Bigger than the Human Genome.
The magnitude of the E8 calculation invites comparison with the Human Genome Project. The human genome, which contains all the genetic information of a cell, is less than a gigabyte in size. The result of the E8 calculation, which contains all the information about E8 and its representations, is 60 gigabytes in size. That is enough space to store 45 days of continuous music in MP3 format. While many scientific projects involve processing large amounts of data, the E8 calculation is very different: the size of the input is comparatively small, but the answer itself is enormous, and very dense.
Like the Human Genome Project, these results are just the beginning. According to project leader Jeffrey Adams, "This is basic research which will have many implications, most of which we don't understand yet. Just as the human genome does not instantly give you a new miracle drug, our results are a basic tool which people will use to advance research in other areas." This could have unforeseen implications in mathematics and physics which do not appear for years.
According to Hermann Nicolai, a director of the Max Planck Institute in Potsdam, Germany (not affiliated with the project), "This is an impressive achievement. While mathematicians have known for a long time about the beauty and the uniqueness of E8, we physicists have come to appreciate its exceptional role only more recently --- yet, in our attempts to unify gravity with the other fundamental forces into a consistent theory of quantum gravity, we now encounter it at almost every corner! Thus, understanding the inner workings of E8 is not only a great advance for pure mathematics, but may also help physicists in their quest for a unified theory."
Beautiful Symmetry
At the most basic level, the E8 calculation is an investigation of symmetry. Mathematicians invented the Lie groups to capture the essence of symmetry: underlying any symmetrical object, such as a sphere, is a Lie group.
Lie groups come in families. The classical groups A1, A2, A3, ... B1, B2, B3, ... C1, C2, C3, ... and D1, D2, D3, ... rise like gentle rolling hills towards the horizon. Jutting out of this mathematical landscape are the jagged peaks of the exceptional groups G2, F4, E6, E7 and, towering above them all, E8. E8 is an extraordinarily complicated group: it is the symmetries of a particular 57-dimensional object, and E8 itself is 248-dimensional!
To describe the new result requires one more level of abstraction. The ways that E8 manifests itself as a symmetry group are called representations. The goal is to describe all the possible representations of E8. These representations are extremely complicated, but mathematicians describe them in terms of basic building blocks. The new result is a complete list of these building blocks for the representations of E8, and a precise description of the relations
There actually are 4 different but related things called E8.
E8 is first of all the largest exceptional root system, which is a set of vectors in an 8-dimensional real vector space satisfying certain properties. Root systems were classified by Wilhelm Killing in the 1890s. He found 4 infinite classes of Lie algebras, labelled An, Bn, Cn, and Dn, where n=1,2,3.... He also found 5 more exceptional ones: G2, F4, E6, E7, and E8.
The E8 root system consists of all vectors (called roots) (a1,a2,a3,a4,a5,a6,a7,a8) where all ai are integers or all ai are integers plus 1/2, the sum is an even integer, and sum of the squares is 2. An example with all integers is (-1,0,1,0,0,0,0,0) (there are 112 of these) and an example with half-integers is (1/2,1/2,-1/2,-1/2,-1/2,1/2,1/2,-1/2) (there are 128 of these). E8 has 240 roots. The 8 refers to the fact that there are 8 coordinates.
Secondly E8 refers to the root lattice obtained by taking all sums (with integral coefficients) of the vectors in the root system. It consists of all vectors above with all ai integers, or all ai integers plus 1/2, and whose sum is even. The integers of squared length 2 are precisely the roots. This lattice, sometimes called the "8-dimensional diamond lattice", has a number of remarkable properties. It gives most efficient sphere-packing in 8 dimensions, and is also the unique even, unimodular lattice in 8 dimensions. This latter property makes it important in string theory.
Next E8 is a semisimple Lie algebra.
A Lie algebra is a vector space, equipped with an operation called the Lie bracket. A simple example is the set of all 2 by 2 matrices. This is a 4-dimensional vector space. The Lie bracket operation is [X,Y]=XY-YX.
E8 is a 248-dimensional Lie algebra. Start with the 8 coordinates above, and add a coordinate for each of the 240 roots of the E8 root system. This vector space has an operation on it, called the Lie bracket: if X,Y are in the Lie algebra so is the Lie bracket [X,Y]. This is like multiplication, except that it is not commutative. Unlike the example of 2 x 2 matrices, it is very hard to write down the formula for the Lie bracket on E8.
This is a complex Lie algebra, i.e. the coordinates are complex numbers. Associated to this Lie algebra is a (complex) Lie group, also called E8. This complex group has (complex) dimension 248. The E8 Lie algebra and group were studied by Elie Cartan in 1894.
Finally E8 is one of three real forms of the the complex Lie group E8. Each of these three real forms has real dimension 248. The group which we are referring to in this web site is the split real form of E8.
Geometric description of the split real form of E8
Consider 16x16 real matrices X satisfying two conditions. First of all X is a rotation matrix, i.e. its rows and columns are orthonormal. Secondly assume X2=-I. The set of all such matrices V0 is a geometric object (a "real algebraic variety"), and it is 56-dimensional. There is a natural way to add a single circle to this to make a 57-dimensional variety V. (V=Spin(16)/SU(8), and is circle bundle over V0, to anyone keeping score). Finally E8 is a group of symmetries of V.E8 and piscis
For well over two decades now String Theory has been the preeminent model for physics beyond the Standard Model. Indeed, String Theory is often touted among physicists as the ultimate Theory of Everything. The basic premise of String Theory is that the most fundamental building blocks of the Universe are not atoms, or even elementary particles like electrons, muons and quarks, but rather exotic string-like objects living in a 26-dimensional space. The great appeal of String Theory to modern physicists is two-fold. First of all, it very deftly circumvents two theoretical obstructions that had long thwarted the unification of Einstein's theory of General Relativity with the quantum field theory of elementary particle physics (viz., the lack of renormalizability and the occurrance of quantization anomalies) . The second is its apparent uniqueness: once one adopts the basic principles of string theory, it can be argued that we live in the universe we live in because it is the only one that is possible.
Actually this uniqueness is not quite complete; there are in fact several competing string models. But the dominant model by far is that of heterotic string theory, and it is there that E8 plays an essential role.
Naturally, the most stringent requirement of a viable string theory of the Universe is that eventually the theory has to make contact with the 4-dimensional space-time in which we (at least appear to) live. In heterotic string theory this passage from 26 dimensions to 4 dimensions occurs in two steps. First of all, 16 of the original 26 dimensions must compactify, or curl up on themselves, in a very nice self-consistent way; and then 6 of the remaining 10 dimensions must compactify nicely as well in order to get down to our apparent 4-dimensional observed universe. E8 arises in heterotic string theory because in order for the initial reduction from 26 to 10 dimensions to procede consistently, one needs to endow a 16-dimensional subspace of the orginal 26-dimensional space with an even, unimodular lattice. It turns out that there are exactly two such lattices in 16 dimensions, one of which is the root lattice of E8+E8. See the beautiful exposition by John Baez for more details of this fascinating story.
The Atlas of Lie Groups Project
The E8 calculation is part of an ambitious project known as the Atlas of Lie Groups and Representations. The goal of the Atlas project is to determine the unitary representations of all the Lie groups. This is one of the great unsolved problems of mathematics, dating from the early 20th century.
The success of the E8 calculation leaves little doubt that the Atlas team will complete their task.
The Atlas team consists of about 20 researchers from the United States and Europe. The core group consists of Jeffrey Adams (University of Maryland), Dan Barbasch (Cornell), John Stembridge (University of Michigan),
Gustavo Canals (University of Oxford), Marc van Leeuwen (Poitiers),
David Vogan (MIT), and Fokko du Cloux (Lyon).
