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We show that the traces of $U_q(\widehat{\mathfrak{sl}}_2)$-intertwiners ofEtingof-Schiffmann-Varchenko valued in the three-dimensional evaluationrepresentation converge in a certain region of parameters and give arepresentation-theoretic construction of Felder-Varchenko's hypergeometricsolutions to the $q$-KZB heat equation. This gives the first proof that such atrace function converges and resolves the first case of the Etingof-Varchenkoconjecture.  As applications, we prove a symmetry property for traces of intertwiners andprove Felder-Varchenko's conjecture that their elliptic Macdonald polynomialsare related to the affine Macdonald polynomials defined as traces overirreducible integrable $U_q(\widehat{\mathfrak{sl}}_2)$-modules byEtingof-Kirillov Jr. In the trigonometric and classical limits, we recoverresults of Etingof-Kirillov Jr. and Etingof-Varchenko. Our method relies on aninterplay between the method of coherent states applied to the free fieldrealization of the $q$-Wakimoto module of Matsuo, convergence properties givenby the theta hypergeometric integrals of Felder-Varchenko, and rationalityproperties originating from the representation-theoretic definition of thetrace function.

Traces of intertwiners for quantum affine sl_2 and Felder-Varchenko functions