Integrable structure of conformal field theory, quantum KdV theory and Thermodynamic Bethe Ansatz

Abstract

We construct the quantum versions of the monodromy matrices of KdV theory. The traces of these quantum monodromy matrices, which will be called as "T-operators", act in highest weight Virasoro modules. The T-operators depend on the spectral parameter λ and their expansion around λ = ∞ generates an infinite set of commuting Hamiltonians of the quantum KdV system. The T-operators can be viewed as the continuous field theory versions of the commuting transfer-matrices of integrable lattice theory. In particular, we show that for the values c = 1 − 3 (2n+1) 2 2n+3 , n = 1, 2, 3... of the Virasoro central charge the eigenvalues of the T-operators satisfy a closed system of functional equations sufficient for determining the spectrum. For the ground-state eigenvalue these functional equations are equivalent to those of massless Thermodynamic Bethe Ansatz for the minimal conformal field theory M 2,2n+3 ; in general they provide a way to generalize the technique of Thermodynamic Bethe Ansatz to the excited states. We discuss a generalization of our approach to the cases of massive field theories obtained by perturbing these Conformal Field Theories with the operator Φ 1,3 . The relation of these T-operators to the boundary states is also briefly described.

Boundary RG flow associated with the AKNS soliton hierarchy

“…In this section we will discuss properties of the vacuum amplitude (142) to reveal some standard integrable structures [11,13,14] of the theory.…”

“…[11] a boundary state associated with integrable boundary flows with conformal bulk can be studied in a framework of Quantum Inverse Scattering method [12][13][14]. To recall the basic idea by the example of IPH model let us assume that the corresponding local IM are hermitian operators and the set { I s } ∞ s=1 is "resolving", i.e., that all eigenspaces of the local IM are one-dimensional and mutually orthogonal (which seems to be the case, see Section 4.2).…”

On the thermodynamic Bethe ansatz equation in the sinh-Gordon model

“…Motivated by the constructions of refs. [14,19] let us introduce slightly rescaled functions t 1 (P ) = T (0) 1 (P ) (Z(p)) −2bQ (8.10)…”

“…In table 1 the values of t 1 (P ) andt 1 (P ) are compared with the following analytic expressions borrowed from refs. [14,19] where these coefficients enter the explicit constructions of the "sin-Gordon" (i.e., related to purely imaginary values of b) analogs of T (θ) Table 2: Numerical values of analytic continuation of the constructions of refs. [14,19] for the ShG or staircase values of the parameter is quite relevant.…”

“…[14,19] where these coefficients enter the explicit constructions of the "sin-Gordon" (i.e., related to purely imaginary values of b) analogs of T (θ) Table 2: Numerical values of analytic continuation of the constructions of refs. [14,19] for the ShG or staircase values of the parameter is quite relevant. The corresponding "CFT integrable structures" will be shown to have a precise relation to the solutions of the massless versions of ShG and staircase TBA equations [18] where the parameter P = P correct is fixed by construction.…”