Symplectic Invariants on Calabi-Yau 3 folds, Modularity and Stability

We discuss techniques to calculate symplectic invariants on CY 3-folds $M$, namely Gromov-Witten (GW) invariants, Pandharipande-Thomas (PT) invariants, and Donaldson-Thomas (DT)  invariants. Physically the latter are closely related to BPS brane bound states in type IIA string compactifications on $M$.  We focus on the rank $r_{\bar 6}=1$  DT invariants  that count $\bar D6-D2-D0$ brane bound states related to PT- and  high genus GW invariants, which are calculable by mirror symmetry and topological string B-model methods modulo certain boundary conditions, and the rank zero DT invariants that count rank $r_4=1$   $D4-D2-D0$ brane bound states. It has been conjectured  by Maldacena, Strominger, Witten and  Yin that  the latter are governed by an index that has modularity properties to due $S-$ duality in string theory and extends to a mock modularity index of higher depth for $r_4>1$. Again the modularity allows to fix the at least the $r_4=1$

index up to boundary conditions fixing their polar terms.  Using Wall crossing formulas obtained by Feyzbakhsh certain PT invariants  close to the Castelnouvo bound can be related to the $r_4=1,2$  $D4-D2-D0$ invariants. 

This provides further boundary conditions for topological string B-model approach as well as for the $D4-D2-D0$ brane indices. The approach allows to prove the Castenouvo bound and  calculate the  $r_{\bar 6}=1$  DT- invariants 

or the GW invariants to higher genus than hitherto possible.