The holography of duality in ${\cal N}=4$ Super-Yang-Mills theory

The space of ${\cal N}=4$ supersymmetric Yang-Mills theories exhibits an intricate structure of global one-form symmetries and $SL(2,\mathbb{Z})$ duality orbits. In this paper we study this structure from the point of view of the holographic dual Type IIB string theory. Generalizing work by Witten, we map the different theories based on the gauge algebras $su(N)$, $so(N)$, and $sp(N)$ to a choice of boundary conditions on bulk gauge fields. We show how the one-form symmetries and their anomalies, as well as the duality properties of the gauge theories, arise in the holographic picture. Along the way we prove that the number of disjoint $SL(2,\mathbb{Z})$ duality orbits for the $su(N)$ theories is given by the number of square divisors of $N$.