500
E. Rabinovici
The n = 0 modes in (8.130) are just the four dimensional graviton, photon and
dilaton. Substituting (8.129) and (8.130) in the action (8.127), integrating over y and
retaining just the n = 0 terms one obtains (dropping the 0 subscripts)
S =
1
2κ 2
d
4 x
√
−g
R −
1
2
∂ μ φ∂
u φ −
1
4
e
−
√
3φ F μν F
μν
(8.132)
where 2πκ 2 = mκ^ 2 and F μν = ∂ μ A ν − ∂ ν A μ . From (8.128), this action is invariant
under general.
coordinate transformations with parameter ξ μ 0 , i.e. (again dropping the 0
subscripts)
δg μν = ∂ μ ξ ρ g ρν + ∂ ν ξ ρ g μρ + ξ ρ ∂ ρ g μν
δ Aμ = ∂ μ ξ ρ A ρ + ξ ρ ∂ ρ A μ
δφ = ξ ρ ∂ ρ ϕ,
(8.133)
local gauge transformations with parameter ξ 4 0
δA μ = ∂ μ ξ
4
(8.134)
and global scale transformations with parameter λ
δA μ = λA μ , δϕ = −2λ/
√
3
(8.135)
The symmetry of a vacuum, determined by the VEVs
< g μν >= η μν , < A μ >= 0, < ϕ >= ϕ 0
(8.136)
is the four-dimensional Poincare group ×R. These expectation values have not been
determined dynamically but let us settle here for them. At this level of the analysis
the masslessness of the graviton is due to unbroken four dimensional general
covariance, the masslessness of the photon is consistent with the four dimensional
gauge invariance and the dilaton seems massless because it is the Goldstone boson
associated with the spontaneous breakdown of the global scale invariance. In fact
taking into account the actual periodicity in y which is not manifested for only the
n = 0 low energy modes the symmetry acting on the scalar field is U(1) rather than
R and is thus not a true scale symmetry. The field ϕ 0 is a pseudo-Goldstone boson
and does not really deserve to called a dilaton. Further analysis uncovers an infinite
tower of charged, massive spin 2 particles with charges e n and masses m n given by
e n = n
√
2km, m n =| n | m
(8.137)
E. Rabinovici
The n = 0 modes in (8.130) are just the four dimensional graviton, photon and
dilaton. Substituting (8.129) and (8.130) in the action (8.127), integrating over y and
retaining just the n = 0 terms one obtains (dropping the 0 subscripts)
S =
1
2κ 2
d
4 x
√
−g
R −
1
2
∂ μ φ∂
u φ −
1
4
e
−
√
3φ F μν F
μν
(8.132)
where 2πκ 2 = mκ^ 2 and F μν = ∂ μ A ν − ∂ ν A μ . From (8.128), this action is invariant
under general.
coordinate transformations with parameter ξ μ 0 , i.e. (again dropping the 0
subscripts)
δg μν = ∂ μ ξ ρ g ρν + ∂ ν ξ ρ g μρ + ξ ρ ∂ ρ g μν
δ Aμ = ∂ μ ξ ρ A ρ + ξ ρ ∂ ρ A μ
δφ = ξ ρ ∂ ρ ϕ,
(8.133)
local gauge transformations with parameter ξ 4 0
δA μ = ∂ μ ξ
4
(8.134)
and global scale transformations with parameter λ
δA μ = λA μ , δϕ = −2λ/
√
3
(8.135)
The symmetry of a vacuum, determined by the VEVs
< g μν >= η μν , < A μ >= 0, < ϕ >= ϕ 0
(8.136)
is the four-dimensional Poincare group ×R. These expectation values have not been
determined dynamically but let us settle here for them. At this level of the analysis
the masslessness of the graviton is due to unbroken four dimensional general
covariance, the masslessness of the photon is consistent with the four dimensional
gauge invariance and the dilaton seems massless because it is the Goldstone boson
associated with the spontaneous breakdown of the global scale invariance. In fact
taking into account the actual periodicity in y which is not manifested for only the
n = 0 low energy modes the symmetry acting on the scalar field is U(1) rather than
R and is thus not a true scale symmetry. The field ϕ 0 is a pseudo-Goldstone boson
and does not really deserve to called a dilaton. Further analysis uncovers an infinite
tower of charged, massive spin 2 particles with charges e n and masses m n given by
e n = n
√
2km, m n =| n | m
(8.137)