Comments on the SN orbifold CFT in the large N-limit

Abstract

We elaborate on various aspects of the conformal field theory of the symmetric orbifold. We collect various results that have appeared in the literature, and we present a coherent picture of the operator content of this CFT, relying on the orbifold extension of the Virasoro algebra. We then focus on the large N -limit of this theory, discuss the OPE of two twist operators, and find various selection rules. We review how to calculate four-point functions of twist operators, and we write down the most general four-point function in the covering space for large N . We show that it depends on some functions that obey a set of algebraic equations, that resemble the scattering equations. Finally, we provide a recipe on how to calculate correlation functions with insertions of the orbifold Virasoro generators.

String theory on AdS3 and the symmetric orbifold of Liouville theory

“…(2.34) has exactly the same form as for the case of a symmetric orbifold, where the L n modes act on the covering space, while the L m generators are those of the seed theory, see e.g. [25][26][27]. The analysis for the other DDF operators is similar.…”

Correlators of the symmetric product orbifold

“…For this purpose, it is useful to employ an explicit parametrisation of the covering map. We choose the parametrisation advocated in [41]. The idea is to determine the derivative of the covering map ∂Γ(t).…”

“…As noted in [41], it is convenient to rewrite the solution (3.20) in terms of these variables, where it takes the form…”

“…In the symmetric orbifold of T 4 , they also have the property that the sphere correlator is the only connected contribution to the full correlator and thus they can be evaluated exactly. 22 There is also a further simplification: the covering map is a polynomial and can be written in closed form using hypergeometric functions [41]. 20 Strictly speaking, [21] discussed that null vectors in the dual CFT correspond to BRST-exact states on the worldsheet only in a bosonic version of the correspondence.…”

“…We then show in Section 4 how to fix this integration constant by imposing factorisation of the four-point function into different OPE channels. In the process we explore a very convenient parametrisation of the covering map in terms of its pole locations [41]. Since the discussion of Sections 3 and 4 is a bit abstract, we then exemplify in Section 5 the construction with the example of the four-point function where all the fields have twist two.…”