Four flavours, triality and bimodular forms

TL;DR

This paper studies the duality symmetries of a specific supersymmetric gauge theory, revealing how duality groups act on different couplings and how order parameters form bimodular structures related to the theory's flavor symmetry.

Contribution

It establishes the structure of duality groups acting separately and simultaneously on UV and IR couplings, and identifies bimodular forms as order parameters in special mass configurations.

Findings

Duality groups can be subgroups of SL(2,Z) for special masses.

Order parameters are bimodular forms with arguments τ and τ_UV.

Duality actions generate orbits of mass configurations related to flavor symmetry.

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.