Abstract
We use machine learning to approximate Calabi-Yau and SU(3)-structure metrics, including for the first time complex structure moduli dependence. Our new methods furthermore improve existing numerical approximations in terms of accuracy and speed. Knowing these metrics has numerous applications, ranging from computations of crucial aspects of the effective field theory of string compactifications such as the canonical normalizations for Yukawa couplings, and the massive string spectrum which plays a crucial role in swampland conjectures, to mirror symmetry and the SYZ conjecture. In the case of SU(3) structure, our machine learning approach allows us to engineer metrics with certain torsion properties. Our methods are demonstrated for Calabi-Yau and SU(3)-structure manifolds based on a one-parameter family of quintic hypersurfaces in ℙ4.
Highlights
Finding numerical approximations to metrics for the compact dimensions of string theory is a subject which has a long history within the literature
Our new methods improve existing numerical approximations in terms of accuracy and speed. Knowing these metrics has numerous applications, ranging from computations of crucial aspects of the effective field theory of string compactifications such as the canonical normalizations for Yukawa couplings, and the massive string spectrum which plays a crucial role in swampland conjectures, to mirror symmetry and the SYZ conjecture
We assess the quality of the neural network (NN) interpolation by comparing the error measure σ obtained by the NN on the test set with the result one would obtain by using the “wrong” Kähler potential computed for a point of the training set that is closest in complex structure moduli space
Summary
Finding numerical approximations to metrics for the compact dimensions of string theory is a subject which has a long history within the literature. Knowledge of such metrics is not always necessary. In the case of Calabi-Yau (CY) compactifications, Yau’s theorem [1] and techniques from algebraic geometry allow many quantities of interest to be computed without explicit knowledge of the Ricci-flat metric on the extra dimensions [2]. The kinetic terms of matter fields, for example, are determined by the Kähler potential. This is a non-holomorphic function that is inaccessible with the methods of algebraic geometry. One needs explicit knowledge of the metric (and often other structures) in order to compute this crucial aspect of the low-energy effective field theory
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