Cao–Maulik–Toda DT–GV correspondence for Calabi–Yau fourfolds

Let $Y$ be a Calabi–Yau $4$-fold, let $\beta\in H_2(Y,\mathbb{Z})$ be a curve class, and let $\mathbf{M}_{\beta}(Y)$ be the moduli space of one-dimensional stable sheaves $F$ on $Y$ satisfying $[F]=\beta$ and $\chi(F)=1$. For $\gamma\in H^{4-2i}(Y,\mathbb{Z})$, define the insertion

$$\tau_i(\gamma)=\pi_{\mathbf{M}*}\left(\pi_Y^*\gamma\cup\operatorname{ch}_{i+3}(\mathcal{F})\right),$$

where $\mathcal{F}$ is the universal sheaf with trivial determinant, and write $\langle\tau_i(\gamma)\rangle_\beta$ for its integral over the virtual class $[\mathbf{M}_\beta(Y)]^{\mathrm{vir}}$. Let $n_{0,\beta}(-)$ and $n_{1,\beta}$ denote the genus-zero and genus-one GV-type invariants, respectively, and let $m_{\beta_1,\beta_2}$ denote the meeting invariants. Under a suitable orientation on $\mathbf{M}_\beta(Y)$, Cao–Maulik–Toda's conjecture.

(a) For $\gamma_0\in H^4(Y,\mathbb{Z})$,

$$n_{0,\beta}(\gamma_0)=\langle\tau_0(\gamma_0)\rangle_\beta.$$

(b) For $\gamma_1\in H^2(Y,\mathbb{Z})$,

$$\langle \tau_1(\gamma_1)\rangle_\beta = \frac{n_{0,\beta}(\gamma_1^2)}{2(\gamma_1\cdot\beta)} - \sum_{\beta_1+\beta_2=\beta} \frac{(\gamma_1\cdot\beta_1)(\gamma_1\cdot\beta_2)}{4(\gamma_1\cdot\beta)}m_{\beta_1,\beta_2} - \sum_{k\geq 1,\\,k|\beta} \frac{\gamma_1\cdot\beta}{k}n_{1,\beta/k}.$$

These identities express the predicted correspondence between DT invariants and GV-type invariants for Calabi–Yau fourfolds, including the genus-one and meeting-invariant contributions. The paper reports that its computations for the local Calabi–Yau fourfold over the Mukai–Umemura variety verify these predictions assuming vanishing of the genus-one GV-type invariants; the general conjecture remains unresolved.