The refinement of cohomology.
“It is noteworthy that already in this mathematical formulation of experimentally well-confirmed fundamental physics the seed of higher differential cohomology is hidden: Dirac had not only identified the electromagnetic field as a line bundle with connection, but he also correctly identified (rephrased in modern language) its underlying cohomological Chern class with the (physically hypothetical but formally inevitable) magnetic charge located in spacetime. But in order to make sense of this, he had to resort to removing the support of the magnetic charge density from the spacetime manifold, because Maxwell’s equations imply that at the support of any magnetic charge the 2-form representing the field strength of the electromagnetic field is in fact not closed and hence in particular not the curvature 2-form of an ordinary connection on an ordinary bundle.
In (Freed) this old argument was improved by refining the model for the electromagnetic field one more step: Dan Freed notices that the charge current 3-form is itself to be regarded as a curvature, but for a connection on a circle 2-bundle with connection – also called a bundle gerbe – , which is a cocycle in degree 3 ordinary differential cohomology. Accordingly, the electromagnetic field is fundamentally not quite a line bundle, but a twisted bundle with connection, with the twist being the magnetic charge 3-cocycle. Freed shows that this perspective is inevitable for understanding the quantum anomaly of the action functional for electromagnetism is the presence of magnetic charge.” (nLab)
Vector bundles and principal bundles have characteristic classes in cohomology; vector bundles with connection and principal bundles with connection have characteristic classes in differential cohomology (ADH 21).
Many issues remain from the examination of “frozen singularities” in M-theory [445] and it would seem the time is ripe for a re-examination using the tools of equivariant differential cohomology, now that differential cohomology is becoming a more familiar tool for physicists. (A Panorama Of Physical Mathematics c. 2022)
Cohesion allow for the expression of differential cohomology.
a cohesive ∞-topos – as opposed to just a cohesive 1-topos – comes canonically with its intrinsic notion of path, hence of process. (here)
Araminta Amabel, Arun Debray, Peter J. Haine, Differential Cohomology: Categories, Characteristic Classes, and Connections (arXiv:2109.12250)
Last revised on November 10, 2022 at 14:01:54. See the history of this page for a list of all contributions to it.