Feynman Path Integral Formulation

52 1 Continuum Formulation
allowed or favored, the manifold is generally chosen so that desirable features are
achieved for the four-dimensional theory; indeed recent work suggests that there
might be infinitely many choices for the vacua in ten dimensions. Generally in these
compactification scenarios the additional unwanted six space dimensions are curled
up in tiny tubes whose spatial extent is of the order of the string scale, although
several alternative scenarios are possible (for a recent review see Font and Theisen,
2005). One of the more speculative points of view asserts that string theory not
only predicts extra dimensions, but that the strength of four-dimensional gravity is
in fact affected by the presence of these extra dimensions by making it stronger in
these higher dimensional directions, an effect which be detectable by future high
precision experiments.
There is one generic feature of all string-theoretic models of gravity, and that is
the appearance of an extra scalar particle called the dilaton. All perturbative string
theories (type I, type II and heterotic) already start out with a dilaton in ten dimensions.
Although different in the details, the low energy effective action for the gravitational
degrees of freedom is similar to the bosonic dilaton actions of Eqs. (1.227)
and (1.228). In particular for the heterotic string one finds (Fradkin and Tseytlin,
1985)
I dil [G,φ,A μν ]=− 1 ∫
16πG
d d X √ Ge −2φ { R + 4(∂ μ φ) 2 −
12 1 F2 μνσ + ... } ,
(1.237)
with R the scalar curvature for the metric G μν , and
√
16πG ∼ gα
′(d−2)/4 , (1.238)
with g the dimensionless string coupling constant. Away from ten dimensions one
also has a cosmological constant contribution which is order one in units of α ′ .
Here and in the following it will be assumed that compactification has occurred
by now, so that the effective low energy theory resides in the physical dimension
d = 4. Furthermore it is a general feature of the string that it contains both massless
and massive modes. In the low energy effective field theory description only the
massless modes are retained, but the effect of the massive modes can in many ways
be regarded as being equivalent to having a cutoff at the string mass scale Λ s =
(α ′ ) −1/2 .
After the Weyl rescaling G μν → G μν exp[4π/(d − 2)] the action coincides with
the corresponding part of N = 2 d = 10 supergravity,
I dil = − 1 ∫
16πG
d d X √ G { R − 1 2 (∂ μφ) 2 − 1
12 F2 μνσ e −φ + ... } . (1.239)
The last term involving the gauge fields contains a dilaton-field dependent gauge
coupling constant, so that the main modification to the matter sector is in the form