Quantum transfer-matrices for the sausage model

Abstract

In this work we revisit the problem of the quantization of the two-dimensional O(3) non-linear sigma model and its one-parameter integrable deformation -the sausage model. Our consideration is based on the so-called ODE/IQFT correspondence, a variant of the Quantum Inverse Scattering Method. The approach allowed us to explore the integrable structures underlying the quantum O(3)/sausage model. Among the obtained results is a system of non-linear integral equations for the computation of the vacuum eigenvalues of the quantum transfer-matrices.

On the Yang–Baxter Poisson algebra in non-ultralocal integrable systems

“…The zero curvature representation for the Fateev model was found in [20] in a gauge which is different but equivalent to that of (4.13) specialized to the case G = SU(2) (the exact relation can be found in Appendix C). In both gauges, the Poisson brackets of the connection do not possess the ultralocal property and it is unknown whether an "ultralocal" gauge actually exists except for the cases with ε 2 /ε 1 = 0, ∞ considered in [11]. Thus, with a view towards first principles quantization, the Poisson algebra generated by the monodromy matrices is of prime interest for the Fateev model and more generally the Klimčík one.…”

“…On the other hand, the redefinition (5.31) has no effect on the monodromy matrix as κ → 0 and both ρ ± → 1 so that the Yang-Baxter algebra is still satisfied but in the form (4.25). Finally, the case ν = 0 with κ ∈ (0, 1) was already considered in the work [11] where it was shown that…”

On dual description of the deformed O(N) sigma model

“…However, this method is known to fail when applied to integrable nonlinear sigma models directly. Only recently a considerable progress has been achieved in this direction (see [9] and discussions therein). Now, let H be the Lie subgroup of G and h be the corresponding Lie algebra, such that the quotient manifold is a symmetric space.…”

Ultralocal Lax connection for para-complex ZT-cosets

“…The most important open problem relating to this class of models is therefore to apply the Quantum Inverse Scattering Method to them. A natural related question is also to determine whether the approach developed by V. Bazhanov, G. Kotousov and S. Lukyanov in [21] can be extended to integrable σ-models on para-complex Z T -coset target spaces.…”

“…One may also wonder if there are other integrable σ-models which admit an ultralocal Lax connection. For instance, there is [21] a generalisation of the ultralocal Lax connection of the O(3) non-linear σ-model for the sausage model, which is a deformation [37] of the former. It would therefore be interesting to investigate if the result of the present article could be extended to one-parameter deformations [38][39][40] of para-complex Z T -cosets.…”