Perturbatively renormalized vertex operator, highest-weight representations of Virasoro algebra, and string dynamics in curved space

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Abstract
String dynamics is considered from the viewpoint of conformally invariant field theory. We discuss the renormalization of a general scalar field in the target space of a two-dimensional nonlinear σ model. Counterterms are constructed to leading order in the slope parameter. The renormalized operator contains exponential factors that give rise to correct scaling properties as a function of the infrared cutoff. The requirement that the operator be a primary conformal field with scaling dimension (h,h¯)=(1,1) (i.e., a vertex operator satisfying the Virasoro gauge conditions) determines the vertex function which describes tachyon emission from the string Iand Rthe equations for the background fields, which in our case are the metric and the dilaton. This then is an explicit, perturbative construction of a representation of the Virasoro algebra. In the Becchi-Rouet-Stora-Tyutin (BRST) formulation the backgrounds and the physical spectrum are solutions of the equation Qψ=0, where Q is the BRST change.

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