Effective String Theory and Nonlinear Lorentz Invariance

TL;DR

The paper analyzes the low-energy effective action for transverse fluctuations of a long string in D-dimensional spacetime within the static gauge, enforcing a nonlinearly realized Lorentz symmetry. It identifies the leading Lorentz-invariant corrections to the Nambu–Goto action: in 2+1 dimensions the correction is a curvature-squared term, while in higher dimensions the leading correction takes a nonlocal form proportional to R(˜Box)^{-1}R, closely related to Polyakov’s determinant. The authors show these completions are unique at the lowest nontrivial orders (classically) and discuss their relation to previous approaches and potential quantum constraints. They also outline broader implications and future directions, including extensions to boundaries, gauge fields, and supersymmetry, as well as connections to lattice results for confining strings.

Abstract

We study the low-energy effective action governing the transverse fluctuations of a long string, such as a confining flux tube in QCD. We work in the static gauge where this action contains only the transverse excitations of the string. The static gauge action is strongly constrained by the requirement that the Lorentz symmetry, that is spontaneously broken by the long string vacuum, is nonlinearly realized on the Nambu-Goldstone bosons. One solution to the constraints (at the classical level) is the Nambu-Goto action, and the general solution contains higher derivative corrections to this. We show that in 2+1 dimensions, the first allowed correction to the Nambu-Goto action is proportional to the squared curvature of the induced metric on the worldsheet. In higher dimensions, there is a more complicated allowed correction that appears at lower order than the curvature squared. We argue that this leading correction is similar to, but not identical to, the one-loop determinant (\sqrt{-h} R \Box^{-1} R) computed by Polyakov for the bosonic fundamental string.