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For a simple, rigid vector bundle $F$ on a Calabi-Yau $3$-fold $Y$, weconstruct a symmetric obstruction theory on the Quot scheme$\textrm{Quot}_Y(F,n)$, and we solve the associated enumerative theory. Wediscuss the case of other $3$-folds. Exploiting the critical structure on$\textrm{Quot}_{\mathbb A^3}(\mathscr O^r,n)$, we construct a virtual motive(in the sense of Behrend-Bryan-Szendr\H{o}i) for $\textrm{Quot}_Y(F,n)$ for anarbitrary vector bundle $F$ on a smooth $3$-fold $Y$. We compute the associatedmotivic partition function. We obtain new examples of higher rank (motivic)Donaldson-Thomas invariants.

Virtual classes and virtual motives of Quot schemes on threefolds