Four flavours, triality and bimodular forms

Abstract

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.

arXiv comment29 pages + Appendices, 7 figures, v2: minor changes, updated Appendix B

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