Three themes in this talk: Doubled-yet-gauged coordinate system - - PowerPoint PPT Presentation

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Three themes in this talk: Doubled-yet-gauged coordinate system - - PowerPoint PPT Presentation

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D OUBLED - YET -G AUGED , S EMI -C OVARIANCE & T WOFOLD S PIN J EONG -H YUCK P ARK Sogang University, Seoul Stringy Geometry, Mainz September 2015 J EONG -H YUCK P ARK D OUBLED - YET -G AUGED , S EMI -C OVARIANCE & T WOFOLD S PIN Three

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SLIDE 1

DOUBLED-YET-GAUGED, SEMI-COVARIANCE & TWOFOLD SPIN

JEONG-HYUCK PARK

Sogang University, Seoul Stringy Geometry, Mainz September 2015

JEONG-HYUCK PARK DOUBLED-YET-GAUGED, SEMI-COVARIANCE & TWOFOLD SPIN

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SLIDE 2

Three themes in this talk: Doubled-yet-gauged coordinate system Semi-covariant formulation of DFT/SDFT Twofold spin and Standard Model

JEONG-HYUCK PARK DOUBLED-YET-GAUGED, SEMI-COVARIANCE & TWOFOLD SPIN

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SLIDE 3

To summarize collaborated works with Imtak Jeon (8), Kanghoon Lee (8), Yoonji Suh (3), Chris Blair (1), Emanuel Malek (1), Wonyoung Cho (1), Jose Fernández-Melgarejo (1), Kang-Sin Choi (1: Phenomenologist), Soo-Jong Rey (1), Woohyun Rim (1), Yuho Sakatani (1), Sung Moon Ko (1), Charles Melby-Thompson (1), Rene Meyér (1).

Differential geometry with a projection: Application to double field theory 1011.1324 Stringy differential geometry, beyond Riemann 1105.6294 Incorporation of fermions into double field theory 1109.2035 Ramond-Ramond Cohomology and O(D,D) T-duality 1206.3478 Supersymmetric Double Field Theory: Stringy Reformulation of Supergravity 1112.0069 Stringy Unification of IIA and IIB Supergravities under N = 2 D= 10 Supersymmetric Double Field Theory 1210.5078 Supersymmetric gauged Double Field Theory: Systematic derivation by virtue of ‘Twist’ 1505.01301 Comments on double field theory and diffeomorphisms 1304.5946 Covariant action for a string in doubled yet gauged spacetime 1307.8377 Double field formulation of Yang-Mills theory ⇒ Standard Model Double Field Theory 1102.0419/1506.05277 O(D, D) Covariant Noether Currents and Global Charges in Double Field Theory 1507.07545 Dynamics of Perturbations in Double Field Theory & Non-Relativistic String Theory 1508.01121 U-geometry: SL(5) ⇒ U-gravity: SL(N) 1302.1652/1402.5027 M-theory and F-theory from a Duality Manifest Action 1311.5109

JEONG-HYUCK PARK DOUBLED-YET-GAUGED, SEMI-COVARIANCE & TWOFOLD SPIN

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Notation 1/2

Capital Latin letters denote the O(D, D) vector indices, A, B, C, · · · , L, M, N, · · · = 1, 2, · · · , D+D . They can be freely raised or lowered by the O(D, D) invariant constant metric, JAB =     1 1     .

JEONG-HYUCK PARK DOUBLED-YET-GAUGED, SEMI-COVARIANCE & TWOFOLD SPIN

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SLIDE 5

Doubled-yet-gauged coordinate system 1304.5946/1307.8377

JEONG-HYUCK PARK DOUBLED-YET-GAUGED, SEMI-COVARIANCE & TWOFOLD SPIN

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SLIDE 6

Section condition in DFT

The section condition in DFT, ∂M∂M = 0 , implies that, arbitrary functions and their arbitrary derivatives, collectively Φ, are invariant under translations generated by a derivative-index-valued vector, Φ0(x + ∆) = Φ0(x) , ∆M = Φ1∂MΦ2 . In fact, the converse is also true. c.f. 1307.8377

JEONG-HYUCK PARK DOUBLED-YET-GAUGED, SEMI-COVARIANCE & TWOFOLD SPIN

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SLIDE 7

Doubled-yet-gauged coordinate system

Start with any D-dimensional coordinate system, xµ, e.g. (t, x, y, z) or (t, r, θ, φ) etc. The doubled coordinates, xM = ( ˜ xµ , xν ) , are then required to be gauged: they are subject to an equivalence relation, xM ∼ xM + Φ1∂MΦ2 , which we call coordinate gauge symmetry. Each equivalence class, or gauge orbit, represents a single physical point. (Strongly constrained) DFT employs such a doubled-yet-gagued coordinate system. Diffeomorphism symmetry means an invariance under arbitrary reparametrizations of the ‘gauge orbits’, rather than ‘points’ in the doubled coordinate space: Hohm-Zwiebach ansatz for finite transformations ≡ exp( ˆ LX ). JHP, Berman-Cederwall-Perry, Hull

JEONG-HYUCK PARK DOUBLED-YET-GAUGED, SEMI-COVARIANCE & TWOFOLD SPIN

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SLIDE 8

Doubled-yet-gauged coordinate system

Start with any D-dimensional coordinate system, xµ, e.g. (t, x, y, z) or (t, r, θ, φ) etc. The doubled coordinates, xM = ( ˜ xµ , xν ) , are then required to be gauged: they are subject to an equivalence relation, xM ∼ xM + Φ1∂MΦ2 , which we call coordinate gauge symmetry. Each equivalence class, or gauge orbit, represents a single physical point. (Strongly constrained) DFT employs such a doubled-yet-gagued coordinate system. Diffeomorphism symmetry means an invariance under arbitrary reparametrizations of the ‘gauge orbits’, rather than ‘points’ in the doubled coordinate space: Hohm-Zwiebach ansatz for finite transformations ≡ exp( ˆ LX ). JHP, Berman-Cederwall-Perry, Hull

JEONG-HYUCK PARK DOUBLED-YET-GAUGED, SEMI-COVARIANCE & TWOFOLD SPIN

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SLIDE 9

Doubled-yet-gauged coordinate system

Start with any D-dimensional coordinate system, xµ, e.g. (t, x, y, z) or (t, r, θ, φ) etc. The doubled coordinates, xM = ( ˜ xµ , xν ) , are then required to be gauged: they are subject to an equivalence relation, xM ∼ xM + Φ1∂MΦ2 , which we call coordinate gauge symmetry. Each equivalence class, or gauge orbit, represents a single physical point. (Strongly constrained) DFT employs such a doubled-yet-gagued coordinate system. Diffeomorphism symmetry means an invariance under arbitrary reparametrizations of the ‘gauge orbits’, rather than ‘points’ in the doubled coordinate space: Hohm-Zwiebach ansatz for finite transformations ≡ exp( ˆ LX ). JHP, Berman-Cederwall-Perry, Hull

JEONG-HYUCK PARK DOUBLED-YET-GAUGED, SEMI-COVARIANCE & TWOFOLD SPIN

slide-10
SLIDE 10

Doubled-yet-gauged coordinate system

Start with any D-dimensional coordinate system, xµ, e.g. (t, x, y, z) or (t, r, θ, φ) etc. The doubled coordinates, xM = ( ˜ xµ , xν ) , are then required to be gauged: they are subject to an equivalence relation, xM ∼ xM + Φ1∂MΦ2 , which we call coordinate gauge symmetry. Each equivalence class, or gauge orbit, represents a single physical point. (Strongly constrained) DFT employs such a doubled-yet-gagued coordinate system. Diffeomorphism symmetry means an invariance under arbitrary reparametrizations of the ‘gauge orbits’, rather than ‘points’ in the doubled coordinate space: Hohm-Zwiebach ansatz for finite transformations ≡ exp( ˆ LX ). JHP, Berman-Cederwall-Perry, Hull

JEONG-HYUCK PARK DOUBLED-YET-GAUGED, SEMI-COVARIANCE & TWOFOLD SPIN

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SLIDE 11

Doubled-yet-gauged coordinate system

Start with any D-dimensional coordinate system, xµ, e.g. (t, x, y, z) or (t, r, θ, φ) etc. The doubled coordinates, xM = ( ˜ xµ , xν ) , are then required to be gauged: they are subject to an equivalence relation, xM ∼ xM + Φ1∂MΦ2 , which we call coordinate gauge symmetry. Each equivalence class, or gauge orbit, represents a single physical point. (Strongly constrained) DFT employs such a doubled-yet-gagued coordinate system. Diffeomorphism symmetry means an invariance under arbitrary reparametrizations of the ‘gauge orbits’, rather than ‘points’ in the doubled coordinate space: Hohm-Zwiebach ansatz for finite transformations ≡ exp( ˆ LX ). JHP, Berman-Cederwall-Perry, Hull

JEONG-HYUCK PARK DOUBLED-YET-GAUGED, SEMI-COVARIANCE & TWOFOLD SPIN

slide-12
SLIDE 12

Doubled-yet-gauged coordinate system

Start with any D-dimensional coordinate system, xµ, e.g. (t, x, y, z) or (t, r, θ, φ) etc. The doubled coordinates, xM = ( ˜ xµ , xν ) , are then required to be gauged: they are subject to an equivalence relation, xM ∼ xM + Φ1∂MΦ2 , which we call coordinate gauge symmetry. Each equivalence class, or gauge orbit, represents a single physical point. (Strongly constrained) DFT employs such a doubled-yet-gagued coordinate system. Diffeomorphism is generated by the generalized Lie derivative Siegel, Courant, Grana ˆ LX TA1···An := X B∂BTA1···An + ωT ∂BX BTA1···An +

n

  • i=1

(∂Ai XB − ∂BXAi )TA1···Ai−1

B Ai+1···An ,

where ωT denotes the weight.

JEONG-HYUCK PARK DOUBLED-YET-GAUGED, SEMI-COVARIANCE & TWOFOLD SPIN

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SLIDE 13

Doubled-yet-gauged coordinate system

Start with any D-dimensional coordinate system, xµ, e.g. (t, x, y, z) or (t, r, θ, φ) etc. The doubled coordinates, xM = ( ˜ xµ , xν ) , are then required to be gauged: they are subject to an equivalence relation, xM ∼ xM + Φ1∂MΦ2 , which we call coordinate gauge symmetry. Each equivalence class, or gauge orbit, represents a single physical point. (Strongly constrained) DFT employs such a doubled-yet-gagued coordinate system. Diffeomorphism is generated by the generalized Lie derivative Siegel, Courant, Grana ˆ LX TA1···An := X B∂BTA1···An + ωT ∂BX BTA1···An +

n

  • i=1

(∂Ai XB − ∂BXAi )TA1···Ai−1

B Ai+1···An ,

where ωT denotes the weight.

JEONG-HYUCK PARK DOUBLED-YET-GAUGED, SEMI-COVARIANCE & TWOFOLD SPIN

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SLIDE 14

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

The usual infinitesimal one-form, dxM, is NOT a covariant vector in 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∂MΦ2) = dxM . These can be all cured by introducing a gauged infinitesimal one-form, DxM := dxM − AM , where AM is the ‘coordinate gauge potential’. Being a derivative-index-valued vector, it satisfies AM∂M = 0, AMAM = 0, or suggestively the ‘gauged section condition’, (∂M + AM)(∂M + AM) = 0 .

JEONG-HYUCK PARK DOUBLED-YET-GAUGED, SEMI-COVARIANCE & TWOFOLD SPIN

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SLIDE 15

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

The usual infinitesimal one-form, dxM, is NOT a covariant vector in 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∂MΦ2) = dxM . These can be all cured by introducing a gauged infinitesimal one-form, DxM := dxM − AM , where AM is the ‘coordinate gauge potential’. Being a derivative-index-valued vector, it satisfies AM∂M = 0, AMAM = 0, or suggestively the ‘gauged section condition’, (∂M + AM)(∂M + AM) = 0 .

JEONG-HYUCK PARK DOUBLED-YET-GAUGED, SEMI-COVARIANCE & TWOFOLD SPIN

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SLIDE 16

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

Under coordinate gauge symmetry, we have the invariance of DxM, xM − → x′M = xM + Φ1∂MΦ2 , AM − → A′M = AM + d(Φ1∂MΦ2) : A′M∂′

M ≡ 0 ,

DxM − → D′x′M = DxM = dxM − AM . Similarly, under (finite) DFT diffeomorphisms à la Hohm-Zwiebach LM N := ∂Mx′N , ¯ L := J LtJ −1 , F := 1

2

  • L¯

L−1 + ¯ L−1L

  • ,

¯ F := J F tJ −1 = 1

2

  • L−1¯

L + ¯ LL−1 = F −1 , we have the covariance, xM − → x′M(x) , HMN(x) − → H′

MN(x′) = ¯

FM K ¯ FN LHKL(x) , AM − → A′M = ANFN M + dX N(L − F)N M : A′M∂′

M ≡ 0 ,

DxM − → D′x′M = DxNFN M .

JEONG-HYUCK PARK DOUBLED-YET-GAUGED, SEMI-COVARIANCE & TWOFOLD SPIN

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SLIDE 17

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

Under coordinate gauge symmetry, we have the invariance of DxM, xM − → x′M = xM + Φ1∂MΦ2 , AM − → A′M = AM + d(Φ1∂MΦ2) : A′M∂′

M ≡ 0 ,

DxM − → D′x′M = DxM = dxM − AM . Similarly, under (finite) DFT diffeomorphisms à la Hohm-Zwiebach LM N := ∂Mx′N , ¯ L := J LtJ −1 , F := 1

2

  • L¯

L−1 + ¯ L−1L

  • ,

¯ F := J F tJ −1 = 1

2

  • L−1¯

L + ¯ LL−1 = F −1 , we have the covariance, xM − → x′M(x) , HMN(x) − → H′

MN(x′) = ¯

FM K ¯ FN LHKL(x) , AM − → A′M = ANFN M + dX N(L − F)N M : A′M∂′

M ≡ 0 ,

DxM − → D′x′M = DxNFN M .

JEONG-HYUCK PARK DOUBLED-YET-GAUGED, SEMI-COVARIANCE & TWOFOLD SPIN

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SLIDE 18

Fixing the coordinate gauge symmetry : conventional choice of the section

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 + Φ1∂µΦ2 , xν . The coordinate gauge potential and the gauged infinitesimal one-form become AM = Aλ∂Mxλ =

  • Aµ , 0
  • ,

DxM =

  • d˜

xµ − Aµ , dxν .

JEONG-HYUCK PARK DOUBLED-YET-GAUGED, SEMI-COVARIANCE & TWOFOLD SPIN

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SLIDE 19

Fixing the coordinate gauge symmetry : conventional choice of the section

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 + Φ1∂µΦ2 , xν . The coordinate gauge potential and the gauged infinitesimal one-form become AM = Aλ∂Mxλ =

  • Aµ , 0
  • ,

DxM =

  • d˜

xµ − Aµ , dxν .

JEONG-HYUCK PARK DOUBLED-YET-GAUGED, SEMI-COVARIANCE & TWOFOLD SPIN

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SLIDE 20

Newton mechanics with doubled-yet-gauged coordinate system

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, xI = (˜ xj, xk), LNewton = 1

2m DtxIDtxJ δIJ − V(x) ,

where I, J = 1, 2, · · · , 6 and the potential, V(x), satisfies the section condition. With the conventional choice of the section, we get LNewton = 1

2 m ˙

xj ˙ xk δjk − V(x) + 1

2 m

˙ ˜ xj − Aj ˙ ˜ xk − Ak

  • δjk .

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

JEONG-HYUCK PARK DOUBLED-YET-GAUGED, SEMI-COVARIANCE & TWOFOLD SPIN

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SLIDE 21

Newton mechanics with doubled-yet-gauged coordinate system

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, xI = (˜ xj, xk), LNewton = 1

2m DtxIDtxJ δIJ − V(x) ,

where I, J = 1, 2, · · · , 6 and the potential, V(x), satisfies the section condition. With the conventional choice of the section, we get LNewton = 1

2 m ˙

xj ˙ xk δjk − V(x) + 1

2 m

˙ ˜ xj − Aj ˙ ˜ xk − Ak

  • δjk .

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

JEONG-HYUCK PARK DOUBLED-YET-GAUGED, SEMI-COVARIANCE & TWOFOLD SPIN

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SLIDE 22

String probes the doubled-yet-gauged spacetime

DFT string action is with DiX M = ∂iX M − AM

i ,

JHP-Lee 2013

1 4πα′

  • d2σ Lstring ,

Lstring = − 1

2

√ −h hijDiX MDjX NHMN(X) − ǫijDiX MAjM , The action is fully symmetric, essentially due to the auxiliary gauge field, AM

i , under

String worldsheet diffeomorphisms plus Weyl symmetry (as usual) O(D, D) T-duality Target spacetime DFT diffeomorphisms The coordinate gauge symmetry

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

JEONG-HYUCK PARK DOUBLED-YET-GAUGED, SEMI-COVARIANCE & TWOFOLD SPIN

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SLIDE 23

String probes the doubled-yet-gauged spacetime

DFT string action is with DiX M = ∂iX M − AM

i ,

JHP-Lee 2013

1 4πα′

  • d2σ Lstring ,

Lstring = − 1

2

√ −h hijDiX MDjX NHMN(X) − ǫijDiX MAjM , HAB(x) is the “generalized metric” which can be defined as a symmetric O(D, D) element, HAB = HBA , HACHBDJCD = JAB , and satisfies the section condition. There are two types of “generalized metric” : Riemannian vs. non-Riemannian.

JEONG-HYUCK PARK DOUBLED-YET-GAUGED, SEMI-COVARIANCE & TWOFOLD SPIN

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SLIDE 24

String probes the doubled-yet-gauged spacetime

DFT string action is with DiX M = ∂iX M − AM

i ,

JHP-Lee 2013

1 4πα′

  • d2σ Lstring ,

Lstring = − 1

2

√ −h hijDiX MDjX NHMN(X) − ǫijDiX MAjM , HAB(x) is the “generalized metric” which can be defined as a symmetric O(D, D) element, HAB = HBA , HACHBDJCD = JAB , and satisfies the section condition. There are two types of “generalized metric” : Riemannian vs. non-Riemannian.

JEONG-HYUCK PARK DOUBLED-YET-GAUGED, SEMI-COVARIANCE & TWOFOLD SPIN

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SLIDE 25

String probes the doubled-yet-gauged spacetime

DFT string action is with DiX M = ∂iX M − AM

i ,

JHP-Lee 2013

1 4πα′

  • d2σ Lstring ,

Lstring = − 1

2

√ −h hijDiX MDjX NHMN(X) − ǫijDiX MAjM , HAB(x) is the “generalized metric” which can be defined as a symmetric O(D, D) element, HAB = HBA , HACHBDJCD = JAB , and satisfies the section condition. There are two types of “generalized metric” : Riemannian vs. non-Riemannian.

JEONG-HYUCK PARK DOUBLED-YET-GAUGED, SEMI-COVARIANCE & TWOFOLD SPIN

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SLIDE 26

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,

1 4πα′ Lstring ≡ 1 2πα′

  • − 1

2

√ −hhij∂iX µ∂jX νGµν(X) + 1

2ǫij∂iX µ∂jX νBµν(X) + 1 2ǫij∂i ˜

Xµ∂jX µ , with the bonus of the topological term introduced by Giveon-Rocek; Hull. The EOM of AM

i

implies self-duality on the full doubled spacetime, HM

NDiX N + 1 √−h ǫijDjX M = 0 .

JEONG-HYUCK PARK DOUBLED-YET-GAUGED, SEMI-COVARIANCE & TWOFOLD SPIN

slide-27
SLIDE 27

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,

1 4πα′ Lstring ≡ 1 2πα′

  • − 1

2

√ −hhij∂iX µ∂jX νGµν(X) + 1

2ǫij∂iX µ∂jX νBµν(X) + 1 2ǫij∂i ˜

Xµ∂jX µ , with the bonus of the topological term introduced by Giveon-Rocek; Hull. The EOM of AM

i

implies self-duality on the full doubled spacetime, HM

NDiX N + 1 √−h ǫijDjX M = 0 .

JEONG-HYUCK PARK DOUBLED-YET-GAUGED, SEMI-COVARIANCE & TWOFOLD SPIN

slide-28
SLIDE 28

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,

1 4πα′ Lstring ≡ 1 2πα′

  • − 1

2

√ −hhij∂iX µ∂jX νGµν(X) + 1

2ǫij∂iX µ∂jX νBµν(X) + 1 2ǫij∂i ˜

Xµ∂jX µ , with the bonus of the topological term introduced by Giveon-Rocek; Hull. The EOM of AM

i

implies self-duality on the full doubled spacetime, HM

NDiX N + 1 √−h ǫijDjX M = 0 .

JEONG-HYUCK PARK DOUBLED-YET-GAUGED, SEMI-COVARIANCE & TWOFOLD SPIN

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SLIDE 29

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. For example, with the decomposition, D = 10 = 2 + 8, HMN =             eαβ δij −eαβ fηαβ δij             , f = 1 + Q

r6 ,

r 2 = 9

i=2(xi)2 .

This is “doublly T-dual”, (t, x1) ⇔ (˜ t, ˜ x1), DFT background to F1 à la Dabholkar-Gibbons-Harvey-Ruiz 1990, c.f. 2D null-wave à la Berkeley-Berman-Rudolf DFT as well as the DFT sigma model is well-defined even for such a non-Riemannian background. In particular, the Gomis-Ooguri ‘non-relativistic’ string theory can be identified precisely as the DFT sigma model on the above non-Riemannian background. Ko-Meyer-Melby-Thompson-JHP 2015

JEONG-HYUCK PARK DOUBLED-YET-GAUGED, SEMI-COVARIANCE & TWOFOLD SPIN

slide-30
SLIDE 30

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. For example, with the decomposition, D = 10 = 2 + 8, HMN =             eαβ δij −eαβ fηαβ δij             , f = 1 + Q

r6 ,

r 2 = 9

i=2(xi)2 .

This is “doublly T-dual”, (t, x1) ⇔ (˜ t, ˜ x1), DFT background to F1 à la Dabholkar-Gibbons-Harvey-Ruiz 1990, c.f. 2D null-wave à la Berkeley-Berman-Rudolf DFT as well as the DFT sigma model is well-defined even for such a non-Riemannian background. In particular, the Gomis-Ooguri ‘non-relativistic’ string theory can be identified precisely as the DFT sigma model on the above non-Riemannian background. Ko-Meyer-Melby-Thompson-JHP 2015

JEONG-HYUCK PARK DOUBLED-YET-GAUGED, SEMI-COVARIANCE & TWOFOLD SPIN

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SLIDE 31

Semi-covariant formulation of DFT/SDFT:

Gravity on doubled-yet-gauged spacetime 1011.1324/1105.6294/· · ·

JEONG-HYUCK PARK DOUBLED-YET-GAUGED, SEMI-COVARIANCE & TWOFOLD SPIN

slide-32
SLIDE 32

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

1 8 (PABPCD − ¯

PAB ¯ PCD)SACBD + 1

2Tr(F ¯

F) − i ¯ ρFρ′ + i ¯ ψ¯

pγqF¯

γ¯

pψ′q

+i 1

2 ¯

ργpD⋆

p ρ − i ¯

ψ¯

pD⋆ ¯ p ρ − i 1 2 ¯

ψ¯

pγqD⋆ q ψ¯ p − i 1 2 ¯

ρ′¯ γ¯

pD′⋆ ¯ p ρ′ + i ¯

ψ′pD′⋆

p ρ′ + i 1 2 ¯

ψ′p¯ γ¯

qD′⋆ ¯ q ψ′p

  • Jeon-Lee-JHP-Suh 2012

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

JEONG-HYUCK PARK DOUBLED-YET-GAUGED, SEMI-COVARIANCE & TWOFOLD SPIN

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SLIDE 33

Notation 2/2

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 ¯ η¯

p¯ q = diag(+ − − · · · −)

¯ α, ¯ β, · · · Spin(D−1, 1)R spinor ¯ C+ ¯

α ¯ β,

(¯ γ¯

p)T = ¯

C+¯ γ¯

p ¯

C−1

+

JEONG-HYUCK PARK DOUBLED-YET-GAUGED, SEMI-COVARIANCE & TWOFOLD SPIN

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SLIDE 34

Field contents of D = 10 Maximal SDFT Bosons NS-NS sector        DFT-dilaton: d DFT-vielbeins: VAp , ¯ VA¯

p

R-R potential: Cα ¯

α

Fermions (Majorana-Weyl) DFT-dilatinos: ρα , ρ′ ¯

α

Gravitinos: ψα

¯ p ,

ψ′ ¯

α p

R-R potential and Fermions carry NOT (D + D)-dimensional BUT undoubled D-dimensional indices.

JEONG-HYUCK PARK DOUBLED-YET-GAUGED, SEMI-COVARIANCE & TWOFOLD SPIN

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SLIDE 35

Field contents of D = 10 Maximal SDFT Bosons NS-NS sector        DFT-dilaton: d DFT-vielbeins: VAp , ¯ VA¯

p

R-R potential: Cα ¯

α

Fermions (Majorana-Weyl) DFT-dilatinos: ρα , ρ′ ¯

α

Gravitinos: ψα

¯ p ,

ψ′ ¯

α p

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

  • f the twofold local Lorentz symmetries.

JEONG-HYUCK PARK DOUBLED-YET-GAUGED, SEMI-COVARIANCE & TWOFOLD SPIN

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SLIDE 36

The DFT-dilaton gives rise to a scalar density with weight one, e−2d . The DFT-vielbeins satisfy four ‘defining’ properties: VApV Aq = ηpq , ¯ VA¯

p ¯

V A¯

q = ¯

η¯

p¯ q ,

VAp ¯ V A¯

q = 0 ,

VApVBp + ¯ VA¯

p ¯

VB

¯ p = JAB .

Naturally, they generate a pair of two-index ‘projectors’, PAB := VApVBp , PABPBC = PAC , ¯ PAB := ¯ VA

¯ p ¯

VB¯

p ,

¯ PAB ¯ PBC = ¯ PAC , which are symmetric, orthogonal and complementary to each other, PAB = PBA , ¯ PAB = ¯ PBA , PAB ¯ PBC = 0 , PAB + ¯ PAB = δAB . Some further projection properties follow PABVBp = VAp , ¯ PAB ¯ VB¯

p = ¯

VA¯

p ,

¯ PABVBp = 0 , PAB ¯ VB¯

p = 0 .

Note also HAB = PAB − ¯

  • PAB. However, our emphasis lies on the ‘projectors’ rather than

the “generalized metric".

JEONG-HYUCK PARK DOUBLED-YET-GAUGED, SEMI-COVARIANCE & TWOFOLD SPIN

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SLIDE 37

The DFT-dilaton gives rise to a scalar density with weight one, e−2d . The DFT-vielbeins satisfy four ‘defining’ properties: VApV Aq = ηpq , ¯ VA¯

p ¯

V A¯

q = ¯

η¯

p¯ q ,

VAp ¯ V A¯

q = 0 ,

VApVBp + ¯ VA¯

p ¯

VB

¯ p = JAB .

Naturally, they generate a pair of two-index ‘projectors’, PAB := VApVBp , PABPBC = PAC , ¯ PAB := ¯ VA

¯ p ¯

VB¯

p ,

¯ PAB ¯ PBC = ¯ PAC , which are symmetric, orthogonal and complementary to each other, PAB = PBA , ¯ PAB = ¯ PBA , PAB ¯ PBC = 0 , PAB + ¯ PAB = δAB . Some further projection properties follow PABVBp = VAp , ¯ PAB ¯ VB¯

p = ¯

VA¯

p ,

¯ PABVBp = 0 , PAB ¯ VB¯

p = 0 .

Note also HAB = PAB − ¯

  • PAB. However, our emphasis lies on the ‘projectors’ rather than

the “generalized metric".

JEONG-HYUCK PARK DOUBLED-YET-GAUGED, SEMI-COVARIANCE & TWOFOLD SPIN

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SLIDE 38

We continue to define a pair of six-index projectors, PCABDEF := PCDP[A[EPB]F] +

2 D−1 PC[APB][EPF]D ,

PCABDEF PDEF GHI = PCABGHI , ¯ PCABDEF := ¯ PCD ¯ P[A[E ¯ PB]F] +

2 D−1 ¯

PC[A ¯ PB][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 , ¯ PAABDEF = 0 , ¯ PAB ¯ PABCDEF = 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 DOUBLED-YET-GAUGED, SEMI-COVARIANCE & TWOFOLD SPIN

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SLIDE 39

Chirality of Spin(1, D−1)L × Spin(D−1, 1)R : γ(D+1)ψ¯

p = c ψ¯ p ,

γ(D+1)ρ = −c ρ , ¯ γ(D+1)ψ′

p = c′ψ′ p ,

¯ γ(D+1)ρ′ = −c′ρ′ , γ(D+1)C¯ γ(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 DOUBLED-YET-GAUGED, SEMI-COVARIANCE & TWOFOLD SPIN

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SLIDE 40

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 DOUBLED-YET-GAUGED, SEMI-COVARIANCE & TWOFOLD SPIN

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SLIDE 41

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 DOUBLED-YET-GAUGED, SEMI-COVARIANCE & TWOFOLD SPIN

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SLIDE 42

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 DOUBLED-YET-GAUGED, SEMI-COVARIANCE & TWOFOLD SPIN

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SLIDE 43

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 − ωΓB

BCTA1A2···An + n

  • i=1

ΓCAi

BTA1···Ai−1BAi+1···An .

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

JEONG-HYUCK PARK DOUBLED-YET-GAUGED, SEMI-COVARIANCE & TWOFOLD SPIN

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SLIDE 44

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 − ωΓB

BCTA1A2···An + n

  • i=1

ΓCAi

BTA1···Ai−1BAi+1···An .

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

JEONG-HYUCK PARK DOUBLED-YET-GAUGED, SEMI-COVARIANCE & TWOFOLD SPIN

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SLIDE 45

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 − ωΓB

BCTA1A2···An + n

  • i=1

ΓCAi

BTA1···Ai−1BAi+1···An .

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

JEONG-HYUCK PARK DOUBLED-YET-GAUGED, SEMI-COVARIANCE & TWOFOLD SPIN

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SLIDE 46

By definition, the master derivative annihilates all the ‘constants’, DAJBC = ∇AJBC = ΓABDJDC + ΓACDJBD = 0 , DAηpq = DAηpq = ΦAprηrq + ΦAqrηpr = 0 , DA¯ η¯

p¯ q = DA¯

η¯

p¯ q = ¯

ΦA¯

p ¯ r ¯

η¯

r¯ q + ¯

ΦA¯

q ¯ r ¯

η¯

p¯ r = 0 ,

DAC+αβ = DAC+αβ = ΦAαδC+δβ + ΦAβδC+αδ = 0 , DA ¯ C+ ¯

α ¯ β = DA ¯

C+ ¯

α ¯ β = ¯

ΦA ¯

α ¯ δ ¯

C+ ¯

δ ¯ β + ¯

ΦA ¯

β ¯ δ ¯

C+ ¯

α¯ δ = 0 ,

including the gamma matrices, DA(γp)αβ = DA(γp)αβ = ΦApq(γq)αβ + ΦAαδ(γp)δβ − (γp)αδΦAδβ = 0 , DA(¯ γ¯

p) ¯ α ¯ β = DA(¯

γ¯

p) ¯ α ¯ β = ¯

ΦA

¯ p¯ q(¯

γ¯

q) ¯ α ¯ β + ¯

ΦA ¯

α ¯ δ(¯

γ¯

p)¯ δ ¯ β − (¯

γ¯

p) ¯ α ¯ δ ¯

ΦA

¯ δ ¯ β = 0 .

JEONG-HYUCK PARK DOUBLED-YET-GAUGED, SEMI-COVARIANCE & TWOFOLD SPIN

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SLIDE 47

It follows then that the connections are all anti-symmetric, ΓABC = −ΓACB , ΦApq = −ΦAqp , ΦAαβ = −ΦAβα , ¯ ΦA¯

p¯ q = −¯

ΦA¯

q¯ p ,

¯ ΦA ¯

α ¯ β = −¯

ΦA ¯

β ¯ α ,

and as usual, ΦAαβ = 1

4 ΦApq(γpq)αβ ,

¯ ΦA ¯

α ¯ β = 1 4 ¯

ΦA¯

p¯ q(¯

γ¯

p¯ q) ¯ α ¯ β .

JEONG-HYUCK PARK DOUBLED-YET-GAUGED, SEMI-COVARIANCE & TWOFOLD SPIN

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SLIDE 48

Further, the master derivative is compatible with the whole NS-NS sector, DAd = ∇Ad := − 1

2 e2d∇A(e−2d) = ∂Ad + 1 2ΓBBA = 0 ,

DAVBp = ∂AVBp + ΓABCVCp + ΦApqVBq = 0 , DA ¯ VB¯

p = ∂A ¯

VB¯

p + ΓABC ¯

VC¯

p + ¯

ΦA¯

p ¯ q ¯

VB¯

q = 0 .

It follows that DAPBC = ∇APBC = 0 , DA ¯ PBC = ∇A ¯ PBC = 0 , and the connections are related to each other, ΓABC = VBpDAVCp + ¯ VB

¯ pDA ¯

VC¯

p ,

ΦApq = V Bp∇AVBq , ¯ ΦA¯

p¯ q = ¯

V B ¯

p∇A ¯

VB¯

q .

JEONG-HYUCK PARK DOUBLED-YET-GAUGED, SEMI-COVARIANCE & TWOFOLD SPIN

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SLIDE 49

Further, the master derivative is compatible with the whole NS-NS sector, DAd = ∇Ad := − 1

2 e2d∇A(e−2d) = ∂Ad + 1 2ΓBBA = 0 ,

DAVBp = ∂AVBp + ΓABCVCp + ΦApqVBq = 0 , DA ¯ VB¯

p = ∂A ¯

VB¯

p + ΓABC ¯

VC¯

p + ¯

ΦA¯

p ¯ q ¯

VB¯

q = 0 .

It follows that DAPBC = ∇APBC = 0 , DA ¯ PBC = ∇A ¯ PBC = 0 , and the connections are related to each other, ΓABC = VBpDAVCp + ¯ VB

¯ pDA ¯

VC¯

p ,

ΦApq = V Bp∇AVBq , ¯ ΦA¯

p¯ q = ¯

V B ¯

p∇A ¯

VB¯

q .

JEONG-HYUCK PARK DOUBLED-YET-GAUGED, SEMI-COVARIANCE & TWOFOLD SPIN

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SLIDE 50

The connections assume the following most general forms: ΓCAB = Γ0

CAB + ∆CpqVApVBq + ¯

∆C¯

p¯ q ¯

VA

¯ p ¯

VB

¯ q ,

ΦApq = Φ0

Apq + ∆Apq ,

¯ ΦA¯

p¯ q = ¯

Φ0

A¯ p¯ q + ¯

∆A¯

p¯ q .

Here Γ0

CAB =

2

  • P∂CP ¯

P

  • [AB] + 2

¯ P[AD ¯ PB]E − P[ADPB]E ∂DPEC −

4 D−1

¯ PC[A ¯ PB]D + PC[APB]D ∂Dd + (P∂EP ¯ P)[ED]

  • ,

Jeon-Lee-JHP 2011 and, with the corresponding derivative, ∇0

A = ∂A + Γ0 A,

Φ0

Apq = V Bp∇0 AVBq = V Bp∂AVBq + Γ0 ABCV BpV Cq ,

¯ Φ0

A¯ p¯ q = ¯

V B ¯

p∇0 A ¯

VB¯

q = ¯

V B ¯

p∂A ¯

VB¯

q + Γ0 ABC ¯

V B ¯

p ¯

V C ¯

q .

JEONG-HYUCK PARK DOUBLED-YET-GAUGED, SEMI-COVARIANCE & TWOFOLD SPIN

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SLIDE 51

The connections assume the following most general forms: ΓCAB = Γ0

CAB + ∆CpqVApVBq + ¯

∆C¯

p¯ q ¯

VA

¯ p ¯

VB

¯ q ,

ΦApq = Φ0

Apq + ∆Apq ,

¯ ΦA¯

p¯ q = ¯

Φ0

A¯ p¯ q + ¯

∆A¯

p¯ q .

The extra pieces, ∆Apq and ¯ ∆A¯

p¯ q, correspond to the torsion of SDFT, which must be

covariant and, in order to maintain DAd = 0, must satisfy ∆ApqV Ap = 0 , ¯ ∆A¯

p¯ q ¯

V A¯

p = 0 .

Otherwise they are arbitrary. As in SUGRA, the torsion can be constructed from the bi-spinorial objects, e.g. ¯ ργpqψA , ¯ ψ¯

pγAψ¯ q ,

¯ ργApqρ , ¯ ψ¯

pγApqψ¯ p ,

where we set ψA = ¯ VA

¯ pψ¯ p, γA = VApγp .

JEONG-HYUCK PARK DOUBLED-YET-GAUGED, SEMI-COVARIANCE & TWOFOLD SPIN

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SLIDE 52

The connections assume the following most general forms: ΓCAB = Γ0

CAB + ∆CpqVApVBq + ¯

∆C¯

p¯ q ¯

VA

¯ p ¯

VB

¯ q ,

ΦApq = Φ0

Apq + ∆Apq ,

¯ ΦA¯

p¯ q = ¯

Φ0

A¯ p¯ q + ¯

∆A¯

p¯ q .

The extra pieces, ∆Apq and ¯ ∆A¯

p¯ q, correspond to the torsion of SDFT, which must be

covariant and, in order to maintain DAd = 0, must satisfy ∆ApqV Ap = 0 , ¯ ∆A¯

p¯ q ¯

V A¯

p = 0 .

Otherwise they are arbitrary. As in SUGRA, the torsion can be constructed from the bi-spinorial objects, e.g. ¯ ργpqψA , ¯ ψ¯

pγAψ¯ q ,

¯ ργApqρ , ¯ ψ¯

pγApqψ¯ p ,

where we set ψA = ¯ VA

¯ pψ¯ p, γA = VApγp .

JEONG-HYUCK PARK DOUBLED-YET-GAUGED, SEMI-COVARIANCE & TWOFOLD SPIN

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SLIDE 53

The connections assume the following most general forms: ΓCAB = Γ0

CAB + ∆CpqVApVBq + ¯

∆C¯

p¯ q ¯

VA

¯ p ¯

VB

¯ q ,

ΦApq = Φ0

Apq + ∆Apq ,

¯ ΦA¯

p¯ q = ¯

Φ0

A¯ p¯ q + ¯

∆A¯

p¯ q .

The extra pieces, ∆Apq and ¯ ∆A¯

p¯ q, correspond to the torsion of SDFT, which must be

covariant and, in order to maintain DAd = 0, must satisfy ∆ApqV Ap = 0 , ¯ ∆A¯

p¯ q ¯

V A¯

p = 0 .

Otherwise they are arbitrary. As in SUGRA, the torsion can be constructed from the bi-spinorial objects, e.g. ¯ ργpqψA , ¯ ψ¯

pγAψ¯ q ,

¯ ργApqρ , ¯ ψ¯

pγApqψ¯ p ,

where we set ψA = ¯ VA

¯ pψ¯ p, γA = VApγp .

JEONG-HYUCK PARK DOUBLED-YET-GAUGED, SEMI-COVARIANCE & TWOFOLD SPIN

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SLIDE 54

The ‘torsionless’ connection, Γ0

CAB =

2

  • P∂CP ¯

P

  • [AB] + 2

¯ P[AD ¯ PB]E − P[ADPB]E ∂DPEC −

4 D−1

¯ PC[A ¯ PB]D + PC[APB]D ∂Dd + (P∂EP ¯ P)[ED]

  • ,

further obeys Γ0

ABC + Γ0 BCA + Γ0 CAB = 0 ,

and PCABDEF Γ0

DEF = 0 ,

¯ PCABDEF Γ0

DEF = 0 .

JEONG-HYUCK PARK DOUBLED-YET-GAUGED, SEMI-COVARIANCE & TWOFOLD SPIN

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SLIDE 55

In fact, the torsionless connection, Γ0

CAB =

2

  • P∂CP ¯

P

  • [AB] + 2

¯ P[AD ¯ PB]E − P[ADPB]E ∂DPEC −

4 D−1

¯ PC[A ¯ PB]D + PC[APB]D ∂Dd + (P∂EP ¯ P)[ED]

  • ,

is the unique solution to the following constraints: ΓCAB + ΓCBA = 0 = ⇒ ∇AJBC = 0 , ∇APBC = ∇A ¯ PBC = 0 , ∇Ad = 0 , ΓABC + ΓCAB + ΓBCA = 0 = ⇒ ˆ L(∂) = ˆ L(∇) , (P + ¯ P)CABDEF ΓDEF = 0 . In this way, Γ0

ABC 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 DOUBLED-YET-GAUGED, SEMI-COVARIANCE & TWOFOLD SPIN

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SLIDE 56

In fact, the torsionless connection, Γ0

CAB =

2

  • P∂CP ¯

P

  • [AB] + 2

¯ P[AD ¯ PB]E − P[ADPB]E ∂DPEC −

4 D−1

¯ PC[A ¯ PB]D + PC[APB]D ∂Dd + (P∂EP ¯ P)[ED]

  • ,

is the unique solution to the following constraints: ΓCAB + ΓCBA = 0 = ⇒ ∇AJBC = 0 , ∇APBC = ∇A ¯ PBC = 0 , ∇Ad = 0 , ΓABC + ΓCAB + ΓBCA = 0 = ⇒ ˆ L(∂) = ˆ L(∇) , (P + ¯ P)CABDEF ΓDEF = 0 . In this way, Γ0

ABC 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 DOUBLED-YET-GAUGED, SEMI-COVARIANCE & TWOFOLD SPIN

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SLIDE 57

Semi-covariant Riemann curvature

The usual curvatures for the three connections, RCDAB = ∂AΓBCD − ∂BΓACD + ΓACEΓBED − ΓBCEΓAED , FABpq = ∂AΦBpq − ∂BΦApq + ΦAprΦBr q − ΦBprΦAr q , ¯ FAB¯

p¯ q = ∂A ¯

ΦB¯

p¯ q − ∂B ¯

ΦA¯

p¯ q + ¯

ΦA¯

p¯ r ¯

ΦB

¯ r ¯ q − ¯

ΦB¯

p¯ r ¯

ΦA

¯ r ¯ q ,

are, from [DA, DB]VCp = 0 and [DA, DB]¯ VC¯

p = 0, related to each other,

RABCD = FCDpqVA

pVB q + ¯

FCD¯

p¯ q ¯

VA

¯ p ¯

VB

¯ q .

However, the crucial object in DFT turns out to be SABCD := 1

2

  • RABCD + RCDAB − ΓE

ABΓECD

  • ,

which we name semi-covariant Riemann curvature.

JEONG-HYUCK PARK DOUBLED-YET-GAUGED, SEMI-COVARIANCE & TWOFOLD SPIN

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SLIDE 58

Semi-covariant Riemann curvature

The usual curvatures for the three connections, RCDAB = ∂AΓBCD − ∂BΓACD + ΓACEΓBED − ΓBCEΓAED , FABpq = ∂AΦBpq − ∂BΦApq + ΦAprΦBr q − ΦBprΦAr q , ¯ FAB¯

p¯ q = ∂A ¯

ΦB¯

p¯ q − ∂B ¯

ΦA¯

p¯ q + ¯

ΦA¯

p¯ r ¯

ΦB

¯ r ¯ q − ¯

ΦB¯

p¯ r ¯

ΦA

¯ r ¯ q ,

are, from [DA, DB]VCp = 0 and [DA, DB]¯ VC¯

p = 0, related to each other,

RABCD = FCDpqVA

pVB q + ¯

FCD¯

p¯ q ¯

VA

¯ p ¯

VB

¯ q .

However, the crucial object in DFT turns out to be SABCD := 1

2

  • RABCD + RCDAB − ΓE

ABΓECD

  • ,

which we name semi-covariant Riemann curvature.

JEONG-HYUCK PARK DOUBLED-YET-GAUGED, SEMI-COVARIANCE & TWOFOLD SPIN

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SLIDE 59

Properties of the semi-covariant curvature

Under arbitrary variation of the connection, δΓABC, it transforms as δS0

ABCD = D[AδΓ0 B]CD + D[CδΓ0 D]AB ,

δSABCD = D[AδΓB]CD + D[CδΓD]AB − 3

2 Γ[ABE]δΓE CD − 3 2Γ[CDE]δΓE AB .

It also satisfies precisely the same symmetric property as the ordinary Riemann curvature, SABCD = 1

2

  • S[AB][CD] + S[CD][AB]
  • ,

S0

[ABC]D = 0 ,

as well as projection property, Sp¯

pq¯ q = SABCDV Ap ¯

V B ¯

pV Cq ¯

V D ¯

q = 0 .

JEONG-HYUCK PARK DOUBLED-YET-GAUGED, SEMI-COVARIANCE & TWOFOLD SPIN

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SLIDE 60

Properties of the semi-covariant curvature

Under arbitrary variation of the connection, δΓABC, it transforms as δS0

ABCD = D[AδΓ0 B]CD + D[CδΓ0 D]AB ,

δSABCD = D[AδΓB]CD + D[CδΓD]AB − 3

2 Γ[ABE]δΓE CD − 3 2Γ[CDE]δΓE AB .

It also satisfies precisely the same symmetric property as the ordinary Riemann curvature, SABCD = 1

2

  • S[AB][CD] + S[CD][AB]
  • ,

S0

[ABC]D = 0 ,

as well as projection property, Sp¯

pq¯ q = SABCDV Ap ¯

V B ¯

pV Cq ¯

V D ¯

q = 0 .

JEONG-HYUCK PARK DOUBLED-YET-GAUGED, SEMI-COVARIANCE & TWOFOLD SPIN

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SLIDE 61

‘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
  • ≡ ˆ

LX

  • ∇CTA1···An
  • +
  • i

2(P+ ¯ P)CAi

BFDE∂F ∂[DXE]T···B··· .

Hence, it is not DFT-diffeomorphism covariant, δX = ˆ 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 DOUBLED-YET-GAUGED, SEMI-COVARIANCE & TWOFOLD SPIN

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SLIDE 62

‘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
  • ≡ ˆ

LX

  • ∇CTA1···An
  • +
  • i

2(P+ ¯ P)CAi

BFDE∂F ∂[DXE]T···B··· .

Hence, it is not DFT-diffeomorphism covariant, δX = ˆ 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 DOUBLED-YET-GAUGED, SEMI-COVARIANCE & TWOFOLD SPIN

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SLIDE 63

Completely covariant derivatives

For O(D, D) tensors: PCD ¯ PA1

B1 ¯

PA2

B2 · · · ¯

PAn

Bn∇DTB1B2···Bn ,

¯ PCDPA1

B1PA2 B2 · · · PAn Bn∇DTB1B2···Bn , PAB ¯ PC1

D1 ¯

PC2

D2 · · · ¯

PCn

Dn ∇ATBD1D2···Dn ,

¯ PABPC1

D1 PC2 D2 · · · PCn Dn ∇ATBD1D2···Dn

       Divergences , PAB ¯ PC1

D1 ¯

PC2

D2 · · · ¯

PCn

Dn ∇A∇BTD1D2···Dn ,

¯ PABPC1

D1 PC2 D2 · · · PCn Dn ∇A∇BTD1D2···Dn

       Laplacians ,

and DAC ¯ PB1

D1 · · · ¯

PBn

DnTCD1···Dn ,

¯ DACPB1

D1 · · · PBn DnTCD1···Dn ,

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

DA

B := (PA BPCD − 2PA DPBC)(∇C∇D − SCD) ,

¯ DA

B := (¯

PA

B ¯

PCD − 2¯ PA

D ¯

PBC)(∇C∇D − SCD) ,

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

JEONG-HYUCK PARK DOUBLED-YET-GAUGED, SEMI-COVARIANCE & TWOFOLD SPIN

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SLIDE 64

Completely covariant derivatives

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

q1¯ q2···¯ qn ,

D¯

pTq1q2···qn ,

DpTp¯

q1¯ q2···¯ qn ,

D¯

pT¯ pq1q2···qn ,

DpDpT¯

q1¯ q2···¯ qn ,

D¯

pD¯ pTq1q2···qn ,

DpqTq¯

p1¯ p2···¯ pn ,

¯ D¯

p ¯ qT¯ qp1p2···pn .

These are the ‘pull-back’ of the previous page using the DFT-vielbeins, such as Dp := V ApDA , D¯

p := ¯

V A¯

pDA .

JEONG-HYUCK PARK DOUBLED-YET-GAUGED, SEMI-COVARIANCE & TWOFOLD SPIN

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SLIDE 65

Completely covariant derivatives

Following the aforementioned general prescription, completely covariant Yang-Mills field strength is given by two opposite projections, or Fp¯

q = V M p ¯

V 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, [gVMg−1 − i(∂Mg)g−1]∂M = 0 , ( ˆ LX VM)∂M = [X N∂NVM + (∂MXN − ∂NX M)VN]∂M = 0 .

JEONG-HYUCK PARK DOUBLED-YET-GAUGED, SEMI-COVARIANCE & TWOFOLD SPIN

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SLIDE 66

Completely covariant derivatives

Following the aforementioned general prescription, completely covariant Yang-Mills field strength is given by two opposite projections, or Fp¯

q = V M p ¯

V 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, [gVMg−1 − i(∂Mg)g−1]∂M = 0 , ( ˆ LX VM)∂M = [X N∂NVM + (∂MXN − ∂NX M)VN]∂M = 0 .

JEONG-HYUCK PARK DOUBLED-YET-GAUGED, SEMI-COVARIANCE & TWOFOLD SPIN

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SLIDE 67

Completely covariant derivatives

O(D, D) covariant Killing equations in DFT: ˆ LX HMN = 0 ⇐ ⇒ (P∇)M(¯ PX)N − (¯ P∇)N(PX)M = 0 , ˆ LX d = 0 ⇐ ⇒ ∇MX M = 0 . JHP-Rey-Rim-Sakatani 2015 Chris Blair 2015

JEONG-HYUCK PARK DOUBLED-YET-GAUGED, SEMI-COVARIANCE & TWOFOLD SPIN

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SLIDE 68

Completely covariant derivatives

Dirac operators for fermions, ρα, ψα

¯ p , ρ′ ¯ α, ψ′ ¯ α p :

γpDpρ = γADAρ , γpDpψ¯

p = γADAψ¯ p ,

D¯

pρ ,

D¯

pψ¯ p = DAψA ,

¯ ψAγp(DAψ¯

q − 1 2 D¯ qψA) ,

¯ γ¯

pD¯ pρ′ = ¯

γADAρ′ , ¯ γ¯

pD¯ pψ′ p = ¯

γADAψ′

p ,

Dpρ′ , Dpψ′p = DAψ′A , ¯ ψ′A¯ γ¯

p(DAψ′ q − 1 2 Dqψ′ A) .

Incorporation of fermions into DFT 1109.2035

JEONG-HYUCK PARK DOUBLED-YET-GAUGED, SEMI-COVARIANCE & TWOFOLD SPIN

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SLIDE 69

Completely covariant derivatives

For R-R potential, Cα ¯

β:

D+C := γADAC + γ(D+1)DAC¯ γA , D−C := γADAC − γ(D+1)DAC¯ γA . 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, Cα ¯

α, 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 DOUBLED-YET-GAUGED, SEMI-COVARIANCE & TWOFOLD SPIN

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SLIDE 70

Completely covariant derivatives

For R-R potential, Cα ¯

β:

D+C := γADAC + γ(D+1)DAC¯ γA , D−C := γADAC − γ(D+1)DAC¯ γA . 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, Cα ¯

α, 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 DOUBLED-YET-GAUGED, SEMI-COVARIANCE & TWOFOLD SPIN

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SLIDE 71

Completely covariant curvatures

Scalar curvature: S := (PABPCD − ¯ PAB ¯ PCD)SACBD c.f. SABAB = 0 . “Ricci” curvature: S0

p¯ q = V Ap ¯

V B ¯

qS0 AB

where we set S0

AB = S0 ACB C .

Further, we have conserved “Einstein” curvature, GAB = 2(PAC ¯ PBD − ¯ PACPBD)SCD − 1

2 J ABS ,

∇AGAB = 0 .

JEONG-HYUCK PARK DOUBLED-YET-GAUGED, SEMI-COVARIANCE & TWOFOLD SPIN

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SLIDE 72

Combining all the results above, we are now ready to spell D = 10 Maximally Supersymmetric Double Field Theory

JEONG-HYUCK PARK DOUBLED-YET-GAUGED, SEMI-COVARIANCE & TWOFOLD SPIN

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SLIDE 73

D = 10 Maximal SDFT [ 1210.5078 ]

Lagrangian: LType II = e−2d

1 8 (PABPCD − ¯

PAB ¯ PCD)SACBD + 1

2Tr(F ¯

F) − i ¯ ρFρ′ + i ¯ ψ¯

pγqF¯

γ¯

pψ′q

+i 1

2 ¯

ργpD⋆

p ρ − i ¯

ψ¯

pD⋆ ¯ p ρ − i 1 2 ¯

ψ¯

pγqD⋆ q ψ¯ p − i 1 2 ¯

ρ′¯ γ¯

pD′⋆ ¯ p ρ′ + i ¯

ψ′pD′⋆

p ρ′ + i 1 2 ¯

ψ′p¯ γ¯

qD′⋆ ¯ 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 DOUBLED-YET-GAUGED, SEMI-COVARIANCE & TWOFOLD SPIN

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SLIDE 74

D = 10 Maximal SDFT [ 1210.5078 ]

Lagrangian: LType II = e−2d

1 8 (PABPCD − ¯

PAB ¯ PCD)SACBD + 1

2Tr(F ¯

F) − i ¯ ρFρ′ + i ¯ ψ¯

pγqF¯

γ¯

pψ′q

+i 1

2 ¯

ργpD⋆

p ρ − i ¯

ψ¯

pD⋆ ¯ p ρ − i 1 2 ¯

ψ¯

pγqD⋆ q ψ¯ p − i 1 2 ¯

ρ′¯ γ¯

pD′⋆ ¯ p ρ′ + i ¯

ψ′pD′⋆

p ρ′ + i 1 2 ¯

ψ′p¯ γ¯

qD′⋆ ¯ 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 DOUBLED-YET-GAUGED, SEMI-COVARIANCE & TWOFOLD SPIN

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SLIDE 75

D = 10 Maximal SDFT [ 1210.5078 ]

Lagrangian: LType II = e−2d

1 8 (PABPCD − ¯

PAB ¯ PCD)SACBD + 1

2Tr(F ¯

F) − i ¯ ρFρ′ + i ¯ ψ¯

pγqF¯

γ¯

pψ′q

+i 1

2 ¯

ργpD⋆

p ρ − i ¯

ψ¯

pD⋆ ¯ p ρ − i 1 2 ¯

ψ¯

pγqD⋆ q ψ¯ p − i 1 2 ¯

ρ′¯ γ¯

pD′⋆ ¯ p ρ′ + i ¯

ψ′pD′⋆

p ρ′ + i 1 2 ¯

ψ′p¯ γ¯

qD′⋆ ¯ q ψ′p

  • .

Torsions: The semi-covariant curvature, SABCD, is given by the connection, ΓABC = Γ0

ABC + i 1 3 ¯

ργABCρ − 2i ¯ ργBCψA − i 1

3 ¯

ψ¯

pγABCψ¯ p + 4i ¯

ψBγAψC + i 1

3 ¯

ρ′¯ γABCρ′ − 2i ¯ ρ′¯ γBCψ′A − i 1

3 ¯

ψ′p¯ γABCψ′p + 4i ¯ ψ′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 11 96 ¯

ργABCρ + i 5

4 ¯

ργBCψA + i 5

24 ¯

ψ¯

pγABCψ¯ p − 2i ¯

ψBγAψC + i 5

2 ¯

ρ′¯ γBCψ′A , Γ′⋆

ABC = ΓABC − i 11 96 ¯

ρ′¯ γABCρ′ + i 5

4 ¯

ρ′¯ γBCψ′A + i 5

24 ¯

ψ′p¯ γABCψ′p − 2i ¯ ψ′B¯ γAψ′C + i 5

2 ¯

ργBCψA .

JEONG-HYUCK PARK DOUBLED-YET-GAUGED, SEMI-COVARIANCE & TWOFOLD SPIN

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SLIDE 76

D = 10 Maximal SDFT [ 1210.5078 ]

Lagrangian: LType II = e−2d

1 8 (PABPCD − ¯

PAB ¯ PCD)SACBD + 1

2Tr(F ¯

F) − i ¯ ρFρ′ + i ¯ ψ¯

pγqF¯

γ¯

pψ′q

+i 1

2 ¯

ργpD⋆

p ρ − i ¯

ψ¯

pD⋆ ¯ p ρ − i 1 2 ¯

ψ¯

pγqD⋆ q ψ¯ p − i 1 2 ¯

ρ′¯ γ¯

pD′⋆ ¯ p ρ′ + i ¯

ψ′pD′⋆

p ρ′ + i 1 2 ¯

ψ′p¯ γ¯

qD′⋆ ¯ q ψ′p

  • .

Torsions: The semi-covariant curvature, SABCD, is given by the connection, ΓABC = Γ0

ABC + i 1 3 ¯

ργABCρ − 2i ¯ ργBCψA − i 1

3 ¯

ψ¯

pγABCψ¯ p + 4i ¯

ψBγAψC + i 1

3 ¯

ρ′¯ γABCρ′ − 2i ¯ ρ′¯ γBCψ′A − i 1

3 ¯

ψ′p¯ γABCψ′p + 4i ¯ ψ′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 11 96 ¯

ργABCρ + i 5

4 ¯

ργBCψA + i 5

24 ¯

ψ¯

pγABCψ¯ p − 2i ¯

ψBγAψC + i 5

2 ¯

ρ′¯ γBCψ′A , Γ′⋆

ABC = ΓABC − i 11 96 ¯

ρ′¯ γABCρ′ + i 5

4 ¯

ρ′¯ γBCψ′A + i 5

24 ¯

ψ′p¯ γABCψ′p − 2i ¯ ψ′B¯ γAψ′C + i 5

2 ¯

ργBCψA .

JEONG-HYUCK PARK DOUBLED-YET-GAUGED, SEMI-COVARIANCE & TWOFOLD SPIN

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SLIDE 77

D = 10 Maximal SDFT [ 1210.5078 ]

Maximal supersymmetry transformation rules are also completely covariant, δεd = −i 1

2 (¯

ερ + ¯ ε′ρ′) , δεVAp = i ¯ VA

¯ q(¯

ε′¯ γ¯

qψ′ p − ¯

εγpψ¯

q) ,

δε ¯ VA¯

p = iVAq(¯

εγqψ¯

p − ¯

ε′¯ γ¯

pψ′ q) ,

δεC = i 1

2(γpε ¯

ψ′

p − ε¯

ρ′ − ψ¯

p ¯

ε′¯ γ¯

p + ρ¯

ε′) + Cδεd − 1

2(¯

V A¯

q δεVAp)γ(d+1)γpC¯

γ¯

q ,

δερ = −γp ˆ Dpε + i 1

2 γpε ¯

ψ′

pρ′ − iγpψ¯ q ¯

ε′¯ γ¯

qψ′ p ,

δερ′ = −¯ γ¯

p ˆ

D′

¯ pε′ + i 1 2 ¯

γ¯

pε′ ¯

ψ¯

pρ − i¯

γ¯

qψ′ p ¯

εγpψ¯

q ,

δεψ¯

p = ˆ

D¯

pε + (F − i 1 2γqρ ¯

ψ′

q + i 1 2 ψ¯ q ¯

ρ′¯ γ¯

q)¯

γ¯

pε′ + i 1 4ε ¯

ψ¯

pρ + i 1 2ψ¯ p ¯

ερ , δεψ′

p = ˆ

D′

pε′ + ( ¯

F − i 1

2 ¯

γ¯

qρ′ ¯

ψ¯

q + i 1 2 ψ′q ¯

ργq)γpε + i 1

4ε′ ¯

ψ′

pρ′ + i 1 2ψ′ p ¯

ε′ρ′ , where ˆ ΓABC = ΓABC − i 17

48 ¯

ργABCρ + i 5

2 ¯

ργBCψA + i 1

4 ¯

ψ¯

pγABCψ¯ p − 3i ¯

ψ′

B¯

γAψ′

C ,

ˆ Γ′

ABC = ΓABC − i 17 48 ¯

ρ′¯ γABCρ′ + i 5

2 ¯

ρ′¯ γBCψ′A + i 1

4 ¯

ψ′p¯ γABCψ′p − 3i ¯ ψBγAψC .

JEONG-HYUCK PARK DOUBLED-YET-GAUGED, SEMI-COVARIANCE & TWOFOLD SPIN

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SLIDE 78

D = 10 Maximal SDFT [ 1210.5078 ]

Lagrangian: LType II = e−2d

1 8 (PABPCD − ¯

PAB ¯ PCD)SACBD + 1

2Tr(F ¯

F) − i ¯ ρFρ′ + i ¯ ψ¯

pγqF¯

γ¯

pψ′q

+i 1

2 ¯

ργpD⋆

p ρ − i ¯

ψ¯

pD⋆ ¯ p ρ − i 1 2 ¯

ψ¯

pγqD⋆ q ψ¯ p − i 1 2 ¯

ρ′¯ γ¯

pD′⋆ ¯ p ρ′ + i ¯

ψ′pD′⋆

p ρ′ + i 1 2 ¯

ψ′p¯ γ¯

qD′⋆ ¯ 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

2 ρ¯

ρ′ + i 1

2 γpψ¯ q ¯

ψ′

p¯

γ¯

q

≡ 0 .

JEONG-HYUCK PARK DOUBLED-YET-GAUGED, SEMI-COVARIANCE & TWOFOLD SPIN

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SLIDE 79

D = 10 Maximal SDFT [ 1210.5078 ]

Lagrangian: LType II = e−2d

1 8 (PABPCD − ¯

PAB ¯ PCD)SACBD + 1

2Tr(F ¯

F) − i ¯ ρFρ′ + i ¯ ψ¯

pγqF¯

γ¯

pψ′q

+i 1

2 ¯

ργpD⋆

p ρ − i ¯

ψ¯

pD⋆ ¯ p ρ − i 1 2 ¯

ψ¯

pγqD⋆ q ψ¯ p − i 1 2 ¯

ρ′¯ γ¯

pD′⋆ ¯ p ρ′ + i ¯

ψ′pD′⋆

p ρ′ + i 1 2 ¯

ψ′p¯ γ¯

qD′⋆ ¯ 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

2 ρ¯

ρ′ + i 1

2 γpψ¯ q ¯

ψ′

p¯

γ¯

q

≡ 0 .

JEONG-HYUCK PARK DOUBLED-YET-GAUGED, SEMI-COVARIANCE & TWOFOLD SPIN

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SLIDE 80

D = 10 Maximal SDFT [ 1210.5078 ]

Under the N = 2 SUSY transformation rule, the Lagrangian transforms as δεLType II = − 1

8 e−2d ¯

V A¯

qδεVApTr

  • γp ˜

F−¯ γ¯

q ˜

F−

  • + total derivatives ,

where the precise self-duality relation appears quadratically, ˜ F− :=

  • 1 − γ(D+1)

F − i 1

2ρ¯

ρ′ + i 1

2γpψ¯ q ¯

ψ′

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, δε ˜ F− = −i

  • ˜

D¯

pρ + γp ˜

Dpψ¯

p − γpF¯

γ¯

pψ′p

  • ¯

ε′¯ γ¯

p − iγpε

  • ˜

D′

p ¯

ρ′ + ˜ D′

¯ p ¯

ψ′

p¯

γ¯

p − ¯

ψ¯

pγpF¯

γ¯

p

.

JEONG-HYUCK PARK DOUBLED-YET-GAUGED, SEMI-COVARIANCE & TWOFOLD SPIN

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SLIDE 81

D = 10 Maximal SDFT [ 1210.5078 ]

Under the N = 2 SUSY transformation rule, the Lagrangian transforms as δεLType II = − 1

8 e−2d ¯

V A¯

qδεVApTr

  • γp ˜

F−¯ γ¯

q ˜

F−

  • + total derivatives ,

where the precise self-duality relation appears quadratically, ˜ F− :=

  • 1 − γ(D+1)

F − i 1

2ρ¯

ρ′ + i 1

2γpψ¯ q ¯

ψ′

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, δε ˜ F− = −i

  • ˜

D¯

pρ + γp ˜

Dpψ¯

p − γpF¯

γ¯

pψ′p

  • ¯

ε′¯ γ¯

p − iγpε

  • ˜

D′

p ¯

ρ′ + ˜ D′

¯ p ¯

ψ′

p¯

γ¯

p − ¯

ψ¯

pγpF¯

γ¯

p

.

JEONG-HYUCK PARK DOUBLED-YET-GAUGED, SEMI-COVARIANCE & TWOFOLD SPIN

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SLIDE 82

Equations of Motion

DFT-vielbein: Sp¯

q+Tr(γpF¯

γ¯

q ¯

F)+i ¯ ργp ˜ D¯

qρ+2i ¯

ψ¯

q ˜

Dpρ−i ¯ ψ¯

pγp ˜

D¯

qψ¯ p+i ¯

ρ′¯ γ¯

q ˜

Dpρ′+2i ¯ ψ′p ˜ D¯

qρ′−i ¯

ψ′q¯ γ¯

q ˜

Dpψ′

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 − iρ¯

ρ′ + iγrψ¯

s ¯

ψ′

r ¯

γ¯

s

= 0 , which is automatically met by the self-duality, together with the nilpotency of D0

+,

D0

−

  • F − iρ¯

ρ′ + iγrψ¯

s ¯

ψ′

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 DOUBLED-YET-GAUGED, SEMI-COVARIANCE & TWOFOLD SPIN

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SLIDE 83

Equations of Motion

DFT-vielbein: Sp¯

q+Tr(γpF¯

γ¯

q ¯

F)+i ¯ ργp ˜ D¯

qρ+2i ¯

ψ¯

q ˜

Dpρ−i ¯ ψ¯

pγp ˜

D¯

qψ¯ p+i ¯

ρ′¯ γ¯

q ˜

Dpρ′+2i ¯ ψ′p ˜ D¯

qρ′−i ¯

ψ′q¯ γ¯

q ˜

Dpψ′

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 − iρ¯

ρ′ + iγrψ¯

s ¯

ψ′

r ¯

γ¯

s

= 0 , which is automatically met by the self-duality, together with the nilpotency of D0

+,

D0

−

  • F − iρ¯

ρ′ + iγrψ¯

s ¯

ψ′

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 DOUBLED-YET-GAUGED, SEMI-COVARIANCE & TWOFOLD SPIN

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SLIDE 84

Equations of Motion

DFT-vielbein: Sp¯

q+Tr(γpF¯

γ¯

q ¯

F)+i ¯ ργp ˜ D¯

qρ+2i ¯

ψ¯

q ˜

Dpρ−i ¯ ψ¯

pγp ˜

D¯

qψ¯ p+i ¯

ρ′¯ γ¯

q ˜

Dpρ′+2i ¯ ψ′p ˜ D¯

qρ′−i ¯

ψ′q¯ γ¯

q ˜

Dpψ′

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 − iρ¯

ρ′ + iγrψ¯

s ¯

ψ′

r ¯

γ¯

s

= 0 , which is automatically met by the self-duality, together with the nilpotency of D0

+,

D0

−

  • F − iρ¯

ρ′ + iγrψ¯

s ¯

ψ′

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 DOUBLED-YET-GAUGED, SEMI-COVARIANCE & TWOFOLD SPIN

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SLIDE 85

Equations of Motion

DFT-vielbein: Sp¯

q+Tr(γpF¯

γ¯

q ¯

F)+i ¯ ργp ˜ D¯

qρ+2i ¯

ψ¯

q ˜

Dpρ−i ¯ ψ¯

pγp ˜

D¯

qψ¯ p+i ¯

ρ′¯ γ¯

q ˜

Dpρ′+2i ¯ ψ′p ˜ D¯

qρ′−i ¯

ψ′q¯ γ¯

q ˜

Dpψ′

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 − iρ¯

ρ′ + iγrψ¯

s ¯

ψ′

r ¯

γ¯

s

= 0 , which is automatically met by the self-duality, together with the nilpotency of D0

+,

D0

−

  • F − iρ¯

ρ′ + iγrψ¯

s ¯

ψ′

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 DOUBLED-YET-GAUGED, SEMI-COVARIANCE & TWOFOLD SPIN

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SLIDE 86

Equations of Motion

DFT-vielbein: Sp¯

q+Tr(γpF¯

γ¯

q ¯

F)+i ¯ ργp ˜ D¯

qρ+2i ¯

ψ¯

q ˜

Dpρ−i ¯ ψ¯

pγp ˜

D¯

qψ¯ p+i ¯

ρ′¯ γ¯

q ˜

Dpρ′+2i ¯ ψ′p ˜ D¯

qρ′−i ¯

ψ′q¯ γ¯

q ˜

Dpψ′

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 − iρ¯

ρ′ + iγrψ¯

s ¯

ψ′

r ¯

γ¯

s

= 0 , which is automatically met by the self-duality, together with the nilpotency of D0

+,

D0

−

  • F − iρ¯

ρ′ + iγrψ¯

s ¯

ψ′

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 DOUBLED-YET-GAUGED, SEMI-COVARIANCE & TWOFOLD SPIN

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SLIDE 87

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

1 8

  • PABPCD − ¯

PAB ¯ PCD SACBD + i 1

2 ¯

ργAD⋆

Aρ − i ¯

ψAD⋆

Aρ − i 1 2 ¯

ψBγAD⋆

AψB

  • .

N = 1 Local SUSY: δεd = −i 1

2 ¯

ερ , δεVAp = −i ¯ εγpψA , δε ¯ VA¯

p

= i ¯ εγAψ¯

p ,

δερ = −γA ˆ DAε , δεψ¯

p

= ¯ V A¯

p ˆ

DAε − i 1

4 (¯

ρψ¯

p)ε + i 1 2 (¯

ερ)ψ¯

p .

JEONG-HYUCK PARK DOUBLED-YET-GAUGED, SEMI-COVARIANCE & TWOFOLD SPIN

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SLIDE 88

N = 1 SUSY Algebra [1112.0069]

Commutator of supersymmetry reads [δε1, δε2] ≡ ˆ LX3 + δε3 + δso(1,9)L + δso(9,1)R + δtrivial . where X A

3 = i ¯

ε1γAε2 , ε3 = i 1

2 [(¯

ε1γpε2)γpρ + (¯ ρε2)ε1 − (¯ ρε1)ε2] , etc. and δtrivial corresponds to the fermionic equations of motion.

JEONG-HYUCK PARK DOUBLED-YET-GAUGED, SEMI-COVARIANCE & TWOFOLD SPIN

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SLIDE 89

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 DOUBLED-YET-GAUGED, SEMI-COVARIANCE & TWOFOLD SPIN

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SLIDE 90

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     , ¯ VA¯

p = 1 √ 2

    (¯ e−1)¯

pµ

(B + ¯ e)ν¯

p

    . Here eµp and ¯ eν ¯

p are two copies of the D-dimensional vielbein corresponding to the

same spacetime metric, eµpeν qηpq = −¯ eµ

¯ p¯

eν

¯ q ¯

η¯

p¯ q = gµν ,

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

p = Bµν(¯

e−1)¯

pν.

JEONG-HYUCK PARK DOUBLED-YET-GAUGED, SEMI-COVARIANCE & TWOFOLD SPIN

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SLIDE 91

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     , ¯ VA¯

p = 1 √ 2

    (¯ e−1)¯

pµ

(B + ¯ e)ν¯

p

    . Here eµp and ¯ eν ¯

p are two copies of the D-dimensional vielbein corresponding to the

same spacetime metric, eµpeν qηpq = −¯ eµ

¯ p¯

eν

¯ q ¯

η¯

p¯ q = gµν ,

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

p = Bµν(¯

e−1)¯

pν.

JEONG-HYUCK PARK DOUBLED-YET-GAUGED, SEMI-COVARIANCE & TWOFOLD SPIN

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SLIDE 92

Parametrization: Reduction to Generalized Geometry

Instead, we may choose an alternative parametrization, VAp =

1 √ 2

    (β + ˜ e)µp (˜ e−1)pν     , ¯ VA

¯ p = 1 √ 2

    (β + ¯ ˜ e)µp (¯ ˜ e−1)pν     , where βµp = βµν(˜ e−1)pν, βµ¯

p = βµν(¯

˜ e−1)pν, and ˜ eµp, ¯ ˜ eµ¯

p correspond to

a pair of T-dual vielbeins for winding modes, ˜ eµp˜ eν qηpq = −¯ ˜ eµ¯

p¯

˜ eν ¯

qη¯ p¯ q = (g − Bg−1B)−1 µν .

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 DOUBLED-YET-GAUGED, SEMI-COVARIANCE & TWOFOLD SPIN

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SLIDE 93

Parametrization: Reduction to Generalized Geometry

Two parametrizations: VAp =

1 √ 2

    (e−1)pµ (B + e)νp     , ¯ VA¯

p = 1 √ 2

    (¯ e−1)¯

pµ

(B + ¯ e)ν¯

p

    versus VAp =

1 √ 2

    (β + ˜ e)µp (˜ e−1)pν     , ¯ VA

¯ p = 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 DOUBLED-YET-GAUGED, SEMI-COVARIANCE & TWOFOLD SPIN

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SLIDE 94

Parametrization: Reduction to Generalized Geometry

Two parametrizations: VAp =

1 √ 2

    (e−1)pµ (B + e)νp     , ¯ VA¯

p = 1 √ 2

    (¯ e−1)¯

pµ

(B + ¯ e)ν¯

p

    versus VAp =

1 √ 2

    (β + ˜ e)µp (˜ e−1)pν     , ¯ VA

¯ p = 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 DOUBLED-YET-GAUGED, SEMI-COVARIANCE & TWOFOLD SPIN

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SLIDE 95

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 DOUBLED-YET-GAUGED, SEMI-COVARIANCE & TWOFOLD SPIN

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SLIDE 96

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

4 ωmnpγnpρ + 1 24Hmnpγnpρ − ∂mφρ

  • ,

√ 2γADAψ¯

p ≡ γm

∂mψ¯

p + 1 4 ωmnpγnpψ¯ p + ¯

ωm¯

p¯ qψ¯ q + 1 24Hmnpγnpψ¯ p + 1 2 Hm¯ p¯ qψ¯ q − ∂mφψ¯ p

  • ,

√ 2¯ V A¯

pDAρ ≡ ∂¯ pρ + 1 4 ω¯ pqrγqrρ + 1 8 H¯ pqrγqrρ ,

√ 2DAψA ≡ ∂¯

pψ¯ p + 1 4 ω¯ pqrγqrψ¯ p + ¯

ω¯

p¯ p¯ qψ¯ q + 1 8H¯ pqrγqrψ¯ p − 2∂¯ pφψ¯ p .

ωµ ± 1

2 Hµ and ωµ ± 1 6 Hµ naturally appear as spin connections. Liu, Minasian

JEONG-HYUCK PARK DOUBLED-YET-GAUGED, SEMI-COVARIANCE & TWOFOLD SPIN

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SLIDE 97

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

4 ωmnpγnpρ + 1 24Hmnpγnpρ − ∂mφρ

  • ,

√ 2γADAψ¯

p ≡ γm

∂mψ¯

p + 1 4 ωmnpγnpψ¯ p + ¯

ωm¯

p¯ qψ¯ q + 1 24Hmnpγnpψ¯ p + 1 2 Hm¯ p¯ qψ¯ q − ∂mφψ¯ p

  • ,

√ 2¯ V A¯

pDAρ ≡ ∂¯ pρ + 1 4 ω¯ pqrγqrρ + 1 8 H¯ pqrγqrρ ,

√ 2DAψA ≡ ∂¯

pψ¯ p + 1 4 ω¯ pqrγqrψ¯ p + ¯

ω¯

p¯ p¯ qψ¯ q + 1 8H¯ pqrγqrψ¯ p − 2∂¯ pφψ¯ p .

ωµ ± 1

2 Hµ and ωµ ± 1 6 Hµ naturally appear as spin connections. Liu, Minasian

JEONG-HYUCK PARK DOUBLED-YET-GAUGED, SEMI-COVARIANCE & TWOFOLD SPIN

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SLIDE 98

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−1¯ e)p

¯ p(e−1¯

e)q

¯ q ¯

η¯

p¯ q = −ηpq .

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

pS−1 e

= γ(D+1)γp(e−1¯ e)p

¯ p ,

such that, in particular, Se¯ γ(D+1)S−1

e

= − det(e−1¯ e)γ(D+1) .

JEONG-HYUCK PARK DOUBLED-YET-GAUGED, SEMI-COVARIANCE & TWOFOLD SPIN

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SLIDE 99

Unification of type IIA and IIB SUGRAs

The D = 10 maximal SDFT ‘Riemannian’ solutions are then classified into two groups, det(e−1¯ e) = +1 : type IIA , det(e−1¯ e) = −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−1¯ e) . Namely, the Pin(D−1, 1)R chirality changes iff det(e−1¯ e) = −1.

JEONG-HYUCK PARK DOUBLED-YET-GAUGED, SEMI-COVARIANCE & TWOFOLD SPIN

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SLIDE 100

Unification of type IIA and IIB SUGRAs

The D = 10 maximal SDFT ‘Riemannian’ solutions are then classified into two groups, det(e−1¯ e) = +1 : type IIA , det(e−1¯ e) = −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−1¯ e) . Namely, the Pin(D−1, 1)R chirality changes iff det(e−1¯ e) = −1.

JEONG-HYUCK PARK DOUBLED-YET-GAUGED, SEMI-COVARIANCE & TWOFOLD SPIN

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SLIDE 101

Unification of type IIA and IIB SUGRAs

The D = 10 maximal SDFT ‘Riemannian’ solutions are then classified into two groups, det(e−1¯ e) = +1 : type IIA , det(e−1¯ e) = −1 : type IIB . That is to say, formulated in terms of the genuine DFT-field variables, i.e. VAp, ¯ VA¯

p,

Cα ¯

α, 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 DOUBLED-YET-GAUGED, SEMI-COVARIANCE & TWOFOLD SPIN

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SLIDE 102

Unification of type IIA and IIB SUGRAs

The D = 10 maximal SDFT ‘Riemannian’ solutions are then classified into two groups, det(e−1¯ e) = +1 : type IIA , det(e−1¯ e) = −1 : type IIB . That is to say, formulated in terms of the genuine DFT-field variables, i.e. VAp, ¯ VA¯

p,

Cα ¯

α, 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 DOUBLED-YET-GAUGED, SEMI-COVARIANCE & TWOFOLD SPIN

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SLIDE 103

Diagonal gauge fixing and Reduction to SUGRA

Setting the diagonal gauge, eµp ≡ ¯ eµ

¯ p

with ηpq = −¯ η¯

p¯ q, ¯

γ¯

p = 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 DOUBLED-YET-GAUGED, SEMI-COVARIANCE & TWOFOLD SPIN

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SLIDE 104

Diagonal gauge fixing and Reduction to SUGRA

Setting the diagonal gauge, eµp ≡ ¯ eµ

¯ p

with ηpq = −¯ η¯

p¯ q, ¯

γ¯

p = 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 DOUBLED-YET-GAUGED, SEMI-COVARIANCE & TWOFOLD SPIN

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SLIDE 105

Diagonal gauge fixing and Reduction to SUGRA

After the diagonal gauge fixing, we may parameterize the R-R potential as C ≡

  • 1

2

D+2

4

′

p 1 p! Ca1a2···apγa1a2···ap

and obtain the field strength, F := D0

+C ≡

  • 1

2

D

4 ′

p 1 (p+1)! Fa1a2···ap+1γa1a2···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] − ∂[a1φ Ca2···ap]
  • +

p! 3!(p−3)! H[a1a2a3Ca4···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 DOUBLED-YET-GAUGED, SEMI-COVARIANCE & TWOFOLD SPIN

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SLIDE 106

Diagonal gauge fixing and Reduction to SUGRA

After the diagonal gauge fixing, we may parameterize the R-R potential as C ≡

  • 1

2

D+2

4

′

p 1 p! Ca1a2···apγa1a2···ap

and obtain the field strength, F := D0

+C ≡

  • 1

2

D

4 ′

p 1 (p+1)! Fa1a2···ap+1γa1a2···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] − ∂[a1φ Ca2···ap]
  • +

p! 3!(p−3)! H[a1a2a3Ca4···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 DOUBLED-YET-GAUGED, SEMI-COVARIANCE & TWOFOLD SPIN

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SLIDE 107

Diagonal gauge fixing and Reduction to SUGRA

In this way,

  • rdinary SUGRA ≡ gauge-fixed SDFT,

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

JEONG-HYUCK PARK DOUBLED-YET-GAUGED, SEMI-COVARIANCE & TWOFOLD SPIN

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SLIDE 108

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 DOUBLED-YET-GAUGED, SEMI-COVARIANCE & TWOFOLD SPIN

slide-109
SLIDE 109

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 DOUBLED-YET-GAUGED, SEMI-COVARIANCE & TWOFOLD SPIN

slide-110
SLIDE 110

Modifying O(D, D) transformation rule

The O(D, D) rotation must accompany a compensating Pin(D−1, 1)R local Lorentz rotation, ¯ L¯

q ¯ p, S¯ L ¯ α ¯ β

which we can construct explicitly, ¯ L = ¯ e−1 at − (g + B)bt at + (g − B)bt−1 ¯ e , ¯ γ¯

q¯

L¯

q ¯ p = S−1 ¯ L

¯ γ¯

pS¯ L ,

where a and b are parameters of a given O(D, D) group element, MA

B =

    aµν bµσ cρν dρσ     .

JEONG-HYUCK PARK DOUBLED-YET-GAUGED, SEMI-COVARIANCE & TWOFOLD SPIN

slide-111
SLIDE 111

Modified O(D, D) Transformation Rule After The Diagonal Gauge Fixing d − → d VAp − → MAB VBp ¯ VA

¯ p

− → MAB ¯ VB

¯ q ¯

L¯

q ¯ p

Cα ¯

α ,

Fα ¯

α

− → Cα ¯

β(S−1 ¯ L

) ¯

β ¯ α ,

Fα ¯

β(S−1 ¯ L

) ¯

β ¯ α

ρα − → ρα ρ′ ¯

α

− → (S¯

L) ¯ α ¯ βρ′ ¯ β

ψα

¯ p

− → (¯ L−1)¯

p ¯ q ψα ¯ q

ψ′ ¯

α p

− → (S¯

L) ¯ α ¯ βψ′ ¯ β p

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 DOUBLED-YET-GAUGED, SEMI-COVARIANCE & TWOFOLD SPIN

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SLIDE 112

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)S¯

L = det(¯

L) S¯

L¯

γ(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 ¯ VA¯

p 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 DOUBLED-YET-GAUGED, SEMI-COVARIANCE & TWOFOLD SPIN

slide-113
SLIDE 113

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)S¯

L = det(¯

L) S¯

L¯

γ(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 ¯ VA¯

p 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 DOUBLED-YET-GAUGED, SEMI-COVARIANCE & TWOFOLD SPIN

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SLIDE 114

Twofold spin and Standard Model 1506.05277

JEONG-HYUCK PARK DOUBLED-YET-GAUGED, SEMI-COVARIANCE & TWOFOLD SPIN

slide-115
SLIDE 115

Standard Model as Double Field Theory 1506.05277

In principle, fermions live on a locally inertial frame. 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 completely covariant DFT-geometry described above, we can couple Standard Model to stringy backgrounds in a covariant way: It is possible to Double Field Theorize the Standard Model, without introducing any extra physical degrees. Doing so, one has to decide the spin group for each fermion (Yukawa coupling). No experimental evidence of proton decay seems to indicate that the quarks and the leptons may belong to different spin groups. If so, this constrains the possible higher order corrections to SM. Further, DFT forbids the theta term and hence solves the strong CP problem.

JEONG-HYUCK PARK DOUBLED-YET-GAUGED, SEMI-COVARIANCE & TWOFOLD SPIN

slide-116
SLIDE 116

Standard Model as Double Field Theory 1506.05277

In principle, fermions live on a locally inertial frame. 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 completely covariant DFT-geometry described above, we can couple Standard Model to stringy backgrounds in a covariant way: It is possible to Double Field Theorize the Standard Model, without introducing any extra physical degrees. Doing so, one has to decide the spin group for each fermion (Yukawa coupling). No experimental evidence of proton decay seems to indicate that the quarks and the leptons may belong to different spin groups. If so, this constrains the possible higher order corrections to SM. Further, DFT forbids the theta term and hence solves the strong CP problem.

JEONG-HYUCK PARK DOUBLED-YET-GAUGED, SEMI-COVARIANCE & TWOFOLD SPIN

slide-117
SLIDE 117

Standard Model as Double Field Theory 1506.05277

In principle, fermions live on a locally inertial frame. 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 completely covariant DFT-geometry described above, we can couple Standard Model to stringy backgrounds in a covariant way: It is possible to Double Field Theorize the Standard Model, without introducing any extra physical degrees. Doing so, one has to decide the spin group for each fermion (Yukawa coupling). No experimental evidence of proton decay seems to indicate that the quarks and the leptons may belong to different spin groups. If so, this constrains the possible higher order corrections to SM. Further, DFT forbids the theta term and hence solves the strong CP problem.

JEONG-HYUCK PARK DOUBLED-YET-GAUGED, SEMI-COVARIANCE & TWOFOLD SPIN

slide-118
SLIDE 118

Standard Model as Double Field Theory 1506.05277

In principle, fermions live on a locally inertial frame. 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 completely covariant DFT-geometry described above, we can couple Standard Model to stringy backgrounds in a covariant way: It is possible to Double Field Theorize the Standard Model, without introducing any extra physical degrees. Doing so, one has to decide the spin group for each fermion (Yukawa coupling). No experimental evidence of proton decay seems to indicate that the quarks and the leptons may belong to different spin groups. If so, this constrains the possible higher order corrections to SM. Further, DFT forbids the theta term and hence solves the strong CP problem.

JEONG-HYUCK PARK DOUBLED-YET-GAUGED, SEMI-COVARIANCE & TWOFOLD SPIN

slide-119
SLIDE 119

Outlook : things to do

JEONG-HYUCK PARK DOUBLED-YET-GAUGED, SEMI-COVARIANCE & TWOFOLD SPIN

slide-120
SLIDE 120

Outlook : things to do

Revist and Double Field Theorize 20th centrury physics covariantly, including string theory itself.

JEONG-HYUCK PARK DOUBLED-YET-GAUGED, SEMI-COVARIANCE & TWOFOLD SPIN