Borisov-Joyce constructed a real virtual cycle on compact
moduli spaces of stable sheaves on Calabi-Yau 4-folds, using derived
differential geometry.
We construct an algebraic virtual cycle. A key step is a localisation
of Edidin-Graham’s square root Euler class for SOp2n, Cq bundles to
the zero locus of an isotropic section, or to the support of an isotropic
cone.
We prove a torus localisation formula, making the invariants computable and extending them to the noncompact case when the fixed
locus is compact.
We give a K-theoretic refinement by defining K-theoretic square root
Euler classes and their localised versions.
In a sequel we prove our invariants reproduce those of Borisov-Joyce.