Orateur
Description
In this talk we discuss some aspects of the connection between the one-dimensional XXZ chain and two-dimensional conformal field theories. Our approach is based on the fermionic basis construction, which allows for the factorisation of the algebraic and physical parts of the model. The generators of this basis are obtained from the representation theory of $U_q(\widehat{\mathfrak{sl}}_2)$ and are independent of any physical data such as temperature, magnetic field or boundary conditions. The physical properties of the model are encoded in two transcendental functions $\rho(\zeta;\kappa,\kappa')$ and $\omega(\zeta,\xi;\kappa,\kappa',\alpha)$. By the Jimbo--Miwa--Smirnov theorem, the expectation values of the fermionic basis operators reduce to a determinant involving these functions, so that the computation of the correlation functions reduces to an asymptotic analysis of $\rho$ and $\omega$.
For equal twists $\kappa=\kappa'$, the Wiener--Hopf method determines the three-point functions of fermionic basis descendants of primary fields only modulo Zamolodchikov's local integrals of motion. We show how the case $\kappa\neq\kappa'$ can be treated at the free fermion point with the help of the so-called master function approach. Using the ODE/IM correspondence, we obtain the master function in closed form and, from it, the coefficients of the large spectral parameter expansion of $\omega$. This yields explicit three-point functions of fermionic basis descendants, now including the full action of the integrals of motion, as well as a link between the fermionic basis and the ODE/IM correspondence. We also discuss possible ways of lifting the result outside of the free fermion point.