The Worldsheet and Conformal Field Theory

The Polyakov viewpoint

The Nambu-Goto action measures the area of the worldsheet, but the Polyakov formulation introduces an independent two-dimensional metric and is better suited to quantization. In schematic form, the action is an integral over the worldsheet metric times the spacetime embedding fields. The embedding fields X describe where each point of the string sits in spacetime.

The worldsheet metric is partly gauge redundancy: different coordinate choices and Weyl rescalings can describe the same physical surface. After gauge fixing, the remaining conformal symmetry becomes central. The quantum theory must preserve that symmetry, or the gauge redundancy becomes anomalous and the spacetime interpretation fails.

Critical dimension

Quantum fluctuations can produce a conformal anomaly. Requiring the total anomaly to cancel fixes the critical dimension. For the bosonic string it is 26. For the superstring it is 10. The extra dimensions are not an optional illustration; they are required by the consistency of the quantized worldsheet theory in the simplest formulations.

Compactification hides those dimensions by making them small and shaped. The low-energy observer sees Kaluza-Klein towers and background fields whose values depend on the geometry. The internal geometry also determines how much supersymmetry survives and which gauge and matter fields appear.

Vertex operators and scattering

A particle state in spacetime is represented on the worldsheet by a vertex operator. Correlation functions of these operators calculate string scattering amplitudes. Because the string interaction is represented by a smooth worldsheet topology rather than a pointlike vertex, many ultraviolet divergences are softened compared with point-particle gravity.

This does not mean every calculation is easy or finite in every background. Perturbative string theory is an expansion in the string coupling, and it can fail when the coupling is large or spacetime curvature reaches the string scale. Dual descriptions are often needed in those regimes.

Backgrounds are dynamical data

A consistent worldsheet sigma model can be viewed as a spacetime background satisfying beta-function equations. At leading order those equations reproduce Einstein-like equations plus additional fields and corrections. In this sense, spacetime geometry is not merely a stage: it is encoded by the condition that the worldsheet quantum theory remain conformal.

The extra-dimension problem is the next layer: choosing a compactification chooses much of the effective four-dimensional physics.