Theory Update 88

The most ubiqitous application of homology in 20th century physics is Stoke's theorem,

$\int_{\partial M} \phi \simeq \int_{M} \textrm{d}\phi$,

where $M$ is usually a manifold with boundary $\partial M$ and $\phi$ a differential form. In the manifold setting, the interplay of pullbacks (categorical limits) and open sets in Euclidean spaces allows us to map differential forms from one manifold to another. Standard homology $1$-functors take chain complexes for $M$ to a category of abelian groups. The groups are abelian because ordinary arithmetic is commutative. Such constructions motivated the first definitions in category theory, by Eilenberg and Mac Lane in 1945.

But today physicists know that spacetimes emerge from quantum gravitational information. For quantum spaces, operator points do not commute. The quaternions (like the Pauli matrices) make more sense than real numbers, although these two number systems may be related by a T duality. With bioctonions we can talk about even richer relations, and nonassociative spaces. A modern physical homology needs to keep track of such relations between numbers, before it blindly decides how a continuum space is really constructed.

So what does Stoke's theorem become? The components of the two sides of the equation differ by one dimension, with homology going down one step to the left and cohomology going up one step to the right. In a world where Space and Algebra are One, homology and cohomology functors operate without moving, like a wary chameleon, shifting one dimension into each other. Without the dualities of M Theory, how could homology possibly make any sense?