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We consider the refined topological vertex of Iqbal et al, as a function of two parameters (x, y), and deform it by introducing Macdonald parameters (q, t), as in the work of Vuletic on plane partitions, to obtain a Macdonald refined topological vertex. In the limit q -> t, we recover the refined topological vertex of Iqbal et al. In the limit x -> y, we obtain a qt-deformation of the topological vertex of Aganagic et al. Copies of the vertex can be glued to obtain qt-deformed 5D instanton partition functions that have well-defined 4D limits and, for generic values of (q, t), contain infinite-towers of poles for every pole in the limit q -> t.
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We test in $(A_{n-1},A_{m-1})$ Argyres-Douglas theories with $mathrm{gcd}(n,m)=1$ the proposal of Songs in arXiv:1612.08956 that the Macdonald index gives a refined character of the dual chiral algebra. In particular, we extend the analysis to higher rank theories and Macdonald indices with surface operator, via the TQFT picture and Gaiotto-Rastelli-Razamats Higgsing method. We establish the prescription for refined characters in higher rank minimal models from the dual $(A_{n-1},A_{m-1})$ theories in the large $m$ limit, and then provide evidence for Songs proposal to hold (at least) in some simple modules (including the vacuum module) at finite $m$. We also discuss some observed mismatch in our approach.
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