An example of non-Kähler Calabi-Yau fourfold

Doubling construction of Calabi-Yau fourfolds from toric Fano fourfolds

We give a differential-geometric construction of Calabi-Yau fourfolds by the `doubling' method, which was introduced in \cite{DY14} to construct Calabi-Yau threefolds. We also give examples of Calabi-Yau fourfolds from toric Fano fourfolds. Ingredients in our construction are \emph{admissible pairs}, which were first dealt with by Kovalev in \cite{K03}. Here in this paper an admissible pair $(\overline{X},D)$ consists of a compact Kähler manifold $\overline{X}$ and a smooth anticanonical divisor $D$ on $\overline{X}$. If two admissible pairs $(\overline{X}_1,D_1)$ and $(\overline{X}_2,D_2)$ wi…

Quantum periods of Calabi-Yau fourfolds

In this work we study the quantum periods together with their Picard-Fuchs differential equations of Calabi-Yau fourfolds. In contrast to Calabi-Yau threefolds, we argue that the large volume points of Calabi-Yau fourfolds generically are regular singular points of the Picard-Fuchs operators of non-maximally unipotent monodromy. We demonstrate this property in explicit examples of Calabi-Yau fourfolds with a single Kahler modulus. For these examples we construct integral quantum periods and study their global properties in the quantum Kahler moduli space with the help of numerical analytic con…

SU(N) transitions in M-theory on Calabi-Yau fourfolds and background fluxes

We study M-theory on a Calabi-Yau fourfold with a smooth surface $S$ of $A_{N-1}$ singularities. The resulting three-dimensional theory has a $\mathcal{N}=2$ $SU(N)$ gauge theory sector, which we obtain from a twisted dimensional reduction of a seven-dimensional $\mathcal{N}=1$ $SU(N)$ gauge theory on the surface $S$. A variant of the Vafa-Witten equations governs the moduli space of the gauge theory, which, for a trivial $SU(N)$ principal bundle over $S$, admits a Coulomb and a Higgs branch. In M-theory these two gauge theory branches arise from a resolution and a deformation to smooth Calabi…

Calabi-Yau 4-folds of Borcea--Voisin type from F-Theory

In this paper, we apply Borcea--Voisin's construction and give new examples of Calabi--Yau fourfolds $Y$, which admit an elliptic fibration onto a smooth threefold $V$, whose singular fibers of type $I_5$ lie above a del Pezzo surface $dP \subset V$. These are relevant models for F-theory according to papers by C. Beasley, J. J. Heckman, C. Vafa. Moreover, at the end of the paper we will give the explicit equations of some of these Calabi--Yau fourfolds and their fibrations.

带对合的不可约全纯辛四维流形的解析挠率，III：与BCOV不变量的关系

Analytic torsion for irreducible holomorphic symplectic fourfolds with involution, III: relation with the BCOV invariant

本文比较了Camere-Garbagnati-Mongardi型Calabi-Yau四维流形的BCOV不变量与对应K3⁽²⁾型带对合流形的不变量，证明特殊情况下前者可表为某Borcherds乘积的Petersson范数。

Prepotentials for local mirror symmetry via Calabi-Yau fourfolds

In this paper, we first derive an intrinsic definition of classical triple intersection numbers of K_S, where S is a complex toric surface, and use this to compute the extended Picard-Fuchs system of K_S of our previous paper, without making use of the instanton expansion. We then extend this formalism to local fourfolds K_X, where X is a complex 3-fold. As a result, we are able to fix the prepotential of local Calabi-Yau threefolds K_S up to polynomial terms of degree 2. We then outline methods of extending the procedure to non canonical bundle cases.

All Weight Systems for Calabi-Yau Fourfolds from Reflexive Polyhedra

For any given dimension $d$, all reflexive $d$-polytopes can be found (in principle) as subpolytopes of a number of maximal polyhedra that are defined in terms of $(d+1)$-tuples of integers (weights), or combinations of $k$-tuples of weights with $k<d+1$. We present the results of a complete classification of sextuples of weights pertaining to the construction of all reflexive polytopes in five dimensions. We find 322 383 760 930 such weight systems. 185 269 499 015 of them give rise directly to reflexive polytopes and thereby to mirror pairs of Calabi-Yau fourfolds. These lead to 532 600 483 …

Aspects of Calabi-Yau Fourfold Compactifications

The study of the geometry of Calabi-Yau fourfolds is relevant for compactifications of string theory, M-theory, and F-theory to various dimensions. In the first part of this thesis, we study the action of mirror symmetry on two-dimensional $\cN=(2,2)$ effective theories obtained by compactifying Type IIA string theory on Calabi-Yau fourfolds. Our focus is on fourfold geometries with non-trivial three-form cohomology. The couplings of the massless zero-modes arising from an expansion of the three-form gauge-potential into these forms depend both on the complex structure deformations and the Käh…

Three-form periods on Calabi-Yau fourfolds: Toric hypersurfaces and F-theory applications

The study of the geometry of Calabi-Yau fourfolds is relevant for compactifications of string theory, M-theory, and F-theory to various dimensions. This work introduces the mathematical machinery to derive the complete moduli dependence of the periods of non-trivial three-forms for fourfolds realized as hypersurfaces in toric ambient spaces. It sets the stage to determine Picard-Fuchs-type differential equations and integral expressions for these forms. The key tool is the observation that non-trivial three-forms on hypersurfaces in toric ambient spaces always stem from divisors that are build…

BPS invariants of symplectic log Calabi-Yau fourfolds

Using the Fredholm setup of [12], we study genus zero (and higher) relative Gromov-Witten invariants with maximum tangency of symplectic log Calabi-Yau fourfolds. In particular, we give a short proof of [23, Conjecture 6.2] that expresses these invariants in terms of certain integral invariants by considering generic almost complex structures to obtain a geometric count. We also revisit the localization calculation of the multiple-cover contributions in [23, Proposition 6.1] and recalculate a few terms differently to provide more details and illustrate the computation of deformation/obstructio…

库仑分支局域化、拟映射与Calabi--Yau四流形中的曲面计数

Coulomb branch localization, quasimaps, and surface counting in Calabi--Yau fourfolds

本文提出了一种基于超对称局域化的弦论方法，用于研究局部Calabi--Yau四流形中的曲面计数，通过PT1稳定对与ADHM箭图扭曲拟映射间的谱对应，导出了K理论拟映射配分函数的留数公式，并消除了等变计算中的符号歧义。

On the Elliptic Calabi-Yau Fourfold with Maximal $h^{1,1}$

In this paper, we explicitly construct the smooth compact base threefold for the elliptic Calabi-Yau fourfold with the largest known $h^{1,1}=303\,148$. It is generated by blowing up a smooth toric "seed" base threefold with $(E_8,E_8,E_8)$ collisions. The 4d F-theory compactification model over it has the largest geometric gauge group, $E_8^{2\,561}\times F_4^{7\,576}\times G_2^{20\,168}\times SU(2)^{30\,200}$, and the largest number of axions, $181\,820$, in the known 4d $\mathcal{N}=1$ supergravity landscape. We also prove that there are at least $1100^{15\,048}\approx 7.5\times 10^{45\,766…

A Note on $h^{2,1}$ of Divisors of CY Fourfolds I

In this note, we prove combinatorial formulas for $h^{2,1}$ of prime toric divisors in an arbitrary toric hypersurface Calabi-Yau fourfold $Y_4.$ We show that it is possible to find a toric hypersurface Calabi-Yau in which there are more than $h^{1,1}(Y_4)$ non-perturbative superpotential terms with trivial intermediate Jacobian. Hodge numbers of divisors in toric CICYs are the subjects of the part two.

Modular Amplitudes and Flux-Superpotentials on elliptic Calabi-Yau fourfolds

We discuss the period geometry and the topological string amplitudes on elliptically fibered Calabi-Yau fourfolds in toric ambient spaces. In particular, we describe a general procedure to fix integral periods. Using some elementary facts from homological mirror symmetry we then obtain Bridgelands involution and its monodromy action on the integral basis for non-singular elliptically fibered fourfolds. The full monodromy group contains a subgroup that acts as PSL(2,Z) on the Kähler modulus of the fiber and we analyze the consequences of this modularity for the genus zero and genus one amplitud…

模Calabi-Yau四流形及其与M-理论流的联系

Modular Calabi-Yau Fourfolds and Connections to M-Theory Fluxes

本文研究了Calabi-Yau四流形的局部zeta函数，发展算术形变技术计算水平4-形式上同调因子，发现其与权重3模形式相关，并据此对M-理论紧化中的背景4-形式流进行分类，将其低能耦合表示为L函数值。

Atiyah class and sheaf counting on local Calabi Yau fourfolds

We discuss Donaldson-Thomas (DT) invariants of torsion sheaves with 2 dimensional support on a smooth projective surface in an ambient non-compact Calabi Yau fourfold given by the total space of a rank 2 bundle on the surface. We prove that in certain cases, when the rank 2 bundle is chosen appropriately, the universal truncated Atiyah class of these codimension 2 sheaves reduces to one, defined over the moduli space of such sheaves realized as torsion codimension 1 sheaves in a noncompact divisor (threefold) embedded in the ambient fourfold. Such reduction property of universal Atiyah class e…

The Vertical, the Horizontal and the Rest: anatomy of the middle cohomology of Calabi-Yau fourfolds and F-theory applications

The four-form field strength in F-theory compactifications on Calabi-Yau fourfolds takes its value in the middle cohomology group $H^4$. The middle cohomology is decomposed into a vertical, a horizontal and a remaining component, all three of which are present in general. We argue that a flux along the remaining or vertical component may break some symmetry, while a purely horizontal flux does not influence the unbroken part of the gauge group or the net chirality of charged matter fields. This makes the decomposition crucial to the counting of flux vacua in the context of F-theory GUTs. We us…

Calabi-Yau modular forms in limit: Elliptic Fibrations

We study the limit of Calabi-Yau modular forms, and in particular, those resulting in classical modular forms. We then study two parameter families of elliptically fibred Calabi-Yau fourfolds and describe the modular forms arising from the degeneracy loci. In the case of elliptically fibred Calabi-Yau threefolds our approach gives a mathematical proof of many observations about modularity properties of topological string amplitudes starting with the work of Candelas, Font, Katz and Morrison. In the case of Calabi-Yau fourfolds we derive new identities not computed before.

Rank 2 ACM bundles on complete intersection Calabi-Yau threefolds

The aim of this paper is to classify indecomposable rank 2 arithmetically Cohen-Macaulay (ACM) bundles on compete intersection Calabi-Yau (CICY) threefolds and prove the existence of some of them. New geometric properties of the curves corresponding to rank 2 ACM bundles (by Serre correspondence) are obtained. These follow from minimal free resolutions of curves in suitably chosen fourfolds (containing Calabi-Yau threefolds as hypersurfaces). Also the existence of an indecomposable vector bundle of higher rank on a CICY threefold of type (2,4) is proved.

卡拉比-丘四维流形的跨壁：框架、工具与应用

Wall-crossing for Calabi-Yau fourfolds: framework, tools, and applications

该研究建立卡拉比-丘四维范畴等变跨壁理论，引入相关形式体系，证明两类情形的跨壁公式，结合合作成果证明一般卡拉比-丘四维跨壁猜想。

Automorphism group of Batyrev Calabi-Yau threefolds

In this paper, we will prove that all Batyrev Calabi-Yau threefolds, arising from a small resolution of a generic hyperplane section of a reflexive Fano-Gorenstein fourfold, have finite automorphism group. Together with Morrison conjecture, this suggests that Batyrev Calabi-Yau threefolds should have a polyhedral Kahler (ample) cone.

论M5-膜的中间雅可比簇

On the intermediate Jacobian of M5-branes

针对带Sen极限的M/F-理论椭圆卡拉比-丘四维紧化中的M5-膜中间雅可比簇问题，提出利用IIB型O3/O7定向形数据结合组合公式分析的方法，含推测修正项，经实例验证与直接计算完全一致。

弦理论中的D(-1)瞬子超势

D(-1)-Instanton Superpotential In String Theory

本文通过引入全局Sen极限下的F理论紧化，证明了底层IIB型紧化的超势在全局Sen极限下不接受D(-1)瞬子贡献，且该结果在$g_s$上是精确的。

晶熔、三重性与托里切利卡拉比-尤康四维的配分函数

Crystal Melting, Triality and Partition Functions for Toric Calabi-Yau Fourfolds

本文研究了托里切利卡拉比-尤康四维的晶熔模型，开发了基于周期性四元组的算法，分析了晶体在三重性下的行为，并提出了稳定变量的概念以稳定配分函数。

Remarks on projective normality for certain Calabi-Yau and hyperkähler varieties

We prove some results on effective very ampleness and projective normality for some varieties with trivial canonical bundle. In the first part we prove an effective projective normality result for an ample line bundle on regular smooth four-folds with trivial canonical bundle. More precisely we show that for a regular smooth fourfold with trivial canonical bundle, $A^{\otimes 15}$ is projectively normal for $A$ ample. In the second part we emphasize on the projective normality of multiples of ample and globally generated line bundles on certain classes of known examples (upto deformation) of p…

Applications of the affine structures on the Teichmüller spaces

We prove the existence of global sections trivializing the Hodge bundles on the Hodge metric completion space of the Torelli space of Calabi--Yau manifolds, a global splitting property of these Hodge bundles. We also prove that a compact Calabi--Yau manifold can not be deformed to its complex conjugate. These results answer certain open questions in the subject. A general result about the period map to be bi-holomorphic from the Hodge metric completion space of the Torelli space of Calabi--Yau type manifolds to their period domains is proved and applied to the cases of K$3$ surfaces, cubic fou…

G-flux in F-theory and algebraic cycles

We construct explicit G4 fluxes in F-theory compactifications. Our method relies on identifying algebraic cycles in the Weierstrass equation of elliptic Calabi-Yau fourfolds. We show how to compute the D3-brane tadpole and the induced chirality indices directly in F-theory. Whenever a weak coupling limit is available, we compare and successfully match our findings to the corresponding results in type IIB string theory. Finally, we present some generalizations of our results which hint at a unified description of the elliptic Calabi-Yau fourfold together with the four-form flux G4 as a coherent…

Wall-crossing for punctual Quot-schemes

We study punctual quot-schemes of torsion-free sheaves $E_Y$ on smooth projective curves, surfaces and Calabi--Yau fourfolds via their virtual geometry. Our goal is to give a complete description of the virtual fundamental classes and their tautological integrals. In the fourfold case, we first construct these classes under additional conditions. We use novel methods relying on the wall-crossing of Joyce. Our results include -the dependence of the cobordism classes on the torsion-free sheaf $E_Y$ where $Y$ is a surface, -relations to the previous results in the literature, which addressed the …

BPS态计数中Calabi-Yau晶体的起源

The Origin of Calabi-Yau Crystals in BPS States Counting

本文研究一般环面Calabi-Yau流形上BPS D-膜计数，通过计算味Witten指数的Jeffrey-Kirwan留数，发现BPS简并度由晶体熔化模型描述；三维情形重现已知模型，四维情形需赋予晶体构型非平凡权重，并将三维与四维统一处理。

Type II/F-theory Superpotentials and Ooguri-Vafa Invariants of Compact Calabi-Yau Threefolds with Three Deformations

We calculate the D-brane superpotentials for two Calabi-Yau manifolds with three deformations by the generalized hypergeometric GKZ systems, which give rise to the flux superpotentials $\mathcal{W}_{GVW}$ of the dual F-theory compactification on the relevant Calabi-Yau fourfolds in the weak decoupling limit. We also compute the Ooguri-Vafa invariants from A-model expansion with mirror symmetry, which are related to the open Gromov-Witten invariants.

零维层的穿墙与 Calabi-Yau 4 维流形上点的 Hilbert 概形

Wall-crossing for zero-dimensional sheaves and Hilbert schemes of points on Calabi-Yau 4-folds

本文在 Gross-Joyce-Tanaka 穿墙猜想下证明 Cao-Kool 的零维层计数猜想，提取 Hilbert 概形虚基本类的完整拓扑信息，并建立通用幂级数公式与 Calabi-Yau 四维流形的 Segre-Verlinde 对应。

非光滑的来自自反多胞形的Calabi-Yau三维流形

Non-smoothable Calabi-Yau threefolds from reflexive polytopes

本文构造了具有孤立Gorenstein典范奇点且不可光滑化的Calabi-Yau三维流形，利用Altmann形变理论与Corti-Filip-Petracci准则，并在Kreuzer-Skarke分类中识别出约8.27%的多胞形存在此阻碍。

String modular motives of mirrors of rigid Calabi-Yau varieties

The modular properties of some higher dimensional varieties of special Fano type are analyzed by computing the L-function of their $Ω-$motives. It is shown that the emerging modular forms are string theoretic in origin, derived from the characters of the underlying rational conformal field theory. The definition of the class of Fano varieties of special type is motivated by the goal to find candidates for a geometric realization of the mirrors of rigid Calabi-Yau varieties. We consider explicitly the cubic sevenfold and the quartic fivefold, and show that their motivic L-functions agree with t…

环面对的形变与Batyrev Calabi-Yau三维流形的光滑化

Deformations of toric pairs and the smoothing of Batyrev Calabi-Yau threefolds

本文构造环面四维流形的形变以光滑化 Calabi-Yau 超曲面的奇点，提出逐面准则，并指出其覆盖 Batyrev--Kreuzer 普查中约 12.5% 的光滑化，其余需跨面关系。

Global magni$4$icence，即：4G Networks

Global magni$4$icence, or: 4G Networks

该研究提出全局壮丽四理论，针对环面Calabi-Yau四折猜想其扭曲Witten指标公式，结合弦论BPS态计数，发展等变K理论DT4理论并构造四价顶点，揭示其与8+1维非对易规范理论超对称局域化的关联。

Calabi-Yau threefolds with large h^{2, 1}

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…

Elliptic Calabi-Yau fivefolds and 2d (0,2) F-theory landscape

In this paper, we initiate the study of the 2d F-theory landscape based on compact elliptic Calabi-Yau fivefolds. In particular, we determine the boundary models of the landscape using Calabi-Yau fivefolds with the largest known Hodge numbers $h^{1,1}$ and $h^{4,1}$. The former gives rise to the largest geometric gauge group in the currently known 2d (0,2) supergravity landscape, which is $E_8^{482\,632\,421}\times F_4^{3\,224\,195\,728}\times G_2^{11\,927\,989\,964}\times SU(2)^{25\,625\,222\,180}$. Besides that, we systematically study the hypersurfaces in weighted projective spaces with sma…

描绘 F-理论复杂结构景观

Charting the Complex Structure Landscape of F-theory

本文研究了 F-理论在复杂结构模空间为三次穿孔球面的 Calabi--Yau 四维流形上的紧致化景观，枚举出 14 个满足准幂幺单值性条件的模型，并分析了无穷远处的四种相及其主导周期和物理耦合。

等质量的三圈和四圈日落积分与局部Calabi--Yau四流形和五流形

The three- and four-loop equal-mass sunset integrals and local Calabi--Yau fourfolds and fivefolds

该研究将等质量的三圈和四圈日落费曼积分与局部Calabi-Yau四流形、五流形的零亏格格罗莫夫-威滕理论关联，推导了积分的显式表达式、归一化及相关性质，还证明了多覆盖变换的整性并给出端点值等结果。

Homological mirror symmetry for generalized Greene-Plesser mirrors

We prove Kontsevich's homological mirror symmetry conjecture for certain mirror pairs arising from Batyrev-Borisov's `dual reflexive Gorenstein cones' construction. In particular we prove HMS for all Greene-Plesser mirror pairs (i.e., Calabi-Yau hypersurfaces in quotients of weighted projective spaces). We also prove it for certain mirror Calabi-Yau complete intersections arising from Borisov's construction via dual nef partitions, and also for certain Calabi-Yau complete intersections which do not have a Calabi-Yau mirror, but instead are mirror to a Calabi-Yau subcategory of the derived cate…

Chiral Four-Dimensional F-Theory Compactifications With SU(5) and Multiple U(1)-Factors

We develop geometric techniques to determine the spectrum and the chiral indices of matter multiplets for four-dimensional F-theory compactifications on elliptic Calabi-Yau fourfolds with rank two Mordell-Weil group. The general elliptic fiber is the Calabi-Yau onefold in dP_2. We classify its resolved elliptic fibrations over a general base B. The study of singularities of these fibrations leads to explicit matter representations, that we determine both for U(1)xU(1) and SU(5)xU(1)xU(1) constructions. We determine for the first time certain matter curves and surfaces using techniques involvin…

Zeta functions of nondegenerate hypersurfaces in toric varieties via controlled reduction in $p$-adic cohomology

We give an interim report on some improvements and generalizations of the Abbott-Kedlaya-Roe method to compute the zeta function of a nondegenerate ample hypersurface in a projectively normal toric variety over $\mathbb{F}_p$ in linear time in $p$. These are illustrated with a number of examples including K3 surfaces, Calabi-Yau threefolds, and a cubic fourfold. The latter example is a non-special cubic fourfold appearing in the Ranestad-Voisin coplanar divisor on moduli space; this verifies that the coplanar divisor is not a Noether-Lefschetz divisor in the sense of Hassett.

锥形流形 $\times$ $\mathbb{C}$ 的规范折纸

Gauge Origami of the Conifold $\times$ $\mathbb{C}$

本文从环面Calabi-Yau四折叠的砖块模型构造规范折纸系统，通过D6/D4膜框架实现三类瞬子扇区，用JK留数定义配分函数，为一般环面四折叠上的规范折纸奠定基础。

Geometric Aspects of Mirror Symmetry (with SYZ for Rigid CY manifolds)

In this article we discuss the geometry of moduli spaces of (1) flat bundles over special Lagrangian submanifolds and (2) deformed Hermitian-Yang-Mills bundles over complex submanifolds in Calabi-Yau manifolds. These moduli spaces reflect the geometry of the Calabi-Yau itself like a mirror. Strominger, Yau and Zaslow conjecture that the mirror Calabi-Yau manifold is such a moduli space and they argue that the mirror symmetry duality is a Fourier-Mukai transformation. We review various aspects of the mirror symmetry conjecture and discuss a geometric approach in proving it. The existence of rig…

辛切割与开/闭弦 II

Symplectic cuts and open/closed strings II

本文将辛切割与开拓扑弦的联系推广到高维 Calabi-Yau 几何，证明等变盘势是特定四重体和五重体的等变周期，并由此导出环面膜的扩展 Picard-Fuchs 方程。

Hyperconifold Transitions, Mirror Symmetry, and String Theory

Multiply-connected Calabi-Yau threefolds are of particular interest for both string theorists and mathematicians. Recently it was pointed out that one of the generic degenerations of these spaces (occurring at codimension one in moduli space) is an isolated singularity which is a finite cyclic quotient of the conifold; these were called hyperconifolds. It was also shown that if the order of the quotient group is even, such singular varieties have projective crepant resolutions, which are therefore smooth Calabi-Yau manifolds. The resulting topological transitions were called hyperconifold tran…

Chern class identities from tadpole matching in type IIB and F-theory

In light of Sen's weak coupling limit of F-theory as a type IIB orientifold, the compatibility of the tadpole conditions leads to a non-trivial identity relating the Euler characteristics of an elliptically fibered Calabi-Yau fourfold and of certain related surfaces. We present the physical argument leading to the identity, and a mathematical derivation of a Chern class identity which confirms it, after taking into account singularities of the relevant loci. This identity of Chern classes holds in arbitrary dimension, and for varieties that are not necessarily Calabi-Yau. Singularities are ess…

具有 nef 反正则丛的 Kähler 流形的 Albanese 映射

Albanese map for Kähler manifolds with nef anticanonical bundle

本文研究具有 nef 反正则丛的 Kähler 流形的 Albanese 映射结构，证明一般纤维为 Calabi-Yau 时流形本身为 Calabi-Yau，并对 Cao–Höring 关于射影空间纤维情形的结果给出更拓扑的证明。

F-theory with hyperelliptic fibrations

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 …

具有$\u2124_2$对称性的Calabi-Yau流形的通量真空与模性

Flux Vacua and Modularity for $\mathbb{Z}_2$ Symmetric Calabi-Yau Manifolds

本研究在具有$\u2124_2$对称性的多参数Calabi-Yau流形的IIB紧致化中找到超对称通量真空连续族，通过扩展方法计算局部zeta函数，验证其分子二次因子对应权2模形式，支持通量模性猜想，并拓展至代数数域参数与相关模形式的研究。