Lattice Moduli and Residual Atoms

Source abstract

We study prescribed holomorphic lattices for regular-singular connections and residual factors of quantum $F$-bundles. For an arbitrary ordered monodromy spectrum, we classify the fully framed local groupoid: each residue fibre is a discrete affine space of dimension $r(r-1)$, with a unique normal form and trivial stabilisers. We then analyse elementary changes of lattice. A strictly positive integral residue difference obstructs the oriented wall; otherwise, in arbitrary rank, the extending normal-form arcs form an affine torsor of relative dimension $r(r-1)$, and specialisation to the shifted residue slice is polynomial and triangular by root height. Adjacent walls exhibit the composite-root correction dictated by the type-$A$ root-subgroup relations, while an example over the dual numbers detects a nilpotent saturation defect. For one-branch $F$-bundles, the scalar branch is recovered from the residual trace, and scalar residualisation yields a symmetric monoidal equivalence with zero-branch $F$-bundles. This construction yields residual atoms, functorial Jordan strata, and a residual chemical formula compatible with blow-ups and projective bundles. In the $A$-model, residualisation removes the unit direction; the rationally marked large-radius residual germ recovers the divisor cubic and distinguishes the marked residual germs of the Pfaffian and Grassmannian Calabi--Yau threefolds.