A Point's Point of View of Stringy Geometry

TL;DR

This paper advances a stringy notion of spacetime points by using the bounded derived category of coherent sheaves and 0-branes as intrinsic probes of target-space topology. It leverages BO-point objects, the Serre functor, and $oldsymbol{oldsymbol{ ext{$oldsymbol{ ext{Pi}}$-stability}}}$ to investigate how flop transitions (via Bridgeland’s framework) affect point-like D-branes, revealing monodromy governed by Fourier–Mukai transforms. Through a concrete flop example, it shows that what counts as a point is region-dependent in Kähler moduli space and that stability conditions are essential to distinguish true points from pre-point objects. The work thus links D-brane charges, central charges, and stability to the evolving topology of the stringy target, highlighting both the power and the limits of deriving geometry from worldsheet data.

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.

Figures (3)

• Figure 1: The correspondence between points in $X$ and $X'$.
• Figure 2: The moduli space of $B+iJ$ for the flop.
• Figure 3: The complex $t$-plane.