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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.
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We present a systematic procedure to extract the dynamics of the low energy soft mode in SYK type models with a single energy scale $J$ and emergent reparametrization symmetry in the IR. This is given in the framework of the perturbation theory scheme of arXiv:1608.07567 based on specific (off-shell) breaking of conformal invariance in the UV, adjusted to yield the exact large-$N$ saddle point. While this breaking formally vanishes on-shell, it has a non-trivial effect on correlation functions and the effective action. In particular, it leads to the Schwarzian action with a specific coupling to bi-local matter. The method is applied to the evaluation of $O(1)$ corrections to the correlation function of bi-locals. As a byproduct we confirm precise agreement with the explicit, symmetry breaking procedure. We provide a verification in the large $q$ limit (Liouville theory), where the correlators can be calculated exactly at all length scales. In this case, our scheme illuminates how the enhanced $O(J)$ and the subleading $O(1)$ contributions originate from the Schwarzian dynamics of the soft mode and its interaction with $h=2$ (bi-local) matter.
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We consider conformal field theories around points of large twist degeneracy. Examples of this are theories with weakly broken higher spin symmetry and perturbations around generalised free fields. At the degenerate point we introduce twist conformal blocks. These are eigenfunctions of certain quartic operators and encode the contribution, to a given four-point correlator, of the whole tower of intermediate operators with a given twist. As we perturb around the degenerate point, the twist degeneracy is lifted. In many situations this breaking is controlled by inverse powers of the spin. In such cases the twist conformal blocks can be decomposed into a sequence of functions which we systematically construct. Decomposing the four-point correlator in this basis turns crossing symmetry into an algebraic problem. Our method can be applied to a wide spectrum of conformal field theories in any number of dimensions and at any order in the breaking parameter. As an example, we compute the spectrum of various theories around generalised free fields. For theories with higher spin symmetry we discuss the relation between twist conformal blocks and higher spin conformal blocks.
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