Bukhvostov–Lipatov model and quantum-classical duality

Abstract

The Bukhvostov-Lipatov model is an exactly soluble model of two interacting Dirac fermions in 1+1 dimensions. The model describes weakly interacting instantons and anti-instantons in the O(3) non-linear sigma model. In our previous work [arXiv:1607.04839] we have proposed an exact formula for the vacuum energy of the Bukhvostov-Lipatov model in terms of special solutions of the classical sinh-Gordon equation, which can be viewed as an example of a remarkable duality between integrable quantum field theories and integrable classical field theories in two dimensions. Here we present a complete derivation of this duality based on the classical inverse scattering transform method, traditional Bethe ansatz techniques and analytic theory of ordinary differential equations. In particular, we show that the Bethe ansatz equations defining the vacuum state of the quantum theory also define connection coefficients of an auxiliary linear problem for the classical sinh-Gordon equation. Moreover, we also present details of the derivation of the non-linear integral equations determining the vacuum energy and other spectral characteristics of the model in the case when the vacuum state is filled by 2-string solutions of the Bethe ansatz equations.

Quantum transfer-matrices for the sausage model

“…[57,58] were focused on the ODE/IQFT correspondence with all three parameters a 1 , a 2 , a 3 positive, in the subsequent work [81] evidence was presented that the correspondence remains valid with minimum modifications to the case a 1 , a 2 > 0 and a 3 < 0. In the recent works [82,83], the same conclusion was reached for a 1 , a 2 > 0 and a 3 = 0. Among the tasks of the current paper is to argue that the ODE/IQFT correspondence remains valid for .…”

Vacuum energy of the Bukhvostov–Lipatov model

“…Following the BL paper [3] here we will impose a straightforward cutoff to the number of pseudoparticles. An alternative and in many respects more efficient lattice-type regularization is considered in our next paper [40].…”

Duality of positive and negative integrable hierarchies via relativistically invariant fields

“…respectively. It is known that K 2n+1 defined in (6) are local symmetries and K −2n−1 defined in (7) are nonlocal symmetries of the PMKdV equation ( 2) for all n = 0, 1, 2, . .…”

“…The study on dualities such as the wave-particle duality in quantum physics [1], the electromagnetic duality between electric and magnetic fields [2], AdS/CFT duality (duality between the Anti-de Sitter gravity and the conformal field theory) [3], the particle-vortex duality in condensed matter physics [4], the boson/fermion dualities in particle physics [5,6] and quantum/classical duality between quantum and classical integrable fields [7] is an important topic in Physics. In this paper, we study the positive/negative hierarchy dualities, the dualities between integrable positive hierarchies and negative hierarchies by means of the relativistically invariant fields, such as the sine-Gordon (sG) field, the Tzitzeica field, the Toda fields and the heavenly equations.…”