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We propose a systematic approach to deriving symmetry generators of quantum field theories in holography. Central to this analysis are the Gauss law constraints in the Hamiltonian quantization of symmetry topological field theories (SymTFTs), which are obtained from supergravity. In turn, we realize the symmetry generators from world-volume theories of D-branes in holography. Our main focus is on noninvertible symmetries, which have emerged in the past year as a new type of symmetry in d≥4 QFTs. We exemplify our proposal in the holographic confinement setup, dual to 4D N=1 Super-Yang Mills. In the brane picture, the fusion of noninvertible symmetries naturally arises from the Myers effect on D-branes. In turn, their action on line defects is modeled by the Hanany-Witten effect.

We determine the d + 1 dimensional topological field theory, which encodes the higher-form symmetries and their 't Hooft anomalies for d -dimensional QFTs obtained by compactifying M-theory on a non-compact space X . The resulting theory, which we call the Symmetry TFT, or SymTFT for short, is derived by reducing the topological sector of 11d supergravity on the boundary ∂ X of the space X . Central to this endeavour is a reformulation of supergravity in terms of differential cohomology, which allows the inclusion of torsion in cohomology of the space ∂ X , which in turn gives rise to the background fields for discrete (in particular higher-form) symmetries. We apply this framework to 7d super-Yang Mills, where X = C 2 / Γ ADE , as well as the Sasaki-Einstein links of Calabi-Yau three-fold cones that give rise to 5d superconformal field theories. This M-theory analysis is complemented with a IIB 5-brane web approach, where we derive the SymTFTs from the asymptotics of the 5-brane webs. Our methods apply to both Lagrangian and non-Lagrangian theories, and allow for many generalisations., Competing Interests: Conflict of interestThe authors have no relevant financial or non-financial interests to disclose. Data sharing not applicable to this article as no datasets were generated or analysed during the current study., (© The Author(s) 2023.)