Sigma model approach to string theory

TL;DR

This work develops a sigma-model framework as a background-independent route to derive the tree-level equations of motion for massless string fields from Weyl invariance, arguing that the sigma-model is more fundamental than the on-shell first-quantized amplitudes. It connects second- and first-quantized perspectives by expressing the quantum effective action and string amplitudes as renormalized sigma-model partition functions across genera, with the central-charge (Weyl-anomaly) coefficient driving the tree-level dynamics. A central result is that Weyl-invariance conditions correspond to stationarity of a central-charge action $S=\int d^D y\sqrt{G}e^{-2\phi}\tilde{\beta}^\phi$, providing a covariant, background-dependent route to the bosonic string effective action and clarifying the role of renormalization-group flow in string vacua. Perturbative calculations of the Weyl anomaly coefficients illustrate how the known $\beta$-functions and the effective action arise in this framework, including nontrivial cases with $B_{\mu\nu}\neq 0$, and support a consistent link between the sigma-model Weyl conditions and the low-energy string dynamics. The conclusions discuss the limitations of integrating over the 2d conformal factor and propose avenues toward a more complete, possibly nonperturbative, sigma-model formulation of string vacua.

Abstract

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.