# Conformal Perturbation Theory for Twisted Fields

## TL;DR

This paper develops a method for second order conformal perturbation theory in 2D orbifold CFTs, using a mapping to the torus cover to evaluate twisted sector correlators and their integrals, revealing how twist fields' marginality changes at second order.

## Contribution

It introduces a regularization scheme and a numerical approach for evaluating twisted sector correlation functions in orbifold CFTs without requiring supersymmetry.

## Key findings

- Twist fields are marginal at first order but become non-marginal at second order in specific orbifold models.
- The method relates crossing symmetry to the modular group via the sphere-to-torus mapping.
- A regularization scheme enables numerical evaluation of integrals in twisted sectors.

## Abstract

We investigate second order conformal perturbation theory for $\mathbb{Z}_2$ orbifolds of conformal field theories in two dimensions. To evaluate the necessary twisted sector correlation functions and their integrals, we map them from the sphere to its torus double cover. We discuss how this relates crossing symmetry to the modular group, and introduce a regularization scheme on the cover that allows to evaluate the integrals numerically. These methods do not require supersymmetry. As an application, we show that in the torus orbifold of 8 and 16 free bosons, $\mathbb{Z}_2$ twist fields are marginal at first order, but stop being marginal at second order.