M-Theory as Mf-Theory

An article that we have written:

Abstract. In the quest for mathematical foundations of M-theory, the Hypothesis H, that fluxes are quantized in Cohomotopy theory, implies, on flat but possibly singular spacetimes, that M-brane charges locally organize into equivariant homotopy groups of spheres.

Here we show how this leads to a correspondence between phenomena conjectured in M-theory and fundamental mathematical concepts/results in stable homotopy, generalized cohomology and Cobordism theory MfM\!f:

In particular, spontaneous K3-reductions with 24 branes are singled out from first principles:

Finally, complex-oriented cohomology emerges from Hypothesis H, connecting it to all previous proposals for brane charge quantization in the chromatic tower: in ordinary cohomology, in K-theory, in elliptic cohomology, etc.:

In fact, we find quaternionic/complex Ravenel-orientations bounded in dimension; and we find the bound to be 10, as befits spacetime dimension 10+1.

Concluding remarks. Since the Pontrjagin theorem identifies Cohomotopy theory with framed Cobordism cohomology, the statement of Hypothesis H is equivalently:

  • M-Brane charge is quantized in framed Cobordism cohomology.

But since stable Cohomotopy/framed Cobordism is the initial multiplicative cohomology theory, every other multiplicative cohomology serves to approximate brane charge quantization in Cohomotopy. Specifically, existence of universal H 3H_3-fluxes near M2-branes forces brane charge observation through quaternionic-orientated – in particular complex-orientated – multiplicative cohomology theories.

Moreover, initial among oriented cohomology theories are again Cobordism cohomology theories, namely quaternionic Coordism MSp and complex cobordism MU, respectively. And indeed, one may recover observations of charges in Cohomotopy, hence in framed Cobordism, from their observations in unitary Cobordism – this is the statement of the Adams-Novikov spectral sequence:

  • M-Brane charge quantization may be observed in complex Cobordism.

Which is to say that, with Hypothesis H, M-brane charge is not only fundamentally in framed Cobordism, but, in the presence of black M2-brane sources, M-brane charges are canonically approximated/perceived through other flavors of Cobordism cohomology MfM\!f, such as quaternionic and unitary cobordism, but also, for instance, special unitary Cobordism MSU, (which connects with Calabi-Yau KK-compactifications).

In summary and in the form of a broad slogan, Hypothesis H implies that:

  • M-brane charge quantization is seen in Cobordism theories.

hence that

  • M-theory is controlled by MfM\!f-theory.

This is a direct analogue – a refinement – of the traditional conjecture that D-brane charge is quantized in topological K-theory. Indeed, if M-brane charge is in Cobordism theory, then perceiving it through the lens of K-orientations means to approximate it by K-theory.

Related articles on Hypothesis H:

Last revised on March 6, 2021 at 09:57:55. See the history of this page for a list of all contributions to it.