On Painlevé VI transcendents related to the Dirac operator on the hyperbolic disk

Abstract

Dirac hamiltonian on the Poincaré disk in the presence of an Aharonov-Bohm flux and a uniform magnetic field admits a one-parameter family of self-adjoint extensions. We determine the spectrum and calculate the resolvent for each element of this family. Explicit expressions for Green functions are then used to find Fredholm determinant representations for the tau function of the Dirac operator with two branch points on the Poincaré disk. Isomonodromic deformation theory for the Dirac equation relates this tau function to a one-parameter class of solutions of the Painlevé VI equation with γ = 0. We analyze long distance behaviour of the tau function, as well as the asymptotics of the corresponding Painlevé VI transcendents as s → 1. Considering the limit of flat space, we also obtain a class of solutions of the Painlevé V equation with β = 0.

Correlation functions of twist fields from Ward identities in the massive Dirac theory

“…It is well known, however, that there exists a set of two-dimensional integrable quantum field theories, having free-fermion representations, where non-trivial vacuum correlation functions can be expressed as solutions to integrable differential equations, the first result of this nature being the twopoint spin-spin correlation function in the thermally perturbed lattice Ising model [24]. Since then it has been found that this result, as well as similar results in the QFT model of Dirac fermions, can be retrieved through alternative methods: holonomic quantum fields (Dirac model) [20], Fredholm determinants (Ising and Dirac models) [1,12,2], determinants of Dirac operators (Dirac model in flat and curved space and with magnetic field) [17,18,13], and doubling of the model (Ising spin chain, and Ising QFT model at zero and non-zero temperature and in curved space) [19,9,6]. The existence of such differential equations is extremely useful, as correlation functions can be evaluated with very high accuracy once the initial conditions have been fixed using conformal perturbation theory and form-factor analysis [3,6,7,13,14].…”

Expectation values of twist fields and universal entanglement saturation of the free massive boson

“…They have been studied in quite some generality in CFT, where they give rise to twisted modules for vertex operator algebras [30,31] (also known as orbifold constructions [11,27,29]) by the operator-state correspondence. Correlation functions of twist fields in massive free-fermion models are related to tau functions of isomonodromic deformation problems [43, 46, 57-59, 62, 63, 70], and the QFT connection problem for twist fields is associated to connection problems for Painlevé equations [32,40,42,53,54]. A fascinating application of the concept of twist fields is to the calculation of measures of entanglement:functions of quantum states that provide a numerical indication of the quantity of entanglement in the state (see for instance [61] for a review).…”

Algebraic solutions of the sixth Painlevé equation

“…Frobenius manifolds [10], symmetry groups of regular polyhedra [11,14], complex reflections [2], Grothendieck's dessins d'enfants and their deformations [1,22,23]. A few families of non-classical solutions have also been constructed in terms of Fredholm determinants, see [7,26].…”