Weighted Euler characteristic of the moduli space of higher rank Joyce-Song pairs

TL;DR

This paper introduces a new method for computing invariants of higher rank Joyce-Song pairs by stratifying moduli spaces and directly calculating weighted Euler characteristics, bypassing complex wall-crossing formulas.

Contribution

It proposes an independent approach to compute rank 2 stable pair invariants without wall-crossing formulas, potentially extendable to higher ranks.

Findings

Successfully computes rank 2 invariants using stratification

Avoids complex combinatorial wall-crossing calculations

Provides a framework for higher rank cases

Abstract

The invariants of rank 2 Joyce-Song semistable pairs over a Calabi-Yau threefold were computed in arXiv:1101.2252, using the wall-crossing formula of Joyce-Song and Kontsevich-Soibelman. Such wall-crossing computations often depend on the combinatorial properties of certain elements of a Hall-algebra (these are the stack functions defined by Joyce). These combinatorial computations become immediately complicated and hard to carry out, when studying higher rank stable pairs with rank$>2$. The main purpose of this article is to introduce an independent approach to computation of rank 2 stable pair invariants, without applying the wallcrossing formula and rather by stratifying their corresponding moduli space and directly computing the weighted Euler characteristics of the strata. This approach may similarly be used to avoid complex combinatorial wallcrossing calculations in rank$>2$ cases.