Refined BPS numbers on compact Calabi-Yau 3-folds from Wilson loops

par Prof. Albrecht Klemm (University of Bonn & University of Sheffield)

vendredi 28 mars 2025, 16:30 → 18:00 Europe/Paris

salle Pierre Grisvard (IHP)

We relate the counting of  refined BPS numbers  on compact elliptically fibred
Calabi-Yau 3-folds $\hat{X}$ to Wilson loop expectations values in the gauge theories that emerge in various rigid local limits  of the 5d supergravity theory defined by M-theory compactification
on $\hat{X}$. In these local limits $X_{\ast }$ the volumes of curves in certain classes go to infinity, the corresponding very massive M2-brane states can be treated as Wilson loop particles and the refined topological string partition
function on $\hat{X}$ becomes a sum of terms proportional to associated refined Wilson loop expectation values. The resulting ansatz for the complete refined topological partition function on $\hat{X}$ is written in terms  of the proportionality coefficients which depend only on the $ϵ$ deformations and the Wilson loop expectations values which
satisfy holomorphic anomaly equations. Since the ansatz is quite restrictive
and can be further constrained by the one-form symmetries and $E$-string type limits for
large base curves, we can efficiently evaluate the refined BPS numbers on $\hat{X}$,  which we  do explicitly for local gauge groups up to rank three and $h_{11}(\hat{X})=5$. These refined BPS numbers pass an impressive number of consistency checks imposed by the direct counting of these numbers using the moduli space of one dimensional stable sheaves on $\hat{X}$ and give us numerical predictions for the complex structure dependency of the refined BPS numbers.