Curve counting and DT/PT correspondence for Calabi-Yau 4-folds

Recently, Cao-Maulik-Toda defined stable pair invariants of a compact Calabi-Yau 4-fold $X$. Their invariants are conjecturally related to the Gopakumar-Vafa type invariants of $X$ defined using Gromov-Witten theory by Klemm-Pandharipande. In this paper, we consider curve counting invariants of $X$ using Hilbert schemes of curves and conjecture a DT/PT correspondence which relates these to stable pair invariants of $X$. After providing evidence in the compact case, we define analogous invariants for toric Calabi-Yau 4-folds using a localization formula. We formulate a vertex formalism for both theories and conjecture a relation between the (fully equivariant) DT/PT vertex, which we check in several cases. This relation implies a DT/PT correspondence for toric Calabi-Yau 4-folds with primary insertions.Comment: 28 pages. Published versio

Orientability for gauge theories on Calabi-Yau manifolds

We study orientability issues of moduli spaces from gauge theories on Calabi-Yau manifolds. Our results generalize and strengthen those for Donaldson-Thomas theory on Calabi-Yau manifolds of dimensions 3 and 4. We also prove a corresponding result in the relative situation which is relevant to the gluing problem in DT theory.Comment: v2: 14 pages, substantially revised and expande

Donaldson-Thomas theory for Calabi-Yau four-folds

Let $X$ be a complex four-dimensional compact Calabi-Yau manifold equipped with a K\"ahler form $\omega$ and a holomorphic four-form $\Omega$. Under certain assumptions, we define Donaldson-Thomas type deformation invariants by studying the moduli space of the solutions of Donaldson-Thomas equations on the given Calabi-Yau manifold. We also study sheaves counting on local Calabi-Yau four-folds. We relate the sheaves countings over $K_{Y}$ with the Donaldson-Thomas invariants for the associated compact three-fold $Y$. In some very special cases, we prove the DT/GW correspondence for $X$. Finally, we compute the Donaldson-Thomas invariants of certain Calabi-Yau four-folds when the moduli spaces are smooth.Comment: 103pages, author's Master thesis, comments are welcom

Orientability of moduli spaces of Spin(7)-instantons and coherent sheaves on Calabi-Yau 4-folds

Suppose $(X,\Omega,g)$ is a compact Spin(7)-manifold, e.g. a Riemannian 8-manifold with holonomy Spin(7), or a Calabi-Yau 4-fold. Let $G$ be U$(m)$ or SU$(m)$, and $P\to X$ be a principal $G$-bundle. We show that the infinite-dimensional moduli space ${\mathcal B}_P$ of all connections on $P$ modulo gauge is orientable, in a certain sense. We deduce that the moduli space ${\mathcal M}_P^{Spin(7)}\subset{\mathcal B}_P$ of irreducible Spin(7)-instanton connections on $P$ modulo gauge, as a manifold or derived manifold, is orientable. This improves theorems of Cao and Leung arXiv:1502.01141 and Mu\~noz and Shahbazi arXiv:1707.02998. If $X$ is a Calabi-Yau 4-fold, the derived moduli stack $\boldsymbol{\mathscr M}$ of (complexes of) coherent sheaves on $X$ is a $-2$-shifted symplectic derived stack $(\boldsymbol{\mathcal M},\omega)$ by Pantev-To\"en-Vaqui\'e-Vezzosi arXiv:1111.3209, and so has a notion of orientation by Borisov-Joyce arXiv:1504.00690. We prove that $(\boldsymbol{\mathscr M},\omega)$ is orientable, by relating algebro-geometric orientations on $(\boldsymbol{\mathscr M},\omega)$ to differential-geometric orientations on ${\mathcal B}_P$ for U$(m)$-bundles $P\to X$, and using orientability of ${\mathcal B}_P$. This has applications to the programme of defining Donaldson-Thomas type invariants counting moduli spaces of (semi)stable coherent sheaves on a Calabi-Yau 4-fold, as in Donaldson and Thomas 1998, Cao and Leung arXiv:1407.7659, and Borisov and Joyce arXiv:1504.00690. This is the third in a series arXiv:1811.01096, arXiv:1811.02405 on orientations of gauge-theoretic moduli spaces.Comment: 57 pages. (v2) Major rewrite: new title, added author, new material on Calabi-Yau manifold