# BPS State Counting in Local Obstructed Curves from Quiver Theory and Seiberg Duality

## TL;DR

This paper develops a quiver-based method utilizing Seiberg duality and localization to compute BPS state indices and Donaldson-Thomas invariants in local obstructed curve geometries, with potential generalizations to other quiver theories.

## Contribution

It introduces a novel quiver description for BPS states in obstructed curve geometries and applies Seiberg duality and localization to compute invariants across stability chambers.

## Key findings

- BPS states have a framed quiver description.
- The method computes generalized Donaldson-Thomas invariants.
- Applicable to affine ADE quiver theories.

## Abstract

In this paper we study the BPS state counting in the geometry of local obstructed curve with normal bundle O+O(-2). We find that the BPS states have a framed quiver description. Using this quiver description along with the Seiberg duality and the localization techniques, we can compute the BPS state indices in different chambers dictated by stability parameter assignments. This provides a well-defined method to compute the generalized Donaldson-Thomas invariants. This method can be generalized to other affine ADE quiver theories.