Alexander Quintero Velez, Alex Boer
In this paper we construct noncommutative resolutions of a certain class of Calabi-Yau threefolds studied by F. Cachazo, S. Katz and C. Vafa. The threefolds under consideration are fibered over a complex plane with the fibers being deformed Kleinian singularities. The construction is in terms of a noncommutative algebra introduced by V. Ginzburg, which we call the "N=1 ADE quiver algebra".
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
The Hodge numbers of generic elliptically fibered Calabi-Yau threefolds over toric base surfaces fill out the "shield" structure previously identified by Kreuzer and Skarke. The connectivity structure of these spaces and bounds on the Hodge numbers are illuminated by considerations from F-theory and the minimal model program. In particular, there is a rigorous bound on the Hodge number h_{21} <= 491 for any elliptically fibered Calabi-Yau threefold. The threefolds with the largest known Hodge numbers are associated with a sequence of blow-ups of toric bases beginning with the Hirzebruch surfac…
This paper classifies a class of holomorphic D-branes, closely related to framed torsion-free sheaves, on threefolds fibered in resolved ADE surfaces over a general curve C, in terms of representations with relations of a twisted Kronheimer--Nakajima-type quiver in the category Coh(C) of coherent sheaves on C. For the local Calabi--Yau case C\cong\A^1 and special choice of framing, one recovers the N=1 ADE quiver studied by Cachazo--Katz--Vafa.
Fibered Calabi-Yau threefolds with relative automorphisms of positive entropy and $c_2$-contractions
本文研究纤维化Calabi-Yau三维流形上保持纤维化的正熵自同构，证明阿贝尔纤维化情形在同构意义下唯一，并给出椭圆纤维化情形的显式结构定理。
K3-fibered Calabi-Yau threefolds I, the twist map
本文提出在加权超曲面背景下通过低维空间乘积的商构造具有纤维结构（特别是 K3 曲面纤维）的 Calabi-Yau 流形的方法，并探讨其在 4 维 Type II-杂化弦对偶性中的意义。
We consider certain $K3$-fibered Calabi--Yau threefolds. One class of such Calabi--Yau threefolds are constructed by Hunt and Schimmrigk using twist maps. They are realized in weighted projective spaces as orbifolds of hypersurfaces. Our main goal of this paper is to investigate arithmetic properties of these Calabi--Yau threefolds. We also consider deformations of our Calabi--Yau threefolds, and we study the variation of the zeta-functions using $p$-adic rigid cohomology theory.
An elliptic approach to Reid's fantasy
通过几何过渡证明非纤维化Calabi-Yau三维流形与纤维化流形相连，为有限性及Reid幻想提供路径。
Complete Calabi-Yau metrics on noncompact abelian fibered threefolds
研究以 $\mathbb{C}$ 为底的非紧阿贝尔纤维化三维流形，构造其上的完备 Calabi-Yau 度量，并给出典范丛为负的紧化。
In part I of this paper we constructed certain fibered Calabi-Yaus by a quotient construction in the context of weighted hypersurfaces. In this paper look at the case of K3 fibrations more closely and study the singular fibers which occur. This differs from previous work since the fibrations we discuss have constant modulus, and the singular fibers have torsion monodromy.
Boundedness of complements for fibered Fano threefolds in positive characteristic
本文在正特征大特征情形下证明了Fano型纤维化的典范丛公式，并解决了具有纤维化结构的Fano型三维配对的Shokurov补集有界性猜想，特别覆盖了$-(K_X+B)$为nef且非big的情形。
We study each of the 16 types of complete intersection Calabi-Yau threefolds in projective n-space times the projective line, for various n, and prove existence of isolated rational curves of bidegree (d,0) for all positive integers d on a general threefold of each of these types. As a corollary we find isolated rational curves in all degrees on a general CICY of each of the 5 possible types in usual projective n-space.
Over a perfect field, we determine the sheaf of $\mathbb{A}^1$-connected components of a class of threefolds given by the Blow-up of a variety admitting a $\mathbb{P}^1$-fibration over either an $\mathbb{A}^1$-rigid or a non-uniruled surface, along a smooth curve. As a consequence, we verify that the sheaf of $\mathbb{A}^1$-connected components for such varieties is $\mathbb{A}^1$-invariant.
Using the techniques introduced by Corvaja and Zannier we solve the non-split case of the geometric Lang-Vojta Conjecture for affine surfaces isomorphic to the complement of a conic and two lines in the projective plane. In this situation we deal with sections of an affine threefold fibered over a curve, whose boundary, in the natural projective completion, is a quartic bundle over the base whose fibers have three irreducible components. We prove that the image of each section has bounded degree in terms of the Euler Characteristic of the base curve.
Elliptic and genus one fibered Calabi-Yau spaces play a prominent role in string theory and mathematics. In this article we discuss a class of genus one fibered Calabi-Yau threefolds with 5-sections from various perspectives. In algebraic geometry, such Calabi-Yaus can be constructed as complete intersections in Grassmannian fibrations and as Pfaffian varieties. These constructions naturally fit into the framework of homological projective duality and lead to dual pairs of Calabi-Yaus. From a physics perspective, these spaces can be realised as low-energy configurations ("phases") of gauged li…
Revisiting the Quantum Geometry of Torus-fibered Calabi-Yau Threefolds
本文证明拓扑弦振幅的模性质等价于相对锥形单值下的波函数性质，并引入全纯模协变版本，揭示GV不变量与DT指数的拟模关系，分析大量新例子。
We study sections of a Calabi-Yau threefold fibered over a curve by K3 surfaces. We show that there exist infinitely many isolated sections on certain K3 fibered Calabi-Yau threefolds and the subgroup of the Néron-Severi group generated by these sections is not finitely generated. This also gives examples of K3 surfaces over the function field F of a complex curve with Zariski dense F-rational points, whose geometric model is Calabi-Yau.
To every elliptic Calabi-Yau threefold with a section $X$ there can be associated a Lie group $G$ and a representation $ρ$ of that group. The group is determined from the Weierstrass model, which has singularities that are generically rational double points; these double points lead to local factors of $G$ which are either the corresponding A-D-E groups or some associated non-simply laced groups. The representation $ρ$ is a sum of representations coming from the local factors of $G$, and of other representations which can be associated to the points at which the singularities are worse than ge…
Let $X$ be a Gorenstein minimal projective 3-fold with at worst locally factorial terminal singularities. Suppose the canonical map is of fiber type. Denote by $F$ a smooth model of a generic irreducible component in fibers of the canonical map of $X$ and so $F$ is a smooth curve or a smooth surface. The main result of the paper is that there is a computable constant $K$ (independent of $X$) such that $g(F)\leq 647$ or $p_g(F)\leq 38$ whenever $p_g(X)\geq K$. The method heavily relies on both a Noether type of inequality and a Miyaoka-Yau inequality and that is the reason we only treat a Goren…
We construct stable bundle extensions on elliptically fibered Calabi-Yau threefolds. We show that these bundles can solve the topological anomaly constraint in heterotic string theory without the need of invoking background fivebranes.
We carry out a systematic analysis of Calabi-Yau threefolds that are elliptically fibered with section ("EFS") and have a large Hodge number h^{2, 1}. EFS Calabi-Yau threefolds live in a single connected space, with regions of moduli space associated with different topologies connected through transitions that can be understood in terms of singular Weierstrass models. We determine the complete set of such threefolds that have h^{2, 1} >= 350 by tuning coefficients in Weierstrass models over Hirzebruch surfaces. The resulting set of Hodge numbers includes those of all known Calabi-Yau threefold…
We study Calabi-Yau threefolds fibered by abelian surfaces, in particular, their arithmetic properties, e.g., Néron models and Zariski density.
We prove a case of the conjecture of Douglas, Reinbacher and Yau about the existence of stable vector bundles with prescribed Chern classes on a Calabi-Yau threefold. For this purpose we prove the existence of certain stable vector bundle extensions over elliptically fibered Calabi-Yau threefolds.
Enumerative geometry and modularity in two-modulus K3-fibered Calabi-Yau threefolds
本文研究了双模数K3纤维化Calabi-Yau三重展开的计数几何与模性，构造了39个Calabi-Yau镜像对，并展示了其模性下的自由能公式。
This paper studies the action of the Fourier-Mukai transform on moduli spaces of vertical torsion sheaves on elliptic Calabi-Yau threefolds in Weierstrass form. Moduli stacks of semistable one dimensional sheaves on such threefolds are identified with open and closed substacksof moduli stacks of vertical semistable two dimensional sheaves on their Fourier-Mukai duals. In particular, this yields explicit conjectural results for Donaldson-Thomas invariants of vertical two dimensional sheaves on K3-fibered elliptic Calabi-Yau threefolds.
We find through a systematic analysis that all but 29,223 of the 473.8 million 4D reflexive polytopes found by Kreuzer and Skarke have a 2D reflexive subpolytope. Such a subpolytope is generally associated with the presence of an elliptic or genus one fibration in the corresponding birational equivalence class of Calabi-Yau threefolds. This extends the growing body of evidence that most Calabi-Yau threefolds have an elliptically fibered phase.
We give a proof of the existence of $G=SU(5)$, stable holomorphic vector bundles on elliptically fibered Calabi--Yau threefolds with fundamental group $\bbz_2$. The bundles we construct have Euler characteristic 3 and an anomaly that can be absorbed by M-theory five-branes. Such bundles provide the basis for constructing the standard model in heterotic M-theory. They are also applicable to vacua of the weakly coupled heterotic string. We explicitly present a class of three family models with gauge group $SU(3)_C\times SU(2)_L\times U(1)_Y$.
A torus fibered Calabi-Yau threefold with first homotopy group Z_3 x Z_3 is constructed as a free quotient of a fiber product of two dP_9 surfaces. Calabi-Yau threefolds of this type admit Z_3 x Z_3 Wilson lines. In conjunction with SU(4) holomorphic vector bundles, such vacua lead to anomaly free, three generation models of particle physics with a right handed neutrino and a U(1)_{B-L} gauge factor, in addition to the SU(3)_C x SU(2)_L x U(1)_Y standard model gauge group. This factor helps to naturally suppress nucleon decay. The moduli space and Dolbeault cohomology of the threefold is also …
We study local, global and local-to-global properties of threefolds with certain singularities. We prove criteria for these threefolds to be rational homology manifolds and conditions for threefolds to satisfy rational Poincaré duality. We relate the topological Euler characteristic of elliptic Calabi-Yau threefolds with $\mathbb Q$-factorial terminal singularities to dimensions of Lie algebras and certain representations, Milnor and Tyurina numbers and other birational invariants of an elliptic fibration. We give an interpretation in terms of complex deformations. We state a conjecture on the…
We develop a combinatorial approach to the construction of general smooth compact base surfaces that support elliptic Calabi-Yau threefolds. This extends previous analyses that have relied on toric or semi-toric structure. The resulting algorithm is used to construct all classes of such base surfaces $S$ with $h^{1, 1} (S) < 8$ and all base surfaces over which there is an elliptically fibered Calabi-Yau threefold $X$ with Hodge number $h^{2, 1} (X) \geq 150$. These two sets can be used todescribe all 6D F-theory models that have fewer than seven tensor multiplets or more than 150 neutral scala…
In this paper we study compactifications of the ${\cal N}=2$ heterotic $E_8\times E_8$ string on $(K3\times T^2)/\mathbb{Z}_3$ with various gauge backgrounds and calculate the topological couplings in the effective supergravity action that arise from one-loop amplitudes. We then identify candidates for dual type IIA compactifications on Calabi-Yau threefolds and compare the heterotic results with the corresponding topological string amplitudes. We find that the dual Calabi-Yau geometries are $K3$ fibrations that are also genus one fibered with three-sections. Moreover, we show that the interse…
In arXiv:1008.1018 it is shown that a given stable vector bundle $V$ on a Calabi-Yau threefold $X$ which satisfies $c_2(X)=c_2(V)$ can be deformed to a solution of the Strominger system and the equations of motion of heterotic string theory. In this note we extend this result to the polystable case and construct explicit examples of polystable bundles on elliptically fibered Calabi-Yau threefolds where it applies. The polystable bundle is given by a spectral cover bundle, for the visible sector, and a suitably chosen bundle, for the hidden sector. This provides a new class of heterotic flux co…
Supersymmetric heterotic string models, built from a Calabi-Yau threefold $X$ endowed with a stable vector bundle $V$, usually lead to an anomaly mismatch between $c_2(V)$ and $c_2(X)$; this leads to the question whether the difference can be realized by a further bundle in the hidden sector. In math.AG/0604597 a conjecture is stated which gives sufficient conditions on cohomology classes on $X$ to be realized as the Chern classes of a stable reflexive sheaf $V$; a weak version of this conjecture predicts the existence of such a $V$ if $c_2(V)$ is of a certain form. In this note we prove that …
An explicit description of the spectral data of stable U(n) vector bundles on elliptically fibered Calabi-Yau threefolds is given, extending previous work of Friedman, Morgan and Witten. The characteristic classes are computed and it is shown that part of the bundle cohomology vanishes. The stability and the dimension of the moduli space of the U(n) bundles are discussed. As an application, it is shown that the U(n) bundles are capable to solve the basic topological constraints imposed by heterotic string theory. Various explicit solutions of the Donaldson-Uhlenbeck-Yau equation are given. The…
Let $Σ$ be a smooth projective complex curve and $\mathfrak{g}$ a simple Lie algebra of type ${\sf ADE}$ with associated adjoint group $G$. For a fixed pair $(Σ, \mathfrak{g})$, we construct a family of quasi-projective Calabi-Yau threefolds parameterized by the base of the Hitchin integrable system associated to $(Σ,\mathfrak{g})$. Our main result establishes an isomorphism between the Calabi-Yau integrable system, whose fibers are the intermediate Jacobians of this family of Calabi-Yau threefolds, and the Hitchin system for $G$, whose fibers are Prym varieties of the corresponding spectral c…
We find that for many Calabi-Yau threefolds with elliptic or genus one fibrations mirror symmetry factorizes between the fiber and the base of the fibration. In the simplest examples, the generic CY elliptic fibration over any toric base surface $B$ that supports an elliptic Calabi-Yau threefold has a mirror that is an elliptic fibration over a dual toric base surface $\tilde{B}$ that is related through toric geometry to the line bundle $-6K_B$. The Kreuzer-Skarke database includes all these examples and gives a wide range of other more complicated constructions where mirror symmetry also fact…
Spectrum bounds in geometry
本文基于椭圆纤维化Calabi-Yau三维流形拓扑类型有限性的已有结果，探讨了其对F理论紧致化无质量谱中几何量有界性的影响，并给出了显式界限。
We use F. Ferrari's methods relating matrix models to Calabi-Yau spaces in order to explain Intriligator and Wecht's ADE classification of $\N=1$ superconformal theories which arise as RG fixed points of $\N = 1$ SQCD theories with adjoints. The connection between matrix models and $\N = 1$ gauge theories can be seen as evidence for the Dijkgraaf--Vafa conjecture. We find that ADE superpotentials in the Intriligator--Wecht classification exactly match matrix model superpotentials obtained from Calabi-Yau's with corresponding ADE singularities. Moreover, in the additional $\Hat{O}, \Hat{A}, \Ha…
CICY3 Families of Nonsingular Codimension two K3
本文在秩2环面环境空间中系统搜索余维2完全交叉Calabi--Yau三流形（CICY3），并对其非奇异及具有孤立奇点的情形进行了分类。
The string landscape satisfies interesting finiteness properties imposed by supersymmetry and string-theoretical consistency conditions. We study N=1 supersymmetric compactifications of Type IIB string theory on smooth elliptically fibered Calabi-Yau threefolds at large volume with magnetized D9-branes and D5-branes. We prove that supersymmetry and tadpole cancellation conditions imply that there is a finite number of such configurations. In particular, we derive an explicitly computable bound on the number of magnetic flux quanta, as well as the number of D5-branes, which is independent of th…
We discuss the role of hyperelliptic fibrations in F-theory. For each even integer $n$ we give a noncompact Calabi--Yau threefold $X$ containing a hyperelliptically fibered surface $Y$, such that $X$ and $Y$ are homotopy equivalent and $c_2(X) = n$. We investigate two distinct cases depending on the position of the hyperelliptic fibration. First, we propose to extend F-theory considering hyperelliptic fibrations, giving an identification between the determinant of the period matrix and the axio-dilaton. Such an identification requires that the curve satisfies an appropriate criterium which we …
We use F. Ferrari's methods relating matrix models to Calabi-Yau spaces in order to explain much of Intriligator and Wecht's ADE classification of $\N=1$ superconformal theories which arise as RG fixed points of $\N = 1$ SQCD theories with adjoints. We find that ADE superpotentials in the Intriligator-Wecht classification exactly match matrix model superpotentials obtained from Calabi-Yaus with corresponding ADE singularities. Moreover, in the additional $\Hat{O}, \Hat{A}, \Hat{D}$ and $\Hat{E}$ cases we find new singular geometries. These `hat' geometries are closely related to their ADE coun…
We describe a family of genus one fibered Calabi-Yau threefolds with fundamental group ${\mathbb Z}/2$. On each Calabi-Yau $Z$ in the family we exhibit a positive dimensional family of Mumford stable bundles whose symmetry group is the Standard Model group $SU(3)\times SU(2)\times U(1)$ and which have $c_{3} = 6$. We also show that for each bundle $V$ in our family, $c_{2}(Z) - c_{2}(V)$ is the class of an effective curve on $Z$. These conditions ensure that $Z$ and $V$ can be used for a phenomenologically relevant compactification of Heterotic M-theory.
A homogeneous roof is a rational homogeneous variety of Picard rank 2 and index $r$ equipped with two different $\mathbb P^{r-1}$-bundle structures. We consider bundles of homogeneous roofs over a smooth projective variety, formulating a relative version of the duality of Calabi--Yau pairs associated to roofs of projective bundles. We discuss how derived equivalence of such pairs can lift to Calabi--Yau fibrations, extending a result of Bridgeland and Maciocia to higher-dimensional cases. We formulate an approach to prove that the $DK$-conjecture holds for a class of simple $K$-equivalent maps…
We construct a class of stable SU(5) bundles on an elliptically fibered Calabi-Yau threefold with two sections, a variant of the ordinary Weierstrass fibration, which admits a free involution. The bundles are invariant under the involution, solve the topological constraint imposed by the heterotic anomaly equation and give three generations of Standard Model fermions after symmetry breaking by Wilson lines of the intermediate SU(5) GUT-group to the Standard Model gauge group. Among the solutions we find some which can be perturbed to solutions of the Strominger system. Thus these solutions pro…
We utilize the deformation theory of algebraic singularities to study charged matter in compactifications of M-theory, F-theory, and type IIa string theory on elliptically fibered Calabi-Yau manifolds. In F-theory, this description is more physical than that of resolution. We describe how two-cycles can be identified and systematically studied after deformation. For ADE singularities, we realize non-trivial ADE representations as sublattices of Z^N, where N is the multiplicity of the codimension one singularity before deformation. We give a method for the determination of Picard-Lefschetz vani…
Folding of ADE-Dynkin diagrams according to graph automorphisms yields irreducible Dynkin diagrams of ABCDEFG-types. This folding procedure allows to trace back the properties of the corresponding simple Lie algebras or groups to those of ADE-type. In this article, we implement the techniques of folding by graph automorphisms for Hitchin integrable systems. We show that the fixed point loci of these automorphisms are isomorphic as algebraic integrable systems to the Hitchin systems of the folded groups away from singular fibers. The latter Hitchin systems are isomorphic to the intermediate Jac…
Any irreducible Dynkin diagram $Δ$ is obtained from an irreducible Dynkin diagram $Δ_h$ of type $\mathrm{ADE}$ by folding via graph automorphisms. For any simple complex Lie group $G$ with Dynkin diagram $Δ$ and compact Riemann surface $Σ$, we give a Lie-theoretic construction of families of quasi-projective Calabi-Yau threefolds together with an action of graph automorphisms over the Hitchin base associated to the pair $(Σ, G)$ . These give rise to Calabi-Yau orbifolds over the same base. Their intermediate Jacobian fibration, constructed in terms of equivariant cohomology, is isomorphic to t…
In previous work, it was argued that the type IIB T^6/Z_2 orientifold with a choice of flux preserving N=2 supersymmetry is dual to a class of purely geometric type IIA compactifications on abelian surface (T^4) fibered Calabi-Yau threefolds. We provide two explicit constructions of the resulting Calabi-Yau duals. The first is a monodromy based description, analogous to F-theory encoding of Calabi-Yau geometry via 7-branes and string junctions, except for T^4 rather than T^2 fibers. The second is an explicit algebro-geometric construction in which the T^4 fibers arise as the Jacobian tori of a…
We initiate a systematic investigation of F-theory on elliptic fibrations with singularities which cannot be resolved without breaking the Calabi-Yau condition, corresponding to $\mathbb Q$-factorial terminal singularities. It is the purpose of this paper to elucidate the physical origin of such non-crepant singularities in codimension two and to systematically analyse F-theory compactifications containing such singularities. The singularities reflect the presence of localised matter states from wrapped M2-branes which are not charged under any massless gauge potential. We identify a class of …
Topological Strings on Non-Commutative Resolutions
本文为无凯勒收缩消解的奇异卡拉比-丘三维流形定义了挠率精细化 GV 不变量，通过非交换消解和 A-模型拓扑弦配分函数给出数学刻画，并以 $\mathbb{P}^3$ 双重覆盖的奇异退化为例阐明其计数几何意义。
The Enumerative Geometry and Arithmetic of Banana Nano-Manifolds
本文构造四个 Hodge 数为 (4,0) 的刚性香蕉纳米 Calabi–Yau 三维流形，并计算其 Donaldson–Thomas、Gromov–Witten 及 Frobenius 迹相关模形式。
One may construct a large class of Calabi-Yau varieties by taking anticanonical hypersurfaces in toric varieties obtained from reflexive polytopes. If the intersection of a reflexive polytope with a hyperplane through the origin yields a lower-dimensional reflexive polytope, then the corresponding Calabi-Yau varieties are fibered by lower-dimensional Calabi-Yau varieties. A top generalizes the idea of splitting a reflexive polytope into two pieces. In contrast to the classification of reflexive polytopes, there are infinite families of equivalence classes of tops. Tops may be used to describe …
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。