Remarks on Geometric Engineering, Symmetry TFTs and Anomalies

Abstract

Geometric engineering is a collection of tools developed to establish dictionaries between local singularities in string theory and (supersymmetric) quantum fields. Extended operators and defects, as well as their higher quantum numbers captured by topological symmetries, can be encoded within geometric engineering dictionaries. In this paper we revisit and clarify aspects of these techniques, with special emphasis on 't Hooft anomalies, interpreted from the SymTFT perspective as obstructions to the existence of Neumann boundary conditions. These obstructions to gauging higher symmetries are captured via higher link correlators for the SymTFT on spheres. In this work, we give the geometric engineering counterpart of this construction in terms of higher links of topological membranes. We provide a consistency check in the context of 5D SCFTs with anomalous 1-form symmetries, where we give two independent derivations of the anomaly in terms of higher links, one purely field theoretical and the other purely geometrical. Along the way, we also recover the construction of non-invertible duality defects in 4D $\mathcal{N}=4$ SYM from a geometric engineering perspective.

v3: 55 pages, 5 figures, reference added, minor corrections, version submitted to JHEP

References in corpus (72)

• Generalized Global Symmetries
• Generalized Symmetries in Condensed Matter
• 6d Conformal Matter
• Kramers-Wannier-like duality defects in (3+1)d gauge theories
• Non-Invertible Duality Defects in 3+1 Dimensions
• Higher Gauging and Non-invertible Condensation Defects
• Comments on One-Form Global Symmetries and Their Gauging in 3d and 4d
• Non-Invertible Higher-Categorical Symmetries
• Non-invertible Condensation, Duality, and Triality Defects in 3+1 Dimensions
• Universal Non-Invertible Symmetries
• Non-Invertible Symmetries from Holography and Branes
• Non-Abelian String and Particle Braiding in Topological Order: Modular SL(3,Z) Representation and 3+1D Twisted Gauge Theory
• The Uncertainty of Fluxes
• Heisenberg Groups and Noncommutative Fluxes
• Non-Invertible Symmetries of $\mathcal{N}=4$ SYM and Twisted Compactification
• The Branes Behind Generalized Symmetry Operators
• Construction of two-dimensional topological field theories with non-invertible symmetries
• Non-invertible Time-reversal Symmetry
• Majorana chain and Ising model -- (non-invertible) translations, anomalies, and emanant symmetries
• Branes and Non-Invertible Symmetries
• Unifying Constructions of Non-Invertible Symmetries
• Exploring Non-Invertible Symmetries in Free Theories
• On Continuous 2-Category Symmetries and Yang-Mills Theory
• Non-invertible Symmetries of Class $\mathcal{S}$ Theories
• Non-Invertible Symmetry Webs
• Non-invertible self-duality defects of Cardy-Rabinovici model and mixed gravitational anomaly
• Higher-form symmetries and 3-group in axion electrodynamics
• Non-invertible symmetries and LSM-type constraints on a tensor product Hilbert space
• Remarks on Boundaries, Anomalies, and Noninvertible Symmetries
• On lattice models of gapped phases with fusion category symmetries
• 2-Group Symmetries in Class S
• Obstructions to Gapped Phases from Non-Invertible Symmetries
• Non-Invertible Defects in 5d, Boundaries and Holography
• Decomposition, condensation defects, and fusion
• Symmetry TFTs for 3d QFTs from M-theory
• Generalized Symmetry Breaking Scales and Weak Gravity Conjectures
• Coulomb and Higgs Branches from Canonical Singularities, Part 1: Hypersurfaces with Smooth Calabi-Yau Resolutions
• 2-Group Symmetries and M-Theory
• Continuous Generalized Symmetries in Three Dimensions
• Anomalies and Symmetry Fractionalization
• Generalized Charges, Part I: Invertible Symmetries and Higher Representations
• On anomalies and gauging of U(1) non-invertible symmetries in 4d QED
• 0-Form, 1-Form and 2-Group Symmetries via Cutting and Gluing of Orbifolds
• Non-Invertible Symmetries from Discrete Gauging and Completeness of the Spectrum
• Mixed Anomalies, Two-groups, Non-Invertible Symmetries, and 3d Superconformal Indices
• Bootstrapping Non-Invertible Symmetries
• Higher Form Symmetries TFT in 6d
• Relative Defects in Relative Theories: Trapped Higher-Form Symmetries and Irregular Punctures in Class S
• Maruyoshi-Song Flows and Defect Groups of $D_p^b(G)$ Theories
• Global 4-group symmetry and 't Hooft anomalies in topological axion electrodynamics
• Solitonic symmetry beyond homotopy: Invertibility from bordism and noninvertibility from topological quantum field theory
• Gauging Noninvertible Defects: A 2-Categorical Perspective
• Topological Defect Lines in Two Dimensional Fermionic CFTs
• Fusion Surface Models: 2+1d Lattice Models from Fusion 2-Categories
• Generalized Symmetries in F-theory and the Topology of Elliptic Fibrations
• Comments on Non-invertible Symmetries in Argyres-Douglas Theories
• SymTFTs and Duality Defects from 6d SCFTs on 4-manifolds
• Notes on gauging noninvertible symmetries, part 1: Multiplicity-free cases
• Quantization of Axion-Gauge Couplings and Non-Invertible Higher Symmetries
• Anomalies of Generalized Symmetries from Solitonic Defects
• The holography of duality in $\mathcal{N}=4$ Super-Yang-Mills theory
• Gauging Discrete Symmetries of $T_N$-theories in Five Dimensions
• Non-invertible symmetries along 4d RG flows
• Non-invertible symmetries and boundaries in four dimensions
• When the moduli space is an orbifold: Spontaneous breaking of continuous non-invertible symmetries
• A fresh view on string orbifolds
• Topology of $SU(N)$ lattice gauge theories coupled with ${\mathbb{Z}}_N$ $2$-form gauge fields
• Reading between the rational sections: Global structures of 4d $\mathcal{N}=2$ KK theories
• Higher-group structure in lattice Abelian gauge theory under instanton-sum modification
• Magnetic operators in 2D compact scalar field theories on the lattice
• Generalized quotients and holographic duals for 5d S-fold SCFTs
• Note on lattice description of generalized symmetries in $SU(N)/{\mathbb{Z}}_N$ gauge theories

Cited by in corpus (4)

• Hasse Diagrams for Gapless SPT and SSB Phases with Non-Invertible Symmetries
• Fermionic Non-Invertible Symmetries in (1+1)d: Gapped and Gapless Phases, Transitions, and Symmetry TFTs
• Generalized Symmetries of Non-SUSY and Discrete Torsion String Backgrounds
• Topological Manipulations On $\mathbb{R}$ Symmetries Of Abelian Gauge Theory