Gravity and Strings

16.5 Toroidal compactification of the heterotic string 471
16.5 Toroidal compactification of the heterotic string
In this section we are going to study toroidal compactifications of the heterotic-string
effective-field theory from ˆd = 10 to d = 10 − n dimensions. Our goal is to find the T and
S dualities that arise in the compactification, especially in d = 4 dimensions. In contrast
to maximal supergravities (32 supercharges, N = 2, d = 10 theories and their toroidal reductions,
for example), after dimensional reduction on an n-torus the supergravity multiplet
becomes reducible into a lower-dimensional supergravity multiplet and n vector multiplets.
We will study how to separate the two kinds of fields. This is important, since matter fields
can always be consistently truncated, but they have to be correctly identified in order to
preserve supersymmetry.
This dimensional reduction was first done in [226] in the Einstein frame and in [675]
in the string frame, in which a stringy interpretation was given to the dualities they
found. Here we repeat what they did, first for pure N = 1, d = 10 supergravity using
our own conventions and emphasizing the relations between ten-dimensional and lowerdimensional
fields that will allow us to relate solutions in different dimensions. Later we
will add Yang–Mills fields in order to have the complete heterotic-string effective-field
theory.
16.5.1 Reduction of the action of pure N = 1, d = 10 supergravity
The Ricci scalar and dilaton terms. We can use the notation and Ansatz we made for the
metric in Section 11.4 and apply immediately Eqs. (11.183)–(11.185), although we have
to insert into the first of them the dilaton prefactor e−2 ˆφ .Ondefining the d = ( ˆd − n)dimensional
dilaton field by
e −2φ ≡ e −2 ˆφ
K, (16.98)
integrating over the n redundant coordinates, and applying again Palatini’s identity to reexpress
the d-dimensional spin-connection coefficients in terms of the Ricci scalar, we
obtain
ˆg 2
16πG (10)
N
=
d 10 ˆx |ˆg| e −2 ˆφ ˆR − 4(∂ ˆφ) 2
g2
d d x |g| e −2φ R − 4(∂φ) 2 + 1
4 F 2 − 1
4∂aGmn∂ aG mn , (16.99)
16πG (d)
N
where the d-dimensional string coupling constant is
g = e φ0 ˆφ0 = e 1
Vn
√ =ˆg
K0 (2πℓs) n
1
2
, Vn = (2π) n R9 ···R(10−n), (16.100)
and the d-dimensional Newton constant G (d)
N
G (d)
N
is related to G(10)
N by
= G(10)
N /Vn. (16.101)