Schreiber Introduction to Hypothesis H

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Lecture notes that we are finalizing:


Abstract. The key open question of contemporary mathematical physics is the elucidation of the currently elusive fundamental laws of strongly-interacting “non-perturbative” quantum states — including bound states as mundane as nucleons but more generally of quarks confined inside hadrons (declared a mathematical “Millennium Problem” by the Clay Math Institute), as well as strongly-correlated topologically ordered quantum materials (currently sought by various laboratories as hardware for topological quantum computation).

The seminal strategy of regarding such systems as located on branes inside a higher dimensional string-theoretic spacetime (the “holographic principle”) shows all signs of promise but has been suffering from the ironic shortcoming that also string theory has only really been defined perturbatively, at small coupling. However, string theory exhibits a web of hints towards the nature of its non-perturbative completion, famous under the working title “M-theory” but still elusive. Thus, mathematically constructing M-theory should imply a mathematical understanding of quantum brane worldvolumes which should solve non-perturbative quantum physics: the M-strategy for attacking the Millennium Problem.

After a time of stagnation in research towards M-theory, we have recently formulated and extensively tested a hypothesis on the precise mathematical nature of at least a core part of the theory: We call this Hypothesis H since it postulates that M-branes are classified by coHomotopy-theory in much the same way that D-branes are expected to be classified by K-theory (a widely held but just as conjectural belief which might analogously be called Hypothesis K). In fact, stabilized coHomotopy is equivalently 𝔽 1\mathbb{F}_1 -K-theory over the “absolute base field with one element”. Last not least, coHomotopy is equivalently framed Cobordism cohomology.

In these lecture notes we try to give an introduction to (1.) the motivation and (2.) some consequences of Hypothesis H, assuming an audience with a little background in electromagnetism, differential geometry and algebraic topology.


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Last revised on October 1, 2024 at 13:29:02. See the history of this page for a list of all contributions to it.

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