Categorified structures over moduli spaces: Anomalies, non-invertible symmetries, and exceptional holonomy

In this note, we propose an extension of the relation between worldsheet global symmetries and structures over moduli spaces of superconformal field theories (SCFTs) to include noninvertible symmetries. The most familiar examples of such structures associated to an ordinary symmetry are the Bagger-Witten line bundles, which arise over the moduli spaces of two-dimensional N=(2,2) SCFTs from a non-anomalous worldsheet U(1)_R symmetry and its associated spectral flow operators. Generalizing this setting, we consider examples involving anomalous worldsheet symmetries, which, despite not being gaugeable, can still give rise to global structures over moduli space -- as illustrated by the momentum/winding symmetries in toroidal compactifications and higher group gauge symmetry structure in spacetime. Motivated by this analogy, we conjecture the existence of a stack of fusion categories over moduli spaces of G_2 and Spin(7) holonomy manifolds which acts as a noninvertible analogue of the Hodge or Bagger-Witten line bundles over Calabi-Yau moduli spaces. This proposal is based on the observation that SCFTs associated with such exceptional holonomy manifolds contain (tricritical) Ising sectors that play a role analogous to the U(1)_R symmetry in N=2 theories. Although these symmetries are not gaugeable, they behave similarly to anomalous invertible symmetries, providing the conceptual foundation for the proposed moduli space structure.