Gravity and Strings

11.1 Classical and quantum mechanics on R 1,3 × S 1 295
(one helicity state), which we will call a KK scalar. The number of helicity states (degrees
of freedom) is conserved in this decomposition. The massless spectrum is, thus,
{gµν, Aµ, k}, (11.14)
and its symmetries are the local version of symmetries of the KK vacuum determined by the
metric Eq. (11.2) plus a vanishing vacuum expectation value for the vector field 〈Aµ〉=0,
i.e. four-dimensional GCTs times local U(1) whose gauge field is Aµ.
The infinite tower of four-dimensional massive modes is constituted by spin-2 particles
(massive gravitons) [829]. They appear as interacting massless 5 gravitons, vectors, and
scalars labeled by an integer:
{g (n)
µν , A(n) µ , k(n) }. (11.15)
As for the n = 0 modes, we will see that these fields are related by an infinite symmetry
group that contains the Virasoro group [324]. These symmetries are spontaneously broken
in the above KK vacuum, and the fields A (n)
µ and k(n) are the corresponding Goldstone
bosons. Owing to the Higgs mechanism, a massless vector and scalar are “eaten” by each
massless graviton, giving rise to the massive gravitons [238, 239, 324]. Observe that the
number of helicity states is also preserved. 6
A brief and approximate description of how the Higgs mechanism works in this case
is worth giving. Some of the symmetries acting on the n = 0sector are massive gauge
transformations, which include shifts of the scalars k (n) by arbitrary functions that are also
standard gauge parameters for the vectors Aµ (n) and shifts of the vectors Aµ (n) by arbitrary
vectors that are standard gauge transformations for the g (n)
µν
s. This means that the gaugeinvariant
field strengths of the scalars and vectors have, very roughly, the structure
∂µk (n) + nA (n)
µ , ∂µ A (n)
ν
+ ng(n) µν . (11.16)
5 Strictly speaking, one cannot speak about the mass of these fields since, due to the interactions, neither of
them is a mass eigenstate [238, 239]. By massless here we simply mean that they enjoy gauge invariances
analogous to those of the massless fields.
6 More generally, in ˆd dimensions the graviton (spin 2) has ˆd( ˆd − 3)/2 helicity states and a massless
(p + 1)-form potential has ( ˆd − 2)!/[(p + 1)!( ˆd − p − 3)!] helicity states. In particular, a spin-1
particle (vector, p = 0) has ˆd − 2 and a spin-0 particle (scalar p =−1) always has one. A massive
graviton (spin-2 particle) has ˆd( ˆd − 1)/2 − 1 helicity states and a massive (p + 1)-form potential has
( ˆd − 1)!/[(p + 1)!( ˆd − p − 2)!] helicity states. In particular, a massive spin-1 particle (a massive vector,
p = 0) has ˆd − 1 helicity states and a massive spin-0 particle (a massive scalar, p =−1) has just one.
Thus, just on the basis of counting helicity states, the ˆd-dimensional graviton can always be decomposed
into a ( ˆd − 1)-dimensional massless graviton, vector, and scalar, and, if the interactions allow it, via the
Higgs mechanism, these massless particles can combine into a ( ˆd − 1)-dimensional massive graviton,
which has the same number of helicity states as the massless ˆd-dimensional one. Analogously, a massless
ˆd-dimensional (p + 1)-form potential gives rise to massless ( ˆd − 1)-dimensional (p + 1)- and p-form
potentials. If the interactions allow it, these two potentials can combine via the Higgs mechanism into a
( ˆd − 1)-dimensional massive (p + 1)-form potential that has the same number of helicity states as the
massless ˆd-dimensional one. Since invariance under GCTs is (see Appendix 3.2) nothing but the gauge
symmetry of the massless spin-2 particle, the theory of the massive graviton cannot have it. However, in the
description of the massive graviton as a coupled system of massless graviton, vector, and scalar field, it is
possible to have invariance under GCTs that is spontaneously broken by the Higgs mechanism.