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.
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.
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 theory – M-brane charge is in rational Cohomotopy in exactly the same way that D-brane charge is in twisted K-theory.
Theorem (FSS 19b,FSS 19c, SS 19) Hypothesis H implies the following list of anomaly cancellation consistency conditions expected in the M-theory folklore:
Hypothesis H is formulated, and first consistency checks were made, in:
Domenico Fiorenza, Hisham Sati, Urs Schreiber:
Domenico Fiorenza, Hisham Sati, Urs Schreiber:
Equivariant Cohomotopy implies orientifold tadpole cancellation
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 theory – M-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
A comprehensive review with pointers to full details is in:
Domenico Fiorenza, Hisham Sati, Urs Schreiber:
The rational higher structure of M-theory,
Proceedings of the LMS-EPSRC Durham Symposium:
Higher Structures in M-Theory, August 2018,
Fortschritte der Physik, 2019
Equivariant Cohomotopy of toroidal orbifolds
talk at Prof. Sadok Kallel‘s group seminar
AUS Sharjah, 2019
The Higher Structure of 11d Supergravity
talk at Souriau 2019
IHP Paris, May 2019
Equivariant Cohomotopy and Branes
talk at String and M-Theory: The New Geometry of the 21st Century
NUS Singapore, 2018
Super p-Brane Theory emerging from Super Homotopy Theory
talk at StringMath2017,
Hamburg, 2017
Twisted Cohomotopy implies M-theory anomaly cancellation
presentation at Strings2019
Brussels, 2019
Equivariant Stable Cohomotopy and Branes
talk at Geometry, Topology and Physics
NYU Abu Dhabi, 2018
Equivariant cohomology of M2/M5-branes
talk at Seminar on Higher Structures
MPI Bonn, 2016
Last revised on October 21, 2019 at 23:03:57. See the history of this page for a list of all contributions to it.