Orthogonal Dualities and Asymptotics of Dynamic Stochastic Higher Spin Vertex Models, using the Drinfeld Twister

Abstract

We introduce a new, algebraic method to construct duality functions for integrable dynamic models. This method will be implemented on dynamic stochastic higher spin vertex models, where we prove the duality functions are the $ _3 \varphi_2$ functions. A degeneration of these duality functions are orthogonal polynomial dualities of Groenevelt--Wagener arXiv:2306.12318. The method involves using the universal twister of $\mathcal{U}_q(\mathfrak{sl}_2)$, viewed as a quasi--triangular, quasi--$^*$--Hopf algebra. The algebraic method is presented very generally and is expected to produce duality functions for other dynamic integrable models. As an application of the duality, we prove that the asymptotic fluctuations of the dynamic stochastic six vertex model with step initial conditions are governed by the Tracy--Widom distribution.