Intermediate Defect Groups, Polarization Pairs, and Non-invertible Duality Defects

TL;DR

This work introduces polarization pairs as a refined description of absolute QFT data in theories with self-dual $(k-1)$-form gauge fields, encoding $(k-1)$-form symmetries, discrete $\theta$-angles, and counterterms via a defect-group framework. It centralizes a symmetry-TFT perspective, showing that gauging, stacking, and automorphisms act naturally on polarization pairs and yield a unified construction of non-invertible duality defects in arbitrary $2k$-dimensional QFTs. The authors apply the formalism to well-known cases (2D Ising, 4D $\mathcal{N}=4$ SYM) and to rich 6D SCFTs (e.g., $D_4$, $A_4\oplus A_4$) and provide a Type IIB-based symmetry TFT embedding, including brane realizations of symmetry operators and polarization data. The framework unifies existing non-invertible duality defects and enables systematic generation of new ones, offering a robust tool to analyze generalized global symmetries in higher-dimensional QFTs with potential string-theoretic interpretations and geometric insights.

Abstract

Within the framework of relative and absolute quantum field theories (QFTs), we present a general formalism for understanding polarizations of the intermediate defect group and constructing non-invertible duality defects in theories in $2k$ spacetime dimensions with self-dual gauge fields. We introduce the polarization pair, which fully specifies absolute QFTs as far as their $(k-1)$-form defect groups are concerned, including their $(k-1)$-form symmetries, global structures (including discrete $θ$-angle), and local counterterms. Using the associated symmetry TFT, we show that the polarization pair is capable of succinctly describing topological manipulations, e.g., gauging $(k-1)$-form global symmetries and stacking counterterms, of absolute QFTs. Furthermore, automorphisms of the $(k-1)$-form charge lattice naturally act on polarization pairs via their action on the defect group; they can be viewed as dualities between absolute QFTs descending from the same relative QFT. Using this formalism, we present a prescription for building non-invertible symmetries of absolute QFTs. A large class of known examples, e.g., non-invertible defects in 4D $\mathcal{N}=4$ super-Yang--Mills, can be reformulated via this prescription. As another class of examples, we identify and investigate in detail a family of non-invertible duality defects in 6D superconformal field theories (SCFTs), including from the perspective of the symmetry TFT derived from Type IIB string theory.

Figures (8)

• Figure 1.1: An illustration of the general construction of non-invertible duality defects by combining the half-space gauging interface $\sigma$, the stacking-counterterm interface $I_r$, and the interface $\mathcal{I}_a$ implementing a charge lattice automorphism. If the absolute theories on the left and the right are the same, for which it is required that the polarization pairs are identical, then the combined defect $\mathcal{N} = I_r \cdot \sigma \cdot \mathcal{I}_a$ is a non-invertible topological duality defect.
• Figure 3.1: The action of non-invertible duality defects in 6D on charged surface operator. $\eta$ is a 3-dimensional topological operator generating the $\overline{L}$ global symmetry.
• Figure 4.1: We show the six different polarization pairs for the relative theory of $\mathcal{N}=4$ SYM with $\mathfrak{su}(2)$ gauge algebra, together with how the topological manipulations and dualities act on the polarization pairs. Any closed loop containing an odd number of topological manipulations associated to gauging the one-form symmetry, i.e., $\sigma$, gives rise to a non-invertible symmetry defect in the theory at the start/end of the loop.
• Figure 5.1: The topological manipulations and Green--Schwarz automorphisms/dualities of the relative 6d $(2,0)$ SCFT of type $D_4$. TOP: gauging 2-form symmetries in magenta, and stacking a counterterm in red. BOTTOM: an order $2$ Green--Schwarz duality fixing $L_{SO} = \langle (1,1) \rangle$ while exchanging $L_{Ss} = \langle (1,0) \rangle$ with $L_{Sc} = \langle (0,1) \rangle$ is represented by blue arrows, and an order $3$ Green--Schwarz duality cyclically permuting $(L_{SO} = \langle(1,1)\rangle, L_{Sc} = \langle(0,1)\rangle, L_{Ss}=\langle(1,0)\rangle)$ is represented by black arrows. After combining these two sets of topological manipulations into a single diagram, any closed loop should be interpreted as a duality defect, which is non-invertible if an odd number of $\sigma$ operations are involved.
• Figure 5.2: All absolute theories descending from the $A_4 \oplus A_4$$\mathcal{N} = (2,0)$ relative theory. Here, each vertical pair of theories can be connected by gauging 2-form symmetry (in magenta arrows) and implementing Green--Schwarz duality (in blue arrows).