Ten-dimensional string cosmology
261
remaining Ne spatial dimensions are static compact dimensions. In the case of
the heterotic string,
N+Nc =9.
(9.62)
Eventually we want N = 3 but we shall leave N general for now. The effective
action for the gravitational field and the dilaton is then (up to a multiplicative
constant)
So = f d N + 1 x JjGje-~(lR + 4G AB 8AI/>8BI/»
(9.63)
where GAB is the metric tensor, G = det[ G AB], 1R is the curvature scalar for
the (N + I )-dimensional space with A = 0, 1, ... , N and I/> is the dilaton. In
the case of N = 3, the connection with the value of the gauge coupling constant
gSUing at the string scale is gsUing = e~ and the connection with the dilaton field of
section 9.3 is Re S = 2e-~. (The antisyrnmetric tensor field in the supergravity
multiplet is being ignored for simplicity.) All spatial dimensions will be taken
toroidal with periodic length ai (t) for i = I, 2, ... , N. Write
ai(t) = eA/{t).
(9.64)
The metric is given by
N
ds 2 = dt 2 - L al(t)(dx i )2.
(9.65)
;=1
Keeping Goo general for the moment, and fixing it to I later so that we do not
lose the field equation obtained by varying with respect to Goo, the curvature
scalar (exercise 1) is
N
N
2
N
N
1R= _G oo [ L(~d+ (L~;) +2L£; -GOOL~1 (9.66)
;=1
1=1
;=1
;=1
It is convenient to define
N
cl> == 21/> - LA;
(9.67)
;=1
which absorbs a factor for the volume of the space because
e-~JiGi = JGooe-().
(9.68)
The action of (9.63) is then (exercise 2)
N
So = -(I dNX) / dtJGooe-()G oo [ ~)i;)2 - ( 1=1