Feynman Path Integral Formulation

1.10 String Theory 47
with [dg ab ] an invariant functional measure over the two-dimensional metrics. The
Euler characteristic term is a constant for a surface of given topology, but adds
a relative weight e −kχ for string contributions with different topology. Eventually
one would like to sum over all topologies for the string (i.e. sum over surfaces
of arbitrary genus h with χ = 2[1 − h)], but what could stand in the way is that
bosonic string perturbation theory seems badly divergent and is not Borel summable,
Z ∼ ∑ h g h h!, contrary to what happens for example in QED, (Gross and Periwal,
1988). Such a result is usually taken as an indication of a serious vacuum instability
and large non-perturbative contributions (Parisi, 1979) (originally this estimate was
in fact viewed as a welcome feature, as there were many aspects of perturbative
string theory that were not shared by the real world!).
The usual treatment of quantum two-dimensional gravity then proceeds to set the
metric in the conformal gauge g ab (x)=e φ (x) ˜g ab , where ˜g ab is a reference metric,
usually taken to be the flat one, δ ab . The conformal gauge fixing for the metric then
implies a non-trivial Faddeev Popov determinant, which when exponentiated results
in the Liouville action for two-dimensional pure gravity in the conformal gauge
I [φ] = 13 ∫
d 2 σ [ 1
24π
2
(∂ a φ) 2 + μ 2 e φ ] , (1.222)
with the μ-term amounting to a renormalization of the bare cosmological constant.
In the language of the conformal gauge, where √ g = e φ and R = e −φ ∂ 2 φ, the preceding
action can in fact be re-written in arbitrary coordinate as a nonlocal contribution.
When the d scalar X fields are coupled to the two-dimensional gravity and integrated
out (since they appear quadratically in the action), the conformal anomaly
contribution modifies the Liouville effective action to
I [φ] = 26 − d
48π
∫
d 2 σ [ 1
2
(∂ a φ) 2 + μ 2 e φ ] , (1.223)
which suggests that the bosonic string only makes sense for embedding dimensions
d < 26, i.e. less than the critical dimension known already from dual models.
So far fluctuations in the string are essentially unconstrained when viewed from
embedding space. But it is possible to add some rigidity to the string by considering
extrinsic curvature terms (Polyakov, 1786). There are a number of equivalent ways
of writing such contributions, the simplest one being of the form
I ex [g,X] = 1 ∫
d 2 σ √ gg ab ∂ a t μν ∂ b t μν (1.224)
2α S
with
t μν = ε ab 1 √ g
∂ a X μ ∂ b X ν , (1.225)
and α a dimensionless coupling. Note that the new term involves higher derivatives
of the “matter field” X μ . A one-loop calculation shows that the renormalization
group behavior for α is α −1 (μ) =α −1 (Λ)+(D/4π)log(μ/Λ) with Λ the