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

TL;DR

The paper shows that Type IIB holography on $AdS_5\times S^5$ and $AdS_5\times \mathbb{R}P^5$ naturally encode the global one-form symmetries and $SL(2,\mathbb{Z})$ dualities of ${\cal N}=4$ SYM with gauge algebras ${\rm su}(N)$, ${\rm so}(2n)$, ${\rm so}(2n+1)$, and $sp(n)$ via boundary conditions on bulk 2-form fields. It develops a precise dictionary between bulk holonomies and boundary line operator spectra, reproduces anomaly inflow from bulk Chern-Simons terms, and identifies duality orbits with boundary data, notably proving that the number of $SL(2,\mathbb{Z})$ orbits for ${\rm su}(N)$ equals the number of square divisors of $N$. The analysis covers brane realizations of lines, interfaces between theta vacua, and axionic Janus solutions, and extends to orientifold backgrounds and the tilde theories. The work sets the stage for future exploration of disconnected groups and non-Lagrangian S-folds, highlighting K-theoretic flux constraints and potential generalizations.

Abstract

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 $N$.

Figures (13)

• Figure 1: Bulk description of line operators in the $SU(N)$ theory: (a) A fundamental Wilson line, (b) $N$ fundamental Wilson lines screened by a gluon, (c) a fundamental 'tHooft line attached to a Dirac-surface.
• Figure 2: Bulk description of electric line operators in the $(SU(N)/\mathbb{Z}_k)_\ell$ theory: (a) $n=k$ is a genuine line, (b) lines with $n<k$ are not genuine, (c) $k'$ genuine lines are screened by a gluon.
• Figure 3: Bulk description of dyonic line operators in the $(SU(N)/\mathbb{Z}_k)_\ell$ theory: (a) $r$$(\ell/r,k'/r)$ strings describe a genuine dyonic line operator ($r\equiv \hbox{gcd}(k',\ell)$). (b) $N/r$ dyonic lines are screened by an $(\ell/r,k'/r)$5-brane on $S^5$. (c) $k$ dyonic lines can turn into $\ell$ electric lines via an NS5-brane on $S^5$.
• Figure 4: The critical case is indicated by the dotted purple curve. The dotted blue curve is the undeformed AdS$_5$ and the solid black curve is a generic case in the range $0<c_0<c_{\ast}$.
• Figure 5: The profile of the axio-dilaton: The axion $C_0=\tau_1$ varies from $-\theta$ at one half of the boundary to $+\theta$ at the other half of the boundary. In the boundary, the axion $C_0$ jumps across the domain wall.