Calabi-Yau 3-folds Research Articles

Combinatorial and algebraic mutations of toric Fano 3-folds and mass deformations of 2d(0,2) quiver gauge theories

We argue that algebraic and combinatorial polytope mutations of Fano 3-folds can be identified with mass deformations of associated 2d (0,2) supersymmetric gauge theories realized by brane brick models. These are type IIA brane configurations that realize a large family of 2d worldvolume theories on probe D1-branes at toric Calabi-Yau 4-folds. We show that brane brick models that are related by mass deformations associated to algebraic and combinatorial polytope mutations of Fano 3-folds have mesonic moduli spaces with the same number of generators. We show that mesonic flavor charges of these generators form convex reflexive lattice polytopes that are dual to the toric diagrams of the Fano 3-folds. The generating function of mesonic gauge invariant operators, also known as the Hilbert series of the mesonic moduli space, appears to be identical for such brane brick models under a particular refinement originating from the U(1)R charges in the brane brick model following the mass deformation. Published by the American Physical Society 2024

Hilbert series of bipartite field theories

We study the algebraic structure of the mesonic moduli spaces of bipartite field theories by computing the Hilbert series. Bipartite field theories form a large family of 4dN\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 1 supersymmetric gauge theories that are defined by bipartite graphs on Riemann surfaces with boundaries. By calculating the Hilbert series, we are able to identify the generators and defining generator relations of the mesonic moduli spaces of these theories. Moreover, we show that certain bipartite field theories exhibit enhanced global symmetries which can be identified through the computation of the corresponding refined Hilbert series. As part of our study, we introduce two one-parameter families of bipartite field theories defined on cylinders whose mesonic moduli spaces are all complete intersection toric Calabi-Yau 3-folds.

The classification of automorphisms and quotients of Calabi-Yau threefolds of type A

The aim of the paper is to investigate the only two families FGA of Calabi-Yau 3-folds A/G with A an abelian 3-fold and G≤Aut(A) a finite group acting freely: one is constructed in [11] and the other is presented here. We provide a complete classification of the automorphism group of X∈FGA. Additionally, we construct and classify the quotients X/ϒ for any ϒ≤Aut(X). Specifically, for those groups ϒ that preserve the volume form of X then X/ϒ admits a desingularization Y which is a Calabi-Yau 3-fold: we compute the Hodge numbers and the fundamental group of these Y, thereby determining all topological in-equivalent Calabi-Yau 3-folds obtained in this way.

Construction and deformations of Calabi–Yau 3-folds in codimension 4

We construct polarized Calabi–Yau 3-folds with at worst isolated canonical orbifold points in codimension 4 that can be described in terms of the equations of the Segre embedding of P2×P2 in P8. We investigate the existence of other deformation families in their Hilbert scheme by either studying Tom and Jerry degenerations or by comparing their Hilbert series with those of existing low codimension Calabi–Yau 3-folds. Among other interesting results, we find a family of Calabi–Yau 3-fold with five distinct Tom and Jerry deformation families, a phenomenon not seen for Q-Fano 3-folds. We compute the Hodge numbers of P2×P2 Calabi–Yau 3-folds and corresponding manifolds obtained by performing crepant resolutions. We obtain a manifold with a pair of Hodge numbers that does not appear in the famously known list of 30108 distinct Hodge pairs of Kruzer–Skarke, in the list of 7890 distinct Hodge pairs corresponding to complete intersections in the product of projective spaces and in Hodge paris obtained from Calabi–Yau 3-folds having low codimension embeddings in weighted projective spaces.

Weyl symmetry for curve counting invariants via spherical twists

We study the curve counting invariants of Calabi–Yau 3-folds via the Weyl reflection along a ruled divisor. We obtain a new rationality result and functional equation for the generating functions of Pandharipande–Thomas invariants. When the divisor arises as resolution of a curve of A 1 A_1 -singularities, our results match the rationality of the associated Calabi–Yau orbifold. The symmetry on generating functions descends from the action of an infinite dihedral group of derived auto-equivalences, which is generated by the derived dual and a spherical twist. Our techniques involve wall-crossing formulas and generalized Donaldson–Thomas invariants for surface-like objects.

Hilbert schemes of points on Calabi–Yau 4-folds via wall-crossing

Gross–Joyce–Tanaka [43] proposed a wall-crossing conjecture for Calabi–Yau fourfolds. Assuming it, we prove the conjecture of Cao–Kool [14] for 0-dimensional sheaf-counting invariants on projective Calabi–Yau 4-folds. From it, we extract the full topological information contained in the virtual fundamental classes of Hilbert schemes of points which turns out to be equivalent to the data of all descendent integrals. As a consequence, we can express many generating series of invariants in terms of explicit universal power series.i)On C4, Nekrasov proposed invariants with a conjectured closed form [72]. We show that an analog of his formula holds for compact Calabi–Yau 4-folds satisfying the wall-crossing conjecture.ii)We notice a relationship to corresponding generating series for Quot schemes on elliptic surfaces which are also governed by a wall-crossing formula. This leads to a Segre–Verlinde correspondence for Calabi–Yau fourfolds.

Thin Monodromy in $$\textrm{O}(5)$$

R\'esum\'eThis article studies the orthogonal hypergeometric groups of degree five. We establish the thinness of 12 out of the 19 hypergeometric groups of type O(3, 2) from [4, Table 6]. Some of these examples are associated with Calabi-Yau 4-folds. We also establish the thinness of 9 out of the 17 hypergeometric groups of type O(4, 1) from [13], where the thinness of 7 other cases was already proven. The O(4, 1) type groups were predicted to be all thin and our result leaves just one case open.

Non-perturbative topological string theory on compact Calabi-Yau 3-folds

We obtain analytic and numerical results for the non-perturbative amplitudes of topological string theory on arbitrary, compact Calabi-Yau manifolds. Our approach is based on the theory of resurgence and extends previous special results to the more general case. In particular, we obtain explicit trans-series solutions of the holomorphic anomaly equations. Our results predict the all orders, large genus asymptotics of the topological string free energies, which we test in detail against high genus perturbative series obtained recently in the compact case. We also provide additional evidence that the Stokes constants appearing in the resurgent structure are closely related to integer BPS invariants.

4d crystal melting, toric Calabi-Yau 4-folds and brane brick models

We introduce a class of 4-dimensional crystal melting models that count the BPS bound state of branes on toric Calabi-Yau 4-folds. The crystalline structure is determined by the brane brick model associated to the Calabi-Yau 4-fold under consideration or, equivalently, its dual periodic quiver. The crystals provide a discretized version of the underlying toric geometries. We introduce various techniques to visualize crystals and their melting configurations, including 3-dimensional slicing and Hasse diagrams. We illustrate the construction with the D0-D8 system on \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb{C}}$$\\end{document}4. Finally, we outline how our proposal generalizes to arbitrary toric CY 4-folds and general brane configurations.

Machine learning regularization for the minimum volume formula of toric Calabi-Yau 3-folds

We present a collection of explicit formulas for the minimum volume of Sasaki-Einstein 5-manifolds. The cone over these 5-manifolds is a toric Calabi-Yau 3-fold. These toric Calabi-Yau 3-folds are associated with an infinite class of 4d N=1 supersymmetric gauge theories, which are realized as world volume theories of D3-branes probing the toric Calabi-Yau 3-folds. Under the AdS/CFT correspondence, the minimum volume of the Sasaki-Einstein base is inversely proportional to the central charge of the corresponding 4d N=1 superconformal field theories. The presented formulas for the minimum volume are in terms of geometric invariants of the toric Calabi-Yau 3-folds. These explicit results are derived by implementing machine learning regularization techniques that advance beyond previous applications of machine learning for determining the minimum volume. Moreover, the use of machine learning regularization allows us to present interpretable and explainable formulas for the minimum volume. Our work confirms that, even for extensive sets of toric Calabi-Yau 3-folds, the proposed formulas approximate the minimum volume with remarkable accuracy. Published by the American Physical Society 2024

Gopakumar–Vafa Type Invariants of Holomorphic Symplectic 4-Folds

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 Gopakumar and Vafa for Calabi–Yau 3-folds, Klemm and Pandharipande for Calabi–Yau 4-folds, and Pandharipande and Zinger for Calabi–Yau 5-folds. We conjecture that our invariants are integers and give a sheaf-theoretic interpretation in terms of reduced 4-dimensional Donaldson–Thomas invariants of one-dimensional stable sheaves. We check our conjectures for the product of two K3 surfaces and for the cotangent bundle of P2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {P}}^2$$\\end{document}. Modulo the conjectural holomorphic anomaly equation, we compute our invariants also for the Hilbert scheme of two points on a K3 surface. This yields a conjectural formula for the number of isolated genus 2 curves of minimal degree on a very general hyperkähler 4-fold of K3[2]\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$K3^{[2]}$$\\end{document}-type. The formula may be viewed as a 4-dimensional analogue of the classical Yau–Zaslow formula concerning counts of rational curves on K3 surfaces. In the course of our computations, we also derive a new closed formula for the Fujiki constants of the Chern classes of tangent bundles of both Hilbert schemes of points on K3 surfaces and generalized Kummer varieties.

The 4-fold Pandharipande–Thomas vertex

We give a conjectural but full and explicit description of the (K-theoretic) equivariant vertex for Pandharipande–Thomas stable pairs on toric Calabi–Yau 4-folds, by identifying torus-fixed loci as certain quiver Grassmannians and prescribing a canonical half of the tangent-obstruction theory. For any number of non-trivial legs, the DT/PT vertex correspondence can then be verified by computer in low degrees.

Bifurcations of embedded curves and toward an extension of Taubes’s Gromov invariant to Calabi–Yau 3-folds

Bifurcations of embedded curves and toward an extension of Taubes’s Gromov invariant to Calabi–Yau 3-folds

Correspondence of Donaldson-Thomas and Gopakumar-Vafa invariants on local Calabi-Yau 4-folds over V5 and V22

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 invariants computation, and use classical results for Hilbert schemes on V5 and V22 for DT invariants computation. From these computations, one can check correspondence between DT and Gopakumar-Vafa (GV) invariants conjectured by Cao-Maulik-Toda in genus 0. Also we can compute genus 1 GV invariants via the conjecture of Cao-Toda, which turned out to be 0. These fit into the fact that there are no smooth elliptic curves in V5 and V22 up to degree 3.

SYZ Geometry for Calabi-Yau 3-folds: Taub-NUT and Ooguri-Vafa Type Metrics

We construct a family of Calabi-Yau metrics on C 3 \mathbb {C}^3 with properties analogous to the Taub-NUT metric on C 2 \mathbb {C}^2 , and construct a family of Calabi-Yau 3-fold metric models on the positive and negative vertices of SYZ fibrations with properties analogous to the Ooguri-Vafa metric.

Mass deformations of brane brick models

We investigate a class of mass deformations that connect pairs of 2d (0, 2) gauge theories associated to different toric Calabi-Yau 4-folds. These deformations are generalizations to 2d of the well-known Klebanov-Witten deformation relating the 4d gauge theories for the ℂ2/ℤ2 × ℂ orbifold and the conifold. We investigate various aspects of these deformations, including their connection to brane brick models and the relation between the change in the geometry and the pattern of symmetry breaking triggered by the deformation. We also explore how the volume of the Sasaki-Einstein 7-manifold at the base of the Calabi-Yau 4-fold varies under deformation, which leads us to conjecture that it quantifies the number of degrees of freedom of the gauge theory and its dependence on the RG scale.

On the master space for brane brick models

We systematically study the master space of brane brick models that represent a large class of 2d (0, 2) quiver gauge theories. These 2d (0, 2) theories are worldvolume theories of D1-branes that probe singular toric Calabi-Yau 4-folds. The master space is the freely generated space of chiral fields subject to the J- and E-terms and the non-abelian part of the gauge symmetry. We investigate several properties of the master space for abelian brane brick models with U(1) gauge groups. For example, we calculate the Hilbert series, which allows us by using the plethystic programme to identify the generators and defining relations of the master space. By studying several explicit examples, we also show that the Hilbert series of the master space can be expressed in terms of characters of irreducible representations of the full global symmetry of the master space.

The asymptotic weak gravity conjecture in M-theory

The tower Weak Gravity Conjecture predicts infinitely many super-extremal states along every ray in the charge lattice of a consistent quantum gravity theory. We show this far-reaching claim in five-dimensional compactifications of M-theory on Calabi-Yau 3-folds for gauge groups with a weak coupling limit. We first characterize the possible weak coupling limits, building on an earlier classification of infinite distance limits in the Kähler moduli space of M-theory compactifications. We find that weakly coupled gauge groups are associated to curves on the compactification space contained in generic fibers or in fibers degenerating at finite distance in their moduli space. These always admit an interpretation as a Kaluza-Klein or winding U(1) in a dual frame or as part of a dual perturbative heterotic gauge group, in agreement with the Emergent String Conjecture. Using the connection between Donaldson-Thomas invariants and Noether-Lefschetz theory, we then show that every ray in the associated charge lattice either supports a tower of BPS states or of non-BPS states, and prove that these satisfy the super-extremality condition, at least in the weak coupling regime.

Counting sheaves on Calabi–Yau 4-folds, I

Borisov and Joyce constructed a real virtual cycle on compact moduli spaces of stable sheaves on Calabi–Yau 4-folds, using derived differential geometry. We construct an algebraic virtual cycle. A key step is a localization of Edidin and Graham’s square root Euler class for SO(2n,C)-bundles to the zero locus of an isotropic section, or to the support of an isotropic cone. We prove a torus localization formula, making the invariants computable and extending them to the noncompact case when the fixed locus is compact. We give a K-theoretic refinement by defining K-theoretic square root Euler classes and their localized versions. In a sequel, we prove that our invariants reproduce those of Borisov and Joyce.

Brane brick models for the Sasaki-Einstein 7-manifolds Yp,k(ℂℙ1× ℂℙ1) and Yp,k(ℂℙ2)

The 2d (0, 2) supersymmetric gauge theories corresponding to the classes of Yp,k(ℂℙ1× ℂℙ1) and Yp,k(ℂℙ2) manifolds are identified. The complex cones over these Sasaki-Einstein 7-manifolds are non-compact toric Calabi-Yau 4-folds. These infinite families of geometries are the largest ones for Sasaki-Einstein 7-manifolds whose metrics, toric diagrams, and volume functions are known explicitly. This work therefore presents the largest list of 2d (0, 2) supersymmetric gauge theories corresponding to Calabi-Yau 4-folds with known metrics.