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

Cao, Yalong; Kool, Martijn

doi:10.1016/j.aim.2020.107371

Cited by 34 publications

(70 citation statements)

References 21 publications

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“…• prove that the critical perfect obstruction theory on Quot A 3 ( ⊕ , ) is T-equivariant (Lemma 3. 13) and that the induced T-fixed obstruction theory on the fixed locus is trivial (Corollary 3.15).…”

Section: Overviewmentioning

confidence: 99%

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Higher rank K-theoretic Donaldson-Thomas Theory of points

Fasola

Monavari

Ricolfi

2021

Forum of Mathematics, Sigma

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We exploit the critical structure on the Quot scheme $\text {Quot}_{{{\mathbb {A}}}^3}({\mathscr {O}}^{\oplus r}\!,n)$ , in particular the associated symmetric obstruction theory, in order to study rank r K-theoretic Donaldson-Thomas (DT) invariants of the local Calabi-Yau $3$ -fold ${{\mathbb {A}}}^3$ . We compute the associated partition function as a plethystic exponential, proving a conjecture proposed in string theory by Awata-Kanno and Benini-Bonelli-Poggi-Tanzini. A crucial step in the proof is the fact, nontrival if $r>1$ , that the invariants do not depend on the equivariant parameters of the framing torus $({{\mathbb {C}}}^\ast )^r$ . Reducing from K-theoretic to cohomological invariants, we compute the corresponding DT invariants, proving a conjecture of Szabo. Reducing further to enumerative DT invariants, we solve the higher rank DT theory of a pair $(X,F)$ , where F is an equivariant exceptional locally free sheaf on a projective toric $3$ -fold X. As a further refinement of the K-theoretic DT invariants, we formulate a mathematical definition of the chiral elliptic genus studied in physics. This allows us to define elliptic DT invariants of ${{\mathbb {A}}}^3$ in arbitrary rank, which we use to tackle a conjecture of Benini-Bonelli-Poggi-Tanzini.

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Section: Overviewmentioning

confidence: 99%

“…However, the theory is much richer than what the bare DT numbers bibliography on the subject. For the latter, see some recent developments after Nekrasov-Okounkov [49], such as [52,1,65] and [48,50,13] for a generalisation to Calabi-Yau 4-folds. In this article we deal with K-theoretic DT theory.…”

Section: Overviewmentioning

confidence: 99%

Higher rank K-theoretic Donaldson-Thomas Theory of points

Fasola

Monavari

Ricolfi

2021

Forum of Mathematics, Sigma

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“…An appearance of 4d melting crystals (also referred to as brane brick models) was already noticed in the literature (see e.g. [54,55]) in applications to enumerative geometry counting problems of toric Calabi-Yau four-folds [56][57][58]. Although CY 3 × T 2 is not a toric CY 4 (for which a color/shape generalization of the solid partition should be relevant [59,60], nevertheless we expect that these two setups may have common features, in their countings for BPS states of D-branes.…”

Section: Jhep02(2022)024mentioning

confidence: 86%

Toroidal and elliptic quiver BPS algebras and beyond

Galakhov

Yamazaki

2022

J. High Energ. Phys.

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The quiver Yangian, an infinite-dimensional algebra introduced recently in [1], is the algebra underlying BPS state counting problems for toric Calabi-Yau three-folds. We introduce trigonometric and elliptic analogues of quiver Yangians, which we call toroidal quiver algebras and elliptic quiver algebras, respectively. We construct the representations of the shifted toroidal and elliptic algebras in terms of the statistical model of crystal melting. We also derive the algebras and their representations from equivariant localization of three-dimensional $$ \mathcal{N} $$ N = 2 supersymmetric quiver gauge theories, and their dimensionally-reduced counterparts. The analysis of supersymmetric gauge theories suggests that there exist even richer classes of algebras associated with higher-genus Riemann surfaces and generalized cohomology theories.

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“…13) inLemma 3.12 and (3.34). Note that the composition(3.42) O X id I −→ RHom X (I, I) λ•lv −−→ RHom X (I, d * J ) # vanishes since we have a commutative diagram O X G G id I d * O D rv•ξ ∨ •id J RHom X (I, I) lv G G RHom X (I, d * J )and an equation λ• (r v • ξ ∨ • id J ) = 0 by (3.33).…”

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confidence: 92%

Virtual pullbacks in Donaldson-Thomas theory of Calabi-Yau 4-folds

Park¹

2021

Preprint

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Recently, Oh and Thomas constructed algebraic virtual cycles for moduli spaces of sheaves on Calabi-Yau 4-folds. The purpose of this paper is to provide a virtual pullback formula between these Oh-Thomas virtual cycles. We find a natural compatibility condition between 3-term symmetric obstruction theories that induces a virtual pullback formula. There are two types of applications.Firstly, we introduce a Lefschetz principle in Donaldson-Thomas theory, which relates the tautological DT4 invariants of a Calabi-Yau 4-fold with the DT3 invariants of its divisor. As corollaries, we prove the Cao-Kool conjecture on the tautological Hilbert scheme invariants for very ample line bundles and the Cao-Kool-Monavari conjecture on the tautological DT/PT correspondence for line bundles with Calabi-Yau divisors when the tautological complexes are vector bundles.Secondly, we present a correspondence between the Oh-Thomas virtual cycles on the moduli spaces of pairs and the moduli spaces of sheaves by combining the virtual pullback formula and a pushforward formula for virtual projective bundles. As corollaries, we prove the Cao-Maulik-Toda conjecture on the primary PT/GV correspondence for irreducible curve classes and the Cao-Toda conjecture on the primary JS/GV correspondence under the coprime condition, assuming the Cao-Maulik-Toda conjecture on the primary Katz/GV correspondence. Moreover, we also prove tautological versions of these two correspondences.Contents Gromov-Witten invariants [46,47] and rationality [51,52], motivic property [3,33], modularity [25,57], etc.).Generalizing Donaldson-Thomas theory to higher-dimensional algebraic varieties is not obvious. The standard method of constructing virtual cycles in [4,43] does not work for higher-dimensional varieties since the natural obstruction theories on the moduli spaces of sheaves are no longer 2-term. In particular, for Calabi-Yau 4-folds, the obstruction theories are 3-term symmetric, which are never 2-term.In the groundbreaking work [8], Borisov and Joyce constructed real virtual cycles for schemes with 3-term symmetric obstruction theories based on the Darboux theorem [9, 5] and derived differential geometry. Thus Donaldson-Thomas invariants for Calabi-Yau 4-folds can be defined via the Borisov-Joyce virtual cycles. However, computation of these DT4 invariants through the Borisov-Joyce virtual cycles is believed to be very difficult.Recently, Oh and Thomas [49] lifted Borisov-Joyce virtual cycles to Chow groups by generalizing Cao-Leung's algebraic approach [15]. The key idea is to localize Edidin-Graham's square root Euler class [23] by an isotropic section via Kiem-Li's cosection-localized Gysin map [34]. This algebraic method enables us to compute DT4 invariants in some cases.Currently, there are three known computational tools in DT4 theory:(1) Reduction to Edidin-Graham/Behrend-Fantechi classes [15];(2) Torus localization [49];(3) Cosection localization [36]. These tools are shown to be effective when they can be applied, i.e., the moduli space is smooth/vir...

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

Cited by 34 publications

References 21 publications

Higher rank K-theoretic Donaldson-Thomas Theory of points

Higher rank K-theoretic Donaldson-Thomas Theory of points

Toroidal and elliptic quiver BPS algebras and beyond

Virtual pullbacks in Donaldson-Thomas theory of Calabi-Yau 4-folds

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