Supergravity 入門

at 美杉ビレッジ 1998 年 11 月 21 日

§1. D = 11 Supergravity

Guess: on-shell mode number counting

e_{μ}^{m} : 11^{2} −_{11}C_{2}(LL) − 2 × 11(GC ghosts) = 44
ψ_{μ} : (11 × 32 + 32(B_{α})) × \frac{1}{2} − 2 × 32(Super ghosts) = 128  (1)

requires

128 − 44 = 84 = 9·8·7/3! = _{9}C_{3} bosons → antisymmetric gauge field A_{μνρ} ! (2)

Lagrangian

L = −\frac{1}{2κ^{2}} eR(e, ω) − \frac{1}{2}eψ̄_{μ}Γ^{μρσ}D_{ρ}(\frac{ω + ω̂}{2} )ψ_{σ} − \frac{1}{2 · 4!}eF_{μνρσ}^{2}

−\frac{\sqrt{2}κ}{16 · 4!}e(ψ̄_{μ}Γ^{μαβγδν}ψ_{ν} + 12ψ̄^{α}Γ^{βγ}ψ^{δ})(F_{αβγδ} + F̂_{αβγδ})

+\frac{\sqrt{2}κ}{6 · (4!)^{2} · 3!}ε^{μ_{1}···μ_{11}}F_{μ_{1}···μ_{4}}F_{μ_{5}···μ_{8}}A_{μ_{9}μ_{10}μ_{11}}  (3)

where

F = dA,  F ≡ \frac{1}{4!}F_{μνρσ}dx^{μ}dx^{ν}dx^{ρ}dx^{σ}, A ≡ \frac{1}{3!}A_{μνρ}dx^{μ}dx^{ν}dx^{ρ}
ω̂_{μ mn} = ω_{μ mn}(e) + \frac{1}{4}(ψ̄_{μ}Γ_{m}ψ_{n} − ψ̄_{μ}Γ_{n}ψ_{m} + ψ̄_{m}Γ_{μ}ψ_{n}) ← supercov’tion of ω(e)
ω_{μ mn} = ω̂_{μ mn} − \frac{1}{8}(ψ̄^{α}Γ_{αμmnβ}ψ^{β}) ← δS/δω = 0 の解
F̂_{μνρσ} = F_{μνρσ} + \frac{1}{8}ψ̄_{[μ}Γ_{νρ}ψ_{σ]} ← (4 × 3 通りの反対称和)
D_{μ}(ω)ψ = (∂_{μ} − \frac{1}{4}ω_{μ}^{mn})Γ_{mn}ψ  (4)

This is invariant under the following (local) SUSY transformation:

δe_{μ}^{m} = \frac{κ}{2}ε̄Γ^{m}ψ_{μ},
δψ_{μ} = \frac{1}{κ}D_{μ}(ω̂)ε + \frac{\sqrt{2}}{12 · 4!}(Γ_{μ}^{αβγδ} − 8δ_{μ}^{α}Γ^{βγδ})ε F̂_{αβγδ},
δA_{μνρ} = −\frac{\sqrt{2}}{4}ε̄Γ_{[μν}ψ_{ρ]}, ← 3 項の反対称和

(5)

SUSY algebra:

[δ_{Q}(ε_{2}), δ_{Q}(ε_{1})] = δ_{GC}(ξ^{μ}) + δ_{Q}(−ξ^{ν}ψ_{ν}) + δ_{M}(λ^{mn}) + δ_{gauge}(Λ_{μν})
where ξ^{μ} = \frac{1}{2}ε̄_{1}Γ^{m}ε_{2} e_{m}^{μ}
λ^{mn} = ξ^{μ}ω̂_{μ}^{mn} + ε̄_{1}(Γ^{mnαβγδ} − 24e^{mα}e^{nβ}Γ^{γδ})F̂_{αβγδ}
Λ_{μν} = −\frac{1}{2}ε̄_{1}Γ_{μν}ε_{2} − ξ^{ρ}A_{ρμν}  with δ_{gauge}A = dΛ.  (6)

§2. Reduction to D = 4 N = 8 SUGRA

Simple Reduction:

Φ(x^{M}=(x^{μ}, x^{i})) → Φ(x^{μ})

e_{M}^{A} = ( \begin{matrix}e_{μ}^{α} & e_{μ}^{a} \\ e_{i}^{α} ≡ 0 & e_{i}^{a} \end{matrix} ) curved index: M = (μ(= 0, 1, 2, 3) , i(= 4, 5, ⋯, 10)), flat index:  A = (α(= 0, 1, 2, 3) , a(= 4, 5, ⋯, 10)). (7)

Now each 11 dimensional index split into a 4 dimensional part and an external part, and the 32 component Majorana spinor in 11 dimensions is decomposed into 8 four-component Majorana spinors in 4 dimensions corresponding to the following tensor product decomposition of 11 γ-matrices Γ^{A}:

\{ \begin{matrix}Γ^{α} & = γ^{α} ⊗ (1)_{A}^{B}, \\ Γ^{a} & = γ_{5} ⊗ (Γ_{(7)}^{a})_{A}^{B}, \end{matrix} A, B = 1, 2, ⋯, 2^{[7/2]} = 8.

Thus this dimensional reduction gives rise to the following field constant in 4 dimensions:

e_{M}^{A}(g_{MN}) → \{ \begin{matrix}e_{μ}^{α}(g_{μν}) & 1 _{(J=2)} \\ e_{μ}^{a}(g_{μi}) ∼ B_{μ}^{i} & 7 \\ e_{i}^{a}(g_{ij}) ∼ ϕ_{ij} & 28 \end{matrix}

A_{MNP} → \{ \begin{matrix}A_{μνρ} ∼ auxiliary & 28 _{(J=1)} 35 _{(J^{P} =0^{+})} \\ A_{μνi} ∼ ϕ_{i} & 7  70 _{(J=0)} \\ A_{μij} ∼ B_{μ}^{ij} & 21 \\ A_{ijk} & _{7}C_{3} = 35 _{(J^{P} =0^{−})} \end{matrix}

ψ_{M} → \{ \begin{matrix}ψ_{μ}^{A} & 8 _{(J=3/2)} \\ ψ_{i}^{A} ∼ λ_{ABC} ∝ (Γ_{(7)}^{i})_{[AB}ψ_{iC]} & _{8}C_{3} = 56 _{(J=1/2)} \end{matrix}  (8)

Now the general coordinate transformation GC(11) and the local Lorentz invariance SO(1, 10)_{LL} in 11 dimensions reduce to

GC(11) → \{ \begin{matrix}GC(4) \\ GL(7, R)_{global} \\ Abelian gauge [U(1)]^{7} \end{matrix}
SO(1, 10)_{LL} → \{ \begin{matrix}SO(1, 3)_{LL} \\ SO(7)_{local}, \end{matrix}  (9)

respectively, in the resultant 4-dimensional theory. Here we should note that the general coordinate transformations in the extra dimensions which survive the dimensional reduction are only those with the transformation parameter ξ^{i}(x^{M}) of the form ξ^{i}(x^{μ}, x^{i}) = Λ_{j}^{i}x^{j} + ξ^{i}(x^{μ}). Λ_{j}^{i} corresponds to the global (x^{μ}-independent) general linear transformation GL(7, R)_{global} and ξ^{i}(x^{μ}) to the abelian gauge transformation [U(1)]^{7}.

So GL(7, R)_{global} × SO(7)_{local} is a manifest internal symmetry of this 4-dimensional theory, N = 8 supergravity. Cremmer and Julia, however, noted that these 7-dimensional symmetries are in fact easily enlarged to the 8-dimensional ones:

GL(7, R)_{global} → SL(8, R)_{global},
SO(7)_{local} → SO(8)_{local}.  (10)

The symptoms of these 8-dimensional symmetries appear in the following facts:

i) The γ-matrix generators Γ^{ab} of SO(7) are combined with Γ^{a} (a = 4, ⋯, 10) to form SO(8) generators.

ii) The vector fields appearing in the above, seven B_{μ}^{i} and twenty one B_{μ}^{ij}, are combined to yield 28 representation of SL(8, R) (or SO(8) ).

iii) The scalar fields, twenty eight ϕ_{ij} and seven ϕ_{i}, are also combined to yield 35 representation of SL(8, R).

Cremmer and Julia have shown at this stage that the lagrangian for the 35 scalar( J^{P} = 0^{+}) sector can be written in the form of nonlinear sigma model based on SL(8, R)/SO(8) [dim(SL(8, R)) − dim(SO(8)) = 63 − 28 = 35].

Proceeding further, they have found that the total scalar field sector, the 35 sector plus 35 pseudo-scalar fields, is just described by the nonlinear sigma model on the coset ^{∗}

E_{7(+7)}/SU(8) (dim(E_{7(+7)}) − dim(SU(8)) = 133 − 63 = 70),

and that the symmetries of N = 8 supergravity are in fact larger ones than (10):

SL(8, R)_{global} → E_{7(+7) global},
SO(8)_{local} → SU(8)_{local}.  (11)

All the fermionic fields are inert under E_{7(+7) global} and J = 1, 2 bosonic fields are inert under SU(8)_{local}. Since the fermionic fields ψ_{μA} and λ_{ABC} are Majorana spinors, the SU(8)_{local} transformation Λ′_{A}^{B} + iΛ′′_{A}^{B} are understood to be Λ′_{A}^{B} + iγ_{5}Λ′′_{A}^{B} on them. [ Since the smallest non-trivial representation of E_{7} is 56 dimensional, the 28 vector fields do not fit in the E_{7} representation and hence the E_{7(+7)} cannot be a lagrangian symmetry of the vector field sector. However, the 28 equations of motion for their field strength and the 28 Bianchiidentities turn out to fall into 56 representation of E_{7(+7)}, and the E_{7(+7)} is therefore a symmetry on-the-mass shell in the vector sector.]

∗ E_{7(+7)} denotes a non-compact form of E_{7}, which has SU(8) as maximum compact subgroup and +7 = 70 − 63 is the signature of the non-compact group ( implying the number of negative metric generators minus positive ones.)

It will be instructive to see explicitly how the 70 scalar (J = 0) fields appear in the N = 8 supergravity lagrangian. The 133 generators of E_{7(+7)} group are given by 63 generators T_{A}^{B} (A, B = 1, 2, ⋯, 8) of its maximal subgroup SU(8) plus 70 generators X_{ABCD} which is totally anti-symmetric and self-dual i.e.,

X̄^{ABCD} = (X_{ABCD})^{∗} = \frac{1}{4!}ϵ^{ABCDEF GH}X_{EF GH}.

The infinitesimal transformation

δ = iΛ_{A}^{B}T_{B}^{A} + \frac{1}{4!} (Σ̄_{ABCD}X̄^{ABCD} + Σ̄^{ABCD}X_{ABCD})

with parameters Λ_{A}^{B} and Σ_{ABCD},

Σ̄^{ABCD} = (Σ_{ABCD})^{∗} = −\frac{1}{4!}ϵ^{ABCDEF GH}Σ_{EF GH}

is expressed as follows in the case of fundamental representation 56 whose basis vector is given by (Z_{AB}, Z̄^{AB}) with Z_{AB} = −Z_{BA}, Z̄^{AB} = (Z_{AB})^{∗} :

δ Z_{AB} = Λ_{A}^{C}Z_{CB} + Λ_{B}^{C}Z_{AC} + Σ_{ABCD}Z̄^{CD},
δ Z̄^{AB} = Λ̄_{C}^{A}Z̄^{CB} + Λ̄_{C}^{B}Z̄^{AC} + Σ̄^{ABCD}Z_{CD}.  (12)

The basic variable ξ in the scalar field sector is the E_{7(+7)} group element

ξ(x) = e^{iϕ_{A}^{B}(x)T_{B}^{A}} · e^{^{\frac{1}{4!}}(ϕ_{ABCD}(x)X̄^{ABCD}+ϕ̄^{ABCD}(x)X_{ABCD})},

which is a 56×56 matrix by taking the matrix representation of T_{A}^{B} and X_{ABCD} in the above fundamental representation 56, and transforms under g ∈ E_{7(+7)global} and h(x) ∈ SU(8)_{local} as

ξ(x) → ξ′(x) = h(x)ξ(x)g^{†}.

Then the Maurer-Cartan 1-form becomes

α_{μ} ≡ \frac{1}{i}∂_{μ}ξ · ξ^{†} = ( \begin{matrix}2Q_{μ[A}^{[C}δ_{B]}^{D]} & P_{μABCD} \\ P̄_{μ}^{ABCD} & 2Q̄_{μ[C}^{[A}δ_{D]}^{B]} \end{matrix} )

where

Q_{μ[A}^{[C}δ_{B]}^{D]} = \frac{1}{2}(Q_{μA}^{C}δ_{B}^{D} − Q_{μA}^{D}δ_{B}^{C} − Q_{μB}^{C}δ_{A}^{D} + Q_{μB}^{D}δ_{A}^{C}).

The Q_{μ} part corresponds to α_{μ||}, proportional to the unbroken” generators T_{A}^{B} of H = SU(8), and the P_{μ} part, to α_{μ⊥}, proportional to the broken” generators X_{ABCD} and X̄^{ABCD} in the terminology of section 4.1. Therefore the scalar field lagrangian invariant under E_{7(+7)global} and SU(8)_{local} is given by

L_{scalar} ∼ tr (α_{μ⊥})^{2} ∼ P_{μABCD}P̄^{μABCD}.

This is exactly the lagrangian of the scalar field sector in N = 8 supergravity found by Cremmer and Julia. The original 70 scalar plus pseudo-scalar fields correspond to ϕ_{ABCD} and ϕ̄^{ABCD} which survive the SU(8)_{local} gauge fixing ϕ_{A}^{B} = 0.

It is also interesting to see that the kinetic term of 8 Rarita-Schwinger fields ψ_{μA} take the following form in the N = 8 supergravity lagrangian:

\frac{1}{2}ϵ^{μνρσ}ψ̄_{μA}γ_{σ}γ_{5}(δ_{A}^{B}D_{ν} − Q_{νA}^{B})ψ_{ρB}.

where D_{μ} is local Lorentz covariant derivative. This form is understandable since α_{μ||} ∼ Q_{μA}^{B} transforms inhomogeneously just like a hidden local SU(8) gauge field and hence δ_{A}^{B}D_{ν} −Q_{νA}^{B} is an SU(8)_{local}-covariantized derivative on ψ_{μA}, transforming as \underline{8} of SU(8)_{local}.

Table I. Supergravity σ-model symmetries.

D  G  H
9  GL(2, IR)  SO(2)
8 SL(3, IR) × SL(2, IR) SO(3) × SO(2)
7  SL(5, IR)  SO(5)
6  SO(5, 5)  SO(5) × SO(5)
5  E_{6(+6)}  USP(8)
4  E_{7(+7)}  SU(8)
3  E_{8(+8)}  SO(16)

E_{7(+7)} is an on-shell symmetry:

Vector B_{μν}^{ij} ∈ 28. The representation 56 of E_{7(+7)} is given by

( \begin{matrix}F_{μν}^{ij} \\ G_{μν}^{ij} \end{matrix} )  F_{μν}^{ij} : field strength of B_{μν}^{ij} G̃_{μν}^{ij} ≡ \frac{4}{e} \frac{δS}{δF_{μν}^{ij}}  (13)

so that

Binachi: ∂_{μ}(e F̃^{μν ij}) = 0 ↔ eq. of motion: ∂_{μ}(e G̃^{μν ij}) = 0.  (14)

→ E_{7(+7)} is a (Cremmer-Julia) duality symmetry.

§3. Superconformal group SU(2, 2|1)

4D Conformal Group: SO(4, 2) ≅ SU(2, 2)

[M_{μν}, M_{ρσ}] = −i(η_{μρ}M_{νσ} − η_{νρ}M_{μσ} − η_{μσ}M_{νρ} + η_{νσ}M_{μρ}),
[P_{ρ}, M_{μν}] = i(η_{ρμ}P_{ν} − η_{ρν}P_{μ}),
[K_{ρ}, M_{μν}] = i(η_{ρμ}K_{ν} − η_{ρν}K_{μ}),
[D, M_{μν}] = [P_{μ}, P_{ν}] = [K_{μ}, K_{ν}] = 0,
[P_{μ}, D] = iP_{μ}, [K_{μ}, D] = −iK_{μ},
[P_{μ}, K_{ν}] = 2i(η_{μν}D − M_{μν}).  (15)

This 4D conformal group is in fact identical with the extended Lorentz group SO(4, 2) in 6 dimensions with metric

η_{ab} = ( \begin{matrix}η_{μν} \\ & −1 \\ & & +1 \end{matrix} ) ,  (16)

for which the generators M_{ab} = −M_{ba} (a, b = 0, 1, ⋯, 5) satisfy

[M_{ab}, M_{cd}] = −i(η_{ac}M_{bd} − η_{bc}M_{ad} − η_{ad}M_{bc} + η_{bd}M_{ac}).  (17)

and

M_{μ4} ≡ \frac{1}{2}(P_{μ} − K_{μ}), M_{μ5} ≡ \frac{1}{2}(P_{μ} + K_{μ}), M_{54} ≡ D.  (18)

By considering the (Weyl) spinor representation, this algebra is also seen to be isomorphic with SU(2, 2). The generators Γ^{a} of the Clifford algebra for SO(4, 2), can be represented, for instance, by the following 8 × 8 matrices:

Γ^{μ} = γ^{μ} ⊗ σ^{1} = ( \begin{matrix}0 & γ^{μ} \\ γ^{μ} & 0 \end{matrix} ) ,
Γ^{4} = iγ_{5} ⊗ σ^{1} = ( \begin{matrix}0 & iγ_{5} \\ iγ_{5} & 0 \end{matrix} ) ,
Γ^{5} = 1_{4} ⊗ (−σ^{2}) = ( \begin{matrix}0 & i1_{4} \\ −i1_{4} & 0 \end{matrix} ) .  (19)

The Lorentz generators M_{ab} of SO(4, 2) are then represented by

M_{ab} = \frac{i}{4}[Γ_{a}, Γ_{b}] = \frac{1}{2} ( \begin{matrix}σ_{ab} & 0 \\ 0 & σ̄_{ab} \end{matrix} ) ,

σ_{a=μ,b=ν} = σ_{μν}, σ_{μ4} = γ_{μ}γ_{5}, σ_{μ5} = γ_{μ}, σ_{54} = iγ_{5}

σ̄_{a=μ,b=ν} = σ_{μν}, σ̄_{μ4} = γ_{μ}γ_{5}, σ̄_{μ5} = −γ_{μ}, σ̄_{54} = −iγ_{5}.  (20)

Clearly the 4-component Weyl spinor gives an irreducible representation of the Lorentz group SO(4, 2), for which the Lorentz group element Λ = exp(\frac{i}{2}ε^{ab}M_{ab}) is represented by

exp(\frac{i}{4}ε^{ab}σ_{ab}).  (21)

These 4 × 4 matrices belong to SU(2, 2) since σ_{ab} are traceless and hermitian under the metric a ≡ γ^{0} (which has two +1 and two −1 eigenvalues). Moreover, Since 6×5/2 = 15 σ_{ab} exist and give a complete set for such traceless and hermitian 4 × 4 matrices, any SU(2, 2) matrix is expressed in the form Eq. (21) (at least in the neighborhood of the identity) and so we have the isomorphism of the algebra SO(4, 2) ≃ SU(2, 2).

We have seen that the 4D conformal algebra SO(4, 2) ≃ SU(2, 2) can be represented by traceless 4 × 4 matrices M_{ab} = \frac{1}{2}σ_{ab} acting on a 4-component spinor ψ. Then it is clear that it can be extended to the superconformal algebra SU(2, 2|1) acting on a (4+1)- component super-spinor (ψ, φ) by adding another single component φ (which should have opposite statistics to the original component ψ). SU(2, 2|1) is defined to be a supergroup consisting of 5 × 5 matrices (of unimodular superdeterminant) which leave the innerproduct

ψ_{1}^{†}γ_{0}ψ_{2} + φ_{1}^{†}φ_{2}  (22)

invariant. Clearly, there are 24 independent generators as a whole, which we can take, for instance,

M_{ab} = \frac{1}{2} ( \begin{matrix}σ_{ab} & 0 \\ 0 & 0 \end{matrix} ) , A = −\frac{1}{4} ( \begin{matrix}1_{4} & 0 \\ 0 & 4 \end{matrix} ) ,

Σ_{α} = 2 ( \begin{matrix}0_{4} & 0 \\ δ_{α}^{j} & 0 \end{matrix} ) ,  Σ̄^{α} = 2 ( \begin{matrix}0_{4} & δ_{i}^{α} \\ 0 & 0 \end{matrix} ) ,  (23)

Note that a diagonal (supertraceless) matrix A appears here. This gives the defining representation of SU(2, 2|1) algebra. From this we can easily find the following albebra written in 6 dimensinal notation:

[Σ, M_{ab}] = \frac{1}{2}σ_{ab}Σ, [Σ̄, M_{ab}] = −\frac{1}{2}Σ̄σ^{ab}
[Σ, A] = +\frac{3}{4}Σ,  [Σ̄, A] = −\frac{3}{4}Σ̄,  [M_{ab}, A] = 0,
{Σ, Σ} = {Σ̄, Σ̄} = 0,  {Σ, Σ̄} = σ^{ab}M_{ab} − 4A.  (24)

This shows that Σ is an SU(2, 2) ≃ SO(4, 2) spinor generator and Σ̄ charge is its conjugate, so that they can be decomposed into two 2-component Weyl spinors in 4-dimension as follows:

Σ = ( Q_{α} ; S̄α̇ ) ,  Σ̄ = Σ^{†}γ_{0} = (S^{α}, Q̄α̇).  (25)

Clearly, these 15+4+4+1 = 24 matrices again span a complete set of 5×5 (supertraceless) matrices and give the whole generators of SU(2, 2|1) superconformal algebra. The SU(2, 2|1) group acts on the 5 component super spinor as

exp i(\frac{1}{2}θ^{ab}M_{ab} + θA + ε̄Σ + Σ̄ε) ( \begin{matrix}ψ \\ φ \end{matrix} )  (26)

(where ψ is an SU(2, 2) spinor field and φ a single component field and they should be fermion and boson (or vice versa), respectively, since the spinor transformation parameter ε is Grassmann odd) with ε̄ = ε^{†}a, which leaves invariant the innerproduct with metric

α ≡ ( \begin{matrix}γ^{0} \\ & 1 \end{matrix} ) .  (27)

Rewriting Eq. (24) into 4 dimensional notation, we find the following algebra in addition to the SO(4, 2) ≃ SU(2, 2) subalgebra:

[( \begin{matrix}Q \\ S \end{matrix} ) , M_{μν}] = \frac{1}{2}σ_{μν} ( \begin{matrix}Q \\ S \end{matrix} ) , [( \begin{matrix}Q \\ S \end{matrix} ) , A] = \frac{3}{4}γ_{5} ( \begin{matrix}Q \\ −S \end{matrix} ) ,

\begin{matrix}[Q, P_{μ}] = 0, & [Q, K_{μ}] = γ_{μ}S, \\ [S, P_{μ}] = γ_{μ}Q, & [S, K_{μ}] = 0, \end{matrix} , [( \begin{matrix}Q \\ S \end{matrix} ) , D] = i\frac{1}{2} ( \begin{matrix}Q \\ −S \end{matrix} ) ,

[A, M_{μν}] = [A, P_{μ}] = [A, K_{μ}] = [A, D] = 0, {Q, Q̄} = 2γ^{μ}P_{μ}, {S, S̄} = 2γ^{μ}K_{μ}, {S, Q̄} = 2iD + σ^{μν}M_{μν} + 4γ_{5}A, ({Q, S̄} = −2iD + σ^{μν}M_{μν} − 4γ_{5}A).

(28)

§4. Yang-Mills theory for superalgebra

Consider a superalgebra whose generators (devided by i), X_{A} = T_{A}/i satisfying

[X_{A}, X_{B}} = f_{AB}^{C}X_{C}.  (29)

For definiteness, we here mean by X_{A} always a certain matrix representation acting a supermultiplet matter field Φ, for which the infinitesimal transformation is given by

δ(ε)Φ = εΦ,  ε ≡ ε^{A}X_{A},  (30)

where ε^{A} are the transformation parameters. Introduce the gauge field by

h_{μ} = h_{μ}^{A}X_{A}.  (31)

The covariant derivative

D_{μ}Φ ≡ (∂_{μ} − h_{μ})Φ  (32)

is defined by a property

δ(ε)(D_{μ}Φ) = ε(D_{μ}Φ),  (33)

from which follows

δ(ε)h_{μ} = ∂_{μ}ε + [ε, h_{μ}], → δ(ε)h_{μ}^{A} = ∂_{μ}ε^{A} + ε^{B}h_{μ}^{C}f_{BC}^{A}.  (34)

The curvature tensor (field strength) is defined by

R_{μν} ≡ [D_{μ}, D_{ν}] = ∂_{ν}h_{μ} − ∂_{μ}h_{ν} − [h_{ν}, h_{μ}],

→ R_{μν}^{A} = ∂_{ν}h_{μ}^{A} − ∂_{μ}h_{ν}^{A} − h_{ν}^{B}h_{μ}^{C}f_{BC}^{A}.  (35)

The curvature tensor is covariant as usual:

δ(ε)R_{μν} = [ε, R_{μν}], → δ(ε)R_{μν}^{A} = ε^{B}R_{μν}^{C}f_{BC}^{A}.  (36)
X_{A} = i^{−1}(P_{m}, Q, M_{mn}, D, A, S, K_{m})
≡ (P_{m}, Q, M_{mn}, D, A, S, K_{m}),
ε^{A}X_{A} = ξ^{m}P_{m} + ε̄Q + \frac{1}{2}λ^{mn}M_{mn} + ρD + θA + ζ̄S + ξ_{K}^{m}K_{m},
h_{μ}^{A}X_{A} = e_{μ}^{m}P_{m} + ψ̄_{μ}Q + \frac{1}{2}ω_{μ}^{mn}M_{mn} + b_{μ}D + A_{μ}A + φ̄_{μ}S + f_{μ}^{m}K_{m}.

(37)

Curvatures:

R_{μν}^{m}(P) = 2∂_{ν}e_{μ}^{m} − 2ω_{ν}^{mn}e_{nμ} + 2b_{ν}e_{μ}^{m} + 2iψ̄_{ν}γ^{m}ψ_{μ},
R_{μν}^{mn}(M) = 2∂_{ν}ω_{μ}^{mn} − 2ω_{ν}^{mc}ω_{μc}^{n} + 4(f_{ν}^{m}e_{μ}^{n} − f_{ν}^{n}e_{μ}^{m}) + 4iψ̄_{ν}σ^{mn}φ_{μ},
R_{μν}(D) = 2∂_{ν}b_{μ} + 4f_{ν}^{n}e_{nμ} + 4ψ̄_{ν}φ_{μ}
R_{μν}(A) = 2∂_{ν}A_{μ} − 8iψ̄_{ν}γ_{5}φ_{μ}
R_{μν}^{m}(K) = 2∂_{ν}f_{μ}^{m} − 2ω_{ν}^{mn}f_{nμ} − 2b_{ν}f_{μ}^{m} + 2iφ̄_{ν}γ^{m}φ_{μ},
R_{μν}(Q) = 2D_{ν}^{ω}ψ_{μ} + b_{ν}ψ_{μ} − \frac{3}{2}iA_{ν}γ_{5}ψ_{μ} − 2iγ_{m}φ_{ν}e_{μ}^{m},
R_{μν}(S) = 2D_{ν}^{ω}φ_{μ} − b_{ν}φ_{μ} + \frac{3}{2}iA_{ν}γ_{5}φ_{μ} − 2iγ_{m}ψ_{ν}f_{μ}^{m},  (38)

with

D_{ν}^{ω}ψ_{μ} ≡ ∂_{ν}ψ_{μ} + \frac{i}{4}ω_{ν}^{mn}σ_{mn}ψ_{μ},  (39)

(and the same for φ_{μ},)

変換則:

δe_{μ}^{m} = ∂_{μ}ξ^{m} + λ^{ml}e_{lμ} − ω_{μ}^{mn}ξ_{n} − ρe_{μ}^{m} + b_{μ}ξ^{m} − 2iε̄γ^{m}ψ_{μ}, δω_{μ}^{mn} = ∂_{μ}λ^{mn} + 2λ^{ml}ω_{μl}^{n} − 2(ξ_{K}^{m}e_{μ}^{n} − ξ_{K}^{n}e_{μ}^{m}) + 2(f_{μ}^{m}ξ^{n} − f_{μ}^{n}ξ^{m})

−2iε̄σ^{mn}φ_{μ} − 2iψ̄_{μ}σ^{mn}ζ,

δb_{μ} = ∂_{μ}ρ − 2ξ_{K}^{n}e_{nμ} + 2f_{μ}^{n}ξ_{n} − 2ε̄φ_{μ} + 2ψ̄_{μ}ζ δA_{μ} = ∂_{μ}θ + 4iε̄γ_{5}φ_{μ} − 4iψ̄_{μ}γ_{5}ζ δf_{μν}^{m} = ∂_{μ}ξ_{K}^{m} + λ ^{mn}f_{nμ} − ω_{μ}^{mn}ξ_{K n} + ρf_{μ}^{m} − b_{μ}ξ_{K}^{m} − 2iζ̄γ^{m}φ_{μ}, δψ_{μ} = D_{μ}^{ω}ε − \frac{i}{4}λ^{mn}σ_{mn}ψ_{μ} − \frac{1}{2}ρψ_{μ} + \frac{1}{2}b_{μ}ε + \frac{3}{4}iθγ_{5}ψ_{μ} − \frac{3}{4}iA_{μ}γ_{5}ε

+ie_{μ}^{m}γ_{m}ζ − iξ^{m}γ_{m}φ_{μ},

δφ_{μ} = D_{μ}^{ω}ζ − \frac{i}{4}λ^{mn}σ_{mn}φ_{μ} + \frac{1}{2}ρφ_{μ} − \frac{1}{2}b_{μ}ζ − \frac{3}{4}iθφ_{μ}γ_{5} + \frac{3}{4}iA_{μ}ζγ_{5}

+if_{μ}^{m}γ_{m}ε − iξ_{K}^{m}γ_{m}ψ_{μ},  (40)

For inverse vierbein,

δe_{m}^{μ} = −e_{n}^{μ}e_{m}^{ν}(δe_{ν}^{n})
= −e_{n}^{μ}∂_{m}ξ^{n} − e_{l}^{μ}λ_{m}^{l} + ω_{m}^{μl}ξ_{l} + ρe_{m}^{μ} − b_{m}ξ^{μ} + 2iε̄γ^{μ}ψ_{m}.  (41)

Curvature の group 変換則 δR_{μν}^{A} は、上の gauge 場の変換則 δh_{μ}^{A} で、 ∂_{μ}ε^{A} を捨て、全ての h_{μ}^{B} を R_{μν}^{B} に置き換えれば良い。

§5. Deformation of the the SU(2, 2|1) algebra

δ_{GC}(ξ^{λ})h_{μ}^{A} = ∂_{μ}ξ^{λ}·h_{λ}^{A} + ξ^{λ}∂_{λ}h_{μ}^{A}
= D_{μ}(ξ^{λ}·h_{λ}^{A}) + ξ^{λ}(∂_{λ}h_{μ}^{A} − D_{μ}h_{λ}^{A})
= [D_{μ}(ξ·h)]^{A} + ξ^{λ}R_{μν}^{A}
= δ(ξ·h) h_{μ}^{A} + ξ^{λ}R_{μν}^{A},  (42)

The last equality is because

δ(ε)h_{μ}^{A} = (D_{μ}ε)^{A}
δ(ξ·h)h_{μ}^{A} = ∂_{μ}(ξ·h^{A}) + (ξ·h)^{B}h_{μ}^{C}f_{BC}^{A}.  (43)

Note that

δ(ξ·h) = δ_{P} (ξ^{m}) + ∑_{A′(≠P)} δ_{A′}(ξ·h^{A′}),

ξ^{m} = ξ^{λ}e_{λ}^{m},
ξ·h^{A′} = ξ^{λ}h_{λ}^{A′} = ξ^{m}h_{m}^{A′}  (44)

Therefore, we have a key relation:

δ_{P} (ξ^{m})h_{μ}^{A} = \underbrace{δ_{GC}(ξ^{λ})h_{μ}^{A} − ∑_{B′} δ_{B′}(ξ·h^{B′})h_{μ}^{A}}_{≡ δ_{P}(ξ^{m})} −ξ^{λ}R_{μλ}^{A}.  (45)

Now, we deform the SU(2, 2|1) algebra by making a replacement

δ_{P} (ξ^{m}) → δ_{P}(ξ^{m}) = δ_{GC}(ξ^{λ}) − ∑_{B′} δ_{B′}(ξ·h^{B′}).  (46)

First we note that, among the commutators [δ_{A′}, δ_{B′}] for A′, B′ ≠ P, the only one yielding δ_{P} in the RHS is [δ_{Q}(ε_{2}), δ_{Q}(ε_{1})] = δ_{P} (−2iε̄_{1}γ^{m}ε_{2}). So we require first that

[δ_{Q}(ε_{2}), δ_{Q}(ε_{1})] = δ_{P}(ξ^{m}), with ξ^{m} ≡ −2iε̄_{1}γ^{m}ε_{2},  (47)

holds on any independent gauge fields, and find constraints necessary for that.

5.1. On e_{μ}^{m} On e_{μ}^{m}, we originary have

[δ_{Q}(ε_{2}), δ_{Q}(ε_{1})]e_{μ}^{m} = δ_{P} (ξ^{m})e_{μ}^{m},
= δ_{P}(ξ^{m})e_{μ}^{m} − ξ^{λ}R_{μλ}^{m}(P).  (48)

So it is necessary and sufficient to impose the constraint:

0 = R_{μν}^{m}(P) = 2∂_{ν}e_{μ}^{m} − 2ω_{ν}^{mn}e_{nμ} + 2b_{ν}e_{μ}^{m} + 2iψ̄_{ν}γ^{m}ψ_{μ}  (49)

This can be solved by the M gauge field ω_{μ}^{mn} and yields

ω_{μ}^{mn} = ω_{μ}^{mn}(e, ψ, b),  (50)

so that ω_{μ}^{mn} is no longer an independent gauge field. However, since the constraint R_{μν}^{m}(P) = 0 is invariant under M_{mn}, D, A, S, K_{m}, ω_{μ}^{mn} still keeps the same transformation law as the original group transformation under M_{mn}, D, A, S, K_{m} transformations. On the other hand, the constraint R_{μν}^{m}(P) = 0 is not invariant under Q transformation, the Q transformation of ω_{μ}^{mn} becomes different from the original group transformation law:

δ_{Q}(ε)ω_{μ}^{mn}(e.ψ, b) = δ_{Q}^{group}(ε)ω_{μ}^{mn} + δ′_{Q}(ε)ω_{μ}^{mn}.  (51)

The difference can be easily found by noting that the constraint R_{μν}^{m}(P) = 0 is of course an identity and Q-invariant if ω_{μ}^{mn} there is replaced by ω_{μ}^{mn}(e, ψ, b), so that we have

0 = δ_{Q}^{group}(ε)R_{μν}^{m}(P) + δ′_{Q}(ε)ω_{μ ν}^{m} − δ′_{Q}(ε)ω_{ν μ}^{m}
= −2iε̄γ^{m}R_{μν}(Q) + δ′_{Q}(ε)ω_{μ ν}^{m} − δ′_{Q}(ε)ω_{ν μ}^{m}.  (52)

(Note that we are anticipating that e_{μ}^{m}, ψ_{μ}, b_{μ} will remain to be independent gauge fields and receive no changes in the Q-transformation laws.) Solving this (in a similar way to solve Christoffel symbol in terms of g_{μν}), we find

δ′_{Q}(ε)ω_{μ mn} = iε̄(γ_{μ}R_{mn}(Q) + γ_{m}R_{μn}(Q) − γ_{n}R_{μm}(Q)) ≡ iε̄R_{μmn}(Q).  (53)

5.2. On ψ_{μ}

Noting

δ_{Q}(ε)ψ_{μ} = (∂_{μ} + \frac{i}{4}ω_{μ}^{mn}σ_{mn} + \frac{1}{2}b_{μ} − \frac{3}{4}iγ_{5}A_{μ})ε  (54)

and that ω_{μ}^{mn} now receives an extra Q transformation δ′_{Q}(ε) in addition to the original group transformation δ_{Q}^{group}(ε), we find that the [δ_{Q}, δ_{Q}] commutator on ψ_{μ} now reads

[δ_{Q}(ε_{2}), δ_{Q}(ε_{1})]ψ_{μ} = [δ_{Q}^{group}(ε_{2}), δ_{Q}^{group}(ε_{1})]ψ_{μ} + \frac{i}{4}(δ′_{Q}(ε_{2})ω_{μ}·σε_{1} − (1 ↔ 2))
= δ_{P}(ξ)ψ_{μ} − ξ^{m}R_{μm}(Q) + \frac{i}{4}(δ′_{Q}(ε_{2})ω_{μ}·σε_{1} − (1 ↔ 2)). (55)

So we see that the condition

\frac{i}{4}((iε̄_{2}R_{μ mn}(Q)) σ^{mn}ε_{1} − (1 ↔ 2)) = −2i(ε̄_{1}γ^{m}ε_{2})R_{μm}(Q)  (56)

should hold. From this, after some some calculations like Fierzing, we find a constraint

γ^{ρ}R_{μρ}(Q) = 0.  (57)

is the necessary and sufficient condition for the [δ_{Q}, δ_{Q}] algebra Eq. (47) hold on ψ_{μ}. The extra Q transformation for ω_{μ}^{mn} now takes a simple form:

δ′_{Q}(ε)ω_{μ mn} = 2iε̄γ_{μ}R_{mn}(Q) (= −2iR̄_{mn}(Q)γ_{μ}ε).  (58)

The constraint (57), γ^{μ}R_{μν}(Q) = 0, is solved by the S-gauge field φ_{μ}:

0 = γ^{μ}R_{μν}(Q) = γ^{μ}[(∂_{ν} + \frac{i}{4}ω_{ν} ·σ + \frac{1}{2}b_{ν} − \frac{3}{4}iγ_{5}A_{ν})ψ_{μ} − (1 ↔ 2)] − iγ^{μ}(γ_{μ}φ_{ν} − γ_{ν}φ_{μ})
⇒ φ_{μ} = φ_{μ}(e, ψ, b, A).  (59)

So φ_{μ} now become dependent gauge field. Since the constraint γ^{μ}R_{μν}(Q) = 0 is M_{mn}, D, A, S, K_{m} invariant but not invariant under Q, the Q-transformation of φ_{μ} is modified:

0 = δ_{Q}(ε)(γ^{μ}R_{μν}(Q)) = γ^{μ}δ_{Q}^{group}(ε)R_{μν}(Q) + (δ_{Q}(ε)e_{m}^{μ})γ^{m}R_{μν}(Q)
+\frac{i}{4}γ^{μ}[(δ′_{Q}(ε)ω_{ν}^{mn})σ_{mn}ψ_{μ} − (μ ↔ ν)] − i(4δ_{ν}^{μ} − γ^{μ}γ_{ν})δ′_{Q}(ε)φ_{μ}  (60)

where

δ_{Q}(ε)e_{m}^{μ} = 2iε̄γ^{μ}ψ_{m}
δ_{Q}^{group}(ε)R_{μν}(Q) = (\frac{i}{4}R_{μν}(M)·σ + \frac{1}{2}R_{μν}(D) − \frac{3}{4}iγ_{5}R_{μν}(A))ε  (61)

After some calculations, we find this leads to:

δ′_{Q}(ε)φ_{μ} = −\frac{i}{2}(δ_{μ}^{ν} − \frac{1}{6}γ_{μ}γ^{ν})R_{ν}ε = −\frac{i}{2}(R_{μ} − \frac{1}{6}γ_{μ}γ·R)ε
R_{μ} ≡ \frac{i}{4}γ^{μ}σ^{mn}εR_{μν mn}^{cov.}(M) + \frac{1}{2}γ^{μ}εR_{μν}(D) − \frac{3}{4}iγ^{μ}γ_{5}εR_{μν}(A).  (62)

This quantity (R_{μ} − \frac{1}{6}γ_{μ}γ·R)ε can be much simplified if we use the Bianchi identity. The Bianchi identity

0 = ε^{μνρσ}[D_{ν}, R_{ρσ}] = ε^{μνρσ}(∂_{ν}R_{ρσ} − [h_{μ}, R_{ρσ}]).
→ ε^{μνρσ}(∂_{ν}R_{ρσ}^{A} − h_{μ}^{B}R_{ρσ}^{C}f_{BC}^{A}) = 0.  (63)

for A = P_{m} leads to identities like

ε_{m}^{abc}R_{na bc}^{cov.}(M) = −2R̃_{mn}(D).  (64)
R_{μν}^{cov.}(M)|_{antisymm. part} ≡ \frac{1}{2}(R_{μν}^{cov.}(M) − R_{νμ}^{cov.}(M)) = −R_{μν}(D),  (65)

where

R_{μν}^{cov.}(M) ≡ R_{μρ}^{cov. mn}(M)e_{m}^{ρ}e_{nν}.  (66)

Using those, we eventually find that the extra Q transformation δ′_{Q}(ε)φ_{μ} is given by

δ′_{Q}(ε)φ_{μ} = −\frac{i}{2}(R_{μ} − \frac{1}{6}γ_{μ}γ·R)ε
= −\frac{i}{2}[γ^{m}ε ( − \frac{1}{12}e_{mμ}R^{cov.}(M)_{ρ}^{ρ} + \frac{1}{2}R_{μm}^{cov.}(M) + \frac{1}{4}R̃_{μm}(A))
+iγ^{m}γ_{5}ε \frac{1}{2}R_{μm}(A)]  (67)

5.3. On A_{μ} and b_{μ} Noting

δ_{Q}(ε)A_{μ} = 4iε̄γ_{5}φ_{μ}
δ_{Q}(ε)b_{μ} = −2ε̄φ_{μ},  (68)

and that φ_{μ} now receives an extra Q transformation δ′_{Q}(ε) in addition to the original group transformation δ_{Q}^{group}(ε), we find that the [δ_{Q}, δ_{Q}] commutator on A_{μ} and b_{μ} now reads

[δ_{Q}(ε_{2}), δ_{Q}(ε_{1})]A_{μ} = [δ_{Q}^{group}(ε_{2}), δ_{Q}^{group}(ε_{1})]A_{μ} + 4i(ε̄_{1}γ_{5}(δ′_{Q}(ε_{2})φ_{μ}) − (1 ↔ 2))
= δ_{P}(ξ)A_{μ} − ξ^{m}R_{μm}(A) + 4i((−\frac{i}{2})(ε̄_{1}γ_{5}iγ^{m}γ_{5}ε_{2}) \frac{1}{2}R_{μm}(A) − (1 ↔ 2))
= δ_{P}(ξ)A_{μ} − ξ^{m}R_{μm}(A) + 4i(−\frac{i}{2})ξ^{m} \frac{1}{2}R_{μm}(A) = δ_{P}(ξ)A_{μ} OK!
[δ_{Q}(ε_{2}), δ_{Q}(ε_{1})]b_{μ} = [δ_{Q}^{group}(ε_{2}), δ_{Q}^{group}(ε_{1})]b_{μ} − 2(ε̄_{1}(δ′_{Q}(ε_{2})φ_{μ}) − (1 ↔ 2))
= δ_{P}(ξ)b_{μ} − ξ^{m}R_{μm}(D)

−2(−\frac{i}{2}) × 2(ε̄_{1}γ^{m}ε_{2})( − \frac{1}{12}e_{mμ}R^{cov.}(M)_{ρ}^{ρ} + \frac{1}{2}R_{μm}^{cov.}(M) + \frac{1}{4}R̃_{μm}(A))

= δ_{P}(ξ)b_{μ} − ξ^{m}R_{μm}(D)
−ξ^{m}( − \frac{1}{12}e_{mμ}R^{cov.}(M)_{ρ}^{ρ} + \frac{1}{2}R_{μm}^{cov.}(M) + \frac{1}{4}R̃_{μm}(A))  (69)

Thus, the [δ_{Q}, δ_{Q}] commutator on A_{μ} requires no constraint but that on b_{μ} requires a condition

− \frac{1}{12}e_{mμ}R^{cov.}(M)_{ρ}^{ρ} + \frac{1}{2}R_{μm}^{cov.}(M) + \frac{1}{4}R̃_{μm}(A) = −R_{μm}(D)  (70)

which leads, by separating the symmetric and antisymmetric parts and using Eq. (65), to

R_{μm}^{cov.}(M)|_{symm. part} = 0,
−\frac{1}{2}R_{μm}(D) + \frac{1}{4}R̃_{μm}(A) = −R_{μm}(D).  (71)

The latter condition is rewritten into

R_{μm}(D) = −\frac{1}{2}R̃_{μm}(A) or → R̃_{μm}(D) = +\frac{1}{2}R_{μm}(A).  (72)

If Eq. (65) is used, these two conditions can be rewritten into a constraint

R_{νμ}^{cov.}(M) + \frac{1}{2}R̃_{μν}(A) = 0.  (73)

This is the necessary and sufficient condition for the [δ_{Q}, δ_{Q}] algebra Eq. (47) to hold on b_{μ}. Then the extra Q transformation Eq. (67) of φ_{μ} is simplified into

δ′_{Q}(ε)φ_{μ} = −\frac{i}{4}γ^{m}(R̃_{μm}(A) + iγ_{5}R_{μm}(A))ε.  (74)

The constraint Eq. (73) can be solved by the K_{m} gauge field f_{μ}^{m}, which now becomes a dependent field:

f_{μ}^{m} = f_{μ}^{m}(e, ψ, b, A).  (75)

Since the constraint Eq. (73) is not Q-invariant and so f_{μ}^{m} also receives an extra Q-transformation, which can be derived in the same way as above:

δ′_{Q}(ε)f_{μ}^{m} = −\frac{i}{2}ε̄(σ^{mν}R_{μν}^{cov.}(S) + e^{mν}R̃_{μν}^{cov.}(S)).  (76)

5.4. Resultant modified SU(2, 2|1) algebra

Now that the M_{mn}, S and K_{m} gauge fields ω_{μ}^{mn}, φ_{μ} and f_{μ}^{m} have become dependent fields, there no longer remain other independent gauge fields. Thus the desired [δ_{Q}, δ_{Q}] algebra (47)

[δ_{Q}(ε_{2}), δ_{Q}(ε_{1})] = δ_{P}(ξ^{m}), with ξ^{m} ≡ −2iε̄_{1}γ^{m}ε_{2}  (77)

already holds on all the independent gauge fields e_{μ}^{m}, ψ_{μ}, A_{μ} and b_{μ}.

This implies that

Proposition: For all the transformations other than P̃_{m} transformation, (which we denote by primed index X′ henceforth, X′ ∈ {Q, M_{mn}, D, A, S, K_{m}}), the commutators

[δ_{Y′}(ε^{Y′}), δ_{X′}(ε^{X′})] = ∑_{C} δ_{C}(ε^{X′}ε^{Y′}f_{X′Y′}^{C})  (78)

of the same form as the original SU(2, 2|1) algebra, still hold. Note that when P_{m} appears in the C sum, it is always understood to stand for P̃_{m}.

Proof) Almost trivial.

Proposition:

[δ_{P}(ξ^{m}), δ_{A′}(ε^{A′})] = ∑_{B all} δ_{B}(ε^{A′}ξ^{m}f_{A′P_{m}}^{B}) + δ_{A′}^{Q} ∑_{B′=M,S,K} δ_{B′}(ξ^{m}δ′_{Q}(ε^{A′})h_{m}^{B′})  (79)

Proof) Straightforward calculation using

δ_{P}(ξ^{m}) = δ_{GC}(ξ^{λ} = ξ^{m}e_{m}^{λ}) − ∑_{B′} δ_{B′}(ξ·h^{B′}).  (80)

Proposition:

[δ_{P}(ξ_{1}), δ_{P}(ξ_{2})] = ∑_{A} δ_{A}(ξ_{1}^{m}ξ_{2}^{n}R_{mn}^{A}) + ∑_{B′=M,S,K} δ_{B′}(δ′_{Q}(ξ_{1}·ψ)ξ_{2}·h^{B′} − δ′_{Q}(ξ_{2}·ψ)ξ_{1}·h^{B′})

(81) or, equivalently,

[δ_{P}(ξ_{1}), δ_{P}(ξ_{2})] = ∑_{A} δ_{A}(ξ_{1}^{m}ξ_{2}^{n}R_{mn}^{cov. A})  (82)

Proof) Straightforward calculation.

Also note

R_{mn}^{cov. A} = R_{mn}^{A} − (δ′_{Q}(ψ_{n})h_{m}^{A} − δ′_{Q}(ψ_{m})h_{n}^{A})  (83)

§6. N = 1 Superconformal Tensor Calculus

6.1. Matter multiplets

The general, or so-called vector, (complex, unconstrained) superconformal multiplet V corresponding to the superfield

V(x, θ) = C(x) + θ̄Z(x) + θ̄θH(x) + θ̄iγ_{5}θK(x) + \frac{1}{2}θ̄iγ^{m}γ_{5}θB_{m}(x)
+(θ̄θ)θ̄(Λ(x) + \frac{1}{2} / ∂Z(x)) + \frac{1}{4}(θ̄θ)^{2}(D(x) + \frac{1}{2} C(x))  (84)

in the case of usual supersymmetry, is now denoted by V = [C, Z, H, K, B_{m}, Λ, D]. (Real vector multiplet is denoted as V = [C, Z, H, K, B_{m}, λ, D], by using the corresponding roman letters.) The basic quantum numbers of the superconformal matter multiplet are Weyl weight w and chiral weight n, which are defined through the transformation law of the first component C:

[δ_{D}(ρ) + δ_{A}(θ)] C(x) = (wρ + \frac{1}{2}inθ) C(x).  (85)

This vector multiplet V exists for any Weyl and chiral weights w, n (and even V_{A} with arbitrary external Lorentz index A = (α_{1}, ⋯, α_{n}; β̇_{1}, ⋯, β̇_{m}). On the contrary, the constrained type multiplets can exist only for particular values of (w, n) (and for particular external Lorentz indices A). For instance, the chiral multiplets exist only when they carry the same values of Weyl and chiral weights, w = n (and only with purely undotted spinor indices A = (α_{1}, ⋯, α_{n})).

Here we do not give the transformation laws for the vector multiplet V, but give those for the chiral multiplet ϕ = [z, χ, h] possessing no external Lorentz index, which is embedded into the vector multiplet as follows:

V(ϕ) = [ z, −iχ_{R}, −h, ih, iD_{m}^{c}z, 0, 0 ].  (86)

The chiral multiplet transforms under Q, S, D and A as

δ_{QSDA} z ≡ (δ_{Q}(ε) + δ_{S}(ζ) + δ_{D}(ρ) + δ_{A}(θ)) z = \frac{1}{2}ε̄_{R}χ_{R} + (wρ + \frac{1}{2}iwθ)z
δ_{QSDA} χ_{R} = \begin{matrix}/ \\ D \end{matrix}^{c}z · ε_{L} + hε_{R} + 2wzζ_{R} + [(w + \frac{1}{2})ρ + i(\frac{1}{2}w − \frac{3}{4})θ]χ_{R}
δ_{QSDA} h = \frac{1}{2}ε̄_{L}\begin{matrix}/ \\ D \end{matrix}^{c}χ_{R} + (1 − w)ζ̄_{R}χ_{R} + [(w + 1)ρ + i(\frac{1}{2}w − \frac{3}{2})θ]h,  (87)

and inert under K_{m}, where D_{m}^{c} denotes conformal covariant derivative:

D_{m}^{c}z = (∂_{m} − wb_{m} − \frac{1}{2}iwA_{m})z − \frac{1}{2}ψ̄_{R}χ_{R}
D_{m}^{c}χ_{R} = (D_{m}^{ω} − (w + \frac{1}{2})b_{m} − i(\frac{1}{2}w − \frac{3}{4})A_{m})χ_{R}
−(\begin{matrix}/ \\ D \end{matrix}^{c}z · ψ_{Lm} + hψ_{Rm}) − 2wzφ_{Rm}  (88)

with local Lorentz covariant derivative D_{m}^{ω}.

6.2. Invariant action formula

F-term formula: applicable to chiral multiplet with weight w = n = 3, ϕ_{w=n=3} = [ z= \frac{1}{2}(A+iB), χ_{R}, h= \frac{1}{2}(F +iG) ]

I_{F} = ∫ d^{4}x [ϕ_{(w=n=3)}]_{F} = ∫ d^{4}x e [h + \frac{1}{2}ψ̄_{Lm}γ^{m}χ_{R} + ψ̄_{Lm}σ^{mn}z ψ_{Ln} + h.c.]
= ∫ d^{4}x e [F + \frac{1}{2}ψ̄_{m}γ^{m}χ + \frac{1}{2}ψ̄_{m}σ^{mn}(A − iγ_{5}B)ψ_{n}]  (89)

The next action formula can be derived from this. Since the chiral projection (analogue of D̄D̄V) of real vector multiplet V with Weyl weight w = 2 gives a chiral multiplet ΠV with weight w = n = 3:

ΠV = [ \frac{1}{2}(H − iK), i\begin{matrix}/ \\ D \end{matrix}^{c}Z_{L} + Λ_{R}, −\frac{1}{2}(D + ^{c}C + iD_{m}^{c}B^{m}) ]  (90)

We can apply the above F-term formula to this chiral multiplet ΠV and obtain D-term formula: applicable to real vector multiplet V = [ C, Z, H, K, B_{m}, λ, D] with weight w = 2 n = 0:

I_{D} = ∫ d^{4}x [V_{(w=2, n=0)}]_{D} = ∫ d^{4}x [−ΠV]_{F}
= ∫ d^{4}x e[D − \frac{1}{2}ψ̄_{m}γ^{m}iγ_{5}λ − φ̄_{m}γ^{m}iγ_{5}Z + \frac{1}{3}C (R + e^{−1}ψ̄_{μ}R^{μ})
+\frac{1}{4}iε^{mnkl}ψ̄_{m}γ_{n}ψ_{k} (B_{l} − A_{l}C − \frac{1}{2}ψ̄_{l}Z)]  (91)

where

R = R_{μν}^{mn}(M)e_{m}^{ν}e_{n}^{μ},  R^{μ} = ε^{μνρσ}γ_{5}γ_{ν}D_{ρ}^{ω}ψ_{σ} .  (92)

6.3. N = 1 SUGRA Lagrangian

One may have wondered in the above why we consider such a superconformal framework possessing rather large local symmetry while we want supergravity which has only local Poincaré invariance. We can now answer to this question. All the possible theories of Poincaré supergravity can be obtained from our superconformal framework simply fixing the gauges for the extraneous gauge symmetries, dilatation D, chiral A, conformal supersymmetry S and special conformal K_{m} symmetries. Then, we need special matter multiplet(s) called compensator, whose component fields are used to fix those extraneous gauges. Choosing different type of multiplet as the compensator yields a different formulation of Poincaré supergravity: namely, chiral multiplet compensator leads to (old) minimal formulation, (real) linear multiplet compensator to new-minimal formulation and complex linear multiplet compensator to Breitenlohner formulation. One of the virtue of the superconformal framework is that all those different formulations of Poincaré supergravity can be dreived in a unified way from this unique framework. There is another and more important advantage in the superconformal tensor calculus actually, which we explain shortly.

We explain only the (old) minimal formulation of Poincaré supergravity. Pure (Poincaré) supergravity Lagrangian is given by

L_{pure SUGRA} = [ΣΣ̄]_{D}  (93)

where Σ is a chiral multiplet with weight w = n = 1, the compensator of the (old) minimal formulation. Denoting the components of this compensator as Σ = [A, ψ_{R}, F], the extraneous D, A, S, K_{m} gauges are fixed by the following conditions:

D : ReA = \sqrt{3}, A : ImA = 0,
S : ψ_{R} = 0, K_{m} : b_{μ} = 0,  (94)

where the last b_{μ} is the Weyl (D) gauge field. Then, writing F = \frac{1}{\sqrt{3}}(S − iP) and A_{μ} = −\frac{2}{3}A_{μ}^{aux}, ΣΣ̄ takes the form

ΣΣ̄ = [ 3, 0, −2S, 2P, −2A_{m}^{aux}, 0, −\frac{1}{3}(S^{2} + P^{2} − A_{m}^{aux 2}) ]  (95)

Substituting this components expression into Eq. (93) and applying the D-term formula, we actually obtain the following action of pure supergravity:

L_{pure SUGRA} = e[R + e^{−1}ψ̄_{μ}R^{μ} − \frac{1}{3}(S^{2} + P^{2} − A_{m}^{aux 2})].  (96)

S, P and A_{μ}^{aux} constitute the well-known minimal set of auxiliarly fields, hence the name of minimal Poincaré supergravity.

If one considers more general matter coupled system, the Lagrangian would take the form

L = [ΣΣ̄ Φ̃(ϕ, ϕ̄)]_{D} + [Σ^{3}W (ϕ)]_{F} ,  (97)

omitting the possible gauge fields. Here ϕ denotes a set of matter multiplets { ϕ_{i} }. Now we can explain another virtue of our superconformal tensor calculus, as promised above.

First, we note that we can eliminate the superpotential term by redefining the compensator as W^{1/3}(ϕ)Σ → Σ, and rewrite the Lagrangian into the following form using Φ ≡ Φ̃/ |W |^{2/3}:

L = [ΣΣ̄ Φ(ϕ, ϕ̄)]_{D} + [Σ^{3}]_{F} ,  (98)

In this matter coupled system, the multiplet ΣΣ̄ Φ(ϕ, ϕ̄) ≡ V in the D-term has the following first two components:

C(V) = |A|^{2} Φ(z, z^{∗})
\frac{1}{2}Z(V) = i |A|^{2} (Φ_{i}χ_{L}^{i} − Φ^{i}χ_{Ri}) + iΦ (Aψ_{L} − A^{∗}ψ_{R}) ,  (99)

with notation Φ^{i} ≡ ∂Φ(z, z^{∗})/∂z_{i}, Φ_{i} ≡ ∂Φ(z, z^{∗})/∂z^{∗ i}. Therefore, to obtain the canonical form of Einstein-Hilbert as well as Rarita-Schwinger action R + e^{−1}ψ̄_{μ}R^{μ}, it would be best to take the gauge conditions for the extraneous gauges D, A, S, K_{m} as

D : ReA = \sqrt{3}Φ^{−1/2}, A : ImA = 0,
S : ψ_{R} = −AΦ^{−1}Φ^{i}χ_{Ri}, K_{m} : b_{μ} = 0.  (100)

Indeed, in this superconformal gauge, we have C(V) = 3 and Z(V) = 0, yielding the desired canonical Einstein-Hilbert and Rarita-Schwinger action R + e^{−1}ψ̄_{μ}R^{μ} from the beginning, as is seen from the D-term action formula. Note that this is really the power of superconformal tensor calculus. In the Poincaré tensor calculus, there is no freedom of choosing those gauges! From the superconformal viewpoint, the Poincaré tensor calculus is just the tensor calculus obtained from the superconformal one by choosing the Poincaré gauge fixing conditions Eq. (94). It is a good gauge conditions for pure supergravity system, but is ridiculous one for the matter coupled system. There is, however, no other way in the Poincaré tensor calculus, since there are no extraneous gauge freedom. Compare this simplification with the big calculation performed by Cremmer, Ferrara, Girardello and Van Proeyen using the Poincaré tensor calculus. The first thing the latter authors had to do was 1) Weyl rescaling of the vierbein and other fields, 2) chiral rotations of the fermion fields, and 3) recombination of Φ̃ and the superpotential W into the Kähler potential \frac{1}{3}K = ln(Φ̃/ |W |^{2/3}). The first and second tasks are simply bypassed here by the above D and A gauge conditions and the third was the task performed in one line already in Eq. (98).

Table II. Kugo-Uehara との換算表 [NP B226 (1983) 49; PTP 73 (1985) 235]

Kugo-Uehara  Ours
x_{m} = (x_{k}, x_{4})  (x^{k}, it) → write x^{μ} or −x_{μ}
∂_{m} = (∂_{k}, ∂_{4})  (∂_{k}, −i∂_{t}) → write ∂_{μ} or −∂^{μ}
∂_{m}∂_{m} ≡  −∂_{μ}∂^{μ} = −
δ_{mn}  −η_{μν}
γ_{m} = (γ_{k}, γ_{4})  (−iγ^{k}, γ^{0}) → write − iγ^{μ} or iγ_{μ}
γ_{m}∂_{m} ≡ / ∂  −iγ^{μ}∂_{μ} = −i/ ∂
γ_{m}B_{m} ≡ \begin{matrix}/ \\ B \end{matrix}  −iγ^{μ}(−V_{μ}) = +i\begin{matrix}/ \\ V \end{matrix}
a_{mn}  ia_{μν}
γ_{5}  γ_{5}
σ_{mn} ≡ (1/4)[γ_{m}, γ_{n}]  (1/4)[−iγ^{μ}, −iγ^{ν}] = (i/2)σ^{μν}
C, Z, H, K, B_{m}, Λ, D  C, χ, N, −M, −V_{μ}, λ, D
A, P_{R}χ, F  φ, \sqrt{2}ψ, F
(ψ̄_{R}, ψ̄_{L}), ( \begin{matrix}ψ_{R} \\ ψ_{L} \end{matrix} )  (ψ, ψ̄), ( ψ ; ψ̄ )
P_{m}, A, D, Q  P_{μ}/i, A/i, D/i, (1/2)Q/i
M_{mn}, K_{m}, S  −M_{μν}/i, −K_{μ}/i, −(1/2)S/i (negative signs!)
x_{m} → x^{μ} or −x_{μ}  の置き換えをする space 優先のルールでは
ε_{mnrs}  −iε_{μνρσ}、 −iε^{μνρσ} (どちらも −i )
super 変換 parameter ε  2α