Lagrangian distributions, shifted symplectic structures and higher dimensional gauge theory on Calabi-Yau $4$-folds

We discuss a generalized construction of a compact moduli space of spin $7$ instantons, or coherent sheaves on Calabi Yau $4$ folds, as a global derived critical locus of a graded potential. We discuss a generalization to Joyces local construction of this moduli space as a derived critical locus. If time permits we report on some applications of this construction. This is report on joint projects with Borisov- Katzarkov, and Borisov-Yau.