Multiple cover formula of generalized DT invariants I: parabolic stable pairs

TL;DR

This paper introduces parabolic stable pairs on Calabi-Yau 3-folds and relates their invariants to generalized Donaldson-Thomas invariants, establishing a connection via wall-crossing formulas and product expansion conjectures.

Contribution

It defines parabolic stable pairs and links their invariants to generalized DT invariants, providing a new perspective on multiple cover formulas.

Findings

Parabolic stable pair invariants relate to generalized DT invariants.

Wall-crossing formulas connect different invariants.

Conjectural multiple cover formula is equivalent to a product expansion.

Abstract

In this paper, we introduce the notion of parabolic stable pairs on Calabi-Yau 3-folds and invariants counting them. By applying the wall-crossing formula developed by Joyce-Song, Kontsevich-Soibelman, we see that they are related to generalized Donaldson-Thomas invariants counting one dimensional semistable sheaves on Calabi-Yau 3-folds. Consequently, the conjectural multiple cover formula of generalized DT invariants is shown to be equivalent to a certain product expansion formula of the generating series of parabolic stable pair invariants. The application of this result to the multiple cover formula will be pursued in the subsequent paper.