Top Down Approach to Topological Duality Defects

TL;DR

The paper develops a top-down framework for constructing 4D duality and triality defects by engineering QFTs with D3-branes at Calabi–Yau cone singularities and deploying 7-branes at infinity that implement SL(2, Z) monodromies. By reducing on the boundary ∂X, a 5D symmetry TFT governs 1-form symmetries and anomaly inflow, while the branch cuts of wrapped 7-branes yield codimension-two defects and 3D TFTs localized on the defect. This approach unifies bottom-up constructions, such as half-space gauging and 1-form symmetry gauging, within a single holographic/top-down narrative and extends from 4D N=4 SYM to N=1 SCFTs. The work also clarifies how different branch-cut realizations map to distinct duality implementations and highlights the role of discrete 0-form symmetries in the broader duality/triality structure. Overall, it provides a coherent geometric mechanism for non-invertible symmetries and their fusion in a wide class of 4D QFTs.

Abstract

Topological duality defects arise as codimension one generalized symmetry operators in quantum field theories (QFTs) with a duality symmetry. Recent investigations have shown that in the case of 4D $\mathcal{N} = 4$ Super Yang-Mills (SYM) theory, an appropriate choice of (complexified) gauge coupling and global form of the gauge group can lead to a rather rich fusion algebra for the associated defects, leading to examples of non-invertible symmetries. In this work we present a top down construction of these duality defects which generalizes to QFTs with lower supersymmetry, where other 0-form symmetries are often present. We realize the QFTs of interest via D3-branes probing $X$ a Calabi-Yau threefold cone with an isolated singularity at the tip of the cone. The IIB duality group descends to dualities of the 4D worldvolume theory. Non-trivial codimension one topological interfaces arise from configurations of 7-branes "at infinity" which implement a suitable $SL(2, \mathbb{Z})$ transformation when they are crossed. Reduction on the boundary topology $\partial X$ results in a 5D symmetry TFT. Different realizations of duality defects, such as the gauging of 1-form symmetries with certain mixed anomalies and half-space gauging constructions, simply amount to distinct choices of where to place the branch cuts in the 5D bulk.

Figures (22)

• Figure 1: Sketch of the symmetry TFT \ref{['eq:5d TFT terms1']}. We depict the half-plane $\mathbb{R}_{\geq 0}\times\mathbb{R}_\perp$ with coordinates $(r,x_\perp)$ where $\mathbb{R}_\perp$ is some direction parallel to the D3-brane worldvolume. The boundary conditions for the symmetry TFT are denoted $\ket{\mathfrak{T}^{(N)}_X},\ket{P,D}$ respectively.
• Figure 2: Boundary conditions and defects for $\mathfrak{T}_X^{(N)}$. We sketch the half-plane $\mathbb{R}_{\geq 0}\times \mathbb{R}_\perp$ parametrized by $(r,x_\perp)$. The polarization $P_1,P_2$ determine that Dirichlet boundary conditions are set for $B_2,C_2$ respectively $B_2|_{\partial X}=D_1$ and $C_2|_{\partial X}=D_2$. Line defects are realized by F1/D1-strings and correspond to Wilson and 't Hooft lines respectively. Our conventions are such that the left, radially outgoing strings are of charge $[0,-1]$ and $[-1,0]$ and the right, incoming strings are of charge $[0,1]$ and $[1,0]$ respectively.
• Figure 3: Case (1), 7-branes wrapped on $M_3\times \partial X$, we sketch the plane $\mathbb{R}_{\geq0}\times \mathbb{R}_\perp$. The topological boundary conditions $\ket{P_1,D_1}$ are the monodromy transform of the boundary conditions $\ket{P_2,D_2}$ and result from stacking the branch cut with the asymptotic boundary. The branch cut is supported on $\mathbb{H}_{\leftarrow}\times \partial X$ and runs parallel to the D3-branes. Conventions are such that the monodromy matrix $\rho$ acts crossing the branch cut top to bottom.
• Figure 4: Case (2), 7-branes wrapped on $M_3\times \partial X$, we sketch the plane $\mathbb{R}_{\geq0}\times \mathbb{R}_\perp$. There is a single set of boundary conditions $\ket{P_1,D_1}$. The branch cut is supported on $\mathbb{H}_{\downarrow}\times \partial X$ and runs perpendicular to the D3-branes. Conventions are such that the monodromy matrix $\rho$ acts crossing the branch cut left to right.
• Figure 5: Case (3), 7-branes wrapped on $M_3\times \partial X$, we sketch the plane $\mathbb{R}_{\geq0}\times \mathbb{R}_\perp$. The 7-brane insertion gives rise to two boundary conditions $\ket{P_1,D_1},\ket{P_2,D_2}$. The branch cut is supported on $\mathbb{H}_{\uparrow} \times \partial X$ and runs perpendicular to the D3-branes. Conventions are such that the monodromy matrix $\rho$ acts crossing the branch cut right to left.