Challenges of $\beta$-deformation

We briefly review problems arising in the study of the beta deformation, which turns out to be the most difficult element in a number of modern problems: the deviation of $\beta$ from unity is connected with the "exit from the free-fermion point" in two-dimensional conformal theories, from the symmetric graviphoton field with $\epsilon_2=-\epsilon_1$ in instanton sums in four-dimensional supersymmetric Yang–Mills theories, with the transition from matrix models to beta ensembles, from HOMFLY polynomials to superpolynomials in the Chern–Simons theory, from quantum groups to elliptic and hyperbolic algebras, and so on. We mainly attend to issues related to the Alday–Gaiotto–Tachikawa correspondence and its possible generalizations.