D-brane inflation
267
where N9 is the number of D9-branes, and NSi is the number of D5;-branes,
i.e. the number of D5-branes which wrap the ith torus (as well as living in fourdimensional spacetime). The potential V9 due to the D9-branes is of the form
V9 = T9RfR~R~
(9.93)
with the D9-brane tension
T9 = a9M:Oe-·
(9.94)
where a9 is a dimensionless constant. Also. the potential due to the D5;-branes is
of the form
VS; = TsR;
(9.95)
with the D5-brane tension
TS = asM6 s e -.
(9.96)
and as a dimension less constant. These potentials can be rewritten in terms of the
real parts of the four-dimensional dilaton S and the T; moduli fields:
s - = ReS - - M6 s e-.R2R2R2 123
(9.97)
t; == ReT; = M;e-.R; fori = 1,2,3.
(9.98)
Then,
3
V = M:[ N9k9S + ?:NSiksit;]
(9.99)
,=1
where k9 and ks; are dimensionless constants of order I. There is also a
contribution to the potential from the exchange of massless bulk states, such as
the graviton, which we are neglecting here. It can be shown that this is small
compared to the retained terms when the moduli are large.
To apply this potential to the study of inflation, it is convenient to recast it
in terms of fields with canonical kinetic terms. After Weyl rescaling to remove
the factor of e-~ in front of the curvature scalar (displayed in (9.63», the kinetic
terms for moduli are
Ckinctic = ~M~Jiiig"'"( a", Insa" Ins + t a",lnti a" In ti) (9.100)
,=1
the Weyl rescaling being
8",11 -
).g",1I
(9.\01)
with
M2e~
). = '8;'
(9.\02)
R2 R2 .
s 1 2 3
These kinetic terms arise from the curvature scalar
IR. == GabRab
(9.\03)