Gravity and Strings

17.5 Consistent truncations and heterotic/type-I duality 499
so one can also consider the S dual of the construction that leads to the type-I theory:
an O9-plane associated with (−1) FL and 32 S duals of the D9-branes (S9-branes).
The result is the heterotic SO(32) superstring theory which arises, then, as the S dual
of the type-I SO(32) superstring. These theories are each other’s strong-coupling
limit [278, 577, 782]. The fields of the effective theories are related by the strong–
weak-coupling transformation
ˆj ˆµˆν = e − ˆφ ˆg ˆµˆν, ˆϕ =−ˆφh, Ĉ (2) ˆµˆν = ˆB ˆµˆν. (17.51)
Since S(−1) F S −1 = (−1) FR , one may also expect to find a (non-supersymmetric)
heterotic dual of the USp(32) superstring.
We can combine the construction of the type-I theory and this heterotic/type-I duality
with our knowledge of type-II T duality. Since we can consider the type-I theory as simply
type IIB with one O9-plane and 32 D9-branes and we know what the T dual of each of
them is (type IIA with an O8-plane and 32 D8-branes), we can immediately say that the T
dual of the type-I theory (called type I ′ [778]) is essentially a nine-dimensional theory with
N = 1 supersymmetry (16 supercharges) and gauge group SO(32)× U(1) 2 . Since
T T −1 = Ix, Ix x =−x, (17.52)
the presence of the O8-plane implies that, instead of R 9 × S 1 ,wehaveR 9 × S 1 /Z2 and we
actually have two O8-planes with RR charge −16 in the two nine-dimensional boundaries.
Introduction of Wilson lines into the compactification separates the D8-branes and one can
obtain different gauge groups [66, 710].
The S-dual version of this T duality is well known to lead from the heterotic SO(32)
theory to the heterotic E8 × E8 theory (up to the possible introduction of Wilson lines)
with one dimension compactified, which is associated with the Hoˇrava–Witten scenario
with one extra dimension compactified, that is, M theory on S 1 × S 1 /Z2. Wehave learned
that type-IIB S duality is a rotation of the 2-torus on which we compactify M theory and
here we are seeing precisely that the type I ′ theory is a rotated version of the heterotic
E8 × E8 theory compactified on a circle and both are related to M theory. Furthermore,
the mysterious objects at the boundaries of the Hoˇrava–Witten scenario, compactified on a
circle, are related to the O8-planes and D8-branes. More consequences of these chains of
dualities were studied in [123].