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

Let $({\bf X},\omega_{\bf X}^*)$ be a separated, $-2$-shifted symplectic
derived $\mathbb C$-scheme, in the sense of Pantev, Toen, Vezzosi and Vaquie
arXiv:1111.3209, of complex virtual dimension ${\rm vdim}_{\mathbb C}{\bf
X}=n\in\mathbb Z$, and $X_{\rm an}$ the underlying complex analytic topological
space. We prove that $X_{\rm an}$ can be given the structure of a derived
smooth manifold ${\bf X}_{\rm dm}$, of real virtual dimension ${\rm
vdim}_{\mathbb R}{\bf X}_{\rm dm}=n$. This ${\bf X}_{\rm dm}$ is not canonical,
but is independent of choices up to bordisms fixing the underlying topological
space $X_{\rm an}$. There is a 1-1 correspondence between orientations on
$({\bf X},\omega_{\bf X}^*)$ and orientations on ${\bf X}_{\rm dm}$.
Because compact, oriented derived manifolds have virtual classes, this means
that proper, oriented $-2$-shifted symplectic derived $\mathbb C$-schemes have
virtual classes, in either homology or bordism. This is surprising, as
conventional algebro-geometric virtual cycle methods fail in this case. Our
virtual classes have half the expected dimension, and from purely complex
algebraic input, can yield a virtual class of odd real 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
$\mathbb C$, which should be unchanged under deformations of $Y$.