Component Expansion

indices off suppressed), one immediately sees that (Of)

0 = (O^M∂_Mf)

0 + (O^abM_abf)

0

=O^M 0∂Mf

0 +O^ab

0Mabf

0 (6.125)

= (O

0f)

0

is different from O

0f

0 =O^m 0∂_mf

0 +O^ab

0M_abf

0. (6.126)

Using pure superspace methods, it is possible (though tedious) to show, that in Wess–Zumino gauge the vector derivative has the following expan-sion,

D_αα˙

0 =∇_αα˙

0 +¹₂Ψ_αα,˙ ^βD_β

0 + ¹₂Ψ¯_αα,˙β˙D¯^β˙

0, (6.127)

with Ψ the gaugino field strength. As a simple example, the expansion of D_αα˙f is given,

(D_αα˙f)

0 = (D_αα˙

0f)

0

=∇αα˙(f

0) + ¹₂Ψαα,˙ β (Dβf)

0

+¹₂Ψ¯αα,β˙ ( ¯D^βf)

0

.

(6.128)

More complicated combination of the derivatives D_α, ¯D_α˙ and D_αα˙ act-ing on a field require rearrangement such that the leftmost derivative is of vector type. Then the above rule (with f containing the remaining derivatives) can be used to recursively reduce the superspace derivatives to space-time covariant derivatives∇_αα˙ until only expressions containing component combinations (6.120) of the spinorial derivatives are left over.

Due to the three-folding caused by each application of (6.128), let alone the required rearrangement of vector derivatives to the left, even terms with a relatively small number of derivatives may grow dramatically. The situation is (slightly) better when one is not interested in terms containing the gaugino field strength. Therefore, the operator

b shall denote space-time projection while simultaneously neglecting all gravitational fermionic and auxiliary fields.

6.5 Component Expansion 117

6.5.2 Supergravity Fields

The derivation of the component expansion in Wess–Zumino gauge is rather involved and only the final expression shall be reproduced here.

The real part of the prepotential W can be gauged away, but requiring instead the condition

exp( ¯Wⁿ∂_n)x^m =x^m+iH^m(x, θ,θ)¯ H^m = ¯H^m (6.129) defines the gravitational Wess–Zumino gauge, also called gravitational su-perfield gauge. In this gauge, the gravitational degrees of freedom are encoded in the gravitational superfield H^m and the chiral compensator

ˆ ϕ(x, θ).

H^m =θσ^aθe¯ _a^m+iθ¯²θ^αψ^m_α−iθ²θ¯_α˙Ψ¯^mα˙ +θ²θ¯²A^m ˆ

ϕ³ =e⁻¹(1−2iθσ_aΨ¯^a+θ²B) ϕˆ= e^−W¯ ϕ (6.130) ˆ¯

ϕ³ =e⁻¹(1−2iθ˜¯σ_aΨ^a+ ¯θ²B¯)

In Wess–Zumino gauge, the spinorial semi-covariant vierbein fields (6.87) coincide with the partial derivatives and can therefore be used to extract the components of the above gravitational superfields just as in flat supersymmetry.

The spinorial semi-covariant vierbein fields ˆEα, ˆE¯α˙ were defined by just pulling out a factor of F from the covariant spinorial derivativesD_α, D¯_α˙. In addition without proof, for the prepotentialF it holds

F

0 = 1, EˆαF =−₂ⁱΨ¯^αβ˙β˙, (6.131) such that

D_αO

0 = ˆE_αO

0,

−¹₄D²O

0 =−¹₄Eˆ²O

0 +₂ⁱΨ¯^αβ,˙

β˙D_αO

0. (6.132) This allows to write down the chiral compensator’s components in terms

of covariant derivatives

ϕ³

0 =e⁻¹, (6.133a)

Dαϕ³

0 =−2ie⁻¹(σ^aΨ¯a)α, (6.133b)

−¹₄D²ϕ³

0 =e⁻¹(B−Ψ˜¯σσΨ),¯ (6.133c) where ¯Ψ˜σσΨ =¯ −Ψ¯^αβ,˙

β˙Ψ¯_αγ,˙^γ˙. In other words ϕ

0 =e^−1/3 (6.134a)

Dαϕ

0 =−2

3ie^−1/3(σ^aΨ¯a)α (6.134b)

−¹₄D²ϕ

0 = 1

3e^−1/3(B− 1 3

Ψ˜¯σσΨ)¯ (6.134c)

For the chiral supertorsion component:

R¯

0 = 1

3B, B =B+ ¹₂Ψ¯^aσ˜aσbΨ¯^b +¹₂Ψ¯^aΨ¯a, (6.135a) D¯_α˙R¯

0 = 4

3 Ψ¯_α˙β,˙

β˙+ i

3BΨ^β_α,β˙ , (6.135b)

D¯²R¯

0 = 2

3(R+ i

2ε^abcdR_abcd) + 8 9

BB¯

−2

9B(Ψ^aσ_aσ˜_bΨ^b + Ψ^aΨ_a) +iD¯_α˙R¯

0(˜σ_bΨ^b)^α˙ +2i

3Ψ^αα,β˙ D{αGβ},α˙

0,

(6.135c)

where R denotes the Ricci scalar, tensor or Riemann tensor, respectively.

For the real supertorsion component:

G_a 0 = 4

3A_a, (6.136a)

A_a=A^a+1

8ε^abcdC_bcd− 1

4(Ψ_aσ_bΨ¯^b + Ψ^bσ_bΨ¯_a)

−1

2Ψ^bσaΨ¯b+ i

8ε^abcdΨbσcΨ¯d,

(6.136b) D¯{α˙G^ββ}˙

0 =−2Ψ_α˙β,˙ β + i

3

B¯Ψ¯^β_{α,˙β}˙ , (6.136c) D¯{α˙D^{γG^δ}β}˙

0 = 2E^γδ_α˙β˙+ 2iΨ^{γ{α,˙ δ}D¯β}˙ R¯

0 −iΨα{α,˙αD^{γG^δ}β}˙

0

+2iΨ¯_α{α,˙β}˙ W^αγδ

0 + 2

3B(˜σ^ab)_α˙β˙Ψ_a^γΨ_b^δ, (6.136d)

6.5 Component Expansion 119

with

E_ab := ¹₄ 2 ˜R_ab+ ₂ⁱ(ε_acdeR^cde_b+ε_bcdeR^cde_a−¹₂η_abε^cdefR_cdef)

, (6.137a) R˜_ab := ¹₂(R_ab+R_ba)−¹₄η_abR=G_{ab}+ ¹₄η_abR. (6.137b)

6.5.3 Full Superspace Integrals

Using the chiral integration rule (6.119), any real superspace integral can be reduced to a chiral one.

S = Z

d⁸z E⁻¹L

= Z

d⁸z E⁻¹ R −¹₄

( ¯D²−4R)L

| {z }

=:Lc

(6.138)

Then the following manipulations, which crucially depend on the semi- density formula covariant vierbein coinciding (6.87) with the partial derivatives in Wess–

Zumino gauge, lead to the density formula S =

Z

d⁶zϕˆ³Lˆc = ¹₄ Z

d⁴x ∂^α∂_α( ˆϕ³Lˆc) =−¹₄ Z

d⁴xEˆ²(ϕ³Lc)

0

=−¹₄ Z

d⁴x ϕ³

0

Eˆ²Lc

0 + 2D^αϕ³

0D_αLc

0 + ˆE²ϕ³

0Lc

0

= Z

d⁴x ϕ³

0(−¹₄D²Lc)

0 − ¹₄D^αϕ³

0DαLc

0 +BLc

0.

(6.139) where B =B− ¹₂Ψ¯^aσ˜_aσ_bΨ¯^b− ¹₂Ψ¯^aΨ¯_a =−¹₄D²ϕ³ +e⁻¹Ψ˜¯σσΨ.¯

I adore simple pleasures. They are the last refuge of the complex.

Oscar Wilde, “The Picture of Dorian Gray”

Chapter 7

Space-Time Dependent Couplings

§7.1Weyl Transformations, 122. §7.1.1Conformal Killing Equation, 122. §7.1.2 Conformal Algebra ind >2, 123. §7.1.3Weyl Transformations of the Riemann Tensor, 124. §7.1.4Weyl Covariant Differential Operators, 125. §7.2 Zamolod-chikov’sc-Theorem in Two Dimensions, 127. §7.3Conformal Anomaly in Four Dimensions, 129. §7.4 Local RG Equation and the c-Theorem, 130. §7.4.1 a-Theorem, 133.

This Chapter is meant to give a short introduction into the space-time dependent couplings technique and its application to a proof of Zamolod-chikov’s c-theorem in two dimensions. Additionally the four dimensional trace anomaly and some of the problems encountered when trying to ex-tend the theorem to four dimensions are discussed.

7.1 Weyl Transformations

7.1.1 Conformal Killing Equation

A Weyl transformation is a rescaling of the metric by a space-time depen-dent factor

g_mn 7→e^−2σg_mn. (7.1)

Upon restriction to flat space these transformations generate the confor-mal group, which locally preserves angles.

Using

δg_mn =−2σg_mn, δx^m =ξ^m,

δdx^m = (∂nξ^m)dxⁿ,

(7.2)

the requirement of invariance of the line element

δ(ds²)= 0 = [−2σg^! _mn+∂_mξ_n+∂_nξ_m]dx^mdxⁿ (7.3) amounts to theconformal Killing vector equation

conformal Killing vector

∂_mξ_n+∂_nξ_m = ²_d∂_kξ^kg_mn, σ= ¹_d∂kξ^k,

(7.4)

where d is the dimension of space time.

Under (7.2), the action transforms as follows, δS =

Z

d^dx δS δg_mnδg_mn

= Z

d^dx

−¹₂T^mn][−2σg_mn],

(7.5)

which demonstrates that for conformal invariance the trace of the energy-momentum tensor has to vanish.

As an aside, in two dimensions after Wick rotation the conformal Killing vector equation becomes the Cauchy–Riemann system, such that

Cauchy–Riemann

conformal transformations are given by holomorphic or antiholomorphic

7.1 Weyl Transformations 123

Conformal Transformations

Name Group Element Generator translations x^a 7→x^a+a^a P_a (Lorentz^∗) rotations x^a 7→Λ^a_bx^b M_ab

dilation x^a 7→λx^a D

SCT^∗∗ x^a 7→ ^x_Ω^a+bax2

SCR(x) K_a

Table 7.1: Finite Conformal Transformations

functions. Decomposing these functions by a Laurent expansion demon-strates that the two dimensional conformal group has infinitely many gen-erators, which form the Witt/Virasoro algebra.

The four dimensional case is generic and will be discussed below.

7.1.2 Conformal Algebra in d > 2

In d >2 dimensions in Minkowski space, infinitesimal conformal transfor-mations are given by

ξ^a(x) = a^a+ω^abxb+λx^a+ (x²b^a−2x^axbb^b) (7.6) with the corresponding generators

δ_C =ia^aP_a+iω^abM_ab+iλD+ib^aK_a, (7.7) which form the conformal algebra

[M_ab, P_c] =−2iP_[aη_b]c, [M_ab, K_c] =−2iK_[aη_b]c, [D, P_a] =−iP_a, [D, K_a] =iK_a,

[D, M_ab] = 0, [P_a, K_b] = 2i(M_ab−η_abD), [M_ab, M_cd] = 2i η_a[cM_d]b −η_b[cM_d]a

.

(7.8)

This can be identified with the algebra so(d,2) by defining a suitable

(d+ 2)×(d+ 2) matrix

M_mˆˆn:=







M_mn ¹₂(K_m−P_m) ¹₂(K_m+P_m)

−¹₂(K_m−P_m) 0 −D

−¹₂(K_m+P_m) D 0





 (7.9)

and choosing η_mˆˆn= diag(η_mn,1,−1) as metric. As an aside, the d-dimen-sional conformal algebra is identical to the (d+ 1)-dimensionaladsalgebra

cf_d≡ads_d+1 ≡so(2, d). (7.10) The finite transformations corresponding to the infinitesimal solutions

finite

transformations (7.6) are shown in Figure7.1, where Ω_SCT(x) := 1−~b·~x+b²~x² is the scale factor Ω of the metric for special conformal transformations, and~a·~b has been used as a short-hand forη_mna^mbⁿ.

7.1.3 Weyl Transformations of the Riemann Tensor

Since superspace supergravity is described using a tangent space formula-tion, which has the additional advantage of a metric δ[η_ab] = 0 invariant under Weyl transformations, the transformational behaviour of the Rie-mann R_abcd and Weyl C_abcd tensor, Ricci tensor R_ab and scalar R, and covariant derivative ∇ under δ[g_mn] =−2σg_mn shall be given in terms of tangent space objects.

δ[e_a^m] =σe_a^m, (7.11a)

δ[p

−detg] =δ[dete⁻¹] =−σdp

−detg =−σddete⁻¹, (7.11b) δ[R^ab_cd] =δ^[a

[c∇^b]∇

d]σ+ 2σR^ab_cd, (7.11c)

δ[R_abcd] =η_[c[a∇_b]∇_d]σ+ 2σR_abcd, (7.11d) δ[R_ab] =η_ab∇²σ+ 2∇_a∇_bσ+ 2σR_ab, (7.11e)

δ[R] = 6∇²σ+ 2σR, (7.11f)

∗Λ^caηcdΛ^db=ηab

∗∗Special Conformal Transformation

7.1 Weyl Transformations 125

δ[G_ab] =δ[R_ab]−¹₂η_abδ[R]

=−2η_ab∇²σ+ 2∇_a∇_bσ+ 2σG_ab, (7.11g)

δ[C_abcd] = 2σC_abcd, (7.11h)

δ[∇_a] =σ∇_a−(∇^bσ)M_ab, M_abV^c =δ_a^cV_b−δ_b^cV_a, (7.11i)

δ[∇_aλ] =σ∇_aλ, (7.11j)

δ[∇²λ] = 2σ(∇²λ) + (2−d)(∇^aσ)(∇_aλ), (7.11k) where d is the space-time dimension, which from now on will be assumed to be equal to four.

7.1.4 Weyl Covariant Differential Operators

By definition a field ψ is denoted conformally covariant if it transforms under Weyl transformations into e^wσψ, that is homogeneously with Weyl weight w. In particular, it is interesting to have invariant expressions of the form

Z

d⁴x e⁻¹χ^∗∆4−2wψ, (7.12) with ∆_4−2w a differential operator of order 4−2w and ψ, χ are assumed to be Lorentz scalars.

The unique local, Weyl covariant differential operator acting on such fields ψ and χ of Weyl weight 1 is given by

∆₂ =∇²− ¹₆R, (7.13)

which can be easily verified using relations (7.11). It is however entertain-ing to derive this expression in a slightly different manner.

General relativity is not invariant under Weyl transformations as can be seen from the Einstein–Hilbert action transforming according to

Z

d⁴x e⁻¹R 7→

Z

d⁴x e⁻¹[e^−2σR+ 6(∇^ae^−σ)(∇_ae^−σ)]. (7.14) Since Weyl transformations form an Abelian group, a parametrisation may be chosen where two consecutive transformations with parameters σ₁ and σ₂ correspond to a single Weyl transformation with parameter

σ₁+σ₂. (Evidently e_a^m 7→ e^σe_a^m is such a parametrisation.) Replacing the parameter of the first transformation by a field φ = e^−σ1 of Weyl weight 1 yields an invariant expression as can be seen from

e^σ1e_a^m =φ⁻¹e_a^m 7→(e^−σ2φ⁻¹)(e^σ2e_a^m) =φ⁻¹e_a^m. (7.15) Therefore, the following action is Weyl invariant

Z

d⁴x e⁻¹[φ²R+ 6(∇^aφ)(∇aφ)] = 6 Z

d⁴x e⁻¹φ[∇²− ¹₆R]φ (7.16) and the operator ∆₂ has been rederived.

In addition the important notion of a compensating field, here φ, has

compensator

been introduced. Compensating fields allow incorporating a symmetry into the formulation of a theory that originally was not part of it. An ana-logue procedure is needed to embed Poincar´e supergravity into the Weyl invariant supergravity algebra by use of a so-calledchiral compensator.

Unfortunately, the elegant method above does not lend itself to gener-alisations and clearly cannot be used to construct a conformally covariant operator for a field of vanishing Weyl weight. However a dimensional anal-ysis can be used to write down a basis for such an operator and determine the prefactors from Weyl variation. The following operator due to Riegert

Riegert operator

[50] is the unique conformally covariant differential operator of fourth or-der, which because of its importance for this work will be given in several equivalent forms,

∆4 :=∇⁴ + 2Gab∇^a∇^b+ ¹₃∇^aR∇a

=∇⁴ + 2G_ab∇^a∇^b+ ¹₃(∇^aR)∇_a+ ¹₃R∇²

=∇⁴ + 2R_ab∇^a∇^b +¹₃(∇^aR)∇_a− ²₃R∇²

=∇⁴ + 2∇^aRab∇^b −²₃(∇^aR)∇a− ²₃R∇²,

(7.17)

or partially integrated,

λ⁰∆₄λ= (∇²λ)(∇²λ⁰)−2G_ab(∇^aλ)(∇^bλ⁰)

− ¹₃R(∇^aλ)(∇_aλ⁰) + (total deriv.). (7.18)

Im Dokument Quantum field theories coupled to supergravity (Seite 151-163)