Supergravities in Ten and Eleven Dimensions — As a Dictionary

TK-NOTE/07-04 since: July 27, 2007 last update: 2014-06-21 13:31

Supergravities in ten and eleven dimensions — as a dictionary —

Tetsuji KIMURA

Yukawa Institute for Theoretical Physics, Kyoto University

Sakyo-ku, Kyoto 606-8502, Japan

Abstract

In this note we start eleven-dimensional supergravity which explicitly contains fermionic terms. We connect it to the type IIA supergravity action, and perform T-duality to obtain the type IIB supergravity. Further we also study the type I and heterotic supergravity actions. In this note we impose the self-dual condition on the Ramond-Ramond ﬁve-form ﬁeld strength F5 by hand in type IIB supergravity. 1 Strategy

We start from the eleven-dimensional supergravity. Along various kinds of duality transformations, we connect five significant supergravity Lagrangian including fermions. We also investigate local supersymmetry transformations in each theory. The reason is that, unfortunately, we only find the bosonic parts of supergravity theories while the local supersymmetry transformation rules are explicitly introduced. This situation is terrible to consider the Killing spinor equations and equations of motion, as well as the Bianchi identity in the presense of fermions. To comple- ment such a difficulty, we will derive the supergravity Lagrangians including fermions and their interactions, explicitly.

It might be quite useful to ﬁx the convention of ﬁelds and transformations for our future work on the investigation of duality transformations in terms of Killing spinors and spinorial geometry.

2 Contents

1 Strategy 2

2 Eleven-dimensional supergravity 5

3 Type IIA supergravity in ten dimensions 8

3.1 Duality transformation rules ...... 8

4 Type IIB supergravity in ten dimensions 9

4.1 Duality transformation rules ...... 9

5 Type I supergravity in ten dimensions 10

6 Heterotic supergravity in ten dimensions 11

7 Heterotic supergravity in ten dimensions, II 12

′ 7.1 Lagrangian with higher-order corrections in α ...... 12

7.2 Local supersymmetry variations ...... 13

A Convention 17

A.1 Contraction rule on antisymmetric tensors ...... 17

A.2 Antisymmetrized symbol ...... 17

A.3 Lorentz algebra ...... 17

A.4 Dirac conjugate and charge conjugate on spinors ...... 18

A.5 Covariant derivatives and curvature tensors ...... 19

A.6 Riemann tensor of Levi-Civita connection ...... 22

A.7 Contorsion ...... 22

A.8 Bismut torsion ...... 24

A.9 Differential forms ...... 25

A.10 Yang-Mills gauge ﬁelds: hermitian variables ...... 27

3 A.11 Yang-Mills gauge ﬁelds: anti-hermitian variables ...... 28

A.12 Chern-Simons three-forms ...... 28

B Conventions 30

C Geometries: complex, hermitian and K¨ahlermanifolds 31

C.1 Complex coordinates ...... 31

C.2 Metric on complex manifold: hermitian metric ...... 31

C.3 General coordinate transformations of connections ...... 33

C.4 Further analysis on hermitian manifold ...... 35

C.5 Additional constraint: Kahler¨ manifold ...... 40

D Introducing torsion 43

4 2 Eleven-dimensional supergravity

Here let us summarize the convantion which appears in the note [4, 7]: ∫ S = d11x L , (2.1a) √ 1√ 1 √ 2κ2 L = β −g R(e, ω) − −g Ψ ΓMNP D [ 1 (ω + ωb)]Ψ − −g F F MNPQ 11 0 11 2 11 M N 2 P 48 11 MNPQ 1 √ + β −g Ψ ΓeMNP QRSΨ (F + Fb) 2 1 11 M N P QRS MNP QRSUV W XY + β2 εˆ FMNPQ FRSUV CWXY , (2.1b)

b where ωbMAB and FMNPQ are the supercovariantizations of ωMAB and FMNPQ, respectively [5, 4]. The supercovariantization means that its supersymmetry variation does not contain derivatives of the supersymmetry parameter. However, such extension introduce fermion bilinear terms in it, which might be irrelevant in this note. Then we truncate such supercovariantized ﬁelds as

b ωbMAB = ωMAB , FMNPQ = FMNPQ , (2.1c)

MNP QRSUW XY where ωMAB is a torsionless spin connection. We should also notice that εˆ is not ··· the invariant tensor but the “antisymmetric symbol”, whose normalization is given as εˆ012 ♮ = 1. This is related to the tensor εMNP QRSUW XY as

MNP QRSUV W XY MNP QRSUV W XY εˆ = g11 ε . (2.1d)

The transformation rule under the local supersymmetry is given as 1 δe A = εΓAΨ , δC = α εΓ Ψ , (2.1e) M 2 M MNP 2 [MN P ] NP QR δΨM = 2DM (ω)ε + 2TM εFNP QR . (2.1f)

In the above formulation, various objects are deﬁned in the following way:

† i Ψ = iΨ Γ0 ,D (ω) = ∂ − ω ΣAB , (2.1g) M M M M 2 MAB eMNP QRS MNP QRS M[P QR S]N Γ = Γ + 12g Γ g , (2.1h) ( ) α1 [N P QR] T NP QR = Γ NP QR − 8δ Γ . (2.1i) M 2 M M

For later convenience, we introduced some unﬁxed coefﬁcients β0, β1 and β2 in the Lagrangian, and α1 and α2 in the supersymmetry transformation rule. These are closely related to each other to preserve symmetry of the action:

2α1β1 2 1 1 β2 = − = β1 = − α1 , α1β1 = − . (2.2a) α2 216 144 96 · 288β0

5 To arrange the usual form of the Einstein-Hilbert action, we ﬁx the constant β0 to be unity: β0 = 1. In this setting we should choose the following explicit solution: 1 3 1 1 α = , α = − , β = − , β = − . (2.2b) 1 144 2 2 1 192 2 (144)2

Let us rewrite the above forms (2.1) to connect other supergravity actions in ten-dimensional spacetime, which appear in the Polchinski’s book [15] and in the book written by Becker, Becker and Schwarz [1]. Let us rescale the gravitino

ΨM → 2ΨM . (2.3)

Substituting (2.1c), (2.2b) and (2.3), we obtain the 11-dimensional supergravity action, which we will mainly use, instead of (2.1). First, the Lagrangian itself is √ √ 1 √ 2κ2 L = −g R(e, ω) − 2 −g Ψ ΓMNP D (ω)Ψ − −g F F MNPQ 11 M N P 48 MNPQ 1 √ − −g Ψ ΓeMNP QRSΨ F 48 M N P QRS 1 − εˆMNP QRSUV W XY F F C . (2.4a) (144)2 MNPQ RSUV WXY

The transformation rule under the local supersymmetry is given as

A A δeM = εΓ ΨM , δCMNP = −3εΓ[MN ΨP ] , (2.5a) NP QR δΨM = DM (ω)ε + TM εFNP QR . (2.5b)

NP QR In addition, the combination of the gamma matrices TM is rewritten in the following way: ( ) 1 [N P QR] T NP QR = Γ NP QR − 8δ Γ . (2.5c) M 288 M M

Here it is worth describing the bosonic part of the action in terms of the differential form to com- pare the conventions in [15, 1]. Let us ﬁrst extract the bosonic part of the action: ∫ ( ) 1 √ 1 S = d11x −g R(e, ω) − F F MNPQ boson 2κ2 48 MNPQ 11 ∫ − 1 11 MNP QRSUV W XY 2 2 d x εˆ FMNPQFRSTU CWXY . (2.6) 2κ11(144)

Notice that the symbol εˆMNP QRSUW XY does not depend on the curved space coordinates. It is just a number. Using the convention in appendix A, the last term is rewritten as ∫ − 1 11 MNP QRSUV W XY 2 d x εˆ FMNPQFRSUV CWXY (144) ∫ 1 = − d11x g εMNP QRSUV W XY F F C (144)2 MNPQ RSUV WXY

6 ∫ − 1 M ∧ N ∧ · · · ∧ Y = 2 dx dx dx FMNPQFRSUV CWXY (144) ∫ ∫ 4!4!3! 1 = − F ∧ F ∧ C = − F ∧ F ∧ C . (2.7) (144)2 4 4 3 6 4 4 3

Then, the bosonic action is simpliﬁed in the following way: ( ) ∫ { } 1 − 1| |2 − 1 ∧ ∧ Sboson = 2 (vol.) R(e, ω) F4 F4 F4 C3 . (2.8) 2κ11 2 6

7 3 Type IIA supergravity in ten dimensions

In this section let us derive the type IIA supergravity from the eleven-dimensional supergravity. The derivation rule has already been investigated very well. Here we follow a convention introduced by [14], because the Lagrangians both in eleven- and in ten-dimensional spacetime are common forms in modern sense, while the fermion transformation rule itself is not explicitly discussed.

3.1 Duality transformation rules

Here let us introduce the duality transformation rules from the eleven-dimensional supergravity to the type IIA supergravity, and vice versa. We only fucus on the bosonic parts. The former rule is given by

− 2 Φ 4 Φ gˆMN = e 3 gMN + e 3 AM AN , CˆMNP = CMNP , (3.1a) ( ) 4 Φ 2 4 gˆ = e 3 A , Cˆ = B , gˆ = exp Φ . (3.1b) M♮ M MN♮ 3 MN ♮♮ 3

Note that the objects with ˆ denote the ones in eleven-dimensions, while the others in ten dimensions. In addition, the symbol ♮ indicates the eleventh direction in the eleven-dimensional spacetime. In the same way, the latter rule is also given by ( ) ( ) 1 gˆ gˆ 2 M♮ ♮N gMN = gˆ♮♮ gˆMN − ,CMNP = CˆMNP , (3.2a) gˆ♮♮ ( ) gˆM♮ 3 3 AM = ,BMN = CˆMN♮ , Φ = log gˆ♮♮ . (3.2b) gˆ♮♮ 2 4

In this note we omit the derivation of the above rules (see section 5.1 in [14]).

8 4 Type IIB supergravity in ten dimensions

In this section let us derive the type IIB supergravity from the type IIA supergravity using the convention introduced by [14].

4.1 Duality transformation rules

Here let us introduce the duality transformation rules from the type IIA supergravity to the type IIB supergravity, and vice versa. We only fucus on the bosonic parts. The former rule is given by ( ) B(1) − 1 (1) (1) 9M 1 gˆMN = gMN gM9g9N + BM9B9N , gˆ9M = , gˆ99 = , (4.1a) g99 g99 g99 ( 2B(2) g ) ˆ 2 (2) 9[M N]9 C9MN = BMN + , (4.1b) 3 g99 εijB(i) B(j) g ˆ 8 ij (i) (j) 9[M |9|N P ]9 CMNP = C9MNP + ε B9[M BNP ] + , (4.1c) 3 g99 2B(1) g ˆ (1) 9[M N]9 ˆ g9M BMN = BMN + , B9M = , (4.1d) g99 g99 ( ) (2) (1) 1 Aˆ = −B + CB , Aˆ = C, Φˆ = Φ − log g . (4.1e) M 9M 9M 9 2 99 Note that the objects with ˆ denote the ones in type IIA, while the others in type IIB. The two dif- (i) (1) NSNS (2) RR ferent two-form ﬁelds BMN implies BMN = BMN and BMN = CMN , respectively. We normalize ε12 = 1. In the same way, the latter rule is also given by ( ) 1 Bˆ9M 1 gMN =g ˆMN − gˆM9gˆ9N + BˆM9Bˆ9N , g9M = , g99 = , (4.2a) gˆ99 ( gˆ99 gˆ)99 gˆ Bˆ Aˆ gˆ Cˆ 3 ˆ ˆ ˆ 9[M NP ] 9 3 9[M NP ]9 C9MNP = CMNP − A[M BNP ] + − , (4.2b) 8 gˆ99 2 gˆ99 2ˆg Bˆ (1) ˆ 9[M N]9 (1) gˆ9M BMN = BMN + ,B9M = , (4.2c) gˆ99 gˆ99 2ˆg Bˆ Aˆ ˆ (2) 3 ˆ − ˆ ˆ 9[M N]9 9 (2) − ˆ A9gˆ9M BMN = CMN9 2A[M BN]9 + ,B9M = AM + , (4.2d) 2 gˆ99 gˆ99 1 ( ) Φ = Φˆ − log gˆ ,C = Aˆ . (4.2e) 2 99 9 In this note we omit the derivation of the above rules (see section 5.1 in [14]). Notice that the

RR four-form ﬁeld CMNPQ without containing the nineth-direction is given by the self-duality condition on its ﬁeld strength:

dC4 = F5 = ∗10F5 . (4.3)

9 5 Type I supergravity in ten dimensions

A The type I supergravity contains the vielbein eM , the dilaton Φ, the one-form anti-hermitian e (2) gauge field A1 , and the Ramond-Ramond two-form field BMN = CMN , as bosonic fields, and the gravitino ΨM , the dilatino λ and the gaugino χ. Here we will follow the convention given by Polchinski [15, 14].

The relation between the type IIB and the type I is as follows: The type I supergravity action is (1) obtained from the type IIB supergravity with truncating the fields BMN , C and CMNPQ, which are not invariant under the Ω-projection, i.e., the worldsheet parity projection. Further, we newly in- e troduce a one-form gauge field A1 from the open string modes, and modify the Ramond-Ramond three-form field strength F3 in the following way:

2 ( ) → ′ − κ10 Y Y ≡ e ∧ e 2 e ∧ e ∧ e F3 F3 = dC2 2 ω3 , ω3 tr A dA + A A A , (5.1) g10 3

Y where g10 and ω3 are called the Yang-Mills coupling and the Yang-Mills Chern-Simons three- e form in ten-dimensional spacetime, respectively. Notice that the one-form gauge ﬁeld A1 couples to anti-hermitian generator of the gauge group1, in the same way as the one in the heterotic supergravity [8], which will appear in the next section. Notice that the Ramond-Ramond three-form ′ F3 should be further modiﬁed via the Green-Schwarz anomaly cancellation mechanism.

1 The one-form gauge field A1 in [14] is given as a hermitian matrix field. See appendix A.12 for the difference between hermitian and anti-hermitian gauge fields.

10 6 Heterotic supergravity in ten dimensions

A The ten-dimensional heterotic supergravity contains the vielbein eM , the NS-NS two-form field e BMN , the dilaton Φ, the anti-hermitian gauge field AM , as bosonic fields, and the gravitino ΨM , the dilatino λ and the gaugino χ. Here we will follow the convention given by Polchinski [15, 14]. In the next section we will expand the heterotic theory of lowest order to the one of higher-order ′ correction in α [2].

The S-duality rule between the type I and the heterotic Lagrangian in ten dimensions is quite simple. We follow the discussion to [14]:

−Φ gˆMN = e gMN , Φˆ = −Φ , (6.1a) ˆ′ ′ ˆ F 3 = H3 , A1 = A1 , (6.1b) where the fields with ˆ denote the ones in the type I, while the ones without hat in the heterotic ′ theory. Notice that the NS-NS three-form field strength H3 is also modified by introducing the Y Yang-Mills Chern-Simons three-form ω3 :

2 ( ) ′ − κ10 Y Y ≡ e ∧ e 2 e ∧ e ∧ e H3 = dB2 2 ω3 , ω3 tr A dA + A A A . (6.2) g10 3 Notice that the NS-NS three-form should be further modiﬁed via the Green-Schwarz anomaly cancellation mechanism.

11 7 Heterotic supergravity in ten dimensions, II

′ The heterotic supergravity with higher-order corrections in α has been well investigated [2]. Thus, in this section, we introduce such a corrected Lagrangian in the heterotic theory given by [2, 8].

7.1 Lagrangian with higher-order corrections in α′

Let us write down the Lagrangian of ten-dimensional heterotic supergravity [2], which is an ex- ′ tended verions of the heterotic theory with higher-order corrections in α :

e2 2 L = L (R) + L (F ) + L (R ) , (7.1a) total [ √ 1 2κ2 L (R) = −g e−2Φ R(ω) − H HMNP + 4(∇ Φ)2 10 12 MNP M

MNP MN − ψM Γ DN (ω)ψP + 8 λ Γ DM (ω)ψN + 16 λD/(ω)λ N M M N + 8 ψM Γ Γ λ (∇N Φ) − 2ψM Γ ψN (∇ Φ) { } 1 P QR [M N] M − H ψM Γ ΓP QRΓ ψN + 8 ψM Γ P QRλ − 16 λΓP QRλ 24 { 1 M ABC N + ψ Γ ψM 2λΓABC λ + λΓABC Γ ψN 48 ] } 1 1 − ψN Γ ψ − ψN Γ Γ ΓP ψ , (7.1b) 4 ABC N 8 N ABC P [ 2 √ ( ) { } ( ) 2 L e2 − κ10 − −2Φ − e eMN − e e e − 1 e ABC e ˆ 2κ10 (F ) = 2 g e tr FMN F 2 tr χD/(ω, A)χ tr χΓ χ HABC 2g10 12 ( ) 1 { ˆ } 2 − tr χeΓM ΓAB(Fe + Fe ) ψ + Γ λ 2 AB AB M 3 M ( ) 1 ABC M M − tr(χeΓ χe) ψM 4ΓABC Γ + 3Γ ΓABC λ 48 ] 1 β + tr(χeΓABC χe)λΓ λ − tr(χeΓABC χe) tr(χeΓ χe) , (7.1c) 12 ABC 96 ABC [ 2 √ 2 L 2 − κ10 − −2Φ − ABMN 2κ10 (R ) = 2 g e RABMN (Ω−)R (Ω−) 2g10

AB 1 AB MNP − 2ψ D/(ω(e, ψ), Ω−)ψ − ψ Γ ψ Hˆ AB 12 AB MNP { }( ) 1 M NP AB AB 2 + ψ Γ Γ R (Ω−) + Rˆ (Ω−) ψ + Γ λ 2 AB NP NP M 3 M ( ) 1 CDE M M − ψABΓ ψAB · ψM 4ΓCDEΓ + 3Γ ΓCDE λ 48 ] 1 ( ) α ( ) + ψABΓCDEψ λΓ λ − ψABΓF GH ψ ψCDΓ ψ . (7.1d) 12 AB CDE 96 AB F GH CD

12 The ten-dimensional gravitational constant κ10, the Yang-Mills coupling g10 are related to

2 ′ κ10 ≡ α 2 . (7.2) g10 4 e Derivatives DM (ω, A) are the covariant derivatives with respect to Lorentz and Yang-Mills gauge transformations. We deﬁne the derivative on fundamental ﬁelds ϕi as

e i i i AB i j e i j i j D (ω, A, Γ)ϕ = ∂ ϕ − ω (Σ ) ϕ + (A ) ϕ + Γ ϕ . (7.3) M M 2 M AB j M j jM

Note that ΣAB is a Lorentz generator whose representation is given by (A.4). Notice that we can always re-deﬁne the gravitino and dilatino via “mixing” with each other such as       ′ ψ 1 b ψM  M  =     , b, c ∈ R , c ≠ 0 , ′ (7.4) ΓM λ 0 c ΓM λ because both ψM and ΓM λ are same chiralities and belong to the gravity multiplet. In the above Lagrangian we do not obtain the gravitino supersymmetry variation including the gradient of the dilaton, dilatino condensation terms and so forth.

7.2 Local supersymmetry variations

Following to [2], let us show the local supersymmetry variations We write δαn (δβm ) for variations n m 2 of order α (β ), while δ0 corresponds to the terms independent of parameters α and β:

A 1 A δ0eM = ϵΓ ψM , (7.5a) (2 ) { } 1 ( ) ( ) ( ) δ ψ = ∂ + Ω ABΓ ϵ + ϵ ψ λ − ψ ϵλ + ΓAλ ψ Γ ϵ , (7.5b) 0 M M 4 +M AB M M M A

δ0BMN = −ϵΓ[M ψN] , (7.5c) ( ) 1 1 1 δ λ = − D/Φ ϵ + ΓABC ϵ − Hˆ − λΓ λ , (7.5d) 0 4 48 ABC 2 ABC

δ0Φ = −ϵλ , (7.5e) e 1 δ A = ϵΓ χe , (7.5f) 0 M 2 M { ( ) ( ) ( )} 1 AB ˆe A δ χe = − Γ ϵ F + ϵ χλe − χe ϵλ + Γ λ χeΓ ϵ , (7.5g) 0 4 AB A 1 1 1 δ ω AB(e, ψ) = − ϵΓ ψAB − ϵΓ[Aψ B] + ϵΓ ψ Hˆ ABC . (7.5h) 0 M 4 M 2 M 4 C M

A Notice that the spin connection ω(e, ψ) is the solution of D[M (ω)eN] = 0, while ω(e) is the solution A 3 of D[M (ω)eN] = 0. Note that various objects such as a spin connection modiﬁed by the H-ﬂux, 2 2 2 Compared to [2] and [2], we assign the expansion parameters α and β to α = β = −κ10/g10. 3If we simply write down the spin connection as ω, this means ω = ω(e).

13 supercovariantizations Hˆ and Fˆ, and so forth:

AB AB 1 AB Ω ≡ ω (e, ψ) Hˆ , (7.6a) M M 2 M 3 Hˆ ≡ H + ψ Γ ψ MNP MNP 2 [M N P ] 2 ( ) 3 − κ10 e e 2 e e e = 3∂[M BNP ] + ψ[M ΓN ψP ] 6 2 tr A[M ∂N AP ] + A[M AN AP ] 2 g10 3 2 ( ) κ10 AB BA 2 AB BC CA + 6 2 Ω−[M ∂N Ω−P ] + Ω−[M Ω−N Ω−P ] , (7.6b) g10 3 2 [ { } ( )] κ10 ∧ − e ∧ e dH = 2 tr R(Ω−) R(Ω−) tr F F , (7.6c) g10 ˆe e F MN ≡ FMN − ψ[M ΓN]χe , (7.6d)

ψMN ≡ DM (Ω+)ψN − DN (Ω+)ψM { ( ) ( ) ( )} P − ψM ψN λ − ψN ψM λ − Γ λ ψM ΓP ψN , (7.6e) 1 1 RˆAB (ω) ≡ RAB (ω) + ψ Γ ψAB + ψ Γ[Aψ B] + ψ ΓC ψ Hˆ AB . (7.6f) MN MN 2 [M N] [M N] 2 [M N] C

We also pick up the supercovariant derivatives DM at hand:

DM Φ = ∇M Φ + ψM λ , (7.7a) ( ) 1 1 1 D (ω)λ = D (ω)λ + D/Φ ψ − ΓABC ψ − Hˆ − λΓ λ , (7.7b) M M 4 M 48 M ABC 2 ABC { ( ) ( ) ( )} e e 1 AB ˆe A D (ω, A)χe = D (ω, A)χe + Γ ψ F − ψ χλe − χe ψ λ + Γ λ χeΓ ψ . (7.7c) M M 4 M AB M M A M

It is both useful and instructive to obtain the supersymmetry algebra from (7.5). The commu- tator of two supersymmetry variations reads [ ] M − M M AB M e δ(ϵ1), δ(ϵ2) = δP(ξ ) + δQ( ξ ψM ) + δL(ξ Ω−M ) + δYM(ξ AM ) √ + δ (− 2 ξ + √1 ξN B ) + δ (ϵ ) + δ (ΛAB) , M 2 M 2 NM Q 3 L (7.8a) 1 ξM = ϵ ΓM ϵ , (7.8b) 2 2 1 7 1 ϵ = − (ϵ ΓAϵ )Γ λ + (ϵ ΓABCDEϵ )Γ λ , (7.8c) 3 8 2 1 A 16 × 120 2 1 ABCDE β ( ) ΛAB = ϵ Γ[AΓ ΓB]ϵ tr χeΓCDEχe . (7.8d) 192 2 CDE 1

On the right-hand side of (7.8a), we encounter all gauge transformations of the ten-dimensional super Yang-Mills theory: δP, δQ, δL, δYM, and δM correspond respectively to “general coordinate”, “supersymmetry”, “local Lorentz”, “Yang-Mills” and “antisymmetric tensor gauge” transformations.

14 The supersymmetry variation of order β are given as follows: β ( ) δ ψ = ΓABC Γ ϵ tr χeΓ χe , (7.9a) β M 192 M ABC { } e e δβBMN = 2β tr A[M δ0AN] , (7.9b) β ( ) δ λ = ΓABC ϵ tr χeΓ χe , (7.9c) β 384 ABC β ( ) δ ω AB(e, ψ) = − ϵΓ[AΓ ΓB]ψ tr χeΓCDEχe . (7.9d) β M 192 CDE M Here the supersymmetry variation of order α are also given such as α δ ψ = ΓCDEΓ ϵ ψABΓ ψ , (7.10a) α M 192 M CDE AB AB AB δαBMN = 2α Ω−[M δ0Ω−N] , (7.10b) α δ λ = ΓCDEΓ ϵ ψABΓ ψ . (7.10c) α 384 M CDE AB

The supersymmetry variations of the supercovariant variables are also obtained. First we write down the zero-th order of α and β. Next the corrections of first order β are described. (Unfortu- nately, there are no descriptions about the corrections of first order α [2].) √ 2 δ (D Φ) = − ϵD (Ω )λ , (7.11a) 0 A 2 A + AB 1 AB δ Ω− = − ϵΓ ψ , (7.11b) 0 M 2 M 3 δ Hˆ = ϵΓ ψ , (7.11c) 0 ABC 2 [A BC] { ( ) ( ) ( )} AB 1 CD AB AB AB C AB δ ψ = Γ ϵ Rˆ (Ω−) + ϵ ψ λ − ψ ϵλ + Γ λ ψ Γ ϵ , (7.11d) 0 4 CD C ê e δ0F AB = −ϵΓ[ADB](Ω+, A)χe , (7.11e) β ( ) δ (D Φ) = − ϵΓ ΓBCDλ tr χeΓ χe , (7.11f) β A 192 A BCD ( ) ˆ 3β ê δβHABC = ϵΓ[A tr χeF BC] , (7.11g) 2[ 3 ( ˆ ˆ ) 1 { } δ ψ = β ΓCDϵ tr Fe Fe + ΓCDEΓ ϵ tr χeΓ D (Ω , Ae)χe β AB 4 [AB CD] 48 [A CDE B] + ( ) 1 CDE GH ˆ − Γ Γ[AΓ ϵ HB]GH tr χeΓCDEχe 3 × 256 ] β ( ) ( ) + ΓCDEΓ ΓF GH Γ ϵ tr χeΓ χe tr χeΓ χe , (7.11h) 96 × 96 [A B] CDE F GH ( ) ê β CDE δ F = ϵΓ Γ Γ χe tr χeΓ χe . (7.11i) β AB 192 [A B] CDE Finally we describe an identity among generalized curvature tensors: 1 R (Ω−) = R (Ω ) − (dH) . (7.12) ABCD CDAB + 2 CDAB

15 The explicit expressions of supersymmetry variations are, of course, just approximate expressions. If you consider not only α, β corrections but also the higher order fermions corrections, the corrections of supersymmetry vairations are also corrected. The supercovariant variables such ˆe AB as Hˆ MNP , F MN and Rˆ MN (ω) are inﬂuenced by the higher order corrections quite sensitively. Thus, you must take care of any calculations when you study the supersymmetry variations and the construction of the Lagrangian of higher order corrections of fermions.

16 Appendix

A Convention

A.1 Contraction rule on antisymmetric tensors

We introduce the following simpliﬁed form:

2 1 M N M N |F | ≡ g 1 1 ··· g p p F ··· F ··· , (A.1) p p! M1 Mp N1 Np

where FM1···Mp is a totally antisymmetric tensor, i.e., the component of a p-form. The coefﬁcient 2 1/p! is adopted to normalize each term appearing in the explicit expansion of |Fp| to unity.

A.2 Antisymmetrized symbol

The totally anti-symmetrized symbol is deﬁned in terms of the square bracket: ( ) 1 T ··· = T ··· − T ··· permutations . (A.2) [M1M2 Mp] p! M1M2 Mp M2M1 Mp

This Gamma matrix is deﬁned by ( ) 1 ΓMNP = Γ[M ΓN ΓP ] = ΓM ΓN ΓP permutations . (A.3) 3!

A.3 Lorentz algebra

The Lorentz symmetry on the tangent space is important to describe vectors, tensors, and spinors in curved spacetime via vielbeins and inverse vielbeins. Let us now deﬁne the Lorentz algebra in

Euclidean space with respect to the Lorentz generators ΣAB such as

i[ΣAB, ΣCD] = ηAC ΣBD + ηBD ΣAC − ηAD ΣBC − ηBC ΣAD , (A.4a) where etaAB is the metric in the local Lorentz frame whose signature is deﬁned by the signature in the original curved space geometry. To show the signature itself, it is very convenient to use the local Lorentz frame metric ηAB in the following way:

− − − ηAB = diag.(| , ,...,{z }; +| , +,...,{z +}) . (A.4b) t s

17 Here t and s denote the number of directions with minus (plus) signatures. Mainly we will discuss the cases as (t, s) = (1,D − 1) or (t, s) = (0,D). The relation between the metrics in the local Lorentz frame and in the curved spacetime will be given later.

Let us go back to the discussions on the local Lorentz generator. The Lorentz generators acting on scalars, vectors (tensors) and spinors are represented as follows:    ΣAB = 0 scalar  ( ) A A A (ΣCD) B = i δ ηBD − δ ηBC vector (A.4c)  C D  i Σ = Γ spinor AB 2 AB where ΓA is the Dirac gamma matrix which satisﬁes the Clifford algebra

{ΓA, ΓB} = 2ηAB . (A.5)

b Here we deﬁne the chirality operator Γ in d = 2k + 2 dimensional spacetime with Lorentz signature:

b −k 0 1 d−1 Γ ≡ i Γ Γ ··· Γ , (A.6) where all superscripts are the local Lorentz coordinate indices, since a spinor can be defined in the local Lorentz frame (or the tangent space of the geometry), in which the Dirac gamma matrix is also defined. On the other hand, the chirality operator in d = 2n dimensional space with Euclidean signature is defined as

b −n 1 2 d Γ ≡ i Γ Γ ··· Γ . (A.7)

The difference between (A.6) and (A.7) mainly comes from the hermiticity on the gamma matrices: in the Lorentzian spacetime, almost all matrices are hermitian except for Γ0, which is anti- hermitian (see the deﬁnition (A.11)), while all the matrices in the Euclidean space are hermitian.

A.4 Dirac conjugate and charge conjugate on spinors

We deﬁne the Dirac conjugate

† ˆ ψ ≡ iψ Γ0 , (A.8)

ˆ where Γ0 lives in the tangent space. Furthermore, we assign the Majorana condition such as

ψ ≡ ψTC, (A.9)

18 where C is called the charge conjugate matrix whose generic properties in D = 2k are

† − † C C = 1 ,C 1 = C = CT = (−1)k+[k/2]C, (A.10a)

A −1 k+[k/2] A T A ···A −1 [ n+1 ] A ···A T C(Γ )C = (−1) (Γ ) ,C(Γ 1 n )C = (−) 2 (Γ 1 n ) , (A.10b)

n+1 { ···} where [ 2 ] = 1, 1, 2, 2, 3, 3, is the Gauss bracket. The hermitian conjugates of gamma matrices are deﬁned by

A † 0ˆ A 0ˆ −1 (Γ ) = ΓA = −Γ Γ (Γ ) . (A.11)

Among the Dirac gamma matrices there exists a useful identity such as

A1A2···Ap Γ ΓB1B2···Bq min(∑p,q) 1 ··· k(2p−k−1) p!q! [A1 Ak Ak+1 Ap] = (−1) 2 δ ··· δ Γ ··· . (A.12) (p − k)!(q − k)!k! [B1 Bk Bk+1 Bq] k=0

A.5 Covariant derivatives and curvature tensors

A M We introduce vielbeins eM and their inverses EA , which come from the spacetime metric gMN A B M N and the metric ηAB on orthogonal frame via gMN = ηAB eM eN and ηAB = gMN EA EB . By using these geometrical variables, let us deﬁne the covariant derivatives DM (ω, Γ) such as

P DM (Γ)AN = ∂M AN − Γ NM AP , (A.13a) N N N P DM (Γ)A = ∂M A + Γ PM A , (A.13b) Q Q DM (Γ)gNP ≡ 0 = ∂M gNP − Γ NM gQP − Γ PM gNQ , (A.13c) NP NP N QP P NQ DM (Γ)g ≡ 0 = ∂M g + Γ QM g + Γ QM g , (A.13d) A A A B P A DM (ω, Γ)eN ≡ 0 = ∂M eN + ωM B eN − Γ NM eP , (A.13e) N N N B N P DM (ω, Γ)EA ≡ 0 = ∂M EA − EB ωM A + Γ PM EA , (A.13f) P P [DM (Γ),DN (Γ)]AQ = −R QMN (Γ)AP + 2T MN DP (Γ)AQ , (A.13g) P P − P P R − P R R QMN (Γ) = ∂M Γ QN ∂N Γ QM + Γ RM Γ QN Γ RN Γ QM . (A.13h)

P Note that AM in the above equations are vector. Γ MN is the affine connection whose two lower P indices are not symmetric in general case. The antisymmetric part of the affine connection Γ [MN] P P is defined as a torsion T MN , while the symmetric part Γ (MN) is given in terms of the Levi-Civita P connection Γ(L)MN and torsion, which we will show from the metricity condition (A.13c). First we prepare the followings:

P P P P P Γ MN = Γ (MN) + Γ [MN] , Γ [NM] = T NM , (A.14a)

19 1 ( ) ΓP = gPQ ∂ g + ∂ g − ∂ g . (A.14b) (L)MN 2 M QN N MQ Q MN

P Next we investigate the symmetric part Γ (MN). The metricity condition gives

Q Q 0 = −DM (Γ)gNP = −∂M gNP + Γ NM gQP + Γ PM gNQ , (A.15a) Q Q 0 = DN (Γ)gPM = ∂N gPM − Γ PN gQM − Γ MN gPQ , (A.15b) Q Q 0 = DP (Γ)gMN = ∂P gMN − Γ MP gQN − Γ NP gMQ . (A.15c)

Summing (A.15a), (A.15b) and (A.15c), we obtain ( ) Q Q Q 0 = ∂P gMN + ∂N gPM − ∂M gNP − 2T MN gPQ − 2T MP gQN − 2Γ (PN)gMQ , ∴ Q Q − Q − Q Γ (PN) = Γ(L)PN TP N TN P . (A.16)

Then the afﬁne connection is also given in terms of the Levi-Civita connection and the other:

P P P P ≡ P − P − P Γ MN = Γ(L)MN + K MN ,K MN T MN TM N TN M . (A.17a)

P The tensor K MN is called the contorsion, which has the following property:

Q KMNP = gMQK NP = TMNP − TNMP − TPMN = −KNMP . (A.17b)

It is worth discussing the Riemann tensor induced from the Levi-Civita connection (A.14b):

P ≡ P − P P L − P L R QMN (Γ(L)) ∂M Γ(L)QN ∂N Γ(L)QM + Γ(L)LM Γ(L)QN Γ(L)LN Γ(L)QM P = −R QNM (Γ(L) . (A.18)

This Riemann tensor has various signiﬁcant (anti)symmetries under exchanges of indices. Let us R carefully analyze them by using RP QMN (Γ(L)) = gPRR QMN (Γ(L)): ( ) ( ) R R L − ↔ RP QMN (Γ(L)) = gPR ∂M Γ(L)QN + Γ(L)LM Γ(L)QN M N { ( )} ( ) 1 RK − R L − ↔ = gPR∂M g ∂QgKN + ∂N gQK ∂K gQN + gPRΓ(L)LM Γ(L)QN M N 2 ( ) ( ) 1 1 = ∂ ∂ g + ∂ g − ∂ g − gKL∂ g ∂ g + ∂ g − ∂ g 2 M Q PN N QP P QN 2 M PL Q KN N QK K QN ( )( ) ( ) 1 LK + g ∂LgPM + ∂M gLP − ∂P gLM ∂QgKN + ∂N gQK − ∂K gQN − M ↔ N 4 ( ) ( ) 1 1 = ∂ ∂ g − ∂ g − gKL∂ g ∂ g + ∂ g − ∂ g 2 M Q PN P QN 2 M PL Q KN N QK K QN ( )( ) ( ) 1 LK + g ∂LgPM + ∂M gLP − ∂P gLM ∂QgKN + ∂N gQK − ∂K gQN − M ↔ N 4 ( ) ( ) 1 1 = ∂M ∂QgPN − ∂P gQN − ∂N ∂QgPM − ∂P gQM 2 ( 2 )( ) 1 + gKL ∂ g + ∂ g − ∂ g ∂ g + ∂ g − ∂ g 4 Q KM M QK K QM P LN N PL L PN

20 ( )( ) 1 KL − g ∂QgKN + ∂N gQK − ∂K gQN ∂P gLM + ∂M gPL − ∂LgPM 4 ( ) ( ) 1 1 = ∂M ∂QgPN − ∂P gQN − ∂N ∂QgPM − ∂P gQM 2 ( 2 ) K L − K L + gKL Γ(L)QM Γ(L)PN Γ(L)QN Γ(L)PM . (A.19a)

Thus we can easily ﬁnd the following remarkable relations:

RP QMN (Γ(L)) = −RP QNM (Γ(L)) = −RQP MN (Γ(L)) = RMNPQ(Γ(L)) . (A.19b)

Here we also describe the ﬁrst and second Bianchi identity on this Riemann tensor:

M M M 1st: 0 = R NPQ(Γ(L)) + R P QN (Γ(L)) + R QNP (Γ(L)) , (A.20a) N N N 2nd: 0 = ∇M R P QR(Γ(L)) + ∇QR PRM (Γ(L)) + ∇RR PMQ(Γ(L)) . (A.20b)

Nest, let us introduce the covariant derivative induced by the local Lorentz transformation acting on a generic ﬁeld ϕi as { } i D (ω)ϕi = δi ∂ − ω AB · (Σ )i ϕj , (A.21) M j M 2 M AB j where ΣAB is the Lorentz generator whose explicit form depends on the representation of the ﬁeld ϕi. The curvature tensor associated with this covariant derivative is given in terms of the spin connection i [D (ω),D (ω)]ϕ = − RAB (ω)Σ ϕ , (A.22a) M N 2 MN AB AB AB AB A CB A CB R MN (ω) = ∂M ωN − ∂N ωM + ωM C ωN − ωN C ωM . (A.22b)

These are closely related via the vielbein and its inverse in the following way:

R R C AB R PMN (Γ) = ηBC EA eP R MN (ω) , (A.23a) R PN R B R A A AC R M (Γ) = g R PMN (Γ) = eM EA R B(ω) ,R B(ω) = R BC (ω) , (A.23b) M A R(Γ) = R M (Γ) = R A(ω) = R(ω) . (A.23c)

It is useful to give a comment on vielbein. We will meet a vielbein with indices at different M M positions such as e A, and so forth. This can be regarded as the inverse EA :

M MN B e A = g δAB eN , (A.24a) A N NP A B NP N A N eM e A = g δAB eM eP = g gMP = δM = eM EA , (A.24b) N N ∴ e A = EA . (A.24c)

A A In the same way, we also prove E M = eM .

21 A.6 Riemann tensor of Levi-Civita connection

Here we analyze the Riemann tensor of the Levi-Civita connection on a hermitian complex man- e ifold. Notice that the “Levi-Civita” connection Γ(L) indicates the torsionless part of the hermitian connection, whose properties are given in (C.29):

e e e e e e e e e e Rpqmn(Γ(L)) = Rpqmn(Γ(L)) = −Rpqnm(Γ(L)) = −Rqpmn(Γ(L)) = Rmnpq(Γ(L)) [ ] [ ] [ ] 1 1 = ∂ ∂ ge − ∂ ge − ∂ ∂ ge − ∂ ge + ge ΓeK ΓeL − ΓeK ΓeL 2 m q pn p qn 2 n q pm p qm KL (L)qm (L)pn (L)qn (L)pm = 0 , (A.25a) e e e e e e e e e e Rpqmn(Γ(L)) = Rpqmn(Γ(L)) = −Rpqnm(Γ(L)) = −Rqpmn(Γ(L)) = Rmnpq(Γ(L)) [ ] [ ] [ ] 1 e − e − 1 e − e e eK eL − eK eL = ∂m ∂qgpn ∂pgqn ∂n ∂qgpm ∂pgqm + gKL Γ(L)qmΓ(L)pn Γ(L)qnΓ(L)pm 2 [ ] 2 [ ] 1 ek el ek el = ∂ ∂ ge − ∂ ge + ge Γ Γ − Γ Γ , (A.25b) 2 m q pn p qn kl (L)qm (L)pn (L)pm (L)qn e e e e e e e e e e Rpqmn(Γ(L)) = Rpqmn(Γ(L)) = −Rpqnm(Γ(L)) = −Rqpmn(Γ(L)) = Rmnpq(Γ(L)) [ ] [ ] 1 1 = ∂m ∂qgepn − ∂pgeqn − ∂n ∂qgepm − ∂pgeqm 2 [ 2 ] e eK eL − eK eL + gKL Γ(L)qmΓ(L)pn Γ(L)qnΓ(L)pm , (A.25c) e e e e e e e e e e Rpqmn(Γ(L)) = Rpqmn(Γ(L)) = −Rpqnm(Γ(L)) = −Rqpmn(Γ(L)) = Rmnpq(Γ(L)) [ ] 1 1 eK eL eK eL = ∂ ∂ ge + ∂ ∂ ge + ge Γ Γ − Γ Γ , (A.25d) 2 m q pn 2 n p qm KL (L)qm (L)pn (L)qn (L)pm e e e e e e e e e e Rpqmn(Γ(L)) = Rpqmn(Γ(L)) = −Rpqnm(Γ(L)) = −Rqpmn(Γ(L)) = Rmnpq(Γ(L)) [ ] 1 1 ek el ek el = ∂ ∂ ge − ∂ ∂ ge + ge Γ Γ − Γ Γ , (A.25e) 2 m q pn 2 n q pm kl (L)qm (L)pn (L)qn (L)pm e e e e e e e e e e Rpqmn(Γ(L)) = Rpqmn(Γ(L)) = −Rpqnm(Γ(L)) = −Rqpmn(Γ(L)) = Rmnpq(Γ(L)) [ ] [ ] [ ] 1 1 = ∂ ∂ ge − ∂ ge − ∂ ∂ ge − ∂ ge + ge ΓeK ΓeL − ΓeK ΓeL 2 m q pn p qn 2 n q pm p qm KL (L)qm (L)pn (L)qn (L)pm = 0 . (A.25f)

Here we used the expression (A.19a).

A.7 Contorsion

The curved space metric gMN has a back-reaction from the (con)torsion if it exists on the manifold.

However, the metricity condition itself is free from this back-reaction. Suppose gMN and gˆMN be the metrics on the manifolds with and without torsion, respectively. These satisfy the following

22 metricity conditions individually:

∇ − Q − Q 0 = M gNP = ∂M gNP Γ(L)NM gQP Γ(L)PM gNQ , (A.26a) 0 = D (Γ)ˆg = ∂ gˆ − ΓQ gˆ − ΓQ gˆ M NP( M NP NM) QP ( PM NQ ) − ˆQ Q − ˆQ Q = ∂M gˆNP Γ(L)NM + K NM gˆQP Γ(L)PM + K PM gˆNQ

= ∇ˆ M gˆNP − KPNM − KNPM . (A.26b)

ˆQ ∇ˆ Note that Γ(L)NM and M are given in terms of the the metric gˆMN . Since the contorsion is antisymmetric KPNM = −KNPM , we ﬁnd that the metricity condition even in the presence of torsion has the same form as the one in the absence of torsion:

0 = ∇ˆ M gˆNP . (A.26c)

Then we need not worry about the existence of torsion when we use the metricity condition (A.26).

The above discussion is quite important when we decompose the spin connection into the A contorsion part and the other. We start from the vielbein postulate 0 = DM (ω, Γ)eN and obtain

ω = −E N ∂ e + ΓP e E N MAB B M NA ( NM PA B ) − N P P N = EB ∂M eNA + Γ(L)NM + K NM ePAEB ≡ (L) ωMAB + KABM , (A.27)

(L) where we defined the spin connection ωMAB given by the Levi-Civita connection Γ(L). The second term in the right-hand side is given by the contorsion tensor, which is, by definition (A.17), antisymmetric under the exchange of the former two indices KABM = −KBAM . Now it is useful to check the antisymmetry of the Levi-Civita spin connection: ( ) (L) − N P N − N Q P N ωMBA = EA ∂M eNB + Γ(L)NM ePBEA = EA ∂M gNQEB + Γ(L)NM ePBEA = −E N E Q∂ g − E ∂ E Q + ΓP e E N A B (M NQ AQ M B (L)) NM PB A − N Q P P − Q P N = EA EB Γ(L)NM gPQ + Γ(L)QM gNP EAQ∂M EB + Γ(L)NM ePBEA Q − P Q = EB ∂M eQA Γ(L)QM ePAEB − (L) = ωMAB , (A.28) where we used (A.24) and the metricity condition (A.26). Substituting (A.17) and (A.28) into (A.27), we confirm that the spin connection with torsion is also antisymmetric:

ωMAB = −ωMBA . (A.29)

23 A.8 Bismut torsion

N Suppose the complex structure JM is covariantly constant with respect to the connection Γ− =

Γ(L) − H with contorsion K = −H:

P P R P P R 0 = DM (Γ−)JN = ∇M JN + H NM JR − H RM JN . (A.30) where we assigned DM (Γ(L)) = ∇M . By using this we express the Nijenhuis tensor such as ( ) Q Q Q R NMNP ≡ JM ∇[QJN]P − JN ∇[QJM]P = HMNP − 3J[M JN HP ]QR , which tells us

Q R HMNP = NMNP + 3J[M JN HP ]QR .

We also show an identitiy in terms of (A.30):

Q Q R J[M ∇|Q|JNP ] = −2J[M JN HP ]QR .

This is nothing but the Bismut torsion deﬁned by

(B) 3 Q R S 3 Q T = J J J ∇ J = − J ∇| |J . MNP 2 M N P [Q RS] 2 [M Q NP ]

Summarizing the above facts, we ﬁnd

N (B) HMNP = MNP + TMNP . (A.31) which denotes that the contorsion in the afﬁne connection corresponds to the Bismut torsion T (B) if the geometry is complex (NMNP = 0):

(B) 3 Q R S 3 Q H = T = J J J ∇ J = − J ∇| |J . (A.32) MNP MNP 2 M N P [Q RS] 2 [M Q NP ]

Especially, in the heterotic superstring theory, the NS-NS three-form ﬂux H appears in the Bismut connection itself [11]. In this scenario we naturally choose a = −1.

N P − P Here we show the explicit computation with using JM JN = δM :

3 Q R S HMNP = JM JN JP ∇[QJRS] , 2 ( ) ( ) 1 1 1 J QJ RJ S∇ J = ∇ J QJ RJ SJ − J ∇ J QJ RJ S 2 M N P Q RS 2 Q M N P RS 2 RS Q M N P ( ) 1 Q = ∇Q JM JNP 2 { } 1 − J (∇ J Q)J RJ S + (∇ J R)J QJ S + (∇ J S)J QJ R 2 RS Q M N P Q N M P Q P M N

24 1 1 1 = J Q(∇ J ) − (∇ J )J Q + J Q∇ J 2 M Q NP 2 Q NP M 2 M Q PN 1 = − J Q(∇ J ) , (A.33a) 2 M Q NP 1 1 1 ∴ H = − J Q∇ J − J Q∇ J − J Q∇ J MNP 2 M Q NP 2 N Q PM 2 P Q MN 3 Q = − J ∇| |J . (A.33b) 2 [M Q NP ] Now let us rewrite this in terms of the (complex) differential forms. Due to the above analysis we have already understood that the NS-ﬂux (or Bismut torsion) is the sum of (2, 1)-form and N (1, 2)-form with respect to the complex structure JM . Then 3 3 H = J QJ RJ S∇ J = J QJ RJ S∂ J , (A.34a) MNP 2 M N P [Q RS] 2 M N P [Q RS] 1 1 H = H dxM ∧ dxN ∧ dxP = J QJ RJ S∂ J dxM ∧ dxN ∧ dxP 3! MNP 4 M N P [Q RS] 1 1 = J QJ RJ S∂ J dzm ∧ dzn ∧ dzp + J QJ RJ S∂ J dzm ∧ dzn ∧ dzp 4 m n p Q RS 4 m n p Q RS 1 1 + J QJ RJ S∂ J dzm ∧ dzn ∧ dzp + J QJ RJ S∂ J dzm ∧ dzn ∧ dzp 4 m n p Q RS 4 m n p Q RS 1 1 + J QJ RJ S∂ J dzm ∧ dzn ∧ dzp + J QJ RJ S∂ J dzm ∧ dzn ∧ dzp 4 m n p Q RS 4 m n p Q RS 1 1 = J QJ RJ S∂ J dzm ∧ dzn ∧ dzp + J QJ RJ S∂ J dzm ∧ dzn ∧ dzp 2 m n p Q RS 2 m n p Q RS 1 1 = i2(−i)∂ J dzm ∧ dzn ∧ dzp + (−i)2i∂ J dzm ∧ dzn ∧ dzp 2 m np 2 m np i ( ) = ∂ − ∂ J, (A.34b) 2 n n n − n Notice that the component of the complex structure is given by Jm = iδm, Jm = iδm and

Jmn = Jmn = 0.

A.9 Differential forms

We define differential forms on D-dimensional geometry (gD = det gmn). For realistic discussions, we will define them in the curved spacetime with signature (t, s) = (1,D − 1) or (0,D), i.e., we will introduce a parameter t = 0, 1 in the following definition which shows whether the spacetime is Lorentzian (t = 1) or Euclidean (t = 0).

1 M M ω ≡ ω ··· dx 1 ∧ · · · ∧ dx p , (A.35a) p p! M1 Mp √ 1 D (vol.) ≡ |gD| dx ∧ · · · ∧ dx . (A.35b)

It is also necessary to introduce a dual form of the p-form via so-called the Hodge dual: √ | | gD M ···M N N ∗ω = ω ··· ε 1 p ··· dx p+1 ∧ · · · ∧ dx D , (A.36a) p p!(D − p)! M1 Mp Np+1 ND

25 √ | | √ gD M M 1 D (∗1) = ε ··· dx 1 ∧ · · · dx D = |g | dx ∧ · · · ∧ dx = (vol.) , (A.36b) D! M1 MD D where the index “1” does not always implies the ﬁrst spatial direction; i.e., dx1 also often implies

M1···MD dt in the negative signature. Notice that εM1···MD and ε are called the invariant tensors whose property is given by

··· M1 Mn M1N1 ··· MnNn ε Mn+1···MD = g g εN1...NnMn+1···MD , (A.37a) ··· − M1M2 MD M1N1 ··· MDND 1 ε = g g εN1N2···ND = gD εN1N2···ND , (A.37b)

12···D 1 ε12···D ≡ 1 , ε = , (A.37c) gD ··· ··· M1 MD M1N1 ··· MDND N1 ND TM1···MD ε = TM1···MD g g εN1···ND = T εN1···ND . (A.37d)

The ﬁnal line is from the deﬁnition that εM1...MD is a tensor. Using the Hodge star operator and the invariant tensor, we can discuss more properties:

p(D−p)+t ∗ ∗ ωp = (−1) ωp , (A.38a)

M1 MD M1···MD 1 D dx ∧ · · · ∧ dx = gD ε dx ∧ · · · ∧ dx , (A.38b)

D 1 D 1 M M d x ≡ dx ∧ · · · ∧ dx = ε ··· dx 1 ∧ · · · ∧ dx D , (A.38c) D! M1 MD

M1···Mp Lp+1···LD Lp+1 LD gD ε N ···N · εM ···M = p!(D − p)! · δ ··· δ . (A.38d) p+1 D 1 p [Np+1 ND]

E We also introduce an inveriant tensor A1···AD in the local Lorentz (or the frame coordinate) system. A A M Introducing the vielbein one-form e = eM dx , we write down in such a way as

− EA1A2...AD A1B1 A2B2 ··· ADBD E 1E = η η η B1B2···BD = ηD B1B2···BD , (A.39a) ··· ··· ··· M1 MD M1 MD A1 ··· AD E A1 AD E T εM1···MD = T eM1 eMD A1···AD = T A1···AD , (A.39b) ··· ··· ··· M1 MD M1 ··· MD EA1 AD EA1 AD TM1···MD ε = TM1···MD EA1 EAD = TA1···AD , (A.39c)

12···D 1 t E12···D = 1 , E = = (−1) , (A.39d) ηD where η ≡ det η = (−1)t, with the number of minus sign in the signature. Furthermore, using √ D AB |ηD| = 1, we also deﬁne the followings:

(vol.) = e1 ∧ · · · ∧ eD , (A.40a)

A1 AD A1···AD 1 D e ∧ · · · ∧ e = ηD E e ∧ · · · ∧ e , (A.40b)

1 D 1 A A e ∧ · · · ∧ e = E ··· e 1 ∧ · · · ∧ e D , (A.40c) D! A1A2 AD

A1···Ap Cp+1···CD Cp+1 CD ηD E B ···B ·EA ···A = p!(D − p)! · δ ··· δ . (A.40d) p+1 D 1 p [Bp+1 BD]

26 A.10 Yang-Mills gauge ﬁelds: hermitian variables

The covariant derivatives with respect to the Yang-Mills transformation on the field ϕi in the fundamental representation, and on the field φa in the adjoint representation, are also defined as

i i i j DM (A)ϕ = ∂M ϕ − i(AM ) j ϕ fundamental representation , (A.41a)   DM (A)χ = ∂M χ − i[AM , χ] adjoint representation . (A.41b) { }a a a b c  DM (A)χ = ∂M χ + f bcAM χ

The ﬁeld strength (i.e., the curvature) is deﬁned as

[DM (A),DM (A)]ϕ = −iFMN ϕ , FMN = ∂M AN − ∂N AM − i[AM ,AN ] , (A.41c) where the gauge ﬁelds AM and the ﬁeld strength FMN are described in terms of the gauge symmetry generators Ta such as

≡ a a a a AM AM T , and FMN = FMN T , (A.42a)

† where T a is a hermitian generator of the group (T a) = T a, which also satisﬁes

a b ab a b ab c c c c tr(T T ) = δ , [T ,T ] = if c T , (Ta)b = ifba = [ad(Ta)]b , (A.42b)

[Ta, [Tb,Tc]] + [Tb, [Tc,Ta]] + [Tc, [Ta,Tb]] = 0 = fbcdfade + fcadfbde + fabdfcde , (A.42c) a a − a a b c FMN = ∂M AN ∂N AM + f bcAM AN . (A.42d)

a Due to the above commutation relation we set the structure constant f bc to be real.

Comment that the trace symbol “tr” in the above deﬁnition is in the fundamental (vector) representation. The exchanging rule between the trace tr in the SO(n) vector and the trace Tr in the SO(n) adjoint representations is given by

Tr(T 2) = (n − 2) tr(T 2) , (A.43a)

Tr(T 4) = (n − 8) tr(T 4) + 3 tr(T 2) tr(T 2) , (A.43b)

Tr(T 6) = (n − 32) tr(T 6) + 15 tr(T 2) tr(T 4) , (A.43c) where T is any linear combination of generators, but this implies the same relations for symmetrized products of different generators.

27 A.11 Yang-Mills gauge ﬁelds: anti-hermitian variables

Here we discuss another definition of the Yang-Mills fields in terms of the “anti-hermitian” gen- e erators T a. The algebra is defined as

ea † ea ea eb ab (T ) = −T , tr(T T ) = −δ , (A.44a) ea eb ab ec e c c e c [T , T ] = f c T , (Ta)b = fba = [ad(Ta)]b , (A.44b) e e e e e e e e e [Ta, [Tb, Tc]] + [Tb, [Tc, Ta]] + [Tc, [Ta, Tb]] = 0 = −ifbcdfade − ifcadfbde − ifabdfcde . (A.44c)

a Note that the structure constant f bc to be real (and to be same as the one in the previous subsec- e tion). The relation between T a and the generators T a is

a ea T = iT . (A.45) e By using this anti-hermitian generators T a we re-deﬁne the gauge ﬁelds

e ≡ a ea e e ≡ a ea e AM AM T with iAM = AM , FMN FMN T with iFMN = FMN . (A.46a) e Here we also described the relations between A and A, which is the hermitian gauge fields defined in the previous subsection. Then the covariant derivatives with respect to the Yang-Mills e transformation on the field ϕi in the fundamental representation, and on the field φe = φaT a in the adjoint representation, are also defined as

e i i e i j DM (A)ϕ = ∂M ϕ + (AM ) j ϕ fundamental representation , (A.47a)  e e  DM (A)χe = ∂M χe + [AM , χe] adjoint representation . (A.47b) { e }a a a b c  DM (A)χ = ∂M χ + f bcAM χ The ﬁeld strength (i.e., the curvature) is deﬁned as e e e e e e e e [DM (A),DM (A)]ϕ = FMN ϕ , FMN = ∂M AN − ∂N AM + [AM , AN ] , (A.47c)

a a where we should notice that the “component ﬁelds” AM and FMN are the real ﬁelds and they corresponds to the ones in the previous subsection, i.e.,

a a − a a b c FMN = ∂M AN ∂N AN + f bc AM AN . (A.47d)

A.12 Chern-Simons three-forms

Here let us introduce two kinds of the Chern-Simons three-forms, i.e., the Lorentz-Chern-Simons L Y three-form ω3 and the Yang-Mills-Chern-Simons three-form ω3 : ( ) 1 2 ωL = ωL dxM ∧ dxN ∧ dxP ≡ ωA ∧ dωB + ωA ∧ ωB ∧ ωC , (A.48a) 3 3! MNP B A 3 B C A

28 ( ) 1 2i Y Y M ∧ N ∧ P ≡ ∧ − ∧ ∧ ω3 = ωMNP dx dx dx tr A dA A A A , (A.48b) (3! ) 3 1 L AB BA 2 AB BC CA ωMNP = ω[M ∂N ωP ] + ω[M ωN ωP ] , (A.48c) 3! ( 3 ) 1 2i ωY = tr A ∂ A − A A A , (A.48d) 3! MNP [M N P ] 3 [M N P ]

A A M a a M where ω B = ωM Bdx and A = AM T dx are ths spin connection and the gauge ﬁelds which satisfy the followings

A A A C R B = dω B + ω C ∧ ω B ,F = dA − iA ∧ A. (A.49)

eY Of course the Yang-Mills Chern-Simons 3-form ω3 with respect to the anti-hermitian generators e T a can be deﬁned as ( ) e e 2 e e e e e e e ωeY ≡ tr A ∧ dA + A ∧ A ∧ A = −ωY with F = dA + A ∧ A. (A.50) 3 3 3

The exterior derivatives of these three-forms are given by

L A ∧ B ∧ dω3 = R B R A = tr(R R) , (A.51a) Y ∧ eY e ∧ e − ∧ − Y dω3 = tr(F F ) , dω3 = tr(F F ) = tr(F F ) = dω3 . (A.51b)

29 B Conventions

Here we show the redeﬁnition rules between the variables in the BdR (φ) and the ones (φ) in this note in heterotic theory:

− 1 ϕ 3 ≡ exp(−2Φ) , ω AB ≡ −ω AB , (B.1a) κ2 M M 10 √ √ e 2 A ≡ −A ,B ≡ + 2B ,H ≡ + H , (B.1b) M M MN MN MNP 3 MNP √ a a a ea χ = χ T ≡ −χ T = −χe , λ ≡ 2λ , (B.1c)

ψM , ϵ keep the same variables , (B.1d) 2 −κ10 α = β = 2 , (B.1e) g10 2 2 κ10 ′ κ10 = 2 , 2 = α . (B.1f) 2g10

30 C Geometries: complex, hermitian and K¨ahlermanifolds

C.1 Complex coordinates

In this section we discuss various conditions on differential manifolds. We start from a Rieman- nian manifold whose coordinates are given in terms of xM , where the indices M runs 1 to D = 2d. ′ ′ First we re-name the coordinates such as xd+m ≡ ym , where m, m = 1, . . . , d, the we deﬁne “(anti-)holomorphic” coordinates in the complex frame such as

1 ( ′ ) 1 ( ′ ) zm ≡ √ xm + iym , zm ≡ √ xm − iym , (C.1a) 2 2 1 ( ) ′ i ( ) xm = √ zm + zm , ym = −√ zm − zm . (C.1b) 2 2

The derivatives ∂/∂zm are given as ( ) ( ) ∂ 1 ∂ ∂ ∂ 1 ∂ ∂ = √ − i ′ , = √ + i ′ , (C.2a) ∂zm 2 ∂xm ∂ym ∂zm 2 ∂xm ∂ym ( ) ( ) ∂ 1 ∂ ∂ ∂ i ∂ ∂ = √ + , ′ = √ − . (C.2b) ∂xm 2 ∂zm ∂zm ∂ym 2 ∂zm ∂zm

C.2 Metric on complex manifold: hermitian metric

We should also re-deﬁne the metric on the geometry. The metric associated with the bosonic ′ operators xM = (xm, ym ) is given as

M N m n m n′ m′ n m′ n′ gMN dx dx = gmn dx dx + gmn′ dx dy + gm′n dy dx + gm′n′ dy dy . (C.3)

This line element is invariant under the rotation with SO(D) group. We also deﬁne the line element given in terms of the complex coordinates zm and zm:

e(z,z) M N e m n e m n gMN dZ dZ = gmn dz dz + gmn dz dz . (C.4)

This line element is invariant under the rotation with U(d) group, the subgroup of SO(2d). We should notice that this rotation group is the structure group. Now let us ﬁnd the relation between gMN in (C.3) and gemn in (C.4) via (C.1). We impose

gemn = gemn = 0 (C.5) on the metric since we assume that the geometry is a complex manifold:

′ ′ ′ ′ ∂xp ∂xq ∂xp ∂yq ∂yp ∂xq ∂yp ∂yq ge = 0 = g + g ′ + g ′ + g ′ ′ mn ∂zm ∂zn pq ∂zm ∂zn pq ∂zm ∂zn p q ∂zm ∂zn p q

31 ( ) 1 p q p q′ p′ q p′ q′ = δ δ g − iδ δ g ′ − iδ δ g ′ − δ δ g ′ ′ , (C.6a) 2 m n pq m n pq m n p q m n p q ′ ′ ′ ′ ∂xp ∂xq ∂xp ∂yq ∂yp ∂xq ∂yp ∂yq e ′ ′ ′ ′ gmn = 0 = m n gpq + m n gpq + m n gp q + m n gp q ∂z( ∂z ∂z ∂z ∂z ∂z ∂z )∂z 1 p q p q′ p′ q p′ q′ = δ δ g + iδ δ g ′ + iδ δ g ′ − δ δ g ′ ′ . (C.6b) 2 m n pq m n pq m n p q m n p q

This indicates a strong condition which is imposed on the metric

gpq − gp′q′ = 0 , gpq′ + gp′q = 0 , (C.7a)

∴ gpq = gp′q′ , gp′q = −gpq′ , gp′p = −gpp′ = 0 , (C.7b) where we used the symmetry gMN = gNM . This decomposes the original structure group SO(2d) to U(d). Notice that the local Lorentz group is not reduced. Let us further investigate it: ( ) ( ) ∂zm ∂zn ∂zm ∂zn 1 g = ge + ge = δmδn ge + δmδn ge pq ∂xp ∂xq mn ∂xp ∂xq mn 2 p q mn p q mn ( ) 1 m n m n e = δp δq + δq δp gmn , (C.8a) (2 ) m n m n ( ) ∂z ∂z ∂z ∂z 1 m n m n g ′ ′ = ge + ge = δ ′ δ ′ ge + δ ′ δ ′ ge p q ∂yp′ ∂yq′ mn ∂yp′ ∂yq′ mn 2 p q mn p q mn ( ) 1 m n m n e = δp′ δq′ + δq′ δp′ gmm , (C.8b) (2 ) m n m n ( ) ∂z ∂z ∂z ∂z i m n m n g ′ = ge + ge = − δ δ ′ ge − δ δ ′ ge pq ∂xp ∂yq′ mn ∂xp ∂yq′ mn 2 p q mn p q mn ( ) − i m n − m n e = δp δq′ δq′ δp gmn , (C.8c) ( 2 ) m n m n ( ) ∂z ∂z ∂z ∂z i m n m n g ′ = ge + ge = δ ′ δ ge − δ ′ δ ge p q ∂yp′ ∂xq mn ∂yp′ ∂xq mn 2 p q mn p q mn ( ) i m n m n = δ ′ δ − δ δ ′ ge , (C.8d) 2 p q q p mn where we used the symmetry gemn = genm. For later discussion, we deﬁne the inverse of gpq as ( ) pq p q q p emn g = δmδn + δmδn g . (C.9)

Then, the determinant of the metric g = det(gMN ) can be represented in such a way as  

′ ( )  gpq gpq  pq g ≡ det gMN = det = det gpq · det gp′q′ − gp′pg gqq′ . (C.10) gp′q gp′q′

The ﬁrst determinant is given by using (C.8): ( ( ) ) 1 m n m n e det gpq = det 2 δp δq + δq δp gmn . (C.11a)

32 Let us consider the second determinant carefully: ( ) ( ) ( ) 1 q p ′ pq ′ m n − m n e p q ers k l − k l e gp pg gqq = δp′ δq δq δp′ gmn δr δs + δr δs g δp δq′ δq′ δp gkl 4( ) 1 k l k l = δ ′ δ ′ + δ ′ δ ′ ge , (C.11b) 4 p q q p kl ( ) ( ( ) ) ∴ ′ ′ − ′ pq ′ 1 k l k l e det gp q gp pg gqq = det 4 δp′ δq′ + δq′ δp′ gkl . (C.11c)

e ≡ e Summarizing the above, we obtain the following expression in terms of g det gmN : ( ( ) ) ( ( ) ) 1 m n m n e · 1 k l k l e g = det 2 δp δq + δq δp gmn det 4 δp′ δq′ + δq′ δp′ gkl { ( )} 1 ( ) 2 1 = det 1 δmδn + δmδn ge = · ge2 . (C.12) 2d 2 p q q p mn 2d

Note that we still use the original expression g in later discussion, if there are no confusions.

C.3 General coordinate transformations of connections

It is worth mentioning the transformation rule of the afﬁne connection under the general coordinate transformation. This transformation can be, for instance, derived from the general coordinate N M N m en transformation of a tensor ∇M A from the (x ; A )-frame to the (xe ; A )-frame: ( )( ) ( ) ∂xem ∂ ∂xN ∂xP ∇ AN = ∂ AN + ΓN AP = Aen + ΓN Aep M M PM ∂xM ∂xem ∂xen PM ∂xep em N ( ) em N ( ) ∂x ∂x e en ∂x ∂x e en en ep ≡ ∇ A = ∂ A + Γ A . (C.13) ∂xM ∂xen m ∂xM ∂xen m pm

N N where A is an arbitrary contravariant vector, and the affine connection Γ PM can contain contorsion. Compared to the first line and the second line in the right-hand side, we easily find the N en relation between Γ PM and Γ pm: P em N em 2 N N ∂x ep ∂x ∂x en ep ∂x ∂ x ep Γ A = Γ A − A , (C.14a) PM ∂xep ∂xM ∂xen pm ∂xM ∂xem∂xep N ep em N 2ep N ∂x ∂x ∂x en ∂x ∂ x ∴ Γ = Γ + . (C.14b) PM ∂xen ∂xP ∂xM pm ∂xep ∂xP ∂xM

Here we also discuss the transformation rule of the spin connection from the xM -coordinate frame to the xem-coordinate frame. To do so, we should also deﬁne the vielbein and its inverse:

A B A N N M B B gMN = δAB eM eN , eM EA = δM , EA eM = δA , (C.15)

A where δAB is the orthogonal metric in the local Lorentz frame. Since the vielbein eM and its M 4 inverse EA are vectors under the general coordinate transformation , they transform in the fol- 4Notice that the local Lorentz coordinates is not transformed under the general coordinate transformations.

33 lowing way: em N A ∂x A N ∂x e n e = e ,E = E . (C.16) M ∂xM m B ∂xen B Focus on the vielbein postulate given by the following equation and its transformation:

0 = D (ω, Γ)e A = ∂ e A + ω A e B − ΓP e A (C.17a) M N (M N M)(B N ) NM P ( ) ∂xem ∂ ∂xen ∂xen = e A + ω A e B ∂xM ∂xem ∂xN n M B ∂xN n ( )( ) P en em P 2ep eq ∂x ∂x ∂x ep ∂x ∂ x ∂x A − Γ + e . (C.17b) ∂xep ∂xN ∂xM nm ∂xep ∂xN ∂xM ∂xP q The equation in the first line is applicable to any Riemannian manifold even in the presence of torsion. Here we used the transformation rule of the affine connection (C.14). This equation behaves as a vector-valued tensor under the general coordinate transformation from the xM -frame to the xem-frame: ( ) ∂xem ∂xen D (ω, Γ)e A = De (ω,e Γ)e e A M N ∂xM ∂xN m n em en ( ) ∂x ∂x e A A B ep A = ∂ e + ωe e − Γ e . (C.18) ∂xM ∂xN m n m B n nm p Comparing the above equations (C.17b) and (C.18), we find that the spin connection transforms as a vector (or a tensor) under the general coordinate transformation: ∂xem ∂xen ∂xen ∂2xen ∂2xen ωe A e B = ω A e B + e A − e A ∂xM ∂xN m B n ∂xN M B n ∂xM ∂xN n ∂xM ∂xN n ∂xen = ω A e B , (C.19a) ∂xN M B n ∂xem ∴ ω A = ωe A . (C.19b) M B ∂xM m B Notice, however, the spin connection does not behave as a vector under the SO(2d) local Lorentz transformation in the same way as the gauge transformation of the non-abelian gauge field.

To impose the condition that the manifold is hermitian (C.22) on the spin connection, we should ﬁnd the relation between the afﬁne connection and the spin connection via the vielbein postulate: ( ) P P A A B Γ NM = EA ∂M eN + ωM B eN . (C.20)

We also note that the spin connection can be described in terms of the vielbein and the afﬁne connection:

A N A P A N ωM B = −EB ∂M eN + Γ NM eP EB . (C.21)

Note that these forms are, of course, applicable in any coordinate frame.

34 C.4 Further analysis on hermitian manifold

Let us further discuss the complex manifold. Actually, the condition (C.5) is nothing but the definition of the hermitian metric. Furthermore, the complex manifold whose metric is given by the hermitian manifold is the hermitian manifold by Yano [17]. However, Nakahara [13] discusses a different definition of the hermitian manifold. We follow the Yano’s definition.

We want to use the formulation of topological invariants on a complex SU(3)-structure manifold in the presence of non-trivial torsion. In heterotic string compactification scenario, the com- pactified six-dimensional manifold is an SU(3)-structure manifold, if we impose an low energy effective theory in four-dimensional spacetime is an N = 1 supersymmetric theory. In the SU(3)- m structure manifold affine connections with non-trivial values are of pure type Γ np and of mixed m type Γ np, and their complex conjugates. This condition is nothing but the condition on a hermitian manifold (for the definitions of the hermitian manifold, see [17], not [13]; for the discussions of the SU(3)-structure manifold, see [16]). Let us impose that the holomorphic covariant derivative along the holomorphic tangent vector should keep the holomorphicity:

e ∂ ep ∂ e ∂ ep ∂ ∇ = Γ , ∇ = Γ , (C.22a) m ∂zn nm ∂zp m ∂zn nm ∂zp e ∂ ep ∂ e ∂ ep ∂ ∇ = Γ , ∇ = Γ , (C.22b) m ∂zn nm ∂zp m ∂zn nm ∂zp ep ep ep ep Γ nm = 0 , Γ nm = 0 , Γ nm = 0 , Γ nm = 0 , (C.22c) where the above afﬁne connections, called the hermitian connections, can contain torsion. Actu- ally, the vanishing afﬁne connection reduces the structure group (or the holonomy group) from e n SO(2d) to U(d) (or SU(d)). Note that ∇m∂/∂z behaves as a tensor. This hermitian connection itself is given by the metricity condition:

e eq eq 0 = ∇mgenp = ∂mgenp − Γ nmgeqp − Γ pmgenq . (C.23)

The Riemann tensor associated with the hermitian connection is also restricted compared to the one on a generic Riemannian manifold. The deﬁnition of the Riemann tensor on a generic Rie- mannian manifold is (see also (A.13))

P P − P P R − P R R QMN (Γ) = ∂M Γ QN ∂N Γ QM + Γ RM Γ QN Γ RN Γ QM , (C.24)

P where the afﬁne connection Γ MN can contain torsion. Applying the hermitian condition on the afﬁne connection (C.22) to the Riemann tensor, the expression becomes quite simple:

ep e ep − ep ep eR − ep eR − ep e R qMN (Γ) = ∂M Γ qN ∂N Γ qM + Γ RM Γ qN Γ RN Γ qM = R qNM (Γ) , (C.25a)

35 ep e ep − ep ep eR − ep eR R qMN (Γ) = ∂M Γ qN ∂N Γ qM + Γ RM Γ qN Γ RN Γ qM = 0 , (C.25b) ep e ep − ep ep eR − ep eR R qMN (Γ) = ∂M Γ qN ∂N Γ qM + Γ RM Γ qN Γ RN Γ qM = 0 , (C.25c) ep e ep − ep ep eR − ep eR − ep e R qMN (Γ) = ∂M Γ qN ∂N Γ qM + Γ RM Γ qN Γ RN Γ qM = R qNM (Γ) , (C.25d) where the capital indices M and N run both holomorphic and anti-holomorphic directions.

The hermitian connection is decomposed into the symmetric and the anti-symmetric hermitian connections, whose explict forms are, for instance, given as

em em em Γ np = Γ (np) + Γ [np] , (C.26a) ( ) ( ) em 1 em em m em 1 em em Γ ≡ Γ + Γ ,T ≡ Γ = Γ − Γ . (C.26b) (np) 2 np pn np [np] 2 np pn

Furthermore, we can decompose the symmetric part of the hermitian connection into the Levi- 5 e Civita connection Γ(L) and the terms depending on the torsion: ( ) em em − m m Γ (np) = Γ(L)np Tn p + Tp n , (C.27a) ∴ em em m m ≡ m − m − m Γ np = Γ(L)np + K np ,K np T np Tn p Tp n , (C.27b)

m where K np is called the contorsion. Here let us explicitly list up all components:

em em m em em m Γ np = Γ(L)np + K np , Γ np = Γ(L)np + K np , (C.28a) em em m em em m Γ np = Γ(L)np + K np , Γ np = Γ(L)np + K np , (C.28b) em em m em em m 0 = Γ np = Γ(L)np + K np , 0 = Γ np = Γ(L)np + K np , (C.28c) em em m em em m 0 = Γ np = Γ(L)np + K np , 0 = Γ np = Γ(L)np + K np . (C.28d)

e It is worth discussing the explicit forms of the Levi-Civita connection Γ(L) on the hermitian manifold: ( ) ( ) 1 1 Γem = gemq ∂ ge + ∂ ge = Γem , Γem = gemq ∂ ge + ∂ ge = Γem , (L)np 2 n qp p nq (L)pn (L)np 2 n qp p nq (L)pn (C.29a)

em em Γ(L)np = 0 , Γ(L)np = 0 , (C.29b) ( ) ( ) em 1 mq em em 1 mq em Γ = ge ∂ ge − ∂ ge = Γ , Γ = ge ∂ ge − ∂ ge = Γ . (C.29c) (L)np 2 p nq q np (L)pn (L)np 2 p nq q np (L)pn

em em Here we should point out that the Levi-Civita connections Γ(L)np and Γ(L)np have non-trivial val- em em ues whereas the hermitian connections of same type vanish: Γ np = Γ np = 0, see (C.22). This

5Strictly speaking, the Levi-Civita connection is deﬁned on a torsionless manifold. Due to this, the metric in the Levi-Civita connection should be given as the metric on the torsionless geometry.

36 indicates that the contorsion of certain types are equal to the Levi-Civita connections in the vanishing hermitian connections: ( ) em em m ∴ m −em −1emq e − e 0 = Γ np = Γ(L)np + K np , K np = Γ(L)np = g ∂ngpq ∂qgpn , (C.30a) 2 ( ) em em m m em 1 mq 0 = Γ = Γ + K , ∴ K = −Γ = − ge ∂ ge − ∂ ge . (C.30b) np (L)np np np (L)np 2 n pq q pn

eP e We should notice that the Riemann tensor R QMN (Γ(L)) should have same properties as the ones P on the Riemann tensor R QMN (Γ0) on a generic real manifold such as

e e e e e e e e RP QMN (Γ(L)) = −RP QNM (Γ(L)) = −RQP MN (Γ(L)) = RMNPQ(Γ(L)) , (C.31a) e e ∇e eN e RP [QMN](Γ(L)) = 0 , [M R|P |QR](Γ(L)) = 0 , (C.31b) where the equations (C.31b) follow the identities on the Riemann tensor associated with the Levi- Civita connection on a generic real manifold (A.19b) and (A.20). Let us further investigate to ﬁnd vanishing components (see, for detail, in appendix A.6):

e e e e e e e e Rpqmn(Γ(L)) = −Rpqnm(Γ(L)) = −Rqpmn(Γ(L)) = Rmnpq(Γ(L)) = 0 , (C.31c) e e e e e e e e Rpqmn(Γ(L)) = −Rpqnm(Γ(L)) = −Rqpmn(Γ(L)) = Rmnpq(Γ(L)) = 0 . (C.31d)

Let us also analyze the reduction of the spin connection. To do so, let us deﬁne the vielbein and its inverse in the complex coordinate frame:

A B A B A B gemn ≡ δAB em en , gemn = 0 = δABem en , gemn = 0 = δABem en , (C.32a) e A e n n e A e n n A e A e p e A e p em EA = δm , em EA = δm , δB = ep EB + ep EB . (C.32b)

Furthermore, since the local Lorentz coordinates coincide with the curved space coordinates in the ﬂat limit, we can set

δab = δba , δab = 0 = δab . (C.32c)

Here we analyze the degrees of freedom of the metrics geMN . First, without any constraints, it 2 2 has (2d) = 4d degrees of freedom. Using the constraints gemn = 0 = gemn, it is reduced to 2 2 2 2 4d − 2 × d = 2d . Furthermore, the symmetries gemn = genm halves the degrees to d , which coincides with the physical degrees of freedom of gemn. The same reduction is also applied to the 2 A metric δAB, which gives d degrees of freedom on δab. The vielbein eM should carries the same number of degrees of freedom as the metrics geMN and δAB, which we will consider. Originally, if A 2 2 there are no constraints, eM has (2d) = 4d degrees. The constraints

e e ae b e ae b e e ae b e ae b 0 = gmn = δab em en + δab em en , 0 = gmn = δab em en + δab em en (C.32d)

37 × 2 e impose 2 d constraints on the vielbein. (The symmetric conditions on gmn and δab do not yield any constraints.) Then, the vielbein has only 4d2 − 2d2 = 2d2 degrees of freeom, which coincides e 2 with the metrics gmn and δab. To realize such a reduction, we set 2d components of the vielbein to be zero. Although, there are actually many ways to do it, the following setting is much useful:

a a em ≡ 0 , em ≡ 0 . (C.32e)

We can conﬁrm the matching of degrees of freedom of the metric and the vielbein. The metric 2 2 has 2 × d degrees of freedom (i.e., both gemn and gemn have d degrees individually), while the a a 2 non-trivial vielbein components {em , en } have 2 × d degrees of freedom. Later we will use the ﬁxing condition (C.32e).

By using the vielbein and its inverse and (C.20), we ﬁnd the relation between the hermitian connection and the spin connection. The non-vanishing hermitian connection give rise to the equations: ( ) ( ) ep e p e a e a e b ep e p e a e a e b Γ mn = Ea ∂nem + ωn b em , Γ mn = Ea ∂nem + ωn b em , (C.33a) ( ) ( ) ep e p e a e a e b ep e p e a e a e b Γ mn = Ea ∂nem + ωn b em , Γ mn = Ea ∂nem + ωn b em , (C.33b) while the vanshing condition of the hermitian connection gives ( ) ep e p e A e A e B e a e b e p 0 = Γ mn = EA ∂nem + ωn B em = ωn b em Ea , (C.33c) ( ) ep e p A A B a b e p 0 = Γ mn = EA ∂nem + ωen B em = ωen b em Ea , (C.33d) ( ) ep e p A A B a b e p 0 = Γ mn = EA ∂nem + ωen B em = ωen b em Ea , (C.33e) ( ) ep e p e A e A e B e a e b e p 0 = Γ mn = EA ∂nem + ωn B em = ωn b em Ea . (C.33f)

Equivalently, we can also discuss the spin connection via (C.21): ( ) a e n a ep a ωem b = −Eb ∂men − Γ nm ep , (C.34a) ( ) e a − e n e a − ep e a ωm b = Eb ∂men Γ nm ep , (C.34b) ( ) a e n a ep a ωem b = −Eb ∂men − Γ nm ep , (C.34c) ( ) e a − e n e a − ep e a ωm b = Eb ∂men Γ nm ep , (C.34d) ( ) a e N a ep a ep a e n ωem b = −Eb ∂meN − Γ Nm ep = Γ nm ep Eb = 0 , (C.34e) ( ) a e N a ep a ep a e n ωem b = −Eb ∂meN − Γ Nm ep = Γ nm ep Eb = 0 , (C.34f) ( ) e a − e N e a − ep e a ep e a e n ωm b = Eb ∂meN Γ Nm ep = Γ nm ep Eb = 0 , (C.34g)

38 ( ) e a − e N e a − ep e a ep e a e n ωm b = Eb ∂meN Γ Nm ep = Γ nm ep Eb = 0 . (C.34h)

e e This indicates that the hermitian spin connections of “pure type” such as ωMab and ωMab vanish on the hermitian manifold. Later we will analyze the spin connection more. The Riemann tensor of the spin connection is given in the following way:

i e AB [D (ωe),D (ωe)] = − R (ωe)Σ , (C.35a) M N 2 ABMN eA A A A C A C R BMN (ωe) = ∂M ωeN B − ∂N ωeM B + ωeM C ωeN B − ωeN C ωeM B . (C.35b)

Here let us explicitly describe the Riemann tensor:

e c c RabMN (ωe) = ∂M ωeNab − ∂N ωeMab + ωeMac ωeN b − ωeNac ωeM b e e = −RabNM (ωe) = −RbaMN (ωe) , (C.35c) e e e − e e e c − e e c RabMN (ω) = ∂M ωNab ∂N ωMab + ωMac ωN b ωNac ωM b − e e − e e = RabNM (ω) = RbaMN (ω) , (C.35d) e e e e RabMN (ω) = 0 , RabMN (ω) = 0 . (C.35e)

Notice that the Riemann tensor is antisymmetric under the exchange between the latter two indices by deﬁnition. We also notice that it is also antisymmetric under the exchange between the former two indices via the antisymmetry of the spin connection ωeMAB = −ωeMBA (A.29).

Here let us again discuss the properties on the Riemann tensor of the hermitian afﬁne connec- e e e e tion RpqMN (Γ) given in (C.25), i.e., tt is worth investigating whether the Riemann tensor RpqMN (Γ) has (anti)symmetries under exchanging of indices. Notice that since the hermitian manifold has, in general, a torsion, then the Riemann tensor might not has all the properties in (A.19b). For- tunately, however, it is related to the Riemann tensor of the spin connection in such a way as e e e RpqMN (Γ) = RabMN (ωe) epaeqb. Then we derive the followings:

e e e e e RpqMN (Γ) = RabMN (ωe) epaeqb = −RabNM (ωe) epaeqb = RbaMN (ωe) epaeqb e e e e = −RpqNM (Γ) = −RqpMN (Γ) . (C.36)

In the same way as the afﬁne connection (C.28), the spin connection ωeMAB should also be decomposed into the Levi-Civita part and the contorsion part via (C.34): ( { } ) e − e n e − ep p e ≡ e(L) ωmab = Eb ∂mena Γ(L)nm + K nm epa ωmab + Kabm , (C.37a) ( { } ) ωe = −Ee n ∂ e − Γep + Kp e ≡ ωe(L) + K , mab b m na (L)nm nm pa mab abm (C.37b) ( { } ) ωe = −Ee n ∂ e − Γep + Kp e ≡ ωe(L) + K , mab b m na (L)nm nm pa mab abm (C.37c)

39 ( { } ) e − e n e − ep p e ≡ e(L) ωmab = Eb ∂mena Γ(L)nm + K nm epa ωmab + Kabm , (C.37d) { } e ep p e e n 0 = ωmab = Γ(L)nm + K nm epaEb = Kabm , (C.37e) { } e ep p e e n ≡ e(L) 0 = ωmab = Γ(L)nm + K nm epaEb ωmab + Kabm , (C.37f) { } e ep p e e n ≡ e(L) 0 = ωmab = Γ (L)nm + K nm epaEb ω + Kabm , (C.37g) { } mab e ep p e e n 0 = ωmab = Γ(L)nm + K nm epaEb = Kabm . (C.37h)

C.5 Additional constraint: K¨ahlermanifold

So far, we have imposed the manifold is hermitian. Now we also introduce the Kahler¨ form Ω

m n m n Ω ≡ igemn dz ∧ dz = Jmn dz ∧ dz ≡ J, (C.38)

p n where the component of this two-form Jmn = Jm gepn is given by the complex structure Jm [13]. Due to this relation, the fundamental two-form J is also interpreted as the Kahler¨ form. The closed condition of the Kahler¨ form, dΩ = 0, is the deﬁnition of the Kahler¨ manifold. This constraint is equivalent to a constraint on the hermitian metric:

dΩ = 0 ↔ ∂mgenp = ∂ngemp and ∂mgenp = ∂pgenm . (C.39)

This is called the Kahler¨ metric. If the metric is Kahler¨ (or equivalently, if there is no (con)torsion), the hermitian connection (C.22) exactly coincides with the Levi-Civita connection of pure type, i.e., the afﬁne connection of mixed type vanishes and that the exchange of the two subscripts of the hermitian connection becomes symmetric, see (C.29):

eM eM M eM Γ NP = Γ0NP + K NP = Γ0NP , (C.40a) em em emq e em em emq e Γ0np = Γ0pn = g ∂ngpq , Γ0np = Γ0pn = g ∂ngqp , (C.40b) em em em em Γ0np = Γ0np = Γ0np = Γ0np = 0 . (C.40c)

This indicates that the metric on the hermitian manifold is affected by the torsion. Actually, the em Levi-Civita connection Γ(L)np on the hermitian manifold vanishes when the torsion disappears on the Kahler¨ manifold. This is nothing but the evidence of the torsion back reaction on the metric! In addition, when we go back to the real coordinate frame, we ﬁnd the equality conditions

P P P P Γ nm = −Γ n′m′ , Γ n′m = Γ nm′ . (C.41)

Futhermore, the closed condition dΩ = 0 also indicates that there is a conserved charge and we can introduce a Kahler¨ potential K(z, z), which yields the metric satisfying the relation (C.39):

gemn ≡ ∂m∂nK(z, z) . (C.42)

40 em Note that a hermitian manifold is a Kahler¨ manifold if and only if the hermitian connection Γ np is torsion free (see p.180 of [12]). Since the hermitian connection on the Kahler¨ manifold is the em Kahler¨ Levi-Civita connection Γ0np (C.40), the Riemann tensor on the Kahler¨ manifold is more symmetric than the one on a generic hermitian manifold. Compared to (C.25), we can see ( ) ( ) em e − em − emr e − emr e em e R npq(Γ0) = ∂qΓ0np = ∂q g ∂pgnr = ∂q g ∂ngpr = R pnq(Γ0) . (C.43a)

Combining it with the properties (C.25), we also obtain

e e e e e e e e Rmnpq(Γ0) = −Rmnqp(Γ0) = −Rnmpq(Γ0) = Rnmqp(Γ0) , (C.43b) em e em e em e em e em e em e R npq(Γ0) = R pnq(Γ0) , R npq(Γ0) = R qpn(Γ0) , R nqp(Γ0) = R pqn(Γ0) . (C.43c)

The spin connection on the Kahler¨ manifold is also more restricted than the one on the hermitian manifold, i.e., the torsionless spin connection ωe0 associated with the Kahler¨ afﬁne connection (C.40) via the relation (C.17a):

e e e a e a − ep e a e a e b 0 = Dm(ω0, Γ0)en = ∂men Γ0nm ep + ω0m b en , (C.44a) e e e a e a e a e b 0 = Dm(ω0, Γ0)en = ∂men + ω0m b en , (C.44b) e a a a b 0 = Dm(ωe0, Γ0)en = ∂men + ωe0m b en , (C.44c) e e e a e a − ep e a e a e b 0 = Dm(ω0, Γ0)en = ∂men Γ0nm ep + ω0m b en . (C.44d)

First, the explicit forms of the spin connection are given from (C.44):

e a − e n e a ep e a e n e a − e n e a ω0m b = Eb ∂men + Γ0nm ep Eb , ω0m b = Eb ∂men , (C.45a) e a − e n e a e a − e n e a ep e a e n ω0m b = Eb ∂men , ω0m b = Eb ∂men + Γ0nm ep Eb . (C.45b)

These are related to each other via the antisymmetry ωeMAB = −ωeMBA in the following way: ( ) ( ) e a − e n e a ep e a e n − e n e eqa epq e e a e n ω0m b = Eb ∂men + Γ0nm ep Eb = Eb ∂m gnq e + g ∂mgnq ep Eb ( ) e n qa e qa qa e n = −Eb ∂mgenq e − Ebq∂me + ∂mgenq e Eb eqa e eaq e ac e q e d = e ∂mEbq = E ∂meqb = δ δbd Ec ∂meq = −δacδ ωe d , (C.46a) bd 0m c ( ) ( ) e a − e n e a ep e a e n − e n e eqa epq e e a e n ω0m b = Eb ∂men + Γ0nm ep Eb = Eb ∂m gnqe + g ∂mgnq ep Eb ( ) − e n e eqa − e eqa e eqa e n = Eb ∂mgnq e Ebq∂me + ∂mgnq e Eb eqa e eaq e ac e q e d = e ∂mEbq = E ∂meqb = δ δbd Ec ∂meq − ac e d = δ δbd ω0m c , (C.46b)

41 e where we used Ebq = eqb given by (A.24). Thus, the following forms are much useful to analyze the Riemann tensor:

e a −e a − ac e d eaq e ω0m b = ω0mb = δbdδ ω0m c = E ∂meqb , (C.47a) e a −e a − ac e d eaq e ω0m b = ω0mb = δ δbd ω0m c = E ∂meqb . (C.47b)

Following these expressions, it turns out the following: ( )( ) ( )( ) a c eap ecq eap ecq 2ωe0[m |c| ωe0n] b = E ∂mepc E ∂neqb − E ∂nepc E ∂meqb ( ) eaq eaq eaq = ∂m E ∂neqb − 2∂mE ∂neqb − E ∂m∂neqb ( ) eaq eaq eaq − ∂n E ∂meqb + 2∂nE ∂meqb + E ∂m∂neqb ( ) ( ) eaq eaq eaq = ∂m E ∂neqb − 2∂n eqb∂mE + 2eqb∂m∂nE ( ) ( ) eaq eaq eaq − ∂n E ∂meqb + 2∂m eqb∂nE − 2eqb∂m∂nE ( ) ( ) eaq eaq a a = −∂m E ∂neqb + ∂n E ∂meqb = −∂mωe0n b + ∂nωe0m b

a = −2∂[mωe0n] b . (C.48)

By using (C.48), we obtain the following remarkable property on the Riemann tensor of the spin connection as well as the one of the afﬁne connection (C.43):

ea a a c R bmn(ωe0) = 2∂[mωe0n] b + 2ωe0[m |c| ωe0n] b = 0 , (C.49a) ea e ea e ea e R bmn(ω0) = 0 , R bmn(ω0) = 0 , R bmn(ω0) = 0 . (C.49b)

Furthermore, since the Riemann tensor of the spin connection is related to the Riemann tensor of the afﬁne connection, we ﬁnd that the non-trivial component of the Riemann tensor is given as

e e e p e q e e Rabmn(ω0) = Ea Eb Rpqmn(Γ0) . (C.49c)

42 D Introducing torsion

We have already known that the NS-NS three-form ﬂux H plays as a totally anti-symmetric contorsion on a compactiﬁed six-dimensional manifold in heterotic string theory [9]. In particular, this manifold is a conformally balanced manifold whose contorsion is given as the Bismut torsion: i ( ) 1 H = ∂ − ∂ J = − dcJ,J = ige dzm ∧ dzn , (D.1) 2 2 mn where J is the fundamental two-form on the manifold, whose component is given by the complex P structure satisfying the covariantly constant condition DM (Γ−)JN = 0, where Γ− = Γ(L) − H. Notice that the fundamental two-form is not closed dJ ≠ 0. See for the detail in appendix A.7. (2,1) i Due to this, the three-form is decomposed into two parts; the (2,1)-form H = 2 ∂J and the (1,2) − i (1, 2)-form H = 2 ∂J, whose components are given as i i H(2,1) = ∂J = ∂ J dzm ∧ dzn ∧ dzp , (D.2a) 2 2 m np (1,2) i i m n p H = − ∂J = − ∂mJnp dz ∧ dz ∧ dz , (D.2b) (2 2 ) ( ) i 1 Hmnp = ∂mJnp − ∂nJmp = − ∂mgenp − ∂ngemp , (D.2c) 2 ( ) 2 ( ) i 1 H = − ∂ J − ∂ J = − ∂ ge − ∂ ge . (D.2d) mnp 2 m np n mp 2 m np n mp

Furthermore, the exterior derivative of the NS-NS three-form is given as i i i dH = (∂ + ∂) (∂ − ∂)J = − ∂∂J + ∂∂J . (D.3) 2 2 2

Then we ﬁnd that dH is (2, 2)-form and there is neither (3, 1)- nor (1, 3)-form in the expansion. Notice that ∂ commutes with ∂ if and only if the manifold is Kahler,¨ which is not the present case. In addition, the NS-NS three-form (D.1) can also be given as 1 1 H(2,1) = H dzm ∧ dzp ∧ dzq ,H(1,2) = H dzm ∧ dzp ∧ dzq . (D.4) 2 mpq 2 mpq

Actually, the NS-NS three-form H is the totally antisymmetric torsion, as well as the totally antisymmetric contorsion on the hermitian manifold (C.26b) with different sign; i.e., K = −H. The difference of sign is from the deﬁnition of the complex structure in the supergravity [11] in such a P way as 0 = DM (Γ−)JN . Following to (C.28) and (C.37), we again describe the hermitian afﬁne connection and the spin connection: ( ) ( ) em ≡ em − m 1emq e e 1emq e − e Γ−np Γ(L)np H np = g ∂ngpq + ∂pgnq + g ∂ngpq ∂pgnq , (D.5a) 2 ( ) 2 ( ) em em m 1 mq 1 mq Γ− ≡ Γ − H = ge ∂ ge + ∂ ge + ge ∂ ge − ∂ ge , (D.5b) np (L)np np 2 n pq p nq 2 n pq p nq

43 ( ) ( ) em ≡ em − m 1emq e − e − 1emq e − e Γ−np Γ(L)np H np = g ∂pgnq ∂qgnp g ∂qgpn ∂pgqn , (D.5c) 2 ( ) 2 ( ) em em m 1 mq 1 mq Γ− ≡ Γ − H = ge ∂ ge − ∂ ge − ge ∂ ge − ∂ ge , (D.5d) np (L)np np 2 p nq q np 2 q pn p qn em em − m − 0 = Γ−np = Γ(L)np H np = 0 0 , (D.5e) em em − m − 0 = Γ−np = Γ(L)np H np = 0 0 , (D.5f) ( ) ( ) em em − m 1emq e − e − 1emq e − e 0 = Γ−np = Γ(L)np H np = g ∂ngqp ∂qgnp g ∂ngqp ∂qgnp , (D.5g) 2 ( ) 2 ( ) em em − m 1emq e − e − 1emq e − e 0 = Γ−np = Γ(L)np H np = g ∂ngqp ∂qgnp g ∂ngqp ∂qgnp , (D.5h) ( 2 ) 2 − { } ωe( ) = −Ee n ∂ e − Γep − Hp e ≡ ωe(L) − H , mab b m na (L)nm nm pa mab mab (D.5i) ( { } ) e(−) − e n e − ep − p e ≡ e(L) − ωmab = Eb ∂mena Γ H nm epa ωmab Hmab , (D.5j) ( (L)nm ) − { } ωe( ) = −Ee n ∂ e − Γep − Hp e ≡ ωe(L) − H , mab b m na (L)nm nm pa mab mab (D.5k) ( { } ) e(−) − e n e − ep − p e ≡ e(L) − ωmab = Eb ∂mena Γ(L)nm H nm epa ωmab Hmab , (D.5l) { } e(−) ep − p e e n − 0 = ωmab = Γ(L)nm H nm epaEb = 0 0 , (D.5m) − { } 0 = ωe( ) = Γep − Hp e Ee n = 0 − 0 , mab (L)nm nm pa b (D.5n) { } e(−) ep − p e e n ≡ e(L) − 0 = ωmab = Γ(L)nm H nm epaEb ωmab Hmab , (D.5o) − { } 0 = ωe( ) = Γep − Hp e Ee n ≡ ωe(L) − H . mab (L)nm nm pa b mab mab (D.5p)

Here we used (C.29) and (D.2). Notice that, under a certain transformation, the connection is not a tensor, while the torsion is. These are easily applied to more genetric spin connection such as

em em m em em m Γ(α)np = Γ(L)np + αH np , Γ(α)np = Γ(L)np + αH np , (D.6a) em em m em em m Γ(α)np = Γ(L)np + αH np , Γ(α)np = Γ(L)np + αH np , (D.6b) em em Γ(α)np = 0 , Γ(α)np = 0 , (D.6c) em m em m Γ(α)np = (1 + α)H np , Γ(α)np = (1 + α)H np , (D.6d) − ωe(α) = ωe(L) + αH = ωe( ) + (1 + α)H , mab mab mab mab mab (D.6e) e(α) e(L) e(−) ωmab = ωmab + αHmab = ωmab + (1 + α)Hmab , (D.6f) − ωe(α) = ωe(L) + αH = ωe( ) + (1 + α)H , mab mab mab mab mab (D.6g) e(α) e(L) e(−) ωmab = ωmab + αHmab = ωmab + (1 + α)Hmab , (D.6h) ωe(α) = 0 , ωe(α) = 0 , mab mab (D.6i) e(α) e(L) ωmab = ωmab + αHmab = (1 + α)Hmab , (D.6j) ωe(α) = ωe(L) + αH = (1 + α)H , mab mab mab mab (D.6k)

44 where the factor α indicates the relation between the contorsion and the NS-NS three-form via K = αH. We should again notice that, under a certain transformation, the connection is not a tensor, while the torsion is. These appear in the equations of motion for fermions in (heterotic) − e(−) − supergravity when α = 1/3, while ωMAB (i.e., α = 1), which is nothing but the hermitian spin connection with contorsion K = −H, appears in the supersymmetry variation of the gravitino in the same supergravity system [11]. Notice that only at the point α = 0, we should set the eqs. (D.6a) are reduced to the Kahler¨ afﬁne connection, while the eqs. (D.6b) disappear, i.e., we should {em em e(α)} {em e } set Γ(L)np, Γ(L)np,H, ω = Γ0np, 0, 0, ω0 , because the manifold becomes Kahler,¨ whose metric satisﬁes (C.39). This point is isolated from the continuous one-parameter line α.

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