Nonlocal Conformal Theories Have State-dependent Central Charges

Using the recently developed fractional Virasoro algebra cite{la_nave_fractional_2019}, we construct a class of nonlocal CFTs with OPEs of the form $T_k(z)Phi(w) sim frac{ h_gamma Phi}{(z-w)^{1+gamma}}+frac{partial_w^gamma Phi}{z-w},$ and $T_k(z)T_k(w) sim frac{ c_kZ_gamma}{(z-w)^{3gamma+1}}+frac{(1+gamma ) T_k(w)}{(z-w)^{1+gamma}}+frac{partial^gamma_w T_k}{z-w}$ 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 theories, can be mapped onto an equivalent local one using a suitable covering/field redefinition. In addition, we discuss various perturbative 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.