Sunday, May 07, 2006

ADE classification, McKay correspondence, and string theory

John Baez starts his Week in Mathematical Physics #230 by several assorted infinite product formulae for "e", "pi", and Euler's "exp(gamma)". He does not know what they mean, whether they are important, and how to prove them, and because I don't know these things either (except that I am completely convinced that it is possible to prove them), let me skip this distracting part of the Week. ;-)

The bulk of the text is dedicated to the ADE classification. The reader of this blog who is not familiar with the Dynkin diagrams is recommended to read Baez's account. To summarize, simply-laced Dynkin diagrams (equivalently, ADE Dynkin diagrams) are graphs with nodes connected by simple links that fall into one of these three groups:
  1. "A_L" = a sequence of "L" nodes connected by "L-1" links
  2. "D_L" = much like "A_{L-1}", except that an additional, the last, "L-th" node is connected to the second node of the "A_{L-1}" chain (usually drawn as the second from the right)
  3. "E_L" = much like "A_{L-1}", except that an additional, the last "L-th" node is connected to the third node (from the right) of the "A_{L-1}" chain; in this (E) case, the number of nodes, always called "L", must be 6 or 7 or 8

These diagrams classify - i.e. they list all possible forms - of various objects in mathematics and theoretical physics. They are relevant for the classification of at least five kinds of objects:

  • @) the planets orbiting the Sun
  • A) lattices inside "R^L" where all the inner products are integer-valued and a basis exists with vectors whose squared length is two
  • B) simply-laced Lie algebras (and Lie groups)
  • C) finite subgroups of the SU(2) group
  • D) two-complex-dimensional singularities
  • E) tame quivers

The link between @) and C) is due to a late German Czech physicist, Johannes Kepler. He has identified five planets (a category that excludes the Earth) with five different Platonic polyhedra - and predicted that there can't exist a seventh planet (Uranus), while he used the radii of orbits and the geometry of the polyhedra as an argument. Because most of us don't believe that this link is true or fundamental, let me skip Kepler's heroic proposal. ;-)

Not Even Wrong would applaud Kepler's astrology because it can't be criticized for being not even wrong. Still, I prefer theories that are not wrong over theories that just fail to be not even wrong. ;-)

The other relations may look comparably spooky as Kepler's conjectured link - but we will see that they are true and they are profound.

John Baez has described the different groups of objects that obey the ADE classification, and I will make a short summary of it, too. However, John Baez can't really explain the relation between A,B,C,D,E) - because it can't be usefully explained or "viscerally understood" without string theory. As Baez himself admits, the relation between A,B) on one side and C,D,E) on the other side remains "spooky black magic" even for him. John Baez believes that the stringy explanation works but as he admits, he does not understand it.

A mathematician who thinks about the ADE classification without string theory is like a proto-human who knows something about the summer and the winter but who can't understand why the seasons change because he does not want to look in the skies and see the motion of the Sun. In this metaphor, an anti-string-theory blogger is another proto-human who eats everyone who dares to look in between the clouds. Be afraid. Be very afraid. :-)

So what are the objects classified by the A,D,E Dynkin diagrams? Please don't be confused about the letters A,B,C,D,E) below - written in bold face - that have nothing to do with A,D,E.

A) Simply-laced lattices

In this application of the Dynkin diagrams, each node represents a basis vector of a lattice equipped with a metric (something that looks like a tilted "Z^L" where "Z" is the set of integers) whose squared length is two. Each two nodes that are disconnected in the diagram correspond to orthogonal basis vectors - whose inner product is zero. Two connected nodes correspond to a pair of basis vectors with the angle 120 degrees in between them - which makes their inner product equal to minus one.

The inner product of two linearly independent vectors whose squared length equals two can only be -1 or 0 or +1, and by appropriate sign flips for the individual basis vectors, you can always make the inner products of the basis vectors to be either -1 or 0. This is why the diagram encodes everything.

Why are the A,D,E Dynkin diagrams the only possibilities? First of all, you can easily show that a diagram that represents linearly independent basis vectors of the lattice can't contain any loops in it. If you denote the vectors in such a hypothetical loop "V1,V2,V3...Vn", you can see that the vector (an element of the lattice)

  • V = V1 + V2 + V3 + ... + Vn


  • V.V = 2-2+2-2 ... +2-2 = 0

This proves that because the inner product was positively definite, "V" must be zero, which proves that "V1,V2,V3...Vn" were not linearly independent. This means that the ADE Dynkin diagram must be a tree graph. Using the same argument with different coefficients defining the critical linear combination, you can also prove that the diagram can't contain a "quartic vertex" - a node with four (or more) links. An analogous argument proves that two (or more) cubic vertices can't coexist. Also, you can prove that if there is a cubic vertex, at least one of the branches that comes from the junction must be shorter than 2 links. A similar argument shows that at most one branch is longer than 2 links. This proves that the "E_L" diagrams are the most complicated ones you can get. Another linear combination "V" leading to "V.V=0" proves that an "E_9" diagram is already impossible for lattices of the kind we want which makes the E-collection finite.

B) Simply laced Lie groups

The reader is supposed to know the relation between Lie algebras and Lie groups. Lie algebras describe the tangent space of the Lie group at a particular point of the group viewed as a manifold - namely the identity.

To see the relevance of the lattices from A) and the relevance of the ADE classification, find a subgroup "U(1)^L" of your group G with a maximal possible value of "L". Call it the Cartan subalgebra (or Cartan subgroup or Cartan torus) and "L" is the rank - which will become the number of nodes of the Dynkin diagram i.e. the dimension of the lattice. The generators of the Cartan subalgebra commute with each other. Therefore, they can be simultaneously diagonalized. The simultaneous eigenvalues of these "L" generators can be combined into an L-dimensional vector, the weight. Each (k-dimensional) representation of your group G or the corresponding Lie algebra has a system of (k) weights.

The weights of the adjoint representation - one in which the algebra acts on itself - are called the roots. ("L" of these weights equal the zero vector, those corresponding to the Cartan generators themselves, and they are usually not counted as roots.) The roots generate a lattice called the root lattice - the set of all integer combinations of the roots.

If you look at it carefully, you will see that the lattice satisfies the conditions studied in A). The inner products are integers, and for a class of groups called the simply-laced groups, there exists a basis of vectors whose squared length is equal to two. This is why the classification of the lattices becomes the crucial step of the classification of the Lie algebras and Lie groups. You repeat the steps of A) again, and find out that the groups that you study through the optics of lattices are called:

  • A_L = SU(L+1)
  • D_L = SO(2L)
  • E_L = E_6, E_7, E_8

The unitary and orthogonal groups (A,D) are well-known. The three groups "E_L" are called the exceptional groups. "E_8" is the largest exceptional group whose dimension is 248 - it's also the dimension of the fundamental representation that coincides with the adjoint representation.

You can construct the "E_8" Lie algebra, much like Green, Schwarz, and Witten in their appendix 6.A of "Superstring Theory", by starting with 120 generators "J_{ij}" of "SO(16)" and adding "128" generators "Q_a" that transform as a chiral real spinor under "SO(16)". Note that 120+128=248. The commutators between "Q_a" and "SO(16)" are determined by the statement that "Q_a" are spinors while the commutators "[Q_a,Q_b]" are also fixed by the "SO(16)" symmetry to be "sigma^{ij}_{ab}.J_{ij}". The only part of the coefficient that can't be absorbed to the normalization of "Q_a" is the sign, and by choosing the two different signs, you either obtain the compact "E_8" group or the noncompact "E_{8(8)}" group.

The Jacobi identity works, mainly due to a small miracle for the "[Q,[Q,Q]] + permutations" sum. The same miracle occurs if you replace "SO(16)" by "SO(9)" which allows you to construct "F_4" instead of "E_8". An analogous construction (adjoint plus spinor) applied on "SO(8)" leads to an "SO(9)" which is not too interesting because it is not a new group: the construction is related by an SO(8)-triality to the usual decomposition of "SO(9)" under the obvious "SO(8)" subgroup.

The smaller groups "E_7" and "E_6" can be defined as centralizers of the minimal "SU(2)" and "SU(3)" subgroups of "E_8", respectively. There is also a "G_2" subgroup inside "SO(7)" inside "SO(16)" inside "E_8", and the centralizer of this "G_2" is an "F_4".

C) Finite subgroups of "SU(2)"

So far we did not need string theory. In fact, the Lie algebras have been known nearly for a century. Things will change now. What are the finite subgroups of "SU(2)"?

Note that "SU(2)" is essentially isomorphic to "SO(3)", the group of rotations in three dimensions, up to the fact that the map is two-to-one. At any rate, you can find a subgroup of "SO(3)" for any subgroup of "SU(2)", just by ignoring the ambiguous sign of the "SU(2)" elements. So what are the possible finite subgroups of rotations in three dimensions?

One of them is the group of rotations around one axis (namely z) by angles that are multiples of "4.pi/L". I wrote "4.pi" instead of "2.pi" so that I get directly the subgroups of "SU(2)" where the identity is distinguished from the rotation by 360 degrees. This simple Abelian group "Z_L" - a whole infinite class of groups - will be classified as "A_L".

Another finite subgroup is much like the rotations above, but you also include the rotation by 180 degrees around another axis, say the x-axis. You draw a regular polygon with "L" vertices on a sheet of paper and you allow the polygon to be rotated and the sheet of paper to be flipped. This sequence of finite groups will be classified as "D_L".

Are there other finite subgroups of "SU(2)" or "SO(3)"? Yes, there are. There are nice regular objects called the Platonic polyhedra and they have rather non-trivial symmetries that can't be reduced to the simple operations described in the previous two paragraphs. You might think that we will have five extra finite subgroups of "SO(3)", namely the symmetries of

  • a tetrahedron (4 faces)
  • a cube (6 faces)
  • an octahedron (8 faces)
  • a dodecahedron (12 faces)
  • an icosahedron (20 faces)

Actually, there are only three new groups because the symmetries of a cube and the symmetries of an octahedron are identical, much like the symmetries of a dodecahedron are identical to the symmetries of an icosahedron. It's because these pairs of polyhedra are "dual" to each other. This "duality" is much simpler than the dualities in string theory, but you can view it as a toy model of the real dualities.

Draw a vertex in the middle of each face of a polyhedron A. Connect all these vertices. You obtain another polyhedron B that is dual to A: for example, the number of vertices of A equals the number of faces of B and vice versa, while the numbers of edges of A,B coincide. (Note that the tetrahedron is self-dual.) Because the symmetries between the faces do not really change if you rename the faces into "vertices", the symmetry groups of the dual polyhedra are identical.

At any rate, we have three new discrete subgroups of "SU(2)" related to

  • tetrahedron: E_6
  • cube or octahedron: E_7
  • dodecahedron or icosahedron (or the pseudo-Platonic 30-hedron): E_8

OK, we still did not need string theory but the relation between A,B) on one side and C) on the other side is spooky without string theory. Let us now define D) as a variation of C), and then show the relation with A,B).

D) HyperKähler singularities

Imagine that you look at a two-complex-dimensional space whose holonomy is "SU(2)" - this is the meaning of the adjective hyperKähler (such four-real-dimensional manifolds admit a Ricci-flat metric) - and that look like flat space at infinity: they are ALE (asymptotically locally Euclidean) spaces. You will find that there are again two infinite families plus three exceptions of such two-complex-dimensional geometries. They can be thought of as "blown-up" (desingularized) versions of the orbifold

  • C^2 / Gamma

Here "Gamma" is one of the subgroups of "SU(2)" described in C). Because Gammas are subgroups of "SU(2)", their elements have a natural action on "C^2". You can identify the points of "C^2" that are related by a transformation from "Gamma", and call the resulting quotient an "orbifold" (a word constructed from the words orbit and manifold). That will be a singular space because the point

  • (0+0i, 0+0i)

is a fixed point of all the transformations because they are linear transformations. At any rate, the singularities can be removed by replacing the singular point with a two-real-dimensional sphere "CP^1". The construction makes the relation of C) and D) manifest. It turns out that these two-complex-dimensional geometries, after you de-singularize them, have topologically non-trivial spheres in them. How much topologically non-trivial is measured, in the zeroth approximation, by the so-called homology. Homology is a linear space - a lattice, in fact. The metric on the lattice is given by the oriented intersection numbers of the two-real-dimensional submanifolds (cycles), and you can argue that you will get a problem that is mathematically equivalent to the classification of the lattices in A).

The appearance of Dynkin diagrams that represent the intersecting spheres of the singular four-real-dimensional manifold is called the geometric McKay correspondence. The McKay correspondence without adjectives is the appearance of Dynkin diagrams in the classification of irreducible representations of the discrete groups - and the decompositions of their tensor products with C^2. The latter can also be visualized in string theory but unlike the geometric McKay correspondence, it won't be done in this article.

The appearance of the same Dynkin diagrams is nice but it does not really explain why particular Lie groups - such as "E_6" - are associated with particular finite subgroups of "SO(3)" - such as the symmetry of the tetrahedron, even though we might understand why the symmetry of the tetrahedron is associated with a particular kind of two-complex-dimensional singularity. If we want to believe that the tetrahedron is unrelated to Mercury, as Kepler argued, what does the 78-dimensional group "E_6" have to do with the same tetrahedron?

The answer is called String theory.

Take type IIA string theory and study its physics on a spacetime background in which four spatial coordinates have the shape of "C^2 / Gamma" where "Gamma" is the finite subgroup of "SU(2)" containing the symmetries of the tetrahedron. The ten-dimensional spacetime is the Cartesian product of this manifold and the "M^{5+1}" Minkowski spacetime. What can you learn about physics? Away from the singularity, string theory can be well approximated by its low-energy limit which is, in this case, type IIA supergravity in ten dimensions.

However, at very short distances, the details of string theory become crucial. What happens near the singularity? Without string theory, all answers to this question follow the GIGO rule: garbage in, garbage out. If you blow up the singularity, you will be able to define the following gauge fields in the 5+1 dimensions that are orthogonal to the "C^2 / Gamma" singularity:

  • A^i_m = integral (Cycle_i) C_{klm}

Here, "i" is the index that goes from "1" to "L" - the different homology classes. The index "m" is a Lorentz index in 5+1 dimensions while the indices "k, l" are vector indices in the "C^2 / Gamma" space and they are contracted against the two-dimensional integral over the "Cycle_i", a topologically non-trivial two-sphere that can be found near the resolved singularities. The object "C_{mnk}" is an anti-symmetric tensor, a three-form, a generalization of the electromagnetic potential from the Ramond-Ramond sector of type IIA string theory.

If you dimensionally reduce along the four dimensions of "C^2 / Gamma", these fields "A^i_m" give us gauge potentials for a "U(1)^L" gauge group in 5+1 dimensions. You can be sure that there will be "L" copies of "electromagnetism". In the case where "Gamma" includes the symmetries of the tetrahedron, you will find "L=6" different topologically non-trivial spheres within the same singularity.

OK, we still have "U(1)^6" only which is not quite "E_6". Does the background also contain the other components of the non-Abelian gauge field? The answer is Yes. The "off-diagonal" (non-Cartan) generators of "E_6" are described by "W bosons", gauge bosons that carry charges with respect to the "U(1)^L" algebra. What does it mean to carry charges with respect to "U(1)^L"? You see that the physical field - the gauge potential - that we used was the Ramond-Ramond three-form, "C_{klm}". The objects that are charged under this field are called the D2-branes, as figured out by Joe Polchinski. They are membranes wrapped on the spheres.

If you desingularize the singularity, these spheres become large and the D2-branes are massive. If you change the shape and return to the singular shape instead, the spheres shrink and the D2-branes become massless. This formula for the masses is just a geometric realization of the Higgs mechanism: the Higgs fields "h" measure how much non-singular the singularity has become. We have "h=0" for a singular "C^2 / Gamma". The masses of the W-bosons are proportional to "h" (the vev), as you know from field theory.

If you count all vector multiplets of particles that are not described by perturbative strings but instead by spherical D2-branes, you will see that they only come from spheres whose self-intersection is minus two, i.e. from the roots of the root lattice (the conventions for the inner product and the intersection number are opposite). The root lattice is the homology. In the case of the tetrahedron singularity, the gauge group is "E_6", and the map works in all other cases, too.

The punch line is that

  • low-energy physics of type IIA string theory on a particular ADE singularity defined using a particular finite subgroup of "SU(2)" is described by the corresponding ADE-group gauge theory localized near the singularity (and propagating in 5+1 dimensions only) in addition to supergravity in the bulk

In fact, this also works for M-theory because D2-branes (and the Ramond-Ramond three-form potential) can simply be replaced by M2-branes (and the M-theoretical three-form potential). Type IIB string theory on the same singularities leads to different physics: new exotic theories with self-dual three-form field strength in six dimensions.

E) Quivers

Without string theory, you would probably be so baffled by the mysterious relations between the ADE objects in A,B) and the ADE objects in C,D) that you would not try to connect them with E) at all. With string theory, it is non-trivial but straightforward. You may also add extra D-branes in the spacetime with the "C^2 / Gamma" singularity. At the singular point, the singularity is an orbifold. This means that each brane must be accompanied by appropriate "mirror images" under the group action.

However, there can exist open strings stretched between various D-branes and their images. Depending on the two D-branes at the endpoints, you get new sectors of the single open string Hilbert space. Some of these sectors will produce new massless particles (the zero-point energy counting is important here) - massless at the orbifold point - and if you look exactly at the spectrum of these new massless open string excitations, you will recover the corresponding ADE quiver diagram (not to be confused with the Dynkin diagram): a diagram in which nodes represent different "U(N)" factors of the gauge group and the links represent additional fields transforming in the "(N,Nbar)" bi-fundamental representation under two "U(N)" groups. See the text about deconstruction.

Overall punch line

The punch line is that we absolutely need string theory to understand what's really going "inside" various physical systems that depend on nice mathematics. For an intelligent person, it is impossible to deny that string theory connects A,B) with C,D) and with E). It allows us to make seemingly obscure but true relations as clear as the sky. String theory allows us to answer questions in each category. It even tells us what are the right questions and what are the wrong questions. It also informs us which similarities, isomorphisms, and identities are deep and which of them are just random coincidences.

The example of the ADE classification was a rather trivial and mathematical one -partly because we have sketched the most important concepts only. But there are dozens of similar examples in string theory and many of them have something to do with physics of gauge theories, gravity, and black holes. With confinement, Higgs mechanism, condensation of magnetic monopoles, with asymptotic freedom, and many other essential physical phenomena. String theory shows that the details of these theories are not just random complex mess but they have an exact - and sometimes solvable - description in terms of an alternative system that can be more geometric or more comprehensible.

Our understanding of such dual systems is necessary if we want to predict what happens behind the borders between our current knowledge and our current ignorance. Sometimes the borders of the hostile empire of ignorance are only defined by our limited calculational skills. Sometimes the borders occur because without string theory, we would have no idea how to define the theory of the physical phenomena in the first place. All questions about the Planckian regime of quantum gravity fall into the latter category.

That's why it is important not only to learn string theory well but also to emphasize that and explain why the permanent critics of string theory are intellectual barbarian cannibals. ;-)


  1. Hmm I was thinking... We know that there are infinite regular polygons, five regular polyhedrae, and only one regular hyperpolyhedra for each dim D>3. This translates to the finite subgroups of SO(2), and SO(3) as you have explained, I am not sure if it also implies that SO(N) in general does not have new finite subgroups besides the trivial permutation of coordinates (the hypercube) Does it?

    And if so, what does it happen for the other group families besides SO(N)? Do they contain finite groups always, or only at low dimensionality?

  2. ___R______C_________H______O