Strings, Junctions and Stability

TL;DR

This paper explores the mathematical and physical structure of string junctions in SU(2) Seiberg-Witten theory, using derived categories and Pi-stability to interpret junctions and transitions.

Contribution

It identifies string junction states as objects in the derived category and interprets transitions between different string configurations via Pi-stability.

Findings

String junctions correspond to triangles in the derived category.

Spiral strings are related to projective resolutions of sheaves.

Transitions between string states are described by Pi-stability grades.

Abstract

Identification of string junction states of pure SU(2) Seiberg-Witten theory as B-branes wrapped on a Calabi-Yau manifold in the geometric engineering limit is discussed. The wrapped branes are known to correspond to objects in the bounded derived category of coherent sheaves on the projective line $\cp{1}$ in this limit. We identify the pronged strings with triangles in the underlying triangulated category using Pi-stability. The spiral strings in the weak coupling region are interpreted as certain projective resolutions of the invertible sheaves. We discuss transitions between the spiral strings and junctions using the grade introduced for Pi-stability through the central charges of the corresponding objects.