Ivancevic_Applied-Diff-Geom

1146 Applied Differential Geometry: A Modern IntroductionThe Nambu–Goto action (6.234) and Polyakov action (6.235) representthe core of the so–called bosonic string theory, the original version ofstring theory, developed in the late 1960s. Although it has many attractivefeatures, it also predicts a particle called the tachyon possessing some unsettlingproperties, and it has no fermions. All of its particles are bosons,the matter particles. The physicists have also calculated that bosonic stringtheory requires 26 space–time dimensions: 25 spatial dimensions and onedimension of time. In the early 1970s, supersymmetry was discovered in thecontext of string theory, and a new version of string theory called superstringtheory (i.e., supersymmetric string theory) became the real focus, asit includes also fermions, the force particles. Nevertheless, bosonic stringtheory remains a very useful ‘toy model’ to understand many general featuresof perturbative string theory (see section 6.7 below).6.5.6 Weyl Invariance and Vertex Operator FormulationThe action S is also invariant under Weyl rescalings of the metric g bya positive function on σ : Σ → R, given by g → e 2σ g. In general, Weylinvariance of the full amplitude may be spoiled by anomalies. AssumingWeyl invariance of the full amplitude, the integral defining Amp may besimplified in two ways.1) The integration over Met(Σ) effectively collapses to an integration overthe moduli space of surfaces, which is finite dimensional, for each genus h.2) The boundary components of Σ — characterizing external string states— may be mapped to regular points on an underlying compact surfacewithout boundary by conformal transformations. The data, such as momentaand other quantum numbers of the external states, are mapped intovertex operators. The amplitudes are now given by the path integralAmp =∞∑∫h=0Met(Σ)∫1D[g]N(g)for suitable vertex operators V 1 , . . . V N .Map(Σ,M)D[x] V 1 . . . V N e −S ,6.5.7 More General ActionsGeneralizations of the action S given above are possible when M carriesextra structure.