Four flavors, 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 contains transformations acting on the UV-coupling $\tau_{\text{UV}}$ as well running coupling $\tau$. establish that subgroups act separately and $\tau$, while a larger acts simultaneously For special choices masses, we find groups can be identified congruence $\text{SL}(2,\mathbb Z)$. demonstrate in such cases, order parameters are instances bimodular forms arguments $\tau$ $\tau_{\text{UV}}$. Since UV triality outer automorphisms flavour symmetry $\text{SO}(8)$, action gives rise to an orbit mass configurations. Consequently, corresponding combine vector-valued Z)$ two couplings.