Hidden symmetries in four-dimensional $\mathcal{N} = 2$ superconformal gauge theories

Abstract

The integrability properties of $\mathcal{N}=4$ Super-Yang-Mills in the planar limit have been studied extensively and are well understood. For certain classes of theories, obtained by orbifolding $\mathcal{N}=4$ Super-Yang-Mills, it was shown that planar integrability is actually inherited and persists at the orbifold point. However, to date, little is known for theories that are deformed away from this fixed line in the marginal couplings. The content of this thesis is the study of global symmetries of the $\mathbb{Z}_2$-orbifold of $\mathcal{N}=4$ Super-Yang-Mills theory and its marginal deformations, with the aim to investigate and describe hidden symmetries appearing in this $\mathcal{N}=2$ superconformal field theory. The process of orbifolding in order to obtain an $\mathcal{N}=2$ theory appears to break the $\mathrm{SU}(4)$ R-symmetry down to $\mathrm{SU}(2)\times\mathrm{SU}(2)\times \mathrm{U}(1)$. We are able to show that the previously broken generators can actually be recovered by moving beyond the Lie algebraic setting and adopting the notion of a Lie algebroid. This remains true even away from the orbifold point after performing a marginal deformation, where we allow for independent variation of the $\mathrm{SU}(N)\times\mathrm{SU}(N)$ gauge couplings.By employing a Drinfeld-type twist of this $\mathrm{SU}(4)$ Lie algebroid, we can capture this marginal deformation. The resulting twist can be read off from the F- and D- terms of the theory, and thus directly from the Lagrangian. Even though at the orbifold point the algebraic structure is associative, it becomes non-associative after the marginal deformation. We explicitly check that the planar Lagrangian of the theory is invariant under this twisted version of the $\mathrm{SU}(4)$ algebroid, and we discuss implications of this hidden symmetry for the spectrum of the $\mathcal{N}=2$ theory.