Virtual fundamental classes for moduli spaces of sheaves on Calabi–Yau four-folds

Abstract

Let $\left(X,{\omega }_X^{\ast }\right)$ be a separated, $-2$–shifted symplectic derived $ℂ$–scheme, in the sense of Pantev, Toën, Vezzosi and Vaquié (2013), of complex virtual dimension ${vdim}_{ℂ}X=n\in \mathbb{Z}$, and $X_{an}$ the underlying complex analytic topological space. We prove that $X_{an}$ can be given the structure of a derived smooth manifold $X_{dm}$, of real virtual dimension ${vdim}_{ℝ}{X}_{dm}=n$. This $X_{dm}$ is not canonical, but is independent of choices up to bordisms fixing the underlying topological space $X_{an}$. There is a one-to-one correspondence between orientations on $\left(X,{\omega }_X^{\ast }\right)$ and orientations on $X_{dm}$.

Because compact, oriented derived manifolds have virtual classes, this means that proper, oriented $-2$–shifted symplectic derived $ℂ$–schemes have virtual classes, in either homology or bordism. This is surprising, as conventional algebrogeometric virtual cycle methods fail in this case. Our virtual classes have half the expected dimension.

Now derived moduli schemes of coherent sheaves on a Calabi–Yau $4$–fold are expected to be $-2$–shifted symplectic (this holds for stacks). We propose to use our virtual classes to define new Donaldson–Thomas style invariants “counting” (semi)stable coherent sheaves on Calabi–Yau $4$–folds $Y$ over $ℂ$, which should be unchanged under deformations of $Y$.