Friday, October 16, 2009

Is M-theory hiding Cayley plane fibers?

Arthur CayleyExceptional algebraic structures are omnipresent in string theory and especially M-theory and F-theory, its (so far) maximally geometrized 11-dimensional and 12-dimensional formulations.

Right now, I plan to write a rather extensive text that should
  • review some basic facts about the exceptional Lie groups and octonions
  • mention some places in M-theory where these structures occur
  • discuss the proposals by Ramond and others that a secret Cayley plane, or a 16-dimensional manifold called "OP^2", is hiding as a fiber in M-theory. After many hours of looking at many attractive possibilities, I became kind of skeptical about this very idea.
Real numbers, complex numbers, quaternions

All readers know the real numbers, R. Most people know the complex numbers, C, of the form "a+ib" where "a,b" are real and "i" is the imaginary unit that satisfies "i^2 = -1", the only thing you need to know.

The complex numbers are crucial in mathematics - for example because every algebraic equation of "n-th" order has "n" complex solutions (some of which may coincide). This "fundamental theorem of algebra" has many more consequences in more abstract and more advanced realms of mathematics.

The complex numbers are also essential in advanced physics - for example, the probability amplitudes in any quantum theory must be complex (e.g. because their phase is obliged to rotate uniformly in time for energy eigenstates, and uniformly in space for momentum eigenstates).

The automorphism group of the complex numbers is "Z_2", a group with two elements. The non-trivial element of this "Z_2" group exchanges "i" and "-i": it is really a matter of convention which of them is declared to be the "preferred" root of "-1".

A smaller percentage of us know the quaternions invented by William Rowan Hamilton (not Atkinson) and called H. They have the form "a+bi+cj+dk" where "a,b,c,d" are real and "i,j,k" are three different imaginary units. Each of them squares to "-1", much like the imaginary unit "i" in the complex numbers (you may imagine it's the same "i" now). And the other multiplication rules are
+ij = k = -ji,
+jk = i = -kj,
+ki = j = -ik.
Note that the ordering of the factors in the product matters. The multiplication table is not "commutative". But each quaternion has its inverse with respect to multiplication. The inverse of "a+bi+cj+dk" is "a-bi-cj-dk" divided by "a^2+b^2+c^2+d^2", as you can easily check by multiplying the two "conjugate" factors (the mixed terms combine to "ij+ji" etc. and they cancel).

While they're not commutative, quaternions H are still associative. We mentioned the "Z_2" automorphism group of the complex numbers C. The quaternions' automorphism group is "SO(3)" because the units "i,j,k" may be rotated by any orthogonal "SO(3)" transformations, without changing anything about their product relationships.

Octonions: Cayley numbers

There exists one more set with a new multiplication table where you can still find the inverse for every nonzero "number", namely the octonions O, also known as the Cayley numbers. You need eight real "coordinates" to describe such a number - which is why the "octo-" root is used in the term. Instead of three imaginary units, we need seven of them. Let's call them
Again, all these seven units square to "-1". And you may imagine that "i,j,k" are the same ones as those in the quaternions: again, an embedding of H to O may be adopted here. In fact, there are many more "triples" that behave like the three quaternionic imaginary units. In other words, the following triples satisfy the same "multiplicative relations" in between each other as "i,j,k" above. As you can check in We Grow Linear Algebra, they are - in the Gardener-Motl conventions
ijk, iAB, iDC, jAC, jBD, kCB, kAD.
Note that for each pair of imaginary units "x,y", you can find "xy" or "yx" in the list above exactly once (even this very simple numerology couldn't work for a different number of imaginary units than 7). That means that you may define the product "xy" as the third unit "z" that appears over there if the cyclic ordering of the entry is "xyz", or "-z" if the cyclic ordering is the opposite, "yxz".

Note that the new four units "A,B,C,D" can be divided to pairs in three ways, and each of them was associated with one unit among "i,j,k". At any rate, the resulting set of octonions is very cool. There is a multiplicative inverse for each nonzero octonion - again, it is proportional to the "complex conjugate" number. The octonions O are non-commutative, like their subset H, but they're also non-associative. For example,
(ij)A = kA = +D ... but ...
i(jA) = iC = -D.
We mentioned that automorphism groups "Z_2" of C, and "SO(3)" of H. What is the automorphism group of the octonions? You might propose "SO(7)", rotating the 7 imaginary units among each other. But besides the fact that the square of each unit is "-1", we also have the nontrivial products of two different imaginary units that must be preserved. Because
im . in = -1 deltamn + fmnp ip,
the automorphism group must preserve both "delta_{mn}" - making it a subgroup of "SO(7)" - but also the antisymmetric tensor with three indices, "f". This constraint is solved by two thirds of the 21 generators of "SO(7)". The resulting group is called "G_2", and it is the smallest exceptional Lie group. Its dimension is 14 and the dimension of its fundamental representation - that you may identify with the space of those 7 imaginary units (the real unity can't mix with them because it's special, being the only one that squares to itself).

Exceptional Lie algebras

When you classify compact simple Lie algebras or groups, and end up with the Dynkin diagrams like those above, you will find the classical sequences, namely SU(L+1), SO(2L+1), USp(2L) = U(L,H), and SO(2L). But you will also find the five exceptional groups,
G2, F4, E6, E7, E8.
The last one, "E_8", is the largest one, with the 248-dimensional adjoint representation being fundamental at the same moment.

All these groups frequently appear in M-theory and they can be constructed in many ways. For example, the "E_8" Lie algebra may be constructed out of the 120 adjoint generators of "SO(16)" and 128 generators that transform as a chiral real spinor of "SO(16)". Note that 120+128=248. The commutators of the type [120,128] are determined by 128's being the spinor, and the [128,128] commutators are determined to be a result proportional to the "120" by the "SO(16)" symmetry, too.

(The Jacobi identity can be proven to hold: the [[128,128],128] part of the verification is the most non-trivial one. The only freedom or choice that leads to non-equivalent Lie algebras is the sign of the [128,128] commutator: we obtain "E_8" or its maximally noncompact cousin, "E_{8(8)}", depending on the sign.)

The other exceptional groups may be defined as various subgroups of "E_8". For example, "E_7" and "E_6" are centralizers of the "simply" embedded groups "SU(2)" and "SU(3)" within "E_8". You may also embed "spin(7)" into "spin(16)" (in the naive, vectorial way) inside "E_8", as described above. Here, "spin(7)" has a "G_2" subgroup. Its centralizer in "E_8" must be "at least" equal to the centralizer of "spin(7)" in "E_8" (or in "spin(16)") - which would obviously be "spin(9)".

However, the centralizer of "G_2" inside "E_8" is actually bigger than "spin(9)". It includes the 36 generators of "spin(9)" but also 16 additional generators that transform as a real spinor of "spin(9)": it's the single "spin(9)" spinor in the decomposition of the "E_8" adjoint (248) that is tensorially multiplied by the "spin(7)" spinor preserved by the "G_2" subgroup.

This may sound convoluted but the conclusion is that "G_2" and "F_4" are centralizers of one another, within "E_8". You may check the decomposition
248 = (14,1) + (1,52) + (7,26).
Replace the parentheses by multiplication and check that at least the dimensionalities of the representations work.

So "F_4" may be constructed by adding a real spinor to the adjoint of "spin(9)", a construction that is analogous to the construction of "E_8" from "spin(16)". This "adjoint plus spinor" construction of a new group only works in one, third, additional example: if you add a real chiral spinor to the adjoint of "spin(8)", you will obtain "spin(9)" in a way that differs from the normal, vectorial way to extend "spin(8)" to "spin(9)" by an "SO(8)" triality transformation.

Cayley plane

But there exists another interesting way to define the "F_4" group. Consider the space "A(3,O)" of 3-by-3 Hermitean matrices over the octonions: the diagonal entries must be real and the off-diagonal ones must be complex conjugate to each other (all 7 imaginary parts revert their signs). Define the product as the half-anticommutator, or the symmetrized product of these 3-by-3 matrices:
A @ B = 1/2 (AB + BA).
The multiplication of the entries in the matrices follows the rules of the octonion multiplication table, in the same order. What is the automorphism group of "A(3,O)"?

Well, let's first see how many parameters matrices in "A(3,O)" have. There are three real numbers on the diagonal and three inequivalent octonions above the diagonal which makes "3 + 3x8 = 27" real components. However, the multiple of the identity matrix must surely map to itself because once again, the identity matrix is the only matrix that squares to itself.

You're left with 26 real components and they will be nontrivially mixing with each other. It means that there will be a 26-dimensional representation of the automorphism group: yes, it is the same number 26 that appeared in the decomposition of "248" a moment ago. To get quickly to the punch line, the automorphism group of the algebra "A(3,O)" is "F_4" and the 26-dimensional representation we got in this way is the fundamental representation.

The real n-dimensional sphere has the isometry "SO(n+1)" - you can get it as the surface of the "n+1"-dimensional ball. Consequently, the n-sphere may be written as the coset "SO(n+1)/SO(n)". Analogously, "SU(n+1)/SU(n)" gives you the real 2n-dimensional or complex n-dimensional projective space, "CP^n", the complex counterpart of the sphere.

You may do the same thing with the "USp(2n) = U(n,H)" groups. The quaternionic complex projective space is "HP^n = U(n+1,H) / U(n,H)". Can you get similar things with the octonions?

Well, the matrices over octonions don't generally satisfy one condition you expect for matrix multiplication: associativity fails because it fails even for the individual octonionic entries. That's why you must be careful and the possible options for interesting "octonionic matrices" are limited. The "symmetrized product" defined for "A(3,O)" above actually seems to be the only interesting example.

Instead of "SO(n+1)" or "SU(n+1)" or "U(n+1,H)" above, we can use the automorphism group of the higher-dimensional multiplicative space "A(3,O)", namely "F_4". And the denominator must be its counterpart for "A(2,O)" which is just "spin(9)".

Note that the coset "F_4 / spin(9)" has 16 real directions - that's exactly how we obtained "F_4" by adding a 16-dimensional spinor to the "spin(9)" adjoint: now we're subtracting the adjoint representation back immediately. And the real dimension 16 is exactly what you expect from an octonionically two-dimensional manifold. That's why the coset "F_4 / spin(9)" may be called the "OP^2", the octonionic projective plane, or the Cayley plane for short.

I will discuss the emergence of the Cayley plane and "F_4" in M-theory at the very end, in the not-yet-well-established part of this article. But let us first look at

Well-known roles of exceptional groups in M-theory

In 1984, the first superstring revolution exploded because the group SO(32) - the symmetry coming from 32 self-adjoint colors of quarks that are allowed at the endpoints of type I open strings - satisfied exactly the right miraculous properties for all the type I gravitational, gauge, and mixed anomalies to cancel.

The dimension of the group was 31x16 = 496, one of the necessary numbers to cancel the gravitational anomalies. However, it was soon realized that there existed another, 248+248 = 496-dimensional group that respected all the constraints (not just the right dimension), namely "E_8 x E_8".

In 1985, the Princeton String Quartet (Gross, Martinec, Harvey, Rohm) found out that one could construct a hybrid of the 26-dimensional bosonic string (for the left-movers) and the 10-dimensional superstring (for the right-movers). The excessive 16 left-moving bosons must live on an even self-dual lattice for the modular invariance (i.e. for the invariance under large diffeomorphisms of the torus) to hold. There are two solutions. One of them is the weight lattice of "spin(32) / Z_2", giving a heterotic string with the "SO(32)" group, as we normally call it.

In the 1990s, it was realized to be the S-dual - i.e. the strong coupling cousin - of the Green-Schwarz "SO(32)" type I theory with open strings. One theory with the coupling constant "g" is equivalent to the other theory with the coupling "G" chosen to be "1/g" - and vice versa (which is the same statement).

The other even self-dual lattice in 16 dimensions is the root lattice of "E_8 x E_8" and in 1985, this second heterotic string theory led to a new 10-dimensional "supergravity plus super-Yang-Mills" coupled system in ten dimensions at low energies whose anomalies had been known to cancel since 1984.

In the late 1980s, it was realized that the two heterotic strings were T-dual to one another (and to themselves). As I mentioned, the strong coupling limit of the "SO(32)" heterotic string is the type I string theory. And in 1995, Hořava and Witten also managed to find the strong coupling limit of the "E_8 x E_8" heterotic string: it is an 11-dimensional theory, M-theory, much like in the case of the strong coupling limit of type IIA string theory.

However, in the heterotic case, the new 11th coordinate is not a circle, like in type IIA, but a circle divided by "Z_2" i.e. a line interval. The line interval has two end points (fixed points of the "Z_2") - producing two 10-dimensional boundaries of the 11-dimensional spacetime. Each of them carried a single copy of the "E_8" gauge supermultiplet - and its gauginos were needed to cancel the gravitational anomaly from the left-right-asymmetrically filtered gravitino components on the boundary. Everything suddenly made sense. The strong coupling limits of all known 10-dimensional superstring theories were found.

In various dual descriptions, "E_8" may also arise as an enhanced symmetry group from classical gauge groups living on D-branes, or a gauge group associated with the "E_8" singularity (the complex two-dimensional space divided by the symmetry group of the dodecahedron or icosahedron, translated into the complex two-dimensional representation). The latter picture plays a prominent role in F-theory.

E8 in the bulk

As Diaconescu, Moore, and Witten pointed out in 2000, the "E_8" group plays a role in the bulk of M-theory, too, although this role is not quite dynamical (like the "E_8" gluons on the M-theoretical boundaries).

They studied the question what are the precise quantization rules for the fluxes of the four-form field strength "F4" in M-theory, as well as "F4 wedge F4" over arbitrary 4-cycles and 8-cycles of arbitrary 11-dimensional manifolds, including the factors of 1/2, 1/3, 1/6, and shifts by 1/2, 1/3, 1/6 away from the integral values. The "k C3 F4 F4" Chern-Simons form in 11 dimensions wasn't obviously invariant, not even up to multiples of "2 pi", and they had to make sure it was.

They found out that the right quantization rules for the fluxes (and/or charges) gives the identical constraints as if you require that it must be possible to write "F4" as the adjoint "E_8" trace of "F2 wedge F2" where "F2" is the ordinary "E_8" gauge (i.e. Yang-Mills) field strength (in the bulk!).

Whether or not the "E_8" plays a deeper role in the bulk - besides this "combinatorial" rule that allows you to see whether various fluxes must be integral or half-integral multiples of "2.pi" or "pi/3" etc. remained unknown. Of course, Witten et al. would surely tell us if they had a complete picture - but Witten would never bother us with an incomplete speculation. Even if the local, propagating modes are affected by such an "E_8" gauge field, the theory describing "E_8" in the bulk has almost certainly nothing to do with the normal Yang-Mills action because the latter would lead to way too many physical degrees of freedom.

It also remained unknown whether the "E_8" playing this combinatorial role in the bulk has anything to do with the "E_8" that appears as the gauge group on the boundaries. It is a very intriguing idea but the identification also seems "immediately wrong" for various reasons that may or may not have loopholes.


The exceptional groups appear in one more place in M-theory which looks totally independent from the dynamical "E_8" on the boundary as well as the combinatorial "E_8" in the bulk: it is the noncompact symmetry group of the toroidal compactifications of M-theory.

It was known that maximally supersymmetric supergravities, those with 32 supercharges, have noncompact symmetry groups such as "E_{8(8)}" in 2+1 dimensions, "E_{7(7)}" in 3+1 dimensions, and similarly in higher dimensions. These noncompact cousins of the "E_8", "E_7", and analogous groups are spontaneously broken down to the maximum compact subgroups, such as "SO(16)", "SU(8)", etc. The moduli spaces of the inequivalent vacua are therefore quotients such as
E8(8) / SO(16)
E7(7) / SU(8)
E6(6) / USp(8)
and so forth, in 3+1, 4+1, 5+1 dimensions, and so on. The dimensions of these moduli spaces are
248 - 120 = 128
133 - 63 = 70
78 - 36 = 42
and so on, which match the numbers of scalar fields in the maximally supersymmetric supergravities in the appropriate dimensions (these numbers may be deduced by studying the spins in the free supermultiplets). These scalar fields are "Goldstone bosons" but all the points in the configuration spaces they span are equivalent to each other, because of the symmetry, at the level of supergravity.

As I mentioned, the numbers of scalars were well-known from supersymmetry applied to linearized/free field theory. In fact, this pure numerology - counting of the fields and requiring the group quotients match these numbers - was (and partially remains) the only way how people could see the emergence of these exceptional symmetries. This fact emphasizes how mysterious the birth of these exceptional symmetries remains, even today.

Of course, we know much more today. The stringy limits of the toroidally compactified M-theories have classical symmetries whose enhancement can be justified or understood in somewhat more satisfactory ways.

At any rate, whenever I talked about the finiteness of supergravity, I was emphasizing that the continuous noncompact symmetry is just an artifact of the low-energy limit. In this limit, the quantization of the charges is not seen - because all electric and magnetic charges are effectively large and quasi-continuous.

However, the Dirac quantization rules demand that all these charges must be discrete. They actually belong to the lattice, and a choice of the lattice physically distinguishes the different vacua. So the continuous noncompact exceptional groups are not real symmetries of the full theories: only their discrete subgroups, such as "E_{7(7)} (Z)" group in the case of 3+1-dimensional N=8 supergravity, remain an exact symmetry - the U-duality group. The points in the moduli space related by these U-duality transformations must be identified.

Again, it seems that these exceptional groups - arising as U-duality groups - have nothing "directly" to do with the "E_8" on the boundaries, or "E_8" in the Diaconescu-Moore-Witten bulk. Nevertheless, the frequent appearance of such exceptional structures suggests that there must exist a common mathematical explanation why these things occur so frequently, and what really governs their activation in physics.

G2 manifolds

The spacetime in M-theory has 11 dimensions. To get realistic 4-dimensional vacua, you need to compactify 7 dimensions. To preserve 1/8 of the original 32 supercharges, i.e. 4 supercharges (i.e. the realistic N=1 in d=4), you need the compact 7-dimensional manifold to have a holonomy group that preserves 1 spinor out of the 8-dimensional spinor representation of "spin(7)": the preserved supersymmetries are proportional to this fixed spinor.

What is the subgroup of "spin(7)" that preserves a fixed spinor? Well, if it preserves "s", it must also preserve the tensor
sT gammaabc s =: fabc
All other tensors that you can define in similar ways - with the invariant gamma matrices - are either zero or they can be expressed in terms of this "f_{abc}" (you may also use "epsilon_{abcdefg}"). So the condition that the subgroup preserves a spinor is actually equivalent to the condition that it preserves the antisymmetric tensor "f_{abc}". Needless to say, the latter may be chosen literally equal to the "f_{abc}" that was used to define the multiplication of the octonions above.

It follows that the holonomy of the 7-dimensional manifolds that preserve 1/8 of the supersymmetry must be "G_2". M-theory phenomenology is the search for realistic manifolds with a "G_2" holonomy. They must be singular if you want to get chiral, left-right asymmetric particle species and interactions.

As you can see, we have seen "G_2, E_6, E_7, E_8" appearing in M-theory, usually at many places. The most underrepresented exceptional group has so far been the "F_4" group. But that's going to change in the rest of this article.

Ramond and the F4 group

In 1998, Pengpan and Ramond noticed some interesting relationships between different representations of "spin(9)". Similar ideas were later studied by various people, including Brink and Ramond, and most recently, Hisham Sati.

Pierre Ramond (the Gentleman who lives in the periodic sector of the NSR superstring) et al. noticed that various "spin(9)" representations can be clustered into triples.

For example, in the light cone gauge formulation of M-theory, there are 9 transverse dimensions. The 256 physical states transform as "44+84+128". The bosonic states, "44" and "84", come from the metric tensor and the three-form, while the fermionic "128" representation describes the spin-3/2 Rarita-Schwinger gravitino. The total number of bosons and fermions match.

As I understand the story, Ramond extended this tripling to infinitely many representations, labeled by four integer-like labels.

Kostant (PDF) summarized these group-theoretical observations in terms of an operator that turned out to be nothing else than a normal Dirac operator with a not-so-normal connection, acting on a quotient space of two groups.

To make the story shorter, these guys end up with the opinion that the 44+84+128 (and similar triplets) of representations can be naturally obtained "at the same moment" from the spectrum of such an operator acting on the coset "F_4 / spin(9)", i.e. acting on the Cayley plane "OP^2"! That would suggest that there is a hidden 16-dimensional fiber, an OP^2, at each point of the 11-dimensional spacetime in M-theory - making the total dimension 27 (like in the notoriously speculative and poorly justified "bosonic M-theory").

Criticism: F4 vs SUSY

Before I return to positive comments about this idea, let me offer you a criticism. Today, I decided to look more critically at the actual evidence that something like that really exists. Well, and I can't get rid of the idea that these people are cheating all the time. ;-)

Is there an "F_4" in M-theory? Well, there's surely the transverse "spin(9)" in the light cone gauge of M-theory. What about the remaining 16 generators of "F_4"? Well, there are surely 16+16 supersymmetry generators. But they have an anticommutator, not a commutator, and its symmetry guarantees that this anticommutator must be a translation in "R^9", not a rotation inside "spin(9)".

So I feel that all the group-theoretical conclusions that they actually derive do follow or must follow from supersymmetry. And they just incorrectly pretend that the additional generators are bosonic, rather than fermionic, and redefine their commutators in the required way (and the supercommutators probably don't matter for the story, anyway).

So I think that they incorrectly overlook the fermionic character of the additional generators and it is supersymmetry, and not a hidden "F_4" symmetry, that implies all of their group-theoretical observations - certainly the clumping of 44, 84, and 128.

Well, I must look more closely whether they have more evidence that there are additional hidden generators which are bosonic. But let me tell you some comments about the cute story if it were right.

The 27-dimensional theory on the Cayley plane

The idea would be that M-theory in 11 dimensions can be represented as a 27-dimensional theory compactified on "OP^2 = F_4 / spin(9)", a real 16-dimensional manifold. Let's say a few more words about it.

The Euler character of this manifold is "3". In Kostant's approach, it's argued that this Euler character is equal to the number of different representations (44, 84, 128) that have to come together. That's an interesting statement I can't immediately derive from supersymmetry - but it only makes sense once we believe the hidden bosonic manifold, so it doesn't yet "explain" any patterns that existed before we added the "OP^2", I think.

There are two classic ways to calculate this Euler character. One of them is to look at the Betti numbers of the 16-dimensional manifold. The only nonzero ones are "b0, b8, b16" which are equal to "+1" each. Obviously, all of the indices are even so we must sum them up. The sum equals three.

Another way to derive the Euler character is to look at the Weyl group - the symmetries of the root lattices. Both the "F_4" and "spin(9)" generate root lattices in 4 dimensions (four is the rank). The "F_4" lattice is denser than the "spin(9)" lattice. It has a greater Weyl group, too. By some tricks, you can also derive the Euler character of the quotient.

If there were this special "OP^2" at each point of M-theory, it could be really funny. For example, in the work by Diaconescu, Moore, and Witten mentioned above, a nice fact about the "E_8" homotopy is used.

Besides "pi_0" which is "Z", the next nontrivial homotopy group is "pi_3" which is also "Z", and then there is nothing... up to "pi_{15}" which is "Z". Of course, 15-dimensional homotopies never appear in normal M-theory whose dimension is safely smaller than 15. But it could become relevant in a 27-dimensional theory. The "pi_{15}" homotopy could have something to do with the fact that the 16 dimensions can be naturally compactified.

Also, the 27-dimensional theory could contain 18-branes. Their compactification on the "OP^2" would give the M2-branes in M-theory - in a way that would be remotely similar to the birth of heterotic strings out of 5-branes wrapped on a K3 surface.

18-branes have a 19-dimensional worldvolume which electrically couples to a 19-form gauge potential. Its 20-form field strength is Hodge-dual to a 7-form magnetic field strength - the usual one coupled to M5-branes in M-theory (that's not a nontrivial check, I just added and re-subtracted 16 from the dimensions).

However, it could be very interesting to study the "E_8" Diaconescu-Moore-Witten bulk gauge field extended to 27 dimensions. Some dualities in which the "OP^2" space would play a similar role as the K3 surface in the heterotic-K3 duality could be found, too. And the 27-dimensional theory could be linked to the 26-dimensional superstring, in a way similar to the notorious "bosonic M-theory".

Unfortunately, this whole picture may be completely fake because all these extra dimensions spanning an "OP^2" could be a result of a confusion, an incorrect attribution of some consequences of the supersymmetry to a bosonic isometry. So the situation is surely not as established as the K3 compactifications. But who knows, maybe the people still know what they're doing...

Heterotic D-branes

By the way, I have finally realized what are the "heterotic D-branes", mysterious objects that are responsible for the "(2g)!" behavior of the high-genus heterotic loop amplitudes - even though no objects whose action/mass/tension goes like "1/g" can be easily seen in the heterotic string.

An analogy between type IIA on a T^3 and heterotic strings on a T^3 is useful. The former is dual to M-theory on a T^4 while the latter is dual to M-theory on a K3 (which breaks one half of the supersymmetry).

The relevant "1/g" objects in the type IIA case are D0-branes and D2-branes wrapping three possible two-cycles of the "T^3". They carry conserved charges and are stable (and BPS). These D0-branes and D2-branes are interpreted as the four momentum modes of M-theory on the T^4.

In the heterotic case, there are no stable objects or central charges. But there still exist momentum modes of M-theory on the K3, although they are unstable. Their width is arguably at most of the same order as their mass, so they can still be viewed as "poles" that do affect the divergence behavior of the high-genus amplitudes. And they hopefully survive in the decompactification limit (both types of heterotic strings in 10 dimensions), too.

And because the parameteric dependence is arguably isomorphic to the type IIA case (just the T^4 is replaced by a K3 on the M-theory side), the mass of these lightest unstable momentum modes on K3 should go like "1 / g_{heterotic}", giving the desired "(2g)!" behavior of the high-genus amplitudes.

Of course, it would be extremely interesting to see the K3 momentum modes arising from the heterotic world sheet. The heterotic world sheet must secretly know that it is actually the world volume of a 5-brane wrapping a K3, and it must know about its K3 inside - which must be ready to decompactify in the strongly coupled limit. The chiral bosons on the heterotic string are integrals of a "B2" over two-cycles of the K3 but it must be possible to construct the local geometry of the K3 from some nonlocal operators involving these chiral bosons, too.

As you can see, this particular memo is so far very incomplete.


  1. Thanks for writing this interesting post.
    As you may recall, back in 2004 you and I (as well as Urs Schreiber and Aaron Bergman) were involved in a sci.physics.research discussion based on a question by John Baez about construction of a realistic string theory,
    the discussion led to a construction in which fermions were represented by orbifolding 8+8 = 16 dimensions of the basic 26 dimensions
    which construction was described in a preprint at CERN-CDS-EXT-2004-031
    As you mention in this post, those 16 fermionic dimensions can be represented by OP2 = F4/Spin(9)
    all this is consistent, in a string theory context, with a type of supersymmetry based on triality
    is also consistent with an overall E8 structure.

    Good luck with your further work along those lines.
    If you think that further discussion with me might be helpful to you, I would be interested,
    since you have clearly expressed a very negative opinion about my abilities, I would understand if you prefer no further discussion.

    Tony Smith

  2. Dear Tony, thanks for your comment: various related things may have been written along these lines but I think it's impossible that I have ever written that "16 fermionic dimensions may represent F4/spin(9)" because it's manifest nonsense.

    The dimensions along F4/spin(9) are ordinary bosonic dimensions - they "almost" commute with each other - so it doesn't seem possible to identify them with fermions.


  3. Lubos, you are correct that you did not write "that "16 fermionic dimensions may represent F4/spin(9)",
    you just participated in a discussion that gave me that idea.
    Since you do not like that idea, I apologize for attempting to give you credit for motivating me to get the idea.

    You are also correct that the F4/Spin(9) can, as part of the 52 adjoint dimensions of F4, be seen as bosonic.

    However, it seems to me that the 16 of F4/Spin(9) also can be seen to have fermionic aspects, as the rest of this message describes (it may be too technical and long for you to want to put it up on your blog, so you are welcome to delete it if you want to do so, but I thought you might find it interesting even though you may not agree with it).

    My view can be seen in statements by Pierre Ramond in hep-th/0112261 where he said:
    "... exceptional algebras relate tensor and spinor representations of their orthogonal subgroups,
    while Spin-Statistics requires them to be treated differently
    all representations of the exceptional group F4 are generated by three sets
    of oscillators transforming as 26. We label each copy of 26 oscillators as
    Ak_0 ,
    Ak_i , i = 1, ... , 9,
    Bk_a , a = 1, ... , 16,
    and their hermitian conjugates,
    and where k = 1, 2, 3.
    Under SO(9), the Ak_i transform as 9,
    Bk_a transform as 16,
    and Ak_0 is a scalar.
    They satisfy the commutation relations of ordinary harmonic oscillators
    Note that the SO(9) spinor operators satisfy Bose-like commutation relations ... both A_0 and B_a ... obey Bose commutation relations
    if both ... A_0 and B_a ... are anticommuting, the F4 algebra is still satisfied
    One can just as easily use a coordinate representation of the oscillators by introducing real coordinates
    ...[ for A_i ]... which transform as transverse space vectors,
    ...[ for A_0 ]... which transform ... as scalars,
    and ...[ for B_a ]... which transform ... as space spinors which satisfy Bose commutation rules ...".

    What I would like to do (which is NOT done by Ramond) is, in Ramond's formalism:

    1 - Let A_0 and B_a be anticommuting, which is consistent with the F4 algebra
    and then
    2 - use coordinates
    ...[ for A_i ]... which transform as transverse space vectors,
    ...[ for A_0 ]... which transform ... as scalars,
    and ...[ for B_a ]... which transform ... as space spinors
    so that
    3 - we have
    B_a space spinors that are anticommuting, so that we can use them to represent fermions in model building.

    If we do that, we have A_0 scalars that are anticommuting, that could physically be related to ghosts.

    The downside of such a construction is that it is not standard,
    but it does seem to be mathematically consistent,
    it does permit construction of a model with 16 fermions,
    8 of which look like fundamental particles
    and 8 of which look like fundamental antiparticles.

    Tony Smith

  4. "It also remained unknown whether the "E_8" playing this combinatorial role in the bulk has anything to do with the "E_8" that appears as the gauge group on the boundaries."

    I used to think (back before I knew much about the subject) that there ought to be a third E8 somewhere in string theory, "because" you get the Leech lattice from three copies of the E8 root lattice. It's an example of the "M is for Monster" theory about what the M in M-theory stands for. :-)

    Also, like many other people, I was keen on the idea of getting the superstring from the bosonic string somehow, e.g. as the worldvolume theory of a 10-brane in the bosonic theory. So it's good to see the more educated version of such ideas described and analysed. Eventually we will know if something like this is the truth.

  5. Dear Mitchell, I surely share your viewpoint that the Leech lattice is "close" to the tripled E8 lattice - and the Monster group could therefore be related to a tripled greatest exceptional E8 group - but I don't know how to do it exactly.

    Concerning the unification of bosonic strings, check e.g. Swanson Hellerman 2006 who find a cosmological solution interpolating between bosonic strings and type 0 theories - which are T-dual to type II superstrings.

    There may exist such links but one surely has to leave the stable flat vacua and go into the risky terrain of cosmology if you want to incorporate the bosonic strings.

    Concerning your 3rd E8, well, there is a 3rd E8 in M-theory. As I said, one of them appears on the boundary (Horava-Witten), another one describes the C-field in the bulk (Diaconescu Moore Witten), and a third, noncompact E8(8) describes the U-duality group of M-theory on an 8-torus.

    All these 3 groups are a priori physically unrelated.

  6. I meant that DMW's E8 in the bulk is a "third E8" in addition to the two gauge E8s on the boundary. (By analogy with monstrous 3D gravity, this is obviously where the "3" in the famous "ln(3)" comes from... Just kidding.)