Feynman Path Integral Formulation

48 1 Continuum Formulation
ultraviolet cutoff. This implies that the coupling α is asymptotically free, and therefore
grows with distance. Since it is 1/α that appears in the action, one would still
recover the Nambu-Goto string in the continuum limit, unless some new unexpected
fixed points emerge to higher order.
Finally it is possible to generalize the bosonic string action of Eq. (1.195) by
including a background X that is not flat (Callan, Friedan, Martinec and Perry, 1985)
I nlsm [g,X,b,φ] = 1
4πα ′
∫
d 2 σ √ g[ √ gg ab g μν (X)∂ a X μ ∂ b X μ
+ g −1/2 ε ab A μν (X)∂ a X μ ∂ b X μ − 1 2 α′ 2 Rφ(X)] ,
(1.226)
where the new fields include the graviton g μν (X), an antisymmetric tensor field
A μν (X), and the dilaton φ(X). These models are usually referred to, for historical
reasons, as non-linear sigma models for strings. Note, from the structure of the last
term in Eq. (1.226) involving 2 R, that the dilaton field is related to the string coupling
“constant”, with the n-loop amplitude involving a factor e −2(1−n)φ (at least for a
slowly varying dilaton field).
In general these theories will no longer be conformally invariant unless one imposes
conditions on the “beta functions for each field”, such as R μν (X)+... = 0,
where the ellipsis refers to contributions from the other two fields φ(X) and A μν (X).
It can be shown that these string consistency equations can be derived from a d-
dimensional action with a rather simple form
I dil = − 1 ∫
16πG
d d X √ Ge φ { R + 4(∇φ) 2 − 1
12 F2 μνσ + ... } , (1.227)
with F μνσ the curl of A μν . After rescaling the d-dimensional metric
G μν → e 4φ/(d−2) G μν one obtains the more familiar form
I dil = − 1 ∫
16πG
d d X √ {
G R − 4
}
d − 2 (∇φ)2 −
12 1 e−8φ(d−2) Fμνσ 2 + ...
(1.228)
which shows the general feature of strings coupled to background gravity: they generally
involve dilaton corrections to Einstein gravity.
,
1.11 Supersymmetric Strings
From the preceding discussion it appears that there are three main problems with
the bosonic string, the first one being that the ground state is a tachyon, a particle
of mass m 2 < 0. The second problem is that the bosonic string is only consistently
defined in d = 26 spacetime dimensions, and the third problem is that it does not
contain fermions which are after all an essential component of ordinary matter.