Gravity and Strings

480 From eleven to four dimensions
on the first 2n indices. Clearly, all the vector fields associated with the n + p negative
eigenvalues of η are matter vector fields. The truncation to pure supergravity now includes
the conditions
a I m = A I µ = 0. (16.150)
The T-duality group is now O(n, n + p) and includes the interchange of KK and winding
vectors with the U(1) gauge vectors. This is not too surprising if we take into account that
the gauge fields of the heterotic string originate from the compactification of the right- or
left-moving part of 16 worldsheet scalars.
16.5.5 Trading the KR 2-form for its dual
As we mentioned in the introduction, in certain dimensions, the symmetry of the compactified
theory can be bigger than O(n, n + p),for instance due to the possibility of dualizing
fields that can be rotated into other already existing fields. Here we are going to see an
important example in d = 4 dimensions, in which the heterotic-string KR 2-form can be
dualized into a pseudoscalar axion field, which, together with the dilaton, parametrizes an
SL(2, R)/SO(2) coset space, Eq. (11.209) (the one present in N = 4, d = 4 supergravity,
studied in Section 12.2). It turns out that the equations of motion (but not the action) of the
full theory are SL(2, R)-covariant because the SL(2, R) transformations involve the dualization
of the vector fields (which are dual to vector fields precisely in d = 4). This new
hidden symmetry of the supergravity theory has been conjectured to be a non-perturbative
S duality of the full heterotic-string theory [397, 803, 842].
We are also interested in the dualization of the KR 2-form in d = 6, in which one obtains
another 2-form potential. A transformation of the dilaton and metric brings the theory into
the form of the theory that one obtains by compactifying N = 2A, d = 10 supergravity on
K3, which is evidence of a strong–weak-coupling duality between the full heterotic-string
theory compactified on T4 and the full type-IIA string theory compactified on K3.
The SO(32) heterotic string is also related by another strong–weak-coupling duality
to the type-I SO(32) superstring but the relation does not involve the dualization of the
(NSNS) KR 2-form, but rather its interchange by a RR 2-form, as we will see in Section
17.5.
Then we are going to perform the dualization of the KR 2-form in arbitrary dimension d.
The general procedure for Poincaré dualizations is explained in Section 8.7.1: we consider
the action as a functional of H and add a Lagrange multiplier term to enforce the Bianchi
identity
E µ1···µ6−n ≡∇µ ⋆ H µµ1···µ6−n
(−1)
+ 6−n
ηF
4
νρ ⋆ F νρµ1···µ6−n = 0. (16.151)
The Lagrange-multiplier term that has to be added to Eq. (16.123) (diagonalized, so L is
replaced by η, F by F, etc.) is
g2 h
16πG (d)
N
d d x |g|
1
(6 − n)! ˜Bµ1···µ6−n E µ1···µ6−n . (16.152)