JEONG-HYUCK PARK - McGill Physicsrhb/DFT/JHPark.pdf · Comments on double ﬁeld theory and...

SEMI-COVARIANT APPROACH TO DOUBLE FIELD THEORY

JEONG-HYUCK PARK

Sogang University, Seoul

Workshop on Double Field Theory

January 20 - 22 2016

ETH Zurich, Switzerland

JEONG-HYUCK PARK SEMI-COVARIANT APPROACH TO DOUBLE FIELD THEORY

Prologue

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

Diffeomorphism: ∂µ −→ ∇µ = ∂µ + Γµ

∇λgµν = 0, Γλ[µν]

= 0 −→ Γλµν = 12 gλρ(∂µgνρ + ∂νgµρ − ∂ρgµν)

Curvature: [∇µ,∇ν ] −→ Rκλµν −→ R

On the other hand, string theory puts gµν , Bµν and φ on an equal footing,

as they – so called NS-NS sector – form a multiplet of T-duality.

This suggests the existence of a novel unifying geometric description of them,

generalizing the above Riemannian formalism.

Basically, Riemannian geometry is for Particle theory. String theory requires a

novel differential geometry which geometrizes the whole NS-NS sector.

Prologue

My talk today aims to introduce such a stringy geometry under the name,

‘Semi-covariant Formalism’

which underlies

Double Field Theory = Stringy Gravity .

Contrary to what it may sound like, the semi-covariant formalism is a completely

covariant approach to DFT, as it manifests simultaneously for every term in

Lagrangians:

O(D,D) T-duality

DFT-diffeomorphisms (generalized Lie derivative)

A pair of local Lorentz symmetries, Spin(1,D−1)L × Spin(D−1, 1)R

c.f. Siegel, Gwak, Hohm, Zwiebach;

Waldram, Coimbra, Strickland-Constable (Generalized Geometry a la Hitchin);

Berman, Blair, Malek, Perry, Grana, Marques, Cederwall ...

My talk today aims to introduce such a stringy geometry under the name,

‘Semi-covariant Formalism’

which underlies

Double Field Theory = Stringy Gravity .

Contrary to what it may sound like, the semi-covariant formalism is a completely

covariant approach to DFT, as it manifests simultaneously for every term in

Lagrangians:

O(D,D) T-duality

c.f. Siegel, Gwak, Hohm, Zwiebach;

Waldram, Coimbra, Strickland-Constable (Generalized Geometry a la Hitchin);

Berman, Blair, Malek, Perry, Grana, Marques, Cederwall ...

Talk based on collaborations with Imtak Jeon (8 papers), Kanghoon Lee (8 papers), Yoonji Suh (3 papers),

Chris Blair, Emanuel Malek, Wonyoung Cho, Jose Fernández-Melgarejo, Soo-Jong Rey, Woohyun Rim, Yuho Sakatani,

Sung Moon Ko, Charles Melby-Thompson, Rene Meyér, and Kang-Sin Choi (Phenomenologist).

Differential geometry with a projection: Application to double field theory 1011.1324 JHEP

Stringy differential geometry, beyond Riemann 1105.6294 PRD

Incorporation of fermions into double field theory 1109.2035 JHEP

Ramond-Ramond Cohomology and O(D,D) T-duality 1206.3478 JHEP

Supersymmetric Double Field Theory: Stringy Reformulation of Supergravity 1112.0069 PRD

Stringy Unification of IIA and IIB Supergravities underN= 2 D= 10 Supersymmetric Double Field Theory

1210.5078 PLB

Supersymmetric gauged Double Field Theory: Systematic derivation by virtue of ‘Twist’ 1505.01301 JHEP

Comments on double field theory and diffeomorphisms 1304.5946 JHEP

Covariant action for a string in doubled yet gauged spacetime 1307.8377 NPB

Double field formulation of Yang-Mills theory 1102.0419 PLB

Standard Model as a Double Field Theory 1506.05277 PRL

O(D,D) Covariant Noether Currents and Global Charges in Double Field Theory 1507.07545 JHEP

Dynamics of Perturbations in Double Field Theory & Non-Relativistic String Theory 1508.01121 JHEP

U-geometry: SL(5) ⇒ U-gravity: SL(N) 1302.1652 JHEP/1402.5027 JHEP

M-theory and Type IIB from a Duality Manifest Action 1311.5109 JHEP

Talk contents

Semi-covariant formulation of DFT

Supersymmetric extension, i.e. SDFT

Worldsheet perspective with ‘doubled-yet-gauged’ coordinates

Phenomenological implications, especially to the Standard Model

Notation

Capital letters denote the O(D,D) vector indices, i.e. A,B,C, · · · = 1, 2, · · · ,D+D,

which can be freely raised or lowered by the O(D,D) invariant constant metric,

Notation

Capital letters denote the O(D,D) vector indices, i.e. A,B,C, · · · = 1, 2, · · · ,D+D,

which can be freely raised or lowered by the O(D,D) invariant constant metric,

Index Representation Metric (raising/lowering indices)

A,B, · · · O(10, 10) & DFT-diffeom. vector JAB

p, q, · · · Spin(1, 9)L vector ηpq = diag(−+ + · · ·+)

α, β, · · · Spin(1, 9)L spinor C+αβ , (γp)T = C+γpC−1+

p, q, · · · Spin(9, 1)R vector ηpq = diag(+−− · · ·−)

α, β, · · · Spin(9, 1)R spinor C+αβ , (γp)T = C+γpC−1+

Doubled-yet-gauged coordinates

The spacetime coordinates are formally doubled, being (D+D)-dimensional.

However, the doubled coordinates need to be gauged: the coordinate space is

equipped with an equivalence relation,

xA ∼ xA + φ∂Aϕ ,

which we call ‘Coordinate Gauge Symmetry’.

In the above, φ and ϕ are arbitrary functions in DFT.

The spacetime coordinates are formally doubled, being (D+D)-dimensional.

However, the doubled coordinates need to be gauged: the coordinate space is

equipped with an equivalence relation,

xA ∼ xA + φ∂Aϕ ,

which we call ‘Coordinate Gauge Symmetry’.

In the above, φ and ϕ are arbitrary functions in DFT.

Each equivalence class, or gauge orbit, represents a single physical point.

Diffeomorphism symmetry means an invariance under arbitrary reparametrizations of

the gauge orbits.

∗ The claim is that D-dimensional spacetime can be better understood in terms of the

doubled-yet-gauged (D+D) number of coordinates, at least for String Theory.

Semi-covariant formulation of Double Field Theory

Realization of the coordinate gauge symmetry.

The equivalence relation is realized in DFT by enforcing that, arbitrary functions and

their arbitrary derivatives, denoted here collectively by Φ, are invariant under the

coordinate gauge symmetry shift,

Φ(x + ∆) = Φ(x) , ∆A = φ∂Aϕ .

Realization of the coordinate gauge symmetry.

The equivalence relation is realized in DFT by enforcing that, arbitrary functions and

their arbitrary derivatives, denoted here collectively by Φ, are invariant under the

coordinate gauge symmetry shift,

Φ(x + ∆) = Φ(x) , ∆A = φ∂Aϕ .

∗ As will be explained later in this talk, the coordinate gauge symmetry can be also

naturally realized on a worldsheet as a conventional gauge symmetry of a string action.

Section condition.

The invariance under the coordinate gauge symmetry can be shown to be equivalent to

the section condition:

Coordinate Gauge Symmetry ⇐⇒ ∂A∂A ≡ 0 .

JHP, Lee-JHP 2013

Section condition.

The invariance under the coordinate gauge symmetry can be shown to be equivalent to

the section condition:

Coordinate Gauge Symmetry ⇐⇒ ∂A∂A ≡ 0 .

JHP, Lee-JHP 2013

Explicitly, acting on arbitrary functions, Φ, Φ′, and their products, the section

condition implies

∂A∂AΦ=0 (weak constraint) ,

∂AΦ∂AΦ′=0 (strong constraint) .

∗ The relaxation of these constraints is also of interest, but this goes beyond the scope of this talk.

c.f. Geissbühler, Grana, Marques, Berman, Lee, Gwak, Hohm, Cho, Fernandez-Melgarejo, Jeon, JHP, ...

Diffeomorphisms.

Diffeomorphism symmetry in O(D,D) DFT is generated by a generalized Lie derivative

Siegel, Courant, Grana

LX TA1···An := X B∂BTA1···An + ωT ∂BX BTA1···An +n∑

(∂Ai XB − ∂BXAi )TA1···Ai−1B

Ai+1···An ,

where ωT denotes the weight.

Diffeomorphisms.

Diffeomorphism symmetry in O(D,D) DFT is generated by a generalized Lie derivative

Siegel, Courant, Grana

LX TA1···An := X B∂BTA1···An + ωT ∂BX BTA1···An +n∑

(∂Ai XB − ∂BXAi )TA1···Ai−1B

Ai+1···An ,

where ωT denotes the weight.

In particular, the generalized Lie derivative of the O(D,D) invariant metric is trivial,

LXJAB = 0 .

The commutator is closed by C-bracket Hull-Zwiebach[LX , LY

]= L[X ,Y ]C , [X ,Y ]AC = X B∂BY A − Y B∂BX A + 1

2 Y B∂AXB − 12 X B∂AYB .

Diffeomorphisms.

Hohm-Zwiebach ansatz for finite transformations:

F := 12

(LL−1 + L−1L

), F := J F tJ−1 = 1

(L−1L + LL−1) = F−1 ,

LMN := ∂M x ′N , L := J LtJ−1 .

Though nice and compact, F does not precisely coincide with exp(LX ).

Yet, up to coordinate gauge symmetry it is possible to show JHP 2013

F ≡ exp(LX )

c.f. Berman-Cederwall-Perry, Hull, Papadopoulos, Sakatani-Rey

Diffeomorphisms.

F := 12

(LL−1 + L−1L

), F := J F tJ−1 = 1

(L−1L + LL−1) = F−1 ,

LMN := ∂M x ′N , L := J LtJ−1 .

F ≡ exp(LX )

Diffeomorphisms.

F := 12

(LL−1 + L−1L

), F := J F tJ−1 = 1

(L−1L + LL−1) = F−1 ,

LMN := ∂M x ′N , L := J LtJ−1 .

F ≡ exp(LX )

Diffeomorphisms.

F := 12

(LL−1 + L−1L

), F := J F tJ−1 = 1

(L−1L + LL−1) = F−1 ,

LMN := ∂M x ′N , L := J LtJ−1 .

F ≡ exp(LX )

Dilaton and a pair of two-index projectors.

The geometric objects in DFT consist of a dilaton, d , and a pair of symmetric

projection operators,

PAB = PBA , PAB = PBA , PABPB

C = P CA , PA

BPBC = P C

Further, the projectors are orthogonal and complementary,

C = 0 , PAB + PAB = JAB .

Dilaton and a pair of two-index projectors.

The geometric objects in DFT consist of a dilaton, d , and a pair of symmetric

projection operators,

C = P CA , PA

BPBC = P C

Further, the projectors are orthogonal and complementary,

C = 0 , PAB + PAB = JAB .

Remark: The difference of the two projectors, PAB − PAB = HAB , corresponds to the

“generalized metric" which can be also independently defined as a symmetric O(D,D)

element, i.e. HAB = HBA, HABHB

C = δ CA . Yet, in supersymmetric double field theories,

it appears that the projectors are more fundamental than the “generalized metric".

Integral measure.

While the projectors are weightless, the dilaton gives rise to the O(D,D) invariant

integral measure with weight one, after exponentiation,

e−2d .

Integral measure.

While the projectors are weightless, the dilaton gives rise to the O(D,D) invariant

integral measure with weight one, after exponentiation,

e−2d .

Naturally the cosmological constant term in DFT is given by

e−2d Λ

which deviates from the conventional one in Riemannian GR. This may provide a new

spin on the cosmological constant problem.

Jeon-Lee-JHP 2011

c.f. Meissner-Veneziano 1991

Semi-covariant derivative and semi-covariant Riemann curvature.

We define a semi-covariant derivative,

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

n∑i=1

ΓCAiBTA1···Ai−1BAi+1···An ,

n∑i=1

and a semi-covariant “Riemann” curvature,

SABCD := 12

(RABCD + RCDAB − ΓE

ABΓECD

Here RABCD denotes the ordinary “field strength" of a connection,

RCDAB = ∂AΓBCD − ∂BΓACD + ΓACE ΓBED − ΓBC

E ΓAED .

n∑i=1

and a semi-covariant “Riemann” curvature,

SABCD := 12

ABΓECD

Here RABCD denotes the ordinary “field strength" of a connection,

E ΓAED .

As will be explained below, one can determine the (torsionelss) connection uniquely:

ΓCAB = 2(P∂CPP

+ 2(P[A

DPB]E − P[A

DPB]E) ∂DPEC

− 4D−1

(PC[APB]

D + PC[APB]D)(∂Dd + (P∂E PP)[ED]

which corresponds to the DFT generalization of the Christoffel connection.

The semi-covariant derivative then obeys the Leibniz rule and annihilates the O(D,D)

invariant constant metric,

∇AJBC = 0 .

A crucial defining property of the semi-covariant “Riemann” curvature is that, under

arbitrary transformation of the connection, it transforms as total derivative,

δSABCD = ∇[AδΓB]CD +∇[CδΓD]AB .

∇AJBC = 0 .

A crucial defining property of the semi-covariant “Riemann” curvature is that, under

arbitrary transformation of the connection, it transforms as total derivative,

δSABCD = ∇[AδΓB]CD +∇[CδΓD]AB .

Further, the semi-covariant “Riemann” curvature satisfies precisely the same symmetric

properties as the ordinary Riemann curvature,

SABCD = S[AB][CD] = SCDAB , S[ABC]D = 0 ,

as well as additional identities involving the projectors,

BPKC PL

DSABCD=0 , PIAPJ

BPKC PL

DSABCD = 0 .

Notably, it turns out that the naive scalar contraction vanishes identically,

SABAB = 0 .

The uniqueness of the torsionless connection.

The connection is the unique solution to the following five constraints:

∇APBC = 0 , ∇APBC = 0 ,

∇Ad = − 12 e2d∇A(e−2d ) = ∂Ad + 1

2 ΓBBA = 0 ,

ΓABC + ΓACB = 0 ,

ΓABC + ΓBCA + ΓCAB = 0 ,

PABCDEF ΓDEF = 0 , PABC

DEF ΓDEF = 0 .

∇APBC = 0 , ∇APBC = 0 ,

∇Ad = − 12 e2d∇A(e−2d ) = ∂Ad + 1

2 ΓBBA = 0 ,

ΓABC + ΓACB = 0 ,

DEF ΓDEF = 0 .

The first two relations are the compatibility conditions with all the geometric

objects , or NS-NS sector, in DFT.

The third constraint is the compatibility condition with the O(D,D) invariant

constant metric, i.e. ∇AJBC = 0.

∇APBC = 0 , ∇APBC = 0 ,

∇Ad = − 12 e2d∇A(e−2d ) = ∂Ad + 1

2 ΓBBA = 0 ,

ΓABC + ΓACB = 0 ,

DEF ΓDEF = 0 .

The next cyclic property makes the semi-covariant derivative compatible with the

generalized Lie derivative as well as with the C-bracket,

LX (∂) = LX (∇) , [X ,Y ]C(∂) = [X ,Y ]C(∇) .

The last formulae are certain projection conditions which we impose intentionally

in order to ensure the uniqueness.

∇APBC = 0 , ∇APBC = 0 ,

∇Ad = − 12 e2d∇A(e−2d ) = ∂Ad + 1

2 ΓBBA = 0 ,

ΓABC + ΓACB = 0 ,

DEF ΓDEF = 0 .

The next cyclic property makes the semi-covariant derivative compatible with the

generalized Lie derivative as well as with the C-bracket,

LX (∂) = LX (∇) , [X ,Y ]C(∂) = [X ,Y ]C(∇) .

The last formulae are certain projection conditions which we impose intentionally

in order to ensure the uniqueness.

Remark: Failure of the Equivalence Principle

Unlike the Christoffel symbol, the DFT-diffeomorphisms cannot transform our connection

to vanish point-wise:

ΓCAB = 2(P∂CPP

+ 2(P[A

DPB]E − P[A

DPB]E) ∂DPEC

− 4D−1

(PC[APB]

)6= 0 .

That is to say, there is no normal coordinate in DFT. This can be viewed as the failure of

the equivalence principle applied to an extended object, i.e. string.

Six-index projection operators.

The six-index projection operators are explicitly,

PCABDEF := PC

DP[A[E PB]

F ] + 2D−1 PC[APB]

[E PF ]D ,

PCABDEF := PC

DP[A[E PB]

F ] + 2D−1 PC[APB]

[E PF ]D ,

which satisfy the ‘projection’ properties,

PABCDEFPDEF

GHI = PABCGHI , PABC

DEF PDEFGHI = PABC

Further, they are symmetric and traceless,

PABCDEF = PDEFABC , PABCDEF = PA[BC]D[EF ] , PABPABCDEF = 0 ,

PABCDEF = PDEFABC , PABCDEF = PA[BC]D[EF ] , PABPABCDEF = 0 .

Crucially, the projection operator dictates the anomalous terms in the diffeomorphic

transformations of the semi-covariant derivative and “Riemann” curvature,

(δX−LX )∇CTA1···An =n∑

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

(δX − LX )SABCD=2∇[A

((P+P)B][CD]

EFG∂E∂F XG

)+ 2∇[C

((P+P)D][AB]

EFG∂E∂F XG

Complete covariantizations.

Both the semi-covariant derivative and the semi-covariant Riemann curvature can be

fully covariantized, through appropriate contractions with the projectors:

PCDPA1

B1 · · · PAnBn∇DTB1···Bn , PC

DPA1B1 · · ·PAn

Bn∇DTB1···Bn ,

PABPC1D1 · · · PCn

Dn∇ATBD1···Dn , PABPC1D1 · · ·PCn

Dn∇ATBD1···Dn (divergences) ,

Dn∇A∇BTD1···Dn , PABPC1D1 · · ·PCn

Dn∇A∇BTD1···Dn (Laplacians) ,

PCDPA1

DPA1B1 · · ·PAn

Bn∇DTB1···Bn ,

PAC PB

DSCEDE (“Ricci” curvature) ,

(PACPBD − PAC PBD)SABCD (scalar curvature) .

PCDPA1

DPA1B1 · · ·PAn

Bn∇DTB1···Bn ,

PAC PB

DSCEDE (“Ricci” curvature) ,

(PACPBD − PAC PBD)SABCD (scalar curvature) .

Combining the curvatures, we also have the conserved “Einstein” curvature:

∇AGAB = 0 , GAB := 2(PAC PBD − PACPBD)SCEDE − 1

2JAB(PCDPEF − PCDPEF )SCEDF .

Further completely covariant examples follow from the above generic prescription:

Completely covariant Yang-Mills field strength is given by two opposite

projections,

PAM PB

NFMN ,

where FMN is the semi-covariant field strength of a YM potential, VM ,

FMN := ∇MVN −∇NVM − i [VM ,VN ] .

Unlike the Riemannian case, the Γ connections are not canceled out.

Jeon-Lee-JHP 2011, Choi-JHP 2015

Completely covariant Killing equations of DFT:

LXHMN = 0 ⇐⇒ (P∇)M (PX)N − (P∇)N (PX)M = 0 ,

LX d = 0 ⇐⇒ ∇M X M = 0 .

JHP-Rey-Rim-Sakatani 2015

Further completely covariant examples follow from the above generic prescription:

Completely covariant Yang-Mills field strength is given by two opposite

projections,

PAM PB

NFMN ,

Unlike the Riemannian case, the Γ connections are not canceled out.

Jeon-Lee-JHP 2011, Choi-JHP 2015

Completely covariant Killing equations of DFT:

LX d = 0 ⇐⇒ ∇M X M = 0 .

Action.

The action of O(D,D) DFT is given by the fully covariant scalar curvature,

∫ΣD

e−2d[

(PACPBD − PAC PBD)SABCD − 2Λ],

where the integral is taken over a section, ΣD , and the DFT-cosmological constant

term has been inserted.

Action.

The action of O(D,D) DFT is given by the fully covariant scalar curvature,

∫ΣD

e−2d[

(PACPBD − PAC PBD)SABCD − 2Λ],

where the integral is taken over a section, ΣD , and the DFT-cosmological constant

term has been inserted.

The dilaton and the projector equations of motion correspond to the vanishing of the

Lagrangian and the “Ricci” curvature respectively.

Section.

Up to O(D,D) duality rotations, the solution to the section condition is unique. It is a

D-dimensional section, ΣD , characterized by the independence of the dual coordinates,

i.e.∂

∂xµ≡ 0 ,

while the whole doubled coordinates are given by

xA = (xµ, xν) ,

where µ, ν are now D-dimensional indices.

Riemannian reduction.

To perform the Riemannian reduction to the D-dimensional section, ΣD , we

parametrize the dilaton and the projectors in terms of D-dimensional Riemannian

metric, gµν , ordinary dilaton, φ, and a Kalb-Ramond two-form potential, Bµν ,

PAB − PAB = HAB =

g−1 −g−1B

Bg−1 g − Bg−1B

, e−2d =√|g|e−2φ .

PAB − PAB = HAB =

g−1 −g−1B

, e−2d =√|g|e−2φ .

Then, the above DFT action reduces to∫ΣD

√−ge−2φ

(Rg + 4(∂φ)2 − 1

12 H2 − 2Λ).

while the DFT-diffeomorphim decomposes into the D-dimensional Riemannian

diffeomorphism plus the B-field gauge symmetry.

As the DFT-cosmological constant term becomes an exponential potential, e−2φ, the

cosmological constant problem is clearly reformulated in DFT.

PAB − PAB = HAB =

g−1 −g−1B

, e−2d =√|g|e−2φ .

PAB − PAB = HAB =

g−1 −g−1B

, e−2d =√|g|e−2φ .

Up to field redefinitions, the above is the most general parametrization of the

“generalized metric", HAB = PAB − PAB , provided its upper left D × D block is

non-degenerate.

Non-Riemannian backgrounds.

When the upper left D × D block of HAB = (P−P)AB is degenerate – where g−1 might

be positioned – the Riemannian metric ceases to exist upon the section, ΣD .

Nevertheless, DFT and a doubled sigma model –which I will discuss later– have no

problem with describing such a non-Riemannian background.

An extreme example of such a non-Riemannian background is the flat background

HAB = (P−P)AB = JAB .

This is a vacuum solution to the bosonic DFT and the corresponding doubled sigma

model reduces to a certain ‘chiral’ sigma model.

Non-Riemannian backgrounds.

When the upper left D × D block of HAB = (P−P)AB is degenerate – where g−1 might

be positioned – the Riemannian metric ceases to exist upon the section, ΣD .

Nevertheless, DFT and a doubled sigma model –which I will discuss later– have no

problem with describing such a non-Riemannian background.

An extreme example of such a non-Riemannian background is the flat background

HAB = (P−P)AB = JAB .

This is a vacuum solution to the bosonic DFT and the corresponding doubled sigma

model reduces to a certain ‘chiral’ sigma model.

Allowing non-Riemannian backgrounds, DFT is NOT a mere reformulation of SUGRA.

It describes a new class of (non-relativistic) string theory backgrounds.

Ko-Melby-Thompson-Meyer-JHP 2015, c.f. Gomis-Ooguri

Now having the semi-covariant formalism at our disposal,we can do much:

Couple to Yang-Mills, fermions 1011.1324, 1109.2035

Double Field Theorize the Standard Model 1506.05277

Incorporate the R-R sector 1206.3478

Supersymmetrizations Half-maximal SDFT 1112.0069

Maximal SDFT 1210.5078

Gauged SDFT 1505.01301

(section condition relaxed)

Noether currents and global charges 1507.07545

Perturbations of δHAB , δd 1508.01121

Application to the worldsheet action 1307.8377

Extensions to U-duality, ‘U-gravity’ 1302.1652, 1402.5027

Supersymmetric Extension

Utilizing the semi-covariant formalism introduced above,

after incorporating fermions and R-R sector,

it is possible to construct, to the ‘full order’ in fermions,

Type II, or N = 2, D = 10 Maximally Supersymmetric Double Field Theory

of which the Lagrangian reads

LType II = e−2d[

18 (PABPCD − PABPCD)SACBD + 1

2Tr(FF)− i ρFρ′ + iψpγqF γpψ′q

+i 12 ργ

pD?pρ− iψpD?pρ− i 12 ψ

pγqD?qψp − i 12 ρ′γpD′?p ρ

′ + iψ′pD′?p ρ′ + i 12 ψ′p γqD′?q ψ

]Jeon-Lee-Suh-JHP 1210.5078

c.f. other approaches: Coimbra, Strickland-Constanble, Waldram

Kwak, Hohm, Zwiebach

Symmetries of N = 2 D = 10 SDFT

O(10, 10) T-duality

Gauge symmetries

1 DFT-diffeomorphism (generalized Lie derivative)

2 A pair of local Lorentz symmetries, Spin(1, 9)L × Spin(9, 1)R

3 local N = 2 SUSY with 32 supercharges.

All the bosonic symmetries are realized manifestly and simultaneously for each term.

For this, it is crucial to have the right field variables:

d , VAp , VAp , Cαα , ρα , ρ′α , ψαp , ψ′αp

which are O(10, 10) covariant genuine DFT-field-variables, and a priori they are NOT

Riemannian, such as metric, B-field, R-R p-forms.

O(10, 10) T-duality

Gauge symmetries

O(10, 10) T-duality

Gauge symmetries

Unificaiton of IIA and IIB

O(10, 10) T-duality

Gauge symmetries

The theory is chiral with respect to both Local Lorentz groups.

Consequently, there is no distinction of IIA and IIB =⇒ Unificaiton of IIA and IIB

While the theory is unique, it contains type IIA and IIB SUGRA backgrounds as

different kind of solutions.

O(10, 10) T-duality

Gauge symmetries

O(10, 10) T-duality

Gauge symmetries

O(10, 10) T-duality

Gauge symmetries

Worldsheet Perspective:

1304.5946/1307.8377

Doubled-yet-gauged coordinates & Gauged infinitesimal one-form

In the doubled-yet-gauged coordinate system,

the usual infinitesimal one-form, dxM , is NOT a covariant vector of DFT:

it does not transform covariantly under DFT-diffeomorphisms (obeying the way the

‘generalized Lie derivative’ would dictate).

Hence, dxMdxNHMN can NOT give a ‘proper length’ in DFT.

Further, it is NOT coordinate gauge symmetry invariant,

dxM −→ d(xM + Φ1∂M Φ2) 6= dxM .

These can be all cured by introducing a gauged infinitesimal one-form,

DxM := dxM −AM .

Being a derivative-index-valued vector, the potential satisfies AM∂M = 0, AMAM = 0,

or suggestively the ‘gauged section condition’,

(∂M +AM )(∂M +AM ) = 0 .

dxM −→ d(xM + Φ1∂M Φ2) 6= dxM .

DxM := dxM −AM .

(∂M +AM )(∂M +AM ) = 0 .

dxM −→ d(xM + Φ1∂M Φ2) 6= dxM .

DxM := dxM −AM .

(∂M +AM )(∂M +AM ) = 0 .

dxM −→ d(xM + Φ1∂M Φ2) 6= dxM .

DxM := dxM −AM .

(∂M +AM )(∂M +AM ) = 0 .

Under coordinate gauge symmetry, we have the invariance of DxM ,

xM −→ x ′M = xM + Φ1∂M Φ2 ,

AM −→ A′M = AM + d(Φ1∂M Φ2) : A′M∂′M ≡ 0 ,

DxM −→ D′x ′M = DxM = dxM −AM .

Similarly, under (finite) DFT diffeomorphisms à la Hohm-Zwiebach

LMN := ∂M x ′N , L := J LtJ−1 ,

F := 12

(LL−1 + L−1L

), F := J F tJ−1 = 1

(L−1L + LL−1) = F−1 ,

we have the covariance,

xM −→ x ′M (x) ,

HMN (x) −→ H′MN (x ′) = FMK FN

LHKL(x) ,

AM −→ A′M = ANFNM + dX N (L− F )N

M : A′M∂′M ≡ 0 ,

DxM −→ D′x ′M = DxNFNM .

Under coordinate gauge symmetry, we have the invariance of DxM ,

xM −→ x ′M = xM + Φ1∂M Φ2 ,

AM −→ A′M = AM + d(Φ1∂M Φ2) : A′M∂′M ≡ 0 ,

DxM −→ D′x ′M = DxM = dxM −AM .

Similarly, under (finite) DFT diffeomorphisms à la Hohm-Zwiebach

LMN := ∂M x ′N , L := J LtJ−1 ,

F := 12

(LL−1 + L−1L

), F := J F tJ−1 = 1

(L−1L + LL−1) = F−1 ,

we have the covariance,

xM −→ x ′M (x) ,

HMN (x) −→ H′MN (x ′) = FMK FN

LHKL(x) ,

AM −→ A′M = ANFNM + dX N (L− F )N

M : A′M∂′M ≡ 0 ,

DxM −→ D′x ′M = DxNFNM .

Fixing the coordinate gauge symmetry

In DFT –unlike EFT or U-gravity– the solution of the section condition, i.e. the

section is unique up to the duality rotations,

∂xM=

∂xµ,∂

∂xν

0 ,∂

∂xν

): Conventional choice of the section

Then, the ‘coordinate gauge symmetry’ reads(xµ , xν

xµ + φ∂µϕ , xν).

The coordinate gauge potential and the gauged infinitesimal one-form become

AM = Aλ∂M xλ =(

Aµ , 0), DxM =

(dxµ − Aµ , dxν

Fixing the coordinate gauge symmetry

In DFT –unlike EFT or U-gravity– the solution of the section condition, i.e. the

section is unique up to the duality rotations,

∂xM=

∂xµ,∂

∂xν

0 ,∂

∂xν

): Conventional choice of the section

Then, the ‘coordinate gauge symmetry’ reads(xµ , xν

xµ + φ∂µϕ , xν).

The coordinate gauge potential and the gauged infinitesimal one-form become

AM = Aλ∂M xλ =(

Aµ , 0), DxM =

(dxµ − Aµ , dxν

Side remark: Newton mechanics with doubled-yet-gauged coordinates

The doubled-yet-gauged coordinates can be applied to any physical system, not

exclusively to DFT.

Newton mechanics can be formulated on the doubled-yet-gauged space, xM = (xm, xn),

LNewton = 12 m Dt xM Dt xN δMN − V (x) ,

where M,N = 1, 2, · · · , 6 and the potential, V (x), satisfies the section condition.

With the conventional choice of the section, we get

LNewton = 12 m xm xn δmn − V (x) + 1

˙xm − Am

)(˙xn − An

)δmn .

Hence, after integrating out Am, we recover the conventional formulation.

Side remark: Newton mechanics with doubled-yet-gauged coordinates

The doubled-yet-gauged coordinates can be applied to any physical system, not

exclusively to DFT.

Newton mechanics can be formulated on the doubled-yet-gauged space, xM = (xm, xn),

LNewton = 12 m Dt xM Dt xN δMN − V (x) ,

where M,N = 1, 2, · · · , 6 and the potential, V (x), satisfies the section condition.

With the conventional choice of the section, we get

LNewton = 12 m xm xn δmn − V (x) + 1

˙xm − Am

)(˙xn − An

)δmn .

Hence, after integrating out Am, we recover the conventional formulation.

String probes the doubled-yet-gauged spacetime

DFT string action is with Di X M = ∂i X M −AMi , JHP-Lee 2013

14πα′

∫d2σ Lstring , Lstring = − 1

√−h hij Di X M Dj X NHMN (X)− εij Di X MAjM ,

The action is fully symmetric for an arbitrary curved generalized metric,

essentially due to the auxiliary coordinate gauge potential, AMi , under

worldsheet diffeomorphisms plus Weyl symmetry

O(D,D) T-duality

target spacetime DFT-diffeomorphisms

the coordinate gauge symmetry : X M ∼ X M + ϕ∂Mϕ′

c.f. Hull; Tseytlin; Copland, Berman, Thompson; Nibbelink, Patalong; Blair, Malek, Routh

14πα′

Recall that the defining property of the generalized metric is that it is a symmetric

O(D,D) element:

HAB = HBA , HACHB

DJCD = JAB .

There are two types of generalized metric : Riemannian vs. non-Riemannian.

14πα′

O(D,D) element:

HAB = HBA , HACHB

DJCD = JAB .

14πα′

O(D,D) element:

HAB = HBA , HACHB

DJCD = JAB .

DFT backgrounds : Riemannian vs. non-Riemannian

W.r.t. the conventional choice of the section, ∂∂xµ≡ 0, Riemannian generalized metric

assumes the well-known form,

G−1 −G−1B

BG−1 G − BG−1B

Up to field redefinition (e.g. β-gravity Andriot-Betz) this is the most general form of a

symmetric O(D,D) element, if the upper left D × D block is ‘non-degenerate’.

The DFT sigma model then reduces to the standard string action,

14πα′Lstring ≡

12πα′

[− 1

√−hhij∂i Xµ∂j XνGµν(X) + 1

2 εij∂i Xµ∂j XνBµν(X) + 1

2 εij∂i Xµ∂j Xµ

with the bonus of the topological term introduced by Giveon-Rocek; Hull.

The EOM of AMi implies self-duality on the full doubled spacetime,

HMNDi X N + 1√

−hεij Dj X M = 0 .

G−1 −G−1B

12πα′

[− 1

HMNDi X N + 1√

G−1 −G−1B

12πα′

[− 1

HMNDi X N + 1√

W.r.t. ∂∂xµ≡ 0 again, the non-Riemannian DFT background is then characterized by

the degenerate upper left D × D block, such that it does not admit any Riemannian

interpretation even locally!

It turns out that, upon non-Riemannian DFT backgrounds the DFT sigma model

reduces to a ‘chiral’ action in general.

An ‘extreme’ example is the case where HAB = JAB . The DFT sigma model becomes

14πα′ ε

ij∂i Xµ∂j Xµ , ∂i Xµ + 1√−hεi

j∂j Xµ = 0 .

W.r.t. ∂∂xµ≡ 0 again, the non-Riemannian DFT background is then characterized by

the degenerate upper left D × D block, such that it does not admit any Riemannian

interpretation even locally!

It turns out that, upon non-Riemannian DFT backgrounds the DFT sigma model

reduces to a ‘chiral’ action in general.

An ‘extreme’ example is the case where HAB = JAB . The DFT sigma model becomes

14πα′ ε

ij∂i Xµ∂j Xµ , ∂i Xµ + 1√−hεi

j∂j Xµ = 0 .

Non-Riemannian DFT background for non-relativistic string theory

Less simple example of a non-Riemannian DFT background, which is supersymmetric,

can be obtained by performing a doubly T-dual rotation, (t , x1)⇔ (t , x1), of the

known F1 background (Dabholkar-Gibbons-Harvey-Ruiz 1990): With D = 10 = 2 + 8,

0 0 eαβ 0

0 δij 0 0

−eαβ 0 fηαβ 0

0 0 0 δij

, f = 1 + Q

r6 , r2 =∑9

i=2(x i )2

c.f. 2D null-wave à la Berkeley-Berman-Rudolf

Upon this non-Riemannian background, the DFT sigma model reduces precisely to

the non-relativistic string theory action by Gomis-Ooguri.

Further, the sigma model spectrum matches with the ‘perturbation’ of DFT.

Ko-Melby-Thompson-Meyer-JHP 2015

Non-Riemannian DFT background for non-relativistic string theory

Less simple example of a non-Riemannian DFT background, which is supersymmetric,

can be obtained by performing a doubly T-dual rotation, (t , x1)⇔ (t , x1), of the

known F1 background (Dabholkar-Gibbons-Harvey-Ruiz 1990): With D = 10 = 2 + 8,

0 0 eαβ 0

0 δij 0 0

−eαβ 0 fηαβ 0

0 0 0 δij

, f = 1 + Q

r6 , r2 =∑9

i=2(x i )2

c.f. 2D null-wave à la Berkeley-Berman-Rudolf

Upon this non-Riemannian background, the DFT sigma model reduces precisely to

the non-relativistic string theory action by Gomis-Ooguri.

Further, the sigma model spectrum matches with the ‘perturbation’ of DFT.

Phenomenological Implication

Standard Model as a Double Field Theory

with Kangsin Choi 1506.05277 PRL

Standard Model as a Double Field Theory : Prediction

In principle, fermions live on a locally inertial frame, and spin is a gauge symmetry.

Physically, this local Lorentz symmetry means the arbitrariness of the locally inertial

frame at each spacetime point.

SDFT manifests twofold local Lorentz symmetries: Spin(1,D−1)L × Spin(D−1, 1)R ,

and as a consequence it unifies type IIA and IIB supergravities.

Left and right string modes perceive/live on two different locally inertial frames. Duff

SDFT predicts the fermions in Standard Model are twofold: Spin(1, 3)L×Spin(3, 1)R .

(Even after Scherk-Schwarz compactification, the spin group remains still twofold.)

Employing the semi-covariant formalism, we can couple the Standard Model to stringy

backgrounds in a completely covariant manner: It is possible to Double Field Theorize

the Standard Model, without introducing any extra physical degree.

Doing so, one has to decide the spin group for each fermion ⇒ Yukawa coupling.

No experimental evidence of proton decay seems to suggest that the quarks and the

leptons may belong to different spin groups.

If so, this constrains the possible higher order corrections to SM which are

experimentally testable.

Conclusion

Equipped with the DFT extension of the Christoffel connection,

ΓCAB= 2(P∂C PP)[AB]+2(

P[AD PB]

E−P[ADPB]

E)∂DPEC− 4

(PC[APB]

D+PC[APB]D)(∂Dd+(P∂E PP)[ED]

the semi-covariant formalism can Double Field Theorize the various conventional

(Riemannian) theories of GR, including SUGRAs and also the Standard Model.

The twofold spin structure, Spin(1, 9)L × Spin(9, 1)R , naturally unifies IIA and IIB.

Moreover, the fact that the spin is twofold can be an experimentally verifiable

prediction of SDFT, or String Theory.

Conclusion

P[AD PB]

E−P[ADPB]

E)∂DPEC− 4

(PC[APB]

Conclusion

P[AD PB]

E−P[ADPB]

E)∂DPEC− 4

(PC[APB]

Conclusion

P[AD PB]

E−P[ADPB]

E)∂DPEC− 4

(PC[APB]

Thank you.

THE END

APPENDIX : SDFT

1011.1324/1105.6294

Contrary to what it may sound like, the semi-covariant formalism is a

completely covariant approach to DFT, as it manifests simultaneously

O(D,D) T-duality

In particular, it makes each term in D = 10 Maximal SDFT completely covariant:

LType II = e−2d[

+i 12 ργ

]Jeon-Lee-JHP-Suh 2012

It also works for SL(N) duality group, N 6= 4 JHP-Suh ‘U-gravity’ 2014

Notation

Index Representation Metric (raising/lowering indices)

A,B, · · · O(D,D) & DFT-diffeom. vector JAB

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

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

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

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

Field contents of D = 10 Maximal SDFT

Bosons

NS-NS sector

DFT-dilaton: d

DFT-vielbeins: VAp , VAp

R-R potential: Cαα

Fermions (Majorana-Weyl)

DFT-dilatinos: ρα , ρ′α

Gravitinos: ψαp , ψ′αp

R-R potential and Fermions carry NOT (D + D)-dimensional

BUT undoubled D-dimensional indices.

Field contents of D = 10 Maximal SDFT

Bosons

NS-NS sector

DFT-dilaton: d

DFT-vielbeins: VAp , VAp

R-R potential: Cαα

Fermions (Majorana-Weyl)

DFT-dilatinos: ρα , ρ′α

Gravitinos: ψαp , ψ′αp

A priori, O(D,D) rotates only the O(D,D) vector indices (capital Roman), and

the R-R sector and all the fermions are O(D,D) T-duality singlet.

The usual IIA⇔ IIB exchange will follow only after the diagonal gauge fixing

of the twofold local Lorentz symmetries.JEONG-HYUCK PARK SEMI-COVARIANT APPROACH TO DOUBLE FIELD THEORY

The DFT-dilaton gives rise to a scalar density with weight one,

e−2d .

The DFT-vielbeins satisfy four ‘defining’ properties:

VApV Aq = ηpq , VApV A

q = ηpq , VApV Aq = 0 , VApVB

p + VApVBp = JAB .

Naturally, they generate a pair of two-index ‘projectors’,

PAB := VApVBp , PA

BPBC = PA

C , PAB := VApVBp , PA

BPBC = PA

which are symmetric, orthogonal and complementary to each other,

C = 0 , PAB + PA

B = δAB .

Some further projection properties follow

PABVBp = VAp , PA

BVBp = VAp , PABVBp = 0 , PA

BVBp = 0 .

Note also HAB = PAB − PAB . However, our emphasis lies on the ‘projectors’ rather than

the “generalized metric".JEONG-HYUCK PARK SEMI-COVARIANT APPROACH TO DOUBLE FIELD THEORY

The DFT-dilaton gives rise to a scalar density with weight one,

e−2d .

The DFT-vielbeins satisfy four ‘defining’ properties:

VApV Aq = ηpq , VApV A

q = ηpq , VApV Aq = 0 , VApVB

p + VApVBp = JAB .

Naturally, they generate a pair of two-index ‘projectors’,

PAB := VApVBp , PA

BPBC = PA

C , PAB := VApVBp , PA

BPBC = PA

which are symmetric, orthogonal and complementary to each other,

C = 0 , PAB + PA

B = δAB .

Some further projection properties follow

PABVBp = VAp , PA

BVBp = VAp , PABVBp = 0 , PA

BVBp = 0 .

Note also HAB = PAB − PAB . However, our emphasis lies on the ‘projectors’ rather than

the “generalized metric".JEONG-HYUCK PARK SEMI-COVARIANT APPROACH TO DOUBLE FIELD THEORY

We continue to define a pair of six-index projectors,

PCABDEF := PC

DP[A[E PB]

F ] + 2D−1 PC[APB]

[E PF ]D , PCABDEFPDEF

GHI = PCABGHI ,

PCABDEF := PC

DP[A[E PB]

F ] + 2D−1 PC[APB]

[E PF ]D , PCABDEF PDEF

GHI = PCABGHI ,

which are symmetric and traceless,

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

PAABDEF = 0 , PABPABCDEF = 0 , PA

ABDEF = 0 , PABPABCDEF = 0 .

As we shall see shorlty, these projectors govern the DFT-diffeomorphic anomaly in the

semi-covariant formalism, which can be then easily projected out.

Chirality of Spin(1,D−1)L × Spin(D−1, 1)R :

γ(D+1)ψp = cψp , γ(D+1)ρ = −c ρ ,

γ(D+1)ψ′p = c′ψ′p , γ(D+1)ρ′ = −c′ρ′ ,

γ(D+1)Cγ(D+1) = cc′ C ,

where c and c′ are arbitrary independent two sign factors, c2 = c′2 = 1.

A priori, all the possible four different sign choices are equivalent up to

Pin(1,D−1)L × Pin(D−1, 1)R rotations.

That is to say, D = 10 maximal SDFT is chiral with respect to both Pin(1,D−1)L and

Pin(D−1, 1)R , and the theory is unique, unlike IIA/IIB SUGRAs.

Hence, without loss of generality, we may safely set

c ≡ c′ ≡ +1 .

Later we shall see that while the theory is unique, it contains type IIA and IIB

supergravity backgrounds as different kind of solutions.

Having all the ‘right’ field-variables prepared, we now discuss their derivatives or

what we call, ‘semi-covariant derivative’.

The meaning of ‘semi-covariane’ will be clear later.

Having all the ‘right’ field-variables prepared, we now discuss their derivatives or

what we call, ‘semi-covariant derivative’.

The meaning of ‘semi-covariane’ will be clear later.

Semi-covariant derivatives

For each gauge symmetry we assign a corresponding connection,

ΓA for the DFT-diffeomorphism (generalized Lie derivative),

ΦA for the ‘unbarred’ local Lorentz symmetry, Spin(1,D−1)L,

ΦA for the ‘barred’ local Lorentz symmetry, Spin(D−1, 1)R .

Combining all of them, we introduce master ‘semi-covariant’ derivative,

DA = ∂A + ΓA + ΦA + ΦA .

It is also useful to set

∇A = ∂A + ΓA , DA = ∂A + ΦA + ΦA .

The former is the ‘semi-covariant’ derivative for the DFT-diffeomorphism (set by the

generalized Lie derivative),

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

n∑i=1

ΓCAiBTA1···Ai−1BAi+1···An .

And the latter is the covariant derivative for the twofold local Lorenz symmetries.

∇A = ∂A + ΓA , DA = ∂A + ΦA + ΦA .

n∑i=1

∇A = ∂A + ΓA , DA = ∂A + ΦA + ΦA .

n∑i=1

By definition, the master derivative annihilates all the ‘constants’,

DAJBC = ∇AJBC = ΓABDJDC + ΓAC

DJBD = 0 ,

DAηpq = DAηpq = ΦAprηrq + ΦAq

rηpr = 0 ,

DAηpq = DAηpq = ΦApr ηr q + ΦAq

r ηpr = 0 ,

DAC+αβ = DAC+αβ = ΦAαδC+δβ + ΦAβ

δC+αδ = 0 ,

DAC+αβ = DAC+αβ = ΦAαδC+δβ + ΦAβ

δC+αδ = 0 ,

including the gamma matrices,

DA(γp)αβ = DA(γp)αβ = ΦAp

q(γq)αβ + ΦAαδ(γp)δβ − (γp)αδΦA

δβ = 0 ,

DA(γp)αβ = DA(γp)αβ = ΦAp

q(γq)αβ + ΦAαδ(γp)δ β − (γp)αδΦA

δβ = 0 .

It follows then that the connections are all anti-symmetric,

ΓABC = −ΓACB ,

ΦApq = −ΦAqp , ΦAαβ = −ΦAβα ,

and as usual,

ΦAαβ = 1

4 ΦApq(γpq)αβ , ΦAαβ = 1

4 ΦApq(γpq)αβ .

Further, the master derivative is compatible with the whole NS-NS sector,

DAd = ∇Ad := − 12 e2d∇A(e−2d ) = ∂Ad + 1

2 ΓBBA = 0 ,

DAVBp = ∂AVBp + ΓABCVCp + ΦAp

qVBq = 0 ,

DAVBp = ∂AVBp + ΓABC VCp + ΦAp

qVBq = 0 .

It follows that

DAPBC = ∇APBC = 0 , DAPBC = ∇APBC = 0 ,

and the connections are related to each other,

ΓABC = VBpDAVCp + VB

pDAVCp ,

ΦApq = V Bp∇AVBq ,

ΦApq = V Bp∇AVBq .

Further, the master derivative is compatible with the whole NS-NS sector,

DAd = ∇Ad := − 12 e2d∇A(e−2d ) = ∂Ad + 1

2 ΓBBA = 0 ,

DAVBp = ∂AVBp + ΓABCVCp + ΦAp

qVBq = 0 ,

DAVBp = ∂AVBp + ΓABC VCp + ΦAp

qVBq = 0 .

It follows that

DAPBC = ∇APBC = 0 , DAPBC = ∇APBC = 0 ,

and the connections are related to each other,

ΓABC = VBpDAVCp + VB

pDAVCp ,

ΦApq = V Bp∇AVBq ,

ΦApq = V Bp∇AVBq .

The connections assume the following most general forms:

ΓCAB = Γ0CAB + ∆CpqVA

pVBq + ∆CpqVA

pVBq ,

ΦApq = Φ0Apq + ∆Apq ,

ΦApq = Φ0Apq + ∆Apq .

Γ0CAB = 2

(P∂CPP

+ 2(P[A

DPB]E − P[A

DPB]E) ∂DPEC

− 4D−1

(PC[APB]

Jeon-Lee-JHP 2011

and, with the corresponding derivative, ∇0A = ∂A + Γ0

Φ0Apq = V B

p∇0AVBq = V B

p∂AVBq + Γ0ABCV B

pV Cq ,

Φ0Apq = V B

p∇0AVBq = V B

p∂AVBq + Γ0ABC V B

pV Cq .

pVBq + ∆CpqVA

pVBq ,

The extra pieces, ∆Apq and ∆Apq , correspond to the torsion of SDFT, which must be

covariant and, in order to maintain DAd = 0, must satisfy

∆ApqV Ap = 0 , ∆ApqV Ap = 0 .

Otherwise they are arbitrary.

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

ργpqψA , ψpγAψq , ργApqρ , ψpγApqψp ,

where we set ψA = VApψp, γA = VA

pγp .

pVBq + ∆CpqVA

pVBq ,

pγp .

pVBq + ∆CpqVA

pVBq ,

pγp .

The ‘torsionless’ connection,

Γ0CAB = 2

(P∂CPP

+ 2(P[A

DPB]E − P[A

DPB]E) ∂DPEC

− 4D−1

(PC[APB]

further obeys

Γ0ABC + Γ0

BCA + Γ0CAB = 0 ,

PCABDEF Γ0

DEF = 0 , PCABDEF Γ0

DEF = 0 .

In fact, the torsionless connection,

Γ0CAB = 2

(P∂CPP

+ 2(P[A

DPB]E − P[A

DPB]E) ∂DPEC

− 4D−1

(PC[APB]

is the unique solution to the following constraints:

ΓCAB + ΓCBA = 0 =⇒ ∇AJBC = 0 ,

∇APBC = ∇APBC = 0 ,

∇Ad = 0 ,

ΓABC + ΓCAB + ΓBCA = 0 =⇒ L(∂) = L(∇) ,

(P + P)CABDEF ΓDEF = 0 .

In this way, Γ0ABC is the DFT analogy of the Christoffel connection.

However, unlike Christoffel symbol, the DFT-diffeomorphism cannot transform it to

vanish point-wise. This can be viewed as the failure of the Equivalence Principle

applied to an extended object, i.e. string.

Precisely the same expression was re-derived by Hohm-Zwiebach.

In fact, the torsionless connection,

Γ0CAB = 2

(P∂CPP

+ 2(P[A

DPB]E − P[A

DPB]E) ∂DPEC

− 4D−1

(PC[APB]

is the unique solution to the following constraints:

ΓCAB + ΓCBA = 0 =⇒ ∇AJBC = 0 ,

∇APBC = ∇APBC = 0 ,

∇Ad = 0 ,

ΓABC + ΓCAB + ΓBCA = 0 =⇒ L(∂) = L(∇) ,

(P + P)CABDEF ΓDEF = 0 .

In this way, Γ0ABC is the DFT analogy of the Christoffel connection.

However, unlike Christoffel symbol, the DFT-diffeomorphism cannot transform it to

vanish point-wise. This can be viewed as the failure of the Equivalence Principle

applied to an extended object, i.e. string.

Precisely the same expression was re-derived by Hohm-Zwiebach.

Semi-covariant Riemann curvature

The usual curvatures for the three connections,

E ΓAED ,

FABpq = ∂AΦBpq − ∂BΦApq + ΦApr ΦBrq − ΦBpr ΦA

are, from [DA,DB ]VCp = 0 and [DA,DB ]VCp = 0, related to each other,

RABCD = FCDpqVApVB

q + FCDpqVApVB

However, the crucial object in DFT turns out to be

SABCD := 12

ABΓECD

which we name semi-covariant Riemann curvature.

Semi-covariant Riemann curvature

The usual curvatures for the three connections,

E ΓAED ,

FABpq = ∂AΦBpq − ∂BΦApq + ΦApr ΦBrq − ΦBpr ΦA

are, from [DA,DB ]VCp = 0 and [DA,DB ]VCp = 0, related to each other,

RABCD = FCDpqVApVB

q + FCDpqVApVB

However, the crucial object in DFT turns out to be

SABCD := 12

ABΓECD

which we name semi-covariant Riemann curvature.

Properties of the semi-covariant curvature

Under arbitrary variation of the connection, δΓABC , it transforms as

δS0ABCD = D[AδΓ0

B]CD +D[CδΓ0D]AB ,

δSABCD = D[AδΓB]CD +D[CδΓD]AB − 32 Γ[ABE ]δΓE

CD − 32 Γ[CDE ]δΓE

It also satisfies precisely the same symmetric property as the ordinary Riemann

curvature,

SABCD = 12

(S[AB][CD] + S[CD][AB]

S0[ABC]D = 0 ,

as well as projection property,

Sppqq = SABCDV ApV B

pV CqV D

q = 0 .

Properties of the semi-covariant curvature

Under arbitrary variation of the connection, δΓABC , it transforms as

δS0ABCD = D[AδΓ0

B]CD +D[CδΓ0D]AB ,

δSABCD = D[AδΓB]CD +D[CδΓD]AB − 32 Γ[ABE ]δΓE

CD − 32 Γ[CDE ]δΓE

It also satisfies precisely the same symmetric property as the ordinary Riemann

curvature,

SABCD = 12

(S[AB][CD] + S[CD][AB]

S0[ABC]D = 0 ,

as well as projection property,

Sppqq = SABCDV ApV B

pV CqV D

q = 0 .

‘Semi-covariance’

Generically, under DFT-diffeomorphisms, the variation of the semi-covariant derivative

carries anomalous terms which are dictated by the six-index projectors,

δX(∇CTA1···An

)≡ LX

(∇CTA1···An

2(P+P)CAiBFDE∂F∂[DXE ]T···B··· .

Hence, it is not DFT-diffeomorphism covariant,

δX 6= LX .

However, the characteristic property of our ‘semi-covariant’ derivative/curvature is

that, the anomaly can be easily projected out, and can thus produce completely

covariant derivatives/curvatures.

‘Semi-covariance’

Generically, under DFT-diffeomorphisms, the variation of the semi-covariant derivative

carries anomalous terms which are dictated by the six-index projectors,

δX(∇CTA1···An

)≡ LX

(∇CTA1···An

2(P+P)CAiBFDE∂F∂[DXE ]T···B··· .

Hence, it is not DFT-diffeomorphism covariant,

δX 6= LX .

However, the characteristic property of our ‘semi-covariant’ derivative/curvature is

that, the anomaly can be easily projected out, and can thus produce completely

covariant derivatives/curvatures.

Completely covariant derivatives

For O(D,D) tensors:

PCDPA1

B1 PA2B2 · · · PAn

Bn∇DTB1B2···Bn , PCDPA1

B1 PA2B2 · · ·PAn

Bn∇DTB1B2···Bn ,

PAB PC1D1 PC2

D2 · · · PCnDn∇ATBD1D2···Dn ,

PABPC1D1 PC2

D2 · · · PCnDn∇ATBD1D2···Dn

Divergences ,

PAB PC1D1 PC2

D2 · · · PCnDn∇A∇BTD1D2···Dn ,

PABPC1D1 PC2

D2 · · · PCnDn∇A∇BTD1D2···Dn

Laplacians ,

DAC PB1

D1 · · · PBnDn TCD1···Dn , DA

CPB1D1 · · ·PBn

Dn TCD1···Dn ,

where we set a pair of semi-covariant second order differential operators,

DAB := (PA

BPCD − 2PADPBC )(∇C∇D − SCD) , DA

B := (PAB PCD − 2PA

D PBC )(∇C∇D − SCD) ,

which are relevant to the DFT fluctuation analysis Ko-Meyer-Melby-Thompson-JHP

For local Lorentz tensors, Spin(1,D−1)L × Spin(D−1, 1)R :

DpTq1q2···qn , DpTq1q2···qn ,

DpTpq1q2···qn , DpTpq1q2···qn ,

DpDpTq1q2···qn , DpDpTq1q2···qn ,

DpqTqp1p2···pn , Dp

qTqp1p2···pn .

These are the ‘pull-back’ of the previous page using the DFT-vielbeins, such as

Dp := V ApDA , Dp := V A

Following the aforementioned general prescription, completely covariant Yang-Mills

field strength is given by two opposite projections, or

Fpq = V MpV N

qFMN ,

Unlike the Riemannian case, the Γ-connections are not canceled out.

Further, we may freely impose “gauged" section condition to halve the off-shell degrees:

(∂M − i VM )(∂M − i VM ) = 0 ,

which implies VM∂M = 0, ∂MVM = 0,VMVM = 0, like the coordinate gauge potential.

For consistency, the above condition is preserved under all the symmetry

transformations: O(D,D) rotations, diffeomorphisms, and the Yang-Mills gauge

symmetry,

[gVM g−1 − i(∂M g)g−1]∂M = 0 , (LXVM )∂M = [X N∂NVM + (∂M XN − ∂NX M )VN ]∂M = 0 .

Following the aforementioned general prescription, completely covariant Yang-Mills

field strength is given by two opposite projections, or

Fpq = V MpV N

qFMN ,

Unlike the Riemannian case, the Γ-connections are not canceled out.

Further, we may freely impose “gauged" section condition to halve the off-shell degrees:

(∂M − i VM )(∂M − i VM ) = 0 ,

which implies VM∂M = 0, ∂MVM = 0,VMVM = 0, like the coordinate gauge potential.

For consistency, the above condition is preserved under all the symmetry

transformations: O(D,D) rotations, diffeomorphisms, and the Yang-Mills gauge

symmetry,

[gVM g−1 − i(∂M g)g−1]∂M = 0 , (LXVM )∂M = [X N∂NVM + (∂M XN − ∂NX M )VN ]∂M = 0 .

O(D,D) covariant Killing equations in DFT:

LX d = 0 ⇐⇒ ∇M X M = 0 .

Chris Blair 2015

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

γpDpρ = γADAρ , γpDpψp = γADAψp ,

Dpρ , Dpψp = DAψ

ψAγp(DAψq − 12DqψA) ,

γpDpρ′ = γADAρ

′ , γpDpψ′p = γADAψ

′p ,

Dpρ′ , Dpψ′p = DAψ′A ,

ψ′Aγp(DAψ′q − 1

2Dqψ′A) .

Incorporation of fermions into DFT 1109.2035

For R-R potential, Cαβ :

D+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 ,

and hence, they define O(D,D) covariant cohomology.

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

F := D0+C .

Thanks to the nilpotency, the R-R gauge symmetry is simply realized

δC = D0+∆ =⇒ δF = D0

+(δC) = (D0+)2∆ = 0 .

For R-R potential, Cαβ :

D+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 ,

and hence, they define O(D,D) covariant cohomology.

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

F := D0+C .

Thanks to the nilpotency, the R-R gauge symmetry is simply realized

δC = D0+∆ =⇒ δF = D0

+(δC) = (D0+)2∆ = 0 .

Completely covariant curvatures

Scalar curvature:

S := (PABPCD − PABPCD)SACBD

c.f. SABAB = 0 .

“Ricci” curvature:

S0pq = V A

pV BqS0

where we set S0AB = S0

ACBC .

Further, we have conserved “Einstein” curvature,

GAB = 2(PAC PBD − PACPBD)SCD − 12J

ABS , ∇AGAB = 0 .

Combining all the results above, we are now ready to spell

D = 10 Maximally Supersymmetric Double Field Theory

D = 10 Maximal SDFT [ 1210.5078 ]

Lagrangian:

LType II = e−2d[

+i 12 ργ

where F αα denotes the charge conjugation, F := C−1+ FT C+.

As they are contracted with the DFT-vielbeins properly,

every term in the Lagrangian is completey covariant.

c.f. Democratic SUGRA à la Bergshoeff-Kallosh-Ortin-Roest-Van Proeyen

& Generalized Geometry à la Coimbra-Strickland-Constable-Waldram

D = 10 Maximal SDFT [ 1210.5078 ]

Lagrangian:

LType II = e−2d[

+i 12 ργ

where F αα denotes the charge conjugation, F := C−1+ FT C+.

As they are contracted with the DFT-vielbeins properly,

every term in the Lagrangian is completey covariant.

c.f. Democratic SUGRA à la Bergshoeff-Kallosh-Ortin-Roest-Van Proeyen

& Generalized Geometry à la Coimbra-Strickland-Constable-Waldram

D = 10 Maximal SDFT [ 1210.5078 ]

Lagrangian:

LType II = e−2d[

+i 12 ργ

Torsions: The semi-covariant curvature, SABCD , is given by the connection,

ΓABC = Γ0ABC + i 1

3 ργABCρ− 2i ργBCψA − i 13 ψ

pγABCψp + 4iψBγAψC

+ i 13 ρ′γABCρ

′ − 2i ρ′γBCψ′A − i 1

3 ψ′p γABCψ

′p + 4iψ′B γAψ

′C ,

which corresponds to the solution for 1.5 formalism.

The master derivatives in the fermionic kinetic terms are twofold:

D?A for the unprimed fermions and D′?A for the primed fermions, set by

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′p γABCψ

′p − 2iψ′B γAψ

′C + i 5

2 ργBCψA .

D = 10 Maximal SDFT [ 1210.5078 ]

Lagrangian:

LType II = e−2d[

+i 12 ργ

Torsions: The semi-covariant curvature, SABCD , is given by the connection,

ΓABC = Γ0ABC + i 1

3 ργABCρ− 2i ργBCψA − i 13 ψ

pγABCψp + 4iψBγAψC

+ i 13 ρ′γABCρ

′ − 2i ρ′γBCψ′A − i 1

3 ψ′p γABCψ

′p + 4iψ′B γAψ

′C ,

which corresponds to the solution for 1.5 formalism.

The master derivatives in the fermionic kinetic terms are twofold:

D?A for the unprimed fermions and D′?A for the primed fermions, set by

Γ?ABC = ΓABC − i 1196 ργABCρ+ i 5

pγABCψp − 2iψBγAψC + i 52 ρ′γBCψ

′A ,

Γ′?ABC = ΓABC − i 1196 ρ′γABCρ

′A + i 5

24 ψ′p γABCψ

′p − 2iψ′B γAψ

′C + i 5

2 ργBCψA .

D = 10 Maximal SDFT [ 1210.5078 ]

Maximal supersymmetry transformation rules are also completely covariant,

δεd = −i 12 (ερ+ ε′ρ′) ,

δεVAp = i VAq(ε′γqψ

′p − εγpψq) ,

δεVAp = iVAq(εγqψp − ε′γpψ

′q) ,

δεC = i 12 (γpεψ′p − ερ′ − ψp ε

′γp + ρε′) + Cδεd − 12 (V A

q δεVAp)γ(d+1)γpCγq ,

δερ = −γpDpε+ i 12γ

pε ψ′pρ′ − iγpψq ε′γqψ

′p ,

δερ′ = −γpD′pε′ + i 1

2 γpε′ ψpρ− i γqψ′p εγ

pψq ,

δεψp = Dpε+ (F − i 12γ

qρ ψ′q + i 12ψ

q ρ′γq)γpε′ + i 1

4 εψpρ+ i 12ψp ερ ,

δεψ′p = D′pε′ + (F − i 12 γ

qρ′ψq + i 12ψ′q ργq)γpε+ i 1

4 ε′ψ′pρ

′ + i 12ψ′p ε′ρ′ ,

ΓABC = ΓABC − i 1748 ργABCρ+ i 5

pγABCψp − 3iψ′B γAψ′C ,

Γ′ABC = ΓABC − i 1748 ρ′γABCρ

′A + i 1

4 ψ′p γABCψ

′p − 3iψBγAψC .

D = 10 Maximal SDFT [ 1210.5078 ]

Lagrangian:

LType II = e−2d[

+i 12 ργ

The Lagrangian is pseudo: It is necessary to impose a self-duality of the R-R field

strength by hand,

F− :=(

1− γ(D+1))(F − i 1

2ρρ′ + i 1

2γpψqψ

′p γ

q)≡ 0 .

D = 10 Maximal SDFT [ 1210.5078 ]

Lagrangian:

LType II = e−2d[

+i 12 ργ

The Lagrangian is pseudo: It is necessary to impose a self-duality of the R-R field

strength by hand,

F− :=(

1− γ(D+1))(F − i 1

2ρρ′ + i 1

2γpψqψ

′p γ

q)≡ 0 .

D = 10 Maximal SDFT [ 1210.5078 ]

Under the N = 2 SUSY transformation rule, the Lagrangian transforms as

δεLType II = − 18 e−2d V A

qδεVApTr(γpF−γqF−

)+ total derivatives ,

where the precise self-duality relation appears quadratically,

F− :=(

1− γ(D+1))(F − i 1

2ρρ′ + i 1

2γpψqψ

′p γ

This verifies, to the full order in fermions, the supersymmetric invariance of the

action, modulo the self-duality.

For a nontrivial consistency check, the supersymmetric variation of the self-duality

relation is precisely closed by the equations of motion for the gravitinos,

δεF− = −i(Dpρ+ γpDpψp − γpF γpψ

)ε′γp − iγpε

(D′p ρ′ + D′pψ

′p γ

p − ψpγpF γp).

D = 10 Maximal SDFT [ 1210.5078 ]

Under the N = 2 SUSY transformation rule, the Lagrangian transforms as

δεLType II = − 18 e−2d V A

qδεVApTr(γpF−γqF−

)+ total derivatives ,

where the precise self-duality relation appears quadratically,

F− :=(

1− γ(D+1))(F − i 1

2ρρ′ + i 1

2γpψqψ

′p γ

This verifies, to the full order in fermions, the supersymmetric invariance of the

action, modulo the self-duality.

For a nontrivial consistency check, the supersymmetric variation of the self-duality

relation is precisely closed by the equations of motion for the gravitinos,

δεF− = −i(Dpρ+ γpDpψp − γpF γpψ

)ε′γp − iγpε

(D′p ρ′ + D′pψ

′p γ

p − ψpγpF γp).

Equations of Motion

DFT-vielbein:

Spq+Tr(γpF γqF)+i ργpDqρ+2iψqDpρ−iψpγpDqψp+i ρ′γqDpρ′+2iψ′pDqρ

′−iψ′q γqDpψ′q= 0.

This is DFT-generalization of Einstein equation.

DFT-dilaton:

LType II = 0 .

Namely, the on-shell Lagrangian vanishes.

R-R potential:

(F − iρρ′ + iγrψsψ

′r γ

which is automatically met by the self-duality, together with the nilpotency of D0+,

′r γ

= D0−

(γ(D+1)F

)= −γ(D+1)D0

+F = −γ(D+1)(D0+)2C = 0 .

The 1.5 formalism works: The variation of the Lagrangian induced by that of the

connection is trivial, δLType II = δΓABC × 0 .

Equations of Motion

DFT-vielbein:

DFT-dilaton:

LType II = 0 .

R-R potential:

′r γ

= D0−

(γ(D+1)F

)= −γ(D+1)D0

+F = −γ(D+1)(D0+)2C = 0 .

Equations of Motion

DFT-vielbein:

DFT-dilaton:

LType II = 0 .

R-R potential:

′r γ

= D0−

(γ(D+1)F

)= −γ(D+1)D0

+F = −γ(D+1)(D0+)2C = 0 .

Equations of Motion

DFT-vielbein:

DFT-dilaton:

LType II = 0 .

R-R potential:

′r γ

= D0−

(γ(D+1)F

)= −γ(D+1)D0

+F = −γ(D+1)(D0+)2C = 0 .

Equations of Motion

DFT-vielbein:

DFT-dilaton:

LType II = 0 .

R-R potential:

′r γ

= D0−

(γ(D+1)F

)= −γ(D+1)D0

+F = −γ(D+1)(D0+)2C = 0 .

Truncation to N = 1 D = 10 SDFT [1112.0069]

Turning off the primed fermions and the R-R sector truncates the N = 2 D = 10

SDFT to N = 1 D = 10 SDFT,

LN=1 = e−2d[

(PABPCD − PABPCD)SACBD + i 1

2 ργAD?Aρ− iψAD?Aρ− i 1

2 ψBγAD?AψB

N = 1 Local SUSY:

δεd = −i 12 ερ ,

δεVAp = −i εγpψA ,

δεVAp = i εγAψp ,

δερ = −γADAε ,

δεψp = V ApDAε− i 1

4 (ρψp)ε+ i 12 (ερ)ψp .

N = 1 SUSY Algebra [1112.0069]

Commutator of supersymmetry reads

[δε1 , δε2 ] ≡ LX3 + δε3 + δso(1,9)L+ δso(9,1)R

+ δtrivial .

X A3 = i ε1γ

Aε2 , ε3 = i 12 [(ε1γ

pε2)γpρ+ (ρε2)ε1 − (ρε1)ε2] , etc.

and δtrivial corresponds to the fermionic equations of motion.

Reduction to SUGRA

Now we are going to

parametrize the DFT-field-variables in terms of Riemannian variables,

discuss the ‘unification’ of IIA and IIB,

choose a diagonal gauge of Spin(1,D−1)L × Spin(D−1, 1)R ,

and reduce SDFT to SUGRAs.

Parametrization: Reduction to Generalized Geometry

As stressed before, one of the characteristic features in our construction of D = 10

maximal SDFT is the usage of the O(D,D) covariant, genuine DFT-field-variables.

However, the relation to an ordinary supergravity can be established only after we

solve the defining algebraic relations of the DFT-vielbeins and parametrize the

solution in terms of Riemannian variables, i.e. zehnbeins and B-field.

Assuming that the upper half blocks are non-degenerate, the DFT-vielbein takes the

general form,

VAp = 1√2

(e−1)pµ

(B + e)νp

, VAp = 1√2

(e−1)pµ

(B + e)νp

Here eµp and eν p are two copies of the D-dimensional vielbein corresponding to the

same spacetime metric,

eµpeνqηpq = −eµ peν q ηpq = gµν ,

and further we set Bµp = Bµν(e−1)pν , Bµp = Bµν(e−1)p

As stressed before, one of the characteristic features in our construction of D = 10

maximal SDFT is the usage of the O(D,D) covariant, genuine DFT-field-variables.

However, the relation to an ordinary supergravity can be established only after we

solve the defining algebraic relations of the DFT-vielbeins and parametrize the

solution in terms of Riemannian variables, i.e. zehnbeins and B-field.

Assuming that the upper half blocks are non-degenerate, the DFT-vielbein takes the

general form,

VAp = 1√2

(e−1)pµ

(B + e)νp

, VAp = 1√2

(e−1)pµ

(B + e)νp

Here eµp and eν p are two copies of the D-dimensional vielbein corresponding to the

same spacetime metric,

eµpeνqηpq = −eµ peν q ηpq = gµν ,

and further we set Bµp = Bµν(e−1)pν , Bµp = Bµν(e−1)p

Instead, we may choose an alternative parametrization,

VAp = 1√

(β + e)µp

(e−1)pν

, VAp = 1√

(β + ¯e)µp

(¯e−1)pν

where βµp = βµν(e−1)pν , βµp = βµν(¯e−1)p

ν , and eµp, ¯eµ p correspond to

a pair of T-dual vielbeins for winding modes,

eµp eνqηpq = −¯eµp

¯eν qηpq = (g − Bg−1B)−1µν .

Note that in the above T-dual winding mode sector, the D-dimensional curved

spacetime indices are all upside-down: xµ, eµp, ¯eµ p, βµν (cf. xµ, eµp , eµ p , Bµν ).

Two parametrizations:

VAp = 1√2

(e−1)pµ

(B + e)νp

, VAp = 1√2

(e−1)pµ

(B + e)νp

versus

VAp = 1√

(β + e)µp

(e−1)pν

, VAp = 1√

(β + ¯e)µp

(¯e−1)pν

In connection to the section condition, ∂A∂A ≡ 0, the former matches well with the

choice, ∂∂xµ≡ 0, while the latter is natural when ∂

∂xµ ≡ 0.

Yet if we consider dimensional reductions from D to lower dimensions,

there is no longer preferred parametrization =⇒ “Non-geometry”

c.f. Other parametrizations: Lüst, Andriot, Betz, Blumenhagen, Fuchs, Sun et al.

Two parametrizations:

VAp = 1√2

(e−1)pµ

(B + e)νp

, VAp = 1√2

(e−1)pµ

(B + e)νp

versus

VAp = 1√

(β + e)µp

(e−1)pν

, VAp = 1√

(β + ¯e)µp

(¯e−1)pν

In connection to the section condition, ∂A∂A ≡ 0, the former matches well with the

choice, ∂∂xµ≡ 0, while the latter is natural when ∂

∂xµ ≡ 0.

Yet if we consider dimensional reductions from D to lower dimensions,

there is no longer preferred parametrization =⇒ “Non-geometry”

c.f. Other parametrizations: Lüst, Andriot, Betz, Blumenhagen, Fuchs, Sun et al.

We re-emphasize that SDFT can describe not only Riemannian (SUGRA)

backgrounds but also novel non-Riemannian (“metric-less") backgrounds.

For example, the Gomis-Ooguri non-relativistic string theory can be readily realized

within DFT on such a non-Riemannian background.

The sigma model spectrum matches with the pertubations of DFT around the

non-Riemannian background.

From now on, let us restrict ourselves to the former parametrization and impose∂∂xµ≡ 0.

This reduces (S)DFT to ‘Generalized Geometry’

Hitchin; Grana, Minasian, Petrini, Waldram

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

√2γADAρ ≡ γm

(∂mρ+ 1

4ωmnpγnpρ+ 124 Hmnpγnpρ− ∂mφρ

√2γADAψp ≡ γm

(∂mψp + 1

4ωmnpγnpψp + ωmpqψq + 1

24 Hmnpγnpψp + 12 Hmpqψ

q − ∂mφψp

√2V A

pDAρ ≡ ∂pρ+ 14ωpqrγ

qrρ+ 18 Hpqrγ

qrρ ,

√2DAψ

A ≡ ∂pψp + 14ωpqrγ

qrψp + ωppqψ

q + 18 Hpqrγ

qrψp − 2∂pφψp .

ωµ ± 12 Hµ and ωµ ± 1

6 Hµ naturally appear as spin connections. Liu, Minasian

From now on, let us restrict ourselves to the former parametrization and impose∂∂xµ≡ 0.

This reduces (S)DFT to ‘Generalized Geometry’

Hitchin; Grana, Minasian, Petrini, Waldram

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

√2γADAρ ≡ γm

(∂mρ+ 1

4ωmnpγnpρ+ 124 Hmnpγnpρ− ∂mφρ

√2γADAψp ≡ γm

(∂mψp + 1

4ωmnpγnpψp + ωmpqψq + 1

24 Hmnpγnpψp + 12 Hmpqψ

q − ∂mφψp

√2V A

pDAρ ≡ ∂pρ+ 14ωpqrγ

qrρ+ 18 Hpqrγ

qrρ ,

√2DAψ

A ≡ ∂pψp + 14ωpqrγ

qrψp + ωppqψ

q + 18 Hpqrγ

qrψp − 2∂pφψp .

ωµ ± 12 Hµ and ωµ ± 1

6 Hµ naturally appear as spin connections. Liu, Minasian

Unification of type IIA and IIB SUGRAs

Since the two zehnbeins correspond to the same spacetime metric, they are related by

a Lorentz rotation,

(e−1e)pp(e−1e)q

q ηpq = −ηpq .

There exists also a spinorial representation for this local Lorentz rotation, Se,

SeγpS−1

e = γ(D+1)γp(e−1e)pp ,

such that, in particular,

Seγ(D+1)S−1

e = − det(e−1e)γ(D+1) .

The D = 10 maximal SDFT ‘Riemannian’ solutions are then classified into two groups,

det(e−1e) = +1 : type IIA ,

det(e−1e) = −1 : type IIB .

This identification with the ordinary IIA/IIB SUGRAs can be established, if we ‘fix’

the two zehnbeins equal to each other,

eµp ≡ eµ p ,

using a Pin(D−1, 1)R local Lorentz rotation which may or may not flip the

Pin(D−1, 1)R chirality,

c′ ≡ +1 −→ c′ = det(e−1e) .

Namely, the Pin(D−1, 1)R chirality changes iff det(e−1e) = −1.

This identification with the ordinary IIA/IIB SUGRAs can be established, if we ‘fix’

the two zehnbeins equal to each other,

eµp ≡ eµ p ,

using a Pin(D−1, 1)R local Lorentz rotation which may or may not flip the

Pin(D−1, 1)R chirality,

c′ ≡ +1 −→ c′ = det(e−1e) .

Namely, the Pin(D−1, 1)R chirality changes iff det(e−1e) = −1.

That is to say, formulated in terms of the genuine DFT-field variables, i.e. VAp, VAp,

Cαα, etc. the D = 10 maximal SDFT is a chiral theory with respect to the pair of local

Lorentz groups. The possible four chirality choices are all equivalent and hence the

theory is unique. We can safely put c ≡ c′ ≡ +1 without loss of generality.

However, the theory contains two ‘types’ of Riemannian solutions, as classified above.

Conversely, any solution in type IIA and type IIB supergravities can be mapped to a

solution of D = 10 maximal SDFT of fixed chirality, i.e. c ≡ c′ ≡ +1.

In conclusion, the single unique D = 10 maximal SDFT unifies type IIA and IIB

SUGRAs. Further it allows non-Riemannian solutions.