# It Was 20 Years Ago Today — the M-theory Conjecture

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**Common Topics:**span, theory, string, mathematics, part

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Part 1: It Was 20 Years Ago Today — the M-theory Conjecture

Part 2: Homotopy Lie-n Algebras in Supergravity

Part 3: Emergence from the Superpoint

Part 4: Why supersymmetry? Because of Deligne’s theorem

Part 5: 11d Gravity from just the Torsion Constraint

Part 6: Spectral Standard Model and String Compactifications

Part 7: Super p-Brane Theory emerging from Super Homotopy Theory

While the world didn’t end, after all, 15 years back at the turn of the millennium, in hindsight it is curious that, almost unnoticed, something grand did come to a halt around that time. Or almost. The 90s had seen a firework of *structural *insight into the mathematical nature of string theory, an unprecedented global confluence of research deeply involving the high energy physics departments with the maths departments. It culminated 20 years ago, in the summer of the annus mirabilis 1995, with the *M-theory conjecture*.

When you ask Google about what happened back then, it will point you to the standard journalistic accounts of the “second superstring revolution“. While this style of discussion is all well in itself, it has trouble to withstand once other journalists shout “untestable!” and “multiverse mania!” and the like. What I am planning to be talking about in a series of articles here is some of the actual substance.

As for what really happened in the 1990s, a serious compilation which everyone should know finished right before y2k, gives a first-hand impression of the history that was written in that decade:

- Mike Duff,
*The world in eleven dimensions: Supergravity, Supermembranes, and M-theory,*IoP 1999 (publisher, GoogleBooks, nLab)

Lay readers will benefit from just looking through the introductions that Duff writes to each chapter. This is the real stuff, from one of the discoverers of the theory. If you gained your knowledge about “M-theory” from the media, or from popular books that don’t get to the point (a maneuver that Duff calls “M-theory without the M“) this is the place to re-gauge.

Duff closes, on p. 330 of this book, like so:

Future historians may judge the period 1984-95 as a time when theorists were like boys playing by the sea shore, and diverting themselves with the smoother pebbles or prettier shells of perturbative ten-dimensiorial superstrings while the great ocean of non-perturbative eleven-dimensional M-theory lay all undiscovered before them.

True as that may be, 20 years down the road, Greg Moore has to remind the boys to “Keep true to the dreams of thy youth”, which is the title of the last section of last year’s “vision talk” at the annual string theory conference

- Gregory Moore,
*Physical Mathematics and the Future,*talk at Strings 2014 (talk slides, companion text pdf, nLab)

where it says:

Work on formulating the fundamental principles underlying M-theory has noticeably waned. […] A good start was given by the Matrix theory approach […but…] it has an important flaw: It has not led to much significant new mathematics. If history is a good guide, then we should expect that anything as profound and far-reaching as a fully satisfactory formulation of M-theory is surely going to lead to new and novel mathematics. Regrettably, it is a problem the community seems to have put aside – temporarily. But, ultimately, Physical Mathematics must return to this grand issue.

At the same time, over in the maths community, they are left wondering, too. There is a community thread on MathOverflow asking members to list the *Most intricate and most beautiful structures in mathematics*. After a while I had looked through the replies that had accumulated and then wondered:

does it strike anyone that many of the candidates for “most intricate and/or beautiful structure in mathematics” proposed here find their natural joint home where they meaningfully relate to each other in… string theory?

- E8 as a GUT group (see the beautiful exposition (Witten 02)), G2 as a structure group, and in fact the whole tower of exceptional generalized geometries inside 11d SuGra;
- the Leech lattice and the monster vertex operator algebra for evident reasons and reasons already mentioned;
- the Grothendieck-Teichmüller tower for obvious reasons (“hence” also the absolute Galois group..);
- geometric Langlands correspondence as but one incarnation of S-duality;
- tmf, being the target of the partition function of the heterotic string (and tmf0(2) as the coefficients for the partition function of the type I superstring; and conjecturally the whole system of Tmf(Γ)s for F-theory)
These are just the most evident. One could go on about how the motivic Galois group also fits in etc., but I don’t want to strain it.

What sometimes makes me wonder is that mathematicians have all that appreciation for all these separate intricate and beautiful phenomena that come out of string theory, one by one, including an impressive list of Fields-awarded work, but that there is little appreciation that closely similar (just vastly “bigger”) to how the Moonshine conjecture qualified as a problem in mathematics, the question “What is string theory?” may be one of the deepest open problems possibly not in phyisics, but in mathematics.

This problem had been anticipated. Edward Witten famously amplified (Nova interview 2003) that:

Back in the early ’70s, the Italian physicist, Daniele Amati reportedly said that string theory was part of 21st-century physics that fell by chance into the 20th century. I think it was a very wise remark.

But meanwhile, the 21st century did arrive. And mathematics itself has seen a major revolution since, revolving around “homotopy theory”, both in its foundational incarnation as homotopy type theory as well as in its geometric incarnation as higher geometry (“higher topos theory”). This is precisely the kind of mathematics that pertains to string theory! I will be speaking about aspects of this in the following articles here. Interested readers may jump ahead and check out the introduction section of this book collection:

- Hisham Sati, Urs Schreiber (eds.)

**Mathematical Foundations of Quantum Field and Perturbative String Theory**

Proceedings of Symposia in Pure Mathematics, volume 83

AMS (2011) (publisher, nLab)

My coauthor Hisham Sati here is the very student of Mike Duff who, 15 years back, was involved in editing “The World in Eleven Dimensions”.

I am a researcher in the department Algebra, Geometry and Mathematical Physics of the Institute of Mathematics at the Czech Academy of the Sciences (CAS) in Prague.

Presently I am on leave at the Max Planck Institute for Mathematics in Bonn.

I am not sure if I am nostalgic or not. Are we?

Nice work Urs! Looking forward to part 2!