Conformal perturbation theory beyond the leading order

TL;DR

The paper develops a systematic framework for conformal perturbation theory beyond leading order in BCFTs, identifying universal, scheme-independent beta-function coefficients and providing explicit expressions for next-to-leading (and cubic) order terms in terms of integrated correlation functions. By comparing a minimal subtraction scheme with a Wilsonian/OPE scheme, it proves quadratic-order renormalizability for bulk–boundary perturbations and demonstrates that universal quantities, such as the boundary dimension shift under bulk moduli, are captured by conformal-block integrals. The formalism is validated through explicit geometric tests: a Neumann brane on a circle reproduces the expected radius dependence of boundary operator dimensions, and branes at angles on a torus yield the correct boundary-changing operator dimensions consistent with the angle. The results yield concrete, computable expressions for cubic universal coefficients in both bulk and boundary sectors, expressed as conformal-block integrals, and generalized to boundary-changing operators, offering a robust world-sheet tool for studying brane stability under bulk deformations.

Abstract

Higher order conformal perturbation theory is studied for theories with and without boundaries. We identify systematically the universal quantities in the beta function equations, and we give explicit formulae for the universal coefficients at next-to-leading order in terms of integrated correlation functions. As an example, we analyse the radius-dependence of the conformal dimension of some boundary operators for the case of a single Neumann brane on a circle, and for an intersecting brane configuration on a torus, reproducing in both cases the expected geometrical answer.