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

Doubled-yet-gauged coordinates & Gauged infinitesimal one-form The usual infinitesimal one-form, d x M , 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, d x M d x N H MN can NOT give a ‘proper length’ in DFT. Further, it is NOT coordinate gauge symmetry invariant, d ( x M + Φ 1 ∂ M Φ 2 ) � = d x M . d x M − → These can be all cured by introducing a gauged infinitesimal one-form , Dx M := d x M − A M , where A M is the ‘coordinate gauge potential’. Being a derivative-index-valued vector, it satisfies A M ∂ M = 0 , A M A M = 0 , or suggestively the ‘gauged section condition’, ( ∂ M + A M )( ∂ M + A M ) = 0 . J EONG -H YUCK P ARK D OUBLED - YET -G AUGED , S EMI -C OVARIANCE & T WOFOLD S PIN

Doubled-yet-gauged coordinates & Gauged infinitesimal one-form The usual infinitesimal one-form, d x M , 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, d x M d x N H MN can NOT give a ‘proper length’ in DFT. Further, it is NOT coordinate gauge symmetry invariant, d ( x M + Φ 1 ∂ M Φ 2 ) � = d x M . d x M − → These can be all cured by introducing a gauged infinitesimal one-form , Dx M := d x M − A M , where A M is the ‘coordinate gauge potential’. Being a derivative-index-valued vector, it satisfies A M ∂ M = 0 , A M A M = 0 , or suggestively the ‘gauged section condition’, ( ∂ M + A M )( ∂ M + A M ) = 0 . J EONG -H YUCK P ARK D OUBLED - YET -G AUGED , S EMI -C OVARIANCE & T WOFOLD S PIN

Doubled-yet-gauged coordinates & Gauged infinitesimal one-form Under coordinate gauge symmetry, we have the invariance of Dx M , x ′ M = x M + Φ 1 ∂ M Φ 2 , x M − → A ′ M = A M + d (Φ 1 ∂ M Φ 2 ) A M A ′ M ∂ ′ − → : M ≡ 0 , D ′ x ′ M = Dx M = d x M − A M . Dx M − → Similarly, under (finite) DFT diffeomorphisms à la Hohm-Zwiebach L M N := ∂ M x ′ N , L := J L t J − 1 , ¯ L ¯ L − 1 + ¯ F := J F t J − 1 = 1 ¯ L − 1 ¯ L + ¯ = F − 1 , F := 1 L − 1 L LL − 1 � � � � , 2 2 we have the covariance, x ′ M ( x ) , x M − → MN ( x ′ ) = ¯ F M K ¯ H ′ F N L H KL ( x ) , H MN ( x ) − → A ′ M = A N F N M + d X N ( L − F ) N M A ′ M ∂ ′ A M − → : M ≡ 0 , D ′ x ′ M = Dx N F N M . Dx M − → J EONG -H YUCK P ARK D OUBLED - YET -G AUGED , S EMI -C OVARIANCE & T WOFOLD S PIN

Doubled-yet-gauged coordinates & Gauged infinitesimal one-form Under coordinate gauge symmetry, we have the invariance of Dx M , x ′ M = x M + Φ 1 ∂ M Φ 2 , x M − → A ′ M = A M + d (Φ 1 ∂ M Φ 2 ) A M A ′ M ∂ ′ − → : M ≡ 0 , D ′ x ′ M = Dx M = d x M − A M . Dx M − → Similarly, under (finite) DFT diffeomorphisms à la Hohm-Zwiebach L M N := ∂ M x ′ N , L := J L t J − 1 , ¯ L ¯ L − 1 + ¯ F := J F t J − 1 = 1 ¯ L − 1 ¯ L + ¯ = F − 1 , F := 1 L − 1 L LL − 1 � � � � , 2 2 we have the covariance, x ′ M ( x ) , x M − → MN ( x ′ ) = ¯ F M K ¯ H ′ F N L H KL ( x ) , H MN ( x ) − → A ′ M = A N F N M + d X N ( L − F ) N M A ′ M ∂ ′ A M − → : M ≡ 0 , D ′ x ′ M = Dx N F N M . Dx M − → J EONG -H YUCK P ARK D OUBLED - YET -G AUGED , S EMI -C OVARIANCE & T WOFOLD S PIN

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, � ∂ � � � ∂ ∂ ∂ ∂ x M = , ≡ 0 , ‘conventional’ choice of the section ∂ ˜ ∂ x ν ∂ x ν x µ Then, the ‘coordinate gauge symmetry’ reads � ˜ � ˜ x µ , x ν � x µ + Φ 1 ∂ µ Φ 2 , x ν � ∼ . The coordinate gauge potential and the gauged infinitesimal one-form become A M = A λ ∂ M x λ = Dx M = � � � d ˜ x µ − A µ , d x ν � A µ , 0 , . J EONG -H YUCK P ARK D OUBLED - YET -G AUGED , S EMI -C OVARIANCE & T WOFOLD S PIN

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, � ∂ � � � ∂ ∂ ∂ ∂ x M = , ≡ 0 , ‘conventional’ choice of the section ∂ ˜ ∂ x ν ∂ x ν x µ Then, the ‘coordinate gauge symmetry’ reads � ˜ � ˜ x µ , x ν � x µ + Φ 1 ∂ µ Φ 2 , x ν � ∼ . The coordinate gauge potential and the gauged infinitesimal one-form become A M = A λ ∂ M x λ = Dx M = � � � d ˜ x µ − A µ , d x ν � A µ , 0 , . J EONG -H YUCK P ARK D OUBLED - YET -G AUGED , S EMI -C OVARIANCE & T WOFOLD S PIN

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, x I = (˜ x j , x k ) , 2 m D t x I D t x J δ IJ − V ( x ) , L Newton = 1 where I , J = 1 , 2 , · · · , 6 and the potential, V ( x ) , satisfies the section condition. With the conventional choice of the section, we get � ˙ � � ˙ x j ˙ x k δ jk − V ( x ) + 1 � δ jk . L Newton = 1 2 m ˙ ˜ ˜ 2 m x j − A j x k − A k Hence, after integrating out A j , we recover the conventional formulation. J EONG -H YUCK P ARK D OUBLED - YET -G AUGED , S EMI -C OVARIANCE & T WOFOLD S PIN

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, x I = (˜ x j , x k ) , 2 m D t x I D t x J δ IJ − V ( x ) , L Newton = 1 where I , J = 1 , 2 , · · · , 6 and the potential, V ( x ) , satisfies the section condition. With the conventional choice of the section, we get � ˙ � � ˙ x j ˙ x k δ jk − V ( x ) + 1 � δ jk . L Newton = 1 2 m ˙ ˜ ˜ 2 m x j − A j x k − A k Hence, after integrating out A j , we recover the conventional formulation. J EONG -H YUCK P ARK D OUBLED - YET -G AUGED , S EMI -C OVARIANCE & T WOFOLD S PIN

String probes the doubled-yet-gauged spacetime DFT string action is with D i X M = ∂ i X M − A M i , JHP-Lee 2013 √ � 1 d 2 σ L string , L string = − 1 − h h ij D i X M D j X N H MN ( X ) − ǫ ij D i X M A jM , 4 πα ′ 2 The action is fully symmetric , essentially due to the auxiliary gauge field, A M 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 J EONG -H YUCK P ARK D OUBLED - YET -G AUGED , S EMI -C OVARIANCE & T WOFOLD S PIN

String probes the doubled-yet-gauged spacetime DFT string action is with D i X M = ∂ i X M − A M i , JHP-Lee 2013 √ � 1 d 2 σ L string , L string = − 1 − h h ij D i X M D j X N H MN ( X ) − ǫ ij D i X M A jM , 4 πα ′ 2 H AB ( x ) is the “generalized metric” which can be defined as a symmetric O ( D , D ) element, H AC H BD J CD = J AB , H AB = H BA , and satisfies the section condition. There are two types of “generalized metric” : Riemannian vs. non-Riemannian . J EONG -H YUCK P ARK D OUBLED - YET -G AUGED , S EMI -C OVARIANCE & T WOFOLD S PIN

String probes the doubled-yet-gauged spacetime DFT string action is with D i X M = ∂ i X M − A M i , JHP-Lee 2013 √ � 1 d 2 σ L string , L string = − 1 − h h ij D i X M D j X N H MN ( X ) − ǫ ij D i X M A jM , 4 πα ′ 2 H AB ( x ) is the “generalized metric” which can be defined as a symmetric O ( D , D ) element, H AC H BD J CD = J AB , H AB = H BA , and satisfies the section condition. There are two types of “generalized metric” : Riemannian vs. non-Riemannian . J EONG -H YUCK P ARK D OUBLED - YET -G AUGED , S EMI -C OVARIANCE & T WOFOLD S PIN

String probes the doubled-yet-gauged spacetime DFT string action is with D i X M = ∂ i X M − A M i , JHP-Lee 2013 √ � 1 d 2 σ L string , L string = − 1 − h h ij D i X M D j X N H MN ( X ) − ǫ ij D i X M A jM , 4 πα ′ 2 H AB ( x ) is the “generalized metric” which can be defined as a symmetric O ( D , D ) element, H AC H BD J CD = J AB , H AB = H BA , and satisfies the section condition. There are two types of “generalized metric” : Riemannian vs. non-Riemannian . J EONG -H YUCK P ARK D OUBLED - YET -G AUGED , S EMI -C OVARIANCE & T WOFOLD S PIN

DFT backgrounds : Riemannian vs. non-Riemannian ∂ x µ ≡ 0 , Riemannian generalized metric W.r.t. the conventional choice of the section, ∂ ˜ assumes the well-known form,   G − 1 − G − 1 B   H AB =  .    BG − 1 G − BG − 1 B 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, √ � X µ ∂ j X µ � 1 1 − 1 − hh ij ∂ i X µ ∂ j X ν G µν ( X ) + 1 2 ǫ ij ∂ i X µ ∂ j X ν B µν ( X ) + 1 2 ǫ ij ∂ i ˜ 4 πα ′ L string ≡ , 2 πα ′ 2 with the bonus of the topological term introduced by Giveon-Rocek; Hull. The EOM of A M implies self-duality on the full doubled spacetime, i N D i X N + √− h ǫ ij D j X M = 0 . H M 1 J EONG -H YUCK P ARK D OUBLED - YET -G AUGED , S EMI -C OVARIANCE & T WOFOLD S PIN

DFT backgrounds : Riemannian vs. non-Riemannian ∂ x µ ≡ 0 , Riemannian generalized metric W.r.t. the conventional choice of the section, ∂ ˜ assumes the well-known form,   G − 1 − G − 1 B   H AB =  .    BG − 1 G − BG − 1 B 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, √ � X µ ∂ j X µ � 1 1 − 1 − hh ij ∂ i X µ ∂ j X ν G µν ( X ) + 1 2 ǫ ij ∂ i X µ ∂ j X ν B µν ( X ) + 1 2 ǫ ij ∂ i ˜ 4 πα ′ L string ≡ , 2 πα ′ 2 with the bonus of the topological term introduced by Giveon-Rocek; Hull. The EOM of A M implies self-duality on the full doubled spacetime, i N D i X N + √− h ǫ ij D j X M = 0 . H M 1 J EONG -H YUCK P ARK D OUBLED - YET -G AUGED , S EMI -C OVARIANCE & T WOFOLD S PIN

DFT backgrounds : Riemannian vs. non-Riemannian ∂ x µ ≡ 0 , Riemannian generalized metric W.r.t. the conventional choice of the section, ∂ ˜ assumes the well-known form,   G − 1 − G − 1 B   H AB =  .    BG − 1 G − BG − 1 B 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, √ � X µ ∂ j X µ � 1 1 − 1 − hh ij ∂ i X µ ∂ j X ν G µν ( X ) + 1 2 ǫ ij ∂ i X µ ∂ j X ν B µν ( X ) + 1 2 ǫ ij ∂ i ˜ 4 πα ′ L string ≡ , 2 πα ′ 2 with the bonus of the topological term introduced by Giveon-Rocek; Hull. The EOM of A M implies self-duality on the full doubled spacetime, i N D i X N + √− h ǫ ij D j X M = 0 . H M 1 J EONG -H YUCK P ARK D OUBLED - YET -G AUGED , S EMI -C OVARIANCE & T WOFOLD S PIN

DFT backgrounds : Riemannian vs. non-Riemannian ∂ W.r.t. x µ ≡ 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 ,   e αβ 0 0 0       δ ij 0 0 0   r 2 = � 9 i = 2 ( x i ) 2 . f = 1 + Q   H MN = , r 6 ,     − e αβ 0 f η αβ 0         0 0 0 δ ij This is “doublly T-dual”, ( t , x 1 ) ⇔ (˜ t , ˜ x 1 ) , DFT background to F1 à la Dabholkar-Gibbons-Harvey-Ruiz 1990, c.f. 2 D 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 J EONG -H YUCK P ARK D OUBLED - YET -G AUGED , S EMI -C OVARIANCE & T WOFOLD S PIN

DFT backgrounds : Riemannian vs. non-Riemannian ∂ W.r.t. x µ ≡ 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 ,   e αβ 0 0 0       δ ij 0 0 0   r 2 = � 9 i = 2 ( x i ) 2 . f = 1 + Q   H MN = , r 6 ,     − e αβ 0 f η αβ 0         0 0 0 δ ij This is “doublly T-dual”, ( t , x 1 ) ⇔ (˜ t , ˜ x 1 ) , DFT background to F1 à la Dabholkar-Gibbons-Harvey-Ruiz 1990, c.f. 2 D 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 J EONG -H YUCK P ARK D OUBLED - YET -G AUGED , S EMI -C OVARIANCE & T WOFOLD S PIN

Semi-covariant formulation of DFT/SDFT: Gravity on doubled-yet-gauged spacetime 1011.1324/1105.6294/ · · · J EONG -H YUCK P ARK D OUBLED - YET -G AUGED , S EMI -C OVARIANCE & T WOFOLD S PIN

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: L Type II = e − 2 d � 8 ( P AB P CD − ¯ P AB ¯ ρ F ρ ′ + i ¯ 2 Tr ( F ¯ γ ¯ 1 P CD ) S ACBD + 1 p ψ ′ q F ) − i ¯ p γ q F ¯ ψ ¯ � p ρ ′ + i ¯ p ρ ′ + i 1 + i 1 ργ p D ⋆ p ρ − i ¯ ψ ¯ p D ⋆ p ρ − i 1 2 ¯ ψ ¯ p γ q D ⋆ p − i 1 ρ ′ ¯ γ ¯ p D ′ ⋆ ψ ′ p D ′ ⋆ 2 ¯ ψ ′ p ¯ γ ¯ q D ′ ⋆ q ψ ′ p 2 ¯ q ψ ¯ 2 ¯ ¯ ¯ ¯ Jeon-Lee-JHP-Suh 2012 It also works for SL ( N ) duality group, N � = 4 JHP-Suh ‘U-gravity’ 2014 J EONG -H YUCK P ARK D OUBLED - YET -G AUGED , S EMI -C OVARIANCE & T WOFOLD S PIN

Notation 2/2 Index Representation Metric (raising/lowering indices) A , B , · · · O ( D , D ) & DFT-diffeom. vector J AB p , q , · · · Spin ( 1 , D − 1 ) L vector η pq = diag ( − + + · · · +) ( γ p ) T = C + γ p C − 1 α, β, · · · Spin ( 1 , D − 1 ) L spinor C + αβ , + ¯ p , ¯ q , · · · Spin ( D − 1 , 1 ) R vector η ¯ ¯ q = diag (+ − − · · · − ) p ¯ p ) T = ¯ p ¯ α, ¯ ¯ γ ¯ γ ¯ C − 1 ¯ β, · · · Spin ( D − 1 , 1 ) R spinor C + ¯ (¯ C + ¯ β , α ¯ + J EONG -H YUCK P ARK D OUBLED - YET -G AUGED , S EMI -C OVARIANCE & T WOFOLD S PIN

Field contents of D = 10 Maximal SDFT Bosons   DFT-dilaton: d   NS-NS sector ¯  DFT-vielbeins: V Ap , V A ¯   p C α ¯ R-R potential: α Fermions (Majorana-Weyl) ρ α , ρ ′ ¯ α DFT-dilatinos: ψ α ψ ′ ¯ α Gravitinos: p , ¯ p R-R potential and Fermions carry NOT ( D + D ) -dimensional BUT undoubled D -dimensional indices. J EONG -H YUCK P ARK D OUBLED - YET -G AUGED , S EMI -C OVARIANCE & T WOFOLD S PIN

Field contents of D = 10 Maximal SDFT Bosons   DFT-dilaton: d   NS-NS sector ¯  DFT-vielbeins: V Ap , V A ¯   p C α ¯ R-R potential: α Fermions (Majorana-Weyl) ρ α , ρ ′ ¯ α DFT-dilatinos: ψ α ψ ′ ¯ α Gravitinos: p , ¯ 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 of the twofold local Lorentz symmetries. J EONG -H YUCK P ARK D OUBLED - YET -G AUGED , S EMI -C OVARIANCE & T WOFOLD S PIN

The DFT-dilaton gives rise to a scalar density with weight one, e − 2 d . The DFT-vielbeins satisfy four ‘defining’ properties: ¯ p ¯ V Ap ¯ V Ap V Bp + ¯ p ¯ p = J AB . V Ap V Aq = η pq , V A ¯ V A ¯ ¯ q = ¯ q = 0 , V A ¯ η ¯ q , V A ¯ V B p ¯ Naturally, they generate a pair of two-index ‘projectors’, P AB P BC = P AC , p ¯ P AB ¯ P BC = ¯ P AC , P AB := V Ap V Bp , P AB := ¯ ¯ ¯ ¯ V A V B ¯ p , which are symmetric, orthogonal and complementary to each other, P AB ¯ P BC = 0 , P AB + ¯ P AB = δ AB . ¯ P AB = ¯ P AB = P BA , P BA , Some further projection properties follow P AB ¯ ¯ p = ¯ ¯ P AB ¯ P AB V Bp = V Ap , P AB V Bp = 0 , V B ¯ V A ¯ p , V B ¯ p = 0 . Note also H AB = P AB − ¯ P AB . However, our emphasis lies on the ‘projectors’ rather than the “generalized metric". J EONG -H YUCK P ARK D OUBLED - YET -G AUGED , S EMI -C OVARIANCE & T WOFOLD S PIN

The DFT-dilaton gives rise to a scalar density with weight one, e − 2 d . The DFT-vielbeins satisfy four ‘defining’ properties: ¯ p ¯ V Ap ¯ V Ap V Bp + ¯ p ¯ p = J AB . V Ap V Aq = η pq , V A ¯ V A ¯ ¯ q = ¯ q = 0 , V A ¯ η ¯ q , V A ¯ V B p ¯ Naturally, they generate a pair of two-index ‘projectors’, P AB P BC = P AC , p ¯ P AB ¯ P BC = ¯ P AC , P AB := V Ap V Bp , P AB := ¯ ¯ ¯ ¯ V A V B ¯ p , which are symmetric, orthogonal and complementary to each other, P AB ¯ P BC = 0 , P AB + ¯ P AB = δ AB . ¯ P AB = ¯ P AB = P BA , P BA , Some further projection properties follow P AB ¯ ¯ p = ¯ ¯ P AB ¯ P AB V Bp = V Ap , P AB V Bp = 0 , V B ¯ V A ¯ p , V B ¯ p = 0 . Note also H AB = P AB − ¯ P AB . However, our emphasis lies on the ‘projectors’ rather than the “generalized metric". J EONG -H YUCK P ARK D OUBLED - YET -G AUGED , S EMI -C OVARIANCE & T WOFOLD S PIN

We continue to define a pair of six-index projectors, P CABDEF := P CD P [ A [ E P B ] F ] + D − 1 P C [ A P B ][ E P F ] D , P CABDEF P DEF GHI = P CABGHI , 2 P CABDEF := ¯ P CD ¯ P [ A [ E ¯ P B ] F ] + D − 1 ¯ P C [ A ¯ P B ][ E ¯ P F ] D , P CABDEF ¯ P DEF GHI = ¯ P CABGHI , ¯ 2 ¯ which are symmetric and traceless, P CABDEF = ¯ ¯ P DEFCAB = ¯ P CABDEF = P DEFCAB = P C [ AB ] D [ EF ] , P C [ AB ] D [ EF ] , P AB ¯ ¯ ¯ P AABDEF = 0 , P AB P ABCDEF = 0 , P AABDEF = 0 , P ABCDEF = 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. J EONG -H YUCK P ARK D OUBLED - YET -G AUGED , S EMI -C OVARIANCE & T WOFOLD S PIN

Chirality of Spin ( 1 , D − 1 ) L × Spin ( D − 1 , 1 ) R : γ ( D + 1 ) ψ ¯ γ ( D + 1 ) ρ = − c ρ , p = c ψ ¯ p , γ ( D + 1 ) ρ ′ = − c ′ ρ ′ , γ ( D + 1 ) ψ ′ p = c ′ ψ ′ ¯ p , ¯ γ ( D + 1 ) = cc ′ C , γ ( D + 1 ) C ¯ where c and c ′ are arbitrary independent two sign factors, c 2 = 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. J EONG -H YUCK P ARK D OUBLED - YET -G AUGED , S EMI -C OVARIANCE & T WOFOLD S PIN

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. J EONG -H YUCK P ARK D OUBLED - YET -G AUGED , S EMI -C OVARIANCE & T WOFOLD S PIN

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. J EONG -H YUCK P ARK D OUBLED - YET -G AUGED , S EMI -C OVARIANCE & T WOFOLD S PIN

Semi-covariant derivatives For each gauge symmetry we assign a corresponding connection, Γ A for the DFT-diffeomorphism (generalized Lie derivative), Φ A for the ‘unbarred’ local Lorentz symmetry, Spin ( 1 , D − 1 ) L , ¯ Φ A for the ‘barred’ local Lorentz symmetry, Spin ( D − 1 , 1 ) R . Combining all of them, we introduce master ‘semi-covariant’ derivative, D A = ∂ A + Γ A + Φ A + ¯ Φ A . J EONG -H YUCK P ARK D OUBLED - YET -G AUGED , S EMI -C OVARIANCE & T WOFOLD S PIN

It is also useful to set D A = ∂ A + Φ A + ¯ ∇ A = ∂ A + Γ A , Φ A . The former is the ‘semi-covariant’ derivative for the DFT-diffeomorphism (set by the generalized Lie derivative), n ∇ C T A 1 A 2 ··· A n := ∂ C T A 1 A 2 ··· A n − ω Γ B � B T A 1 ··· A i − 1 BA i + 1 ··· A n . BC T A 1 A 2 ··· A n + Γ CA i i = 1 And the latter is the covariant derivative for the twofold local Lorenz symmetries. J EONG -H YUCK P ARK D OUBLED - YET -G AUGED , S EMI -C OVARIANCE & T WOFOLD S PIN

It is also useful to set D A = ∂ A + Φ A + ¯ ∇ A = ∂ A + Γ A , Φ A . The former is the ‘semi-covariant’ derivative for the DFT-diffeomorphism (set by the generalized Lie derivative), n ∇ C T A 1 A 2 ··· A n := ∂ C T A 1 A 2 ··· A n − ω Γ B � B T A 1 ··· A i − 1 BA i + 1 ··· A n . BC T A 1 A 2 ··· A n + Γ CA i i = 1 And the latter is the covariant derivative for the twofold local Lorenz symmetries. J EONG -H YUCK P ARK D OUBLED - YET -G AUGED , S EMI -C OVARIANCE & T WOFOLD S PIN

It is also useful to set D A = ∂ A + Φ A + ¯ ∇ A = ∂ A + Γ A , Φ A . The former is the ‘semi-covariant’ derivative for the DFT-diffeomorphism (set by the generalized Lie derivative), n ∇ C T A 1 A 2 ··· A n := ∂ C T A 1 A 2 ··· A n − ω Γ B � B T A 1 ··· A i − 1 BA i + 1 ··· A n . BC T A 1 A 2 ··· A n + Γ CA i i = 1 And the latter is the covariant derivative for the twofold local Lorenz symmetries. J EONG -H YUCK P ARK D OUBLED - YET -G AUGED , S EMI -C OVARIANCE & T WOFOLD S PIN

By definition, the master derivative annihilates all the ‘constants’, D A J BC = ∇ A J BC = Γ ABD J DC + Γ ACD J BD = 0 , D A η pq = D A η pq = Φ Apr η rq + Φ Aqr η pr = 0 , r ¯ r ¯ q = ¯ ¯ q + ¯ ¯ D A ¯ q = D A ¯ Φ A ¯ Φ A ¯ r = 0 , η ¯ η ¯ η ¯ η ¯ p ¯ p ¯ p r ¯ q p ¯ D A C + αβ = D A C + αβ = Φ A αδ C + δβ + Φ A βδ C + αδ = 0 , δ ¯ ¯ δ ¯ ¯ D A ¯ β = D A ¯ β = ¯ β + ¯ C + ¯ C + ¯ Φ A ¯ C + ¯ Φ A ¯ C + ¯ δ = 0 , α ¯ α ¯ α δ ¯ α ¯ β including the gamma matrices, D A ( γ p ) αβ = D A ( γ p ) αβ = Φ Apq ( γ q ) αβ + Φ A αδ ( γ p ) δβ − ( γ p ) αδ Φ A δβ = 0 , p ) ¯ ¯ γ ¯ p ) ¯ α ¯ γ ¯ p ) ¯ α ¯ β = ¯ ¯ γ ¯ q ) ¯ α ¯ β + ¯ Φ A ¯ α ¯ γ ¯ δ ¯ γ ¯ p ) ¯ α ¯ δ ¯ δ ¯ p ¯ D A (¯ β = D A (¯ Φ A q (¯ δ (¯ β − (¯ Φ A β = 0 . J EONG -H YUCK P ARK D OUBLED - YET -G AUGED , S EMI -C OVARIANCE & T WOFOLD S PIN

It follows then that the connections are all anti-symmetric, Γ ABC = − Γ ACB , Φ Apq = − Φ Aqp , Φ A αβ = − Φ A βα , ¯ q = − ¯ ¯ β = − ¯ Φ A ¯ Φ A ¯ Φ A ¯ Φ A ¯ p , α , p ¯ q ¯ α ¯ β ¯ and as usual, Φ A αβ = 1 ¯ β = 1 4 ¯ γ ¯ p ¯ 4 Φ Apq ( γ pq ) αβ , Φ A ¯ α ¯ q ) ¯ α ¯ Φ A ¯ q (¯ β . p ¯ J EONG -H YUCK P ARK D OUBLED - YET -G AUGED , S EMI -C OVARIANCE & T WOFOLD S PIN

Further, the master derivative is compatible with the whole NS-NS sector, D A d = ∇ A d := − 1 2 e 2 d ∇ A ( e − 2 d ) = ∂ A d + 1 2 Γ BBA = 0 , D A V Bp = ∂ A V Bp + Γ ABC V Cp + Φ Apq V Bq = 0 , p + Γ ABC ¯ q ¯ D A ¯ p = ∂ A ¯ p + ¯ ¯ V B ¯ V B ¯ V C ¯ Φ A ¯ V B ¯ q = 0 . p It follows that D A ¯ P BC = ∇ A ¯ D A P BC = ∇ A P BC = 0 , P BC = 0 , and the connections are related to each other, Γ ABC = V Bp D A V Cp + ¯ ¯ p D A ¯ V B V C ¯ p , Φ Apq = V Bp ∇ A V Bq , ¯ q = ¯ p ∇ A ¯ V B ¯ Φ A ¯ V B ¯ q . p ¯ J EONG -H YUCK P ARK D OUBLED - YET -G AUGED , S EMI -C OVARIANCE & T WOFOLD S PIN

Further, the master derivative is compatible with the whole NS-NS sector, D A d = ∇ A d := − 1 2 e 2 d ∇ A ( e − 2 d ) = ∂ A d + 1 2 Γ BBA = 0 , D A V Bp = ∂ A V Bp + Γ ABC V Cp + Φ Apq V Bq = 0 , p + Γ ABC ¯ q ¯ D A ¯ p = ∂ A ¯ p + ¯ ¯ V B ¯ V B ¯ V C ¯ Φ A ¯ V B ¯ q = 0 . p It follows that D A ¯ P BC = ∇ A ¯ D A P BC = ∇ A P BC = 0 , P BC = 0 , and the connections are related to each other, Γ ABC = V Bp D A V Cp + ¯ ¯ p D A ¯ V B V C ¯ p , Φ Apq = V Bp ∇ A V Bq , ¯ q = ¯ p ∇ A ¯ V B ¯ Φ A ¯ V B ¯ q . p ¯ J EONG -H YUCK P ARK D OUBLED - YET -G AUGED , S EMI -C OVARIANCE & T WOFOLD S PIN

The connections assume the following most general forms: CAB + ∆ Cpq V Ap V Bq + ¯ p ¯ q ¯ ¯ ¯ q , Γ CAB = Γ 0 ∆ C ¯ V A V B p ¯ Φ Apq = Φ 0 Apq + ∆ Apq , ¯ q = ¯ q + ¯ Φ A ¯ Φ 0 ∆ A ¯ q . p ¯ p ¯ A ¯ p ¯ Here P [ AD ¯ P B ] E − P [ AD P B ] E � P ∂ C P ¯ � ¯ Γ 0 � � CAB = 2 P [ AB ] + 2 ∂ D P EC P B ] D + P C [ A P B ] D �� � ¯ P C [ A ¯ ∂ D d + ( P ∂ E P ¯ 4 � − P ) [ ED ] , D − 1 Jeon-Lee-JHP 2011 and, with the corresponding derivative, ∇ 0 A = ∂ A + Γ 0 A , Apq = V Bp ∇ 0 A V Bq = V Bp ∂ A V Bq + Γ 0 ABC V Bp V Cq , Φ 0 ¯ q = ¯ A ¯ q = ¯ p ∂ A ¯ ABC ¯ p ¯ Φ 0 V B ¯ p ∇ 0 V B ¯ q + Γ 0 V B ¯ V C ¯ V B ¯ V B ¯ q . A ¯ p ¯ J EONG -H YUCK P ARK D OUBLED - YET -G AUGED , S EMI -C OVARIANCE & T WOFOLD S PIN

The connections assume the following most general forms: CAB + ∆ Cpq V Ap V Bq + ¯ p ¯ q , q ¯ ¯ ¯ Γ CAB = Γ 0 ∆ C ¯ V A V B p ¯ Φ Apq = Φ 0 Apq + ∆ Apq , ¯ q = ¯ q + ¯ Φ 0 Φ A ¯ ∆ A ¯ q . p ¯ p ¯ A ¯ p ¯ The extra pieces, ∆ Apq and ¯ ∆ A ¯ q , correspond to the torsion of SDFT, which must be p ¯ covariant and, in order to maintain D A d = 0 , must satisfy ∆ Apq V Ap = 0 , p = 0 . ¯ q ¯ V A ¯ ∆ A ¯ p ¯ Otherwise they are arbitrary. As in SUGRA, the torsion can be constructed from the bi-spinorial objects, e.g. ¯ ¯ p γ Apq ψ ¯ p , ¯ ¯ ργ pq ψ A , ψ ¯ p γ A ψ ¯ q , ργ Apq ρ , ψ ¯ where we set ψ A = ¯ ¯ p ψ ¯ p , γ A = V Ap γ p . V A J EONG -H YUCK P ARK D OUBLED - YET -G AUGED , S EMI -C OVARIANCE & T WOFOLD S PIN

The connections assume the following most general forms: CAB + ∆ Cpq V Ap V Bq + ¯ p ¯ q , q ¯ ¯ ¯ Γ CAB = Γ 0 ∆ C ¯ V A V B p ¯ Φ Apq = Φ 0 Apq + ∆ Apq , ¯ q = ¯ q + ¯ Φ 0 Φ A ¯ ∆ A ¯ q . p ¯ p ¯ A ¯ p ¯ The extra pieces, ∆ Apq and ¯ ∆ A ¯ q , correspond to the torsion of SDFT, which must be p ¯ covariant and, in order to maintain D A d = 0 , must satisfy ∆ Apq V Ap = 0 , p = 0 . ¯ q ¯ V A ¯ ∆ A ¯ p ¯ Otherwise they are arbitrary. As in SUGRA, the torsion can be constructed from the bi-spinorial objects, e.g. ¯ ¯ p γ Apq ψ ¯ p , ¯ ¯ ργ pq ψ A , ψ ¯ p γ A ψ ¯ q , ργ Apq ρ , ψ ¯ where we set ψ A = ¯ ¯ p ψ ¯ p , γ A = V Ap γ p . V A J EONG -H YUCK P ARK D OUBLED - YET -G AUGED , S EMI -C OVARIANCE & T WOFOLD S PIN

The connections assume the following most general forms: CAB + ∆ Cpq V Ap V Bq + ¯ p ¯ q , q ¯ ¯ ¯ Γ CAB = Γ 0 ∆ C ¯ V A V B p ¯ Φ Apq = Φ 0 Apq + ∆ Apq , ¯ q = ¯ q + ¯ Φ 0 Φ A ¯ ∆ A ¯ q . p ¯ p ¯ A ¯ p ¯ The extra pieces, ∆ Apq and ¯ ∆ A ¯ q , correspond to the torsion of SDFT, which must be p ¯ covariant and, in order to maintain D A d = 0 , must satisfy ∆ Apq V Ap = 0 , p = 0 . ¯ q ¯ V A ¯ ∆ A ¯ p ¯ Otherwise they are arbitrary. As in SUGRA, the torsion can be constructed from the bi-spinorial objects, e.g. ¯ ¯ p γ Apq ψ ¯ p , ¯ ¯ ργ pq ψ A , ψ ¯ p γ A ψ ¯ q , ργ Apq ρ , ψ ¯ where we set ψ A = ¯ ¯ p ψ ¯ p , γ A = V Ap γ p . V A J EONG -H YUCK P ARK D OUBLED - YET -G AUGED , S EMI -C OVARIANCE & T WOFOLD S PIN

The ‘torsionless’ connection, P ∂ C P ¯ � ¯ P [ AD ¯ P B ] E − P [ AD P B ] E � Γ 0 � � CAB = 2 P [ AB ] + 2 ∂ D P EC P B ] D + P C [ A P B ] D �� 4 � ¯ P C [ A ¯ ∂ D d + ( P ∂ E P ¯ � − P ) [ ED ] , D − 1 further obeys Γ 0 ABC + Γ 0 BCA + Γ 0 CAB = 0 , and P CABDEF Γ 0 P CABDEF Γ 0 ¯ DEF = 0 , DEF = 0 . J EONG -H YUCK P ARK D OUBLED - YET -G AUGED , S EMI -C OVARIANCE & T WOFOLD S PIN

In fact, the torsionless connection, P [ AD ¯ P B ] E − P [ AD P B ] E � P ∂ C P ¯ � ¯ � � Γ 0 CAB = 2 P [ AB ] + 2 ∂ D P EC P B ] D + P C [ A P B ] D �� 4 � ¯ P C [ A ¯ ∂ D d + ( P ∂ E P ¯ � − P ) [ ED ] , D − 1 is the unique solution to the following constraints: Γ CAB + Γ CBA = 0 = ⇒ ∇ A J BC = 0 , ∇ A P BC = ∇ A ¯ P BC = 0 , ∇ A d = 0 , L ( ∂ ) = ˆ ˆ Γ ABC + Γ CAB + Γ BCA = 0 = ⇒ L ( ∇ ) , P ) CABDEF Γ DEF = 0 . ( P + ¯ In this way, Γ 0 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. J EONG -H YUCK P ARK D OUBLED - YET -G AUGED , S EMI -C OVARIANCE & T WOFOLD S PIN

In fact, the torsionless connection, P [ AD ¯ P B ] E − P [ AD P B ] E � P ∂ C P ¯ � ¯ � � Γ 0 CAB = 2 P [ AB ] + 2 ∂ D P EC P B ] D + P C [ A P B ] D �� 4 � ¯ P C [ A ¯ ∂ D d + ( P ∂ E P ¯ � − P ) [ ED ] , D − 1 is the unique solution to the following constraints: Γ CAB + Γ CBA = 0 = ⇒ ∇ A J BC = 0 , ∇ A P BC = ∇ A ¯ P BC = 0 , ∇ A d = 0 , L ( ∂ ) = ˆ ˆ Γ ABC + Γ CAB + Γ BCA = 0 = ⇒ L ( ∇ ) , P ) CABDEF Γ DEF = 0 . ( P + ¯ In this way, Γ 0 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. J EONG -H YUCK P ARK D OUBLED - YET -G AUGED , S EMI -C OVARIANCE & T WOFOLD S PIN

Semi-covariant Riemann curvature The usual curvatures for the three connections, R CDAB = ∂ A Γ BCD − ∂ B Γ ACD + Γ ACE Γ BED − Γ BCE Γ AED , F ABpq = ∂ A Φ Bpq − ∂ B Φ Apq + Φ Apr Φ Br q − Φ Bpr Φ Ar q , ¯ q = ∂ A ¯ q − ∂ B ¯ q + ¯ r ¯ ¯ q − ¯ r ¯ ¯ r ¯ r ¯ F AB ¯ Φ B ¯ Φ A ¯ Φ A ¯ Φ B Φ B ¯ Φ A q , p ¯ p ¯ p ¯ p ¯ p ¯ are, from [ D A , D B ] V Cp = 0 and [ D A , D B ]¯ V C ¯ p = 0 , related to each other, q + ¯ p ¯ q . p V B q ¯ ¯ ¯ R ABCD = F CDpq V A F CD ¯ V A V B p ¯ However, the crucial object in DFT turns out to be � � S ABCD := 1 R ABCD + R CDAB − Γ E AB Γ ECD , 2 which we name semi-covariant Riemann curvature. J EONG -H YUCK P ARK D OUBLED - YET -G AUGED , S EMI -C OVARIANCE & T WOFOLD S PIN

Semi-covariant Riemann curvature The usual curvatures for the three connections, R CDAB = ∂ A Γ BCD − ∂ B Γ ACD + Γ ACE Γ BED − Γ BCE Γ AED , F ABpq = ∂ A Φ Bpq − ∂ B Φ Apq + Φ Apr Φ Br q − Φ Bpr Φ Ar q , ¯ q = ∂ A ¯ q − ∂ B ¯ q + ¯ r ¯ ¯ q − ¯ r ¯ ¯ r ¯ r ¯ F AB ¯ Φ B ¯ Φ A ¯ Φ A ¯ Φ B Φ B ¯ Φ A q , p ¯ p ¯ p ¯ p ¯ p ¯ are, from [ D A , D B ] V Cp = 0 and [ D A , D B ]¯ V C ¯ p = 0 , related to each other, q + ¯ p ¯ q . p V B q ¯ ¯ ¯ R ABCD = F CDpq V A F CD ¯ V A V B p ¯ However, the crucial object in DFT turns out to be � � S ABCD := 1 R ABCD + R CDAB − Γ E AB Γ ECD , 2 which we name semi-covariant Riemann curvature. J EONG -H YUCK P ARK D OUBLED - YET -G AUGED , S EMI -C OVARIANCE & T WOFOLD S PIN

Properties of the semi-covariant curvature Under arbitrary variation of the connection, δ Γ ABC , it transforms as δ S 0 ABCD = D [ A δ Γ 0 B ] CD + D [ C δ Γ 0 D ] AB , δ S ABCD = D [ A δ Γ B ] CD + D [ C δ Γ D ] AB − 3 2 Γ [ ABE ] δ Γ E CD − 3 2 Γ [ CDE ] δ Γ E AB . It also satisfies precisely the same symmetric property as the ordinary Riemann curvature, S ABCD = 1 � � S [ AB ][ CD ] + S [ CD ][ AB ] , 2 S 0 [ ABC ] D = 0 , as well as projection property, q = S ABCD V Ap ¯ V B ¯ p V Cq ¯ V D ¯ S p ¯ q = 0 . pq ¯ J EONG -H YUCK P ARK D OUBLED - YET -G AUGED , S EMI -C OVARIANCE & T WOFOLD S PIN

Properties of the semi-covariant curvature Under arbitrary variation of the connection, δ Γ ABC , it transforms as δ S 0 ABCD = D [ A δ Γ 0 B ] CD + D [ C δ Γ 0 D ] AB , δ S ABCD = D [ A δ Γ B ] CD + D [ C δ Γ D ] AB − 3 2 Γ [ ABE ] δ Γ E CD − 3 2 Γ [ CDE ] δ Γ E AB . It also satisfies precisely the same symmetric property as the ordinary Riemann curvature, S ABCD = 1 � � S [ AB ][ CD ] + S [ CD ][ AB ] , 2 S 0 [ ABC ] D = 0 , as well as projection property, q = S ABCD V Ap ¯ V B ¯ p V Cq ¯ V D ¯ S p ¯ q = 0 . pq ¯ J EONG -H YUCK P ARK D OUBLED - YET -G AUGED , S EMI -C OVARIANCE & T WOFOLD S PIN

‘Semi-covariance’ Generically, under DFT-diffeomorphisms, the variation of the semi-covariant derivative carries anomalous terms which are dictated by the six-index projectors, ≡ ˆ � 2 ( P + ¯ BFDE ∂ F ∂ [ D X E ] T ··· B ··· . � ∇ C T A 1 ··· A n � L X � ∇ C T A 1 ··· A n � P ) CA i δ X + i Hence, it is not DFT-diffeomorphism covariant, ˆ δ X � = L X . 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. J EONG -H YUCK P ARK D OUBLED - YET -G AUGED , S EMI -C OVARIANCE & T WOFOLD S PIN

‘Semi-covariance’ Generically, under DFT-diffeomorphisms, the variation of the semi-covariant derivative carries anomalous terms which are dictated by the six-index projectors, ≡ ˆ � 2 ( P + ¯ BFDE ∂ F ∂ [ D X E ] T ··· B ··· . � ∇ C T A 1 ··· A n � L X � ∇ C T A 1 ··· A n � P ) CA i δ X + i Hence, it is not DFT-diffeomorphism covariant, ˆ δ X � = L X . 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. J EONG -H YUCK P ARK D OUBLED - YET -G AUGED , S EMI -C OVARIANCE & T WOFOLD S PIN

Completely covariant derivatives For O ( D , D ) tensors: P CD ¯ B 1 ¯ B 2 · · · ¯ ¯ B n ∇ D T B 1 B 2 ··· B n , P CD P A 1 B 1 P A 2 B 2 · · · P A n B n ∇ D T B 1 B 2 ··· B n , P A 1 P A 2 P A n  P AB ¯ D 1 ¯ D 2 · · · ¯ Dn ∇ A T BD 1 D 2 ··· Dn , P C 1 P C 2 P Cn    Divergences , P AB P C 1 ¯ D 1 P C 2 D 2 · · · P Cn Dn ∇ A T BD 1 D 2 ··· Dn     P AB ¯ D 1 ¯ D 2 · · · ¯ Dn ∇ A ∇ B T D 1 D 2 ··· Dn , P C 1 P C 2 P Cn    Laplacians , P AB P C 1 ¯ D 1 P C 2 D 2 · · · P Cn Dn ∇ A ∇ B T D 1 D 2 ··· Dn    and D AC ¯ D 1 · · · ¯ D n T CD 1 ··· D n , D AC P B 1 ¯ D 1 · · · P B n D n T CD 1 ··· D n , P B 1 P B n where we set a pair of semi-covariant second order differential operators, B := ( P A B P CD − 2 P A B := (¯ B ¯ P CD − 2 ¯ D ¯ D P BC )( ∇ C ∇ D − S CD ) , ¯ P BC )( ∇ C ∇ D − S CD ) , P A P A D A D A which are relevant to the DFT fluctuation analysis Ko-Meyer-Melby-Thompson-JHP J EONG -H YUCK P ARK D OUBLED - YET -G AUGED , S EMI -C OVARIANCE & T WOFOLD S PIN

Completely covariant derivatives For local Lorentz tensors, Spin ( 1 , D − 1 ) L × Spin ( D − 1 , 1 ) R : D p T ¯ q n , D ¯ p T q 1 q 2 ··· q n , q 1 ¯ q 2 ··· ¯ D p T p ¯ D ¯ p T ¯ q n , pq 1 q 2 ··· q n , q 1 ¯ q 2 ··· ¯ p D ¯ D p D p T ¯ p T q 1 q 2 ··· q n , q n , D ¯ q 1 ¯ q 2 ··· ¯ D pq T q ¯ ¯ ¯ q T ¯ p n , D ¯ qp 1 p 2 ··· p n . p 1 ¯ p 2 ··· ¯ p These are the ‘pull-back’ of the previous page using the DFT-vielbeins, such as p := ¯ D p := V Ap D A , V A ¯ D ¯ p D A . J EONG -H YUCK P ARK D OUBLED - YET -G AUGED , S EMI -C OVARIANCE & T WOFOLD S PIN

Completely covariant derivatives Following the aforementioned general prescription, completely covariant Yang-Mills field strength is given by two opposite projections, or q = V M p ¯ V N ¯ F p ¯ q F MN , where F MN is the semi-covariant field strength of a YM potential, V M , F MN := ∇ M V N − ∇ N V M − i [ V M , V N ] . 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 V M )( ∂ M − i V M ) = 0 , which implies V M ∂ M = 0 , ∂ M V M = 0 , V M V M = 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, [ g V M g − 1 − i ( ∂ M g ) g − 1 ] ∂ M = 0 , L X V M ) ∂ M = [ X N ∂ N V M + ( ∂ M X N − ∂ N X M ) V N ] ∂ M = 0 . ( ˆ J EONG -H YUCK P ARK D OUBLED - YET -G AUGED , S EMI -C OVARIANCE & T WOFOLD S PIN

Completely covariant derivatives Following the aforementioned general prescription, completely covariant Yang-Mills field strength is given by two opposite projections, or q = V M p ¯ V N ¯ F p ¯ q F MN , where F MN is the semi-covariant field strength of a YM potential, V M , F MN := ∇ M V N − ∇ N V M − i [ V M , V N ] . 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 V M )( ∂ M − i V M ) = 0 , which implies V M ∂ M = 0 , ∂ M V M = 0 , V M V M = 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, [ g V M g − 1 − i ( ∂ M g ) g − 1 ] ∂ M = 0 , L X V M ) ∂ M = [ X N ∂ N V M + ( ∂ M X N − ∂ N X M ) V N ] ∂ M = 0 . ( ˆ J EONG -H YUCK P ARK D OUBLED - YET -G AUGED , S EMI -C OVARIANCE & T WOFOLD S PIN

Completely covariant derivatives O ( D , D ) covariant Killing equations in DFT: ˆ ( P ∇ ) M (¯ PX ) N − (¯ L X H MN = 0 ⇐ ⇒ P ∇ ) N ( PX ) M = 0 , ∇ M X M = 0 . ˆ L X d = 0 ⇐ ⇒ JHP-Rey-Rim-Sakatani 2015 Chris Blair 2015 J EONG -H YUCK P ARK D OUBLED - YET -G AUGED , S EMI -C OVARIANCE & T WOFOLD S PIN

Completely covariant derivatives Dirac operators for fermions, ρ α , ψ α p , ρ ′ ¯ α , ψ ′ ¯ α p : ¯ γ p D p ρ = γ A D A ρ , γ p D p ψ ¯ p = γ A D A ψ ¯ p , p = D A ψ A , p ψ ¯ D ¯ D ¯ p ρ , ¯ q − 1 ψ A γ p ( D A ψ ¯ 2 D ¯ q ψ A ) , p ρ ′ = ¯ γ A D A ρ ′ , γ ¯ p D ¯ γ ¯ p D ¯ p ψ ′ γ A D A ψ ′ ¯ ¯ p = ¯ p , D p ψ ′ p = D A ψ ′ A , D p ρ ′ , ψ ′ A ¯ ¯ p ( D A ψ ′ q − 1 2 D q ψ ′ γ ¯ A ) . Incorporation of fermions into DFT 1109.2035 J EONG -H YUCK P ARK D OUBLED - YET -G AUGED , S EMI -C OVARIANCE & T WOFOLD S PIN

Completely covariant derivatives For R-R potential, C α ¯ β : γ A , D + C := γ A D A C + γ ( D + 1 ) D A C ¯ γ A . D − C := γ A D A C − γ ( D + 1 ) D A C ¯ Especially for the torsionless case, the corresponding operators are nilpotent, + ) 2 C = 0 , − ) 2 C = 0 , ( D 0 ( D 0 and hence, they define O ( D , D ) covariant cohomology. The field strength of the R-R potential, C α ¯ α , is then defined by F := D 0 + C . Thanks to the nilpotency, the R-R gauge symmetry is simply realized δ C = D 0 δ F = D 0 + ( δ C ) = ( D 0 + ) 2 ∆ = 0 . + ∆ = ⇒ J EONG -H YUCK P ARK D OUBLED - YET -G AUGED , S EMI -C OVARIANCE & T WOFOLD S PIN

Completely covariant derivatives For R-R potential, C α ¯ β : γ A , D + C := γ A D A C + γ ( D + 1 ) D A C ¯ γ A . D − C := γ A D A C − γ ( D + 1 ) D A C ¯ Especially for the torsionless case, the corresponding operators are nilpotent, + ) 2 C = 0 , − ) 2 C = 0 , ( D 0 ( D 0 and hence, they define O ( D , D ) covariant cohomology. The field strength of the R-R potential, C α ¯ α , is then defined by F := D 0 + C . Thanks to the nilpotency, the R-R gauge symmetry is simply realized δ C = D 0 δ F = D 0 + ( δ C ) = ( D 0 + ) 2 ∆ = 0 . + ∆ = ⇒ J EONG -H YUCK P ARK D OUBLED - YET -G AUGED , S EMI -C OVARIANCE & T WOFOLD S PIN

Completely covariant curvatures Scalar curvature: S := ( P AB P CD − ¯ P AB ¯ P CD ) S ACBD c.f. S ABAB = 0 . “Ricci” curvature: q = V Ap ¯ S 0 V B ¯ q S 0 p ¯ AB C . where we set S 0 AB = S 0 ACB Further, we have conserved “Einstein” curvature, P BD − ¯ G AB = 2 ( P AC ¯ ∇ A G AB = 0 . P AC P BD ) S CD − 1 2 J AB S , J EONG -H YUCK P ARK D OUBLED - YET -G AUGED , S EMI -C OVARIANCE & T WOFOLD S PIN

Combining all the results above, we are now ready to spell D = 10 Maximally Supersymmetric Double Field Theory J EONG -H YUCK P ARK D OUBLED - YET -G AUGED , S EMI -C OVARIANCE & T WOFOLD S PIN

D = 10 Maximal SDFT [ 1210.5078 ] Lagrangian : L Type II = e − 2 d � 8 ( P AB P CD − ¯ P AB ¯ ρ F ρ ′ + i ¯ 1 P CD ) S ACBD + 1 2 Tr ( F ¯ γ ¯ p ψ ′ q F ) − i ¯ ψ ¯ p γ q F ¯ � p ρ − i ¯ 2 ¯ p ρ ′ + i ¯ p ρ ′ + i 1 2 ¯ + i 1 ργ p D ⋆ ψ ¯ p D ⋆ p ρ − i 1 ψ ¯ p γ q D ⋆ p − i 1 ρ ′ ¯ γ ¯ p D ′ ⋆ ψ ′ p D ′ ⋆ ψ ′ p ¯ γ ¯ q D ′ ⋆ q ψ ′ p 2 ¯ 2 ¯ q ψ ¯ . ¯ ¯ ¯ where ¯ αα denotes the charge conjugation, ¯ F := ¯ C − 1 + F T C + . F ¯ 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 J EONG -H YUCK P ARK D OUBLED - YET -G AUGED , S EMI -C OVARIANCE & T WOFOLD S PIN

D = 10 Maximal SDFT [ 1210.5078 ] Lagrangian : L Type II = e − 2 d � 8 ( P AB P CD − ¯ P AB ¯ ρ F ρ ′ + i ¯ 1 P CD ) S ACBD + 1 2 Tr ( F ¯ γ ¯ p ψ ′ q F ) − i ¯ ψ ¯ p γ q F ¯ � p ρ − i ¯ 2 ¯ p ρ ′ + i ¯ p ρ ′ + i 1 2 ¯ + i 1 ργ p D ⋆ ψ ¯ p D ⋆ p ρ − i 1 ψ ¯ p γ q D ⋆ p − i 1 ρ ′ ¯ γ ¯ p D ′ ⋆ ψ ′ p D ′ ⋆ ψ ′ p ¯ γ ¯ q D ′ ⋆ q ψ ′ p 2 ¯ 2 ¯ q ψ ¯ . ¯ ¯ ¯ where ¯ αα denotes the charge conjugation, ¯ F := ¯ C − 1 + F T C + . F ¯ 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 J EONG -H YUCK P ARK D OUBLED - YET -G AUGED , S EMI -C OVARIANCE & T WOFOLD S PIN

D = 10 Maximal SDFT [ 1210.5078 ] Lagrangian : L Type II = e − 2 d � 8 ( P AB P CD − ¯ P AB ¯ ρ F ρ ′ + i ¯ 2 Tr ( F ¯ γ ¯ 1 P CD ) S ACBD + 1 p ψ ′ q F ) − i ¯ p γ q F ¯ ψ ¯ p ρ ′ + i ¯ p ρ ′ + i 1 � + i 1 p ρ − i ¯ ψ ¯ p ρ − i 1 2 ¯ ψ ¯ p − i 1 ρ ′ ¯ γ ¯ p D ′ ⋆ ψ ′ p D ′ ⋆ 2 ¯ ψ ′ p ¯ γ ¯ q D ′ ⋆ q ψ ′ p ργ p D ⋆ p D ⋆ p γ q D ⋆ 2 ¯ q ψ ¯ 2 ¯ . ¯ ¯ ¯ Torsions: The semi-covariant curvature, S ABCD , is given by the connection, 3 ¯ ψ ¯ p + 4 i ¯ ABC + i 1 ργ BC ψ A − i 1 p γ ABC ψ ¯ Γ ABC = Γ 0 3 ¯ ργ ABC ρ − 2 i ¯ ψ B γ A ψ C γ ABC ρ ′ − 2 i ¯ + i 1 ρ ′ ¯ ρ ′ ¯ γ BC ψ ′ A − i 1 3 ¯ ψ ′ p ¯ γ ABC ψ ′ p + 4 i ¯ ψ ′ B ¯ γ A ψ ′ C , 3 ¯ which corresponds to the solution for 1.5 formalism. The master derivatives in the fermionic kinetic terms are twofold: A for the unprimed fermions and D ′ ⋆ D ⋆ A for the primed fermions, set by ABC = Γ ABC − i 11 ργ ABC ρ + i 5 ργ BC ψ A + i 5 24 ¯ ψ ¯ p − 2 i ¯ ψ B γ A ψ C + i 5 Γ ⋆ p γ ABC ψ ¯ ρ ′ ¯ γ BC ψ ′ A , 96 ¯ 4 ¯ 2 ¯ γ ABC ρ ′ + i 5 Γ ′ ⋆ ABC = Γ ABC − i 11 ρ ′ ¯ ρ ′ ¯ γ BC ψ ′ A + i 5 24 ¯ ψ ′ p ¯ γ ABC ψ ′ p − 2 i ¯ ψ ′ B ¯ γ A ψ ′ C + i 5 96 ¯ 4 ¯ 2 ¯ ργ BC ψ A . J EONG -H YUCK P ARK D OUBLED - YET -G AUGED , S EMI -C OVARIANCE & T WOFOLD S PIN

D = 10 Maximal SDFT [ 1210.5078 ] Lagrangian : L Type II = e − 2 d � 8 ( P AB P CD − ¯ P AB ¯ ρ F ρ ′ + i ¯ 2 Tr ( F ¯ γ ¯ 1 P CD ) S ACBD + 1 p ψ ′ q F ) − i ¯ p γ q F ¯ ψ ¯ p ρ ′ + i ¯ p ρ ′ + i 1 � + i 1 p ρ − i ¯ ψ ¯ p ρ − i 1 2 ¯ ψ ¯ p − i 1 ρ ′ ¯ γ ¯ p D ′ ⋆ ψ ′ p D ′ ⋆ 2 ¯ ψ ′ p ¯ γ ¯ q D ′ ⋆ q ψ ′ p ργ p D ⋆ p D ⋆ p γ q D ⋆ 2 ¯ q ψ ¯ 2 ¯ . ¯ ¯ ¯ Torsions: The semi-covariant curvature, S ABCD , is given by the connection, 3 ¯ ψ ¯ p + 4 i ¯ ABC + i 1 ργ BC ψ A − i 1 p γ ABC ψ ¯ Γ ABC = Γ 0 3 ¯ ργ ABC ρ − 2 i ¯ ψ B γ A ψ C γ ABC ρ ′ − 2 i ¯ + i 1 ρ ′ ¯ ρ ′ ¯ γ BC ψ ′ A − i 1 3 ¯ ψ ′ p ¯ γ ABC ψ ′ p + 4 i ¯ ψ ′ B ¯ γ A ψ ′ C , 3 ¯ which corresponds to the solution for 1.5 formalism. The master derivatives in the fermionic kinetic terms are twofold: A for the unprimed fermions and D ′ ⋆ D ⋆ A for the primed fermions, set by ABC = Γ ABC − i 11 ργ ABC ρ + i 5 ργ BC ψ A + i 5 24 ¯ ψ ¯ p − 2 i ¯ ψ B γ A ψ C + i 5 Γ ⋆ p γ ABC ψ ¯ ρ ′ ¯ γ BC ψ ′ A , 96 ¯ 4 ¯ 2 ¯ γ ABC ρ ′ + i 5 Γ ′ ⋆ ABC = Γ ABC − i 11 ρ ′ ¯ ρ ′ ¯ γ BC ψ ′ A + i 5 24 ¯ ψ ′ p ¯ γ ABC ψ ′ p − 2 i ¯ ψ ′ B ¯ γ A ψ ′ C + i 5 96 ¯ 4 ¯ 2 ¯ ργ BC ψ A . J EONG -H YUCK P ARK D OUBLED - YET -G AUGED , S EMI -C OVARIANCE & T WOFOLD S PIN

D = 10 Maximal SDFT [ 1210.5078 ] Maximal supersymmetry transformation rules are also completely covariant, δ ε d = − i 1 ε ′ ρ ′ ) , 2 (¯ ερ + ¯ δ ε V Ap = i ¯ ¯ q (¯ ε ′ ¯ q ψ ′ V A γ ¯ p − ¯ εγ p ψ ¯ q ) , δ ε ¯ p = iV Aq (¯ ε ′ ¯ p ψ ′ p − ¯ V A ¯ εγ q ψ ¯ γ ¯ q ) , ρ ′ − ψ ¯ p + ρ ¯ δ ε C = i 1 2 ( γ p ε ¯ γ ¯ ε ′ ) + C δ ε d − 1 2 (¯ γ ¯ q , ψ ′ ε ′ ¯ V A ¯ q δ ε V Ap ) γ ( d + 1 ) γ p C ¯ p − ε ¯ p ¯ δ ε ρ = − γ p ˆ p ρ ′ − i γ p ψ ¯ q ¯ D p ε + i 1 2 γ p ε ¯ ψ ′ ε ′ ¯ q ψ ′ γ ¯ p , δ ε ρ ′ = − ¯ p ˆ p ε ′ + i 1 p ε ′ ¯ γ ¯ γ ¯ γ ¯ D ′ q ψ ′ εγ p ψ ¯ 2 ¯ ψ ¯ p ρ − i ¯ p ¯ q , ¯ q ¯ p ε ′ + i 1 p = ˆ p ε + ( F − i 1 2 γ q ρ ¯ ψ ′ q + i 1 2 ψ ¯ ρ ′ ¯ 4 ε ¯ p ρ + i 1 D ¯ δ ε ψ ¯ γ ¯ q )¯ γ ¯ ψ ¯ 2 ψ ¯ p ¯ ερ , p ε ′ + ( ¯ q ρ ′ ¯ 2 ψ ′ q ¯ 4 ε ′ ¯ p ρ ′ + i 1 p = ˆ F − i 1 γ ¯ q + i 1 ργ q ) γ p ε + i 1 ε ′ ρ ′ , δ ε ψ ′ D ′ ψ ′ 2 ψ ′ 2 ¯ ψ ¯ p ¯ where Γ ABC = Γ ABC − i 17 ˆ ργ ABC ρ + i 5 ργ BC ψ A + i 1 4 ¯ ψ ¯ p − 3 i ¯ p γ ABC ψ ¯ ψ ′ γ A ψ ′ 48 ¯ 2 ¯ B ¯ C , γ ABC ρ ′ + i 5 Γ ′ ˆ ABC = Γ ABC − i 17 ρ ′ ¯ ρ ′ ¯ γ BC ψ ′ A + i 1 4 ¯ ψ ′ p ¯ γ ABC ψ ′ p − 3 i ¯ 48 ¯ 2 ¯ ψ B γ A ψ C . J EONG -H YUCK P ARK D OUBLED - YET -G AUGED , S EMI -C OVARIANCE & T WOFOLD S PIN

D = 10 Maximal SDFT [ 1210.5078 ] Lagrangian : L Type II = e − 2 d � 8 ( P AB P CD − ¯ P AB ¯ ρ F ρ ′ + i ¯ 1 P CD ) S ACBD + 1 2 Tr ( F ¯ γ ¯ p ψ ′ q F ) − i ¯ ψ ¯ p γ q F ¯ � p ρ − i ¯ 2 ¯ p ρ ′ + i ¯ p ρ ′ + i 1 2 ¯ + i 1 ργ p D ⋆ ψ ¯ p D ⋆ p ρ − i 1 ψ ¯ p γ q D ⋆ p − i 1 ρ ′ ¯ γ ¯ p D ′ ⋆ ψ ′ p D ′ ⋆ ψ ′ p ¯ γ ¯ q D ′ ⋆ q ψ ′ p 2 ¯ 2 ¯ q ψ ¯ . ¯ ¯ ¯ The Lagrangian is pseudo: It is necessary to impose a self-duality of the R-R field strength by hand, � 1 − γ ( D + 1 ) � � ρ ′ + i 1 q � ˜ q ¯ ψ ′ γ ¯ F − i 1 2 γ p ψ ¯ F − := 2 ρ ¯ p ¯ ≡ 0 . J EONG -H YUCK P ARK D OUBLED - YET -G AUGED , S EMI -C OVARIANCE & T WOFOLD S PIN

D = 10 Maximal SDFT [ 1210.5078 ] Lagrangian : L Type II = e − 2 d � 8 ( P AB P CD − ¯ P AB ¯ ρ F ρ ′ + i ¯ 1 P CD ) S ACBD + 1 2 Tr ( F ¯ γ ¯ p ψ ′ q F ) − i ¯ ψ ¯ p γ q F ¯ � p ρ − i ¯ 2 ¯ p ρ ′ + i ¯ p ρ ′ + i 1 2 ¯ + i 1 ργ p D ⋆ ψ ¯ p D ⋆ p ρ − i 1 ψ ¯ p γ q D ⋆ p − i 1 ρ ′ ¯ γ ¯ p D ′ ⋆ ψ ′ p D ′ ⋆ ψ ′ p ¯ γ ¯ q D ′ ⋆ q ψ ′ p 2 ¯ 2 ¯ q ψ ¯ . ¯ ¯ ¯ The Lagrangian is pseudo: It is necessary to impose a self-duality of the R-R field strength by hand, � 1 − γ ( D + 1 ) � � ρ ′ + i 1 q � ˜ q ¯ ψ ′ γ ¯ F − i 1 2 γ p ψ ¯ F − := 2 ρ ¯ p ¯ ≡ 0 . J EONG -H YUCK P ARK D OUBLED - YET -G AUGED , S EMI -C OVARIANCE & T WOFOLD S PIN

D = 10 Maximal SDFT [ 1210.5078 ] Under the N = 2 SUSY transformation rule, the Lagrangian transforms as � � 8 e − 2 d ¯ γ p ˜ q ˜ δ ε L Type II = − 1 V A ¯ γ ¯ q δ ε V Ap Tr F − ¯ F − + total derivatives , where the precise self-duality relation appears quadratically, � 1 − γ ( D + 1 ) � � ρ ′ + i 1 q � q ¯ ψ ′ γ ¯ ˜ F − i 1 2 γ p ψ ¯ F − := 2 ρ ¯ p ¯ . 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, � p ρ + γ p ˜ � p − i γ p ε � ρ ′ + ˜ p − ¯ p � p ψ ′ p ε ′ ¯ γ ¯ D ′ D ′ p ¯ ψ ′ γ ¯ γ ¯ δ ε ˜ ˜ p − γ p F ¯ ˜ F − = − i D ¯ D p ψ ¯ ¯ p ¯ p ¯ p γ p F ¯ γ ¯ ψ ¯ . ¯ J EONG -H YUCK P ARK D OUBLED - YET -G AUGED , S EMI -C OVARIANCE & T WOFOLD S PIN

D = 10 Maximal SDFT [ 1210.5078 ] Under the N = 2 SUSY transformation rule, the Lagrangian transforms as � � 8 e − 2 d ¯ γ p ˜ q ˜ δ ε L Type II = − 1 V A ¯ γ ¯ q δ ε V Ap Tr F − ¯ F − + total derivatives , where the precise self-duality relation appears quadratically, � 1 − γ ( D + 1 ) � � ρ ′ + i 1 q � q ¯ ψ ′ γ ¯ ˜ F − i 1 2 γ p ψ ¯ F − := 2 ρ ¯ p ¯ . 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, � p ρ + γ p ˜ � p − i γ p ε � ρ ′ + ˜ p − ¯ p � p ψ ′ p ε ′ ¯ γ ¯ D ′ D ′ p ¯ ψ ′ γ ¯ γ ¯ δ ε ˜ ˜ p − γ p F ¯ ˜ F − = − i D ¯ D p ψ ¯ ¯ p ¯ p ¯ p γ p F ¯ γ ¯ ψ ¯ . ¯ J EONG -H YUCK P ARK D OUBLED - YET -G AUGED , S EMI -C OVARIANCE & T WOFOLD S PIN

Equations of Motion DFT-vielbein: q ¯ ργ p ˜ q ρ + 2 i ¯ q ˜ D p ρ − i ¯ ψ ¯ p γ p ˜ ρ ′ ¯ q ˜ D p ρ ′ + 2 i ¯ ψ ′ p ˜ q ρ ′ − i ¯ ψ ′ q ¯ q ˜ D p ψ ′ S p ¯ q + Tr ( γ p F ¯ γ ¯ F )+ i ¯ D ¯ ψ ¯ D ¯ q ψ ¯ p + i ¯ γ ¯ D ¯ γ ¯ q = 0 . This is DFT-generalization of Einstein equation. DFT-dilaton: L Type II = 0 . Namely, the on-shell Lagrangian vanishes. R-R potential: � ρ ′ + i γ r ψ ¯ s � γ ¯ s ¯ ψ ′ D 0 F − i ρ ¯ r ¯ = 0 , − which is automatically met by the self-duality, together with the nilpotency of D 0 + , � ρ ′ + i γ r ψ ¯ s � � � s ¯ ψ ′ γ ¯ D 0 = D 0 γ ( D + 1 ) F = − γ ( D + 1 ) D 0 + F = − γ ( D + 1 ) ( D 0 + ) 2 C = 0 . F − i ρ ¯ r ¯ − − The 1.5 formalism works: The variation of the Lagrangian induced by that of the connection is trivial, δ L Type II = δ Γ ABC × 0 . J EONG -H YUCK P ARK D OUBLED - YET -G AUGED , S EMI -C OVARIANCE & T WOFOLD S PIN

Equations of Motion DFT-vielbein: q ¯ ργ p ˜ q ρ + 2 i ¯ q ˜ D p ρ − i ¯ ψ ¯ p γ p ˜ ρ ′ ¯ q ˜ D p ρ ′ + 2 i ¯ ψ ′ p ˜ q ρ ′ − i ¯ ψ ′ q ¯ q ˜ D p ψ ′ S p ¯ q + Tr ( γ p F ¯ γ ¯ F )+ i ¯ D ¯ ψ ¯ D ¯ q ψ ¯ p + i ¯ γ ¯ D ¯ γ ¯ q = 0 . This is DFT-generalization of Einstein equation. DFT-dilaton: L Type II = 0 . Namely, the on-shell Lagrangian vanishes. R-R potential: � ρ ′ + i γ r ψ ¯ s � γ ¯ s ¯ ψ ′ D 0 F − i ρ ¯ r ¯ = 0 , − which is automatically met by the self-duality, together with the nilpotency of D 0 + , � ρ ′ + i γ r ψ ¯ s � � � s ¯ ψ ′ γ ¯ D 0 = D 0 γ ( D + 1 ) F = − γ ( D + 1 ) D 0 + F = − γ ( D + 1 ) ( D 0 + ) 2 C = 0 . F − i ρ ¯ r ¯ − − The 1.5 formalism works: The variation of the Lagrangian induced by that of the connection is trivial, δ L Type II = δ Γ ABC × 0 . J EONG -H YUCK P ARK D OUBLED - YET -G AUGED , S EMI -C OVARIANCE & T WOFOLD S PIN

Equations of Motion DFT-vielbein: q ¯ ργ p ˜ q ρ + 2 i ¯ q ˜ D p ρ − i ¯ ψ ¯ p γ p ˜ ρ ′ ¯ q ˜ D p ρ ′ + 2 i ¯ ψ ′ p ˜ q ρ ′ − i ¯ ψ ′ q ¯ q ˜ D p ψ ′ S p ¯ q + Tr ( γ p F ¯ γ ¯ F )+ i ¯ D ¯ ψ ¯ D ¯ q ψ ¯ p + i ¯ γ ¯ D ¯ γ ¯ q = 0 . This is DFT-generalization of Einstein equation. DFT-dilaton: L Type II = 0 . Namely, the on-shell Lagrangian vanishes. R-R potential: � ρ ′ + i γ r ψ ¯ s � γ ¯ s ¯ ψ ′ D 0 F − i ρ ¯ r ¯ = 0 , − which is automatically met by the self-duality, together with the nilpotency of D 0 + , � ρ ′ + i γ r ψ ¯ s � � � s ¯ ψ ′ γ ¯ D 0 = D 0 γ ( D + 1 ) F = − γ ( D + 1 ) D 0 + F = − γ ( D + 1 ) ( D 0 + ) 2 C = 0 . F − i ρ ¯ r ¯ − − The 1.5 formalism works: The variation of the Lagrangian induced by that of the connection is trivial, δ L Type II = δ Γ ABC × 0 . J EONG -H YUCK P ARK D OUBLED - YET -G AUGED , S EMI -C OVARIANCE & T WOFOLD S PIN

Equations of Motion DFT-vielbein: q ¯ ργ p ˜ q ρ + 2 i ¯ q ˜ D p ρ − i ¯ ψ ¯ p γ p ˜ ρ ′ ¯ q ˜ D p ρ ′ + 2 i ¯ ψ ′ p ˜ q ρ ′ − i ¯ ψ ′ q ¯ q ˜ D p ψ ′ S p ¯ q + Tr ( γ p F ¯ γ ¯ F )+ i ¯ D ¯ ψ ¯ D ¯ q ψ ¯ p + i ¯ γ ¯ D ¯ γ ¯ q = 0 . This is DFT-generalization of Einstein equation. DFT-dilaton: L Type II = 0 . Namely, the on-shell Lagrangian vanishes. R-R potential: � ρ ′ + i γ r ψ ¯ s � γ ¯ s ¯ ψ ′ D 0 F − i ρ ¯ r ¯ = 0 , − which is automatically met by the self-duality, together with the nilpotency of D 0 + , � ρ ′ + i γ r ψ ¯ s � � � s ¯ ψ ′ γ ¯ D 0 = D 0 γ ( D + 1 ) F = − γ ( D + 1 ) D 0 + F = − γ ( D + 1 ) ( D 0 + ) 2 C = 0 . F − i ρ ¯ r ¯ − − The 1.5 formalism works: The variation of the Lagrangian induced by that of the connection is trivial, δ L Type II = δ Γ ABC × 0 . J EONG -H YUCK P ARK D OUBLED - YET -G AUGED , S EMI -C OVARIANCE & T WOFOLD S PIN

Equations of Motion DFT-vielbein: q ¯ ργ p ˜ q ρ + 2 i ¯ q ˜ D p ρ − i ¯ ψ ¯ p γ p ˜ ρ ′ ¯ q ˜ D p ρ ′ + 2 i ¯ ψ ′ p ˜ q ρ ′ − i ¯ ψ ′ q ¯ q ˜ D p ψ ′ S p ¯ q + Tr ( γ p F ¯ γ ¯ F )+ i ¯ D ¯ ψ ¯ D ¯ q ψ ¯ p + i ¯ γ ¯ D ¯ γ ¯ q = 0 . This is DFT-generalization of Einstein equation. DFT-dilaton: L Type II = 0 . Namely, the on-shell Lagrangian vanishes. R-R potential: � ρ ′ + i γ r ψ ¯ s � γ ¯ s ¯ ψ ′ D 0 F − i ρ ¯ r ¯ = 0 , − which is automatically met by the self-duality, together with the nilpotency of D 0 + , � ρ ′ + i γ r ψ ¯ s � � � s ¯ ψ ′ γ ¯ D 0 = D 0 γ ( D + 1 ) F = − γ ( D + 1 ) D 0 + F = − γ ( D + 1 ) ( D 0 + ) 2 C = 0 . F − i ρ ¯ r ¯ − − The 1.5 formalism works: The variation of the Lagrangian induced by that of the connection is trivial, δ L Type II = δ Γ ABC × 0 . J EONG -H YUCK P ARK D OUBLED - YET -G AUGED , S EMI -C OVARIANCE & T WOFOLD S PIN

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, L N = 1 = e − 2 d � P AB P CD − ¯ � P AB ¯ A ρ − i ¯ 2 ¯ 1 P CD � S ACBD + i 1 ργ A D ⋆ ψ A D ⋆ A ρ − i 1 ψ B γ A D ⋆ � 2 ¯ A ψ B . 8 N = 1 Local SUSY: = − i 1 δ ε d 2 ¯ ερ , = − i ¯ δ ε V Ap εγ p ψ A , δ ε ¯ V A ¯ = i ¯ εγ A ψ ¯ p , p = − γ A ˆ D A ε , δ ε ρ = ¯ p ˆ D A ε − i 1 p ) ε + i 1 V A ¯ δ ε ψ ¯ 4 (¯ ρψ ¯ 2 (¯ ερ ) ψ ¯ p . p J EONG -H YUCK P ARK D OUBLED - YET -G AUGED , S EMI -C OVARIANCE & T WOFOLD S PIN

N = 1 SUSY Algebra [1112.0069] Commutator of supersymmetry reads [ δ ε 1 , δ ε 2 ] ≡ ˆ L X 3 + δ ε 3 + δ so ( 1 , 9 ) L + δ so ( 9 , 1 ) R + δ trivial . where X A ε 1 γ A ε 2 , ε 3 = i 1 ε 1 γ p ε 2 ) γ p ρ + (¯ ρε 2 ) ε 1 − (¯ etc. 3 = i ¯ 2 [(¯ ρε 1 ) ε 2 ] , and δ trivial corresponds to the fermionic equations of motion. J EONG -H YUCK P ARK D OUBLED - YET -G AUGED , S EMI -C OVARIANCE & T WOFOLD S PIN

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. J EONG -H YUCK P ARK D OUBLED - YET -G AUGED , S EMI -C OVARIANCE & T WOFOLD S PIN

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,     ( e − 1 ) p µ e − 1 ) ¯ p µ (¯ 1 ¯ 1     V Ap =  , V A ¯ p =  . √ √     2 2   ( B + ¯ ( B + e ) ν p e ) ν ¯ p Here e µ p and ¯ p are two copies of the D -dimensional vielbein corresponding to the e ν ¯ same spacetime metric, q ¯ e µ p e ν q η pq = − ¯ ¯ p ¯ ¯ e µ e ν η ¯ q = g µν , p ¯ and further we set B µ p = B µν ( e − 1 ) p ν , B µ ¯ p = B µν (¯ e − 1 ) ¯ p ν . J EONG -H YUCK P ARK D OUBLED - YET -G AUGED , S EMI -C OVARIANCE & T WOFOLD S PIN

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,     ( e − 1 ) p µ e − 1 ) ¯ p µ (¯ 1 ¯ 1     V Ap =  , V A ¯ p =  . √ √     2 2   ( B + ¯ ( B + e ) ν p e ) ν ¯ p Here e µ p and ¯ p are two copies of the D -dimensional vielbein corresponding to the e ν ¯ same spacetime metric, q ¯ e µ p e ν q η pq = − ¯ ¯ p ¯ ¯ e µ e ν η ¯ q = g µν , p ¯ and further we set B µ p = B µν ( e − 1 ) p ν , B µ ¯ p = B µν (¯ e − 1 ) ¯ p ν . J EONG -H YUCK P ARK D OUBLED - YET -G AUGED , S EMI -C OVARIANCE & T WOFOLD S PIN

Parametrization: Reduction to Generalized Geometry Instead, we may choose an alternative parametrization,     ( β + ¯ e ) µ p e ) µ p ( β + ˜ ˜ V Ap = p = 1 ¯ ¯ 1      , V A  , √ √     2 2   (¯ e − 1 ) p ν e − 1 ) p ν (˜ ˜ where β µ p = β µν (˜ p = β µν (¯ e µ p , ¯ e − 1 ) p ν , β µ ¯ e − 1 ) p ν , and ˜ e µ ¯ ˜ ˜ p correspond to a pair of T-dual vielbeins for winding modes, e ν q η pq = − ¯ q = ( g − Bg − 1 B ) − 1 µν . p ¯ q η ¯ p ¯ e µ p ˜ e µ ¯ e ν ¯ ˜ ˜ ˜ Note that in the above T-dual winding mode sector, the D -dimensional curved e µ p , ¯ p , β µν ( cf. x µ , e µ p , ¯ e µ ¯ e µ ¯ p , B µν ). spacetime indices are all upside-down: ˜ x µ , ˜ ˜ J EONG -H YUCK P ARK D OUBLED - YET -G AUGED , S EMI -C OVARIANCE & T WOFOLD S PIN

Parametrization: Reduction to Generalized Geometry Two parametrizations:     ( e − 1 ) p µ e − 1 ) ¯ (¯ p µ ¯ 1   1   V Ap = p = √  , V A ¯ √     2 2    ( B + ¯ ( B + e ) ν p e ) ν ¯ p versus     ( β + ¯ ( β + ˜ e ) µ p e ) µ p ˜ V Ap = p = 1 ¯ ¯ 1     √  , V A √  .     2 2   (¯ e − 1 ) p ν e − 1 ) p ν (˜ ˜ In connection to the section condition, ∂ A ∂ A ≡ 0 , the former matches well with the ∂ ∂ choice, x µ ≡ 0 , while the latter is natural when ∂ x µ ≡ 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. J EONG -H YUCK P ARK D OUBLED - YET -G AUGED , S EMI -C OVARIANCE & T WOFOLD S PIN

Parametrization: Reduction to Generalized Geometry Two parametrizations:     ( e − 1 ) p µ e − 1 ) ¯ (¯ p µ ¯ 1   1   V Ap = p = √  , V A ¯ √     2 2    ( B + ¯ ( B + e ) ν p e ) ν ¯ p versus     ( β + ¯ ( β + ˜ e ) µ p e ) µ p ˜ V Ap = p = 1 ¯ ¯ 1     √  , V A √  .     2 2   (¯ e − 1 ) p ν e − 1 ) p ν (˜ ˜ In connection to the section condition, ∂ A ∂ A ≡ 0 , the former matches well with the ∂ ∂ choice, x µ ≡ 0 , while the latter is natural when ∂ x µ ≡ 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. J EONG -H YUCK P ARK D OUBLED - YET -G AUGED , S EMI -C OVARIANCE & T WOFOLD S PIN

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 J EONG -H YUCK P ARK D OUBLED - YET -G AUGED , S EMI -C OVARIANCE & T WOFOLD S PIN

Parametrization: Reduction to Generalized Geometry From now on, let us restrict ourselves to the former parametrization and impose ∂ x µ ≡ 0 . ∂ ˜ This reduces (S)DFT to ‘Generalized Geometry’ Hitchin; Grana, Minasian, Petrini, Waldram For example, the O ( D , D ) covariant Dirac operators become √ 2 γ A D A ρ ≡ γ m � � ∂ m ρ + 1 4 ω mnp γ np ρ + 1 24 H mnp γ np ρ − ∂ m φρ , √ p ≡ γ m � q + 1 q − ∂ m φψ ¯ � p + 1 q ψ ¯ p + 1 q ψ ¯ 2 γ A D A ψ ¯ 4 ω mnp γ np ψ ¯ 24 H mnp γ np ψ ¯ ∂ m ψ ¯ p + ¯ ω m ¯ 2 H m ¯ , p ¯ p ¯ p √ 2 ¯ V A ¯ p ρ + 1 pqr γ qr ρ + 1 pqr γ qr ρ , p D A ρ ≡ ∂ ¯ 4 ω ¯ 8 H ¯ √ 2 D A ψ A ≡ ∂ ¯ p + ¯ q + 1 p − 2 ∂ ¯ p + 1 pqr γ qr ψ ¯ ω ¯ q ψ ¯ pqr γ qr ψ ¯ p φψ ¯ p . p ψ ¯ p ¯ 4 ω ¯ 8 H ¯ p ¯ ω µ ± 1 2 H µ and ω µ ± 1 6 H µ naturally appear as spin connections. Liu, Minasian J EONG -H YUCK P ARK D OUBLED - YET -G AUGED , S EMI -C OVARIANCE & T WOFOLD S PIN

Parametrization: Reduction to Generalized Geometry From now on, let us restrict ourselves to the former parametrization and impose ∂ x µ ≡ 0 . ∂ ˜ This reduces (S)DFT to ‘Generalized Geometry’ Hitchin; Grana, Minasian, Petrini, Waldram For example, the O ( D , D ) covariant Dirac operators become √ 2 γ A D A ρ ≡ γ m � � ∂ m ρ + 1 4 ω mnp γ np ρ + 1 24 H mnp γ np ρ − ∂ m φρ , √ p ≡ γ m � q + 1 q − ∂ m φψ ¯ � p + 1 q ψ ¯ p + 1 q ψ ¯ 2 γ A D A ψ ¯ 4 ω mnp γ np ψ ¯ 24 H mnp γ np ψ ¯ ∂ m ψ ¯ p + ¯ ω m ¯ 2 H m ¯ , p ¯ p ¯ p √ 2 ¯ V A ¯ p ρ + 1 pqr γ qr ρ + 1 pqr γ qr ρ , p D A ρ ≡ ∂ ¯ 4 ω ¯ 8 H ¯ √ 2 D A ψ A ≡ ∂ ¯ p + ¯ q + 1 p − 2 ∂ ¯ p + 1 pqr γ qr ψ ¯ ω ¯ q ψ ¯ pqr γ qr ψ ¯ p φψ ¯ p . p ψ ¯ p ¯ 4 ω ¯ 8 H ¯ p ¯ ω µ ± 1 2 H µ and ω µ ± 1 6 H µ naturally appear as spin connections. Liu, Minasian J EONG -H YUCK P ARK D OUBLED - YET -G AUGED , S EMI -C OVARIANCE & T WOFOLD S PIN

Unification of type IIA and IIB SUGRAs Since the two zehnbeins correspond to the same spacetime metric, they are related by a Lorentz rotation, q ¯ ( e − 1 ¯ ¯ p ( e − 1 ¯ ¯ e ) p e ) q q = − η pq . η ¯ p ¯ There exists also a spinorial representation for this local Lorentz rotation, S e , p , γ ¯ p S − 1 = γ ( D + 1 ) γ p ( e − 1 ¯ ¯ S e ¯ e ) p e such that, in particular, e ) γ ( D + 1 ) . γ ( D + 1 ) S − 1 = − det ( e − 1 ¯ S e ¯ e J EONG -H YUCK P ARK D OUBLED - YET -G AUGED , S EMI -C OVARIANCE & T WOFOLD S PIN

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, e µ p ≡ ¯ p , ¯ e µ 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 . J EONG -H YUCK P ARK D OUBLED - YET -G AUGED , S EMI -C OVARIANCE & T WOFOLD S PIN

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, e µ p ≡ ¯ p , ¯ e µ 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 . J EONG -H YUCK P ARK D OUBLED - YET -G AUGED , S EMI -C OVARIANCE & T WOFOLD S PIN