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.
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:
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
download: pdf
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
Lift of fractional D-brane charge to equivariant Cohomotopy theory
Differential Cohomotopy implies intersecting brane observables
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
and was developed in
Domenico Fiorenza, Hisham Sati, Urs Schreiber
The WZW term of the M5-brane and differential cohomotopy
J. Math. Phys. 56, 102301 (2015)
Domenico Fiorenza, Hisham Sati, Urs Schreiber,
Rational sphere valued supercocycles in M-theory and type IIA string theory
Journal of Geometry and Physics, Volume 114, Pages 91-108 April 2017
A comprehensive review 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
The following earlier article in the string theory literature touched on aspects of Hypothesis H:
Kenneth Intriligator, Anomaly Matching and a Hopf-Wess-Zumino Term in 6d, N=(2,0) Field Theories, Nucl. Phys. B581 (2000) 257-273 (arXiv:hep-th/0001205)
In the context of discussion of the Hopf-Wess-Zumino term of the M5-brane sigma-model, the bit of equation (2.5), text below (2.5) and text around (5.3) in Intriligator 00 considers C-field flux which is classified by smooth Cohomotopy classes, hence by smooth functions to $S^4$. See FSS 19c, p. 2 for discussion.
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 Cohomotopy and Branes
talk at String and M-Theory: The New Geometry of the 21st Century
NUS Singapore, 2018
Microscopic brane physics from Cohomotopy theory
talk at M-Theory and Mathematics
NYU Abu Dhabi, 2020
The Higher Structure of 11d Supergravity
talk at Souriau 2019
IHP Paris, 2019
Equivariant Cohomotopy of toroidal orbifolds
talk at Prof. Sadok Kallel‘s group seminar
AUS Sharjah, 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 January 16, 2020 at 03:48:00. See the history of this page for a list of all contributions to it.