Calabi-Yau Models Research Articles

Krylov complexity in Calabi–Yau quantum mechanics

Recently, a novel measure for the complexity of operator growth is proposed based on Lanczos algorithm and Krylov recursion method. We study this Krylov complexity in quantum mechanical systems derived from some well-known local toric Calabi–Yau geometries, as well as some nonrelativistic models. We find that for the Calabi–Yau models, the Lanczos coefficients grow slower than linearly for small [Formula: see text]’s, consistent with the behavior of integrable models. On the other hand, for the nonrelativistic models, the Lanczos coefficients initially grow linearly for small [Formula: see text]’s, then reach a plateau. Although this looks like the behavior of a chaotic system, it is mostly likely due to saddle-dominated scrambling effects instead, as argued in the literature. In our cases, the slopes of linearly growing Lanczos coefficients almost saturate a bound by the temperature. During our study, we also provide an alternative general derivation of the bound for the slope.

LG/CY Correspondence Between $tt^∗$ bGeometries

The concept of $tt^*$ geometric structure was introduced by physicists (see \cite{CV1, BCOV} and references therein) , and then studied firstly in mathematics by C. Hertling \cite{Het1}. It is believed that the $tt^*$ geometric structure contains the whole genus $0$ information of a two dimensional topological field theory. In this paper, we propose the LG/CY correspondence conjecture for $tt^*$ geometry and obtain the following result. Let $f\in\mathbb{C}[z_0, \dots, z_{n+2}]$ be a nondegenerate homogeneous polynomialof degree $n+2$, then it defines a Calabi-Yau model represented by a Calabi-Yau hypersurface $X_f$ in $\mathbb{CP}^{n+1}$ or a Landau-Ginzburg model represented by a hypersurface singularity $(\mathbb{C}^{n+2}, f)$, both can be written as a $tt^*$ structure. We proved that there exists a $tt^*$ substructure on Landau-Ginzburg side, which should correspond to the $tt^*$ structure from variation of Hodge structures in Calabi-Yau side. We build the isomorphism of almost all structures in $tt^*$ geometries between these two models except the isomorphism between real structures.

A Generalized Construction of Calabi-Yau Models and Mirror Symmetry

We extend the construction of Calabi-Yau manifolds to hypersurfaces in non-Fano toric varieties, requiring the use of certain Laurent defining polynomials, and explore the phases of the corresponding gauged linear sigma models. The associated non-reflexive and non-convex polytopes provide a generalization of Batyrev’s original work, allowing us to construct novel pairs of mirror models. We showcase our proposal for this generalization by examining Calabi-Yau hypersurfaces in Hirzebruch n-folds, focusing on n=3,4 sequences, and outline the more general class of so-defined geometries.

Yukawa unification in heterotic string theory

We analyze Yukawa unification in the the context of $E_8\times E_8$ heterotic Calabi-Yau models which rely on breaking to a GUT theory via a non-flat gauge bundle and subsequent Wilson line breaking to the standard model. Our focus is on underlying GUT theories with gauge group $SU(5)$ or $SO(10)$. We provide a detailed analysis of the fact that, in contrast to traditional field theory GUTs, the underlying GUT symmetry of these models does not enforce Yukawa unification. Using this formalism, we present various scenarios where Yukawa unification can occur as a consequence of additional symmetries. These additional symmetries arise naturally in some heterotic constructions and we present an explicit heterotic line bundle model which realizes one of these scenarios.

Some ball quotients with a Calabi–Yau model

Recently we determined explicitly a Picard modular variety of general type. On the regular locus of this variety there are holomorphic three forms which have been constructed as Borcherds products. Resolutions of quotients of this variety, such that the zero divisors are in the branch locus, are candidates for Calabi–Yau manifolds. Here we treat one distinguished example for this. In fact we shall recover a known variety given by the equations \[ X_0X_1X_2=X_3X_4X_5, X_0^3+X_1^3+X_2^3=X_3^3+X_4^3+X_5^3. \] as a Picard modular variety. This variety has a projective small resolution which is a rigid Calabi–Yau manifold ($h^{12}=0$) with Euler number $72$.

Hypercharge flux in heterotic compactifications

We study heterotic Calabi-Yau models with hypercharge flux breaking, where the visible E8 gauge group is directly broken to the standard model group by a non-flat gauge bundle, rather than by a two-step process involving an intermediate grand unified theory and a Wilson line. It is shown that the required alternative E8 embeddings of hypercharge, normalized as required for gauge unification, can be found and we classify these possibilities. However, for all but one of these embeddings we prove a general no-go theorem which asserts that no suitable geometry and vector bundle leading to a standard model spectrum can be found. Intuitively, this happens due to the large number of index conditions which have to be imposed in order to obtain a correct physical spectrum in the absence of an underlying grand unified theory.

Topological strings and quantum spectral problems

We consider certain quantum spectral problems appearing in the study of local Calabi-Yau geometries. The quantum spectrum can be computed by the Bohr-Sommerfeld quantization condition for a period integral. For the case of small Planck constant, the periods are computed perturbatively by deformation of the Omega background parameters in the Nekrasov-Shatashvili limit. We compare the calculations with the results from the standard perturbation theory for the quantum Hamiltonian. There have been proposals in the literature for the non-perturbative contributions based on singularity cancellation with the perturbative contributions. We compute the quantum spectrum numerically with some high precisions for many cases of Planck constant. We find that there are also some higher order non-singular non-perturbative contributions, which are not captured by the singularity cancellation mechanism. We fix the first few orders formulas of such corrections for some well known local Calabi-Yau models.

Heterotic Calabi-Yau compactifications with flux

Compactifications of the heterotic string with NS flux normally require non Calabi-Yau internal spaces which are complex but no longer K\"ahler. We point out that this conclusion rests on the assumption of a maximally symmetric four-dimensional space-time and can be avoided if this assumption is relaxed. Specifically, it is shown that an internal Calabi-Yau manifold is consistent with the presence of NS flux provided four-dimensional space-time is taken to be a domain wall. These Calabi-Yau domain wall solutions can still be associated with a covariant four-dimensional N=1 supergravity. In this four-dimensional context, the domain wall arises as the "simplest" solution to the effective supergravity due to the presence of a flux potential with a runaway direction. Our main message is that NS flux is a legitimate ingredient for moduli stabilization in heterotic Calabi-Yau models. Ultimately, the success of such models depends on the ability to stabilize the runaway direction and thereby "lift" the domain wall to a maximally supersymmetric vacuum.

Some Siegel Threefolds with a Calabi-Yau Model II

In a previous paper, we described some Siegel modular threefolds which ad- mit a Calabi-Yau model. Using a different method we give in this paper an enlarged list of such varieties. Basic for this method is a paper of van Geemen and Nygaard. They study a variety X that is the complete intersection of four quadrics in P 7 (C). This is biholomorphic equivalent to the Satake compactification of H2=Γ ' for a certain subgroup Γ ' Sp(2;Z) and it will be the starting point of our investigation. It has been pointed out that a (projective) small resolution of this variety is a rigid Calabi-Yau manifold ˜ X. Then we will consider the action of quotients of modular groups on X and study possible resolutions that admits a Calabi-Yau model in the category of complex spaces.

THE GEOMETRY AND ARITHMETIC OF A CALABI–YAU SIEGEL THREEFOLD

In this paper we treat in details a Siegel modular variety [Formula: see text] that has a Calabi–Yau model, [Formula: see text]. We shall describe the structure of the ring of modular forms and its geometry. We shall illustrate two different methods of producing the Hodge numbers. The first uses the definition of [Formula: see text] as the quotient of another known Calabi–Yau variety [Formula: see text]. In this case we will get the Hodge numbers considering the action of the group on a crepant resolution [Formula: see text] of [Formula: see text]. The second, purely algebraic geometric, uses the equations derived from the ring of modular forms and is based on determining explicitly the Calabi–Yau model [Formula: see text] and computing the Picard group and the Euler characteristic.

Sequestering by global symmetries in Calabi–Yau string models

We study the possibility of realizing an effective sequestering between visible and hidden sectors in generic heterotic string models, generalizing previous work on orbifold constructions to smooth Calabi–Yau compactifications. In these theories, genuine sequestering is spoiled by interactions mixing chiral multiplets of the two sectors in the effective Kähler potential. These effective interactions however have a specific current–current-like structure and can be interpreted from an M-theory viewpoint as coming from the exchange of heavy vector multiplets. One may then attempt to inhibit the emergence of generic soft scalar masses in the visible sector by postulating a suitable global symmetry in the dynamics of the hidden sector. This mechanism is however not straightforward to implement, because the structure of the effective contact terms and the possible global symmetries is a priori model-dependent. To assess whether there is any robust and generic option, we study the full dependence of the Kähler potential on the moduli and the matter fields. This is well known for orbifold models, where it always leads to a symmetric scalar manifold, but much less understood for Calabi–Yau models, where it generically leads to a non-symmetric scalar manifold. We then examine the possibility of an effective sequestering by global symmetries, and argue that whereas for orbifold models this can be put at work rather naturally, for Calabi–Yau models it can only be implemented in rather peculiar circumstances.

On Siegel three-folds with a projective Calabi–Yau model

In the papers [FS1], [FS2] we described some Siegel modular threefolds which admit a weak Calabi–Yau model.*) Not all of them admit a projective model. In fact, Bert van Geemen, in a private communication, pointed out a significative example which cannot admit a projective model. His comment was a starting motivation for this paper. We mention that a weak Calabi–Yau threefold is projective if, and only if, it admits a Kaehler metric. The purpose of this paper is to exhibit criteria for the projectivity, to treat several examples, and to compute their Hodge numbers. Some of these Calabi–Yau manifolds seem to be new. For example, we obtain a rigid Calabi–Yau manifold with Euler number 4. Basic for our examples is a certain complete intersection X of four quadrics introduced the paper [GN] of van Geemen and Nygaard:

Some Siegel threefolds with a Calabi-Yau model

Some Siegel threefolds with a Calabi-Yau model

Heterotic models from vector bundles on toric Calabi-Yau manifolds

We systematically approach the construction of heterotic E_8 X E_8 Calabi-Yau models, based on compact Calabi-Yau three-folds arising from toric geometry and vector bundles on these manifolds. We focus on a simple class of 101 such three-folds with smooth ambient spaces, on which we perform an exhaustive scan and find all positive monad bundles with SU(N), N=3,4,5 structure groups, subject to the heterotic anomaly cancellation constraint. We find that anomaly-free positive monads exist on only 11 of these toric three-folds with a total number of bundles of about 2000. Only 21 of these models, all of them on three-folds realizable as hypersurfaces in products of projective spaces, allow for three families of quarks and leptons. We also perform a preliminary scan over the much larger class of semi-positive monads which leads to about 44000 bundles with 280 of them satisfying the three-family constraint. These 280 models provide a starting point for heterotic model building based on toric three-folds.

Moduli webs and superpotentials for five-branes

We investigate the one-parameter Calabi-Yau models and identify families of D5-branes which are associated to lines embedded in these manifolds. The moduli spaces are given by sets of Riemann curves, which form a web whose intersection points are described by permutation branes. We arrive at a geometric interpretation for bulk-boundary correlators as holomorphic differentials on the moduli space and use this to compute effective open-closed superpotentials to all orders in the open string couplings. The fixed points of D5-brane moduli under bulk deformations are determined.

Enriques and octonionic magic supergravity models

We reconsider the Enriques Calabi Yau (FHSV) model and its string derivation and argue that the Octonionic magic supergravity theory admits a string interpretation closely related to the Enriques model. The uplift to D = 6 of the Octonionic magic model has 16 abelian vectors related to the rank of Type I and Heterotic strings.

Moduli Potentials in Type IIA Compactifications with RR and NS Flux

We describe a simple class of type-IIA string compactifications on Calabi-Yau manifolds where background fluxes generate a potential for the complex structure moduli, the dilaton, and the Kähler moduli. This class of models corresponds to gauged = 2 supergravities, and the potential is completely determined by a choice of gauging and by data of the = 2 Calabi-Yau model — the prepotential for vector multiplets and the quaternionic metric on the hypermultiplet moduli space. Using mirror symmetry, one can determine many (though not all) of the quantum corrections which are relevant in these models.

The modularity of the Barth-Nieto quintic and its relatives

The moduli space of O1; 3U-polarized abelian surfaces with full level-2 structure is birational to a double cover of the Barth-Nieto quintic. Barth and Nieto have shown that these varieties have Calabi-Yau models Z and Y, respectively. In this paper we apply the Weil conjectures to show that Y and Z are rigid and we prove that the L-function of their common third etale cohomology group is modular, as predicted by a conjecture of Fontaine and Mazur. The corresponding modular form is the unique normalized cusp form of weight 4 for the group G1O6U. By Tate's conjecture, this should imply that Y, the fibred square of the universal elliptic curve S1O6U, and Verrill's rigid Calabi-Yau ZA3 , which all have the same L-function, are in correspondence over Q. We show that this is indeed the case by giving explicit maps.

D-branes and bundles on elliptic fibrations

We study the D-brane spectrum on a two-parameter Calabi–Yau model. The analysis is based on different tools in distinct regions of the moduli space: wrapped brane configurations on elliptic fibrations near the large radius limit, and SCFT boundary states at the Gepner point. We develop an explicit correspondence between these two classes of objects, suggesting that boundary states are natural quantum generalizations of bundles. We also find interesting D-brane dynamics in deep stringy regimes. The most striking example is, perhaps, that non-supersymmetric D6–D0 and D4–D2 large radius configurations become stable BPS states at the Gepner point.

The modular form of the Barth-Nieto quintic

Barth and Nieto have found a remarkable quintic threefold which parametrizes Heisenberg invariant Kummer surfaces which belong to abelian surfaces with a (1,3)-polarization and a lecel 2 structure. A double cover of this quintic, which is also a Calabi-Yau variety, is birationally equivalent to the moduli space {\cal A}_3(2) of abelian surfaces with a (1,3)-polarization and a level 2 structure. As a consequence the corresponding paramodular group \Gamma_3(2) has a unique cusp form of weight 3. In this paper we find this cusp form which is \Delta_1^3. The form \Delta_1 is a remarkable weight 1 cusp form with a character with respect to the paramodular group \Gamma_3. It has several interesting properties. One is that it admits an infinite product representation, the other is that it vanishes of order 1 along the diagonal in Siegel space. In fact \Delta_1 is an element of a short series of modular forms with this last property. Using the fact that \Delta_1 is a weight 3 cusp form with respect to the group \Gamma_3(2) we give an independent construction of a smooth projective Calabi-Yau model of the moduli space {\cal A}_3(2).