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.
Highlights
The N 1⁄4 2 supersymmetric Yang-Mills field theory with gauge group SU(2) and Nf 1⁄4 4 fundamental hypermultiplets is distinguished for various reasons [1], including (i) The theory is superconformal up to mass terms for the hypermultiplets, and is a benchmark for fourdimensional superconformal field theories (SCFTs) with N 1⁄4 2 supersymmetry [2,3,4,5]
We have demonstrated that for generic masses this function has branch points, with the consequence that the fundamental domain for τ is in general not of the form ΓnH for a congruence subgroup Γ ⊂ SLð2; ZÞ
For specific values of the masses, such as those giving rise to Argyres-Douglas (AD) points, the order parameter is holomorphic as function of τ, and the fundamental domain is that of a congruence subgroup
Summary
The N 1⁄4 2 supersymmetric Yang-Mills field theory with gauge group SU(2) and Nf 1⁄4 4 fundamental hypermultiplets is distinguished for various reasons [1], including (i) The theory is superconformal up to mass terms for the hypermultiplets, and is a benchmark for fourdimensional superconformal field theories (SCFTs) with N 1⁄4 2 supersymmetry [2,3,4,5]. (iii) The theory exhibits an intriguing electric-magnetic duality group including triality [1] This duality group acts on the UV coupling τUV and running coupling constant τ, and contains elements which act simultaneously on the two couplings as well as separately. There are specific mass configurations with enhanced global symmetry that are invariant under subgroups of the triality group, for which the orbits collapse, either to three elements or to a single element. We study four such configurations in detail, and show that their order parameters, periods, and discriminants are bimodular forms for subgroups of SLð2; ZÞ. Appendix D provides expressions for the limits to the asymptotically free theories
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have