Machine learning Calabi–Yau three-folds, four-folds, and five-folds

We study the geometry of $3$-codimensional smooth subvarieties of the complex projective space. In particular, we classify all quasi-Buchsbaum Calabi--Yau threefolds in projective $6$-space. Moreover, we prove that this classification includes all Calabi--Yau threefolds contained in a possibly singular 5-dimensional quadric as well as all Calabi--Yau threefolds of degree at most $14$ in $\mathbb{P}^6$.

The aim of this paper is to report on recent progress in understanding mirror symmetry for some non-complete intersection Calabi-Yau threefolds.We first construct four new smooth non-complete intersection Calabi-Yau threefolds with h 1,1 = 1, whose existence was previously conjectured by C. van Enckevort and D. van Straten in [19].We then compute the period integrals of candidate mirror families of F. Tonoli's degree 13 Calabi-Yau threefold and three of the new Calabi-Yau threefolds.The Picard-Fuchs equations coincide with the expected Calabi-Yau equations listed in [18,19].Some of the mirror families turn out to have two maximally unipotent monodromy points.

In this paper we discuss four methods of proving of Calabi--Yau threefolds with $h^{12}=1$: existence of elliptic ruled surfaces inside (Hulek-Verrill), correspondence with a product of an elliptic curve and a K3 surface (Livn\'e-Yui), correspondence with a (modular) rigid Calabi-Yau threefold, and existence of an involution splitting the fourdimensional representation into twodimensional subrepresentations. We apply these methods to prove of 17 out of 18 double octic Calabi-Yau threefolds for which numerical evidence of modularity was found in the second Author's thesis. We observe that holds for those elements in a pencil having some additional geometric properties. In the proofs we use representations of the considered Calabi-Yau threefolds as a Kummer fibration associated to a fiber product of rational elliptic fibrations.

In this article we present a pheno-inspired classification for the divisor topologies of the favorable Calabi Yau (CY) threefolds with 1 ≤ h1,1(CY) ≤ 5 arising from the four-dimensional reflexive polytopes of the Kreuzer-Skarke database. Based on some empirical observations we conjecture that the topologies of the so-called coordinate divisors can be classified into two categories: (i). χh (D) ≥ 1 with Hodge numbers given by {h0,0 = 1, h1,0 = 0, h2,0 = χh (D) − 1, h1,1 = χ(D) − 2χh (D)} and (ii). χh (D) ≤ 1 with Hodge numbers given by {h0,0 = 1, h1,0 = 1 −χh (D), h2,0 = 0, h1,1 = χ(D) + 2 − 4χh (D)}, where χh (D) denotes the Arithmetic genus while χ(D) denotes the Euler characteristic of the divisor D. We present the Hodge numbers of around 140000 coordinate divisors corresponding to all the CY threefolds with 1 ≤ h1,1(CY) ≤ 5 which corresponds to a total of nearly 16000 distinct CY geometries. Subsequently we argue that our conjecture can help in “bypassing” the need of cohomCalg for computing Hodge numbers of coordinate divisors, and hence can be significantly useful for studying the divisor topologies of CY threefolds with higher h1,1 for which cohomCalg gets too slow and sometimes even breaks as well. We also demonstrate how these scanning results can be directly used for phenomenological model building, e.g. in estimating the D3-brane tadpole charge (under reflection involutions) which is a central ingredient for constructing explicit global models due to several different reasons/interests such as the de-Sitter uplifting through anti-D3 brane and (flat) flux vacua searches.

In this article we study combinatorial degenerations of minimal surfaces of\nKodaira dimension 0 over local fields, and in particular show that the `type'\nof the degeneration can be read off from the monodromy operator acting on a\nsuitable cohomology group. This can be viewed as an arithmetic analogue of\nresults of Persson and Kulikov on degenerations of complex surfaces, and\nextends various particular cases studied by Matsumoto, Liedtke/Matsumoto and\nHern\\'andez-Mada. We also study `maximally unipotent' degenerations of\nCalabi--Yau threefolds, following Koll\\'ar/Xu, showing in this case that the\ndual intersection graph is a 3-sphere.\n

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 $SU(3)_C\times SU(2)_L\times U(1)_Y$.

Expressions for the number of moduli of arbitrary SU(n) vector bundles constructed via Fourier-Mukai transforms of spectral data over Calabi- Yau threefolds are derived and presented. This is done within the context of simply connected, elliptic Calabi-Yau threefolds with base Fr, but the methods have wider applicability. The condition for a vector bundle to possess the minimal number of moduli for fixed r and n is discussed and an explicit formula for the minimal number of moduli is presented. In addition, transition moduli for small instanton phase transitions involving non-positive spectral covers are defined, enumerated and given a geometrical interpretation.

Recently Calabi–Yau threefolds have been studied intensively by physicists and mathematicians. They are used as physical models of superstring theory [Y] and they are one of the building blocks in the classification of complex threefolds [KMM]. These are three dimensional analogues of K3 surfaces. However, there is a fundamental difference as is to be expected. For K3 surfaces, the moduli space N of K3 surfaces is irreducible of dimension 20, inside which a countable number of families Ng with g [ges ] 2 of algebraic K3 surfaces of dimension 19 lie as a dense subset. More explicitly, an element in Ng is (S, H), where S is a K3 surface and H is a primitive ample divisor on S with H2 = 2g − 2. For a generic (S, H), Pic (S) is generated by H, so that the rank of the Picard group of S is 1. A generic surface S in N is not algebraic and it has Pic (S) = 0, but dim N = h1(S, TS) = 20 [BPV]. It is quite an interesting problem whether or not the moduli space M of all Calabi–Yau threefolds is irreducible in some sense [R]. A Calabi–Yau threefold is algebraic if and only if it is Kaehler, while every non-algebraic K3 surface is still Kaehler. Inspired by the K3 case, we define Mh,d to be {(X, H)[mid ]H3 = h, c2(X) · H = d}, where H is a primitive ample divisor on a smooth Calabi–Yau threefold X. There are two parameters h, d for algebraic Calabi–Yau threefolds, while there is only one parameter g for algebraic K3 surfaces. (Note that c2(S) = 24 for every K3 surface.) We know that Ng is of dimension 19 for every g and is irreducible but we do not know the dimension of Mh,d and whether or not Mh,d is irreducible. In fact, the dimension of Mh,d = h1(X, TX), where (X, H) ∈ Mh,d. Furthermore, it is well known that χ(X) = 2 (rank of Pic (X) − h1(X, TX)), where χ(X) is the topological Euler characteristic of X. Calabi–Yau threefolds with Picard rank one are primitive [G] and play an important role in the moduli spaces of all Calabi–Yau threefolds. In this paper we give a bound on c3 of Calabi–Yau threefolds with Picard rank 1.

In this paper, we apply machine learning to the problem of finding numerical Calabi–Yau metrics. We extend previous work on learning approximate Ricci-flat metrics calculated using Donaldson’s algorithm to the much more accurate “optimal” metrics of Headrick and Nassar. We show that machine learning is able to predict the Kähler potential of a Calabi–Yau metric having seen only a small sample of training data.

If X is a nonsingular curve in a Calabi--Yau threefold Y whose normal bundle N_{X/Y} is a generic semistable bundle, are the local Gromov-Witten invariants of X well defined? For X of genus two or higher, the issues are subtle. We will formulate a precise line of inquiry and present some results, some positive and some negative.

ON ALGEBRAIC FIBER SPACE STRUCTURES ON A CALABI-YAU 3-FOLD

The moduli spaces of Calabi–Yau threefolds are conjectured to be connected by the combination of birational contraction maps and flat deformations. In this context, it is important to calculate dim Def(X) from dim Def(˜X) in terms of certain geometric information of f, when we are given a birational morphism f:˜X→X from a smooth Calabi–Yau threefold ˜X to a singular Calabi–Yau threefold X. A typical case of this problem is a conjecture of Morrison-Seiberg which originally came from physics. In this paper we give a mathematical proof to this conjecture. Moreover, by using output of this conjecture, we prove that certain Calabi–Yau threefolds with nonisolated singularities have flat deformations to smooth Calabi–Yau threefolds. We shall use invariants of singularities closely related to Du Bois's work to calculate dim Def(X) from dim Def(˜X).

The moduli stack of Mixed Spin P-fields (MSP) provides an effective algorithm to evaluate all genus Gromov–Witten (GW) invariants of the quintic Calabi–Yau (CY) three-folds. This paper is to apply the algorithm to the genus-one case. We use the localization formula, the proposed algorithm in [ 10, 11], and Zinger’s packaging technique to compute the genus-one GW invariants of the quintic CY three-folds. Our approach to the formula suggests a correspondence between each type of MSP graphs with each physics’ phase: CY, Landau–Ginzburg, or conifold point. In this process, new differential relations among Givental’s I-functions are also discovered.

Motivated by a conjecture [23, 27] on quantum cohomology, Li and Ruan studied the transformation of Gromov–Witten (GW) invariants of projective Calabi–Yau (CY) threefolds under flops using symplectic approach [16]. The algebra-geometric approach was pursued in [18]. The same problem for Donaldson–Thomas invariants was studied in [9], and this may be related since there is a conjecture that Donaldson–Thomas invariants and GW invariants are related at the level of generating functions [21, 22]. In this paper, we study the behavior of GW invariants of toric Calabi–Yau (TCY) threefolds (which are noncompact) under a flop based on the method of the topological vertex. It is a formalism which expresses the partition functions of GW invariants of TCY threefolds in terms of symmetric functions [1]. (In this paper, the partition function of GW invariants means the exponential of the generating function.) Although its original argument was based on the duality to the Chern– Simons theory, a mathematical theory including a definition of GW invariants for

In this article, we summarize combinatorial description of complete intersection Calabi-Yau threefolds in Hibi toric varieties. Such Calabi-Yau threefolds have at worst conifold singularities, and are often smoothable to non-singular Calabi-Yau threefolds. We focus on such non-singular Calabi-Yau threefolds of Picard number one, and illustrate the calculation of topological invariants, using new motivating examples.