Post on 09-Jun-2020
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
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.
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.
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.
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.
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.
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.
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.
JEONG-HYUCK PARK SEMI-COVARIANT APPROACH TO DOUBLE FIELD THEORY
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 ...
JEONG-HYUCK PARK SEMI-COVARIANT APPROACH TO DOUBLE FIELD THEORY
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 ...
JEONG-HYUCK PARK SEMI-COVARIANT APPROACH TO DOUBLE FIELD THEORY
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
JEONG-HYUCK PARK SEMI-COVARIANT APPROACH TO DOUBLE FIELD THEORY
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
JEONG-HYUCK PARK SEMI-COVARIANT APPROACH TO DOUBLE FIELD THEORY
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,
JAB =
0 1
1 0
JEONG-HYUCK PARK SEMI-COVARIANT APPROACH TO DOUBLE FIELD THEORY
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,
JAB =
0 1
1 0
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+
JEONG-HYUCK PARK SEMI-COVARIANT APPROACH TO DOUBLE FIELD THEORY
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.
JEONG-HYUCK PARK SEMI-COVARIANT APPROACH TO DOUBLE FIELD THEORY
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.
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.
JEONG-HYUCK PARK SEMI-COVARIANT APPROACH TO DOUBLE FIELD 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 .
JEONG-HYUCK PARK SEMI-COVARIANT APPROACH TO DOUBLE FIELD 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 .
∗ 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.
JEONG-HYUCK PARK SEMI-COVARIANT APPROACH TO DOUBLE FIELD THEORY
Semi-covariant formulation of Double Field Theory
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
JEONG-HYUCK PARK SEMI-COVARIANT APPROACH TO DOUBLE FIELD THEORY
Semi-covariant formulation of Double Field Theory
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, ...
JEONG-HYUCK PARK SEMI-COVARIANT APPROACH TO DOUBLE FIELD THEORY
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∑
i=1
(∂Ai XB − ∂BXAi )TA1···Ai−1B
Ai+1···An ,
where ωT denotes the weight.
JEONG-HYUCK PARK SEMI-COVARIANT APPROACH TO DOUBLE FIELD THEORY
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∑
i=1
(∂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 .
JEONG-HYUCK PARK SEMI-COVARIANT APPROACH TO DOUBLE FIELD THEORY
Diffeomorphisms.
Hohm-Zwiebach ansatz for finite transformations:
F := 12
(LL−1 + L−1L
), F := J F tJ−1 = 1
2
(L−1L + LL−1) = F−1 ,
where
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
JEONG-HYUCK PARK SEMI-COVARIANT APPROACH TO DOUBLE FIELD THEORY
Diffeomorphisms.
Hohm-Zwiebach ansatz for finite transformations:
F := 12
(LL−1 + L−1L
), F := J F tJ−1 = 1
2
(L−1L + LL−1) = F−1 ,
where
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
JEONG-HYUCK PARK SEMI-COVARIANT APPROACH TO DOUBLE FIELD THEORY
Diffeomorphisms.
Hohm-Zwiebach ansatz for finite transformations:
F := 12
(LL−1 + L−1L
), F := J F tJ−1 = 1
2
(L−1L + LL−1) = F−1 ,
where
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
JEONG-HYUCK PARK SEMI-COVARIANT APPROACH TO DOUBLE FIELD THEORY
Diffeomorphisms.
Hohm-Zwiebach ansatz for finite transformations:
F := 12
(LL−1 + L−1L
), F := J F tJ−1 = 1
2
(L−1L + LL−1) = F−1 ,
where
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
JEONG-HYUCK PARK SEMI-COVARIANT APPROACH TO DOUBLE FIELD THEORY
Semi-covariant formulation of Double Field Theory
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
A .
Further, the projectors are orthogonal and complementary,
PABPB
C = 0 , PAB + PAB = JAB .
JEONG-HYUCK PARK SEMI-COVARIANT APPROACH TO DOUBLE FIELD THEORY
Semi-covariant formulation of Double Field Theory
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
A .
Further, the projectors are orthogonal and complementary,
PABPB
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".
JEONG-HYUCK PARK SEMI-COVARIANT APPROACH TO DOUBLE FIELD THEORY
Semi-covariant formulation of Double Field Theory
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 .
JEONG-HYUCK PARK SEMI-COVARIANT APPROACH TO DOUBLE FIELD THEORY
Semi-covariant formulation of Double Field Theory
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
JEONG-HYUCK PARK SEMI-COVARIANT APPROACH TO DOUBLE FIELD THEORY
Semi-covariant formulation of Double Field Theory
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 ,
and
JEONG-HYUCK PARK SEMI-COVARIANT APPROACH TO DOUBLE FIELD THEORY
Semi-covariant formulation of Double Field Theory
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 ,
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 .
JEONG-HYUCK PARK SEMI-COVARIANT APPROACH TO DOUBLE FIELD THEORY
Semi-covariant formulation of Double Field Theory
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 ,
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 .
As will be explained below, one can determine the (torsionelss) connection uniquely:
ΓCAB = 2(P∂CPP
)[AB]
+ 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.
JEONG-HYUCK PARK SEMI-COVARIANT APPROACH TO DOUBLE FIELD THEORY
Semi-covariant formulation of Double Field Theory
The semi-covariant derivative then obeys the Leibniz rule and annihilates the O(D,D)
invariant constant metric,
∇AJBC = 0 .
JEONG-HYUCK PARK SEMI-COVARIANT APPROACH TO DOUBLE FIELD THEORY
Semi-covariant formulation of Double Field Theory
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 .
JEONG-HYUCK PARK SEMI-COVARIANT APPROACH TO DOUBLE FIELD THEORY
Semi-covariant formulation of Double Field Theory
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 .
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,
PIAPJ
BPKC PL
DSABCD=0 , PIAPJ
BPKC PL
DSABCD = 0 .
Notably, it turns out that the naive scalar contraction vanishes identically,
SABAB = 0 .
JEONG-HYUCK PARK SEMI-COVARIANT APPROACH TO DOUBLE FIELD THEORY
Semi-covariant formulation of Double Field Theory
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 .
JEONG-HYUCK PARK SEMI-COVARIANT APPROACH TO DOUBLE FIELD THEORY
Semi-covariant formulation of Double Field Theory
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 .
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.
JEONG-HYUCK PARK SEMI-COVARIANT APPROACH TO DOUBLE FIELD THEORY
Semi-covariant formulation of Double Field Theory
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 .
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.
JEONG-HYUCK PARK SEMI-COVARIANT APPROACH TO DOUBLE FIELD THEORY
Semi-covariant formulation of Double Field Theory
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 .
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.
JEONG-HYUCK PARK SEMI-COVARIANT APPROACH TO DOUBLE FIELD THEORY
Semi-covariant formulation of Double Field Theory
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
)[AB]
+ 2(P[A
DPB]E − P[A
DPB]E) ∂DPEC
− 4D−1
(PC[APB]
D + PC[APB]D)(∂Dd + (P∂E PP)[ED]
)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.
JEONG-HYUCK PARK SEMI-COVARIANT APPROACH TO DOUBLE FIELD THEORY
Semi-covariant formulation of Double Field Theory
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
GHI .
Further, they are symmetric and traceless,
PABCDEF = PDEFABC , PABCDEF = PA[BC]D[EF ] , PABPABCDEF = 0 ,
PABCDEF = PDEFABC , PABCDEF = PA[BC]D[EF ] , PABPABCDEF = 0 .
JEONG-HYUCK PARK SEMI-COVARIANT APPROACH TO DOUBLE FIELD THEORY
Semi-covariant formulation of Double Field Theory
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∑
i=1
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
).
JEONG-HYUCK PARK SEMI-COVARIANT APPROACH TO DOUBLE FIELD THEORY
Semi-covariant formulation of Double Field Theory
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) ,
PABPC1D1 · · · PCn
Dn∇A∇BTD1···Dn , PABPC1D1 · · ·PCn
Dn∇A∇BTD1···Dn (Laplacians) ,
and
JEONG-HYUCK PARK SEMI-COVARIANT APPROACH TO DOUBLE FIELD THEORY
Semi-covariant formulation of Double Field Theory
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) ,
PABPC1D1 · · · PCn
Dn∇A∇BTD1···Dn , PABPC1D1 · · ·PCn
Dn∇A∇BTD1···Dn (Laplacians) ,
and
PAC PB
DSCEDE (“Ricci” curvature) ,
(PACPBD − PAC PBD)SABCD (scalar curvature) .
JEONG-HYUCK PARK SEMI-COVARIANT APPROACH TO DOUBLE FIELD THEORY
Semi-covariant formulation of Double Field Theory
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) ,
PABPC1D1 · · · PCn
Dn∇A∇BTD1···Dn , PABPC1D1 · · ·PCn
Dn∇A∇BTD1···Dn (Laplacians) ,
and
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 .
JEONG-HYUCK PARK SEMI-COVARIANT APPROACH TO DOUBLE FIELD THEORY
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
JEONG-HYUCK PARK SEMI-COVARIANT APPROACH TO DOUBLE FIELD THEORY
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
JEONG-HYUCK PARK SEMI-COVARIANT APPROACH TO DOUBLE FIELD THEORY
Semi-covariant formulation of Double Field Theory
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.
JEONG-HYUCK PARK SEMI-COVARIANT APPROACH TO DOUBLE FIELD THEORY
Semi-covariant formulation of Double Field Theory
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.
JEONG-HYUCK PARK SEMI-COVARIANT APPROACH TO DOUBLE FIELD THEORY
Semi-covariant formulation of Double Field Theory
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.
JEONG-HYUCK PARK SEMI-COVARIANT APPROACH TO DOUBLE FIELD THEORY
Semi-covariant formulation of Double Field Theory
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 .
JEONG-HYUCK PARK SEMI-COVARIANT APPROACH TO DOUBLE FIELD THEORY
Semi-covariant formulation of Double Field Theory
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 .
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.
JEONG-HYUCK PARK SEMI-COVARIANT APPROACH TO DOUBLE FIELD THEORY
Semi-covariant formulation of Double Field Theory
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 .
JEONG-HYUCK PARK SEMI-COVARIANT APPROACH TO DOUBLE FIELD THEORY
Semi-covariant formulation of Double Field Theory
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 .
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.
JEONG-HYUCK PARK SEMI-COVARIANT APPROACH TO DOUBLE FIELD THEORY
Semi-covariant formulation of Double Field Theory
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
where
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.
JEONG-HYUCK PARK SEMI-COVARIANT APPROACH TO DOUBLE FIELD THEORY
Semi-covariant formulation of Double Field Theory
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
where
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
JEONG-HYUCK PARK SEMI-COVARIANT APPROACH TO DOUBLE FIELD THEORY
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
JEONG-HYUCK PARK SEMI-COVARIANT APPROACH TO DOUBLE FIELD THEORY
Supersymmetric Extension
JEONG-HYUCK PARK SEMI-COVARIANT APPROACH TO DOUBLE FIELD THEORY
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
′p
]Jeon-Lee-Suh-JHP 1210.5078
c.f. other approaches: Coimbra, Strickland-Constanble, Waldram
Kwak, Hohm, Zwiebach
JEONG-HYUCK PARK SEMI-COVARIANT APPROACH TO DOUBLE FIELD THEORY
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.
JEONG-HYUCK PARK SEMI-COVARIANT APPROACH TO DOUBLE FIELD THEORY
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.
JEONG-HYUCK PARK SEMI-COVARIANT APPROACH TO DOUBLE FIELD THEORY
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.
JEONG-HYUCK PARK SEMI-COVARIANT APPROACH TO DOUBLE FIELD THEORY
Unificaiton of IIA and IIB
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.
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.
JEONG-HYUCK PARK SEMI-COVARIANT APPROACH TO DOUBLE FIELD THEORY
Unificaiton of IIA and IIB
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.
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.
JEONG-HYUCK PARK SEMI-COVARIANT APPROACH TO DOUBLE FIELD THEORY
Unificaiton of IIA and IIB
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.
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.
JEONG-HYUCK PARK SEMI-COVARIANT APPROACH TO DOUBLE FIELD THEORY
Unificaiton of IIA and IIB
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.
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.
JEONG-HYUCK PARK SEMI-COVARIANT APPROACH TO DOUBLE FIELD THEORY
Worldsheet Perspective:
Doubled-yet-gauged coordinates
1304.5946/1307.8377
JEONG-HYUCK PARK SEMI-COVARIANT APPROACH TO DOUBLE FIELD THEORY
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 .
JEONG-HYUCK PARK SEMI-COVARIANT APPROACH TO DOUBLE FIELD THEORY
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 .
JEONG-HYUCK PARK SEMI-COVARIANT APPROACH TO DOUBLE FIELD THEORY
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 .
JEONG-HYUCK PARK SEMI-COVARIANT APPROACH TO DOUBLE FIELD THEORY
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 .
JEONG-HYUCK PARK SEMI-COVARIANT APPROACH TO DOUBLE FIELD THEORY
Doubled-yet-gauged coordinates & Gauged infinitesimal one-form
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
2
(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 .
JEONG-HYUCK PARK SEMI-COVARIANT APPROACH TO DOUBLE FIELD THEORY
Doubled-yet-gauged coordinates & Gauged infinitesimal one-form
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
2
(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 .
JEONG-HYUCK PARK SEMI-COVARIANT APPROACH TO DOUBLE FIELD THEORY
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
).
JEONG-HYUCK PARK SEMI-COVARIANT APPROACH TO DOUBLE FIELD THEORY
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
).
JEONG-HYUCK PARK SEMI-COVARIANT APPROACH TO DOUBLE FIELD THEORY
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
2 m(
˙xm − Am
)(˙xn − An
)δmn .
Hence, after integrating out Am, we recover the conventional formulation.
JEONG-HYUCK PARK SEMI-COVARIANT APPROACH TO DOUBLE FIELD THEORY
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
2 m(
˙xm − Am
)(˙xn − An
)δmn .
Hence, after integrating out Am, we recover the conventional formulation.
JEONG-HYUCK PARK SEMI-COVARIANT APPROACH TO DOUBLE FIELD THEORY
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
2
√−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
JEONG-HYUCK PARK SEMI-COVARIANT APPROACH TO DOUBLE FIELD THEORY
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
2
√−h hij Di X M Dj X NHMN (X)− εij Di X MAjM ,
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.
JEONG-HYUCK PARK SEMI-COVARIANT APPROACH TO DOUBLE FIELD THEORY
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
2
√−h hij Di X M Dj X NHMN (X)− εij Di X MAjM ,
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.
JEONG-HYUCK PARK SEMI-COVARIANT APPROACH TO DOUBLE FIELD THEORY
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
2
√−h hij Di X M Dj X NHMN (X)− εij Di X MAjM ,
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.
JEONG-HYUCK PARK SEMI-COVARIANT APPROACH TO DOUBLE FIELD THEORY
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,
HAB =
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
2
√−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 .
JEONG-HYUCK PARK SEMI-COVARIANT APPROACH TO DOUBLE FIELD THEORY
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,
HAB =
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
2
√−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 .
JEONG-HYUCK PARK SEMI-COVARIANT APPROACH TO DOUBLE FIELD THEORY
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,
HAB =
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
2
√−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 .
JEONG-HYUCK PARK SEMI-COVARIANT APPROACH TO DOUBLE FIELD THEORY
DFT backgrounds : Riemannian vs. non-Riemannian
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′Lstring ≡
14πα′ ε
ij∂i Xµ∂j Xµ , ∂i Xµ + 1√−hεi
j∂j Xµ = 0 .
JEONG-HYUCK PARK SEMI-COVARIANT APPROACH TO DOUBLE FIELD THEORY
DFT backgrounds : Riemannian vs. non-Riemannian
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′Lstring ≡
14πα′ ε
ij∂i Xµ∂j Xµ , ∂i Xµ + 1√−hεi
j∂j Xµ = 0 .
JEONG-HYUCK PARK SEMI-COVARIANT APPROACH TO DOUBLE FIELD THEORY
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,
HMN =
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
JEONG-HYUCK PARK SEMI-COVARIANT APPROACH TO DOUBLE FIELD THEORY
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,
HMN =
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
JEONG-HYUCK PARK SEMI-COVARIANT APPROACH TO DOUBLE FIELD THEORY
Phenomenological Implication
Standard Model as a Double Field Theory
with Kangsin Choi 1506.05277 PRL
JEONG-HYUCK PARK SEMI-COVARIANT APPROACH TO DOUBLE FIELD THEORY
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.
JEONG-HYUCK PARK SEMI-COVARIANT APPROACH TO DOUBLE FIELD THEORY
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.
JEONG-HYUCK PARK SEMI-COVARIANT APPROACH TO DOUBLE FIELD THEORY
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.
JEONG-HYUCK PARK SEMI-COVARIANT APPROACH TO DOUBLE FIELD THEORY
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.
JEONG-HYUCK PARK SEMI-COVARIANT APPROACH TO DOUBLE FIELD THEORY
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
D−1
(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.
JEONG-HYUCK PARK SEMI-COVARIANT APPROACH TO DOUBLE FIELD THEORY
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
D−1
(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.
JEONG-HYUCK PARK SEMI-COVARIANT APPROACH TO DOUBLE FIELD THEORY
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
D−1
(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.
JEONG-HYUCK PARK SEMI-COVARIANT APPROACH TO DOUBLE FIELD THEORY
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
D−1
(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.
Thank you.
JEONG-HYUCK PARK SEMI-COVARIANT APPROACH TO DOUBLE FIELD THEORY
THE END
JEONG-HYUCK PARK SEMI-COVARIANT APPROACH TO DOUBLE FIELD THEORY
APPENDIX : SDFT
1011.1324/1105.6294
JEONG-HYUCK PARK SEMI-COVARIANT APPROACH TO DOUBLE FIELD THEORY
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
DFT-diffeomorphisms (generalized Lie derivative)
A pair of local Lorentz symmetries, Spin(1,D−1)L × Spin(D−1, 1)R
In particular, it makes each term in D = 10 Maximal SDFT completely covariant:
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
′p
]Jeon-Lee-JHP-Suh 2012
It also works for SL(N) duality group, N 6= 4 JHP-Suh ‘U-gravity’ 2014
JEONG-HYUCK PARK SEMI-COVARIANT APPROACH TO DOUBLE FIELD THEORY
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+
JEONG-HYUCK PARK SEMI-COVARIANT APPROACH TO DOUBLE FIELD THEORY
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.
JEONG-HYUCK PARK SEMI-COVARIANT APPROACH TO DOUBLE FIELD THEORY
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
C ,
which are symmetric, orthogonal and complementary to each other,
PAB = PBA , PAB = PBA , PABPB
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
C ,
which are symmetric, orthogonal and complementary to each other,
PAB = PBA , PAB = PBA , PABPB
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.
JEONG-HYUCK PARK SEMI-COVARIANT APPROACH TO DOUBLE FIELD THEORY
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.
JEONG-HYUCK PARK SEMI-COVARIANT APPROACH TO DOUBLE FIELD THEORY
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.
JEONG-HYUCK PARK SEMI-COVARIANT APPROACH TO DOUBLE FIELD THEORY
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.
JEONG-HYUCK PARK SEMI-COVARIANT APPROACH TO DOUBLE FIELD THEORY
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 .
JEONG-HYUCK PARK SEMI-COVARIANT APPROACH TO DOUBLE FIELD THEORY
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.
JEONG-HYUCK PARK SEMI-COVARIANT APPROACH TO DOUBLE FIELD THEORY
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.
JEONG-HYUCK PARK SEMI-COVARIANT APPROACH TO DOUBLE FIELD THEORY
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.
JEONG-HYUCK PARK SEMI-COVARIANT APPROACH TO DOUBLE FIELD THEORY
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 .
JEONG-HYUCK PARK SEMI-COVARIANT APPROACH TO DOUBLE FIELD THEORY
It follows then that the connections are all anti-symmetric,
ΓABC = −ΓACB ,
ΦApq = −ΦAqp , ΦAαβ = −ΦAβα ,
ΦApq = −ΦAqp , ΦAαβ = −ΦAβα ,
and as usual,
ΦAαβ = 1
4 ΦApq(γpq)αβ , ΦAαβ = 1
4 ΦApq(γpq)αβ .
JEONG-HYUCK PARK SEMI-COVARIANT APPROACH TO DOUBLE FIELD THEORY
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 .
JEONG-HYUCK PARK SEMI-COVARIANT APPROACH TO DOUBLE FIELD THEORY
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 .
JEONG-HYUCK PARK SEMI-COVARIANT APPROACH TO DOUBLE FIELD THEORY
The connections assume the following most general forms:
ΓCAB = Γ0CAB + ∆CpqVA
pVBq + ∆CpqVA
pVBq ,
ΦApq = Φ0Apq + ∆Apq ,
ΦApq = Φ0Apq + ∆Apq .
Here
Γ0CAB = 2
(P∂CPP
)[AB]
+ 2(P[A
DPB]E − P[A
DPB]E) ∂DPEC
− 4D−1
(PC[APB]
D + PC[APB]D)(∂Dd + (P∂E PP)[ED]
),
Jeon-Lee-JHP 2011
and, with the corresponding derivative, ∇0A = ∂A + Γ0
A,
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 .
JEONG-HYUCK PARK SEMI-COVARIANT APPROACH TO DOUBLE FIELD THEORY
The connections assume the following most general forms:
ΓCAB = Γ0CAB + ∆CpqVA
pVBq + ∆CpqVA
pVBq ,
ΦApq = Φ0Apq + ∆Apq ,
ΦApq = Φ0Apq + ∆Apq .
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 .
JEONG-HYUCK PARK SEMI-COVARIANT APPROACH TO DOUBLE FIELD THEORY
The connections assume the following most general forms:
ΓCAB = Γ0CAB + ∆CpqVA
pVBq + ∆CpqVA
pVBq ,
ΦApq = Φ0Apq + ∆Apq ,
ΦApq = Φ0Apq + ∆Apq .
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 .
JEONG-HYUCK PARK SEMI-COVARIANT APPROACH TO DOUBLE FIELD THEORY
The connections assume the following most general forms:
ΓCAB = Γ0CAB + ∆CpqVA
pVBq + ∆CpqVA
pVBq ,
ΦApq = Φ0Apq + ∆Apq ,
ΦApq = Φ0Apq + ∆Apq .
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 .
JEONG-HYUCK PARK SEMI-COVARIANT APPROACH TO DOUBLE FIELD THEORY
The ‘torsionless’ connection,
Γ0CAB = 2
(P∂CPP
)[AB]
+ 2(P[A
DPB]E − P[A
DPB]E) ∂DPEC
− 4D−1
(PC[APB]
D + PC[APB]D)(∂Dd + (P∂E PP)[ED]
),
further obeys
Γ0ABC + Γ0
BCA + Γ0CAB = 0 ,
and
PCABDEF Γ0
DEF = 0 , PCABDEF Γ0
DEF = 0 .
JEONG-HYUCK PARK SEMI-COVARIANT APPROACH TO DOUBLE FIELD THEORY
In fact, the torsionless connection,
Γ0CAB = 2
(P∂CPP
)[AB]
+ 2(P[A
DPB]E − P[A
DPB]E) ∂DPEC
− 4D−1
(PC[APB]
D + PC[APB]D)(∂Dd + (P∂E PP)[ED]
),
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.
JEONG-HYUCK PARK SEMI-COVARIANT APPROACH TO DOUBLE FIELD THEORY
In fact, the torsionless connection,
Γ0CAB = 2
(P∂CPP
)[AB]
+ 2(P[A
DPB]E − P[A
DPB]E) ∂DPEC
− 4D−1
(PC[APB]
D + PC[APB]D)(∂Dd + (P∂E PP)[ED]
),
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.
JEONG-HYUCK PARK SEMI-COVARIANT APPROACH TO DOUBLE FIELD THEORY
Semi-covariant Riemann curvature
The usual curvatures for the three connections,
RCDAB = ∂AΓBCD − ∂BΓACD + ΓACE ΓBED − ΓBC
E ΓAED ,
FABpq = ∂AΦBpq − ∂BΦApq + ΦApr ΦBrq − ΦBpr ΦA
rq ,
FABpq = ∂AΦBpq − ∂BΦApq + ΦApr ΦBrq − ΦBpr ΦA
rq ,
are, from [DA,DB ]VCp = 0 and [DA,DB ]VCp = 0, related to each other,
RABCD = FCDpqVApVB
q + FCDpqVApVB
q .
However, the crucial object in DFT turns out to be
SABCD := 12
(RABCD + RCDAB − ΓE
ABΓECD
),
which we name semi-covariant Riemann curvature.
JEONG-HYUCK PARK SEMI-COVARIANT APPROACH TO DOUBLE FIELD THEORY
Semi-covariant Riemann curvature
The usual curvatures for the three connections,
RCDAB = ∂AΓBCD − ∂BΓACD + ΓACE ΓBED − ΓBC
E ΓAED ,
FABpq = ∂AΦBpq − ∂BΦApq + ΦApr ΦBrq − ΦBpr ΦA
rq ,
FABpq = ∂AΦBpq − ∂BΦApq + ΦApr ΦBrq − ΦBpr ΦA
rq ,
are, from [DA,DB ]VCp = 0 and [DA,DB ]VCp = 0, related to each other,
RABCD = FCDpqVApVB
q + FCDpqVApVB
q .
However, the crucial object in DFT turns out to be
SABCD := 12
(RABCD + RCDAB − ΓE
ABΓECD
),
which we name semi-covariant Riemann curvature.
JEONG-HYUCK PARK SEMI-COVARIANT APPROACH TO DOUBLE FIELD THEORY
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
AB .
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 .
JEONG-HYUCK PARK SEMI-COVARIANT APPROACH TO DOUBLE FIELD THEORY
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
AB .
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 .
JEONG-HYUCK PARK SEMI-COVARIANT APPROACH TO DOUBLE FIELD THEORY
‘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
)+∑
i
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.
JEONG-HYUCK PARK SEMI-COVARIANT APPROACH TO DOUBLE FIELD THEORY
‘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
)+∑
i
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.
JEONG-HYUCK PARK SEMI-COVARIANT APPROACH TO DOUBLE FIELD THEORY
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 ,
and
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
JEONG-HYUCK PARK SEMI-COVARIANT APPROACH TO DOUBLE FIELD THEORY
Completely covariant derivatives
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
pDA .
JEONG-HYUCK PARK SEMI-COVARIANT APPROACH TO DOUBLE FIELD THEORY
Completely covariant derivatives
Following the aforementioned general prescription, completely covariant Yang-Mills
field strength is given by two opposite projections, or
Fpq = V MpV N
qFMN ,
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.
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 .
JEONG-HYUCK PARK SEMI-COVARIANT APPROACH TO DOUBLE FIELD THEORY
Completely covariant derivatives
Following the aforementioned general prescription, completely covariant Yang-Mills
field strength is given by two opposite projections, or
Fpq = V MpV N
qFMN ,
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.
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 .
JEONG-HYUCK PARK SEMI-COVARIANT APPROACH TO DOUBLE FIELD THEORY
Completely covariant derivatives
O(D,D) covariant Killing equations in DFT:
LXHMN = 0 ⇐⇒ (P∇)M (PX)N − (P∇)N (PX)M = 0 ,
LX d = 0 ⇐⇒ ∇M X M = 0 .
JHP-Rey-Rim-Sakatani 2015
Chris Blair 2015
JEONG-HYUCK PARK SEMI-COVARIANT APPROACH TO DOUBLE FIELD THEORY
Completely covariant derivatives
Dirac operators for fermions, ρα, ψαp , ρ′α, ψ′αp :
γpDpρ = γADAρ , γpDpψp = γADAψp ,
Dpρ , Dpψp = DAψ
A ,
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
JEONG-HYUCK PARK SEMI-COVARIANT APPROACH TO DOUBLE FIELD THEORY
Completely covariant derivatives
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 .
JEONG-HYUCK PARK SEMI-COVARIANT APPROACH TO DOUBLE FIELD THEORY
Completely covariant derivatives
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 .
JEONG-HYUCK PARK SEMI-COVARIANT APPROACH TO DOUBLE FIELD THEORY
Completely covariant curvatures
Scalar curvature:
S := (PABPCD − PABPCD)SACBD
c.f. SABAB = 0 .
“Ricci” curvature:
S0pq = V A
pV BqS0
AB
where we set S0AB = S0
ACBC .
Further, we have conserved “Einstein” curvature,
GAB = 2(PAC PBD − PACPBD)SCD − 12J
ABS , ∇AGAB = 0 .
JEONG-HYUCK PARK SEMI-COVARIANT APPROACH TO DOUBLE FIELD THEORY
Combining all the results above, we are now ready to spell
D = 10 Maximally Supersymmetric Double Field Theory
JEONG-HYUCK PARK SEMI-COVARIANT APPROACH TO DOUBLE FIELD THEORY
D = 10 Maximal SDFT [ 1210.5078 ]
Lagrangian:
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
′p
].
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
JEONG-HYUCK PARK SEMI-COVARIANT APPROACH TO DOUBLE FIELD THEORY
D = 10 Maximal SDFT [ 1210.5078 ]
Lagrangian:
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
′p
].
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
JEONG-HYUCK PARK SEMI-COVARIANT APPROACH TO DOUBLE FIELD THEORY
D = 10 Maximal SDFT [ 1210.5078 ]
Lagrangian:
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
′p
].
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 .
JEONG-HYUCK PARK SEMI-COVARIANT APPROACH TO DOUBLE FIELD THEORY
D = 10 Maximal SDFT [ 1210.5078 ]
Lagrangian:
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
′p
].
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 .
JEONG-HYUCK PARK SEMI-COVARIANT APPROACH TO DOUBLE FIELD THEORY
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′ ,
where
ΓABC = ΓABC − i 1748 ργABCρ+ i 5
2 ργBCψA + i 14 ψ
pγABCψp − 3iψ′B γAψ′C ,
Γ′ABC = ΓABC − i 1748 ρ′γABCρ
′ + i 52 ρ′γBCψ
′A + i 1
4 ψ′p γABCψ
′p − 3iψBγAψC .
JEONG-HYUCK PARK SEMI-COVARIANT APPROACH TO DOUBLE FIELD THEORY
D = 10 Maximal SDFT [ 1210.5078 ]
Lagrangian:
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
′p
].
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 .
JEONG-HYUCK PARK SEMI-COVARIANT APPROACH TO DOUBLE FIELD THEORY
D = 10 Maximal SDFT [ 1210.5078 ]
Lagrangian:
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
′p
].
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 .
JEONG-HYUCK PARK SEMI-COVARIANT APPROACH TO DOUBLE FIELD THEORY
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
q).
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
′p
)ε′γp − iγpε
(D′p ρ′ + D′pψ
′p γ
p − ψpγpF γp).
JEONG-HYUCK PARK SEMI-COVARIANT APPROACH TO DOUBLE FIELD THEORY
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
q).
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
′p
)ε′γp − iγpε
(D′p ρ′ + D′pψ
′p γ
p − ψpγpF γp).
JEONG-HYUCK PARK SEMI-COVARIANT APPROACH TO DOUBLE FIELD THEORY
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:
D0−
(F − iρρ′ + iγrψsψ
′r γ
s)
= 0 ,
which is automatically met by the self-duality, together with the nilpotency of D0+,
D0−
(F − iρρ′ + iγrψsψ
′r γ
s)
= 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 .
JEONG-HYUCK PARK SEMI-COVARIANT APPROACH TO DOUBLE FIELD THEORY
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:
D0−
(F − iρρ′ + iγrψsψ
′r γ
s)
= 0 ,
which is automatically met by the self-duality, together with the nilpotency of D0+,
D0−
(F − iρρ′ + iγrψsψ
′r γ
s)
= 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 .
JEONG-HYUCK PARK SEMI-COVARIANT APPROACH TO DOUBLE FIELD THEORY
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:
D0−
(F − iρρ′ + iγrψsψ
′r γ
s)
= 0 ,
which is automatically met by the self-duality, together with the nilpotency of D0+,
D0−
(F − iρρ′ + iγrψsψ
′r γ
s)
= 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 .
JEONG-HYUCK PARK SEMI-COVARIANT APPROACH TO DOUBLE FIELD THEORY
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:
D0−
(F − iρρ′ + iγrψsψ
′r γ
s)
= 0 ,
which is automatically met by the self-duality, together with the nilpotency of D0+,
D0−
(F − iρρ′ + iγrψsψ
′r γ
s)
= 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 .
JEONG-HYUCK PARK SEMI-COVARIANT APPROACH TO DOUBLE FIELD THEORY
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:
D0−
(F − iρρ′ + iγrψsψ
′r γ
s)
= 0 ,
which is automatically met by the self-duality, together with the nilpotency of D0+,
D0−
(F − iρρ′ + iγrψsψ
′r γ
s)
= 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 .
JEONG-HYUCK PARK SEMI-COVARIANT APPROACH TO DOUBLE FIELD THEORY
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[
18
(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 .
JEONG-HYUCK PARK SEMI-COVARIANT APPROACH TO DOUBLE FIELD THEORY
N = 1 SUSY Algebra [1112.0069]
Commutator of supersymmetry reads
[δε1 , δε2 ] ≡ LX3 + δε3 + δso(1,9)L+ δso(9,1)R
+ δtrivial .
where
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.
JEONG-HYUCK PARK SEMI-COVARIANT APPROACH TO DOUBLE FIELD THEORY
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.
JEONG-HYUCK PARK SEMI-COVARIANT APPROACH TO DOUBLE FIELD THEORY
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
ν .
JEONG-HYUCK PARK SEMI-COVARIANT APPROACH TO DOUBLE FIELD THEORY
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
ν .
JEONG-HYUCK PARK SEMI-COVARIANT APPROACH TO DOUBLE FIELD THEORY
Parametrization: Reduction to Generalized Geometry
Instead, we may choose an alternative parametrization,
VAp = 1√
2
(β + e)µp
(e−1)pν
, VAp = 1√
2
(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 ).
JEONG-HYUCK PARK SEMI-COVARIANT APPROACH TO DOUBLE FIELD THEORY
Parametrization: Reduction to Generalized Geometry
Two parametrizations:
VAp = 1√2
(e−1)pµ
(B + e)νp
, VAp = 1√2
(e−1)pµ
(B + e)νp
versus
VAp = 1√
2
(β + e)µp
(e−1)pν
, VAp = 1√
2
(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.
JEONG-HYUCK PARK SEMI-COVARIANT APPROACH TO DOUBLE FIELD THEORY
Parametrization: Reduction to Generalized Geometry
Two parametrizations:
VAp = 1√2
(e−1)pµ
(B + e)νp
, VAp = 1√2
(e−1)pµ
(B + e)νp
versus
VAp = 1√
2
(β + e)µp
(e−1)pν
, VAp = 1√
2
(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.
JEONG-HYUCK PARK SEMI-COVARIANT APPROACH TO DOUBLE FIELD THEORY
Parametrization: Reduction to Generalized Geometry
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.
Ko-Melby-Thompson-Meyer-JHP 2015
JEONG-HYUCK PARK SEMI-COVARIANT APPROACH TO DOUBLE FIELD THEORY
Parametrization: Reduction to Generalized Geometry
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
JEONG-HYUCK PARK SEMI-COVARIANT APPROACH TO DOUBLE FIELD THEORY
Parametrization: Reduction to Generalized Geometry
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
JEONG-HYUCK PARK SEMI-COVARIANT APPROACH TO DOUBLE FIELD THEORY
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) .
JEONG-HYUCK PARK SEMI-COVARIANT APPROACH TO DOUBLE FIELD THEORY
Unification of type IIA and IIB SUGRAs
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.
JEONG-HYUCK PARK SEMI-COVARIANT APPROACH TO DOUBLE FIELD THEORY
Unification of type IIA and IIB SUGRAs
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.
JEONG-HYUCK PARK SEMI-COVARIANT APPROACH TO DOUBLE FIELD THEORY
Unification of type IIA and IIB SUGRAs
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 .
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.
JEONG-HYUCK PARK SEMI-COVARIANT APPROACH TO DOUBLE FIELD THEORY
Unification of type IIA and IIB SUGRAs
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 .
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.
JEONG-HYUCK PARK SEMI-COVARIANT APPROACH TO DOUBLE FIELD THEORY
Diagonal gauge fixing and Reduction to SUGRA
Setting the diagonal gauge,
eµp ≡ eµp
with ηpq = −ηpq , γp = γ(D+1)γp, γ(D+1) = −γ(D+1), breaks the local Lorentz
symmetry,
Spin(1,D−1)L × Spin(D−1, 1)R =⇒ Spin(1,D−1)D .
And it reduces SDFT to SUGRA:
N = 2 D = 10 SDFT =⇒ 10D Type II democratic SUGRA
Bergshoeff, et al.; Coimbra, Strickland-Constable, Waldram
N = 1 D = 10 SDFT =⇒ 10D minimal SUGRA Chamseddine; Bergshoeff et al.
JEONG-HYUCK PARK SEMI-COVARIANT APPROACH TO DOUBLE FIELD THEORY
Diagonal gauge fixing and Reduction to SUGRA
Setting the diagonal gauge,
eµp ≡ eµp
with ηpq = −ηpq , γp = γ(D+1)γp, γ(D+1) = −γ(D+1), breaks the local Lorentz
symmetry,
Spin(1,D−1)L × Spin(D−1, 1)R =⇒ Spin(1,D−1)D .
And it reduces SDFT to SUGRA:
N = 2 D = 10 SDFT =⇒ 10D Type II democratic SUGRA
Bergshoeff, et al.; Coimbra, Strickland-Constable, Waldram
N = 1 D = 10 SDFT =⇒ 10D minimal SUGRA Chamseddine; Bergshoeff et al.
JEONG-HYUCK PARK SEMI-COVARIANT APPROACH TO DOUBLE FIELD THEORY
Diagonal gauge fixing and Reduction to SUGRA
After the diagonal gauge fixing, we may parameterize the R-R potential as
C ≡(
12
) D+24 ∑′
p1p!Ca1a2···apγ
a1a2···ap
and obtain the field strength,
F := D0+C ≡
(12
) D4 ∑′
p1
(p+1)!Fa1a2···ap+1γ
a1a2···ap+1
where∑′
p denotes the odd p sum for Type IIA and even p sum for Type IIB, and
Fa1a2···ap = p(
D[a1Ca2···ap ] − ∂[a1
φ Ca2···ap ]
)+ p!
3!(p−3)!H[a1a2a3
Ca4···ap ]
The pair of nilpotent differential operators, D0+ and D0
−, reduce to a ‘twisted K-theory’
exterior derivative and its dual, after the diagonal gauge fixing,
D0+ =⇒ d + (H − dφ) ∧
D0− =⇒ ∗ [ d + (H − dφ) ∧ ] ∗
JEONG-HYUCK PARK SEMI-COVARIANT APPROACH TO DOUBLE FIELD THEORY
Diagonal gauge fixing and Reduction to SUGRA
After the diagonal gauge fixing, we may parameterize the R-R potential as
C ≡(
12
) D+24 ∑′
p1p!Ca1a2···apγ
a1a2···ap
and obtain the field strength,
F := D0+C ≡
(12
) D4 ∑′
p1
(p+1)!Fa1a2···ap+1γ
a1a2···ap+1
where∑′
p denotes the odd p sum for Type IIA and even p sum for Type IIB, and
Fa1a2···ap = p(
D[a1Ca2···ap ] − ∂[a1
φ Ca2···ap ]
)+ p!
3!(p−3)!H[a1a2a3
Ca4···ap ]
The pair of nilpotent differential operators, D0+ and D0
−, reduce to a ‘twisted K-theory’
exterior derivative and its dual, after the diagonal gauge fixing,
D0+ =⇒ d + (H − dφ) ∧
D0− =⇒ ∗ [ d + (H − dφ) ∧ ] ∗
JEONG-HYUCK PARK SEMI-COVARIANT APPROACH TO DOUBLE FIELD THEORY
Diagonal gauge fixing and Reduction to SUGRA
In this way, ordinary SUGRA ≡ gauge-fixed SDFT,
Spin(1,D−1)L × Spin(D−1, 1)R =⇒ Spin(1,D−1)D .
JEONG-HYUCK PARK SEMI-COVARIANT APPROACH TO DOUBLE FIELD THEORY
Modifying O(D,D) transformation rule
The diagonal gauge, eµp ≡ eµ p, is incompatible with the vectorial O(D,D)
transformation rule of the DFT-vielbein.
In order to preserve the diagonal gauge, it is necessary to modify the O(D,D)
transformation rule.
JEONG-HYUCK PARK SEMI-COVARIANT APPROACH TO DOUBLE FIELD THEORY
Modifying O(D,D) transformation rule
The diagonal gauge, eµp ≡ eµ p, is incompatible with the vectorial O(D,D)
transformation rule of the DFT-vielbein.
In order to preserve the diagonal gauge, it is necessary to modify the O(D,D)
transformation rule.
JEONG-HYUCK PARK SEMI-COVARIANT APPROACH TO DOUBLE FIELD THEORY
Modifying O(D,D) transformation rule
The O(D,D) rotation must accompany a compensating Pin(D−1, 1)R local Lorentz
rotation, Lqp, SL
αβ which we can construct explicitly,
L = e−1 [at − (g + B)bt] [at + (g − B)bt]−1 e , γq Lqp = S−1
LγpSL ,
where a and b are parameters of a given O(D,D) group element,
MAB =
aµν bµσ
cρν dρσ
.
JEONG-HYUCK PARK SEMI-COVARIANT APPROACH TO DOUBLE FIELD THEORY
Modified O(D,D) Transformation Rule After The Diagonal Gauge Fixing
d −→ d
VAp −→ MA
B VBp
VAp −→ MA
B VBq Lq
p
Cαα , Fαα −→ Cαβ(S−1L
)β α , Fαβ(S−1L
)β α
ρα −→ ρα
ρ′α −→ (SL)αβρ′β
ψαp −→ (L−1)pq ψαq
ψ′αp −→ (SL)αβψ′βp
All the barred indices are now to be rotated. Consistent with Hassan
The R-R sector can be also mapped to O(D,D) spinors.
Fukuma, Oota Tanaka; Hohm, Kwak, Zwiebach
JEONG-HYUCK PARK SEMI-COVARIANT APPROACH TO DOUBLE FIELD THEORY
Flipping the chirality: IIA ⇔ IIB
If and only if det(L) = −1, the modified O(D,D) rotation flips the chirality of the
theory, since
γ(D+1)SL = det(L) SLγ(D+1) .
Thus, the mechanism above naturally realizes the exchange of Type IIA and IIB
supergravities under O(D,D) T-duality.
However, since L explicitly depends on the parametrization of VAp and VAp in terms of
gµν and Bµν , it is impossible to impose the modified O(D,D) transformation rule from
the beginning on the parametrization-independent covariant formalism.
It is an artifact of the diagonal gauge fixing.
JEONG-HYUCK PARK SEMI-COVARIANT APPROACH TO DOUBLE FIELD THEORY
Flipping the chirality: IIA ⇔ IIB
If and only if det(L) = −1, the modified O(D,D) rotation flips the chirality of the
theory, since
γ(D+1)SL = det(L) SLγ(D+1) .
Thus, the mechanism above naturally realizes the exchange of Type IIA and IIB
supergravities under O(D,D) T-duality.
However, since L explicitly depends on the parametrization of VAp and VAp in terms of
gµν and Bµν , it is impossible to impose the modified O(D,D) transformation rule from
the beginning on the parametrization-independent covariant formalism.
It is an artifact of the diagonal gauge fixing.
JEONG-HYUCK PARK SEMI-COVARIANT APPROACH TO DOUBLE FIELD THEORY