Derived categories and stacks in physics

TL;DR

<3-5 sentence high-level summary> Sharpe surveys how derived categories and stacks enter physics, highlighting that equivalences are realized by renormalization group flow: boundary RG flow implements localization on quasi-isomorphisms in the derived-category picture, while worldsheet RG flow yields presentation-independence for stacks. The discussion covers the mapping of D-branes to complexes via tachyon condensation, the role of $Q_{BRST}$ deformations, and how physical massless spectra are counted by $Ext^n$ groups, with the Cardy condition equated to the Hirzebruch–Riemann–Roch index. The note also develops the physical understanding of stacks, including gerbes and their decomposition into components with twisted $B$ fields, and outlines mirror symmetry and quantum cohomology for toric stacks. Collectively, these ideas provide a RG-based organizing principle for how categorical and stacky structures appear in string theory, while highlighting present limitations and open problems.

Abstract

In this note we review how both derived categories and stacks enter physics. The physical realization of each has many formal similarities. For example, in both cases, equivalences are realized via renormalization group flow: in the case of derived categories, (boundary) renormalization group flow realizes the mathematical procedure of localization on quasi-isomorphisms, and in the case of stacks, worldsheet renormalization group flow realizes presentation-independence. For both, we outline current technical issues and applications.