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We consider $\mathcal{N}=2$ supersymmetric $\text{SU}(2)$ gauge theory with$N_f=4$ massive hypermultiplets. The duality group of this theory containstransformations acting on the UV-coupling $\tau_{\text{UV}}$ as well as on therunning coupling $\tau$. We establish that subgroups of the duality group actseparately on $\tau_{\text{UV}}$ and $\tau$, while a larger group actssimultaneously on $\tau_{\text{UV}}$ and $\tau$. For special choices of themasses, we find that the duality groups can be identified with congruencesubgroups of $\text{SL}(2,\mathbb Z)$. We demonstrate that in such cases, theorder parameters are instances of bimodular forms with arguments $\tau$ and$\tau_{\text{UV}}$. Since the UV duality group of the theory contains thetriality 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 formswith $\text{SL}(2,\mathbb Z)$ acting simultaneously on the two couplings.

Four flavours, triality and bimodular forms