The Bailey lemma and 3d mirror symmetry

In this talk, we discuss the relationship between the integral Bailey lemma and mirror symmetry for three-dimensional supersymmetric gauge theories. Three-dimensional mirror symmetry relates infrared fixed points of a certain class of quiver gauge theories. The simplest example of such a duality is the equivalence of 3d $\mathcal N=2$ supersymmetric quantum electrodynamics and the theory of three chiral multiplets $X, Y, Z$ with a superpotential $W=XYZ$. One can check these dualities by computing supersymmetric partition functions. It happens that in some cases, starting with the partition function identity for a certain mirror duality one can construct a family of duality via the integral analog of the Bailey lemma.