## Tuesday, November 30, 2021

### Metastrings and mysterious triality

Two new neat stringy papers

I want to mention two new hep-th preprints. First, Berglund, Hübsch, and Minić wrote
Mirror Symmetry, Born Geometry and String Theory
where they sell their love for the "bosonic string as the parent of the superstring" and the "doubled degrees of freedom for T-duality" including the new term "metastring theory", not to mention the "Born geometry" (a structure on the doubled tori, with some symplectic structure and modular group), but the real new beef of the 5-page-short paper seems to be their ability to get a manifest mirror symmetry out of the doubled starting point. Some non-commutative generalizations of the Calabi-Yaus are automatically included.

The last hep-th paper is
Mysterious Triality
by Sati and Voronov. I've been in love with the mysterious duality by Vafa, Iqbal, and Neitzke, and debated the authors when they were making and releasing the paper. OK, the exceptional Lie groups' lattices which are relevant in the U-duality of M-theory on tori seem to be reproduced, along with other quantities, in the del Pezzo surfaces $${\mathbb B}_k$$.

I've tried to complete a theory defined on the del Pezzos that could produce the string/M-theory spacetime as the "target space of the target space". Now these two authors have a third dual description i.e. the duality is extended to a triality. Their third descrpition involves
the Sullivan minimal models of $${\mathcal L}_c^k S^4$$, the iterated cyclip loop space of the four sphere.
So start with the four-sphere $$S^4$$ and find its loop space, i.e. the space of all maps from a circle $$S^1$$ to the space. The space of all strings within $$S^4$$, or the space of all strings in the space of all strings within $$S^4$$, and so on. ;-) When you do it roughly 8 times, to get to $$E_8$$, you will get a loopo-loopo-loopo-loopo-loopo-loopo-loopo-loopo-space of the four-sphere, and that will manifest the $$E_8$$ symmetry and some other nice structures found in the U-duality of M-theory on tori, too.

Mysterious Duality Launch Video, press to play: I guess that even Vafa et al. fail to know it! ;-)

Needless to say, the idea of the "strings in the space of strings in the space of strings..." is even more aligned with my general program of "iteratively stringizing" the spaces, it is pretty much exactly what I always found relevant in these mysterious duality constructions in order to extend the pattern to a full-blown theory of everything in all forms. It is still mysterious for me why their sequence starts with a four-sphere.

Note that $$S^4$$ is a real-four-dimensional compact manifold, and so is $${\mathbb C \mathbb P}^2$$, the simplest del Pezzo surface (or Forefather Del Pezzo). But these two manifolds are distinct. In particular, the four-sphere isn't a complex manifold, doesn't admit complex coordinates. There are of course various ways to relate them or obtain them from one another but I have no idea why these seemingly non-unique constructions could be relevant for getting something as unique as M-theory, or even a semi-unique thing like M-theory on tori.

At any rate, my program would be that you define some appropriate enough string theory on $$S^4$$, get some new target space out of it, define string theory in the same way on that, get something again, ... repeat it about 8 times, and then you get a complete description of M-theory on an eight-torus with the $$E_8$$ U-duality group that is manifest.

You probably ask what the Sullivan model is. Roughly speaking, the Sullivan model is a rational homotopy model based on cohomology while the Quillen model is based on homology, so they are dual in some sense; see [Ta83][Maj00][FHT01]. I surely explained to you what a clipboard is now! ;-) Yes, the terminology is way too mathematical for me.