Four flavours, triality and bimodular forms

We consider $\mathcal{N}=2$ supersymmetric $\text{SU}(2)$ gauge theory with $N_f=4$ massive hypermultiplets. The duality group of this theory contains transformations acting on the UV-coupling $\tau_{\text{UV}}$ as well as on the running coupling $\tau$. We establish that subgroups of the duality group act separately on $\tau_{\text{UV}}$ and $\tau$, while a larger group acts simultaneously on $\tau_{\text{UV}}$ and $\tau$. For special choices of the masses, we find that the duality groups can be identified with congruence subgroups of $\text{SL}(2,\mathbb Z)$. We demonstrate that in such cases, the order parameters are instances of bimodular forms with arguments $\tau$ and $\tau_{\text{UV}}$. Since the UV duality group of the theory contains the triality group of outer automorphisms of the flavour symmetry $\text{SO}(8)$, the duality action gives rise to an orbit of mass configurations. Consequently, the corresponding order parameters combine to vector-valued bimodular forms with $\text{SL}(2,\mathbb Z)$ acting simultaneously on the two couplings.