This talk concerns two approaches for studying a family of special functions occurring in the study of the $q$-Knizhnik-Zamolodchikov-Bernard $(q-KZB)$ equation. The philosophy of $KZ$-type equations predicts that it admits solutions via (1) traces of intertwining operators between representations of quantum affine algebras produced by Etingof-Schiffmann-Varchenko and (2) certain theta hypergeometric integrals we term Felder-Varchenko functions. In a series of papers in the early 2000’s, Etingof-Varchenko conjectured that these families of solutions are related by a simple renormalization; in the trigonometric limit, they proved such a link and used it to study these functions.
In recent work, I resolve the first case of the Etingof-Varchenko conjecture by showing that the traces of quantum affine $sl_2$-intertwiners of Etingof-Schiffmann-Varchenko valued in the $3$-dimensional evaluation representation converge in a certain region of parameters and give a representation-theoretic construction of Felder-Varchenko functions. I will explain the two constructions of solutions, the methods used to relate them, and connections to the Felder-Varchenko conjecture on the $q-KZB$ heat operator and corresponding $SL(3,Z)$-action, affine Macdonald theory, and recent results in geometric representation theory.
This talk is based on the preprint arXiv:1508.03918.