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Andre Lukas

Publications and source records attributed to Andre Lukas.

At least 19 recordsLinked to original sources
arXiv2026

Calabi-Yau Metrics with Full Moduli Dependence

Recent advances in numerical and machine-learning methods have enabled highly accurate constructions of Ricci-flat metrics on compact Calabi-Yau three-folds. For phenomenological applications it is crucial to understand how these metrics vary across moduli space. In this work, we construct approximate analytic expressions for Ricci-flat Calabi-Yau metrics with explicit complex-structure and K\"ahler moduli dependence by combining machine-learned numerical data with symbolic regression. Our approach is based on an explicit Ansatz for the K\"ahler potential with moduli-dependent coefficients. Fitting this Ansatz to numerical data and applying symbolic regression allows us to reconstruct analytic formulae for these coefficients, thereby obtaining approximate Ricci-flat metrics with explicit moduli dependence. We apply the construction to a one-parameter family of bi-cubic three-folds in $\mathbb{P}^2 \times \mathbb{P}^2$, achieving percent-level agreement with the underlying numerical data.

hep-th↗
arXiv2026

CIPro Package: Complete Intersections in Products of Projective Spaces and Line Bundles

CIPro is a Mathematica package for constructing and analyzing complete intersections in products of projective spaces and line bundles over such varieties. It computes properties of complete intersections, such as Chern classes, Hilbert series, GV invariants and symmetries, as well as properties of line bundles on complete intersections, including their cohomology groups. The package also consolidates a number of data sets available in the literature into a single system, including the lists of complete intersection Calabi-Yau three- and four-folds. This short tutorial introduces the package, provides a brief discussion of some of the mathematical background underlying its computations, and gives a series of examples to illustrate its use. These tools are of utility for many computations in string compactifications, especially for Calabi-Yau geometries appearing in Heterotic and Type II constructions. Many tools apply beyond the Calabi-Yau context, including for example, almost Fano bases in F-theory.

hep-th↗
arXiv2026

Heterotic Flux Vacua with a Small Superpotential

We study heterotic Calabi--Yau compactifications with NSNS three-form flux in view of moduli stabilisation and investigate whether the value $|W_0|$ of the flux superpotential evaluated at supersymmetric minima can be small. Unlike in type IIB string theory, heterotic compactifications lack a no-scale structure, so that a non-vanishing flux superpotential generically induces a tree-level scalar potential for all moduli. Controlled moduli stabilisation therefore requires the flux superpotential to be sufficiently small in order to compete with non-perturbative effects. Working within a four-dimensional effective field theory and exploiting the special geometry of Calabi--Yau complex structure moduli spaces, we analyse the complex structure F-term equations and derive two no-go theorems: (1) supersymmetric vacua with vanishing $|W_0|$, which would lead to a vanishing tree-level scalar potential as in type IIB, occur only at singular loci in moduli space, and (2) no supersymmetric vacua with small $|W_0|$ exist in the large complex structure limit. Motivated by these results, we analyse explicit models away from the large complex structure regime using exact period expressions and identify one- and two-parameter examples in which suitable flux choices allow for values of $|W_0|$ of order unity, with some one-parameter models admitting values moderately below unity. Our findings show that small heterotic flux superpotentials are highly constrained, with the values of $|W_0|$ found in the examples studied being only marginally compatible with moduli stabilisation.

hep-th↗
arXiv2026

Sven: Singular Value Descent as a Computationally Efficient Natural Gradient Method

We introduce Sven (Singular Value dEsceNt), a new optimization algorithm for neural networks that exploits the natural decomposition of loss functions into a sum over individual data points, rather than reducing the full loss to a single scalar before computing a parameter update. Sven treats each data point's residual as a separate condition to be satisfied simultaneously, using the Moore-Penrose pseudoinverse of the loss Jacobian to find the minimum-norm parameter update that best satisfies all conditions at once. In practice, this pseudoinverse is approximated via a truncated singular value decomposition, retaining only the $k$ most significant directions and incurring a computational overhead of only a factor of $k$ relative to stochastic gradient descent. This is in comparison to traditional natural gradient methods, which scale as the square of the number of parameters. We show that Sven can be understood as a natural gradient method generalized to the over-parametrized regime, recovering natural gradient descent in the under-parametrized limit. On regression tasks, Sven significantly outperforms standard first-order methods including Adam, converging faster and to a lower final loss, while remaining competitive with LBFGS at a fraction of the wall-time cost. We discuss the primary challenge to scaling, namely memory overhead, and propose mitigation strategies. Beyond standard machine learning benchmarks, we anticipate that Sven will find natural application in scientific computing settings where custom loss functions decompose into several conditions.

cs.LG↗
arXiv2026

Calabi-Yau Metrics with K\"ahler Moduli Dependence

We present a method to construct approximate analytic expressions for Ricci-flat K\"ahler metrics on Calabi-Yau threefolds with explicit dependence on the K\"ahler moduli. Our strategy combines numerical data obtained from machine learning with an explicit analytic Ansatz for the K\"ahler potential and symbolic regression methods. Specifically, we use neural networks to learn the K\"ahler potential at selected points in K\"ahler moduli space, fit this data to analytic expressions with K\"ahler moduli-dependent parameters, and determine an analytic form of these coefficients as functions of the K\"ahler moduli using symbolic regression. In this way, we reconstruct closed-form approximations to the Ricci-flat metric that retain explicit K\"ahler-moduli dependence. We apply this method to two Calabi-Yau threefolds with $h^{1,1}=2$, namely a bicubic hypersurface in $\mathbb{P}^2 \times \mathbb{P}^2$ and a bi-degree $(2,4)$ hypersurface in $\mathbb{P}^1 \times \mathbb{P}^3$, both of which admit nontrivial discrete symmetry groups that simplify the structure of the metric. In both cases, the resulting analytic expressions reproduce the numerically learned K\"ahler potentials with percent-level accuracy and yield a Ricci-flatness measure that remains sufficiently small across the sampled region. Our results represent a concrete bridge between purely numerical results for Calabi-Yau metrics and analytic constructions, opening the door to a systematic study of their dependence on K\"ahler moduli.

hep-th↗
arXiv2025

Reproducing Standard Model Fermion Masses and Mixing in String Theory: A Heterotic Line Bundle Study

Deriving the Yukawa couplings and the resulting fermion masses and mixing angles of the Standard Model (SM) from a more fundamental theory remains one of the central outstanding problems in theoretical high-energy physics. It has long been recognised that string theory provides a framework within which this question can, at least in principle, be addressed. While substantial progress has been made in studying flavour physics in string compactifications over the past few decades, a concrete string construction that reproduces the full set of observed SM flavour parameters remains unknown. Here, we take a significant step in this direction by identifying two explicit $E_8 \times E_8$ heterotic string models compactified on a Calabi-Yau threefold with abelian, holomorphic, and poly-stable vector bundles with an (MS)SM spectrum. Subject to reasonable assumptions about the moduli, we show that these models reproduce the correct values of the quark and charged lepton masses, as well as the quark mixing parameters, at some point in their moduli spaces. The resulting four-dimensional theories are $\mathcal{N}=1$ supersymmetric, contain no exotic fields and realise a $\mu$-term suppressed to the electroweak scale. While the issues of moduli stabilisation and supersymmetry breaking are not addressed here, our main result constitutes a proof of principle: there exist choices of topology and moduli within heterotic string compactifications which allow for an MSSM spectrum with the correct flavour parameters.

hep-th↗
arXiv2025

Approximate Ricci-flat Metrics for Calabi-Yau Manifolds

We outline a method to determine analytic K\"ahler potentials with associated approximately Ricci-flat K\"ahler metrics on Calabi-Yau manifolds. Key ingredients are numerically calculating Ricci-flat K\"ahler potentials via machine learning techniques and fitting the numerical results to Donaldson's Ansatz. We apply this method to the Dwork family of quintic hypersurfaces in $\mathbb{P}^4$ and an analogous one-parameter family of bi-cubic CY hypersurfaces in $\mathbb{P}^2\times\mathbb{P}^2$. In each case, a relatively simple analytic expression is obtained for the approximately Ricci-flat K\"ahler potentials, including the explicit dependence on the complex structure parameter. We find that these K\"ahler potentials only depend on the modulus of the complex structure parameter.

hep-th↗
arXiv2024

Fermion Masses and Mixing in String-Inspired Models

We study a class of supersymmetric Froggatt-Nielsen (FN) models with multiple U(1) symmetries and Standard Model (SM) singlets inspired by heterotic string compactifications on Calabi-Yau threefolds. The string-theoretic origin imposes a particular charge pattern on the SM fields and FN singlets, dividing the latter into perturbative and non-perturbative types. Employing systematic and heuristic search strategies, such as genetic algorithms, we identify charge assignments and singlet VEVs that replicate the observed mass and mixing hierarchies in the quark sector, and subsequently refine the Yukawa matrix coefficients to accurately match the observed values for the Higgs VEV, the quark and charged lepton masses and the CKM matrix. This bottom-up approach complements top-down string constructions and our results demonstrate that string FN models possess a sufficiently rich structure to account for flavour physics. On the other hand, the limited number of distinct viable charge patterns identified here indicates that flavour physics imposes tight constraints on string theory models, adding new constraints on particle spectra that are essential for achieving a realistic phenomenology.

hep-th↗
arXiv2024

Computation of Quark Masses from String Theory

We present a numerical computation, based on neural network techniques, of the physical Yukawa couplings in a heterotic string theory compactification on a smooth Calabi-Yau threefold with non-standard embedding. The model belongs to a large class of heterotic line bundle models that have previously been identified and whose low-energy spectrum precisely matches that of the MSSM plus fields uncharged under the Standard Model group. The relevant quantities for the calculation, that is, the Ricci-flat Calabi-Yau metric, the Hermitian Yang-Mills bundle metrics and the harmonic bundle-valued forms, are all computed by training suitable neural networks. For illustration, we consider a one-parameter family in complex structure moduli space. The computation at each point along this locus takes about half a day on a single twelve-core CPU. Our results for the Yukawa couplings are estimated to be within 10% of the expected analytic result. We find that the effect of the matter field normalisation can be significant and can contribute towards generating hierarchical couplings. We also demonstrate that a zeroth order, semi-analytic calculation, based on the Fubini-Study metric and its counterparts for the bundle metric and the bundle-valued forms, leads to roughly correct results, about 25% away from the numerical ones. The method can be applied to other heterotic line bundle models and generalised to other constructions, including to F-theory models.

hep-th↗
arXiv2023

Enumerating Calabi-Yau Manifolds: Placing bounds on the number of diffeomorphism classes in the Kreuzer-Skarke list

The diffeomorphism class of simply-connected smooth Calabi-Yau threefolds with torsion-free cohomology is determined via certain basic topological invariants: the Hodge numbers, the triple intersection form, and the second Chern class. In the present paper, we shed some light on this classification by placing bounds on the number of diffeomorphism classes present in the set of smooth Calabi-Yau threefolds constructed from the Kreuzer-Skarke list of reflexive polytopes up to Picard number six. The main difficulty arises from the comparison of triple intersection numbers and divisor integrals of the second Chern class up to basis transformations. By using certain basis-independent invariants, some of which appear here for the first time, we are able to place lower bounds on the number of classes. Upper bounds are obtained by explicitly identifying basis transformations, using constraints related to the index of line bundles. Extrapolating our results, we conjecture that the favourable entries of the Kreuzer-Skarke list of reflexive polytopes leads to some $10^{400}$ diffeomorphically distinct Calabi-Yau threefolds.

hep-th↗
arXiv2023

New Calabi-Yau Manifolds from Genetic Algorithms

Calabi-Yau manifolds can be obtained as hypersurfaces in toric varieties built from reflexive polytopes. We generate reflexive polytopes in various dimensions using a genetic algorithm. As a proof of principle, we demonstrate that our algorithm reproduces the full set of reflexive polytopes in two and three dimensions, and in four dimensions with a small number of vertices and points. Motivated by this result, we construct five-dimensional reflexive polytopes with the lowest number of vertices and points. By calculating the normal form of the polytopes, we establish that many of these are not in existing datasets and therefore give rise to new Calabi-Yau four-folds. In some instances, the Hodge numbers we compute are new as well.

hep-th↗
arXiv2023

Decoding Nature with Nature's Tools: Heterotic Line Bundle Models of Particle Physics with Genetic Algorithms and Quantum Annealing

The string theory landscape may include a multitude of ultraviolet embeddings of the Standard Model, but identifying these has proven difficult due to the enormous number of available string compactifications. Genetic Algorithms (GAs) represent a powerful class of discrete optimisation techniques that can efficiently deal with the immensity of the string landscape, especially when enhanced with input from quantum annealers. In this letter we focus on geometric compactifications of the $E_8\times E_8$ heterotic string theory compactified on smooth Calabi-Yau threefolds with Abelian bundles. We make use of analytic formulae for bundle-valued cohomology to impose the entire range of spectrum requirements, something that has not been possible so far. For manifolds with a relatively low number of Kahler parameters we compare the GA search results with results from previous systematic scans, showing that GAs can find nearly all the viable solutions while visiting only a tiny fraction of the solution space. Moreover, we carry out GA searches on manifolds with a larger numbers of Kahler parameters where systematic searches are not feasible.

hep-th↗
arXiv2023

Cosmic Inflation and Genetic Algorithms

Large classes of standard single-field slow-roll inflationary models consistent with the required number of e-folds, the current bounds on the spectral index of scalar perturbations, the tensor-to-scalar ratio, and the scale of inflation can be efficiently constructed using genetic algorithms. The setup is modular and can be easily adapted to include further phenomenological constraints. A semi-comprehensive search for sextic polynomial potentials results in roughly O(300,000) viable models for inflation. The analysis of this dataset reveals a preference for models with a tensor-to-scalar ratio in the range 0.0001 < r < 0.0004. We also consider potentials that involve cosine and exponential terms. In the last part we explore more complex methods of search relying on reinforcement learning and genetic programming. While reinforcement learning proves more difficult to use in this context, the genetic programming approach has the potential to uncover a multitude of viable inflationary models with new functional forms.

hep-th↗
arXiv2023

Symmetries of Calabi-Yau Prepotentials with Isomorphic Flops

Calabi-Yau threefolds with infinitely many flops to isomorphic manifolds have an extended Kahler cone made up from an infinite number of individual Kahler cones. These cones are related by reflection symmetries across flop walls. We study the implications of this cone structure for mirror symmetry, by considering the instanton part of the prepotential in Calabi-Yau threefolds. We show that such isomorphic flops across facets of the Kahler cone boundary give rise to symmetry groups isomorphic to Coxeter groups. In the dual Mori cone, non-flopping curve classes that are identified under these groups have the same Gopakumar-Vafa invariants. This leads to instanton prepotentials invariant under Coxeter groups, which we make manifest by introducing appropriate invariant functions. For some cases, these functions can be expressed in terms of theta functions whose appearance can be linked to an elliptic fibration structure of the Calabi-Yau manifold.

hep-th↗
arXiv2022

Spatially Homogeneous Universes with Late-Time Anisotropy

The cosmological principle asserts that on sufficiently large scales the Universe is homogeneous and isotropic on spatial slices. To deviate from this principle requires a departure from the FLRW ansatz. In this paper we analyze the cosmological evolution of two spatially homogeneous but anisotropic universes, namely the spatially closed Kantowski-Sachs Universe and the open axisymmetric Bianchi type III Universe. These models are characterized by two scale factors and we study their evolution in universes with radiation, matter and a cosmological constant. In all cases, the two scale factors evolve differently and this anisotropy leads to a lensing effect in the propagation of light. We derive explicit formulae for computing redshifts, angular diameter distances and luminosity distances and discuss the predictions of these models in relation to observations for type Ia supernovae and the CMB. We comment on the possibility of explaining the observed luminosity distance plot for type Ia supernovae within the context of cosmologies featuring late-time anisotropy and a vanishing cosmological constant.

astro-ph.CO↗
arXiv2022

Numerical Metrics for Complete Intersection and Kreuzer-Skarke Calabi-Yau Manifolds

We introduce neural networks to compute numerical Ricci-flat CY metrics for complete intersection and Kreuzer-Skarke Calabi-Yau manifolds at any point in Kähler and complex structure moduli space, and introduce the package cymetric which provides computation realizations of these techniques. In particular, we develop and computationally realize methods for point-sampling on these manifolds. The training for the neural networks is carried out subject to a custom loss function. The Kähler class is fixed by adding to the loss a component which enforces the slopes of certain line bundles to match with topological computations. Our methods are applied to various manifolds, including the quintic manifold, the bi-cubic manifold and a Kreuzer-Skarke manifold with Picard number two. We show that volumes and line bundle slopes can be reliably computed from the resulting Ricci-flat metrics. We also apply our results to compute an approximate Hermitian-Yang-Mills connection on a specific line bundle on the bi-cubic.

hep-th↗
arXiv2022

Recent Developments in Line Bundle Cohomology and Applications to String Phenomenology

Vector bundle cohomology represents a key ingredient for string phenomenology, being associated with the massless spectrum arising in string compactifications on smooth compact manifolds. Although standard algorithmic techniques exist for performing cohomology calculations, they are laborious and ill-suited for scanning over large sets of string compactifications to find those most relevant to particle physics. In this article we review some recent progress in deriving closed-form expressions for line bundle cohomology and discuss some applications to string phenomenology.

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