Let $X$ be a Calabi-Yau threefold. Ordinary Donaldson-Thomas theory is formulated as a virtual count of ideal sheaves $\mathcal{I}$ with discrete invariants $\text{ch}(\mathcal{I}) = (1,0, -\beta, -n)$, which is equivalent to counting one-dimensional subschemes $Y \subseteq X$ with $[Y] = \beta \in H_{2}(X, \mathbb{Z})$ and $\chi(Y)=n$. In the derived category, we have the equivalence $\mathcal{I} \cong [\mathcal{O}_{X} \to \mathcal{O}_{Y}]$. For this reason, in the physical literature, they describe the DT theory as enumerating "bound states of D0-D2 branes with a single D6-brane. This is because $\text{ch}(\mathcal{O}_{X})=(1,0,0,0)$ represents a single D6-brane while $\mathcal{O}_{Y}$ represents the D0-D2 branes.

I was thinking it would be nice if divisors (or what a physicist would call a D4-brane) could be placed on the same footing and incorporated into the DT partition function as well. Mathematically, I feel like it would be nice to sort of have all the holomorphic sub-geometry of $X$ in one generating function and physically, it would be preferable to have all the possible D-branes in the Topological B-model (D0,D2,D4,D6) branes sort of on the same footing.

Of course divisors are special as they're given by vanishing of sections of line bundles, so I'm wondering if it's possible to write something like

$$\text{Hilb}_{D, \beta, n}(X) \cong \text{Pic}_{D}(X) \times \text{Hilb}_{\beta' n'}(X)$$

where $\text{Hilb}_{D, \beta, n}(X)$ would be the Hilbert scheme of points, curves, and divisors. *Note that $\beta$ and $n$ will be different from $\beta'$ and $n'$.*

We have a deformation/obstruction theory for $\text{Hilb}_{\beta' n'}(X)$ with a virtual class. Since $\text{Pic}_{D}(X)$ is smooth, can we compute $[\text{Hilb}_{D, \beta, n}(X)]^{\text{vir}}$ to define enumerative invariants? I was thinking that since the Picard factor is a torus, maybe by some localization argument it wouldn't contribute to an integral over the moduli space?

Perhaps what I'm asking for is nonsense, but even if it's possible, I guess it would probably destroy modularity properties of the partition function? And would almost certainly lose connection to the Gromov-Witten theory via the MNOP conjecture.

This post imported from StackExchange MathOverflow at 2017-05-19 14:25 (UTC), posted by SE-user Stephen Pietromonaco