On the effective theory of long open strings

TL;DR

This work derives the long-string effective action for open strings, showing that the leading correction to Nambu-Goto energies appears at $O(1/R^4)$ and is controlled by a single boundary coefficient for Dirichlet boundaries and by a similar boundary term for Neumann boundaries. By enforcing Lorentz invariance in the worldsheet theory, the authors constrain bulk couplings to the Nambu-Goto values and identify the distinct boundary operators that contribute at this order, with $b_2$ (Dirichlet) and $a_2$ (Neumann) remaining as the primary free parameters. They compute cylinder partition functions in both boundary conditions, extract energy corrections to open-string levels, and confirm constraints via a general Lorentz-noninvariant-free approach and a holographic realization. The holographic example with strings between D-branes demonstrates nonzero boundary coefficients, matching the field-theoretic expectations and illustrating the method's applicability to confining gauge theories. Overall, the paper clarifies how boundary dynamics modify the open-string spectrum and provides a practical framework for evaluating these effects in strongly coupled settings.

Abstract

We study the general low-energy effective action on long open strings, such as confining strings in pure gauge theories. Using Lorentz invariance, we find that for a string of length R, the leading deviation from the Nambu-Goto energy levels generically occurs at order 1/R^4 (including a correction to the ground state energy), as opposed to 1/R^5 for excited closed strings in four dimensions, and 1/R^7 for closed strings in three dimensions. This is true both for Dirichlet and for Neumann boundary conditions for the transverse directions, though the worldsheet boundary actions are different. The Dirichlet case is relevant (for instance) for the force between external quarks in a confining gauge theory, and the Neumann case for a string stretched between domain walls. In the specific case of confining gauge theories with a weakly curved holographic dual, we compute the coefficient of the leading correction when the open string ends on two D-branes, and find a non-vanishing result.