### Mysterious Triality

#### Posted by David Corfield

When we started this blog back in 2006 my co-founders were both interested in higher gauge theory. Their paths diverged as Urs looked to adapt these constructions to formulate the elusive M-theory.

Over the years I’ve been following this work, which has taken up a proposal by Hisham Sati in Framed M-branes, corners, and topological invariants, Sec 2.5 that M-theory be understood in terms of 4-cohomotopy, culminating in what they call Hypothesis H. I even chipped in sufficiently to one article to be included with them as an author:

- David Corfield, Hisham Sati, Urs Schreiber, Fundamental weight systems are quantum states.

Philosophically speaking, I’ve been intrigued by the idea that the novel mathematical framework of twisted equivariant differential cohomology theory, required for Hypothesis H, may be formulated via modal homotopy type theory. This is the line of thought I mentioned a few weeks ago in Dynamics of Reason Revisited.

But I’ve also been thinking that you can’t add something important to fundamental physics without it causing ripples through mathematics. So I was interested to see appear yesterday:

- Hisham Sati, Alexander Voronov,
*Mysterious triality*(arXiv:2111.14810).

The authors take mysterious duality which links del Pezzo surfaces with compactifications of M-theory, and throw in a third pole, iterated cyclic loop spaces of the 4-sphere, motivated by Hypothesis H. Now we have a mathematical mystery linking the algebraic geometry of del Pezzo surfaces with the algebraic topology of iterated cyclic loop spaces.

I’m sure there will be many further mathematical ramifications, but it’s intriguing to see in this paper two elements of an association I first heard of in a talk by John McKay that I reported here in which he related the triple of exceptional Lie groups $(E_8, E_7, E_6)$ to the triple (120 tritangents on sextic of genus 4, 28 bitangents on quartic, 27 lines on cubic). This was written up in

- Yang-Hui He, John McKay,
*Sporadic and Exceptional*, (arXiv:1505.06742).

Since exceptional structures are so tightly inter-related, perhaps we should expect McKay’s triple of sporadic simple groups, (Monster, Baby monster, Fischer group $Fi_{24}$), to appear.

## Re: Mysterious Triality

Everyone is aware that the diffeomorphism group of the 4-sphere is very interesting, cf eg

Watanabe, https://arxiv.org/abs/1812.02448

or for ex

Gay, https://arxiv.org/abs/2102.12890,

right?