Machine learning of Calabi-Yau volumes

Abstract

We employ machine learning techniques to investigate the volume minimum of Sasaki-Einstein base manifolds of noncompact toric Calabi-Yau three-folds. We find that the minimum volume can be approximated via a second-order multiple linear regression on standard topological quantities obtained from the corresponding toric diagram. The approximation improves further after invoking a convolutional neural network with the full toric diagram of the Calabi-Yau three-folds as the input. We are thereby able to circumvent any minimization procedure that was previously necessary and find an explicit mapping between the minimum volume and the topological quantities of the toric diagram. Under the AdS/CFT correspondence, the minimum volumes of Sasaki-Einstein manifolds correspond to central charges of a class of $4d$ $\mathcal{N}=1$ superconformal field theories. We therefore find empirical evidence for a function that gives values of central charges without the usual extremization procedure.