Twisted cohomology (2401.03966v1)

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.

• The paper introduces a unified framework that extends traditional cohomology to encompass twisted settings and K-theory.
• It details how twisting via local coefficients and closed forms leads to innovative concepts such as twisted de Rham cohomology and equivariant generalizations.
• It demonstrates practical applications in mathematical physics, including D-brane charge classification and topological phase analysis in condensed matter.

Insightful Overview of "Twisted Cohomology"

The paper "Twisted Cohomology" by Jonathan Rosenberg explores the concept of twisted cohomology and its applications across various fields, notably in mathematical physics. The paper addresses not only ordinary cohomology but extends the discussion to $K$-theory and other exceptional cohomology theories. It emphasizes the expansive nature of twisted theories and the myriad contexts they can be applied in, providing readers with guidance for further exploration.

Twisted Cohomology: Foundations and Variants

Twisted cohomology serves as a generalized framework to extend the classical notion of cohomology, accommodating scenarios where local coefficients and more complex algebraic structures are involved. The concept initiates from cohomology with local coefficients, leveraging the Leray-Serre spectral sequence to account for non-simply connected base spaces. This foundational idea is further expanded with twisted de Rham cohomology, where differential forms are compounded with a closed $k$-form, leading to a new cohomological perspective.

Generalized Cohomology Theories

A significant portion of the paper is dedicated to twisted generalized cohomology theories, with a primary focus on $K$-theory. Twisted $K$-theory has emerged as an essential tool in mathematical physics, providing a comprehensive framework for dealing with cohomological invariants in contexts that involve continuous-trace algebras or nontrivial $PU(H)$ bundles. The twisting often relates to a class in $H^3(X;\mathbb{Z})$, recognizing the pivotal role of the Dixmier-Douady class in classifying $C^*$-algebraic phenomena.

The paper traces the evolution of "classical" twisted $K$-theory from Karoubi and Donovan's work, addressing Azumaya algebras and their connection to Brauer group classes. This perspective provides deep insights into how twisting reflects the nontrivial topology of the underlying space. Innovations in twisted $K$-theory include equivariant variations, $KR$-theory twists considering Real spaces, and exotic twists arising from more complex global structures, such as $BGL_1(K)$.

Physical Implications and Applications

The implications of twisted cohomology in physics are profound, offering novel frameworks for understanding dualities and charge quantization in string theories. For instance, twisted $K$-theory explains the classification of D-brane charges in the presence of background $H$-flux, establishing profound connections between topological invariants and physical phenomena. T-duality in string theory, a striking manifestation of this connection, relies heavily on twisted cohomological perspectives to address the equivalence of various string theory vacua.

The paper extends into other domains, such as condensed matter physics, where twisted cohomology aids in classifying topological phases of matter. The tenfold way and its extensions showcase how topological and symplectic structures are captured by twisted and equivariant cohomological frameworks.

Speculation on Future Developments

The landscape of twisted cohomology is poised for continued expansion, with potential developments in several directions. The theoretical underpinnings of twisted theories could provide deeper insights into phenomena with higher structure, such as those emerging in quantum field theories and higher algebraic constructs. Moreover, the interplay between twisted theories and emerging areas like quantum topology could uncover further applications of these concepts in both mathematical physics and beyond.

In conclusion, the paper by Rosenberg not only highlights the utility of twisted cohomology and $K$-theory across various disciplines but also underscores a fertile ground for future exploration and application in increasingly complex mathematical and physical systems. As research in these areas progresses, the fundamental principles elucidated in twisted cohomology will likely continue to play a central role in advancing understanding in both mathematics and theoretical physics.