Notation and Conventions

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Appendix A Notation and Conventions

We use natural units, in which the speed of light is c = 1 and Planck’s constant is = 1. We also take Newton’s gravitational constant as 16πG = 1. In this system of units the Einstein term for the gravitational field in the Lagrangian has a coefficient 1. One can recover c, and G in other system of units by dimensional analysis. We use μ, ν, ρ, . . . = 0, 1, 2,...,D − 1 for world indices of D-dimensional spacetime and a, b, c,... = 0, 1, 2,...,D − 1 for local Lorentz indices. The flat metric in a local Lorentz frame is

ab ηab = diag(−1, +1,...,+1) = η . (A.1)

μ μ ...μ ε 1 2 D ε The Levi-Civita symbols and μ1μ2...μD represent totally antisymmetric tensor densities and have components

012...D−1 ε =+1,ε012...D−1 =−1. (A.2)

... εa1a2 aD ε We also use the Levi-Civita symbols and a1a2...aD with local Lorentz indices, which are totally antisymmetric and have the same components as (A.2). Symmetrization and antisymmetrization of n indices with unit strength are de- noted as 1 T( ... ) = T ( ... ), a1 an n! P a1 an P 1 P T[ ... ] = (−1) T ( ... ), (A.3) a1 an n! P a1 an P ( ... ) ... where P is a sum over permutations P a1 an of the indices a1a2 an.The sign factor (−1)P is +1 for even permutations and −1 for odd permutations. For instance, 1 1 T( ) = (Tab + Tba) , T[ab] = (Tab − Tba) . (A.4) ab 2 2

Y. Tanii, Introduction to Supergravity, SpringerBriefs in Mathematical Physics, 121 DOI: 10.1007/978-4-431-54828-7, © The Author(s) 2014 122 Appendix A: Notation and Conventions

For conventions of geometric quantities such as curvatures see Sect. 1.2.

Groups Here, we summarize deﬁnitions of groups appearing in the text. In the following m and n denote natural numbers, and we deﬁne −1 × 0 0 1 × η = m m = ηT ,Ω= n n =−ΩT , (A.5) 0 1n×n −1n×n 0 where 1n×n is the n × n unit matrix. • GL(n, R): general linear group The group of n×n real matrices with a non-vanishing determinant. A non-compact group of dimension n2. GL(1, R)+ = R+ is a group of positive real numbers. • SL(n, R): special linear group The group of n × n real matrices with a unit determinant. A non-compact group of dimension n2 − 1. • SO(m, n): special orthogonal group The group of (m + n) × (m + n) real matrices O satisfying

OT η O = η, det O = 1. (A.6)

SO(0, n) = SO(n, 0) = SO(n) is the n-dimensional rotation group, and SO(1, D − 1) is the D-dimensional Lorentz group. SO(n) is compact and ( , )( , = ) 1 ( + )( + − ) SO m n m n 0 is non-compact. The dimension is 2 m n m n 1 . • SU(m, n): special unitary group The group of n × n complex matrices U satisfying

U †ηU = η, det U = 1. (A.7)

SU(0, n) = SU(n, 0) = SU(n) is a compact special unitary group. SU(m, n) (m, n = 0) is non-compact. Dimension is (m + n)2 − 1. When the condition det U = 1 is not imposed, the group is called U(m, n). • Sp(2n, R): real symplectic group The group of 2n × 2n real matrices M satisfying

M T Ω M = Ω. (A.8)

A non-compact group of dimension n(2n + 1). • USp(2n): unitary symplectic group The group of 2n × 2n complex matrices U satisfying

U T ΩU = Ω, U †U = 1. (A.9)

A compact group of dimension n(2n + 1). Appendix B Formulae of Gamma Matrices

Antisymmetrized Products

The antisymmetrized products of n gamma matrices of SO(t, s) (t + s = D)are deﬁned as ... [ ] γ a1a2 an = γ a1 γ a2 ...γan (n = 1, 2,...,D). (B.1)

a ...a a a a When all the indices a1,...,an are different, γ 1 n = γ 1 γ 2 ...γ n .Theyare ... traceless tr γ a1a2 an = 0.

Duality Relations

When D is even, the antisymmetrized products satisfy duality relations

a ...a 1 (s−t)+ 1 n(n−1) 1 a ...a γ 1 n = (−1) 4 2 ε 1 D γ ... γ,¯ (B.2) (D − n)! an+1 aD

... where εa1 aD is the totally antisymmetric Levi-Civita symbol and γ¯ is the chirality matrix (3.11). When D is odd, they satisfy

a ...a 1 (s+3t−1)+ 1 n(n−1) 1 a ...a γ 1 n = (−1) 4 2 ε 1 D γ ... . (B.3) (D − n)! an+1 aD

Products

A product of two antisymmetrized products can be expressed as a linear combination of antisymmetrized products as

Y. Tanii, Introduction to Supergravity, SpringerBriefs in Mathematical Physics, 123 DOI: 10.1007/978-4-431-54828-7, © The Author(s) 2014 124 Appendix B: Formulae of Gamma Matrices

a ...a a ...a [a a ...a − ] γ 1 m γ ... = γ 1 m ... + mn δ m γ 1 m 1 ... ] b1 bn b1 bn [b1 b2 bn [a am−1 a ...a − ] + 2 C C δ m δ γ 1 m 2 ... ] m 2 n 2 [b1 b2 b3 bn [a am−1 am−2 a ...a − ] + 3! C C δ m δ δ γ 1 m 3 ... ] +··· . (B.4) m 3 n 3 [b1 b2 b3 b4 bn

The last term contains an antisymmetrized product of |m − n| gamma matrices. For instance, γ abcγ = γ abc + γ [ab δc] + γ [aδc δb]. de de 6 [e d] 6 [d e] (B.5)

The ﬁrst term of (B.4) corresponds to the case in which a1,...,am and b1,...,bn do not contain the same number, the second term corresponds to the case in which only one of a1,...,am is the same as one of b1,...,bn, the third term corresponds to the case in which only two of a1,...,am are the same as two of b1,...,bn, and so on. The coefﬁcients can be found by counting combinatorial numbers. From (B.4) we can obtain formulae such as

ba1a2...an a1a2...an a1a2...anb γbγ = (D − n)γ = γ γb,

b a1a2...an n a1a2...an γ γ γb = (−1) (D − 2n)γ . (B.6)

Completeness

When D is even, a set of matrices

... 1,γa1 ,γa1a2 ,...,γa1a2 aD , (B.7) where a1 < a2 < ··· for each matrix, are linearly independent and form a complete set of 2D/2 × 2D/2 complex matrices. Namely, an arbitrary 2D/2 × 2D/2 complex matrix can be expressed uniquely as a linear combination of these matrices. To show this, notice that the number of the matrices in (B.7)is

D D DC0 + DC1 + DC2 +···+ DC D = (1 + 1) = 2 , (B.8) which is equal to the number of components of a 2D/2 × 2D/2 matrix. Furthermore, the matrices in (B.7) are orthogonal with respective to the trace D/2 δa1 ...δam ( = ) a ...a 2 m n tr γ 1 m γ ... = b1 bm , (B.9) bn b1 0 (m = n) where a1 < a2 < ··· < am, b1 < b2 < ··· < bn. Therefore, the matrices in (B.7) are a set of 2D linearly independent matrices and hence form a complete set of 2D/2 × 2D/2 matrices. When D is odd, a set of matrices Appendix B: Formulae of Gamma Matrices 125

a a ...a a a a 1 2 1 (D−1) 1,γ,γ1 2 ,...,γ 2 (B.10) are linearly independent and form a complete set of 2(D−1)/2 × 2(D−1)/2 matrices. This can be shown by the above even dimensional result and the fact that odd D- dimensional gamma matrices can be constructed using even (D − 1)-dimensional ... γ a1 an > 1 ( − ) ones (See Sect. 3.1). Note that (B.10) does not contain for n 2 D 1 , ≤ 1 ( − ) which are related to those for n 2 D 1 as in (B.3) and are not independent.

Bilinear Forms

From two spinors ψ, λ we can construct a bilinear form

... ψγ¯ a1 an λ, (B.11) which transforms as an n-th rank tensor under SO(t, s) transformations in (3.6). Here, ψ¯ is the Dirac conjugate of ψ defined in (3.9). Using the definition of the charge conjugation (3.22) and the properties of the gamma matrices (3.8), (3.23) we find

¯ a ...a † tn+ 1 n(n−1) ¯ a ...a (ψγ 1 n λ) = (−1) 2 λγ 1 n ψ, ¯ a ...a t+n 1 t(t+1)+tn+ 1 n(n−1) ¯ c a ...a c ψγ 1 n λ =−(±1) (−1) 2 2 ε±λ γ 1 n ψ . (B.12)

Here, the sign factor ±1 on the right-hand side of the second equation is +1 when the charge conjugation matrix C+ is used and −1 when C− is used. Components of the spinors are treated as anticommuting Grassmann numbers. For instance, for Majorana spinors ψ = ψc, λ = λc of the four-dimensional Lorentz group SO(1, 3) we ﬁnd

(ψλ)¯ † = λψ¯ = ψλ,¯ (ψγ¯ aλ)† =−λγ¯ aψ = ψγ¯ aλ, (ψγ¯ abλ)† =−λγ¯ abψ = ψγ¯ abλ, (ψγ¯ abcλ)† = λγ¯ abcψ = ψγ¯ abcλ, (ψγ¯ abcdλ)† = λγ¯ abcdψ = ψγ¯ abcdλ. (B.13)

For Weyl spinors ψ±, λ± of SO(t, s) (t + s = even) we ﬁnd

¯ a ...a ψ±γ 1 n λ± = 0 (n + t = odd), ¯ a ...a ψ±γ 1 n λ∓ = 0 (n + t = even), (B.14)

...... which can be shown by (3.13), (3.14) and γγ¯ a1 an = (−1)nγ a1 an γ¯. 126 Appendix B: Formulae of Gamma Matrices

Fierz Identities

¯ ¯ A product of two bilinears ψ1ψ2 and ψ3ψ4 can be rearranged as

− D ¯ ¯ 2 ¯ ¯ ¯ ¯ a 1 ¯ ¯ ba ψ1ψ2ψ3ψ4 =−2 ψ1ψ4ψ3ψ2 + ψ1γaψ4ψ3γ ψ2 + ψ1γabψ4ψ3γ ψ2 2 1 1 ¯ ¯ cba ¯ ¯ aN ...a1 + ψ γabcψ ψ γ ψ +···+ ψ γa ...a ψ ψ γ ψ , 3! 1 4 3 2 N! 1 1 N 4 3 2 (B.15)

= = 1 ( − ) where N D when D is even and N 2 D 1 when D is odd. An identity like (B.15), which exchanges the combinations of the spinors, is called the Fierz identity. The Fierz identity (B.15) can be shown as follows. First, the left-hand side can be written as ψ¯ ψ ψ¯ ψ = ψ¯ αψ ψ¯ γ ψ δβ δδ , 1 2 3 4 1 2β 3 4δ α γ (B.16)

[D/2] β δ where α, β, γ, δ = 1, 2,...,2 are spinor indices. Regard δα δγ on the right- hand side of (B.16)asthe(γβ) component of a 2[D/2] × 2[D/2] matrix for ﬁxed α, δ. This matrix can be written as a linear combination of (B.7)or(B.10)

δβ δδ = δδβ + a δ(γ ) β + ab δ(γ ) β +··· . α γ C0α γ C1 α a γ C2 α ab γ (B.17)

... a1 an δ (γ ) γ The coefﬁcients Cn α can be found by multiplying a1...an β on the both-hand sides and using (B.9). In this way we ﬁnd

− D β δ 2 δ β δ a β 1 δ ba β δα δγ = 2 δαδγ + (γa)α (γ )γ + (γab)α (γ )γ 2 1 1 δ cba β δ aN ...a1 β + (γabc)α (γ )γ +···+ (γa ...a )α (γ )γ . (B.18) 3! N! 1 N

Substituting this into (B.16) and exchanging positions of the spinors we obtain (B.15). The minus sign on the right-hand side of (B.15) appears because of the Grassmann nature of the spinors. Similarly, we can consider Fierz identities for other types of bilinears such as ¯ a ¯ bc ψ1γ ψ2ψ3γ ψ4. The identity in this case can be found by making replacements a ¯ ¯ bc ψ2 → γ ψ2, ψ3 → ψ3γ in (B.15). Index

A Coset space, 53 Action, 2, 91 Cosmological constant, 3, 27, 33 AdS spacetime, 27, 33, 111, 114 Cosmological term, 3, 26, 33, 111 AdS supergravity, 27, 33, 111 Covariant derivative, 3, 6, 7, 9, 22, 33, 54, D = 4, N = 1, 26 112, 116 D = 4, N = 2, 32 CPT theorem, 19, 29 Anomaly cancellation, 84 Anti de Sitter spacetime, see AdS spacetime Anti de Sitter supergravity, see AdS super- D gravity D-brane, 1 Antisymmetric tensor field, 9, 46, 58, 71, 96, D8 brane, 119 116 De Sitter spacetime, 27 Antisymmetrization, 121 Dilaton, 75, 82, 85 Automorphism, 43, 51 Dimensional reduction, 64, 90, 115 Auxiliary field, 20, 23 Dirac conjugate, 39 Dirac spinor, 39 Dual field, 11, 101–103, 106, 108, 116 B Duality symmetry, 10, 32, 57 Bianchi identity, 9, 11, 32, 57, 58 Duality transformation, 57, 100, 102, 103, Bilinear form, 125 108

C E Cartan subalgebra, 96, 105 E7(+7), 64, 65, 112 Central charge, 28, 31, 43, 51, 73 Einstein equation, 3 Charge conjugation, 17, 40, 44, 65 Einstein frame, 76, 82, 86 Charge conjugation matrix, 17, 41 Einstein term, 3, 76 Chern–Simons term, 72, 84, 98, 104 Electric–magnetic duality, 57 Chern–Simons type, 13, 117 Embedding tensor, 111 Chiral multiplet, 19, 20 Energy–momentum tensor, 4, 11, 60 Chirality, 39, 46, 73, 76, 83 Extended super Poincaré algebra, 28 Chirality matrix, 18, 39 Extended supergravity, 29, 33 Christoffel symbol, 3 Extended supersymmetry, 28 Commutator algebra, 20, 23, 25, 27, 30, 33, 72, 75, 78, 85, 113, 119 Compactiﬁcation, 89, 115, 117 F Compensating transformation, 54, 81 Field strength, 6, 8, 9

Y. Tanii, Introduction to Supergravity, SpringerBriefs in Mathematical Physics, 127 DOI: 10.1007/978-4-431-54828-7, © The Author(s) 2014 128 Index

Fierz identity, 25, 126 M M theory, 2, 47, 72 Majorana condition, 17, 40 G Majorana spinor, 17, 40, 41 Gamma matrix, 18, 37, 123 Majorana–Weyl spinor, 42 Gauge coupling constant, 8, 33, 111, 112 Massive supergravity, 111, 118 Gauge ﬁxing, 54, 56, 80, 115 D = 10, N = (1, 1), 118 Gauged supergravity, 33, 111, 112, 116, 118 Matter supermultiplet, 24, 46, 83 D = 4, N = 8, 112 Maximal compact subgroup, 34, 53, 61, 62, D = 5, N = 8, 117 64 D = 7, N = 4, 116 Maximal supergravity, 29, 46, 98 General coordinate transformation, 1, 4, 21, Maximally symmetric, 27 22 Maxwell type, 10, 13, 116 Ghost, 50, 53, 56 Maxwell’s equations, 57 ( , R) GL n , 122 Metric, 3 Global symmetry, 23, 31, 34, 53, 64, 71 Metric formulation, 3 Grassmann number, 8, 125 Metricity condition, 3, 6 Gravitational ﬁeld, 2 Minimal coupling, 9, 30, 33, 111, 112 Gravitino, 1, 21 Minkowski spacetime, 18, 24, 71, 73, 119 Green–Schwarz mechanism, 84

H N Helicity, 19, 29, 46 Natural units, 121 Heterotic string, 49, 83, 85, 107 Neveu–Schwarz sector, see NS sector Higgs mechanism, 119 Neveu–Schwarz–Ramond formalism, 48 Hodge dual, 9, 31, 57, 58 Newton’s gravitational constant, 3, 121 Non-compact Lie group, 53 Non-compact symmetry, 34 I Non-linear sigma model, 34, 53, 62 Inönü–Wigner contraction, 51 E6(+6)/USp(8), 102 Integrability condition, 28, 114 E7(+7)/SU(8), 65, 103 Inverse radius of AdS spacetime, 27, 51, 114 SL(2, R)/SO(2), 55, 76, 100, 108 Isometry, 27, 114, 117 SL(3, R)/SO(3), 100 SL(5, R)/SO(5), 101 SL(d, R)/SO(d), 94 K SO(5, 5)/[SO(5) × SO(5)], 102 Kaluza–Klein mode, 90, 115 SO(d, d + n)/[SO(d) × SO(d + n)], Killing form, 53 107 Killing spinor, 28, 114 SU(1, 1)/U(1), 76 Killing vector, 27 NS sector, 48 NS–NS sector, 48, 75, 82, 85 NS–R sector, 48 L Lagrangian, 2 Levi-Civita symbol, 10, 24, 121 Linear fractional transformation, 56, 81 O Local Lorentz index, 5, 121 Off-shell closure, 23 Local Lorentz transformation, 5, 22 Off-shell formulation, 23 Local supersymmetry, 1, 21 On-shell closure, 23 Local supertransformation, 23 Orthogonal complement, 53, 66 Lorentz generator, 17, 43 OSp(1|4),28 Lorentz group, 37, 122 OSp(8|4), 115 Index 129

P String coupling constant, 75, 83, 86, 99 Pauli matrix, 18 String duality, 1 Pauli term, 30 String frame, 76, 82, 85 Pauli’s fundamental theorem, 38 SU(8), 64 Physical degrees of freedom, 13, 46, 117 SU(m, n), 122 Poincaré algebra, 17 Super AdS algebra, 28, 50, 115 Poincaré supergravity, 24, 71 Super anti de Sitter algebra, see super AdS D = 4, N = 1, 22 algebra D = 4, N = 2, 30, 62 Super de Sitter algebra, 50 D = 4, N = 8, 103, 64 Super Poincaré algebra, 17, 24, 43 D = 5, N = 8, 102 Super Yang–Mills theory, 49, 83, 105 D = 6, N = (4, 4), 101 Superalgebra, 17 D = 7, N = 4, 100 Supercharge, 17, 43 D = 8, N = 2, 100 Superconformal algebra, 50 D = 9, N = 2, 99 Supercovariant, 23, 30 D = 10, N = (1, 0), 49, 83, 105 Superfield, 20 D = 10, N = (1, 1), 48, 73, 99 Supergravity, 1, 21 D = 10, N = (2, 0), 49, 76, 103 Supergravity multiplet, 19, 29, 46 D = 11, N = 1, 48, 72, 98 Supermultiplet, 17, 18, 28, 46 Poincaré transformation, 17 Superpotential, 20 Proca type, 13 Superstring theory, 1, 47 Pseudo Majorana spinor, 41 type I, 49, 83, 85 Pseudo Majorana–Weyl spinor, 42 type IIA, 48, 73, 75, 99, 105 type IIB, 49, 76, 82, 105 Supersymmetry, 1, 17 R Supertransformation, 17, 20 R sector, 48 Supertransformation of spin connection, 26 R+, 74, 84, 94, 122 Symmetrization, 121 R–NS sector, 48 Symplectic Majorana spinor, 43 Symplectic Majorana–Weyl spinor, 43 R–R sector, 48, 75, 82, 85 Symplectic pseudo Majorana spinor, 43 Ramond sector, see R sector Symplectic pseudo Majorana–Weyl spinor, Rarita–Schwinger field, 1, 21 43 Ricci tensor, 3 Riemann tensor, 3, 6, 24, 27 T T duality, 105 S T tensor, 113 S duality, 83, 86 Tetrad, 5 Scalar curvature, 3 Toroidal compactification, 90 Scalar field potential, 34, 111, 114 Torsion, 22 Self-duality, 12, 46, 79, 104 Torsionless condition, 3, 6 Self-duality in odd dimensions, 13 Translation generator, 17, 43 SL(2, Z), 83 Truncation, 32, 83 SL(d, R), 94, 122 SO(1, D − 1), 37 SO(D), 37 U SO(M, M), 60 U(m, n), 122 SO(t, s), 37, 122 USp(2n), 122 Sp(2M, R), 60, 122 Spin connection, 6, 22 Spinor representation, 37 V Spontaneous compactification, 90 Vector multiplet, 19, 20 Stückelberg mechanism, 119 Vielbein, 5 130 Index

Vielbein formulation, 4 X Vierbein, 5 ξ α transformation, 94, 95, 98

W Weyl condition, 40 Weyl spinor, 39 Y Weyl transformation, 7, 75 Yang–Mills ﬁeld, 8, 34, 83, 95 World index, 5, 121 Yukawa coupling, 34, 111, 113