Joint Harvard-CUHK-YMSC Differential Geometry: The Gopakumar-Vafa finiteness conjecture

SEMINARS

Speaker:

Professor Thomas Walpuski - Humboldt-Universitaet zu Berlin

In 1998, using arguments from M-theory, Gopakumar and Vafa argued that there are integer BPS invariants of symplectic Calabi-Yau 3-folds. Unfortunately, they did not give a direct mathematical definition of their BPS invariants, but they predicted that they are related to the Gromov-Witten invariants by a transformation of the generating series. The Gopakumar-Vafa conjecture asserts that if one defines the BPS invariants indirectly through this procedure, then they satisfy an integrality and a (genus) finiteness condition.

The integrality conjecture has been resolved by Ionel and Parker. A key innovation of their proof is the introduction of the cluster formalism: an ingenious device to side-step questions regarding multiple covers and super-rigidity. Their argument could not resolve the finiteness conjecture, however. The reason for this is that it relies on Gromov’s compactness theorem for pseudo-holomorphic maps which requires an a priori genus bound. It turns out, however, that rather powerful tools from geometric measure theory imply a compactness theorem for pseudo-holomorphic cycles. This can be used to upgrade Ionel and Parker’s cluster formalism and prove both the integrality and finiteness conjecture.

Zoom Link: https://cuhk.zoom.us/j/95974708402

Meeting ID: 959 7470 8402

Passcode: 20220309