The holography of non-invertible self-duality symmetries

TL;DR

This work develops a holographic framework for non-invertible self-duality symmetries by identifying a discrete emergent G gauge sector in type IIB string theory at special self-dual moduli τ, and by encoding global boundary data via a 5d Chern-Simons-like theory with 2-form fields. Twisted sectors and boundary defects are analyzed through both Lagrangian and DW-theoretic formalisms, yielding explicit fusion rules for self-duality defects and their boundary realizations. The authors then gauge a discrete subgroup G⊂SL(2,ℤ_N) in the bulk, producing dressed twist defects 𝔻[𝒯] whose fusion on gapped boundaries reproduces the known duality and triality defects of 4d N=4 SYM, and providing a general method applicable to class S theories. The approach clarifies how non-invertible categorical symmetries arise in holography, links bulk topological data to boundary non-invertible fusion, and offers a versatile template for exploring similar structures in other holographic CFTs. The work thus bridges the symmetry TFT perspective with holographic duality to elucidate higher-form and non-invertible symmetries in quantum gravity contexts, with potential broad applicability beyond N=4 SYM.

Abstract

We study how non-invertible self-duality defects arise in theories with a holographic dual. We focus on the paradigmatic example of $\mathfrak{su}(N)$ $\mathcal{N} = 4$ SYM. The theory is known to have non-invertible duality and triality defects at $τ=i$ and $τ= e^{2 πi /3}$, respectively. At these points in the gravitational moduli space, the gauged $SL(2,\mathbb{Z})$ duality symmetry of type IIB string theory is spontaneously broken to a finite subgroup $G$, giving rise to a discrete emergent $G$ gauge field. After reduction on the internal manifold, the low-energy physics is dominated by an interesting 5d Chern-Simons theory, further gauged by $G$, that we analyze and which gives rise to the self-duality defects in the boundary theory. Using the five-dimensional bulk theory, we compute the fusion rules of those defects in detail. The methods presented here are general and may be used to investigate such symmetries in other theories with a gravity dual.

Figures (11)

• Figure 1: Left: a topological duality interface $I_g$. Right: the definition of a local topological defect $\mathfrak{D}_g$ at fixed points in the conformal manifold.
• Figure 2: Left: Antisymmetric braiding $B_{mm'}$ between 2-dimensional defects $U_m$ in 5d Chern-Simons theory. Right: Induced braiding between 2-dimensional defects $\widehat{U}_t$ and line defects $\partial U_{l \in \mathcal{L}}$ on gapped boundaries $\rho(\mathcal{L})$.
• Figure 3: Action of $PSL(2,\mathbb{Z}_2) \cong S_3$ (left) and $PSL(2,\mathbb{Z}_3) \cong A_4$ (right) on Lagrangian subgroups (gapped boundaries).
• Figure 4: The condensation defect on $\Sigma = T^2 \times S^2$ with its network of 2d defects (left) surrounds a topological defect $U_l$ placed on $T^2 \times \{0 \}$ (center), where $0\in B^3$ is the center of the 3-ball whose boundary is $S^2$. Up to a phase (\ref{['normal ordering phase']}), the network can be resolved into a collection of closed surfaces with no junctions (right).
• Figure 5: The 4d symmetry defect $V[\mathcal{T}]$ induces a discontinuity in the gauge field $\mathcal{B}$ across its surface. Compared with the setup of Figure \ref{['fig: cylinder']} center, the region $r<0$ is the interior of the cylinder while $r>0$ is the exterior.