Modular properties of correlators in generalised Gibbs ensembles

We study the modular properties of torus correlators in a generalised Gibbs ensemble (GGE) of a two-dimensional conformal field theory. For the free fermion with KdV charges, we compute the exact two-point function and derive its modular $S$-transformation as a formal power series in the chemical potential. The transformed correlator admits a factorised description: an effective charge deforms the ensemble, while a local dressing operator acts on the external fields. Both are determined by nested operator product expansions involving the deforming current. Motivated by this result, we conjecture an analogous transformation law for $n$-point functions in rational chiral conformal field theories. Using Zhu's recursion, we establish the conjecture at the lowest explicit power of the modular parameter to all orders in the chemical potential, thereby determining the dressing operator. We also verify the full conjectured transformation through cubic order in the chemical potential, retaining all explicit powers of the modular parameter. These results extend the nested-OPE description of modular transformations of GGE partition functions to local correlation functions.