# Stable Pair Invariants of Local Calabi–Yau 4-folds

@article{Cao2020StablePI,
title={Stable Pair Invariants of Local Calabi–Yau 4-folds},
author={Yalong Cao and Martijn Kool and Sergej Monavari},
journal={International Mathematics Research Notices},
year={2020}
}
• Published 20 April 2020
• Mathematics
• International Mathematics Research Notices
In 2008, Klemm–Pandharipande defined Gopakumar–Vafa type invariants of a Calabi–Yau 4-folds $X$ using Gromov–Witten theory. Recently, Cao–Maulik–Toda proposed a conjectural description of these invariants in terms of stable pair theory. When $X$ is the total space of the sum of two line bundles over a surface $S$, and all stable pairs are scheme theoretically supported on the zero section, we express stable pair invariants in terms of intersection numbers on Hilbert schemes of points on $S… 15 Citations Orientations on the moduli stack of compactly supported perfect complexes over a non-compact Calabi--Yau 4-fold We consider a Calabi--Yau 4-fold$(X,\omega)$, where$X$is quasi-projective and$\omega$is a nowhere vanishing section of its canonical bundle$K_X$. The (derived) moduli stack of compactly Canonical vertex formalism in DT theory of toric Calabi-Yau 4-folds Gopakumar-Vafa type invariants of holomorphic symplectic 4-folds • Mathematics • 2022 Using reduced Gromov-Witten theory, we define new invariants which capture the enumerative geometry of curves on holomorphic symplectic 4-folds. The invariants are analogous to the BPS counts of K-Theoretic DT/PT Correspondence for Toric Calabi–Yau 4-Folds • Mathematics Communications in Mathematical Physics • 2022 Recently, Nekrasov discovered a new “genus” for Hilbert schemes of points on $${\mathbb {C}}^4$$ C 4 . We extend its definition to Hilbert schemes of curves and moduli spaces of stable Correspondence of Donaldson-Thomas and Gopakumar-Vafa invariants on local Calabi-Yau 4-folds over V_5 and V_22 • Mathematics • 2021 We compute Gromov-Witten (GW) and Donaldson-Thomas (DT) invariants (and also descendant invariants) for local CY 4-folds over Fano 3-folds, V5 and V22 up to degree 3. We use torus localization for GW Wall-crossing for zero-dimensional sheaves and Hilbert schemes of points on Calabi--Yau 4-folds We define a vertex algebra on the moduli stack of the auxiliary abelian category of pairs to give a precise formulation of the wall-crossing conjecture for Calabi–Yau 4-folds proposed by Stable maps to Looijenga pairs: orbifold examples • Mathematics • 2021 In [15] we established a series of correspondences relating five enumerative theories of log Calabi–Yau surfaces, i.e. pairs (Y,D) with Y a smooth projective complex surface and D = D1 + · · ·+Dl an ## References SHOWING 1-10 OF 48 REFERENCES Virtual fundamental classes for moduli spaces of sheaves on Calabi-Yau four-folds • Mathematics • 2015 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
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