Abstract
We study conformal twist field four-point functions on a ℤN orbifold. We examine in detail the case N = 3 and analyze theories obtained by replicated N-times a minimal model with central charge c < 1. A fastly convergent expansion of the twist field correlation function in terms of sphere conformal blocks with central charge Nc is obtained by exploiting covering map techniques. We discuss extensive applications of the formalism to the entanglement of two disjoint intervals in CFT, in particular we propose a conformal block expansion for the partially transposed reduced density matrix. Finally, we refine the bounds on the structure constants of unitary CFTs determined previously by the genus two modular bootstrap.
Highlights
In the last thirty years, the bootstrap technique allowed calculating exactly or with great precision correlation functions of many two-dimensional critical statistical models [1]
Partition functions of CFTs on Riemann surfaces with ZN symmetry can be interpreted as powers of reduced density matrices for subsystems embedded into an extended quantum state, either pure or mixed [12, 13]
We propose a systematic expansion of the Z3-orbifold conformal blocks that allows building crossing symmetric twist field four-point functions with significantly better accuracy than previous attempts [31, 32]
Summary
Let us consider a CFT denoted by C, with central charge c, defined on a Riemann surface. The partition function with a flat metric everywhere on the surface can be derived from the orbifold C⊗N /ZN [21]. In this theory, there exist ZN twist and anti-twist fields σN and σN , which are spinless primary fields of conformal dimension [22, 23]. When inserted on the complex plane at the branch points of the algebraic curve in eq (2.1), they implement the multivaluedness of correlation functions under the analytic continuation (z − zb) → (z − zb)e2πi. For the class of surfaces Σg(x), the moduli space is one-dimensional and modular invariance implies the crossing symmetry of the twist field four-point correlation function [27]. Eq (2.6) can be extended analytically to complex values of x, see for instance section 5.2
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