Symmetries Beyond Branes: Geometric Engineering and Isometries

In this work we consider the relation between finite isometries of the internal space and symmetries of the transverse field theory in Geometric Engineering. On top of the established relation between branes wrapping torsional cycles and topological defects, we study other symmetries of the field theory that are not captured by branes wrapped at infinity. Isometries of the engineering geometry have a representations on the field theory, encoded by their non-trivial actions on exceptional and torsional cycles. In this work we describe such action in general terms. As examples, we focus on three classes of geometric engineering spaces: Du Val singularities, toric Calabi-Yau threefolds and $G_2$ manifolds obtained as fibrations of Du Val singularities over $\mathbb S^3$. In the latter case, we check explicitly that M-theory on Calabi-Yau or $G_2$ manifold with finite isometries reproduces correctly the higher group structures and non-invertible symmetries of the transverse field theories.