Contents

# Contents

## The open problem of formulating M-Theory

It is an open problem to mathematically formulate M-theory: Supposedly a single coherent theory whose various limiting cases reproduce the zoo of perturbative string theories (HET/I/IIA/IIB&F/11d SuGra) and the expected dualities relating them; but which also makes mathematical sense of non-perturbative D-/M-brane physics, and hence solves, via intersecting D-brane models of confined quantum Yang-Mills theory (holographic QCD), the last of the Millenium Problems.

In the physics-minded string theory community, parts of the literature tends to forget or downplay the conjectural nature of many assumptions and leaps of faiths that are being made when it comes to discussion of D-brane/M-brane dynamics and generally of non-perturbative effects in string theory. More widely recognized is the fact that the theory of coincident M5-branes is missing, together with the formulation of its expected decoupling limit to the D=6 N=(2,0) SCFT – but of course these major open problems are but one aspect of full M-theory. For string-theoretic references that do highlight the open problem of formulating M-theory see at M-theory – The open problem.

In mathematics, it is rare but not unfamiliar that an open problem is not the proof of a theorem within an established theory, but is the establishing of a theory in the first place. A famous example of such a situation is the search for a theory of “absolute geometry” over the “field with one element”. In this analogy, the various perturbative string theories (HET, I, IIA, IIB and their KK-compactified perturbative string theory vacua) correspond to arithmetic geometries over base prime field $\mathbb{F}_p$ for $p \geq 2$, and the would-be M-theory corresponds to a theory of a “field with one element” that unifies and completes all this, by describing it at a deeper level (literally: a deeper base).

Hence the task here is to conjure a mathematical theory $\mathcal{X}$, hypothesize that and explain how $\mathcal{X}$ is the putative M-theory, and then rigorously work out the mathematical implications of $\mathcal{X}$ in order to check to which extent they include the required design criterion “$\mathcal{X} \Rightarrow ST$”. To the extent that $\mathcal{X}$ implies known or expected phenomena in perturbative string theory the hypothesis that $\mathcal{X}$ is M-theory finds support, to the extent that it doesn’t $\mathcal{X}$ needs to be modified to or be replaced by some $\mathcal{X}'$, and the process re-started.

## Hypothesis H

### Statement

In the limit of D=11 supergravity, the covariant phase space of M-theory must consist of super-torsion free super orbi $\mathbb{R}^{10,1\vert \mathbf{32}}$-folds equipped with a suitable higher gauge field: the C-field. The first ingredient of a non-perturbative quantization of this phase space must be a choice of Dirac charge quantization-condition for the C-field.

Hypothesis H: (FSS 19b,FSS 19c) The C-field is charge quantized in J-twisted Cohomotopy cohomology theory.

### Motivation from analysis of $D_p$-brane super WZ terms

This Hypothesis H is motivated by analysis (based on Sati 13, Sec. 2.5 see FSS 19a for comprehensive review) of super p-brane WZ terms in super homotopy theory, which proves that – in the approximation of rational homotopy theoryM-brane charge is in rational Cohomotopy in exactly the same way that D-brane charge is in twisted K-theory.

### Implication of list of anomaly cancellation conditions

Theorem (FSS 19b,FSS 19c, SS 19) Hypothesis H implies the following list of anomaly cancellation consistency conditions expected in the M-theory folklore:

## References

### Survey

Here is a quick survey of the implications of Hypothesis H on expected M-theory anomaly cancellation conditions for M-theory on 8-manifolds:

• Cohomotopy and M-theory on 8-Manifolds – Introduction and Survey

### Formulation in full cohomotopy theory

Hypothesis H is formulated, and first consistency checks were made, in:

### Motivation by results in rational Cohomotopy

Hypothesis H is motivated by analysis of super p-brane WZ terms in super homotopy theory, which proves that – in the approximation of rational homotopy theoryM-brane charge is in rational Cohomotopy in exactly the same way that D-brane charge is in twisted K-theory.

This observation goes back to section 2.5 of:

• Framed M-branes, corners, and topological invariants,

J. Math. Phys. 59 (2018), 062304

arXiv:1310.1060

and was developed in

A comprehensive review is in: