Generalized Symmetries in F-theory and the Topology of Elliptic Fibrations

TL;DR

This work presents a comprehensive geometric framework for generalized (higher-form) symmetries in F-/M-theory using non-compact elliptic fibrations. By constructing Lefschetz thimbles for discriminant loci and introducing Pontryagin-dual center divisors, the authors derive 1-form (and higher) symmetry data directly from boundary topology and relative cycles, including explicit SymTFT couplings for various elliptic fibrations and NHC quivers. The analysis spans local K3s, higher-codimension fibers, Mordell–Weil torsion, and box-graph techniques to determine defect groups and their screenings, yielding concrete 1-form symmetry groups and mixed anomalies across 5d/6d theories. The methodology unifies boundary topology with a cycle-based realization of charges, enabling systematic study of global forms, anomalies, and potential 2-group structures in F-theory compactifications, with applications to conformal matter and quiver theories. The results provide a robust toolkit for understanding how geometry encodes and screens higher-form symmetries in string-theoretic QFTs and offer a path to exploring 2-group symmetries and higher-form dynamics in broader dimensional settings.

Abstract

We realize higher-form symmetries in F-theory compactifications on non-compact elliptically fibered Calabi-Yau manifolds. Central to this endeavour is the topology of the boundary of the non-compact elliptic fibration, as well as the explicit construction of relative 2-cycles in terms of Lefschetz thimbles. We apply the analysis to a variety of elliptic fibrations, including geometries where the discriminant of the elliptic fibration intersects the boundary. We provide a concrete realization of the 1-form symmetry group by constructing the associated charged line operator from the elliptic fibration. As an application we compute the symmetry topological field theories in the case of elliptic three-folds, which correspond to mixed anomalies in 5d and 6d theories.

Figures (10)

• Figure 1: Elliptic Calabi-Yau an $n$-fold $\bm{X}\rightarrow B$ with discriminant $\Delta$ containing a compact and non-compact components denoted $\Delta_i$ and $\Delta_j^{}$ respectively. The Lefschetz thimble $\mathfrak{T}_i\in \mathfrak{h}_{(2)}$ (dotted red) is fibered by a vanishing cycle of $\Delta_i$ and intersects the boundary in $\gamma_i\in H_1(\partial \bm{X})$. It projects to a semi-infinite path in the base (red) starting at the discriminant and ending at the boundary of the base.
• Figure 2: Picture of a representative for the thimble $\mathfrak{T}_i( \gamma) \in\mathfrak{h}_{(2)}$. The thimble projects to the path $\Gamma\subset B$ terminating at $z_i\in \Delta_i$. We depict the fiber $\pi^{-1}(z_i)$ as standard in algebraic geometry with each straight line denoting a rational curve introduced by the resolution.
• Figure 3: Picture of two homologous relative 2-cycles $\mathfrak{T}'_\Gamma ( \gamma,z_i,\{C_k \})$ and $\mathfrak{T}'_{\Gamma'} ( \gamma,z_i',\{C_k \})$. Sliding the thimbles vertically along the discriminant component $\Delta_i$ establishes the homotopy. As a consequence they project to the same thimble $\mathfrak{T}_i( \gamma)$.
• Figure 4: Picture of topological manipulations permitted on (three) thimbles in $\mathfrak{h}_{f,(2)}$ whenever these are linearly dependent. We show the projections of the deformation to the base. The initial configuration (left) are three non-compact thimbles attaching to three disconnected discriminant components. The final configuration (right) lifts to a compact 2-cycle and a non-compact 2-cycle which can be further deformed into $\partial B$.
• Figure 5: Picture of a compact 2-cycle $\Sigma$ connecting to multiple components of the ramification locus and projecting to a graph.