Free fermions, KdV charges, generalised Gibbs ensembles and modular transforms

Abstract

In this paper we consider the modular properties of generalised Gibbs ensembles in the Ising model, realised as a theory of one free massless fermion. The Gibbs ensembles are given by adding chemical potentials to chiral charges corresponding to the KdV conserved quantities. (They can also be thought of as simple models for extended characters for the W-algebras). The eigenvalues and Gibbs ensembles for the charges can be easily calculated exactly using their expression as bilinears in the fermion fields. We re-derive the constant term in the charges, previously found by zeta-function regularisation, from modular properties. We expand the Gibbs ensembles as a power series in the chemical potentials and find the modular properties of the corresponding expectation values of polynomials of KdV charges. This leads us to an asymptotic expansion of the Gibbs ensemble calculated in the opposite channel. We obtain the same asymptotic expansion using Dijkgraaf’s results for chiral partition functions. By considering the corresponding TBA calculation, we are led to a conjecture for the exact closed-form expression of the GGE in the opposite channel. This has the form of a trace over multiple copies of the fermion Fock space. We give analytic and numerical evidence supporting our conjecture.

Spectrum of quantum KdV hierarchy in the semiclassical limit

“…This may indicate that in the presence of higher KdV charges modular invariance of Z GGE is not mathematically natural. Similar conclusion is recently reached in [35], which evaluated Z GGE explicitly in the case of c = 1/2 free fermion model. They found that to reproduce Z GGE in the dual channel, one needs to sum over not one but three fermion Hilbert spaces, schematically Z GGE (t 1 , t 3 ) ∝ Z 1 (t 1 )Z 2 (t 1 )Z 3 (t 1 ), a mathematical observation (conjecture), which so far has no physical interpretation.…”

Manifest Modular Invariance in the Near-Critical Ising Model

“…As already noted, the τ 2 in intermediate steps may also seem inconvenient. The purpose of this section is not to be convenient, but to show using elementary methods that the possibly unfamiliar (34) gives the same results as found earlier. The next section describes a more efficient alternative calculation, by residue calculus.…”

“…A clear alternative exposition of this is given in [34], appendix D. The zero-point energies in eq. ( 9) are integral representations over a variable λ, where the sum over n is manifestly convergent before integration.…”

“…where 8πc α,µ τ 2 in (33) provides the c = 0 terms in (34). Starting from the "one-dimensional lattice sum" (30) over n, it is conceivable to guess the form of the two-dimensional lattice sum (34) over c and d, recalling m = µ/τ 2 .…”

“…The paper [34], that constructs tensor currents in the two-dimensional theory of free massive fermions, was communicated to me by the authors.…”