Challenges of β-deformation(Review)

摘要

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 β from unity is connected with the "exit from the free-fermion point" in two-dimensional conformal theories, from the symmetric graviphoton field with ∈ _2 = -∈ _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.