Metric Flows with Neural Networks

There are no known nontrivial compact Calabi-Yau metrics, objects of central importance in string theory and algebraic geometry, despite decades of study. The essence of the problem is that theorems by Calabi [1] and Yau [2, 3] guarantee the existence of a Ricci-flat Kähler metric (Calabi-Yau metric) when certain criteria are satisfied, but Yau's proof is non-constructive. It is not for lack of examples satisfying the criteria, since topological constructions ensure the existence of an exponentially large number of examples [4-6]. The problem also does not prevent certain types of progress in string theory, since aspects of Calabi-Yau manifolds can be studied without knowing the metric. For instance, much is known about volumes of calibrated submanifolds [7], an artifact of supersymmetry and the existence of BPS objects, as well a metric deformations the preserve Ricci-flatness, the (in)famous moduli spaces [8].