We study the complex structure moduli dependence of the scalar Laplacian eigenmodes for oneparameter families of CalabiYau
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A<sc>bstract</sc> n folds in ℙ^{n+1}. It was previously observed that some eigenmodes get lighter while others get heavier as a function of these moduli, which leads to eigenvalue crossing. We identify the cause for this behavior for the torus. We then show that at points in a sublocus of complex structure moduli space where Laplacian eigenmodes cross, the torus has complex multiplication. We speculate that the generalization to arbitrary CalabiYau manifolds could be that level crossing is related to rank one attractor points. To test this, we compute the eigenmodes numerically for the quartic K3 and the quintic threefold, and match crossings to CM and attractor points in these varieties. To quantify the error of our numerical methods, we also study the dependence of the numerical spectrum on the quality of the CalabiYau metric approximation, the number of points sampled from the CalabiYau variety, the truncation of the eigenbasis, and the distance from degeneration points in complex structure moduli space. 
Despite their successes, machine learning techniques are often stochastic, errorprone and blackbox. How could they then be used in fields such as theoretical physics and pure mathematics for which errorfree results and deep understanding are a must? In this Perspective, we discuss techniques for obtaining zeroerror results with machine learning, with a focus on theoretical physics and pure mathematics. Nonrigorous methods can enable rigorous results via conjecture generation or verification by reinforcement learning. We survey applications of these techniquesforrigor ranging from string theory to the smooth 4D Poincaré conjecture in lowdimensional topology. We also discuss connections between machine learning theory and mathematics or theoretical physics such as a new approach to field theory motivated by neural network theory, and a theory of Riemannian metric flows induced by neural network gradient descent, which encompasses Perelman’s formulation of the Ricci flow that was used to solve the 3D Poincaré conjecture.more » « lessFree, publiclyaccessible full text available May 1, 2025

A bstract CalabiYau threefolds with infinitely many flops to isomorphic manifolds have an extended Kähler cone made up from an infinite number of individual Kähler 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 CalabiYau threefolds. We show that such isomorphic flops across facets of the Kähler cone boundary give rise to symmetry groups isomorphic to Coxeter groups. In the dual Mori cone, nonflopping curve classes that are identified under these groups have the same GopakumarVafa 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 CalabiYau manifold.more » « less