Supergravities in ten and eleven dimensions TK …tetsuji.kimura/NOTEs/various...1 Strategy We start from the eleven-dimensional supergravity. Along various kinds of duality transforma-tions,

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

[email protected]

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 transforma-

tions, we connect five significant supergravity Lagrangian including fermions. We also inves-

tigate 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 Kahler manifolds 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 =

∫d11xL , (2.1a)

2κ211 L = β0√−g11R(e, ω)−

1

2

√−g11ΨMΓMNPDN [12(ω + ω)]ΨP − 1

48

√−g11 FMNPQF

MNPQ

+1

2β1√−g11ΨM ΓMNPQRSΨN (F + F )PQRS

+ β2 εMNPQRSUVWXY FMNPQ FRSUV CWXY , (2.1b)

where ωMAB 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

ωMAB = ωMAB , FMNPQ = FMNPQ , (2.1c)

where ωMAB is a torsionless spin connection. We should also notice that εMNPQRSUWXY is not

the invariant tensor but the “antisymmetric symbol”, whose normalization is given as ε012··· = 1.

This is related to the tensor εMNPQRSUWXY as

εMNPQRSUVWXY = g11 εMNPQRSUVWXY . (2.1d)

The transformation rule under the local supersymmetry is given as

δeMA =

1

2εΓAΨM , δCMNP = α2 εΓ[MNΨP ] , (2.1e)

δΨM = 2DM (ω)ε+ 2TMNPQR εFNPQR . (2.1f)

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

ΨM = iΨ†MΓ0 , DM (ω) = ∂M − i

2ωMABΣ

AB , (2.1g)

ΓMNPQRS = ΓMNPQRS + 12gM [PΓQRgS]N , (2.1h)

TMNPQR =

α1

2

(ΓM

NPQR − 8δ[NM Γ

PQR]). (2.1i)

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:

β2 = −2α1β1α2

=2

216β1 = − 1

144α1 , α1β1 = − 1

96 · 288β0. (2.2a)

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:

α1 =1

144, α2 = −3

2, β1 = − 1

192, β2 = − 1

(144)2. (2.2b)

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

2κ211 L =√−g R(e, ω)− 2

√−gΨMΓMNPDN (ω)ΨP − 1

48

√−g FMNPQF

MNPQ

− 1

48

√−gΨM ΓMNPQRSΨN FPQRS

− 1

(144)2εMNPQRSUVWXY FMNPQ FRSUV CWXY . (2.4a)

The transformation rule under the local supersymmetry is given as

δeMA = εΓAΨM , δCMNP = −3εΓ[MNΨP ] , (2.5a)

δΨM = DM (ω)ε+ TMNPQR εFNPQR . (2.5b)

In addition, the combination of the gamma matrices TMNPQR is rewritten in the following way:

TMNPQR =

1

288

(ΓM

NPQR − 8δ[NM Γ

PQR]). (2.5c)

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:

Sboson =1

2κ211

∫d11x

√−g

(R(e, ω)− 1

48FMNPQF

MNPQ

)

− 1

2κ211(144)2

∫d11x εMNPQRSUVWXY FMNPQFRSTUCWXY . (2.6)

Notice that the symbol εMNPQRSUWXY 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

(144)2

∫d11x εMNPQRSUVWXY FMNPQFRSUV CWXY

= − 1

(144)2

∫d11x g εMNPQRSUVWXY FMNPQFRSUV CWXY

6

= − 1

(144)2

∫dxM ∧ dxN ∧ · · · ∧ dxY FMNPQFRSUV CWXY

= − 4!4!3!

(144)2

∫F4 ∧ F4 ∧ C3 = −1

6

∫F4 ∧ F4 ∧ C3 . (2.7)

Then, the bosonic action is simplified in the following way:

Sboson =1

2κ211

∫ ((vol.)

R(e, ω)− 1

2|F4 |2

− 1

6F4 ∧ F4 ∧ C3

). (2.8)

7

3 Type IIA supergravity in ten dimensions

In this section let us derive the type IIA supergravity from the eleven-dimensional supergrav-

ity. 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

gMN = e−23ΦgMN + e

43ΦAMAN , CMNP = CMNP , (3.1a)

gM = e43ΦAM , CMN =

2

3BMN , g = exp

(43Φ). (3.1b)

Note that the objects with ˆ denote the ones in eleven-dimensions, while the others in ten di-

mensions. In addition, the symbol indicates the eleventh direction in the eleven-dimensional

spacetime. In the same way, the latter rule is also given by

gMN =(g) 1

2

(gMN −

gM gNg

), CMNP = CMNP , (3.2a)

AM =gM

g, BMN =

3

2CMN , Φ =

3

4log(g). (3.2b)

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

gMN = gMN − 1

g99

(gM9g9N +B

(1)M9B

(1)9N

), g9M =

B(1)9M

g99, g99 =

1

g99, (4.1a)

C9MN =2

3

(B

(2)MN +

2B(2)9[MgN ]9

g99

), (4.1b)

CMNP =8

3C9MNP + εijB

(i)9[MB

(j)NP ] +

εijB(i)9[MB

(j)|9|NgP ]9

g99, (4.1c)

BMN = B(1)MN +

2B(1)9[MgN ]9

g99, B9M =

g9Mg99

, (4.1d)

AM = −B(2)9M + CB

(1)9M , A9 = C , Φ = Φ− 1

2log(g99). (4.1e)

Note that the objects with ˆ denote the ones in type IIA, while the others in type IIB. The two dif-

ferent two-form fields B(i)MN implies B(1)

MN = BNSNSMN and B(2)

MN = CRRMN , respectively. We normalize

ε12 = 1. In the same way, the latter rule is also given by

gMN = gMN − 1

g99

(gM9g9N + BM9B9N

), g9M =

B9M

g99, g99 =

1

g99, (4.2a)

C9MNP =3

8

(CMNP − A[M BNP ] +

g9[M BNP ]A9

g99− 3

2

g9[M CNP ]9

g99

), (4.2b)

B(1)MN = BMN +

2g9[M BN ]9

g99, B

(1)9M =

g9Mg99

, (4.2c)

B(2)MN =

3

2CMN9 − 2A[M BN ]9 +

2g9[M BN ]9A9

g99, B

(2)9M = −AM +

A9g9Mg99

, (4.2d)

Φ = Φ− 1

2log(g99), C = A9 . (4.2e)

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

The type I supergravity contains the vielbein eMA, the dilaton Φ, the one-form anti-hermitian

gauge field A1 , and the Ramond-Ramond two-form field B(2)MN = 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

obtained from the type IIB supergravity with truncating the fieldsB(1)MN , C and CMNPQ, which are

not invariant under the Ω-projection, i.e., the worldsheet parity projection. Further, we newly in-

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:

F3 → F ′3 = dC2 − κ210

g210ωY3 , ωY

3 ≡ tr(A ∧ dA+

2

3A ∧ A ∧ A

), (5.1)

where g10 and ωY3 are called the Yang-Mills coupling and the Yang-Mills Chern-Simons three-

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 su-

pergravity [8], which will appear in the next section. Notice that the Ramond-Ramond three-form

F ′3 should be further modified via the Green-Schwarz anomaly cancellation mechanism.

1The 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

The ten-dimensional heterotic supergravity contains the vielbein eMA, the NS-NS two-form field

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]:

gMN = e−ΦgMN , Φ = −Φ , (6.1a)

F ′3 = H ′

3 , 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 H ′3 is also modified by introducing the

Yang-Mills Chern-Simons three-form ωY3 :

H ′3 = dB2 − κ210

g210ωY3 , ωY

3 ≡ tr(A ∧ dA+

2

3A ∧ A ∧ A

). (6.2)

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′:

Ltotal = L (R) + L (F 2) + L (R2) , (7.1a)

2κ210L (R) =√−g e−2Φ

[R(ω)− 1

12HMNPH

MNP + 4(∇MΦ)2

− ψMΓMNPDN (ω)ψP + 8λΓMNDM (ω)ψN + 16λ /D(ω)λ

+ 8ψMΓNΓMλ (∇NΦ)− 2ψMΓMψN (∇NΦ)

− 1

24HPQR

ψMΓ[MΓPQRΓ

N ]ψN + 8ψMΓMPQRλ− 16λΓPQRλ

+

1

48ψMΓABCψM

2λΓABCλ+ λΓABCΓ

NψN

− 1

4ψNΓABCψN − 1

8ψNΓNΓABCΓ

PψP

], (7.1b)

2κ210L (F 2) = − κ2102g210

√−g e−2Φ

[− tr

(FMN F

MN)− 2 tr

χ /D(ω, A)χ

− 1

12tr(χΓABC χ

)HABC

− 1

2trχΓMΓAB(FAB +

ˆFAB)

(ψM +

2

3ΓMλ

)− 1

48tr(χΓABC χ)ψM

(4ΓABCΓ

M + 3ΓMΓABC

)λ

+1

12tr(χΓABC χ)λΓABCλ− β

96tr(χΓABC χ) tr(χΓABC χ)

], (7.1c)

2κ210L (R2) = − κ2102g210

√−g e−2Φ

[−RABMN (Ω−)R

ABMN (Ω−)

− 2ψAB /D(ω(e, ψ),Ω−)ψAB − 1

12ψABΓMNPψABHMNP

+1

2ψABΓ

MΓNPRAB

NP (Ω−) + RABNP (Ω−)

(ψM +

2

3ΓMλ

)− 1

48ψABΓ

CDEψAB · ψM

(4ΓCDEΓ

M + 3ΓMΓCDE

)λ

+1

12ψABΓCDEψAB

(λΓCDEλ

)− α

96ψABΓFGHψAB

(ψCDΓFGHψCD

)]. (7.1d)

12

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

κ210g210

≡ α′

4. (7.2)

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

DM (ω, A,Γ)ϕi = ∂Mϕi − i

2ωM

AB(ΣAB)ij ϕ

j + (AM )ij ϕj + Γi

jM ϕj . (7.3)

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′M

ΓMλ′

=

1 b

0 c

ψM

ΓMλ

, b, c ∈ R , c = 0 , (7.4)

because both ψM and ΓMλ 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

of order αn (βm), while δ0 corresponds to the terms independent of parameters2 α and β:

δ0eMA =

1

2ϵΓAψM , (7.5a)

δ0ψM =(∂M +

1

4Ω+M

ABΓAB

)ϵ+

ϵ(ψMλ

)− ψM

(ϵλ)+ ΓAλ

(ψMΓAϵ

), (7.5b)

δ0BMN = −ϵΓ[MψN ] , (7.5c)

δ0λ = −1

4/DΦ ϵ+

1

48ΓABCϵ

(− HABC − 1

2λΓABCλ

), (7.5d)

δ0Φ = −ϵλ , (7.5e)

δ0AM =1

2ϵΓM χ , (7.5f)

δ0χ = −1

4ΓABϵ

ˆFAB +

ϵ(χλ)− χ

(ϵλ)+ ΓAλ

(χΓAϵ

), (7.5g)

δ0ωMAB(e, ψ) = −1

4ϵΓMψ

AB − 1

2ϵΓ[AψM

B] +1

4ϵΓCψM HABC . (7.5h)

Notice that the spin connection ω(e, ψ) is the solution of D[M (ω)eN ]A = 0, while ω(e) is the solution

of D[M (ω)eN ]A = 0. Note that various objects such as a spin connection3 modified by the H-flux,

2Compared to [2] and [2], we assign the expansion parameters α and β to α = β = −κ210/g

210.

3If we simply write down the spin connection as ω, this means ω = ω(e).

13

supercovariantizations H and F , and so forth:

Ω±MAB ≡ ωM

AB(e, ψ)± 1

2HM

AB , (7.6a)

HMNP ≡ HMNP +3

2ψ[MΓNψP ]

= 3∂[MBNP ] +3

2ψ[MΓNψP ] − 6

κ210g210

tr(A[M∂N AP ] +

2

3A[M AN AP ]

)+ 6

κ210g210

(Ω−[M

AB ∂NΩ−P ]BA +

2

3Ω−[M

AB Ω−NBC Ω−P ]

CA), (7.6b)

dH =κ210g210

[trR(Ω−) ∧R(Ω−)

− tr

(F ∧ F

)], (7.6c)

ˆFMN ≡ FMN − ψ[MΓN ]χ , (7.6d)

ψMN ≡ DM (Ω+)ψN −DN (Ω+)ψM

−ψM

(ψNλ

)− ψN

(ψMλ

)− ΓPλ

(ψMΓPψN

), (7.6e)

RABMN (ω) ≡ RAB

MN (ω) +1

2ψ[MΓN ]ψ

AB + ψ[MΓ[AψN ]B] +

1

2ψ[MΓCψN ] H

ABC . (7.6f)

We also pick up the supercovariant derivatives DM at hand:

DMΦ = ∇MΦ+ ψMλ , (7.7a)

DM (ω)λ = DM (ω)λ+1

4/DΦψM − 1

48ΓABCψM

(− HABC − 1

2λΓABCλ

), (7.7b)

DM (ω, A)χ = DM (ω, A)χ+1

4ΓABψM

ˆFAB −

ψM

(χλ)− χ

(ψMλ

)+ ΓAλ

(χΓAψM

). (7.7c)

It is both useful and instructive to obtain the supersymmetry algebra from (7.5). The commu-

tator of two supersymmetry variations reads

[δ(ϵ1), δ(ϵ2)

]= δP(ξ

M ) + δQ(−ξMψM ) + δL(ξMΩ−M

AB) + δYM(ξM AM )

+ δM(−√22 ξM + 1√

2ξNBNM ) + δQ(ϵ3) + δL(Λ

AB) , (7.8a)

ξM =1

2ϵ2Γ

M ϵ1 , (7.8b)

ϵ3 = −7

8(ϵ2Γ

Aϵ1) ΓAλ+1

16× 120(ϵ2Γ

ABCDEϵ1) ΓABCDEλ , (7.8c)

ΛAB =β

192ϵ2Γ

[AΓCDEΓB]ϵ1 tr

(χΓCDEχ

). (7.8d)

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” transforma-

tions.

14

The supersymmetry variation of order β are given as follows:

δβψM =β

192ΓABCΓM ϵ tr

(χΓABC χ

), (7.9a)

δβBMN = 2β trA[Mδ0AN ]

, (7.9b)

δβλ =β

384ΓABCϵ tr

(χΓABC χ

), (7.9c)

δβωMAB(e, ψ) = − β

192ϵΓ[AΓCDEΓ

B]ψM tr(χΓCDEχ

). (7.9d)

Here the supersymmetry variation of order α are also given such as

δαψM =α

192ΓCDEΓM ϵ ψ

ABΓCDEψAB , (7.10a)

δαBMN = 2αΩ−[MAB δ0Ω−N ]

AB , (7.10b)

δαλ =α

384ΓCDEΓM ϵ ψ

ABΓCDEψAB . (7.10c)

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].)

δ0(DAΦ) = −√2

2ϵDA(Ω+)λ , (7.11a)

δ0Ω−MAB = −1

2ϵΓMψ

AB , (7.11b)

δ0HABC =3

2ϵΓ[AψBC] , (7.11c)

δ0ψAB =

1

4ΓCDϵ RAB

CD(Ω−) +ϵ(ψABλ

)− ψAB

(ϵλ)+ ΓCλ

(ψABΓCϵ

), (7.11d)

δ0ˆFAB = −ϵΓ[ADB](Ω+, A)χ , (7.11e)

δβ(DAΦ) = − β

192ϵΓAΓ

BCDλ tr(χΓBCDχ

), (7.11f)

δβHABC =3β

2ϵΓ[A tr

(χˆFBC]

), (7.11g)

δβψAB = β

[3

4ΓCDϵ tr

( ˆF [AB

ˆFCD]

)+

1

48ΓCDEΓ[Aϵ tr

χΓCDEDB](Ω+, A)χ

− 1

3× 256ΓCDEΓ[AΓ

GHϵ HB]GH tr(χΓCDEχ

)+

β

96× 96ΓCDEΓ[AΓ

FGHΓB]ϵ tr(χΓCDEχ

)tr(χΓFGH χ

)], (7.11h)

δβˆFAB =

β

192ϵΓ[AΓ

CDEΓB]χ tr(χΓCDEχ

). (7.11i)

Finally we describe an identity among generalized curvature tensors:

RABCD(Ω−) = RCDAB(Ω+)−1

2(dH)CDAB . (7.12)

15

The explicit expressions of supersymmetry variations are, of course, just approximate expres-

sions. 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

as HMNP , ˆFMN and RAB

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:

|Fp|2 ≡ 1

p!gM1N1 · · · gMpNp FM1···MpFN1···Np , (A.1)

where FM1···Mp is a totally antisymmetric tensor, i.e., the component of a p-form. The coefficient

1/p! is adopted to normalize each term appearing in the explicit expansion of |Fp|2 to unity.

A.2 Antisymmetrized symbol

The totally anti-symmetrized symbol is defined in terms of the square bracket:

T[M1M2···Mp] =1

p!

(TM1M2···Mp − TM2M1···Mp ± permutations

). (A.2)

This Gamma matrix is defined by

ΓMNP = Γ[MΓNΓP ] =1

3!

(ΓMΓNΓP ± permutations

). (A.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.(−,−, . . . ,−︸︷︷︸t

; +,+, . . . ,+︸︷︷︸s

) . (A.4b)

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

(ΣCD)AB = i

(δAC ηBD − δAD ηBC

)vector

ΣAB =i

2ΓAB spinor

(A.4c)

where ΓA is the Dirac gamma matrix which satisfies the Clifford algebra

ΓA,ΓB = 2ηAB . (A.5)

Here we define the chirality operator Γ in d = 2k + 2 dimensional spacetime with Lorentz signa-

ture:

Γ ≡ i−kΓ0Γ1 · · ·Γd−1 , (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

Γ ≡ i−nΓ1Γ2 · · ·Γd . (A.7)

The difference between (A.6) and (A.7) mainly comes from the hermiticity on the gamma matri-

ces: 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

ψ ≡ iψ†Γ0 , (A.8)

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)

C(ΓA)C−1 = (−1)k+[k/2](ΓA)T , C(ΓA1···An)C−1 = (−)[n+12

](ΓA1···An)T , (A.10b)

where [n+12 ] = 1, 1, 2, 2, 3, 3, · · · is the Gauss bracket. The hermitian conjugates of gamma

matrices are defined by

(ΓA)† = ΓA = −Γ0ΓA(Γ0)−1 . (A.11)

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

ΓA1A2···ApΓB1B2···Bq

=

min(p,q)∑k=0

(−1)12k(2p−k−1) p!q!

(p− k)!(q − k)!k!δ[A1

[B1· · · δAk

BkΓAk+1···Ap]

Bk+1···Bq ]. (A.12)

A.5 Covariant derivatives and curvature tensors

We introduce vielbeins eMA and their inverses EAM , which come from the spacetime metric gMN

and the metric ηAB on orthogonal frame via gMN = ηAB eMA eN

B and ηAB = gMN EAM EB

N . By

using these geometrical variables, let us define the covariant derivatives DM (ω,Γ) such as

DM (Γ)AN = ∂MAN − ΓPNMAP , (A.13a)

DM (Γ)AN = ∂MAN + ΓN

PMAP , (A.13b)

DM (Γ)gNP ≡ 0 = ∂MgNP − ΓQNMgQP − ΓQ

PMgNQ , (A.13c)

DM (Γ)gNP ≡ 0 = ∂MgNP + ΓN

QMgQP + ΓP

QMgNQ , (A.13d)

DM (ω,Γ)eNA ≡ 0 = ∂MeN

A + ωMAB eN

B − ΓPNM eP

A , (A.13e)

DM (ω,Γ)EAN ≡ 0 = ∂MEA

N − EBN ωM

BA + ΓN

PM EAP , (A.13f)

[DM (Γ), DN (Γ)]AQ = −RPQMN (Γ)AP + 2TP

MN DP (Γ)AQ , (A.13g)

RPQMN (Γ) = ∂MΓP

QN − ∂NΓPQM + ΓP

RMΓRQN − ΓP

RNΓRQM . (A.13h)

Note that AM in the above equations are vector. ΓPMN is the affine connection whose two lower

indices are not symmetric in general case. The antisymmetric part of the affine connection ΓP[MN ]

is defined as a torsion TPMN , while the symmetric part ΓP

(MN) is given in terms of the Levi-Civita

connection ΓP(L)MN and torsion, which we will show from the metricity condition (A.13c). First we

prepare the followings:

ΓPMN = ΓP

(MN) + ΓP[MN ] , ΓP

[NM ] = TPNM , (A.14a)

19

ΓP(L)MN =

1

2gPQ

(∂MgQN + ∂NgMQ − ∂QgMN

). (A.14b)

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

0 = −DM (Γ)gNP = −∂MgNP + ΓQNMgQP + ΓQ

PMgNQ , (A.15a)

0 = DN (Γ)gPM = ∂NgPM − ΓQPNgQM − ΓQ

MNgPQ , (A.15b)

0 = DP (Γ)gMN = ∂P gMN − ΓQMP gQN − ΓQ

NP gMQ . (A.15c)

Summing (A.15a), (A.15b) and (A.15c), we obtain

0 =(∂P gMN + ∂NgPM − ∂MgNP

)− 2TQ

MNgPQ − 2TQMP gQN − 2ΓQ

(PN)gMQ ,

∴ ΓQ(PN) = ΓQ

(L)PN − TPQN − TN

QP . (A.16)

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

ΓPMN = ΓP

(L)MN +KPMN , KP

MN ≡ TPMN − TM

PN − TN

PM . (A.17a)

The tensor KPMN is called the contorsion, which has the following property:

KMNP = gMQKQNP = TMNP − TNMP − TPMN = −KNMP . (A.17b)

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

RPQMN (Γ(L)) ≡ ∂MΓP

(L)QN − ∂NΓP(L)QM + ΓP

(L)LMΓL(L)QN − ΓP

(L)LNΓL(L)QM

= −RPQNM (Γ(L) . (A.18)

This Riemann tensor has various significant (anti)symmetries under exchanges of indices. Let us

carefully analyze them by using RPQMN (Γ(L)) = gPRRRQMN (Γ(L)):

RPQMN (Γ(L)) = gPR

(∂MΓR

(L)QN + ΓR(L)LMΓL

(L)QN

)−(M ↔ N

)=

1

2gPR∂M

gRK

(∂QgKN + ∂NgQK − ∂KgQN

)+ gPRΓ

R(L)LMΓL

(L)QN −(M ↔ N

)=

1

2∂M

(∂QgPN + ∂NgQP − ∂P gQN

)− 1

2gKL∂MgPL


)+

1

4gLK

(∂LgPM + ∂MgLP − ∂P gLM

)(∂QgKN + ∂NgQK − ∂KgQN

)−(M ↔ N

)=

1

2∂M

(∂QgPN − ∂P gQN

)− 1

2gKL∂MgPL


)+

1

4gLK

(∂LgPM + ∂MgLP − ∂P gLM

)(∂QgKN + ∂NgQK − ∂KgQN

)−(M ↔ N

)=

1

2∂M


)− 1

2∂N

(∂QgPM − ∂P gQM

)+

1

4gKL

(∂QgKM + ∂MgQK − ∂KgQM

)(∂P gLN + ∂NgPL − ∂LgPN

)20

− 1

4gKL


)(∂P gLM + ∂MgPL − ∂LgPM

)=

1

2∂M


)− 1

2∂N

(∂QgPM − ∂P gQM

)+ gKL

(ΓK(L)QMΓL

(L)PN − ΓK(L)QNΓL

(L)PM

). (A.19a)

Thus we can easily find the following remarkable relations:

RPQMN (Γ(L)) = −RPQNM (Γ(L)) = −RQPMN (Γ(L)) = RMNPQ(Γ(L)) . (A.19b)

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

1st: 0 = RMNPQ(Γ(L)) +RM

PQN (Γ(L)) +RMQNP (Γ(L)) , (A.20a)

2nd: 0 = ∇MRN

PQR(Γ(L)) +∇QRN

PRM (Γ(L)) +∇RRN

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

DM (ω)ϕi =δij ∂M − i

2ωM

AB · (ΣAB)ij

ϕj , (A.21)

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

[DM (ω), DN (ω)]ϕ = − i

2RAB

MN (ω)ΣABϕ , (A.22a)

RABMN (ω) = ∂MωN

AB − ∂NωMAB + ωM

AC ωN

CB − ωNAC ωM

CB . (A.22b)

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

RRPMN (Γ) = ηBC EA

R ePC RAB

MN (ω) , (A.23a)

RRM (Γ) = gPNRR

PMN (Γ) = eMB EA

RRAB(ω) , RA

B(ω) = RACBC(ω) , (A.23b)

R(Γ) = RMM (Γ) = RA

A(ω) = R(ω) . (A.23c)

It is useful to give a comment on vielbein. We will meet a vielbein with indices at different

positions such as eMA, and so forth. This can be regarded as the inverse EAM :

eMA = gMNδAB eNB , (A.24a)

eMA eNA = gNP δAB eM

AePB = gNP gMP = δNM = eM

AEAN , (A.24b)

∴ eNA = EAN . (A.24c)

In the same way, we also prove EAM = eM

A.

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-

ifold. Notice that the “Levi-Civita” connection Γ(L) indicates the torsionless part of the hermitian

connection, whose properties are given in (C.29):

Rpqmn(Γ(L)) = Rpqmn(Γ(L)) = −Rpqnm(Γ(L)) = −Rqpmn(Γ(L)) = Rmnpq(Γ(L))

=1

2∂m

[∂qgpn − ∂pgqn

]− 1

2∂n

[∂qgpm − ∂pgqm

]+ gKL

[ΓK(L)qmΓL

(L)pn − ΓK(L)qnΓ

L(L)pm

]= 0 , (A.25a)


=1

2∂m


]− 1

2∂n


]+ gKL

[ΓK(L)qmΓL


L(L)pm

]=

1

2∂m


]+ gkl

[Γk(L)qmΓl

(L)pn − Γk(L)pmΓl

(L)qn

], (A.25b)


=1

2∂m


]− 1

2∂n


]+ gKL

[ΓK(L)qmΓL


L(L)pm

], (A.25c)


=1

2∂m∂q gpn +

1

2∂n∂pgqm + gKL

[ΓK(L)qmΓL


L(L)pm

], (A.25d)


=1

2∂m∂q gpn − 1

2∂n∂qgpm + gkl

[Γk(L)qmΓl

(L)pn − Γk(L)qnΓ

l(L)pm

], (A.25e)


=1

2∂m


]− 1

2∂n


]+ gKL

[ΓK(L)qmΓL


L(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 gMN be

the metrics on the manifolds with and without torsion, respectively. These satisfy the following

22

metricity conditions individually:

0 = ∇MgNP = ∂MgNP − ΓQ(L)NMgQP − ΓQ

(L)PMgNQ , (A.26a)

0 = DM (Γ)gNP = ∂M gNP − ΓQNM gQP − ΓQ

PM gNQ

= ∂M gNP −(ΓQ(L)NM +KQ

NM

)gQP −

(ΓQ(L)PM +KQ

PM

)gNQ

= ∇M gNP −KPNM −KNPM . (A.26b)

Note that ΓQ(L)NM and ∇M are given in terms of the the metric gMN . Since the contorsion is anti-

symmetric 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 = ∇M gNP . (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

contorsion part and the other. We start from the vielbein postulate 0 = DM (ω,Γ)eNA and obtain

ωMAB = −EBN∂MeNA + ΓP

NM ePAEBN

= −EBN∂MeNA +

(ΓP(L)NM +KP

NM

)ePAEB

N

≡ ω(L)MAB +KABM , (A.27)

where we defined the spin connection ω(L)MAB given by the Levi-Civita connection Γ(L). The sec-

ond 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:

ω(L)MBA = −EA

N∂MeNB + ΓP(L)NM ePBEA

N = −EAN∂M

(gNQEB

Q)+ ΓP

(L)NM ePBEAN

= −EANEB

Q∂MgNQ − EAQ∂MEBQ + ΓP

(L)NM ePBEAN

= −EANEB

Q(ΓP(L)NMgPQ + ΓP

(L)QMgNP

)− EAQ∂MEB

Q + ΓP(L)NM ePBEA

N

= EBQ∂MeQA − ΓP

(L)QM ePAEBQ

= −ω(L)MAB , (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

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

Γ(L) −H with contorsion K = −H :

0 = DM (Γ−)JNP = ∇MJN

P +HRNMJR

P −HPRMJN

R . (A.30)

where we assigned DM (Γ(L)) = ∇M . By using this we express the Nijenhuis tensor such as

NMNP ≡ JMQ∇[QJN ]P − JN

Q∇[QJM ]P =(HMNP − 3J[M

QJNRHP ]QR

),

which tells us

HMNP = NMNP + 3J[MQJN

RHP ]QR .

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

J[MQ∇|Q|JNP ] = −2J[M

QJNRHP ]QR .

This is nothing but the Bismut torsion defined by

T(B)MNP =

3

2JM

QJNRJP

S∇[QJRS] = −3

2J[M

Q∇|Q|JNP ] .

Summarizing the above facts, we find

HMNP = NMNP + T(B)MNP . (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):

HMNP = T(B)MNP =

3

2JM

QJNRJP

S∇[QJRS] = −3

2J[M

Q∇|Q|JNP ] . (A.32)

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.

Here we show the explicit computation with using JMNJNP = −δPM :

HMNP =3

2JM

QJNRJP

S∇[QJRS] ,

1

2JM

QJNRJP

S∇QJRS =1

2∇Q

(JM

QJNRJP

SJRS

)− 1

2JRS∇Q

(JM

QJNRJP

S)

=1

2∇Q

(JM

QJNP

)− 1

2JRS

(∇QJM

Q)JNRJP

S + (∇QJNR)JM

QJPS + (∇QJP

S)JMQJN

R

24

=1

2JM

Q(∇QJNP )−1

2(∇QJNP )JM

Q +1

2JM

Q∇QJPN

= −1

2JM

Q(∇QJNP ) , (A.33a)

∴ HMNP = −1

2JM

Q∇QJNP − 1

2JN

Q∇QJPM − 1

2JP

Q∇QJMN

= −3

2J[M

Q∇|Q|JNP ] . (A.33b)

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

(1, 2)-form with respect to the complex structure JMN . Then

HMNP =3

2JM

QJNRJP

S∇[QJRS] =3

2JM

QJNRJP

S∂[QJRS] , (A.34a)

H =1

3!HMNP dxM ∧ dxN ∧ dxP =

1

4JM

QJNRJP

S∂[QJRS] dxM ∧ dxN ∧ dxP

=1

4Jm

QJnRJp

S∂QJRS dzm ∧ dzn ∧ dzp +1

4Jm

QJnRJp

S∂QJRS dzm ∧ dzn ∧ dzp

+1

4Jm

QJnRJp


4Jm

QJnRJp


+1

4Jm

QJnRJp


4Jm

QJnRJp


=1

2Jm

QJnRJp


2Jm

QJnRJp


=1

2i2(−i)∂mJnp dzm ∧ dzn ∧ dzp +

1

2(−i)2i∂mJnp dzm ∧ dzn ∧ dzp

=i

2

(∂ − ∂

)J , (A.34b)

Notice that the component of the complex structure is given by Jmn = iδnm, Jmn = −iδnm 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).

ωp ≡ 1

p!ωM1···Mp dx

M1 ∧ · · · ∧ dxMp , (A.35a)

(vol.) ≡√

|gD| dx1 ∧ · · · ∧ dxD . (A.35b)

It is also necessary to introduce a dual form of the p-form via so-called the Hodge dual:

∗ωp =

√|gD|

p!(D − p)!ωM1···Mp ε

M1···MpNp+1···ND

dxNp+1 ∧ · · · ∧ dxND , (A.36a)

25

(∗1) =

√|gD|D!

εM1···MDdxM1 ∧ · · · dxMD =

√|gD| dx1 ∧ · · · ∧ dxD = (vol.) , (A.36b)

where the index “1” does not always implies the first spatial direction; i.e., dx1 also often implies

dt in the negative signature. Notice that εM1···MDand εM1···MD are called the invariant tensors

whose property is given by

εM1···MnMn+1···MD

= gM1N1 · · · gMnNn εN1...NnMn+1···MD, (A.37a)

εM1M2···MD = gM1N1 · · · gMDND εN1N2···ND= g−1

D εN1N2···ND, (A.37b)

ε12···D ≡ 1 , ε12···D =1

gD, (A.37c)

TM1···MDεM1···MD = TM1···MD

gM1N1 · · · gMDNDεN1···ND= TN1···ND εN1···ND

. (A.37d)

The final line is from the definition that εM1...MDis a tensor. Using the Hodge star operator and

the invariant tensor, we can discuss more properties:

∗ ∗ ωp = (−1)p(D−p)+t ωp , (A.38a)

dxM1 ∧ · · · ∧ dxMD = gD εM1···MD dx1 ∧ · · · ∧ dxD , (A.38b)

dDx ≡ dx1 ∧ · · · ∧ dxD =1

D!εM1···MD

dxM1 ∧ · · · ∧ dxMD , (A.38c)

gD εM1···Mp

Np+1···ND· εM1···Mp

Lp+1···LD = p!(D − p)! · δLp+1

[Np+1· · · δLD

ND] . (A.38d)

We also introduce an inveriant tensor EA1···ADin the local Lorentz (or the frame coordinate) system.

Introducing the vielbein one-form eA = eMA dxM , we write down in such a way as

EA1A2...AD = ηA1B1ηA2B2 · · · ηADBD EB1B2···BD= η−1

D EB1B2···BD, (A.39a)

TM1···MD εM1···MD= TM1···MD eM1

A1 · · · eMD

AD EA1···AD= TA1···AD EA1···AD

, (A.39b)

TM1···MDεM1···MD = TM1···MD

EA1M1 · · ·EAD

MD EA1···AD = TA1···ADEA1···AD , (A.39c)

E12···D = 1 , E12···D =1

ηD= (−1)t , (A.39d)

where ηD ≡ det ηAB = (−1)t, with the number of minus sign in the signature. Furthermore, using√|ηD| = 1, we also define the followings:

(vol.) = e1 ∧ · · · ∧ eD , (A.40a)

eA1 ∧ · · · ∧ eAD = ηD EA1···AD e1 ∧ · · · ∧ eD , (A.40b)

e1 ∧ · · · ∧ eD =1

D!EA1A2···AD

eA1 ∧ · · · ∧ eAD , (A.40c)

ηD EA1···ApBp+1···BD

· EA1···ApCp+1···CD = p!(D − p)! · δCp+1

[Bp+1· · · δCD

BD] . (A.40d)

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 fun-

damental representation, and on the field φa in the adjoint representation, are also defined as

DM (A)ϕi = ∂Mϕi − i(AM )ij ϕ

j fundamental representation , (A.41a)

DM (A)χ = ∂Mχ− i[AM , χ]

DM (A)χa = ∂Mχa + fabcA

bM χc

adjoint representation . (A.41b)

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

[DM (A), DM (A)]ϕ = −iFMN ϕ , FMN = ∂MAN − ∂NAM − i[AM , AN ] , (A.41c)

where the gauge fields AM and the field strength FMN are described in terms of the gauge sym-

metry generators Ta such as

AM ≡ AaMT

a , and FMN = F aMNT

a , (A.42a)

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

tr(T aT b) = δab , [T a, T b] = ifabc Tc , (Ta)b

c = ifbac = [ad(Ta)]b

c , (A.42b)

[Ta, [Tb, Tc]] + [Tb, [Tc, Ta]] + [Tc, [Ta, Tb]] = 0 = fbcdfade + fcadfbde + fabdfcde , (A.42c)

F aMN = ∂MA

aN − ∂NA

aM + fabcA

bMA

cN . (A.42d)

Due to the above commutation relation we set the structure constant fabc 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 sym-

metrized 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-

erators T a. The algebra is defined as

(T a)† = −T a , tr(T aT b) = −δab , (A.44a)

[T a, T b] = fabc Tc , (Ta)b

c = fbac = [ad(Ta)]b

c , (A.44b)

[Ta, [Tb, Tc]] + [Tb, [Tc, Ta]] + [Tc, [Ta, Tb]] = 0 = −ifbcdfade − ifcadfbde − ifabdfcde . (A.44c)

Note that the structure constant fabc to be real (and to be same as the one in the previous subsec-

tion). The relation between T a and the generators T a is

T a = iT a . (A.45)

By using this anti-hermitian generators T a we re-define the gauge fields

AM ≡ AaM T

a with iAM = AM , FMN ≡ F aMN T

a with iFMN = FMN . (A.46a)

Here we also described the relations between A and A, which is the hermitian gauge fields de-

fined in the previous subsection. Then the covariant derivatives with respect to the Yang-Mills

transformation on the field ϕi in the fundamental representation, and on the field φ = φaT a in the

adjoint representation, are also defined as

DM (A)ϕi = ∂Mϕi + (AM )ij ϕ

j fundamental representation , (A.47a)

DM (A)χ = ∂M χ+ [AM , χ]

DM (A)χa = ∂Mχa + fabcA

bM χc

adjoint representation . (A.47b)

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

[DM (A), DM (A)]ϕ = FMN ϕ , FMN = ∂M AN − ∂N AM + [AM , AN ] , (A.47c)

where we should notice that the “component fields” AaM and F a

MN are the real fields and they

corresponds to the ones in the previous subsection, i.e.,

F aMN = ∂MA

aN − ∂NA

aN + fabcA

bMA

cN . (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

three-form ωL3 and the Yang-Mills-Chern-Simons three-form ωY

3 :

ωL3 =

1

3!ωLMNP dxM ∧ dxN ∧ dxP ≡

(ωA

B ∧ dωBA +

2

3ωA

B ∧ ωBC ∧ ωC

A

), (A.48a)

28

ωY3 =

1

3!ωYMNP dxM ∧ dxN ∧ dxP ≡ tr

(A ∧ dA− 2i

3A ∧A ∧A

), (A.48b)

1

3!ωLMNP =

(ω[M

AB∂NωP ]BA +

2

3ω[M

ABωNBCωP ]

CA), (A.48c)

1

3!ωYMNP = tr

(A[M∂NAP ] −

2i

3A[MANAP ]

), (A.48d)

where ωAB = ωM

ABdx

M and A = AaMT

adxM are ths spin connection and the gauge fields which

satisfy the followings

RAB = dωA

B + ωAC ∧ ωC

B , F = dA− iA ∧A . (A.49)

Of course the Yang-Mills Chern-Simons 3-form ωY3 with respect to the anti-hermitian generators

T a can be defined as

ωY3 ≡ tr

(A ∧ dA+

2

3A ∧ A ∧ A

)= −ωY

3 with F = dA+ A ∧ A . (A.50)

The exterior derivatives of these three-forms are given by

dωL3 = RA

B ∧RBA = tr(R ∧R) , (A.51a)

dωY3 = tr(F ∧ F ) , dωY

3 = tr(F ∧ F ) = −tr(F ∧ F ) = −dωY3 . (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:

ϕ−3 ≡ 1

κ210exp(−2Φ) , ωM

AB ≡ −ωMAB , (B.1a)

AM ≡ −AM , BMN ≡ +√2BMN , HMNP ≡ +

√2

3HMNP , (B.1b)

χ = χaT a ≡ −χaT a = −χ , λ ≡√2λ , (B.1c)

ψM , ϵ keep the same variables , (B.1d)

α = β = −κ210

g210, (B.1e)

κ210 = 2 ,κ2102g210

= α′ . (B.1f)

30

C Geometries: complex, hermitian and Kahler manifolds

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

zm ≡ 1√2

(xm + iym

′), zm ≡ 1√

2

(xm − iym

′), (C.1a)

xm =1√2

(zm + zm

), ym

′= − i√

2

(zm − zm

). (C.1b)

The derivatives ∂/∂zm are given as

∂

∂zm=

1√2

( ∂

∂xm− i

∂

∂ym′

),

∂

∂zm=

1√2

( ∂

∂xm+ i

∂

∂ym′

), (C.2a)

∂

∂xm=

1√2

( ∂

∂zm+

∂

∂zm

),

∂

∂ym′ =i√2

( ∂

∂zm− ∂

∂zm

). (C.2b)

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

gMN dxMdxN = gmn dxmdxn + gmn′ dxmdyn

′+ gm′n dy

m′dxn + gm′n′ dym

′dyn

′. (C.3)

This line element is invariant under the rotation with SO(D) group. We also define the line ele-

ment given in terms of the complex coordinates zm and zm:

g(z,z)MN dZMdZN = gmn dz

mdzn + gmn dzmdzn . (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 gmn in (C.4) via (C.1). We impose

gmn = gmn = 0 (C.5)

on the metric since we assume that the geometry is a complex manifold:

gmn = 0 =∂xp

∂zm∂xq

∂zngpq +

∂xp

∂zm∂yq

′

∂zngpq′ +

∂yp′

∂zm∂xq

∂zngp′q +

∂yp′

∂zm∂yq

′

∂zngp′q′

31

=1

2

(δpmδ

qn gpq − iδpmδ

q′n gpq′ − iδp

′mδ

qn gp′q − δp

′mδ

q′n gp′q′

), (C.6a)

gmn = 0 =∂xp

∂zm∂xq

∂zngpq +

∂xp

∂zm∂yq

′

∂zngpq′ +

∂yp′

∂zm∂xq

∂zngp′q +

∂yp′

∂zm∂yq

′

∂zngp′q′

=1

2

(δpmδ

qn gpq + iδpmδ

q′n gpq′ + iδp

′mδ

qn gp′q − δp

′mδ

q′n gp′q′

). (C.6b)

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:

gpq =

(∂zm

∂xp∂zn

∂xqgmn +

∂zm

∂xp∂zn

∂xqgmn

)=

1

2

(δmp δ

nq gmn + δmp δ

nq gmn

)=

1

2

(δmp δ

nq + δmq δ

np

)gmn , (C.8a)

gp′q′ =

(∂zm

∂yp′∂zn

∂yq′gmn +

∂zm

∂yp′∂zn

∂yq′gmn

)=

1

2

(δmp′ δ

nq′ gmn + δmp′ δ

nq′ gmn

)=

1

2

(δmp′ δ

nq′ + δmq′ δ

np′

)gmm , (C.8b)

gpq′ =

(∂zm

∂xp∂zn

∂yq′gmn +

∂zm

∂xp∂zn

∂yq′gmn

)= − i

2

(δmp δ

nq′ gmn − δmp δ

nq′ gmn

)= − i

2

(δmp δ

nq′ − δmq′ δ

np

)gmn , (C.8c)

gp′q =

(∂zm

∂yp′∂zn

∂xqgmn +

∂zm

∂yp′∂zn

∂xqgmn

)=

i

2

(δmp′ δ

nq gmn − δmp′ δ

nq gmn

)=

i

2

(δmp′ δ

nq − δmq δ

np′

)gmn , (C.8d)

where we used the symmetry gmn = gnm. For later discussion, we define the inverse of gpq as

gpq =(δpmδ

qn + δqmδ

pn

)gmn . (C.9)

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

g ≡ det gMN = det

gpq gpq′

gp′q gp′q′

= det gpq · det(gp′q′ − gp′pg

pqgqq′). (C.10)

The first determinant is given by using (C.8):

det gpq = det(12

(δmp δ

nq + δmq δ

np

)gmn

). (C.11a)

32

Let us consider the second determinant carefully:

gp′pgpqgqq′ =

1

4

(δmp′ δ

nq − δmq δ

np′

)gmn

(δprδ

qs + δqrδ

ps

)grs(δkpδ

lq′ − δkq′δ

lp

)gkl

=1

4

(δkp′δ

lq′ + δkq′δ

lp′

)gkl , (C.11b)

∴ det(gp′q′ − gp′pg

pqgqq′)= det

(14

(δkp′δ

lq′ + δkq′δ

lp′)gkl

). (C.11c)

Summarizing the above, we obtain the following expression in terms of g ≡ det gmN :

g = det(12

(δmp δ

nq + δmq δ

np

)gmn

)· det

(14

(δkp′δ

lq′ + δkq′δ

lp′)gkl

)=

1

2d

det(12

(δmp δ

nq + δmq δ

np

)gmn

)2=

1

2d· g2 . (C.12)

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 coordi-

nate transformation. This transformation can be, for instance, derived from the general coordinate

transformation of a tensor ∇MAN from the (xM ;AN )-frame to the (xm; An)-frame:

∇MAN = ∂MA

N + ΓNPMA

P =

(∂xm

∂xM∂

∂xm

)(∂xN

∂xnAn

)+ ΓN

PM

(∂xP

∂xpAp

)≡ ∂xm

∂xM∂xN

∂xn

(∇mA

n)

=∂xm

∂xM∂xN

∂xn

(∂mA

n + ΓnpmA

p). (C.13)

where AN is an arbitrary contravariant vector, and the affine connection ΓNPM can contain con-

torsion. Compared to the first line and the second line in the right-hand side, we easily find the

relation between ΓNPM and Γn

pm:

ΓNPM

∂xP

∂xpAp =

∂xm

∂xM∂xN

∂xnΓn

pmAp − ∂xm

∂xM∂2xN

∂xm∂xpAp , (C.14a)

∴ ΓNPM =

∂xN

∂xn∂xp

∂xP∂xm

∂xMΓn

pm +∂xN

∂xp∂2xp

∂xP∂xM. (C.14b)

Here we also discuss the transformation rule of the spin connection from the xM -coordinate

frame to the xm-coordinate frame. To do so, we should also define the vielbein and its inverse:

gMN = δAB eMA eN

B , eMAEA

N = δNM , EAM eM

B = δBA , (C.15)

where δAB is the orthogonal metric in the local Lorentz frame. Since the vielbein eMA and its

inverse EAM are vectors under the general coordinate transformation4, they transform in the fol-

4Notice that the local Lorentz coordinates is not transformed under the general coordinate transformations.

33

lowing way:

eMA =

∂xm

∂xMem

A , EBN =

∂xN

∂xnEB

n . (C.16)

Focus on the vielbein postulate given by the following equation and its transformation:

0 = DM (ω,Γ)eNA = ∂MeN

A + ωMAB eN

B − ΓPNM eP

A (C.17a)

=

(∂xm

∂xM∂

∂xm

)(∂xn

∂xNen

A

)+ ωM

AB

(∂xn

∂xNen

B

)−(∂xP

∂xp∂xn

∂xN∂xm

∂xMΓp

nm +∂xP

∂xp∂2xp

∂xN∂xM

)(∂xq

∂xPeq

A

). (C.17b)

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 xm-frame:

DM (ω,Γ)eNA =

∂xm

∂xM∂xn

∂xN

(Dm(ω, Γ)en

A)

=∂xm

∂xM∂xn

∂xN

(∂men

A + ωmAB en

B − Γpnm ep

A). (C.18)

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:

∂xm

∂xM∂xn

∂xNωm

AB en

B =∂xn

∂xNωM

AB en

B +∂2xn

∂xM∂xNen

A − ∂2xn

∂xM∂xNen

A

=∂xn

∂xNωM

AB en

B , (C.19a)

∴ ωMAB =

∂xm

∂xMωm

AB . (C.19b)

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:

ΓPNM = EA

P(∂MeN

A + ωMAB eN

B). (C.20)

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

connection:

ωMAB = −EB

N∂MeNA + ΓP

NMePAEB

N . (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 mani-

fold 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)-

structure manifold affine connections with non-trivial values are of pure type Γmnp and of mixed

type Γmnp, and their complex conjugates. This condition is nothing but the condition on a hermi-

tian 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:

∇m∂

∂zn= Γp

nm∂

∂zp, ∇m

∂

∂zn= Γp

nm∂

∂zp, (C.22a)

∇m∂

∂zn= Γp

nm∂

∂zp, ∇m

∂

∂zn= Γp

nm∂

∂zp, (C.22b)

Γpnm = 0 , Γp

nm = 0 , Γpnm = 0 , Γp

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

SO(2d) to U(d) (or SU(d)). Note that ∇m∂/∂zn behaves as a tensor. This hermitian connection

itself is given by the metricity condition:

0 = ∇mgnp = ∂mgnp − Γqnmgqp − Γq

pmgnq . (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))

RPQMN (Γ) = ∂MΓP

QN − ∂NΓPQM + ΓP

RMΓRQN − ΓP

RNΓRQM , (C.24)

where the affine connection ΓPMN can contain torsion. Applying the hermitian condition on the

affine connection (C.22) to the Riemann tensor, the expression becomes quite simple:

RpqMN (Γ) = ∂M Γp

qN − ∂N ΓpqM + Γp

RM ΓRqN − Γp

RN ΓRqM = −Rp

qNM (Γ) , (C.25a)

35



RM ΓRqN − Γp

RN ΓRqM = 0 , (C.25b)



RM ΓRqN − Γp

RN ΓRqM = 0 , (C.25c)



RM ΓRqN − Γp

RN ΓRqM = −Rp

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

Γmnp = Γm

(np) + Γm[np] , (C.26a)

Γm(np) ≡ 1

2

(Γm

np + Γmpn

), Tm

np ≡ Γm[np] =

1

2

(Γm

np − Γmpn

). (C.26b)

Furthermore, we can decompose the symmetric part of the hermitian connection into the Levi-

Civita connection5 Γ(L) and the terms depending on the torsion:

Γm(np) = Γm

(L)np −(Tn

mp + Tp

mn

), (C.27a)

∴ Γmnp = Γm

(L)np +Kmnp , Km

np ≡ Tmnp − Tn

mp − Tp

mn , (C.27b)

where Kmnp is called the contorsion. Here let us explicitly list up all components:

Γmnp = Γm

(L)np +Kmnp , Γm

np = Γm(L)np +Km

np , (C.28a)

Γmnp = Γm

(L)np +Kmnp , Γm

np = Γm(L)np +Km

np , (C.28b)

0 = Γmnp = Γm

(L)np +Kmnp , 0 = Γm

np = Γm(L)np +Km

np , (C.28c)

0 = Γmnp = Γm

(L)np +Kmnp , 0 = Γm

np = Γm(L)np +Km

np . (C.28d)

It is worth discussing the explicit forms of the Levi-Civita connection Γ(L) on the hermitian mani-

fold:

Γm(L)np =

1

2gmq

(∂ngqp + ∂pgnq

)= Γm

(L)pn , Γm(L)np =

1

2gmq

(∂ngqp + ∂pgnq

)= Γm

(L)pn ,

(C.29a)

Γm(L)np = 0 , Γm

(L)np = 0 , (C.29b)

Γm(L)np =

1

2gmq

(∂pgnq − ∂q gnp

)= Γm

(L)pn , Γm(L)np =

1

2gmq

(∂pgnq − ∂qgnp

)= Γm

(L)pn . (C.29c)

Here we should point out that the Levi-Civita connections Γm(L)np and Γm

(L)np have non-trivial val-

ues whereas the hermitian connections of same type vanish: Γmnp = Γm

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 van-

ishing hermitian connections:

0 = Γmnp = Γm

(L)np +Kmnp , ∴ Km

np = −Γm(L)np = −1

2gmq

(∂ngpq − ∂qgpn

), (C.30a)

0 = Γmnp = Γm

(L)np +Kmnp , ∴ Km

np = −Γm(L)np = −1

2gmq

(∂ngpq − ∂qgpn

). (C.30b)

We should notice that the Riemann tensor RPQMN (Γ(L)) should have same properties as the ones

on the Riemann tensor RPQMN (Γ0) on a generic real manifold such as

RPQMN (Γ(L)) = −RPQNM (Γ(L)) = −RQPMN (Γ(L)) = RMNPQ(Γ(L)) , (C.31a)

RP [QMN ](Γ(L)) = 0 , ∇[M RN|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):

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

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:

gmn ≡ δAB emA en

B , gmn = 0 = δAB emA en

B , gmn = 0 = δAB emA en

B , (C.32a)

emA EA

n = δnm , emA EA

n = δnm , δAB = epAEB

p + epAEB

p . (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 gMN . First, without any constraints, it

has (2d)2 = 4d2 degrees of freedom. Using the constraints gmn = 0 = gmn, it is reduced to

4d2 − 2 × d2 = 2d2. Furthermore, the symmetries gmn = gnm halves the degrees to d2, which

coincides with the physical degrees of freedom of gmn. The same reduction is also applied to the

metric δAB , which gives d2 degrees of freedom on δab. The vielbein eMA should carries the same

number of degrees of freedom as the metrics gMN and δAB , which we will consider. Originally, if

there are no constraints, eMA has (2d)2 = 4d2 degrees. The constraints

0 = gmn = δab emaen

b + δab emaen

b , 0 = gmn = δab emaen

b + δab emaen

b (C.32d)

37

impose 2× d2 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

with the metrics gmn and δab. To realize such a reduction, we set 2d2 components of the vielbein

to be zero. Although, there are actually many ways to do it, the following setting is much useful:

ema ≡ 0 , em

a ≡ 0 . (C.32e)

We can confirm the matching of degrees of freedom of the metric and the vielbein. The metric

has 2 × d2 degrees of freedom (i.e., both gmn and gmn have d2 degrees individually), while the

non-trivial vielbein components ema, ena have 2× d2 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:

Γpmn = Ea

p(∂nem

a + ωnab em

b), Γp

mn = Eap(∂nem

a + ωnab em

b), (C.33a)

Γpmn = Ea

p(∂nem

a + ωnab em

b), Γp

mn = Eap(∂nem

a + ωnab em

b), (C.33b)

while the vanshing condition of the hermitian connection gives

0 = Γpmn = EA

p(∂nem

A + ωnAB em

B)

= ωnab em

bEap , (C.33c)

0 = Γpmn = EA

p(∂nem

A + ωnAB em

B)

= ωnab em

bEap , (C.33d)

0 = Γpmn = EA

p(∂nem

A + ωnAB em

B)

= ωnab em

bEap , (C.33e)

0 = Γpmn = EA

p(∂nem

A + ωnAB em

B)

= ωnab em

bEap . (C.33f)

Equivalently, we can also discuss the spin connection via (C.21):

ωmab = −Eb

n(∂men

a − Γpnm ep

a), (C.34a)

ωmab = −Eb

n(∂men

a − Γpnm ep

a), (C.34b)

ωmab = −Eb

n(∂men

a − Γpnm ep

a), (C.34c)

ωmab = −Eb

n(∂men

a − Γpnm ep

a), (C.34d)

ωmab = −Eb

N(∂meN

a − ΓpNm ep

a)

= Γpnm ep

aEbn = 0 , (C.34e)

ωmab = −Eb

N(∂meN

a − ΓpNm ep

a)

= Γpnm ep

aEbn = 0 , (C.34f)

ωmab = −Eb

N(∂meN

a − ΓpNm ep

a)

= Γpnm ep

aEbn = 0 , (C.34g)

38

ωmab = −Eb

N(∂meN

a − ΓpNm ep

a)

= Γpnm ep

aEbn = 0 . (C.34h)

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:

[DM (ω), DN (ω)] = − i

2RABMN (ω)ΣAB , (C.35a)

RABMN (ω) = ∂M ωN

AB − ∂N ωM

AB + ωM

AC ωN

CB − ωN

AC ωM

CB . (C.35b)

Here let us explicitly describe the Riemann tensor:

RabMN (ω) = ∂M ωNab − ∂N ωMab + ωMac ωNcb − ωNac ωM

cb

= −RabNM (ω) = −RbaMN (ω) , (C.35c)

RabMN (ω) = ∂M ωNab − ∂N ωMab + ωMac ωNcb − ωNac ωM

cb

= −RabNM (ω) = −RbaMN (ω) , (C.35d)

RabMN (ω) = 0 , RabMN (ω) = 0 . (C.35e)

Notice that the Riemann tensor is antisymmetric under the exchange between the latter two in-

dices 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 ωMAB = −ωMBA (A.29).

Here let us again discuss the properties on the Riemann tensor of the hermitian affine connec-

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

RpqMN (Γ) = RabMN (ω) epaeqb. Then we derive the followings:

RpqMN (Γ) = RabMN (ω) epaeqb = −RabNM (ω) epaeqb = RbaMN (ω) epaeqb

= −RpqNM (Γ) = −RqpMN (Γ) . (C.36)

In the same way as the affine connection (C.28), the spin connection ωMAB should also be

decomposed into the Levi-Civita part and the contorsion part via (C.34):

ωmab = −Ebn(∂mena −

Γp(L)nm +Kp

nm

epa

)≡ ω

(L)mab +Kabm , (C.37a)


Γp(L)nm +Kp

nm

epa

)≡ ω

(L)

mab+Kabm , (C.37b)


Γp(L)nm +Kp

nm

epa

)≡ ω

(L)

mab+Kabm , (C.37c)

39


Γp(L)nm +Kp

nm

epa

)≡ ω

(L)mab +Kabm , (C.37d)

0 = ωmab =Γp(L)nm +Kp

nm

epaEb

n = Kabm , (C.37e)


nm

epaEb

n ≡ ω(L)mab +Kabm , (C.37f)

0 = ωmab =Γp

(L)nm +Kpnm

epaEb

n ≡ ω(L)

mab+Kabm , (C.37g)


nm

epaEb

n = Kabm . (C.37h)

C.5 Additional constraint: Kahler manifold

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

Ω ≡ igmn dzm ∧ dzn = Jmn dz

m ∧ dzn ≡ J , (C.38)

where the component of this two-form Jmn = Jmp gpn is given by the complex structure Jmn [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 ↔ ∂mgnp = ∂ngmp and ∂mgnp = ∂pgnm . (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):

ΓMNP = ΓM

0NP +KMNP = ΓM

0NP , (C.40a)

Γm0np = Γm

0pn = gmq∂ngpq , Γm0np = Γm

0pn = gmq∂ngqp , (C.40b)

Γm0np = Γm

0np = Γm0np = Γm

0np = 0 . (C.40c)

This indicates that the metric on the hermitian manifold is affected by the torsion. Actually, the

Levi-Civita connection Γm(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

ΓPnm = −ΓP

n′m′ , ΓPn′m = ΓP

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):

gmn ≡ ∂m∂nK(z, z) . (C.42)

40

Note that a hermitian manifold is a Kahler manifold if and only if the hermitian connection Γmnp

is torsion free (see p.180 of [12]). Since the hermitian connection on the Kahler manifold is the

Kahler Levi-Civita connection Γm0np (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

Rmnpq(Γ0) = −∂qΓm

0np = −∂q(gmr∂pgnr

)= −∂q

(gmr∂ngpr

)= Rm

pnq(Γ0) . (C.43a)

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

Rmnpq(Γ0) = −Rmnqp(Γ0) = −Rnmpq(Γ0) = Rnmqp(Γ0) , (C.43b)

Rmnpq(Γ0) = Rm

pnq(Γ0) , Rmnpq(Γ0) = Rm

qpn(Γ0) , Rmnqp(Γ0) = Rm

pqn(Γ0) . (C.43c)

The spin connection on the Kahler manifold is also more restricted than the one on the hermi-

tian manifold, i.e., the torsionless spin connection ω0 associated with the Kahler affine connection

(C.40) via the relation (C.17a):

0 = Dm(ω0, Γ0)ena = ∂men

a − Γp0nm ep

a + ω0mab en

b , (C.44a)


a + ω0mab en

b , (C.44b)


a + ω0mab en

b , (C.44c)


a − Γp0nm ep

a + ω0mab en

b . (C.44d)

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

ω0mab = −Eb

n∂mena + Γp

0nm epaEb

n , ω0mab = −Eb

n∂mena , (C.45a)

ω0mab = −Eb

n∂mena , ω0m

ab = −Eb

n∂mena + Γp

0nm epaEb

n . (C.45b)

These are related to each other via the antisymmetry ωMAB = −ωMBA in the following way:

ω0mab = −Eb

n∂mena + Γp

0nm epaEb

n = −Ebn∂m

(gnq e

qa)+(gpq∂mgnq

)ep

aEbn

= −Ebn(∂mgnq

)eqa − Ebq∂me

qa + ∂mgnq eqaEb

n

= eqa∂mEbq = Eaq∂meqb = δacδbd Ecq∂meq

d

= −δacδbd ω0mdc , (C.46a)

ω0mab = −Eb

n∂mena + Γp

0nm epaEb

n = −Ebn∂m

(gnq e

qa)+(gpq∂mgnq

)ep

aEbn

= −Ebn(∂mgnq

)eqa − Ebq∂me

qa + ∂mgnq eqaEb

n

= eqa∂mEbq = Eaq∂meqb = δacδbd Ecq∂meq

d

= −δacδbd ω0mdc , (C.46b)

41

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

the Riemann tensor:

ω0mab = −ω0mb

a = −δbdδac ω0m

dc = Eaq∂meqb , (C.47a)

ω0mab = −ω0mb

a = −δacδbd ω0mdc = Eaq∂meqb . (C.47b)

Following these expressions, it turns out the following:

2ω0[ma|c| ω0n]

cb =

(Eap∂mepc

)(Ecq∂neqb

)−(Eap∂nepc

)(Ecq∂meqb

)= ∂m

(Eaq∂neqb

)− 2∂mE

aq∂neqb − Eaq∂m∂neqb

− ∂n

(Eaq∂meqb

)+ 2∂nE

aq∂meqb + Eaq∂m∂neqb

= ∂m

(Eaq∂neqb

)− 2∂n

(eqb∂mE

aq)+ 2eqb∂m∂nE

aq

− ∂n

(Eaq∂meqb

)+ 2∂m

(eqb∂nE

aq)− 2eqb∂m∂nE

aq

= −∂m(Eaq∂neqb

)+ ∂n

(Eaq∂meqb

)= −∂mω0n

ab + ∂nω0m

ab

= −2∂[mω0n]ab . (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):

Rabmn(ω0) = 2∂[mω0n]

ab + 2ω0[m

a|c| ω0n]

cb = 0 , (C.49a)

Rabmn(ω0) = 0 , Ra

bmn(ω0) = 0 , Rabmn(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

Rabmn(ω0) = EapEb

q 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 con-

torsion 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:

H =i

2

(∂ − ∂

)J = −1

2dcJ , J = igmn dz

m ∧ dzn , (D.1)

where J is the fundamental two-form on the manifold, whose component is given by the complex

structure satisfying the covariantly constant condition DM (Γ−)JNP = 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.

Due to this, the three-form is decomposed into two parts; the (2,1)-form H(2,1) = i2∂J and the

(1, 2)-form H(1,2) = − i2∂J , whose components are given as

H(2,1) =i

2∂J =

i

2∂mJnp dz

m ∧ dzn ∧ dzp , (D.2a)

H(1,2) = − i

2∂J = − i

2∂mJnp dz

m ∧ dzn ∧ dzp , (D.2b)

Hmnp =i

2

(∂mJnp − ∂nJmp

)= −1

2

(∂mgnp − ∂ngmp

), (D.2c)

Hmnp = − i

2

(∂mJnp − ∂nJmp

)= −1

2

(∂mgnp − ∂ngmp

). (D.2d)

Furthermore, the exterior derivative of the NS-NS three-form is given as

dH = (∂ + ∂)i

2(∂ − ∂)J = − i

2∂∂J +

i

2∂∂J . (D.3)

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

H(2,1) =1

2Hmpq dz

m ∧ dzp ∧ dzq , H(1,2) =1

2Hmpq dz

m ∧ dzp ∧ dzq . (D.4)

Actually, the NS-NS three-form H is the totally antisymmetric torsion, as well as the totally anti-

symmetric 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

way as 0 = DM (Γ−)JNP . Following to (C.28) and (C.37), we again describe the hermitian affine

connection and the spin connection:

Γm−np ≡ Γm

(L)np −Hmnp =

1

2gmq

(∂ngpq + ∂pgnq

)+

1

2gmq

(∂ngpq − ∂pgnq

), (D.5a)

Γm−np ≡ Γm

(L)np −Hmnp =

1

2gmq

(∂ngpq + ∂pgnq

)+

1

2gmq

(∂ngpq − ∂pgnq

), (D.5b)

43

Γm−np ≡ Γm

(L)np −Hmnp =

1

2gmq


)− 1

2gmq

(∂q gpn − ∂pgqn

), (D.5c)

Γm−np ≡ Γm

(L)np −Hmnp =

1

2gmq


)− 1

2gmq

(∂q gpn − ∂pgqn

), (D.5d)

0 = Γm−np = Γm

(L)np −Hmnp = 0− 0 , (D.5e)

0 = Γm−np = Γm

(L)np −Hmnp = 0− 0 , (D.5f)

0 = Γm−np = Γm

(L)np −Hmnp =

1

2gmq

(∂ngqp − ∂qgnp

)− 1

2gmq


), (D.5g)

0 = Γm−np = Γm

(L)np −Hmnp =

1

2gmq


)− 1

2gmq


), (D.5h)

ω(−)

mab= −Eb

n(∂mena −

Γp(L)nm −Hp

nm

epa

)≡ ω

(L)

mab−Hmab , (D.5i)

ω(−)mab = −Eb

n(∂mena −

Γp(L)nm −Hp

nm

epa

)≡ ω

(L)mab −Hmab , (D.5j)

ω(−)

mab= −Eb

n(∂mena −

Γp(L)nm −Hp

nm

epa

)≡ ω

(L)

mab−Hmab , (D.5k)

ω(−)mab = −Eb

n(∂mena −

Γp(L)nm −Hp

nm

epa

)≡ ω

(L)mab −Hmab , (D.5l)

0 = ω(−)mab =

Γp(L)nm −Hp

nm

epaEb

n = 0− 0 , (D.5m)

0 = ω(−)

mab=Γp(L)nm −Hp

nm

epaEb

n = 0− 0 , (D.5n)

0 = ω(−)mab =

Γp(L)nm −Hp

nm

epaEb

n ≡ ω(L)mab −Hmab , (D.5o)

0 = ω(−)

mab=Γp

(L)nm −Hpnm

epaEb

n ≡ ω(L)

mab−Hmab . (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

Γm(α)np = Γm

(L)np + αHmnp , Γm

(α)np = Γm(L)np + αHm

np , (D.6a)

Γm(α)np = Γm

(L)np + αHmnp , Γm

(α)np = Γm(L)np + αHm

np , (D.6b)

Γm(α)np = 0 , Γm

(α)np = 0 , (D.6c)

Γm(α)np = (1 + α)Hm

np , Γm(α)np = (1 + α)Hm

np , (D.6d)

ω(α)

mab= ω

(L)

mab+ αHmab = ω

(−)

mab+ (1 + α)Hmab , (D.6e)

ω(α)mab = ω

(L)mab + αHmab = ω

(−)mab + (1 + α)Hmab , (D.6f)

ω(α)

mab= ω

(L)

mab+ αHmab = ω

(−)

mab+ (1 + α)Hmab , (D.6g)

ω(α)mab = ω

(L)mab + αHmab = ω

(−)mab + (1 + α)Hmab , (D.6h)

ω(α)mab = 0 , ω

(α)

mab= 0 , (D.6i)

ω(α)mab = ω

(L)mab + αHmab = (1 + α)Hmab , (D.6j)

ω(α)

mab= ω

(L)

mab+ αHmab = (1 + α)Hmab , (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)

supergravity when α = −1/3, while ω(−)MAB (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

set Γm(L)np, Γ

m(L)np,H, ω

(α) = Γm0np, 0, 0, ω0, because the manifold becomes Kahler, whose metric

satisfies (C.39). This point is isolated from the continuous one-parameter line α.

45

References

[1] K. Becker, M. Becker and J.H. Schwarz, “String Theory and M-theory, a modern introduc-

tion,” Cambridge University Press.

[2] E.A. Bergshoeff and M. de Roo, “The Quartic Effective Action of the Heterotic String and

Supersymmetry,” Nucl. Phys. B 328 (1989) 439.

[3] B.A. Bertlmann, “Anomalies in quantum field theory,” Oxford University Press.

[4] 藤井保憲, “超重力理論入門,”マグロウヒル出版,産業図書.

[5] M. B. Green, J. H. Schwarz and E. Witten, “Superstring Theory,” Cambridge University Press

(1987).

[6] 今村洋介, “超重力理論ノート”.

[7] T. Kimura, “Eleven-dimensional supergravities on maximally supersymmetric back-

grounds,” TK-NOTE/03-11, on

http://www2.yukawa.kyoto-u.ac.jp/˜tetsuji/NOTEs/index.html .

[8] T. Kimura, “Note on the quartic effective action of heterotic string,” TK-NOTE/06-03.

[9] T. Kimura, “Heterotic string with Neveu-Schwarz fluxes,” TK-NOTE/06-05.

[10] T. Kimura, “Index theorems on torsional geometries,” arXiv:0704.2111, to appear in JHEP.

[11] T. Kimura and P. Yi, “Comments on heterotic flux compactifications,” JHEP 0607 (2006) 030

[arXiv:hep-th/0605247].

[12] S. Kobayashi and K. Nomizu, “Foundations of Differential geometry,” vol.2, Interscinence

Publishers (1969).

[13] M. Nakahara, “Geometry, Topology and Physics,” Institute of Physics Publishing (1990), Bris-

tol.

[14] 太田信義, “超弦理論・ブレイン・Ｍ理論,”シュプリンガー・フェアラーク東京.

[15] J. Polchinski, “String Theory”, Cambridge University Press.

[16] A. Strominger, “Superstrings with Torsion,” Nucl. Phys. B 274 (1986) 253.

[17] K. Yano, “Differential geometry on complex and almost complex spaces,” Pergamon Press

(1965).

46

Documents

Supergravities in ten and eleven dimensions TK …tetsuji.kimura/NOTEs/various...1 Strategy We start from the eleven-dimensional supergravity. Along various kinds of duality transforma-tions,