  The partition functions of three-dimensional N=2 supersymmetric gauge
theories on different manifolds can be expressed as q-hypergeometric integrals.
By comparing the partition functions of three-dimensional mirror dual theories,
one finds complicated integral identities. In some cases, these identities can
be written in the form of pentagon relations. Such identities often have an
interpretation as the Pachner's 3-2 move for triangulated manifolds via the
so-called 3d-3d correspondence. From the physics perspective, another important
application of pentagon identities is that they may be used to construct new
solutions to the quantum Yang-Baxter equation.
