Four flavors, triality, and bimodular forms

Abstract

We consider $\mathcal{N}=2$ supersymmetric SU(2) gauge theory with ${N}_{f}=4$ massive hypermultiplets. The duality group of this theory contains transformations acting on the UV coupling ${\ensuremath{\tau}}_{\mathrm{UV}}$ as well as on the running coupling $\ensuremath{\tau}$. We establish that subgroups of the duality group act separately on ${\ensuremath{\tau}}_{\mathrm{UV}}$ and $\ensuremath{\tau}$, while a larger group acts simultaneously on ${\ensuremath{\tau}}_{\mathrm{UV}}$ and $\ensuremath{\tau}$. For special choices of the masses, we find that the duality groups can be identified with congruence subgroups of $\mathrm{SL}(2,\mathbb{Z})$. We demonstrate that in such cases, the order parameters are instances of bimodular forms with arguments $\ensuremath{\tau}$ and ${\ensuremath{\tau}}_{\mathrm{UV}}$. Since the UV duality group of the theory contains the triality group of outer automorphisms of the flavor symmetry 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 $\mathrm{SL}(2,\mathbb{Z})$ acting simultaneously on the two couplings.