Twisted cohomology

TL;DR

This survey develops the landscape of twisted cohomology, from ordinary cohomology with local coefficients to twisted de Rham cohomology and beyond, framing twists within a unified homotopy-theoretic perspective. It then systematically analyzes twisted generalized cohomology theories, with a focus on twisted $K$-theory: its classification via Dixmier–Douady data, computational tools (twisted AHSS and Khorami’s theorem), and Thom isomorphisms under spin$^c$ structures, including exotic twists beyond $H^3$. The physical applications section highlights how twists encode D-brane charges, topological T-duality, Verlinde theory via Freed–Hopkins–Teleman, branes in WZW models, orientifolds, and condensed-matter topological phases, illustrating the deep interplay between topology and high-energy/condensed-matter physics. The paper also notes extensions to twisted KR-theory, equivariant twists, and twists of other cohomology theories (e.g., twisted Morava K-theory and cohomotopy), underscoring the broad potential of twisted frameworks in mathematics and physics.

Abstract

We discuss twisted cohomology, not just for ordinary cohomology but also for $K$-theory and other exceptional cohomology theories, and discuss several of the applications of these in mathematical physics. Our list of applications is by no means exhaustive, but we are hoping that it is extensive enough to give the reader a feel for the possible applications of twisted theories in many different contexts. We also give many suggestions for further reading, but this subject has now expanded to the point where the bibliography is necessarily very incomplete.