The 3-fold vertex via stable pairs
Donaldson-Thomas invariants, which count stable sheaves on projective threefolds, are important invariants, especially for Calabi-Yau threefolds. They are conjecturally equivalent to Gromov-Witten invariants of all genera on Calabi-Yau threefolds. In order to take away the contribution from embedded points inside curves in Calabi-Yau threefolds, we need to normalize the Donaldson-Thomas invariant. Pandharipande and Thomas found a more effective way that should capture these normalized Donaldson-Thomas invariant. Instead they count the number of stable pairs \([O_{X}\overset {s} \rightarrow F ]\) in a threefold \(X\). Here \(F\) is a pure sheaf supported on a curve in \(X\) and the cokernel of \(s\) has zero dimensional support. In this article, the authors compute these invariants via the localization method when \(X\) is toric. In the toric Calabi-Yau case, they are expressed in terms of pure box counting.
scientific article