arXiv:2011.04662v3 [hep-th] 16 Sep 2021 ✐ ✐ “atmp-fractional-cft-final” — 2021/9/17 — 0:44 — page 1 — #1 ✐ ✐ ✐ ✐ ✐ ✐ ADV. THEOR. MATH. PHYS. Volume , Number , 1–31, Nonlocal conformal theories have state-dependent central charges Bora Basa 1 , Gabriele La Nave 2 and Philip Phillips 1 Using the recently developed fractional Virasoro algebra [27], we construct a class of nonlocal CFTs with OPE’s of the form T k (z )Φ(w) ∼ hγ Φ (z-w) 1+γ + ∂ γ w Φ z-w , and T k (z )T k (w) ∼ c k Zγ (z-w) 3γ+1 + (1+γ)T k (w) (z-w) 1+γ + ∂ γ w T k z-w which naturally results in a central charge, c k , that is state-dependent, with k indexing a particular grading. Our work indicates that only those theories which are nonlocal have state-dependent central charges, regardless of the pseudo-differential operator content of their action. All others, including certain fractional Laplacian the- ories, can be mapped onto an equivalent local one using a suitable covering/field redefinition. In addition, we discuss various pertur- bative implications of deformations of fractional CFTs that realize a fractional Virasoro algebra through the lense of a degree/state- dependent refinement of the 2 dimensional C-theorem. 1. Introduction While the locality of the action is a key tenet of field theory, there are nu- merous settings in which nonlocal operators appear explicitly. A mathemat- ically precise example comes from a physical interpretation of the Caffarelli- Silvestre (CS) extension theorem [7]: Local bulk dynamics can have corre- sponding boundary dynamics governed by a nonlocal operator, the Lapla- cian raised to a non-integer power. We will focus exclusively on the fractional Laplacian, but more generally, nonlocal operators of the form f (−Δ) appear naturally in many effective theories, for instance, in the context of gravity and cosmology [2, 11]. Quite generally, the fractional Laplacian (−Δ) γ (or its conformal ex- tension, the Panietz operator[9, 16, 17]) on a function f in R n provides a Dirichlet-to-Neumann map for a function φ in R n+1 that satisfies a local second-order elliptic differential equation. Paulos, et al. [35], simultaneous with ours [26], noted that the extension theorem allows one to equate the action of a free massive theory in d +1 dimensions in a spacetime such as 1