Heisenberg double, pentagon relation, supersymmetric dualities

In this thesis, we explore the pentagon identity as it appears in Quantum Groups within the Heisenberg double algebra and non-perturbative results derived using the superconformal index technique in three-dimensional supersymmetric gauge theories with four supercharges. The pentagon identity provides insights into the interplay between geometry, topology, and quantum field theory, note that it appears particularly through the $3 \mathrm{~d}-3 \mathrm{~d}$ correspondence. In the first part, we discuss the operator form of the pentagon identity, which is related to quantum groups. In the second part, we show interesting integral identities that appear in index calculations through mirror symmetry, satisfied quantum dilogarithm functions or variants of it, and their relation to supersymmetric dualities in three dimensions. We examine known pentagon identities on $S^2 \times S, S^3, S_b^3, S^3 / Z_r$, and $S_b^3 / Z_r$, and present a new pentagon identity on the orbifold $S_b^3 / Z_r$.