David Corfield cohomology in physics

Idea

In that cohomology is an important part of the story of higher category theory as revolution, we should expect it to show itself in the development of current physics.

Cohomology plays a fundamental role in modern physics. (Zeidler, Quantum Field Theory, 2006, Volume 1, p. 14).

Fundamental physics is all controled by cohomology. (Schreiber) See his The use of nonabelian differential cohomology

“Is there much in fundamental physics that does not originate in natural constructions in nonabelian differential cohomology?”

For the cutting edge, see the introduction to

• Domenico Fiorenza, Hisham Sati, Urs Schreiber, The Character Map in Twisted Non-Abelian Cohomology

Cohomology plays a fundamental role in modern mathematics and physics (Kouneiher, Topological Foundations of Physics, PhilPapers) (“We want to explore thus the cohomological aspect of physics, more precisely to show the cohomological foundations of the physics.” 246-7)

Stasheff, A Survey of Cohomological Physics.

Cohomological physics is a phrase I introduced sometime ago in the context of anomalies in gauge theory, but it all began with Gauss in 1833, if not sooner.

This survey is limited to the years before 2001 since there has been an explosion of cohomological applications in theoretical physics (even of K-theory) in the new century. Since 1931 but especially toward the end of the XXth century, there has been increased use of cohomological and more recently homotopy theoretical techniques in mathematical physics.

See John Baez’s course Quantization and Cohomology.

See the meeting Generalized Cohomology and Physics.

Quotations

The notions of homology and cohomology are deeply rooted in electrodynamics….the most simple approach to homology and cohomology groups is related to electrical circuits. This generalizes then to the Maxwell equations in electrodynamics by using differential forms and the de Rham cohomology. (Zeidler II, 227)

Cohomology is deeply routed (sic) in the following topics: Gauss’ surface theory, the Kirchhoff-Weyl theory of electrical networks, and Maxwell’s theory of electromagnetism. Cohomology lies at the heart of both modern differential topology and modern quantum field theory (the BRST approach).

Note: “Both the classical roots and the modern extensions will be thoroughly studies in Vol III on gauge theory and in Vol IV on quantum mathematics. (Zeidler II, 305)

The de Rham cohomology is a far-reaching generalization of Maxwell’s theory for the electromagnetic field. If the open set $\mathcal{O}$ is contractible to a point, then the existence of the electric (resp. magnetic) potantial is based on the local constraint $\mathbf{curl E} = 0$ (resp. div $\mathbf{B}$ = 0) for the electric field $\mathbf{E}$ (resp. the magnetic field $\mathbf{B}$). The number of global constraints for the existence of electric (resp. magnetic) potentials depends on the first Betti number $\beta_1$ (resp. the second Betti number $\beta_2$) of the open set $\mathcal{O}$. The Betti number $\beta_1$ (resp. $\beta_2$) measures the number of essential 1-cycles (resp. 2-cycles) which are not boundaries. The Betti numbers are homotopical invariants, and hence the number of linearly independent constraints is also a homotopical invariant. (Zeidler III: 1039)

Zeidler IV will treat cohomology at length

(Note his idea of cohomology from energy levels of atom. See also Vol I 16.8).

• Joe Davighi, Ben Gripaios, Oscar Randal-Williams, Differential cohomology and topological actions in physics (arXiv:2011.05768)

Topological actions have come to play an important role in physics. Examples include the Aharonov–Bohm term and the Dirac monopole in quantum mechanics, Chern–Simons terms and theta terms in gauge theories, and the Wess–Zumino–Novikov–Witten (WZNW) terms occurring in hadronic physics and elsewhere. Such actions have hitherto mostly been described in an ad hoc fashion. We will show how all of the above examples, and many more besides, can be described using differential cohomology, a mathematical gadget which is a diffeomorphism invariant of a manifold that refines integral cohomology by information about differential forms, thus merging topological data about the manifold (or rather its homotopy type)with geometrical information in an intricate way.

Mass in classical mechanics

Mass “has a cohomological significance, it parametrizes the extensions of the Galileo group.” (Santiago Garcıa, hep-th/9306040). From discussion:

“Briefly: in classical mechanics, the Galilei group acts on the symplectic manifold of states of a free particle. But in quantum mechanics, we only have a projective representation of this group on the Hilbert space of states of the free particle. The cocycle is the particle’s mass.

Switching to a much more lowbrow way of talking: you can’t see the mass of a free classical particle by just watching its trajectory, since it goes along a straight line at constant velocity no matter what it’s mass is. But you can see the mass of a free quantum particle, because its wavefunction smears out faster if it’s lighter! So there’s some difference between classical and quantum mechanics. Ultimately this arises from the fact that the latter involves an extra constant, Planck’s constant.“ (John Baez)

“In slight disguise, one can see this cocycle also control already the classical free non-relativistic particle, in the sense that its action functional is of the form of a 1d WZW model with that cocycle being the ”WZW term“ that however comes down to be the ordinary free action.” (Urs Schreiber)

String theory and cohomology

H. Sati, Geometric and topological structures related to M-branes

“We consider the topological and geometric structures associated with cohomological and homological objects in M-theory.”

Edward Witten, Elliptic Genera And Quantum Field Theory , Commun.Math.Phys. 109 525 (1987) (euclid:1104117076)

on the elliptic genus/Witten genus as the partition function of the type II superstring/heterotic superstring

“A properly developed theory of elliptic cohomology is likely to shed some light on what string theory really means.”

Minasian and Moore (1997) suggested that D-brane charges are classified by K-theory and not just by homology as was proposed first. (S. Fredenhagen, Basic Bundle Theory and K-Cohomology Invariants, p. 1)