The d-critical structure on the Quot scheme of points of a Calabi-Yau 3-fold

The Artin stack M n of zero-dimensional sheaves of length n on A(3) carries two natural d-critical structures in the sense of Joyce. One comes from its description as a quotient stack [crit(fn)/ GLn], another comes from derived deformation theory of sheaves. We show that these d-critical structures agree. We use this result to prove the analogous statement for the Quot scheme of points Quot A(3) (O. r, n) = crit(fr,(n)), which is a global critical locus for every r > 0, and also carries a derived-in-flavor d-critical structure besides the one induced by the potential fr,n. Again, we show these two d-critical structures agree. Moreover, we prove that they locally model the d-critical structure on QuotX(F, n), where F is a locally free sheaf of rank r on a projective Calabi-Yau 3-fold X. Finally, we prove that the perfect obstruction theory on Hilbn A(3) = crit(f1,n) induced by the Atiyah class of the universal ideal agrees with the critical obstruction theory induced by the Hessian of the potential f1,(n).