Modular Properties of Symplectic Fermion Generalised Gibbs Ensemble

TL;DR

This paper investigates the modular properties of symplectic fermion conformal field theory, deriving conserved charges, constructing GGEs, and analyzing their modular transformations and asymptotic behaviors.

Contribution

It provides explicit expressions for conserved charges, modular S-transforms, and connects GGEs to known integrable hierarchies and defect models in symplectic fermion theory.

Findings

Derived conserved charges on the cylinder for symplectic fermion.

Obtained exact modular S-transform expressions for GGEs.

Confirmed asymptotic behaviors match conjectures and known hierarchies.

Abstract

The symplectic fermion is a much-studied non-unitary conformal field theory with $c=-2$ and is known to contain an infinite family of mutually commuting conserved charges. We derive expressions for the conserved charges on the cylinder and use these to construct Generalised Gibbs Ensembles (GGEs) in the particular case of the ${W}(1,2)$ triplet model. We derive exact expressions for the modular $S$-transforms in each sector of the symplectic fermion (and so of the whole GGE) and further extract the expressions in the asymptotic regime where the chemical potentials go to zero. Subsets of the conserved charges are known to reproduce the KdV and Boussinesq hierarchies. For the case in which the charge is identified with the zero mode $W_0$ of the $W_3$ algebra, we obtain asymptotic behaviour in precise agreement with the conjecture proposed in our companion paper [1]; for the KdV subset we obtain results which exactly mirror the case for a single free fermion. Finally we identify the GGE with a translation invariant and purely transmitting defect for the symplectic fermion fields, and make some comments on the relation to other $W_n$ algebras.