Higher-Form Symmetries and Their Anomalies in M-/F-Theory Duality

TL;DR

The paper develops a unified framework for higher-form symmetries in M-/F-theory compactifications on elliptic fibrations, tying 1-form center symmetries to the topology of asymptotic boundaries through relative homology and Mordell–Weil torsion. It constructs a precise dictionary between M-theory boundary data, string junctions, and F-theory torsional sections, showing how the global gauge-group structure is encoded in the defect group and boundary fluxes. By dimensional reduction of the 11d Chern–Simons term in the presence of torsional boundary G_4-fluxes, it uncovers fractional shifts in instanton densities that realize mixed ’t Hooft anomalies between 1-form symmetries and gauge/large-gauge transformations across 7d, 5d, and 8d, with explicit SU(N) examples and extensions to 6d/5d UV completions. The results provide insight into the interplay between geometry (elliptic fibrations, torsion, and junctions) and quantum field-theoretic anomalies, highlighting non-perturbative corrections near UV fixed points and potential topological counterterms that can alter the anomaly structure.

Abstract

We explore higher-form symmetries of M- and F-theory compactified on elliptic fibrations, determined by the topology of their asymptotic boundaries. The underlying geometric structures are shown to be equivalent to known characterizations of the gauge group topology in F-theory via Mordell--Weil torsion and string junctions. We further study dimensional reductions of the 11d Chern--Simons term in the presence of torsional boundary $G_4$-fluxes, which encode background gauge fields of center 1-form symmetries in the lower-dimensional effective gauge theory. We find contributions that can be interpreted as 't Hooft anomalies involving the 1-form symmetry which originate from a fractionalization of the instanton number of non-Abelian gauge theories in F-/M-theory compactifications to 8d/7d and 6d/5d.

Figures (6)

• Figure 1: Singular geometry with fiber singularity in the interior that induces an $SL(2,\mathbb{Z})$ monodromy around the boundary circle.
• Figure 2: Cartoon of a Higgsing transition, where one of the singular fibers is moved outside of the disk, thereby changing the monodromy.
• Figure 3: Transformation properties of a general $(r,s)$-string passing through the branch-cut of a $[p,q]$-brane.
• Figure 4: A local null junction (left) obtained from encircling a brane stack with a string. In general, it carries an asymptotic charge. Via a Hanany--Witten transition, the null junction can also be presented as joining prongs from the constituent branes of the stack (middle). One can connect local null junctions via their asymptotic charges; the global null junctions on a compact $\mathbb{P}^1$ have no net asymptotic charge (schematically on the right).
• Figure 5: Schematic depiction of a local contribution to a fractional null junction in terms of a physical asymptotic junction and a fractional combination of root junctions for $A^4$ stack (which we depicted as separated for convenience).