Vacuum energy of the Bukhvostov–Lipatov model

Abstract

Bukhvostov and Lipatov have shown that weakly interacting instantons and anti-instantons in the O(3) non-linear sigma model in two dimensions are described by an exactly soluble model containing two coupled Dirac fermions. We propose an exact formula for the vacuum energy of the model for twisted boundary conditions, expressing it through a special solution of the classical sinh-Gordon equation. The formula perfectly matches predictions of the standard renormalized perturbation theory at weak couplings as well as the conformal perturbation theory at short distances. Our results also agree with the Bethe ansatz solution of the model. A complete proof the proposed expression for the vacuum energy based on a combination of the Bethe ansatz techniques and the classical inverse scattering transform method is presented in the second part of this work [40].

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 .…”

Bukhvostov–Lipatov model and quantum-classical duality

“…This model provides a remarkable illustration to the idea of the exact instanton counting. Indeed, as explained in [2], the model can be reformulated as a bosonic QFT with two interacting bosons, which upon an analytic continuation into a strong coupling regime, becomes equivalent to the originating O(3) non-linear sigma model. Our interest to the BL model is also motivated by the study of an intriguing correspondence between Integrable Quantum Field Theories (IQFT) and Integrable Classical Field Theories in two dimensions, which cannot be expected from the standard correspondence principle.…”

“…As in [2] we consider the theory (1.1) in a finite volume imposing twisted (quasiperiodic) boundary conditions on the fundamental fermion fields, Ψ ± (t, x + R) = −e 2πik ± Ψ ± (t, x) ,Ψ ± (t, x + R) = −e −2πik ±Ψ ± (t, x) .…”

“…In [2] the scaling function (1.3) was computed in a variety of different ways. Let us briefly review this work and then describe new considerations contained in this paper.…”

“…Further, in [2] we also used the conformal perturbation theory and computed the small-r expansion of the scaling function (1.3) to within the forth order terms O(r 4 ), inclusively. Then, applying rather non-trivial integral identities for the hypergeometric function, we discovered that the result of these perturbative calculations can be expressed through a particular solution of the Liouville equation.…”