M-theory, Octonions and Tricategories

October 5, 2014

M-theory, Octonions and Tricategories

Posted by John Baez

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Quite a witches’ brew, eh?

Amazingly, they seem to be deeply related. John Huerta has just finished a paper connecting them… and this concludes a series of papers that makes me very happy, because it fulfills a long-held dream: to connect physics, division algebras, and higher categories.

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Let me start with a very simple sketchy explanation. Experts should please forgive the inaccuracies in this first section: it’s hard to tell a story that’s completely accurate without getting bogged down in detail!

The rough idea

You’ve probably heard rumors that superstring theory lives in 10 dimensions and something more mysterious called M-theory lives in 11. You may have wondered why.

In fact, there’s a nice way to write down theories of superstrings in dimensions 3, 4, 6, and 10 — at least before you take quantum mechanics into account. Of these theories, it seems you can only consistently quantize the 10-dimensional version. But never mind that. What’s so great about the numbers 3, 4, 6 and 10?

What’s so great is that they’re 2 more than 1, 2, 4, and 8.

There are only normed division algebras in dimensions 1, 2, 4, and 8. The real numbers are 1-dimensional. The complex numbers are 2-dimensional. There are also more esoteric options: the quaternions are 4-dimensional, and the octonions are 8-dimensional. When you try to go beyond these, you lose the law that

$|x y| = |x| |y|$

and things aren’t so nice.

I’ve spent decades studying the quaternions and octonions, just because they’re weird and interesting. Why do the dimensions double each time in this game? There’s a nice answer. What happens if you go further, to dimension 16? I’ve learned a bit about that too, though I bet there are big mysteries still lurking here.

Most important, what — if anything — do normed division algebras have to do with physics? The jury is still out on this one, but there are some huge clues. Most fundamentally, a normed division algebra of dimension $n$ gives a nice unified way to describe both spin-1 and spin-1/2 particles in $(n+2)$ -dimensional spacetime! The gauge bosons in nature are spin-1 particles, while the fermions are spin-1/2 particles. We’d definitely like a good theory of physics to fit these together somehow.

One cool thing is this. A string is a curve, so it’s 1-dimensional, but as time passes it traces out a 2-dimensional surface. So, if we have a string floating around in some spacetime, we’ve got a 2d surface together with some extra dimensions of spacetime. It turns out to be very good to put complex coordinates on that 2d surface. Then you can describe how the string wiggles in the extra dimensions using equations that have symmetry under conformal transformations.

But for the string to be ‘super’ — for it to have supersymmetry, a symmetry between bosons and fermions — we need a certain special identity to hold, called the 3- $\psi$ ’s rule. And this holds precisely when we can take the extra dimensions and think of them as forming a normed division algebra!

So, we need 1, 2, 4 or 8 extra dimensions. So the total dimension of spacetime needs to be 3, 4, 6, or 10. Not at all coincidentally, these are also the dimensions where spin-1 and spin-1/2 particles can be described using a normed division algebra.

(This is a very rough sketch of a complicated argument, of course. I’m leaving out the details, but later I’ll show you where to find them.)

We can also look at theories of ‘branes’, which are like strings but higher-dimensional. Instead of a curve, a 2-brane is a 2-dimensional surface. As time passes, it traces out a 3-dimensional surface. So, if we have a 2-brane floating around in some spacetime, we’ve got a 3-dimensional surface together with some extra dimensions of spacetime. And it turns out that 2-branes can also have supersymmetry when the extra dimensions can be seen as a normed division algebra!

So now the total dimension of spacetime needs to be 3 more than 1, 2, 4, and 8. It needs to be 4, 5, 7 or 11.

When we take quantum mechanics into account it seems that the 11-dimensional theory works best… but the quantum aspects are still mysterious, murky and messy compared to superstring theory, so it’s called M-theory.

In his new paper, John Huerta has shown that using the octonions we can build a ‘super-3-group’, an algebraic structure that seems just right for understanding the symmetries of supersymmetric 2-branes in 11 dimensions.

I could say a lot more, but if you want more explanation without too much fancy math, try this:

John Baez and John Huerta, The strangest numbers in string theory.

This is a fun and easy article about this stuff, which we wrote for Scientific American.

The details

The detailed story has four parts.

John Baez and John Huerta, Division algebras and supersymmetry I, in Superstrings, Geometry, Topology, and C*-Algebras, eds. Robert Doran, Greg Friedman and Jonathan Rosenberg, Proc. Symp. Pure Math. 81, AMS, Providence, 2010, pp. 65–80.

Here we explain how to use normed division algebras to describe vectors (and thus spin-1 particles) and spinors (and thus spin-1/2 particles) in spacetimes of dimensions 3, 4, 6 and 10. We use this description to derive the 3- $\psi$ ’s rule, an identity obeyed by three spinors only in these special dimensions. We also explain how the 3- $\psi$ ’s rule is important in supersymmetric Yang–Mills theory. This stuff was known before, but not explained all in one place.

John Baez and John Huerta, Division algebras and supersymmetry II, Adv. Math. Theor. Phys. 15 (2011), 1373–1410.

Here go up a dimension and use normed division algebras to derive a special identity that is obeyed by 4 spinors in dimensions 4, 5, 7 and 11. This is called the 4- $\Psi$ ’s rule, and it’s important for supersymmetric 2-branes.

More importantly, we start studying how the symmetries of superstrings and super-2-branes arise from the normed division algebras. Mathematicians and physicists use Lie algebras to study symmetry, as well as generalizations called ‘Lie superalgebras’, which describe symmetries that mix bosons and fermions. Here we study categorified versions called ‘Lie 2-superalgebras’ and ‘Lie 3-superalgebras’. It turns out that the 3- $\psi$ ’s rule is a ‘3-cocycle condition’ — just the thing you need to build a Lie 2-superalgebra extending the Poincaré Lie superalgebra! Similarly, the 4- $\Psi$ ’s rule is a ‘4-cocycle condition’ which lets you build a Lie 2-superalgebra extending the Poincaré Lie superalgebra.

Next, try this:

John Huerta, Division algebras and supersymmetry III.

At this point John Huerta sailed off on his own!

In this paper John cooked up the ‘Lie 2-supergroups’ that govern classical superstrings in dimensions 3, 4, 6 and 10. Just as a group is a category with one object and with all its morphisms being invertible, a 2-group is a bicategory with one object and with all its morphisms and 2-morphisms being weakly invertible. A Lie 2-supergroup is a bicategory internal to the category of supermanifolds. John shows how to derive the pentagon identity for this bicategory from the 3- $\psi$ ’s rule!

And here’s his new paper, the last of the series:

John Huerta, Division algebras and supersymmetry IV.

Here John built the ‘Lie 3-supergroups’ that govern classical super-2-branes in dimensions 3, 4, 6 and 10. A 3-group is a tricategory with one object and with all its morphisms, 2-morphisms and 3-morphisms being weakly invertible. John shows how to derive the ‘pentagonator identity’ — that is, a commutative diagram shaped like the 3d Stasheff polytope — from the 4- $\Psi$ ’s rule.

In case you’re wondering: I believe this game stops here. I’m pretty sure there isn’t a nontrivial 5-cocycle (valued in the trivial representation) which gives a Lie 4-superalgebra extending the Poincaré superalgebra in 12 dimensions. But I hope someone proves this, or has proved it already!

Of course, Urs Schreiber and collaborators have done vastly more general things using a more intensely modern point of view. For example:

Hisham Sati, Urs Schreiber and Jim Stasheff, L_∞ algebra connections and applications to String- and Chern-Simons n-transport.
Domenico Fiorenza, Hisham Sati, Urs Schreiber, Super Lie n-algebra extensions, higher WZW models, and super p-branes with tensor multiplet fields.
Urs Schreiber, Differential cohomology in a cohesive infinity-topos.

But one thing the ‘Division Algebras and Supersymmetry’ series has to offer is a focus on the way normed division algebras help create the exceptional higher algebraic structures that underlie superstring and super-2-brane theories. And with the completion of this series, I can now relax and forget all about these ideas, confident that at this point, the minds of a younger generation will do much better things with them than I could.

I should add that Layra Idarani has been ‘live-blogging’ his reading of John Huerta’s new paper:

Layra Idarani, Liveblogging “Division Algebras and Supersymmetry IV” by John Huerta, part 0 .
Layra Idarani, Liveblogging “Division Algebras and Supersymmetry IV” by John Huerta, part 1 .
Layra Idarani, Liveblogging “Division Algebras and Supersymmetry IV” by John Huerta, part 2 .

So, you can get another perspective there.

Posted at October 5, 2014 10:25 PM UTC

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Re: M-theory, Octonions and Tricategories

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But one thing the ‘Division Algebras and Supersymmetry’ series has to offer is a focus on the way normed division algebras help create the exceptional higher algebraic structures that underlie superstring and super-2-brane theories.

three cheers for this! Now let’s look for spaces with interesting bundles for these structures ;-)

Posted by: David Roberts on October 7, 2014 1:42 AM | Permalink | Reply to this

Re: M-theory, Octonions and Tricategories

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I don’t know anything about this, so let me see if I’ve got this right: there are gadgets called Lie $n$ -superalgebras, and in particular there is a Lie 1-superalgebra called the Poincare superalgebra. A Lie n-algebra can be said to extend a Lie 1-superalgebra (if the 1-superalgebra is some sort of quotient of the $n$ -superalgebra?) and these extensions correspond to $(n+1)$ -cocycles. And it turns out that the Lie $n$ -superalgebra extensions of the Poincare Lie 1-superalgebra are $(n+k)$ -dimensional, where $k$ is the dimension of a normed division algebra, for $n=1,2$ – but for larger $n$ there are expected to be no such extensions.

I guess this is sort of orthogonal to the intent of the papers, but is it possible to say a few words about why the possibilities are so limited? In particular, why shouldn’t there be any extensions for larger $n$ ?

Have extensions like this been investigated over other Lie 1-superalgebras? For example, maybe there would be some physical significance to looking at just the Lorentz group rather than the full Poincare group?

Posted by: Tim Campion on October 7, 2014 10:35 PM | Permalink | Reply to this

Re: M-theory, Octonions and Tricategories

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Tim wrote:

I don’t know anything about this, so let me see if I’ve got this right…

Good, I’m glad you’re asking questions. I imagine this stuff sounds rather intimidating to most people; that’s one reason I’m not working on it anymore—I prefer to be a bit closer to the ground. But it’s fun.

there are gadgets called Lie $n$ -superalgebras, and in particular there is a Lie 1-superalgebra called the Poincare superalgebra.

Right. A Lie 1-superalgbra is usually called just a Lie superalgebra, just as a Lie 1-algebra is a Lie algebra and a Lie 1-group is just a Lie group. ‘1’ means that we’re working with familiar mathematical objects that are just sets, not categories or 2-categories or… A 1-thing has just one ‘layer’: its just a set of elements, which you can visualize as dots. A 2-thing has two ‘layers’: objects and morphisms. You can visualize these as dots and arrows. And so on: an $n$ -thing has $n$ layers.

A Lie $n$ -algebra can be said to extend a Lie 1-superalgebra (if the 1-superalgebra is some sort of quotient of the $n$ -superalgebra?)…

Yes, that’s exactly right: if you know about any sort of extensions (like group extensions or Lie algebra extensions), this is a lot like that. The only difference is that we’re extending a 1-thing up to an $n$ thing. So the quotient map ‘squashes down’ the $n$ -thing to a mere 1-thing.

… and these extensions correspond to $(n+1)$ -cocycles.

Right. At least, that’s true for extensions where we are taking a 1-thing and extending it to an $n$ -thing by putting on new stuff only at the very top layer. There’s a more general story that applies to more general extensions, but it involves a more sophisticated concept of cocycle than the kind you meet in school.

If you’re used to ordinary extensions, like extending a group to a group and getting a new bigger group, that’s the case $n = 1$ . It works kinda the same for groups, Lie algebras, Lie superalgebras, and many other gizmos. In the case $n = 1$ the ‘top layer’ is the same as the ‘bottom layer’. So, it’s a sort of degenerate case.

The first really interesting case—where I’m denigrating the usual fascinating theory of group extensions as uninteresting—is $n = 2$ . Then we’re extending something like a group (which is just a set with extra structure) to something like a 2-group (which is a category with extra structure) by throwing in some morphisms. The 3-cocycle says precisely how to do this. It’s like the ‘jam’ that we stick in the ‘layer cake’ to make the top layer stick to the bottom layer.

And it turns out that the Lie $n$ -superalgebra extensions of the Poincare Lie 1-superalgebra are $(n+k)$ -dimensional, where $k$ is the dimension of a normed division algebra, for $n = 1,2$ – but for larger $n$ there are expected to be no such extensions.

This isn’t quite right. As you probably know there’s not just one Poincaré group, there are lots, depending on the dimension of spacetime. Let me call that $d$ . If you think there’s just one Poincaré group from taking physics course, that’s the case $d = 4$ .

Similarly, there’s one Poincaré Lie algebra for each dimension $d = 1,2,\dots$ .

The Poincaré Lie superalgebra is a bigger thing containing the Poincaré Lie algebra but also ‘supertranslations’ which go in ‘odd’ or ‘spinorial’ directions, new weird directions that have to do with spin-1/2 particles. There’s more than one Poincaré Lie superalgebra for each dimension $d$ , because there’s more than one way to define these spinorial directions.

So, for each $d$ and each choice of Poincaré Lie superalgebra, one can ask whether it admits a nontrivial extension to a Lie $n$ -superalgebra. If we restrict attention to extensions where we only slap on a new layer ‘at the very top’, this is a question we can answer if we know the cohomology of the Poincaré Lie superalgebra.

But actually it’s lots of questions, because there’s another choice to make: what we slap on as the new top layer! Different choices of the top layer will be classified by cohomology with different ‘coefficients’. In John Huerta’s papers, he is only using a copy of $\mathbb{R}$ as the new top layer. This is the simplest choice.

So: what we need is for someone to systematically go through all dimensions $d$ , all the Poincaré Lie superalgebras that arise for each dimension, and all $n$ , and compute the $n$ th cohomology of these Poincaré Lie superalgebras using lots of different coefficients… or at the very least, using $\mathbb{R}$ as coefficients.

This sounds very tiring, but mathematicians actually enjoy work of this sort, and they get paid for it. The best I’ve seen so far is this:

M.V. Movshev, A. Schwarz and Renjun Xu, Homology of Lie algebra of supersymmetries and of super Poincare Lie algebra.

Abstract: We study the homology and cohomology groups of the super Lie algebra of supersymmetries and of super Poincaré Lie algebras in various dimensions. We give complete answers for (non-extended) supersymmetry in all dimensions $d \le 11$ . For dimensions $d = 10,11$ we describe also the cohomology of reduction of supersymmetry Lie algebra to lower dimensions. Our methods can be applied to extended supersymmetry algebras.

Finally, to actually answer your question: in practice getting nontrivial cocycles on Poincaré Lie superalgbras means to proving certain identities involving vectors and spinors, called ‘Fierz identities’. These are surprisingly tricky, and the only ones I understand arise from normed division algebras. Perhaps there are patterns having to do with the dimension $d$ of spacetime mod 8. Or perhaps things get boring above a certain dimension—just like there are no normed division algebras after the octonions.

I just don’t know. I haven’t even stared hard at the data up to $d = 11$ and tried to see if some period-8 repetition starts showing up.

My remark “I believe the game stops here” was quite limited in scope. John Huerta and I showed how to use the octonions to get a Lie 2-superalgebra extending the Poincaré superalgebra in dimension $d = 10$ , and a Lie 3-superalgebra extending the Poincaré superalgebra in dimension $d = 11$ . I was just betting that this trick fails to give a Lie 4-superalgebra extending the Poincaré superalgebra in dimension $d = 12$ . But I have no proof, and someone needs to compute some cohomologies for $d = 12$ to answer this question.

Have extensions like this been investigated over other Lie 1-superalgebras? For example, maybe there would be some physical significance to looking at just the Lorentz group rather than the full Poincaré group?

People probably understand cohomologies of Lorenz and Poincaré Lie algebras and groups to their heart’s content. The tricky stuff starts happening when you make them ‘super’, and that only kicks in when you include translations. There’s no ‘Lorentz Lie superalgebra’, at least that I know of.

If you want to simplify life, it makes more sense to focus on the ‘translation Lie superalgebra’, which contains ordinary translations and also supertranslations. This is something John Huerta talks about, and I think this is the ‘super Lie algebra of supersymmetries’ mentioned in that paper abstract.

Posted by: John Baez on October 8, 2014 9:41 PM | Permalink | Reply to this

Re: M-theory, Octonions and Tricategories

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There is no a Lie superalgebra whose bosonic part is the Lorentz Lie algebra, and the reason is that a natural susy-generalization of o(n) is osp(n|2m) and this superalgebra has the bosonic subalgebra given by so(n) (+) sp(2m). This problem is circumvented in string theory by considering AdS algebras o(d-1,2), whose subalgebra is the d-dim Lorentz algebra so(d-1,1), so that one can use the superalgebras of the type o(n|2m) or su(n|m). For example, for strings in the space AdS_5 x S^5, the bosonic Lie group is SO(4,2)x SO(6), which can be immersed into SU(2,2|4).

Posted by: Aleksandar Mikovic on October 11, 2014 11:59 AM | Permalink | Reply to this

Re: M-theory, Octonions and Tricategories

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The reason why there is no a Lie superalgebra for the Lorentz group is that a natural supersymmetric generalization is a Lie superalgebra osp(n|2m), whose bosonic subalgebra is o(n) + sp(2m).

This problem is circumvented in string theory by using the anti-de Sitter group SO(d-1,2), which contains the Lorentz subgroup SO(d-1,1). For example, strings in the space AdS_5 x S^5 have a symmetry group SO(4,2) x SO(6), which is the bosonic subgroup of the supergroup SU(2,2|4).

Posted by: Aleksandar Mikovic on October 11, 2014 12:25 PM | Permalink | Reply to this

Re: M-theory, Octonions and Tricategories

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It’s a bit too early to hang up your octonionic gloves, John. Someone still has to relate this work to generalized spin networks and foams.

Duff et al recently used octonionic 4-spinors to re-write the M-algebra, giving an octonionic description of M-theory in D=11. However, as you know, octonionic 4-spinors require 4x4 matrices, and one loses nice Jordan algebraic properties, as well as a projective space interpretation.

Also, Duff’s team has used the ‘gravity as squared Yang-Mills’ to fill out a magic pyramid of such division algebraic would-be theories. John Schwarz, while at Caltech, was quite delighted to see such a pyramid, as nonassociativity was thought to be a game-killer, back in the late 70’s. So now the game, so to speak, is back on. Where does one go from here?

Eventually, one must confront the number theoretic signs staring everyone in the face, coming from the Moonshine relation. We at least know Conway’s group $Co_0$ can be absorbed by F4, thanks to the octonionic Leech lattice. As $Co_0$ is in the Monster, it seems the Monster group is actually doing a lot of work describing symmetries in D=24,26,27. So why the restriction to integers, one may ask? Well, physically one is considering a charge space in D=27, so charge quantization is forcing one to use the integers. One may also want to go further and study motives for Hermitian symmetric domains, which are the moduli spaces for the extremal black holes that have such a D=27 charge space. However, such exceptional domains are not yet fully understood mathematically, as one would like to continue the Langland’s programme here and tabulate L-functions for motives in D=24,26,27. The intent here is to generate such spaces from the automorphic data, alone.

And of course, lastly, there’s scattering amplitudes. Even John Schwarz asked about octonionic scattering. The division algebraic approach naturally leads one to the picture of using 2x2 octonionic lightlike bispinors to construct amplitudes with SO(9,1) symmetry.

There’s still so much to do.

Posted by: Metatron on October 11, 2014 5:09 PM | Permalink | Reply to this

Re: M-theory, Octonions and Tricategories

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Duff et al recently used^{[citation needed]}

We at least know Conway’s group^{[citation needed]}

construct amplitudes with $SO(9,1)$ symmetry^{[citation needed]}

Not trolling, just curious to see it!

Posted by: David Roberts on October 13, 2014 12:05 AM | Permalink | Reply to this

Re: M-theory, Octonions and Tricategories

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Here are some references for David Roberts, to help flesh out Metatron’s remarks:

“Duff et al recently used octonionic 4-spinors to re-write the M-algebra, giving an octonionic description of M-theory in D=11.”

I think that’s

A. Anastasiou, L. Borsten, M. J. Duff, L. J. Hughes and S. Nagy, An octonionic formulation of the M-theory algebra.

“Also, Duff’s team has used the ‘gravity as squared Yang-Mills’ to fill out a magic pyramid of such division algebraic would-be theories.”

I think that’s

A. Anastasiou, L. Borsten, M. J. Duff, L. J. Hughes and S. Nagy, A magic pyramid of supergravities.

“We at least know Conway’s group $\mathrm{Co}_0$ can be absorbed by $\mathrm{F}_4$ , thanks to the octonionic Leech lattice.”

I think that’s

Michal Rios, U-Duality and the Leech lattice.
Robert A. Wilson, Octonions and the Leech lattice.

Posted by: John Baez on October 13, 2014 6:09 AM | Permalink | Reply to this

Re: M-theory, Octonions and Tricategories

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Thanks John. I remember looking at that first one now and being blinded by the onslaught of formulas with little (for a mathematician) proofs.

Posted by: David Roberts on October 13, 2014 6:28 AM | Permalink | Reply to this

Re: M-theory, Octonions and Tricategories

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Metatron wrote:

There’s still so much to do.

Yes indeed! I’m bowing out not because everything has been figured out, but because I think these topics have a kind of inevitability to them: the necessary work will get done regardless of whether I’m involved or not—as long as civilization survives. Actually I feel this way about $n$ -categories, higher gauge theory and also the exceptional mathematical structures related to the octonions, $\mathrm{E}_8$ , the Leech lattice and the Monster. They sing a siren song that mathematicians are unable to resist! They captivate us with their beauty.

On the other hand, the kind of math we need to help our civilization survive seems more iffy. I’ve written about it here. I feel it’s struggling to be born and not enough people are helping it. So I feel a kind of duty to see what I can do about that. This becomes more and more true as I get older.

But I hope people working on the topics you list write good review articles, because I want to know what happens.

Posted by: John Baez on October 13, 2014 5:57 AM | Permalink | Reply to this

Re: M-theory, Octonions and Tricategories

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I regret derailing the discussion, but do you really think that working on mathematics is the “optimal” use of ones resources to save the planet from ecological disaster?

I think that new mathematics is so far down the list of useful things to do, that it is almost irrelevant. High up on the list are

Politcal activism. We need massive regulations, on a global scale, and we need them yesterday.
Profitable models of sustainable agriculture. For example, the work that The Land Institute is doing. I also have high hopes for perennial leaf protein crops, like mulberry, made into a leaf protein concentrate. More ideas like this, which conserve soil, fix carbon, and support biodiversity are essential. Having the ideas is not enough, we need this to be profitable enough to spur massive development and implementation.
Engineering green energy sources. High tech like solar panels, fusion generators, nanotech, and low tech like rocket mass stoves.
Space colonization! We need self sustaining colonies on other planets if we want to survive. Probably the first place to start is with a good biodome project here on Earth, preferably in a desert or underground.

When we talk about “saving the planet”, I also hope we all realize we are really talking about saving humanity. If we go extinct, even if we take down 99% of the other species with us, life will survive and bloom into a similar level of complexity in another couple million years. We have had similar mass extinction events before, and life on Earth has recovered just fine. So it is really about saving a planet which is familiar to us, and in which we could live.

My personal approach to these problems is to try and teach mathematics well. I am starting an online free interactive math website which will hopefully give a lot of people access to free high quality math instruction. This just generates more engineers and scientists hopefully. I also want to grow most of my own calories and protein in the form of woody perennials, namely chestnuts and hazelnuts, as a personal step toward environmental sustainability.

I am not trying to belittle what you are doing (this weeks finds was/is a huge inspiration to me as a young mathematician), but I am genuinely curious about your thoughts on the potential of new mathematics to have a big impact on environmental problems. Can you “change my view”?

Posted by: Steven Gubkin on October 13, 2014 4:29 PM | Permalink | Reply to this

Re: M-theory, Octonions and Tricategories

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I also realize that my own actions are pretty far down on the list of “optimal use of energy”. I kind of wish I had majored in plant science and gone to work for the Land Institute, but that is not the path I took. I feel that, given my life path, I am doing the best I can with the knowledge and skills I have developed. Maybe you feel similarly?

Posted by: Steven Gubkin on October 13, 2014 4:37 PM | Permalink | Reply to this

Re: M-theory, Octonions and Tricategories

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Steven Gubkin wrote:

I regret derailing the discussion, but do you really think that working on mathematics is the “optimal” use of ones resources to save the planet from ecological disaster?

It depends who “one” is. I don’t think the question makes sense in the abstract.

Believe me, this question has vexed me immeasurably ever since I decided to quit working on fancy-schmancy pure mathematics and theoretical physics. To me mathematics is quite obviously not the optimal thing for an arbitrary person to do if they want to save the planet from ecological disaster. If you were going to raise a child in a way that would make them into an ecological superhero, you would not spend a lot of time teaching them about $n$ -categories, quantum gravity, exceptional algebraic structures and the like. Unfortunately, I had already spent about 35 years studying this stuff before I decided to do something more useful. At first I hoped I could make a radical shift in my career, but I found — to my dismay — that I was not capable of it. So, I decided to make the best of the brain I had built for myself.

Even under this constraint, I’m never sure I’m doing the right thing. I keep vacillating between more ‘applied’ activities and more ‘theoretical’ ones: alas, I feel much more comfortable in highly theoretical territory, having spent my adult life there. For example, I’d like to work on El Niño prediction, but my heart keeps pulling me toward the abstract delights of network theory… and sometimes I even get sick of that, and indulge in a dose of pure mathematics, mainly as a kind of hobby now.

On the bright side, this sort of confusion makes life much more interesting than back when I thought I had it all planned out. I’ve read that switching career directions in middle age has a way of revitalizing people, bringing back the creativity associated with youth. It’s true! But they forgot to say that it also brings back the insecurity and bewilderment of youth. Not the pimples, luckily; and I have tenure now.

The benefits of highly abstract math take quite a while to be felt; often half a century or more. If it helps at all, it can sometimes help a lot — but only after a lot of work. For example, Gödel’s work on logic and computability helped inspire Turing, who helped develop ideas that led to the computer I’m typing on now. I hope that civilization lasts long enough that work on ‘green mathematics’ will someday have an equally big effect.

Don’t get me wrong: I don’t flatter myself that I’m another Gödel or Turing. We probably need a bunch of people to try hard and fall flat on their face before someone gets the really good ideas. I’m trying hard, and I will probably fall flat on my face.

Posted by: John Baez on October 13, 2014 11:02 PM | Permalink | Reply to this

Re: M-theory, Octonions and Tricategories

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I understand. Well, it’s still a struggle but for anyone who has read your octonions survey, the beauty of division and composition algebras is motivation enough to keep going.

Let’s suppose the division algebras are indeed behind the mathematical structure of the universe. This would have profound implications, since so many structures in mathematics seem to be related to the octonions. In a general sense, you’d be right about inevitability, as any sufficiently intelligent civilization in this universe would also have discovered the division algebra structure behind the universe after studying quantum gravity. In this mathematical sense, we’d all have reached a climactic milestone, scientifically, when our mathematical and physical disciplines become grand unified.

The technological advances coming from such a unification would be unimaginable.

Posted by: Metatron on October 13, 2014 8:35 PM | Permalink | Reply to this

Re: M-theory, Octonions and Tricategories

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Metatron wrote:

The technological advances coming from such a unification would be unimaginable.

It’s true that I can’t imagine them. We are currently unable to build interesting technologies using what we know of general relativity: for example, we don’t command enough energy to make black holes. Even the Standard Model has lots of features that we don’t know how to use, because it’s hard to attain the extreme conditions where these features become important. For example, we can make a quark-gluon plasma for a very short time by smashing ions into each other, but we can’t do much with it. The only practical application I know of particles in the second generation of fermions is muon spin spectroscopy. The third generation doesn’t help us at all yet.

So, it seems quite possible that a beautiful, unified theory of all particles and forces won’t change our technological abilities at all in the short term (meaning, say, a hundred years or so). Maybe this theory would in principle let us create new ‘baby universes’ or other wonderful things — but not in practice, because the energies required are higher than we can muster.

Posted by: John Baez on October 16, 2014 8:23 PM | Permalink | Reply to this

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October 5, 2014