A classical model for derived critical loci

Understand

Let $f:U\to{\mathbb A}^1$ be a regular function on a smooth scheme $U$ over a field $\mathbb K$.

• Pantev, Toen, Vaquie and Vezzosi (arXiv:1111.3209, arXiv:1109.5213) define the "derived critical locus" Crit$(f)$, an example of a new class of spaces in derived algebraic geometry, which they call "$-1$-shifted symplectic derived schemes".
• They show that intersections of algebraic Lagrangians in a smooth symplectic $\mathbb K$-scheme, and stable moduli schemes of coherent sheaves on a Calabi-Yau 3-fold over $\mathbb K$, are also $-1$-shifted symplectic derived schemes.
• Thus, their theory may have applications in algebraic symplectic geometry, and in Donaldson-Thomas theory of Calabi-Yau 3-folds.