Quantum interaction models discussed here are the (asymmetric) quantum Rabi model (QRM) and non-commutative harmonic oscillator (NCHO). The QRM is the most fundamental model describing the interaction between a photon and two-level atoms. The NCHO can be considered as a covering model of the QRM, and recently, the eigenvalue problems of NCHO and two-photon QRM (2pQRM) are shown to be equivalent. Spectral degeneracy can occur in models, but correspondingly there is a hidden symmetry relates geometrical nature described by hyperelliptic curves. In addition, the analytical formula for the heat kernel (propagator)/partition function of the QRM is described as a discrete path integral and gives the meromorphic continuation of its spectral zeta function (SZF). This discrete path integral can be interpreted to the irreducible decomposition of the infinite symmetric group $\mathfrak{S}_\infty$ naturally acting on $\mathbb{F}_2^\infty$, $\mathbb{F}_2$ being the binary field. Moreover, from the special values of the SZF of NCHO, an analogue of the Apéry numbers is naturally appearing, and their generating functions are, e.g., given by modular forms, Eichler integrals of a congruence subgroup. The talk overviews those above and present questions which are open.

[pre-talk at 3:00PM]