266
Superstring cosmology
orbifold fixed point in the compact dimensions. We shall assume that this orbifold
is a product of three two-dimensional tori with points on the tori identified under
the action of a discrete ZN or ZM x ZN group.
Models with the gauge fields of the standard model (up to some U (I) factors)
and the massless matter content of the standard model (up to some vector-like
matter) have been obtained by employing D3-branes and D7-branes located at
fixed points. The presence of D7-branes as well as D3-branes is necessary to
satisfy certain 'twisted tadpole' cancellation conditions, which are required by
a consistent string theory and, among other things, ensure non-Abelian gauge
anomaly cancellation. The models also contain D3-antibranes and D7-antibranes
which are needed to satisfy untwisted tadpole cancellation conditions. The chiral
matter states are associated with open strings with their endpoints on D3-branes
or with one endpoint on a D3-brane and the other on a D7 -brane. In such theories,
the string scale and the compactificatioD scale do not necessarily coincide and low
string scales are possible.
An important aspect of such constructions is that the D p-branes and Dpantibranes can be prevented from moving from the fixed points in some models
by the requirement that the twisted-tadpole conditions are always satisfied. Then,
the brane-antibrane separations are fixed except to the extent that they share in
the contraction or expansion of a toroidal space when the radius of space varies.
Thus, a modulus field in the associated low-energy supergravity theory whose
expectation value is a brane-antibrane separation is not a candidate inflaton.
However, possible candidates are the moduli scalars T;, i = 1, 2, 3, the real parts
of which are associated with the radii Ri of the three tori in the form
ti == Re T; = e'M; Rl
(9.89)
where Ms is the string scale and tP is the ten-dimensional dilaton. The fourdimensional dilaton S is also a candidate. Models of inflation have been
constructed [13] in which S or one of the T; provides the inflaton while the other
moduli (T; or S) are frozen by some unidentified mechanism.
To discuss such models of inflation, we now require the form of the effective
potential V as a function of the unfrozen modulus field. It is convenient to use
T-duality with respect to all directions simultaneously:
a'
i=I,2,3
(9.90)
Ri -+ Ri
where
2
a' = M s -
(9.91)
to map D3-branes into D9-branes and D7-branes into D5-branes. We need the
potential due to the tension in the branes. This is proportional to the volume of
the branes and, for a theory of D9-branes and D5-branes, is of the form
3
V = N9V9+ ENs/Vs/
(9.92)
;=1
Superstring cosmology
orbifold fixed point in the compact dimensions. We shall assume that this orbifold
is a product of three two-dimensional tori with points on the tori identified under
the action of a discrete ZN or ZM x ZN group.
Models with the gauge fields of the standard model (up to some U (I) factors)
and the massless matter content of the standard model (up to some vector-like
matter) have been obtained by employing D3-branes and D7-branes located at
fixed points. The presence of D7-branes as well as D3-branes is necessary to
satisfy certain 'twisted tadpole' cancellation conditions, which are required by
a consistent string theory and, among other things, ensure non-Abelian gauge
anomaly cancellation. The models also contain D3-antibranes and D7-antibranes
which are needed to satisfy untwisted tadpole cancellation conditions. The chiral
matter states are associated with open strings with their endpoints on D3-branes
or with one endpoint on a D3-brane and the other on a D7 -brane. In such theories,
the string scale and the compactificatioD scale do not necessarily coincide and low
string scales are possible.
An important aspect of such constructions is that the D p-branes and Dpantibranes can be prevented from moving from the fixed points in some models
by the requirement that the twisted-tadpole conditions are always satisfied. Then,
the brane-antibrane separations are fixed except to the extent that they share in
the contraction or expansion of a toroidal space when the radius of space varies.
Thus, a modulus field in the associated low-energy supergravity theory whose
expectation value is a brane-antibrane separation is not a candidate inflaton.
However, possible candidates are the moduli scalars T;, i = 1, 2, 3, the real parts
of which are associated with the radii Ri of the three tori in the form
ti == Re T; = e'M; Rl
(9.89)
where Ms is the string scale and tP is the ten-dimensional dilaton. The fourdimensional dilaton S is also a candidate. Models of inflation have been
constructed [13] in which S or one of the T; provides the inflaton while the other
moduli (T; or S) are frozen by some unidentified mechanism.
To discuss such models of inflation, we now require the form of the effective
potential V as a function of the unfrozen modulus field. It is convenient to use
T-duality with respect to all directions simultaneously:
a'
i=I,2,3
(9.90)
Ri -+ Ri
where
2
a' = M s -
(9.91)
to map D3-branes into D9-branes and D7-branes into D5-branes. We need the
potential due to the tension in the branes. This is proportional to the volume of
the branes and, for a theory of D9-branes and D5-branes, is of the form
3
V = N9V9+ ENs/Vs/
(9.92)
;=1