α

) a√a2+b2 dr2 + (r − α)(r + β)dΩ2]

which has four parameters, α, β, a, h, while

b2 = (α+ β)2 − a2 ≥ h2 , γ± = 12

(1±

√1− h2/b2

).

If b = h = 0, it reduces to Schwarzschild geometry.

• Inside the star, EDFE fix all the constants, α, β, a, h, in terms of TAB , for example

a =

∫ rc

0dr∫ π

0dϑ∫ 2π

0dϕ e−2d

[1

4πHrϑϕH rϑϕ + 2G(Kr

r + Kϑϑ + Kϕϕ − Ktt − T(0)

)].

Namely, various components of TAB enrich the spherical geometry of Stringy Gravity.


Solution: stringy ‘star’ of radius rc

• Outside the star, r ≥ rc, the vacuum geometry is given by [Burgess-Myers-Quevedo ’94]

e2φ = γ+

(r−αr+β

) b√a2+b2 + γ−

(r+βr−α

) b√a2+b2 , H(3) = h sinϑ dt ∧ dϑ ∧ dϕ ,

ds2 = e2φ

[−(

r−αr+β

) a√a2+b2 dt2 +

(r+βr−α

) a√a2+b2 dr2 + (r − α)(r + β)dΩ2]

which has four parameters, α, β, a, h, while

b2 = (α+ β)2 − a2 ≥ h2 , γ± = 12

(1±

√1− h2/b2

).

If b = h = 0, it reduces to Schwarzschild geometry.

• Inside the star, EDFE fix all the constants, α, β, a, h, in terms of TAB , for example

a =

∫ rc

0dr∫ π

0dϑ∫ 2π

0dϕ e−2d

[1

4πHrϑϕH rϑϕ + 2G(Kr

r + Kϑϑ + Kϕϕ − Ktt − T(0)

)].

Namely, various components of TAB enrich the spherical geometry of Stringy Gravity.


Stringy Gravity modifies GR when RMG is small (e.g. it can be repulsive)

• In terms of Areal Radius, R : ds2 = gttdt2 + gRRdR2 + R2dΩ2, gravitational potential reads

ΦNewton = − 12 (1 + gtt ) = −MG

R +

(2b2−h2+2ab

√1−h2/b2

a2+b2−h2+2ab√

1−h2/b2

)(MGR

)2+ · · ·

where the ellipses denote higher order terms in MGR which is ‘dimensionless’, and

MG = 12

(a + b

√1− h2/b2

)=

∫ ∞0

dr∫ π

0dϑ∫ 2π

0dϕ e−2d

(−2GKt

t + 18π

∣∣HtϑϕH tϑϕ∣∣) .

• Since B-field does not couple to particle geodesics, from the mass formula above, we mightspeculate that electric H-flux is dark matter, while Kt

t represents ordinary matter (baryonic).




ΦNewton = − 12 (1 + gtt ) = −MG

R +

(2b2−h2+2ab

√1−h2/b2

a2+b2−h2+2ab√

1−h2/b2

)(MGR

)2+ · · ·


MG = 12

(a + b

√1− h2/b2

)=

∫ ∞0

dr∫ π

0dϑ∫ 2π

0dϕ e−2d

(−2GKt

t + 18π







ΦNewton = − 12 (1 + gtt ) = −MG

R +

(2b2−h2+2ab

√1−h2/b2

a2+b2−h2+2ab√

1−h2/b2

)(MGR

)2+ · · ·


MG = 12

(a + b

√1− h2/b2

)=

∫ ∞0

dr∫ π

0dϑ∫ 2π

0dϕ e−2d

(−2GKt

t + 18π




• Intriguingly, dark matter and energy problems arise from small RMG observations:

The observations of stars/galaxies far away may reveal the short-distance nature of gravity.




ΦNewton = − 12 (1 + gtt ) = −MG

R +

(2b2−h2+2ab

√1−h2/b2

a2+b2−h2+2ab√

1−h2/b2

)(MGR

)2+ · · ·


MG = 12

(a + b

√1− h2/b2

)=

∫ ∞0

dr∫ π

0dϑ∫ 2π

0dϕ e−2d

(−2GKt

t + 18π




• Intriguingly, dark matter and energy problems arise from small RMG observations:

The observations of stars/galaxies far away may reveal the short-distance nature of gravity.

Thank you


Stringy Gravity: O D completion of General Relativity

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