Luca :
I’m no expert in this specific thing so please if any other mathematician/physicist that knows better is around feel free to correct me. Regarding the Weil-Petersson metric: you can construct geometric spaces, you can think of them as surfaces, such that each point on those spaces correspond to essentially different objects. For example, I can construct a space for which each point represents a substantially different (meaning that if I have 2 triangles such that one is the reflection of the other, I call them substantially the same triangle) right-angle triangles (or more general triangles), by this I mean is that if I pick a point on this space I can give you a specific triangle (i.e. with specific length for each side/specific angle for each corner). These spaces are called Moduli Spaces (or Teichmüller space, but that’s beside the point). One can construct a Moduli space for Calabi-Yau manifolds, each point representing different types of Calabi-Yau. On this space we can define a (Kähler) metric, a mathematical object that records di distance between different points, you can think of it as a way of measuring how much different are distinct Calabi-Yau. However, there are some kind of Calabi-Yau manifolds, on the boundary of the Moduli space, that are “bad” and the metric (called Weil-Petersson) is ill-behaved there, maybe divergent.
2025-08-06 12:17:53