String Theory and M-Theory

178 Strings with space-time supersymmetry
and
YC = −16 L(R/4). (5.129)
This decomposition has a simple interpretation in terms of string world
sheets. YB is the boundary – or D-brane – contribution. It carries all the
dependence on the gauge fields. YC is the cross-cap – or orientifold plane –
contribution. Note that
I ′ = Y 2 = Y 2 B + 2YBYC + Y 2 C
(5.130)
displays the anomaly contributions arising from distinct world-sheet topologies:
the cylinder, the Moebius strip, and the Klein bottle, as shown in
Fig. 5.3.
Fig. 5.3. World-sheet topologies contributing to the anomaly in type I superstring
theory. Opposite edges with arrows are identified with the arrow aligned.
Cancellation of the anomaly requires a local counterterm, Sct, with the
property that
δSct = − G10, (5.131)
where G10 is the anomaly ten-form that follows, via the descent equations,
from [I]12 = 2Y4Y8. As was mentioned earlier, there are inconsequential
ambiguities in the determination of G10 from [I]12. A convenient choice in
the present case is
G10 = 2G2Y8, (5.132)
where G2 is a two-form that is related to Y4 by the descent equations Y4 =
dω3 and δω3 = dG2. This works because Y8 is closed and gauge invariant.
Specifically, for the normalizations given here,
Y4 = 1
4 (trR2 − trF 2 ) (5.133)