Four flavors, triality, and bimodular forms

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 ${\tau }_{\mathrm{UV}}$ as well as on the running coupling $\tau$. We establish that subgroups of the duality group act separately on ${\tau }_{\mathrm{UV}}$ and $\tau$, while a larger group acts simultaneously on ${\tau }_{\mathrm{UV}}$ and $\tau$. 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 $\tau$ and ${\tau }_{\mathrm{UV}}$. 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.

Figure 1

Dynkin diagram of ${\mathfrak{d}}_4=\mathrm{Lie}(\text{Spin}(8))$. The group $𝒯\cong S_3$ of outer isomorphisms acts by permutations on the three conjugacy classes of irreducible representations $v$, $s$, and $\overline{s}$ attached to the nodes of the diagram. The 28-dimensional adjoint representation is left invariant by $𝒯$.

Figure 2

Action of $\mathrm{SL}(2,\mathbb{Z})$ on $\mathrm{SL}(2,\mathbb{Z})/\mathrm{\Gamma }(2)\cong S_3$.

Figure 3

Relation among ${\mathit{m}}_{\mathrm{B}}=(m,m,m,m)$, ${\mathit{m}}_{\mathrm{C}}=(2m,0,0,0)$, and ${\mathit{m}}_{\mathrm{D}}=(m,m,m,-m)$.

Figure 4

Orbit of the mass vector $\mathit{m}=(0,0,0,2m)$ under $\mathcal{T}$ and $\mathcal{S}$.

Figure 5

The loci (2.20) with nontrivial stabilizer groups on the subspace $m_2=m_3=0$ in $ℳ$. They all mutually intersect in the locus ${\mathcal{L}}_1$ of triality invariant masses.

Figure 6

Fundamental domain of $N_f=4$ case A with $\mathit{m}=(m,m,0,0)$.

Figure 7

Decoupling the two massive hypermultiplets in $N_f=4$ case A gives the domain of massless $N_f=2$ (blue). Two of the differing regions (gray) are the regions near $\tau =\frac{1}{2}$, which are mapped (orange) to $i\infty$. The remaining two are merely mapped to $\mathrm{\Gamma }(2)$ equivalent regions near the same cusp such that the resulting domain is connected. Alternatively, the fundamental domain in this figure is the image of $S{T}^{-1}STS$ acting on the domain in Fig. 6.