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Modular Properties of Symplectic Fermion Generalised Gibbs Ensemble

Faisal Karimi, Gérard M. T. Watts

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.

Modular Properties of Symplectic Fermion Generalised Gibbs Ensemble

Abstract

The symplectic fermion is a much-studied non-unitary conformal field theory with 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 triplet model. We derive exact expressions for the modular -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 of the 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 algebras.
Paper Structure (31 sections, 285 equations, 3 figures)

This paper contains 31 sections, 285 equations, 3 figures.

Table of Contents

  1. Introduction
  2. The Symplectic Fermion and Representations
  3. The Triplet Algebra
  4. Hierarchy of Bilinear Fields and Charges
  5. Quasi-Primary Bilinear Fields
  6. Charges associated to quasi-primary fields
  7. Expressions for the Charges in the Bilinear Hierarchy
  8. The Way of Modular Forms
  9. Even Weight Fields and Odd Spin Charges
  10. Odd Weight Fields and Even Spin Charges
  11. The Way of the Zeta Function
  12. The Symplectic Fermion GGE
  13. Exact Transformation of Symplectic Fermion GGE
  14. Asymptotic Analysis of Symplectic Fermion GGE
  15. Asymptotic Check with $\mathbf{Q}_{2}$
  16. ...and 16 more sections

Figures (3)

  • Figure 1: (a) The cylinder of length $\beta$, circumference $R$ with a translation through $\gamma$ before the identification of the ends can be realised as a quotient of the plane (b) and (c), and is equivalent (after rescaling) to a torus (d) with modular parameter $\tau$ and an annulus (e) with coordinates $u,v,v',z$ respectively. The solid points are identified in each picture. This figure has been taken and modified from Downing:2021mfw.
  • Figure 2: Ground state structure of the chiral indecomposable module $\mathbb{S}_0$ of the Symplectic Fermion.
  • Figure 3: Illustration of (a) the current $J_n$ being integrated on a fixed spatial slice of an infinitely long cylinder being equivalent to (b) a line integral in the plane of $J_n(w)$ and (c) a contour integral of a local field $[z^{L_0} \cdot \hat{J}_n](z)$ on the plane. The green arrow indicates the time direction.