A Point's Point of View of Stringy Geometry

TL;DR

This paper explores the concept of points in string theory using the derived category of D-branes, analyzing flops and Pi-stability to understand the intrinsic role of points in spacetime topology.

Contribution

It introduces a novel approach to defining points in string theory through derived categories and Pi-stability, linking geometric transitions to stability conditions.

Findings

Pi-stability relates to flop transitions in string theory.

Monodromy computations align with previous conjectures.

Derived categories provide insights into the intrinsic nature of points.

Abstract

The notion of a "point" is essential to describe the topology of spacetime. Despite this, a point probably does not play a particularly distinguished role in any intrinsic formulation of string theory. We discuss one way to try to determine the notion of a point from a worldsheet point of view. The derived category description of D-branes is the key tool. The case of a flop is analyzed and Pi-stability in this context is tied in to some ideas of Bridgeland. Monodromy associated to the flop is also computed via Pi-stability and shown to be consistent with previous conjectures.