Covariant Quantization of Supergravity Theory

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Covariant Quantization of Supergravity Theory

Yuki Wakimoto

^∗

February 24, 2014

Abstract

The quantization of gravity is one of the aims of modern physics.

Supergravity, the Einstein’s gravity with supersymmetry, is born in this context. Recently, modified supergravity has been applied to Starobin- sky inflationary model [11–13]. On the other hand, to write the quantum theory covariantly is important to see the theory clearly. In this paper, I introduce the method to quantize supergravity covariantly based on the Nieuwenhuizen’s text [17]. Supergravity is similar to Yang-Mills theory but has some special points which obstruct the ordinary way of covariant quantization, the Becci-Rouet-Stora-Tyutin method. de Wit and van Holten expanded this method to the cases where the BRST method does not apply [6].

1 Introduction

Why Supergravity?

We have two credible theories, quantum field theory and general relativity. The Standard Model based on quantum field theory describes microscopic phenom- ena which are dominated by electroweak force and strong one. The general relativity treat the dynamics of our macroscopic universe and the fourth force, gravity. Going together, they are inconsistent because gravity is not renormalizable. We can not get data from quantum field theory with gravity because of the impenetrable divergences.

Hence, we expect the existence of new physics beyond the Standard Model , such that it is able to understand four forces together. It is not visible now.

A new physics may have the supersymmetry, the invariance under a boson- fermion mixing transformation. And the supergravity is the Einstein gravity with the local supersymmetry. We expect that the supergravity is the foothold to find the new physics. For example, the supersymmetrization of the f(R) gravity which is an extension of the Einstein gravity was studied to explain inflation [11–13].

Why Covariant Quantization?

The canonical quantization, the consistent way to get the quantum theory is defined by the replacement of the classical Poisson bracket between the field φ^a and its momentum πa ≡ ∂L/∂φ˙^a by a commutator by the corresponding operators,

{φ^a, π_b}P.B.

quantization

−−−−−−−−→ 1

i~[ ˆφ^a,πˆ_b]. (1) But the Yang-Mills LagrangianLhas no time derivative of the zero component of the gauge field A^α_µ. In this case, constraint conditions appear such that φ¹_α =π_α = 0 and φ²_α = ˙π_α|_π_α₌₀ = 0. And there is the invariance under the transformation generated by these conditions. ¹ This means the gauge theory has extra freedom.

#{off-shell d.o.f.}= #{gauge fields} −#{gauge parameters}. How do we treat this extra freedom? There are five histories.

1. Dirac(1950) [7]

2. Faddeev-Popov(1967) [8]

3. Becci-Rouet-Stora-Tyutin(1975) [4] [5] [16]

4. de Wit-van Holten(1978) [6]

1The first class constraint becomes a generator of gauge transformation. [14]

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5. Batalin-Vilkoviski(1981) [2] [1]

The first is the Dirac method of the canonical quantization of singular sys- tems [7]. Introducing the Dirac bracket, we get the quantum theory. The Dirac bracket is the “constrained Poisson bracket” on the partial phase space given by the constraint condition. This method is constructive and has unitarity. But the Lorentz covariance is not evident. The Standard Model is based on the Lorentz covariant quantum field theory, so it is the problem.

Unitarity is clear, but Lorentz covariance is not.

The second method is the Faddeev-Popov prescription, which adds the gauge fixing term and the ghost fields to the ordinary Lagrangian in the path inte- gral formalism. The Faddeev-Popov quantization is equivalent to the canonical quantization for the Yang-Mills theory. This method is able to clarify the Lorentz covariance and the loop unitarity is ensured by the Faddeev-Popov ghosts. But the ghosts have odd spin-statistics relation, so they are non- physical. We must define the physical states |physiin our Hilbert space which excludes ghosts. S. N. Gupta and K. Bleuler gave the condition to the physical state on Feynman gauge and N. Nakanishi and B. Lautrup generalized it, but their method is applicable toU(1) gauge theory only.

Unitarity and Lorentz covariance are evident, but ghosts appear.

After Faddeev and Popov, four physicists, C. Becci, A. Rouet, R. Stora and I.V. Tyutin gave a systematic method to get the full Lagrangian which contains the gauge fixing term and the Faddeev-Popov ghosts. And they found the Lagrangian has invariance under the “BRST transformation”. Furthermore, T.

Kugo and I. Ojima found the general condition to get physical states

δBRST|physi= 0 (2)

which is called the Kugo-Ojima condition. This means the BRST invariant spaces are physical.

Unitarity, Lorentz covariance and physical space of states are evident.

On the other hand, many new theories are similar to gauge theory, but they often violate some BRST’s assumptions. For example, supersymmetry in supergravity violates the closedness of gauge algebra. The BRST method does not apply to them. But the extension, given by de Wit and van Holten, resolves this problem.

We can apply BRST method to various gauge theories.

Finally, but I. A. Batalin and G. A. Vilkovisky found the simple formulation in which all symmetries are manifest. They gave the “master equation” whose so- lutions have a symmetry which becomes the BRST one in the Yang-Mills theory.

Therefore, the Batalin-Vilkovisky formalism contains the BRST quantization, but it is more general. This formalism is applied to string theory, for example.

I am going to review the way to construct the quantum supergravity action based on the de Wit and van Holten method by using the text of Neuwenhuizen [17]. I will not treat Batalin-Vilkoviski formalism in this paper. The application of their formalism to supergravity is discussed in [3].

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2 Notation

• We consider Euclidean space E⁴ instead of Minkowski spacetime. So the signature is (+,+,+,+). It appears to be more convenient in the quantum field theory since the one-loop level.

• The supplementary information about the gauge theories and the gravity theory is in the appendices A and B.

symbols meaning

[A^m]µ A^mequal to [A^m]µdx^µ locally

[AB]ⁱ_ij matrix notationAⁱ_ikB_kj^k (position left-right⇔suffix down-up) e^m, dx^µ local frame fields and holonomic basis

[e^m]µ,[e^µ]m vierbein and its inverse

∂_m, ∂_µ ∂_µ=∂/∂x^µ= [e^m]_µ∂_m

|e| determinant of vierbein _4!¹abcd|e^a∧e^b∧e^c∧e^d|=|e|d⁴x γ0, γ1, γ2, γ3 Gamma matrices

γ4, γ5 iγ0, γ1γ2γ3γ4

γ^µνρ γ^[µγ^νγ^ρ]/3! =^µνρσγσγ5

σmn 1/4[γm, γn]

P_± Chiral projection ¹^±₂^γ⁵

[T^a]ⁱ_ij (i, j) component of the representation matrixT^a

(∂A)² η^mn∂_mA∂_nA

λ¯ λ^†γ⁴

∂ γ^m∂_m

C,ˆ Charge conjugation; ˆα=^βCβα

O_j a basis for 4×4 matrices (1, γ_m,2iσ_mn, iγ₅γ_m, γ₅)_j Fα,j Right derivativeδFα(φ)/δφ^j, (FαFβ),j = (−)^|^F^β^||^φ^j^|Fα,jFβ+...

δ_QF transformation ofF by Q with parameter; [Q, F]

|φⁱ| 0 to boson and 1 to fermion,i means|φⁱ|

S^cl the gauge action without gauge fixing and Faddeev-Popov term

3 Global Supersymmetry

Supergravity haslocalsupersymmetry whose parameters are spacetime-dependent.

Here I introduce the simple example which has globalsupersymmetry, before considering supergravity.

The Wess-Zumino model for two bosons and one Majorana fermion is [17]

S_WT[A, B, λ] =−1 2 Z

d⁴x

(∂A)²+ (∂B)²+ ¯λ∂λ

. (3)

The equation of motions are

A=B=∂λ =−∂λ¯= 0. (4)

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This action has a global symmetry with infinitesimal fermionic parameter δ_QA = 1

2¯λ (5)

δ_QB = −i

2¯γ5λ, (6)

δ_Qλ = 1

2∂(A −iγ5B) (7)

δ_Qλ¯ = −1

2¯∂(A +iγ₅B)

. In fact,

δ_QS_WT[Ψ] = S_WT[Ψ +δ_QΨ]−S_WT[Ψ]

= Z

d⁴x∂_mK^m= 0

where K^m=−¹₄¯γ^m[∂(A −iγ₅B)]λand Ψ = (A, B, λ). The symmetry of the boson-fermion mixing transformation is calledsupersymmetry.

The algebra ofQis not closed but open. For example, h

δ_Q¹, δ_Q² i

λ=1

2(¯₂γ^m₁)∂_mλ−1

4(¯₂γⁿ₁)γ_n∂λ. (8) The first term on the right hand side is a translation, but the second term vanishes only on-shell (i.e. on the λ-equation of motion).

The irreducible representation of this transformation (5-7) is (A+iB, P₊λ) or (A−iB, P₋λ). It is a “supersymmetry doublet”.

Closed Algebra

The auxiliary fields to Wess-Zumino model are given by [17]

S_WT+aux[A, B, λF, G] =S_WT[A, B, λ] +1 2

Z d⁴x

F²+G²

. (9)

On shell,F andGvanish. The total action has a symmetry under δ_QA = 1

2¯λ, (10)

δ_QB = −i

2¯γ₅λ, (11)

δ_QF = 1

2¯∂λ, (12)

δ_QG = i

2¯γ₅∂λ, (13)

δ_Qλ = 1

2∂(A−iγ₅B)+1

2(F+iγ₅G). (14)

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And the generatorsQ^αwith translation and Lorentz rotation satisfy the closed algebra

[Pm, Pn] = 0 (15)

[M_rs, P_m] = η_rmP_s−η_smP_r (16)

[Mmn, Mrs] = ηmsMnr+ηnrMms−ηmrMns−ηnsMmr (17)

[Q^α, Pm] = 0 (18)

[Q^α, M_mn] = [σ_mn]^α_αβQ^β (19)

Q^α, Q^β = 1

2[γ^mC⁻¹]^αβPm. (20)

(20) is equivalent to

Qα,Q¯α˙ = 1/2(σ^m)αα˙Pmin the two-components formalism. For example, we can get by using (10-14) that

h δ_Q¹, δ_Q²

i

λ = 1

2(¯₂γ^m₁)∂_mλ (21)

= δ⁽_P²^γ^m¹⁾λ. (22)

In general, this algebra (15-20) is called theN = 1 super Poincar´e algebra and the symmetry under the generators Q is calledN = 1 supersymmetry. N=1 SUSY stands for this.

In a gauge theory which has alocalsymmetry, closed algebra is needed to apply the method of ordinary covariant quantization, the BRST method. To use open algebra directly, we must extend the method. We discuss both methods in the supergravity section 4.

4 Supergravity

4.1 Action and its properties

The action of simple supergravity is given by [17]

SSG[e, ψ, ω] = Z

d⁴x

− 1

2κ²|e|R−1 2

ψ¯µγ^µρνDρψν.

(23) The SUSY transformations are [17]

δ_Qe^m_mµ = κ

2¯γ^mψµ (24)

δ_Qψµ = 1

κDµ (25)

where is a spacetime dependent fermionic parameter of local supersymmetry transformation. So the spin ³₂ particleψ_µis a gauge field to supersymmetry and is named gravitino. δ_Qω is irrelevant because we use the “1.5 order formalism”

here, so that the factorδS_SG/δωin front of theδωinδS_SGvanishes,S_SG/δω= 0.

(c.f. eq. (33) in the section “1.5 order formalism”)

The equations of motion for the spin connection are similar to (134) , [ωmn]µ = 1

2(Rµn,m−Rµm,n+Rmn,µ) (e, ψ) (26) R_µν,m = −∂_µe_mν+∂_νe_mµ+κ²

2

ψ¯_µγ_mψ_ν. (27)

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And gravitino obey the massless Rarita-Schwinger equation in curved spacetime

R^µ≡γ^µρνD_ρψ_ν = 0 (28)

whereDρψν =∂ρψν+¹₂[ω^mn]ρσmnψσ. Torsion in Supergravity

The third term of (27) gives the nonzero torsion tensor [T^α]µν ≡ ¹₂([Γ^µα_µ ]ν− [Γ^µα_ν ]µ)

[T^α]µν =−κ²

4 ( ¯ψµγ^αψν) (29)

in the vierbein postulate (128), the spacetime with supergravity has a torsion.

The 1.5 order Formalism

The formalism which treats three fields,e, ψ andω, as independent, δS[e, ψ, ω]≡δeδS

δe +δψδS

δψ +δωδS

δω, (30)

is called the first order formalism or Palatini formalism. On the other hand, it is called the second order formalism when ω is Levi-Civita connection derived by the metric conservation and torsion free condition. The 1.5 order formalism uses the relation (26) as the definition of ω=ω(e, ψ) ;

δS[e, ψ, ω(e, ψ)]≡δeδS δe

ω(e,ψ)

+δψδS δψ

ω(e,ψ)

+

δeδω

δe +δψδω δψ

δS δω (31) whereδe(δS/δe)|ω(e,ψ) means the variation without the fieldseinω(e, ψ). The third term vanishes becauseω(e, ψ) is defined byδS/δω= 0. Therefore, we can derive the relation between the first and the 1.5 order formalism,

δS[e, ψ, ω(e, ψ)]

ω(e,ψ)

=δS[e, ψ, ω]

ω=ω(e,ψ)

. (32)

The right hand side is the first order formalism butω is classical, on-shell.

It is not trivial that the two formalisms return the same result in quantum theory. The second order formalism is more proper by analogy with Yang-Mills theory, but Einstein-Hilbert action is special: the connection is not dynamical.

Therefore, in this paper, we decide to make our discussion based on the 1.5 order formalism.

order connection

first independent field 1.5 defined by equation of motion second Levi-Civita connection In the 1.5 order formalism,

δ_Q[ω_mn]_µ=−κ

4(¯γ_mψ_µn−¯γ_nψ_µm+ ¯γ_µψ_mn) (33) because

δ_Q[R_m]_µν =−κ

2¯γ_mψ_µν (34)

whereψµν =Dµψν−Dνψµ.

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Supercovariant derivative

When the supertransformation of D^cov_µ Aν =∂µAν+· · · has no ∂ term, this operator is called a supercovariant derivative. We are able to get it by;

D_µ^covAν =∂µAν+· · ·+δQ(κψµ)A. (35) For example,

D^cov_µ ωâb_abν =∂_µωâb_abν+ωâ_acµω^cb_cbν−1

4( ¯ψ_µγ^aψ_ν^νb−ψ¯_µγ^bψ^νa_ν + ¯ψ_µγ_νψ^ab) (36) where thisψ is redefined byκψ. Using supercovariant derivative, supergravity action is equivalent to

Z

dx⁴|e|eâµ_a e^bν_b (D^cov_µ ωâb_abν−D^cov_ν ω_abµâb ) (37) under the gauge γ·ψ= 0. (c.f. (46)). Now I useγνψ^νµ= R^µ−¹₂γ^µγ·R.

Open Algebra

An algebra of the transformations (24,25 and 33) is closed only on-shell, similar to (8) ,

h δ_Q¹, δ_Q²

i

ψµ = 1 2

δ_G^ξ +δ_Q +δ_L^λ

ψµ

+1

4(¯1γ^α2)VµαβR^β+1

4(¯1σ^ρσ2)TµρσλR^λ (38) whereδG,δLandδQare the general coordinate transformation, Lorentz rotation and SUSY transformation: δ^ξ_Gψµ =ξ^ν∂νψµ+ (∂µξ^ν)ψν, δ_L^λψµ = ¹₂σ^mnλmnψµ

and (25). Infinitesimal parametersξ, λ, are defined by ξ^µ = ¯2γ^µ1, λ^mn = ξ^µ[ω^mn]µ and=−ξ^µψµ now. VµαρandTµρσλare called non closure functions.

They are given by

|e|Vµαρ = 1

4gµργα+1

2|e|γµαρ (39)

|e|Tµρσλ = gµρgσλ+1

2gµλσρσ−1

2|e|ρσµλγ5. (40) Other transformations are closed,

[δ_G(η^α), δ_G(ξ^α)] = δ_G(ξ^α∂_αη^β−η^α∂_αξ^β) (41) [δL(θ^mn), δG(η^α)] = δL(η^α∂αθ^mn) (42) [δL(θ^mn), δL(ϑ^mn)] = δL(−θ^m_mkϑ^kn+ϑ^m_mkθ^kn) (43)

δQ(^a), δG(η^β)

= δQ(η^β∂β) (44)

[δQ(^a), δL(θ^mn)] = δQ(1/2θ^mnσmn). (45) Gauge fixing terms and Faddeev-Popov ghosts

One of gauge fixings is given by Fα= (−∂µ(√

gg^µν), eaµ−ebνδ_µ^bδ^ν_α,−γ^νψν) = 0 (46)

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so the action is Sfix[e, ψ] =

Z dx⁴

−1 4(∂µ√

gg^µν)²+α(eaµ−ebνδ_µ^bδ^ν_a)²+1 4

ψ¯^µγµ∂(γ^νψν)

. (47) The first term is the “general covariant Lorentz gauge” for a metric fieldgµν = eaµe^a_aν and this is called the Fock-de Donder gauge. The last term corresponds to an irreducible representation. Vectorical spinor is (¹₂,¹₂)⊗((¹₂,0) + (0,¹₂)) representation, and it has a decomposition

1 2,1

2

⊗ 1

2,0

+

0,1 2

=

1,1 2

+

1 2,1

+

1 2,0

+

0,1

2

. (48) The last two terms are “trace parts” of spinor,γ·ψ=

ψ¯^α^β^˙

αβ˙β˙, ψ^αββ_αβ_˙^˙

. This part vanishes by the gauge fixing above.

We have three local symmetries: general coordinate, local Lorentz, and local supersymmetry. So we need Faddeev-Popov ghosts and anti-ghosts corresponding to each symmetry ,c^ν,c^∗_ν,c^mn,c^∗_mn,c^aandc^∗_a. They have an odd statistics to the corresponding gauge field,|c^α|=|φ^α|+ 1, so the spinor ghostsc^a andc^∗_a are bosonic.

4.2 Action with Auxiliary fields and closed algebra

The supergravity action with auxiliary fields is given by [17]

SSG+aux[e, ψ, S, P, A] =SSG[e, ψ] + Z

d⁴x h−e

3 S²+P²−A²_mi

. (49) This total action is invariant under the transformations [17]

δ_Qe^m_mµ = κ

2¯γ^mψµ (50)

δ_Qψµ = 1 κ

Dµ+iκ 2 Aµγ5

−1

2γµη (51)

δ_QS = 1

4¯γ·R^cov (52)

δ_QP = −i

4¯γ5γ·R^cov (53)

δ_QAm = i3 4¯γ5

R^cov_m −1

3γmγ·R^cov

(54) whereη=−¹₃(S−iγ₅P−iAγ₅) and R^µ,cov=γ^µρσ(D_ρψ_σ−i¹₂A_σγ₅ψ_ρ+¹₂γ_σηψ_ρ).

This R^covis a covariant version of massless Rarita-Schwinger equation (28). And the situation is same to the Wess-Zumino model with auxiliary fields (12-14):

the transformation of auxiliary fields is proportional to the equation of motion of fermionic field.

These transformations are closed even if the fields are off-shell because extra terms in (38) vanish by auxiliary fields. [δQ(1), δQ(2)] is given by

δG(ξ^α) +δQ(−ξ^αψα) +δL

ξ^µ

[ωab]µ− i

3µabmA^m

+1

3¯2σ^mn(S−iγ5P)1

(55) whereξ^α= ¹₂¯2γ^α1.

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4.3 Quantization with Auxiliary Fields

Supergravity with auxiliary fields has a local supersymmetry but it is closed, so we are able to apply the method as for the Yang-Mills theory (see section A).

The action of ghosts is given byc^∗^a(δ^cF)a for all generatorsδ^c=δ_G^c +δ_L^c +δ^c_Q SFP[e, ψ, c, c^∗] =

Z dx⁴

c^∗_ν∂µδ^c(√

gg^µν)−¯cδ^c(γ·ψ) +c^∗^ane^µ_µnδ^c(eaµ−ebνδ_µ^bδ^ν_a) (56)

δ^c(√

gg^µν) = −√

g[(∂αc^µ)g^αν+ (∂αc^ν)g^µα] +∂α(c^α√ gg^µν)

−1 2

√g(¯cγ^µψ^ν+ ¯cγ^νψ^µ) +1 2¯cγ·ψ√

gγ^µν (57) δ^c(γ·ψ) = c^α∂α(γ·ψ) +

D+ i

2Aγ5−2η

c +1

2c^mnσmn(γ·ψ) (58)

δ(eaµ−ebνδ_µ^bδ^ν_a) =

(∂µc^β)eaβ+c^β∂βeaµ+cabe^b_bµ+1 2¯cγaψµ

−(a↔b, µ↔ν)δ^b_µδ_a^ν (59) The kinetic terms of ghosts are

c^∗_νc^ν−¯cDc+ 2c^∗^ab(c_ab+∂_bc^νe_aν) (60) So Lorentz ghosts does not propagate.

The terms related to the auxiliary fields in total action are

−1

3|e|(S²+P²−A²_m)−¯c i

2Aγ5+2

3(S−iγ5P−iAγ5)

c (61) They do not propagate, so we are able to eliminate them by using the equations

|e|S=−¯cc, |e|P = ¯eiγ₅c, |e|A_m= 1

4¯ciγ₅γ_mc. (62) Therefore the (61) gives the four-ghost term of SUSY ghosts,

1 3|e|

(¯cc)²−(¯cγ5c)²+ 1

16(¯cγ5γmc)²

=− 5

32|e|(¯cγmˆ¯c)(ˆcγ^mc). (63) Such term is absent in the naive BRST method.

4.4 Quantization with Open Algebra

In the cases where algebra is open, (8) and (38) , the nonclosure term are proportional to the equation of motion. So, in general, gauge commutator of such cases is given by

Rⁱ_iα,kR_kβ^k −(−)^αβRⁱ_iβ,kR^k_kα=Rⁱ_iγf_γαβ^γ +S_cl,j^cl η_ijαβ^ij . (64)

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The final solution to quantize supergravity is given by the following action and the extended BRST transformation [10]:

Stotal = S^cl+1

2FαE^αβFβ+ ¯FjR_jβ^j c^β

−1 4

F¯jF¯kη^kj_kjγδc^δc^γ(−)^(k+1)j+γ+1, (65) δφ = Rⁱ_iαc^αΛ−1

2

F¯jη^ji_jiβγc^γΛc^β(−)^(j+1)i, (66) δc^α = −1

2

f_αβγ^α + ¯FjX_jαγβδ^jα c^δ

c^γΛc^β, (67) where ¯Fj=c^∗^αFα,j and|F¯j|=j+ 1. I construct them below.

If nonclosure functionη vanishes, gauge algebra closes and the total action is BRST invariant δS= 0. But ifη is nonzero, an extra term appears,

δS = ¯F_j1 2

S_cl,k^cl η_jkγβ^jk c^βΛ

c^γ (68)

because the variation of Faddeev-Popov term δ( ¯FjR^j_jkc^k) contains the terms which vanish by closedness of algebra in the transformation of the last two factors,

F¯j

R^j_jβ,kR_kγ^k c^γΛc^β−1

2R_jβ^j f_βδγ^β c^γΛc^δ

. (69)

(Other terms vanish by themselves.) So the action is not invariant under the ordinary BRST transformation. Adding an extra term to the transformation δφ, this term is canceled byδ_exS^cl=S_cl,j^cl δ_exφⁱ

δexφⁱ =−1 2

F¯jη_jiβγ^ji c^γΛc^β(−)^(j+1)i. (70) This term appear by the Faddeev-Popov term δ(c^∗^αFα). Therefore, the new extended action is given by (65). But the new action is not invariant under the new BRST transformation. Now, we specialize the situation to supergravity.

The nonclosure functionη_ijαβ^ij appear in the variation of gravitino only, so the indices i, j, α, β do not refer to general coordinate ^µ and Lorentz rotation

mn. And it is given byV andT, (39) and (40), so they depend on vierbein only.

Therefore,

η_ijαβ^ij =

η^ij_ijαβ(e) (i, j, α, β refer to spinorial indices)

0 (otherwise) . (71)

This property gives

η^kl_klγδ,iη_jiρσ^ji = 0. (72)

Now, the first three term of commutator (64) is the symmetry of classical action, so δ⁰φ=S_cl,j^cl η^ij_ijαβη^βξ^αis also. Therefore,

δ⁰S^cl=S_cl,i^cl S_cl,j^cl η^ij_ijαβη^βξ^α= 0⇔

η^ij = (−)^ij+1η^ji

∃X s.t. δ⁰φⁱ=Rⁱ_iλ(X_λjαβ^λj S_cl,j^cl )η^βξ^α . (73)

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In our case, such a matrixX vanishes. So we choose the upper condition. (71) and (73) giveη^ij =η^ji. AndF_α,j does not depend on the gravitino field in our gauge choice.

Our nonclosure term of the new variation is F¯jR^j_jβ,l

−1 2

F¯kη^kl_klρσc^σΛc^ρ

c^β−1 4

F¯jF¯kη_klγδ,l^kl (R^l_lλc^λΛ)c^δc^γ

−1 4

F¯jF¯kη_kjρσ^kj

−1

2f_σγδ^σ c^δΛc^γ

c^ρ. (74) The first term becomes

−1 4

F¯_jF¯_k

R^j_jβ,lη_lkρσ^lk + (−)^(j+1)(k+1)(j↔k)

(−)(j+β+l)(k+1)c^σΛc^ρc^β (75) but nowk=l= 1, so the total extra term is written by

δexS=−1 4

F¯lF¯iP_ilγβα^il c^αΛc^βc^γ (76) where

P_ilγβαîl =ηîl_ilγβ,kR^k_kα+R^l_lγ,kη^ki_kiβα+Rⁱ_iγ,kη^kl_klβα−ηîl_ilγδf_δβα^δ . (77) If Pîl = (−)(i+1)(l+1)+1P^li, this term vanishes automatically. But the above definition denies this case. So, our next aim is to cancel this term, or find a matrixX such that P_ilγβαîl ∝Rⁱ_iλX_iλαβγîλ exist and extend the variation of the ghost fields similar to (73).

This expression of P is constructed by δ⁰ and [δ, δ]. So we are going to consider the Jacobi identity

δ^ζ, δ^η, δ^ξ

+...= 0. (78)

The Jacobi identity becomes

R_iλⁱ A^λ+S_cl,l^cl B^li+...= 0 (79) where

A^λ= (−f_λαβ^λ f_δβγ^δ +f_λαβ,k^λ R^k_kγ)ζ^γη^βξ^α (80) and

B^li = −η^il_ilγδf_δαβ^δ η^βξ^αζ^γ+η^il_ilαβ,kR^k_kγζ^γη^βξ^α

−R_lγ,k^l ζ^γη^ik_ikαβη^βξ^α(−)^γk+R_iγ,kⁱ η^kl_klαβη^βξ^αζ^γ(−)^(i+γ)l. (81) The third term in B is given by

0 = (δS^cl),k= (S_cl,l^cl R^l_lγξ^γ),k=S_cl,lk^cl R^l_lγξ^γ(−)^lk+S^cl_cl,lR^l_lα,kξ^γ(−)^kγ. (82) To consider

F¯iF¯lB^li, (83) we writeη^β =c^βΛ⁰, ξ^α=c^αΛ⁰⁰andζ^γ =c^γΛ. Cycling of them gives

η^βξ^αζ^γ+...=c^βΛc^αc^γ(−)^γ(3Λ⁰⁰Λ⁰). (84)

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In our case,|η^βξ^α|= 0 whoseαandβcontract withηandηîk(−)^γk=ηîk(−)^γi. They give Bîl = B^li(−)îl+1. So the term which contributes to (83) satisfies i+l= 0 andil= 1. We get the result

−F¯iF¯lP_liγβα^li c^αΛc^βc^γ = ¯FiF¯lB^li. (85) Therefore we can consider P to solve (79).

The second term of (79) vanishes on-shell but the first term does not. It follows ∃A^λjA^λ =S_cl,j^cl A^λj. Hence (79) equal toS_cl,l^cl (Rⁱ_iλA^λl+B^li+...) = 0.

This means∃X or∃M :

Rⁱ_iλA^λl+B^li+...=R^l_lλXîλ+S^cl_cl,jM^jli (86) where M is a matrix such that S_cl,l^cl S_cl,j^cl M^jli = 0. In our case, B^li does not contain∂ψ or∂∂eso suchM vanishes. Contracting both side withS_cl,i^cl , we get B^li=R^l_lλXîλ+ (−)îl+1(i↔l) (87) because the first term of (86) does not contribute. Therefore, the remaining term (70) is

δexS= 1 4

F¯lF¯i(2R^l_lλX_iλγβα^iλ )c^αΛc^βc^γ (88) and this term is canceled by the extra term in (67). If the matrixM in (86) does not vanish, we must go on to get another extra term and extend the quantum action again.

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5 Conclusion

I reviewed the BRST quantization of supergravity theory with auxiliary fields and without them. We considered supergravity as a local supersymmetry theory which is the gauge theory with two gauge fields, vierbein and gravitino, related by super transformation. Supergravity has the difference between the first order formalism and the second one, so we use the 1.5 order formalism which defines the connection to satisfy the own equation of motion. We introduced ad hoc auxiliary fields because the gauge algebra is not closed off-shell. And we applied the naive BRST method to the closed algebra with auxiliary fields and saw the existence of the four-ghost term which is absent in ordinary gauge theory.

Finally, we reviewed the extended method of BRST quantization and got the quantum action without the auxiliary fields, and saw the origin of the four-ghost term again.

The quantized Einstein gravity it is non-renormalizable [9]. Supergravity is expected to be non-renormalizable too. Nevertheless, it makes sense to quantize gravity and supergravity, by using the ultra-violet cut-off ΛUV.Mpl, i.e.

restricting all Euclidean (loop) momenta as p²≤ΛUV.

6 Outlook

The covariant quantization of the standard N = 1 supergravity above can be also applied to the modified supergravity theories which were recently used to describe the (Starobinsky) inflation in the context of supergravity [11–13]. The modified supergravities [11–13] are the higher-derivative supergravity theories where the auxiliary fields become dynamical. I am going to covariantly quantize the modified supergravities in my next research project in a future.

A Yang-Mills theory and its covariant quantiza- tion

A.1 Vector bundle, connection and curvature

The theory of the vector bundles applies to the Yang-Mills theory and the general relativity. The general relativity is related to the tangent bundle, and the Yang-Mills theory is related to a vector bundle.

Connection

A vector bundle is a space which is isomorphic to V ×U locally where V is a vector space andU is a local area (open set). Each point has a vector space and a relation to its neighbour’s one is given by a connection. The connection∇ is represented by the 1-form matrix Γ^m_mn = [Γ^m_mn]_µdx^µ to use local frame filede_n

∇en =emΓ^m_mn, (89) or

∇ξe_n=e_m[Γ^m_mn]_µξ^µ (90) if the directionξ^µ is given.

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∇satisfy the Leibniz rule

∇(f v) =df⊗v+f∇v (91)

wheref is a function andv is a vector field. Hence the operation tov=v^mem

is

∇v = dv^me_m+v^m∇e_m

= (dv^m+vⁿΓ^m_mn)em, (92) and the coordinate transformation is given by

Γ⁰ =U⁻¹ΓU+U⁻¹dU (93)

because

∇e⁰ = e⁰Γ⁰

∇(eU) = eUΓ⁰ e(dU+ ΓU) = eUΓ⁰. Remarks

Using the matrix basis Tâ, Γ^m_mn is written by [Γâ]_µ([Tâ]^m_mn ⊗dx^µ) and the coefficient [Γâ]_µ is called a gauge field. We must be careful in the dependence between theTâanddx^µ. [Tâ]^m_mn⊗dx^µbecomesTâ⊗(∂_νξ^µdx^ν)+(ξ^µ[Γâ_ab]_µT^b)⊗ dx^µ by the infinitesimal translation by ξ^µ. Now the basis [f^c]â_ab such that Γâ_ab = Γ^c[f^c]â_ab is the adjoint representations of the Lie algebra of a group G which has the vector space V as the representation space. Such group G is called the structure group or the gauge group.

Curvature

The flat condition of the vector bundle is∃U; Γ =U⁻¹dU⇔ ∃ coordinate such that Γ vanishes. This is equivalent to

dU =UΓ (94)

⇔ dU∧Γ +U dΓ = 0 (95)

⇔ U(Γ∧Γ +dΓ) = 0. (96)

So if Ω ≡ dΓ + Γ∧Γ vanishes, our vector bundle is flat. This tensor Ω, the

“obstruction” to flatness, is called the curvature tensor. Conversely, the curved vector bundle has a non-zero curvature tensor.

The coefficients to the curvature Ωmn=¹₂[Ωmn]µνdx^µ∧dx^ν are given by [Ωmn]µν=∂µ[Γmn]ν−∂ν[Γmn]µ+ [Γmk]µ[Γ^k_kn]ν−[Γmk]ν[Γ^k_kn]µ. (97) In the matrix notation it reads

Ω_µν =∂_µΓ_ν−∂_νΓ_µ+ [Γ_µ,Γ_ν]. (98) The connection Γ is transformed by (93) so it is not a tensor, and not a representation ofG. But the curvature Ω is transformed by Ω⁰_µν=U⁻¹ΩµνU, hence, it is in the adjoint representation ofG.

Covariant Quantization of Supergravity Theory