Integrability of Bukhvostov-Lipatov model and ODE/IQFT correspondence

We consider Bukhvostov-Lipatov model, an integrable quantum field theory in two dimensions that arises as an approximation to O(3) NLSM. We compute its vacuum energy on a cylinder with twisted boundary conditions in weak coupling limit using renormalized perturbation theory, and in the short distance limit using conformal perturbation theory. The exact solution of this model via coordinate Bethe ansatz is provided. Two different regularizations of Bethe ansatz equations are constructed. The vacuum state is constructed and the vacuum energy is computed within both regularizations using numerical methods. Bethe ansatz equations governing the vacuum state are shown to coincide with functional relations between connection coefficients of auxiliary linear problem for an integrable classical PDE known as modified sinh-Gordon equation. Based on this correspondence the system of nonlinear integral equations equivalent to the full system of Bethe ansatz equations is derived for massive and conformal cases. This system of NLIE was solved numerically. We have also used it to investigate analytically the properties of solution describing vacuum state. Finally, we have derived a formula that expresses the vacuum energy of Bukhvostov-Lipatov model in terms of the regularized area of constant mean curvature surface embedded into ADS3 space

Vacuum energy of the Bukhvostov-Lipatov model

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].Comment: 28 pages, 10 figure

Bukhvostov-Lipatov model and quantum-classical duality

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.Comment: 49 pages, 8 figure

On different approaches to integrable lattice models

Interaction-Round the Face (IRF) models are two-dimensional lattice models of statistical mechanics defined by an affine Lie algebra and admissibility conditions depending on a choice of representation of that affine Lie algebra. Integrable IRF models, i.e., the models the Boltzmann weights of which satisfy the quantum Yang-Baxter equation, are of particular interest. In this paper, we investigate trigonometric Boltzmann weights of integrable IRF models. By using an ansatz proposed by one of the authors in some previous works, the Boltzmann weights of the restricted IRF models based on the affine Lie algebras $\mathfrak{su}(2)_k$ and $\mathfrak{su}(3)_k$ are computed for fundamental and adjoint representations for some fixed levels $k$. New solutions for the Boltzmann weights are obtained. We also study the vertex-IRF correspondence in the context of an unrestricted IRF model based on $\mathfrak {su}(3)_k$ (for general $k$) and discuss how it can be used to find Boltzmann weights in terms of the quantum $\hat{R}$ matrix when the adjoint representation defines the admissibility conditions

Bukhvostov-Lipatov model and quantum-classical duality

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.Research of S.L. was supported by the NSF under grant number NSF-PHY-1404056

Spectral Duality Between Heisenberg Chain and Gaudin Model

In our recent paper we described relationships between integrable systems inspired by the AGT conjecture. On the gauge theory side an integrable spin chain naturally emerges while on the conformal field theory side one obtains some special reduced Gaudin model. Two types of integrable systems were shown to be related by the spectral duality. In this paper we extend the spectral duality to the case of higher spin chains. It is proved that the N-site GL(k) Heisenberg chain is dual to the special reduced k+2-points gl(N) Gaudin model. Moreover, we construct an explicit Poisson map between the models at the classical level by performing the Dirac reduction procedure and applying the AHH duality transformation.Comment: 36 page