# Crossing, Modular Averages and $N \leftrightarrow k $ in WZW Models

## TL;DR

This paper explores the construction of genus zero correlators in SU(N)_k WZW models using modular averaging, revealing dualities and relationships that simplify calculations and connect to holographic theories.

## Contribution

It introduces a novel modular averaging approach for WZW correlators, establishing dualities and analytical relations between OPE coefficients in level rank dual theories.

## Key findings

- Modular averaging reproduces exact correlators in finite orbit cases.
- Numerical modular averaging matches known exact results.
- Dual theories exhibit a one-to-one correspondence in their conformal block orbits.

## 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.