*Science*:

Quantum Criticality in an Ising Chain: Experimental Evidence for Emergent E8 Symmetry (abstract)

They used an effectively one-dimensional piece of a very cold ferromagnet, cobalt niobate (Co Nb2 O6), and adjusted the strong transverse magnetic field to get close to a critical point. At one side of the phase transition, they observe two sharp mesons with the correct, predicted golden ratio (1.618=1/.618). See also Physics World and Google News.

The existence of the group was first realized by Wilhelm Killing in 1887: see his picture. Between 1888 and 1890, he made sure that everyone else knew that he had discovered it. New key insights were added by Élie Joseph Cartan in 1894.

What is the group?

If you have a D-dimensional Euclidean space (or a ball), it doesn't change when you rotate it. The set of rotations is known as the SO(D) group. It's compact (if you go/rotate in the same direction, you will eventually return close to your starting point, like Magellan) and simple (the group can't be written as two groups acting on two different objects, roughly speaking). There exist two additional infinite sequences of such groups - rotations in complex and quaternionic spaces, namely SU(N) and USp(2N) - which are also compact and simple.

Are there other abstract, compact and simple Lie groups, that don't belong to the sequences above? Yes, there are. There are exactly five "exceptional" Lie groups: G2, F4, E6, E7, E8. The E8 group is the largest one and contains the other four exceptional groups as subgroups. Its dimension is 248 - and the fundamental representation has the same dimension.

You may understand the group as a set of rotations of a 248-dimensional Euclidean space that are constrained in a very special way so that what remains is neither too much, nor too little: it is a beautiful group. Or you can construct the group as an extension of SO(16) by additional 128 generators transforming as a chiral spinor of SO(16) which nicely commute to the original SO(16) and peacefully co-exist. Or... there are many other ways to build or define E8. As I mentioned, there are only a few mathematical jewels of this kind, and E8 is the largest one.

The mathematical beauty - and importance in geometry and algebra - has been known for quite some time. But just like most fancy mathematical structures, it has been thought to be irrelevant in physics. After all, almost all Lie groups were thought to be irrelevant in physics just a century ago. Try to appreciate how important they're in the Standard Model today.

**String theory enters the scene**

String theory has completely changed the scene. Let's return 25 years into the past.

In the 1970s, realistic versions of string theory were thought to suffer from anomalies - inconsistencies that spoil the gauge symmetries and prevent us from decoupling the unphysical modes of the gauge bosons and gravitons in spacetime.

As we discussed half a year ago, Green and Schwarz have profoundly changed the situation. The other people's arguments that the anomalies had to persist were sloppy. In Aspen 1984, Green and Schwarz did the full calculation and found that the anomalies exactly canceled for the SO(32) gauge group. One only adjusts one number, 32 half-colors, and lots of independent sums of complicated rational numbers miraculously add up to zero. All the anomalies disappear.

One of the purely gravitational anomalies that would destroy the the diffeomorphism symmetry - and hurt the graviton and the equivalence principle - if they persisted is canceled for all 496-dimensional groups. Note that 31 x 16 = 496. However, there are many other terms in the ten-dimensional spacetime anomalies that have to cancel, too. And for SO(32), they do cancel because of some special properties of the adjoint representation of SO(32).

However, it was soon realized that there was actually another pretty solution where all the anomalies cancel. The E8 x E8 group - two copies of the largest exceptional group - has the dimension equal to 248+248 = 496, too. While it's much harder to show (but you should try to do it!), all the other conditions are also satisfied. The E8 x E8 group is anomaly-free, too!

And a theory with an E8 x E8 group would actually be a much better starting point to build realistic theories because E8 contains e.g. the E6 group as a subgroup - and the latter may be used as a grand unified group (E6 is the only exceptional group that has complex representations needed for chiral fermions etc.). Imagine: people had one version of string theory - type I string theory with open and closed unoriented superstrings - but arguments about group theory have convinced them that there is another nice group where everything works.

A problem was that they didn't know any version of string theory that actually had E8 x E8 group as its gauge group in spacetime.

**Heterotic strings**

This problem dramatically evaporated just one year later, in 1985. The so-called Princeton String Quartet, i.e. David Gross, Jeff Harvey, Emil Martinec, and Ryan Rohm, finally published their heterotic string papers.

I think that only these papers have ignited the "industry of beauty" and a "miracle-building factory" in string theory.

These four authors have constructed a completely new kind of string theory that also has the SO(32) gauge group - as well as its cousin whose gauge group is E8 x E8: the two ten-dimensional heterotic string theories.

The old type I string theory of Green and Schwarz was "totally supersymmetric". Both left-moving and right-moving bosonic excitations along the strings lived in 10 dimensions (8 transverse bosons) - which is the right, "critical" number of spacetime dimensions in perturbative supersymmetric string theory - and were teamed up with fermions (8 of them) to get supersymmetry both on the worldsheet and in spacetime.

However, the heterotic strings are different. They employ the heterosis, or "hybrid vigor" i.e. "outbreeding enhancement", that you obtain by convincing the 10-dimensional supersymmetric theory to have sex with the old 26-dimensional bosonic (fermion-free) string theory. What you get out of this intercourse are interesting children.

They inherit supersymmetry after the supersymmetric mother - even though it's just 1/2 of the supersymmetry relatively to type II superstring theories. The spacetime supersymmetry also guarantees that these theories don't inherit the ugly tachyon from the bosonic father (a "superluminal" sign of the instability of spacetime): the energies can't go negative in supersymmetric theories. And the limited number of supersymmetries, as guaranteed by the father, is actually better to build chiral theories with large enough gauge groups.

The fundamental strings in these new theories - the heterotic strings - are left-right asymmetric. If you look at the oscillations that move in one direction along the string - we may call it the "clock-wise direction" although I will stick to the conventional "right movers" - they think that the string propagates in 10 spacetime dimensions. However, the "counter-clockwise oscillations" along the string, the "left movers", think that the spacetime is 26-dimensional!

This may look like a contradiction but it's not because the left-movers and right-movers are pretty much separated from each other at the level of strings so one may construct similar "hybrids". That's one of the general magics of two-dimensional conformal field theories. When you look at Euclidean world sheets, this property is linked to the segregation of holomorphic and antiholomorphic functions.

**Finding two possibilities**

Fine, so how many kinds of heterotic strings can you construct? Note that 26-10 = 16, so there seem to be 16 excessive spacetime dimensions that are purely left-moving. I won't quite explain the exact details but it turns out that while purely left-moving dimensions are possible, their geometry is not completely arbitrary: it is restricted. The bosons must parameterize an even self-dual lattice.

Why? It's because only the left-moving bosons are allowed to exist. In particular, the left-moving "zero mode" or "center-of-mass degree of freedom" is the difference between the momentum and the winding number, "p-w" (sorry if the sign convention is the opposide one). However, the momentum is dual to the position in quantum mechanics. So if the position parameterizes a torus, which means that the winding numbers are in the corresponding lattice, the momentum must belong to the dual lattice (the lattice Lambda* of points such that the inner products of a site in Lambda* with all sites in Lambda are integers: note that it you "inflate" one, the dual one shrinks).

But because "p+w" must be zero (the forbidden right-moving part of the boson), these two lattices must actually be subsets of each other. The lattice (and the corresponding torus) must actually be self-dual (identical to its dual lattice): the inner product of any pair of lattice vectors must be integer while the fundamental cell must have its volume equal to one. And additional constraints in the two-dimensional theory require that the lattices must also be even: the squared length of any lattice vector must be even.

It's a mathematical fact that in 16 dimensions, there are only two even self-dual lattices, Lambda16 and Lambda8+Lambda8. The first one is the weight lattice of spin(32)/Z2 - which is the more accurate name for the SO(32) gauge group we have spent so much time with. And the second one is the root=weight lattice of E8 x E8, two copies of the largest exceptional Lie group. Correspondingly, the two gauge groups and their gauge bosons are generated from the strings in spacetime.

**The fermionic construction**

The previous paragraphs using the lattices described the "bosonic" construction. But there exists a completely equivalent description of the very same physics that uses 32 left-moving fermions instead of 16 left-moving bosons on a lattice.

The stringy consistency criteria force you to only allow the same periodic and/or antiperiodic boundary conditions for whole groups of 16 fermions (you can't treat the conditions for the separate fermions too independently) - and you must also apply the corresponding proper "GSO-like projections" to the spectrum.

When you realize how much space to maneuver is left (and eliminate some would-be consistent but sick theories with groups of 8 fermions only), you will see that there are two possibilities what you can do with the fermions: you either allow the periodic and/or antiperiodic boundary conditions only for the whole set of 32 fermions (two sectors - universally periodic P and universally antiperiodic A), or you divide the fermions to two groups of 16 fermions each, and allow four sectors (AA, AP, PA, PP), together with a new GSO-like projection.

These two solutions - possible theories of a string - end up to be identical to the two solutions found with the lattices above. The first one will produce the SO(32) group in spacetime - clearly, the group comes from rotations of those 32 fermions into each other. The other theory produces a string theory with E8 x E8 gauge bosons: the SO(16) x SO(16) subgroup should be manifest but each of those SO(16) groups is actually enhanced to a larger group, an E8, by additional massless states in the AP and PA sectors, respectively. The new states always transform as 128, a real chiral spinor of spin(16), under one of those E8 groups.

It's quite a general fact that one boson in 2 dimensions is equivalent to two fermions. The equivalence and consistency of these string theories extends at the level of the interactions, too. There also exist other ways to construct the same E8 conformal field theory - including a method with 248 bosons spanning the E8 group manifold - but I have said enough.

**Dualities**

In the 1990s, it was realized that these solutions - possible different "string theories" - are not as isolated from each other as previously thought. They're actually expansions of the same theory - one string theory, and that's it - around different points in the configuration space of scalars (the moduli space).

First, the two heterotic string theories can be continuously deformed into one another once you compactify one dimension on a circle i.e. as soon as you go from 10 large dimensions to 9 large dimensions. If you use the bosonic description, the compactification means that instead of having 16+0 left-moving plus right-moving dimensions on a lattice, you will have 17+1 dimensions.

There is actually exactly one even self-dual lattice in 17+1 dimensions (or any number of dimensions of the form (8M+P)+(8N+P) for positive P) - only one heterotic string theory in 9 spacetime dimensions. If you look at it from different angles, you get different points of the moduli space. That corresponds to different lengths of the circle in spacetime, different B-fields around it (not in 9 dimensions), and different Wilson lines around the circle that generically break any of the two gauge groups to U(1)^16 (times two more copies of U(1) that were not there in 10 dimensions).

There exist special points of the moduli space - special angles to look at the lattice - where the U(1)^18 gauge group gets enhanced (by new massless states, the non-Abelian gauge bosons) to SO(32) x U(1)^2, and/or E8 x E8 x U(1)^2. Note that none of the gauge groups contains the other one. E8 x E8 is as big as SO(32). They're just different. You can't say that there was one "master" gauge group that was broken - what the "master" group is depends on your position in the moduli space. That's one of the stringy proofs that the gauge groups aren't fundamental - they're physical redundancies that depend on your description.

Also, the two spin(32)/Z2 - or "SO(32)", if you allow me to be sloppy - theories are equivalent to each other. Type I string theory may be obtained as the strong coupling limit of the SO(32) heterotic string theory (if you instruct the strings to interact very strongly), and vice versa.

The strong coupling limit of the E8 x E8 heterotic string theory was the last one to be understood, in October 1995. Edward Witten and Petr Hořava showed that if the string coupling constant in this theory gets stronger, the originally 10-dimensional theory (counting the infinite dimensions only) develops a new, 11th dimension that is getting arbitrarily large. It is becoming M-theory, much like in the case of type IIA string theory.

The only difference is that the shape of the new, 11th dimension is not a circle: it is a time interval with two end points i.e. two 10-dimensional boundaries of the 11-dimensional spacetime. Each of them carries one E8 group (you have a product of two of them because a line interval has two end points!) - and 11-dimensional anomaly cancellation at the end of the world can actually be used to prove that the only acceptable gauge group at the boundary is E8. Things fit together perfectly. Note that many facts are being geometrized in diverse ways. Pretty.

There are other ways to get the E8 in string theory - the most important one that has not yet been discussed is a compactification on the E8 singularity. If you read my article about the ADE classification, you will learn what it is.

The "simply-laced" (with Dynkin diagrams only containing simple links, not double or triple ones) regular and exceptional Lie groups, namely SU(N), SO(2N), and E6, E7, E8, are in one-to-one correspondence with finite subgroups of SO(3) - or SU(2), if you wish. This is not just a coincidence or an

*ad hoc*map relating objects that have nothing to do with each other. String theory gives the fastest tools to understand why this miraculous correspondence exists.

In particular, the "most complex" or the "most non-Abelian" subgroup of SO(3) is the symmetry of the icosahedron - the largest Platonic polyhedron, in some counting, which is the same as the symmetry of the dual dodecahedron, which is the largest in other respects. (They're dual because if you create a "new vertex" at the center of each face, and connect the "new vertices", you get the dual Platonic polyhedron: both of them are compounded at the picture below.)

Now, if you translate this discrete subgroup Gamma(E8) of SO(3) into a subgroup of SU(2), interpreted as a group of transformations on a 2-complex-dimensional or 4-real-dimensional space, if you then orbifold the space by this group, and if you allow M-theory or type II string theory to probe the resulting singular quotient space, you will find out that there will be new E8 gauge bosons living exactly at the singular point, the fixed point, of the orbifold! (The point is stretched in 6-7 additional spacetime dimensions unaffected by the orbifolding.) This works for all ADE groups, too.

**The E8 conformal field theory**

Let's return back to the old-fashioned worldsheet which was already beautiful and amazing enough.

The two-dimensional conformal field theory with the E8 gauge group is pretty special and important - and of course, mathematically oriented physicists have studied it for quite some time. Now, a string with this extended symmetry has been constructed in the lab. And some properties of the spectrum - like the golden ratio relating two peaks in the spectrum (see e.g. this preprint) - have been experimentally checked.

It's kind of amazing that some seemingly ordinary pieces of material were enough to construct an object that has a sort of 248-dimensional rotational symmetry of a restricted type that most people (even most people who studied maths in the college) could never think of (the surfer dude did!). In some sense, they did construct a 248-dimensional monster in the lab. ;-)

The E8 group squeezes all states - and local operators - into representations of the group. And they're quite big. There are not too many singlets (one-dimensional representation that never transforms) so let's omit them. The fundamental representation (which is the same as adjoint for this particular group and no other) is 248-dimensional - it's the smallest one that does transform in some non-trivial way. If you wonder what is the "next" representation, it is 3875-dimensional. Then you already get to tens of thousands. So these are pretty big families of the states!

On the other hand, the E8 symmetry is even more powerful at the level of the interacting string theory. Unfortunately, they haven't allowed the ferromagnets to interact in the stringy way yet. :-) The strings they isolated would have to be given the freedom to split and join and that's too much to ask.

At any rate, many properties of the theories describing the stringy world sheets - two dimensional conformal field theories - may now be measured in the lab. Of course, the "strings" produced in the lab are "fundamental strings" of string theory: unlike their true fundamental counterparts from string theory, the building blocks of the Ising model are composed out of cobalt and other atoms.

However, when you look how naturally such strings with all their big symmetries and other nice features arise in condensed matter physics, and if you're a rational person, you have to be reinforced in your belief that these mathematical structures are bound to be important in physics. Mother Nature recycles robust mechanisms - and "critical points" (conformal theories) - at many places (other systems naturally flow to them) and it would be silly not to use these things at the fundamental level - especially if we know that such a fundamental string generates the right gravitons, grand unified gauge bosons, as well as fermions to match the spectrum of particles in the reality.

And that's the memo.

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