Crossing, modular averages and N ↔ k in WZW models

Abstract

We consider the construction of genus zero correlators of SU (N ) k WZW models involving two Kac Moody primaries in the fundamental and two in the anti-fundamental representation from modular averaging of the contribution of the vacuum conformal block. In cases where we find the orbit of the vacuum conformal block to be finite, modular averaging reproduces the exact result for the correlators. In other cases, we perform the modular averaging numerically, the results are in agreement with the exact answers. We find a close relationship between the modular averaging sums of the theories related by level rank duality. We establish a one to one correspondence between elements of the orbits of the vacuum conformal blocks of dual theories. The contributions of paired terms to their respective correlators are simply related. One consequence of this is that the ratio between the OPE coefficients associated with dual correlators can be obtained analytically without performing the sums involved in the modular averagings. The pairing of terms in the modular averaging sums for dual theories suggests an interesting connection between level rank duality and semi-classical holographic computations of the correlators in the theories.

Modular linear differential equations for four-point sphere conformal blocks

“…This paper discusses the construction of differential equation for the conformal blocks that appear in the expansion of a crossing symmetric 4-point sphere correlator. Crossing symmetry translates as certain modular properties of the correlator under a well-known map [6] between the cross-ratio space and the complex upper half plane H + [7][8][9]. This allows us to write modular linear differential equations (MLDEs) in τ ∈ H + w.r.t.…”

“…In this section, we review the necessary elements of this relationship. For more details, see [6][7][8][9].…”