Contents

# Contents

## The open problem of formulating M-Theory

Much like the open problem of formulating a theory of “absolute geometry over the field with one element”, 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.

As familiar, by analogy, from proposals for a would-be theory of “absolute geometry over the field with one element”, 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 $\mathcal{X}$ to check that 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

### 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

A comprehensive review with pointers to full details is in:

### Exposition and talk notes

Last revised on October 21, 2019 at 23:03:57. See the history of this page for a list of all contributions to it.