3d field theory, plane partitions and triple Macdonald polynomials

Abstract

We argue that MacMahon representation of Ding-Iohara-Miki (DIM) algebra spanned by plane partitions is closely related to the Hilbert space of a 3d field theory. Using affine matrix model we propose a generalization of Bethe equations associated to DIM algebra with solutions also labelled by plane partitions. In a certain limit we identify the eigenstates of the Bethe system as new triple Macdonald polynomials depending on an infinite number of families of time variables. We interpret these results as first hints of the existence of an integrable 3d field theory, in which DIM algebra plays the same role as affine algebras in 2d WZNW models.