Four flavors, triality, and bimodular forms

We consider 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 τ_UV as well as on the running coupling τ. We establish that subgroups of the duality group act separately on τ_UV and τ, while a larger group acts simultaneously on τ_UV and τ. For special choices of the masses, we find that the duality groups can be identified with congruence subgroups of SL(2, ℤ) . We demonstrate that in such cases, the order parameters are instances of bimodular forms with arguments τ and τ_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 SL(2, ℤ) acting simultaneously on the two couplings.