Emergent non-invertible symmetries in $\mathcal{N}=4$ Super-Yang-Mills theory

One of the simplest examples of non-invertible symmetries in higher dimensions appears in 4d Maxwell theory, where its $SL(2,\mathbb{Z})$ duality group can be combined with gauging subgroups of its electric and magnetic 1-form symmetries to yield such defects at many different values of the coupling. Even though $\mathcal{N}=4$ Super-Yang-Mills (SYM) theory also has an $SL(2,\mathbb{Z})$ duality group, it only seems to share two types of such non-invertible defects with Maxwell theory (known as duality and triality defects). Motivated by this apparent difference, we begin our investigation of the fate of these symmetries by studying the case of 4d $\mathcal{N}=4$ $U(1)$ gauge theory which contains Maxwell theory in its content. Surprisingly, we find that the non-invertible defects of Maxwell theory give rise, when combined with the standard $U(1)$ symmetry acting on the free fermions, to defects which act on local operators as elements of the $U(1)$ outer-automorphism of the $\mathcal{N}=4$ superconformal algebra, an operation that was referred to in the past as the "bonus symmetry". Turning to the nonabelian case of $\mathcal{N}=4$ SYM, the bonus symmetry is not an exact symmetry of the theory but is known to emerge at the supergravity limit. Based on this observation we study this limit and show that if it is taken in a certain way, non-invertible defects that realize different elements of the bonus symmetry emerge as approximate symmetries, in analogy to the abelian case.