Normalization Factors, Reflection Amplitudes and Integrable Systems

TL;DR

The work develops a unified framework connecting ultraviolet CFT data, namely normalization factors and reflection amplitudes, to infrared observables in integrable 2D QFTs based on affine and non-affine Toda theories. It provides explicit formulas for normalization factors $N(a)$ and reflection amplitudes $R_s(a)$, derives nonperturbative vacuum expectation values $G(a)$ of exponential fields, and applies these to order-parameter VEVs in statistical models such as XY, $Z_n$-Ising, and Ashkin–Teller. The analysis extends to boundary Toda theories, where boundary reflection amplitudes and one-point functions are computed via screening charges and the ${f G}(x)$ function, and a conjectured quantum boundary ground state energy is proposed. The paper further explores integrable deformations and dualities, including Sine-Toda representations and dual non-simply laced theories, illustrating a cohesive picture where ultraviolet CFT data, exact nonperturbative quantities, and dual descriptions inform both bulk and boundary dynamics in Toda-type integrable systems.

Abstract

We calculate normalization factors and reflection amplitudes in the W-invariant conformal quantum field theories. Using these CFT data we derive vacuum expectation values of exponential fields in affine Toda theories and related perturbed conformal field theories. We apply these results to evaluate explicitly the expectation values of order parameters in the field theories associated with statistical systems, like XY, Z_n-Ising and Ashkin-Teller models. The same results are used for the calculation of the asymptotics of cylindrically symmetric solutions of the classical Toda equations which appear in topological field theories. The integrable boundary Toda theories are considered. We derive boundary reflection amplitudes in non-affine case and boundary one point functions in affine Toda theories. The boundary ground state energies are cojectured. In the last section we describe the duality properties and calculate the reflection amplitudes in integrable deformed Toda theories.