What Are the Implications of the Leech Lattice and Strings in String Theory?

[arivero]

I am ashamed that no one of the textbooks on string theory take care to refer to the Leech lattice or to its (n+1,1) kin, even if only of a footnote when they use selfdual lorentzian lattices.

I wonder if it is ignorance or intentional misinformation. Because the singularity of the (25,1) self dual even lattice is a clasical, mostly grouptheoretical if you wish, result, and then they should be forced to ackowledge than their "critical dimension" is not exclusive of the realm of String Theory.

 

[R.X.]

I am ashamed that no one of the textbooks on string theory take care to refer to the Leech lattice or to its (n+1,1) kin, even if only of a footnote when they use selfdual lorentzian lattices.

I wonder if it is ignorance or intentional misinformation. Because the singularity of the (25,1) self dual even lattice is a clasical, mostly grouptheoretical if you wish, result, and then they should be forced to ackowledge than their "critical dimension" is not exclusive of the realm of String Theory.

Ashamed about what? There were lots of papers about the 24 dimensional leech lattice in the 1980's, and its lorentzian extension. Some of the major proponents of those ideas were Olive and Goddard. All sorts of games have been played (eg compactifications of bosonic strings on the Leech lattice), but nothing substantial or particular interesting came out. That's why it is not mentioned anywhere in textbooks today - superfluous ballast for beginners. Or do you think that crucial information is intentionally kept secret in order that poor grad students don't see the light??

There is an abundance of unique or exceptional mathematical structures, and there is often the hope that some of this should show up in physics. Everyone dreams at some point of connecting the standard model to the monster group, to octonions, some other wonderful algebraic structure. But in most cases this is a blind, or unfruitful alley.

Sometimes it does work out, though. The number 24 indeed shows also up as the light cone dimension of the bosonic string, relatedly as a property of the eta function, and then again relatedly as the Euler number of K3. The ADE classification of del Pezzo surfaces, singularities on K3 manifolds and related self-dual lattices, exceptional groups, yes all this turned up in string theory. All these connections are there, but while interesting or even fascinating, they are not really useful for particle physics. They tend to show up in very specific circumstances (eg in higher dimensions or in theories with extended susy), and not in more realistic models.

 

[arivero]

Ashamed about what? There were lots of papers about the 24 dimensional leech lattice in the 1980's, and its lorentzian extension. Some of the major proponents of those ideas were Olive and Goddard.

And indeed I like their "Algebras, Strings and Lattices" paper.

But no is that there is a lot. Looking for Leech Lattice in the arxiv physics fulltext search shows about 70 papers. Most of them, as you point out, play around the compatification game. The real point about Leech lattice is the critical dimension of the bosonic string, via the vertex algebra and associated machinary.

There are above one hundred eprints in the math section, particularly Borcherds his thesis http://arxiv.org/abs/math.NT/9911195 is there. But the are mostly pure math papers.

I can not detect any trace of Olive and Goddard paper really used in the main textbooks. The textbooks invoke lorentizan lattices in the right place when speaking of the Heterotic strings, but it is only because the original papers invoke it, and they do not go beyond the case II(17,1) they need for the heterotical string.

Or do you think that crucial information is intentionally kept secret in order that poor grad students don't see the light??

I think that string theoretists are proud of the way they come to discover the critical dimension and they prefer to explain it on its way, call it the historical way. But the path from lattice classification to vertex algebra and then Kac-Moody etc should be the one in textbooks, as it stress the mathematical structure.

All these connections are there, .

The simple fact you consider it "connections", and not root fundations, underlines the point I was raising. It is sort of telling that Lie Groups are "connected" to Gauge Fields.

 

[arivero]

Just to illuminate the point, some excerps from Baez old week95:

even unimodular lattices are only possible in certain dimensions - namely, dimensions divisible by 8.
In dimension 8 there is only one
In dimension 16 there are only two even unimodular lattices ... give us the two kinds of heterotic string theory!
in dimension 24, there are 24 even unimodular lattices,

let's think about Lorentzian lattices
It turns out that the only even unimodular Lorentzian lattices occur in dimensions of the form 8k + 2. There is only one in each of those dimensions

the fundamental roots of the even unimodular Lorentzian lattices in dimensions 10, 18, and 26 are the vectors r with r.r = 2 and r.v = -1, where the "Weyl vector" v is

... the fact that the Weyl vector is also lightlike makes a lot of magic happen in 26 dimensions. For example, it turns out that in 26 dimensions there are *infinitely many* fundamental roots, unlike in the two lower dimensional cases.

Just to add mystery upon mystery, Conway notes that in higher dimensions there is no vector v for which all the fundamental roots r have r.v equal to some constant. So the pattern above does not continue.

Also Goddard laudatio of Borcherds, math/9808136, is interesting reading.

 

[R.X.]

I can not detect any trace of Olive and Goddard paper really used in the main textbooks. The textbooks invoke lorentizan lattices in the right place when speaking of the Heterotic strings, but it is only because the original papers invoke it, and they do not go beyond the case II(17,1) they need for the heterotical string.

Well I guess the Narain lattices of the form (20,4) or (22,6) must be mentioned in textbooks, because they play an important role in six and four dimensional theories. Higher dimensional lattices don't play a big role because a lattice refers to a compact torus, and so all what you can do eg with the 24d Leech lattice or its (25,1) cousin, is to compactify the bosonic string upon it. This may give a mathematically interesting structure in two or zero spacetime dimensions, but physically these theories are not very interesting or important. Nevertheless, I fully agree on that the influential work of Goddard and Olive would have deserved a citation in a textbook.

I think that string theoretists are proud of the way they come to discover the critical dimension and they prefer to explain it on its way, call it the historical way. But the path from lattice classification to vertex algebra and then Kac-Moody etc should be the one in textbooks, as it stress the mathematical structure.

While the number 24 or 26 seems magical from a variety of perspectives, it is however not clear whether there are always physically significant relations between those. In particular
I don't see how the critical dimension of the bosonic string, which lives in 26 (ie, (25,1)) _uncompactified_ dimensions, could have been derived from the classification of lattices which has to do with compactified dimensions; there is certainly a lot of selfdual lattices also in other, eg 32 dimensions, but I am not aware of any relevance of this for a physical model.

Vertex operators and Borcherds algebras indeed show up in physics, but typically in a way that is not so canonical and useful as one may have wished. Eg the BPS states related to any given Calabi-Yau manifold form an algebra of that sort, but this algebra is different for every Calabi-Yau, and thus not canonical and interesting (see papers by Harvey and Moore for details). The more canonical or distinguished objects tend to show up in theories with more supersymmetries, because those are more constrained. In theories with less supersymmetries which are less constrained (while more close to the physics we want to describe), distinguished algebraic objects play little role as far as I know.

.. the fact that the Weyl vector is also lightlike makes a lot of magic happen in 26 dimensions. For example, it turns out that in 26 dimensions there are *infinitely many* fundamental roots, unlike in the two lower dimensional cases.

Just to add mystery upon mystery, Conway notes that in higher dimensions there is no vector v for which all the fundamental roots r have r.v equal to some constant. So the pattern above does not continue.

This magic is fascinating but again, what do you conclude from it for physics. Be assured that a lot of people have pondered upon this kind of questions since ages, and the reason why you don't read about them in textbooks, is that not much significant physics came out.

Actually this is a good occasion for a retort against the omnipresent accusations that string theorists would be too much focused on abstract mathematics for the sake of it. In fact this is an evolving subject where many ideas and paths have been and are being followed, and when a direction turns out not to be fruitful, it is abandoned and consequently not mentioned in textbooks. As far as I can tell, most string physicists have written off algebraic constructions such as lattices and vertex algebras precisely because of that, despite these things being mathematically very very sweet.

 

[Thomas Larsson]

There were lots of papers about the 24 dimensional leech lattice in the 1980's, and its lorentzian extension. Some of the major proponents of those ideas were Olive and Goddard. All sorts of games have been played (eg compactifications of bosonic strings on the Leech lattice), but nothing substantial or particular interesting came out. That's why it is not mentioned anywhere in textbooks today - superfluous ballast for beginners.

Some of the relevant structures, in particular non-affine Kac-Moody algebras, appear in recent works by Nicolai/West and others on E10/E11. Not so many people seem interested, though.

There is an abundance of unique or exceptional mathematical structures, and there is often the hope that some of this should show up in physics. Everyone dreams at some point of connecting the standard model to the monster group, to octonions, some other wonderful algebraic structure. But in most cases this is a blind, or unfruitful alley.

I agree. Subjective beauty is not enough, even when some mathematicians start to chant magic and mystery. A physicist should know why it is inevitable that a proposed algebraic structure must arise.

Let us consider a well-understood example. The reason why the (super) Virasoro algebra is important to perturbative (super) string theory is not its intrinsic beauty (it is a quite cool algebraic structure, although I showed that it is not as exceptional as people thought), but because it is the correct quantum form of the correct constraint algebra. The word "correct" here means that it arises in lowest-energy, i.e. quantum, representations.

Analogously, the reason why the generalization of the Virasoro algebra to four dimensions must be relevant to gravity is not that it is exceptionally beautiful. It is indeed strikingly beautiful (but beauty is always in the eyes of the beholder), but this is not the reason why it is relevant to quantum gravity. Instead, it follows from the following argument. We know that general relativity is the correct classical theory of gravity. We also know that the correct constraint algebra of GR is the full spacetime diffeomorphism algebra (in covariant formulations). Finally, we know the correct quantum form of the spacetime diffeomorphism algebra acquires an abelian extension, making it into a multidimensional generalization of the Virasoro algebra.

This is the reason why I desided to discover the multidimensional Virasoro algebra and its representation theory, many years ago, cf. my 1999 manifesto in http://www.arxiv.org/abs/gr-qc/9909039 .

 

[arivero]

(eg compactifications of bosonic strings on the Leech lattice), .

parenthesis: Can you expand on this concrete game?

 

[R.X.]

parenthesis: Can you expand on this concrete game?

It's been a while and I don't have the papers nearby, but what I recall is that since the Leech lattice has no roots of length two (in contrast to four), there are no non-abelian gauge symmetries generated, so this seems less interesting from the physics point of view. On the other hand one may ponder about whether the monster group is realized on the massive spectrum - I don't remember the details any more. There are also super extensions of this which may be interesting as well; there is a whole pile of math literature, check in particular also Lepowsky.

Some of the relevant structures, in particular non-affine Kac-Moody algebras, appear in recent works by Nicolai/West and others on E10/E11. Not so many people seem interested, though.

This is definitely very interesting stuff. In a way the most exceptional (in some sense extremal) objects are tied to 11d supergravity, which is also physicswise extremal (the highest dimensional susy theory in the classification of Nahm), and still today people are working on uncovering exceptional symmetries in this theory.

But again...while those theories with maximal supersymmetry have the nicest mathematical properties, the theories we really want to describe are at the opposite end, and for those no unique exceptional or otherwise distinguished structures are known (at least to me). That's in a nutshell part of the dilemma!

This is the reason why I desided to discover the multidimensional Virasoro algebra and its representation theory, many years ago, cf. my 1999 manifesto in http://www.arxiv.org/abs/gr-qc/9909039 .

This seems nice. And what is the significance for physics - is there any model you can solve with it, or is there otherwise a physical quantity you can compute with it?