Sigma model approach to string theory

A review of the $σ$-model approach to derivation of effective string equations of motion for the massless fields is presented. We limit our consideration to the case of the tree approximation in the closed bosonic string theory.

💡 Research Summary

The paper “Sigma model approach to string theory” by A.A. Tseytlin and P.N. Lebedev (1988) is a comprehensive review of how two‑dimensional sigma‑models can be used to derive the effective equations of motion for the massless fields of closed bosonic string theory, restricting the discussion to tree‑level (genus‑zero) amplitudes. The authors begin by pointing out that a non‑perturbative dynamical principle for string theory is still missing; what is known is that conformally invariant two‑dimensional field theories correspond to perturbative (tree) string vacua. They argue that a generalized sigma‑model, defined for an arbitrary number of space‑time dimensions D and arbitrary background couplings (metric G_{\mu\nu}, antisymmetric tensor B_{\mu\nu}, dilaton φ), provides a background‑independent framework that could eventually lead to a non‑perturbative definition of a “critical” string theory.

The paper proceeds in several logical steps:

1. Sigma‑model and second‑quantized string field theory – By drawing an analogy with ordinary field theory (e.g., a φ³ theory), the authors show how the propagator of a string field can be represented as a sum over cylindrical world‑surfaces, and how the cubic interaction vertex corresponds to an infinitesimally short three‑tube surface. This construction mirrors the way Polyakov’s path integral builds up higher‑genus surfaces from elementary building blocks. The background dependence of the string field theory is encoded in the propagator N = (Δ + K·\barΦ)^{-1}, where K is the three‑vertex functional and \barΦ is a background functional. The authors argue that the full quantum effective action Γ_q of the string field theory can be written as a sum over world‑surfaces weighted by the sigma‑model action I = I_0 + I_int, with the Euler characteristic χ = 2 – 2g controlling the genus expansion.

2. Sigma‑model and first‑quantized string theory – The authors reinterpret the first‑quantized Polyakov path integral as a sigma‑model defined on arbitrary world‑sheet metrics, but with the conformal factor (the Weyl mode) fixed, i.e., a choice of Weyl gauge. Different choices of gauge correspond to different off‑shell extensions of the first‑quantized theory. By “gluing” three disc‑like surfaces through the cubic vertex, they recover the closed‑string sphere amplitude and obtain an effective action of the form S = (1/g²)∫