arXiv subjects

Viktor Mirjanić

Publications and source records attributed to Viktor Mirjanić.

4 recordsLinked to original sources

arXiv2026

Balanced Metrics Know About SYZ

Numerical Ricci-flat metrics on Calabi-Yau manifolds are becoming increasingly accurate. However, they often lack the interpretability required to extract theoretical insights. In this paper, we introduce a novel variant of Donaldson's algorithm based on the Moore-Penrose pseudo-inverse that operates on the global sections of the ambient space rather than the manifold itself. This approach allows us to use the canonical monomial basis to compute interpretable balanced metrics even at large degrees $k$. Applying our ambient algorithm to multiple families, including the Dwork family and complete intersection Calabi-Yau manifolds, we discover that the metric parameters obey novel power laws near the Large Complex Structure Limit (LCSL). We connect these to the Gromov-Hausdorff metric collapse predicted by the SYZ conjecture.

hep-th↗

arXiv2026

The Sharp Edges of Calabi-Yau Manifolds: Designing Symmetric Models for Ricci-flat Metrics

Computing Ricci-flat metrics on Calabi-Yau manifolds is challenging since no closed-form solutions are known. However, these computations are needed in order to make physical predictions in heterotic string theory, such as the masses of quarks and Yukawa couplings. In this manuscript, we present an overview of relevant literature for learning about Calabi-Yau manifolds, as ML researchers often face a steep learning curve when entering the field. Furthermore, we survey the impact of the manifold's symmetries on machine learning approximations to these flat metrics. We also characterise the isometries of Ricci-flat metrics, a result frequently omitted or used without proof. Then, we address symmetry breaking in point sampling and introduce a novel formula for computing volume ratios on general CICY manifolds. We conclude by presenting a new symmetry-aware model built using graph neural networks that avoids pathological behaviour witnessed in some other models.

hep-th↗

arXiv2025

A remark on weighted average multiplicities in prime factorisation

We study a generalisation of the quality of an ABC triple that we call the weighted average multiplicity (WAM), in which the logarithmic heights of prime factors are raised to a complex exponent s. The WAM is connected to the standard ABC conjecture at s=1. We show that for real part of s less than 1, WAM is unbounded over ABC triples both for integers and polynomials. For real part greater than 1, we characterise a boundary beyond which WAM is holomorphic and bounded. In this region, we show that WAM is related to the multiplicity of the largest prime factor of the triple, a quantity that we connect with the original ABC conjecture and whose distribution we explore computationally.

math.NT↗

arXiv2025

Symbolic Approximations to Ricci-flat Metrics Via Extrinsic Symmetries of Calabi-Yau Hypersurfaces

Ever since Yau's non-constructive existence proof of Ricci-flat metrics on Calabi-Yau manifolds, finding their explicit construction remains a major obstacle to development of both string theory and algebraic geometry. Recent computational approaches employ machine learning to create novel neural representations for approximating these metrics, offering high accuracy but limited interpretability. In this paper, we analyse machine learning approximations to flat metrics of Fermat Calabi-Yau n-folds and some of their one-parameter deformations in three dimensions in order to discover their new properties. We formalise cases in which the flat metric has more symmetries than the underlying manifold, and prove that these symmetries imply that the flat metric admits a surprisingly compact representation for certain choices of complex structure moduli. We show that such symmetries uniquely determine the flat metric on certain loci, for which we present an analytic form. We also incorporate our theoretical results into neural networks to reduce Ricci curvature for multiple Calabi--Yau manifolds compared to previous machine learning approaches. We conclude by distilling the ML models to obtain for the first time closed form expressions for Kahler metrics with near-zero scalar curvature.

hep-th↗