Feynman Path Integral Formulation

48 1 Continuum Formulationultraviolet cutoff. This implies that the coupling α is asymptotically free, and thereforegrows with distance. Since it is 1/α that appears in the action, one would stillrecover the Nambu-Goto string in the continuum limit, unless some new unexpectedfixed points emerge to higher order.Finally it is possible to generalize the bosonic string action of Eq. (1.195) byincluding a background X that is not flat (Callan, Friedan, Martinec and Perry, 1985)I nlsm [g,X,b,φ] = 14πα ′∫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 fieldA μν (X), and the dilaton φ(X). These models are usually referred to, for historicalreasons, as non-linear sigma models for strings. Note, from the structure of the lastterm 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 aslowly varying dilaton field).In general these theories will no longer be conformally invariant unless one imposesconditions 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 formI dil = − 1 ∫16πGd d X √ Ge φ { R + 4(∇φ) 2 − 112 F2 μνσ + ... } , (1.227)with F μνσ the curl of A μν . After rescaling the d-dimensional metricG μν → e 4φ/(d−2) G μν one obtains the more familiar formI dil = − 1 ∫16πGd 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 generallyinvolve dilaton corrections to Einstein gravity.,1.11 Supersymmetric StringsFrom the preceding discussion it appears that there are three main problems withthe bosonic string, the first one being that the ground state is a tachyon, a particleof mass m 2 < 0. The second problem is that the bosonic string is only consistentlydefined in d = 26 spacetime dimensions, and the third problem is that it does notcontain fermions which are after all an essential component of ordinary matter.