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Mathematics

A beauty and a beast

The mapping of the largest exceptional Lie group, E8, is a milestone for enthusiasts for the aesthetics of mathematics. But this embodiment of complex symmetry could be of interest to fundamental physics, too.

Symmetry and beauty are often interlinked. This is so not only in art and music — one need only think of the intricate symmetry of a Bach fugue — but also in mathematics and, at its most fundamental level, physics. In a recent breakthrough1,2, which involved 18 researchers and was four years in the making, mathematicians have mapped out one of the most mysterious and fascinating of all mathematical objects: the 'exceptional Lie group' E8. In view of the magnitude of the computation and the sheer amount of data involved, the achievement has been likened to the mapping of the human genome2. But seeing the beauty in this complex beast can be hard: certainly more difficult than appreciating a Bach fugue without knowing the rules of counterpoint.

Credit: AM. INST. MATH./P. MCMULLEN/J. STEMBRIDGE

In mathematics, symmetries are usually associated with operations that leave a geometrical object invariant. A sphere, for instance, remains the same under continuous rotations in space. The collection of such operations forms a mathematical 'group'. The mathematical description of continuous symmetries (as opposed to discrete symmetries, such as those that leave a crystal lattice invariant) is codified in the notion of a Lie group, named after the Norwegian mathematician Sophus Lie. Finite-dimensional Lie groups were classified more than a century ago, by Wilhelm Killing and Elie Cartan, by dint of considering only group elements infinitesimally close to identity: that is, to 'rotations' by arbitrarily small angles. Simply put3, they identified four infinite series of such groups, labelled An, Bn, Cn and Dn for n = 1, 2, 3..., which essentially correspond to linear transformations in spaces of arbitrary dimension that leave certain quadratic expressions invariant. There are also five exceptional groups that do not fit into these categories, designated G2, F4, E6, E7 and E8.

Visualizing rotations in three-dimensional space is straightforward (as it is, with some training in mathematics, in higher dimensions!), but the 'visualization' of exceptional symmetries and their action on geometrical objects is much harder. The results of such attempts are often collectively (and jokingly) referred to as the 'botany' of these Lie groups. For instance G2, by far the 'easiest' of the exceptional groups, can be defined as the group that leaves invariant the multiplication table of a system of hypercomplex numbers known as octonions. E8 stands out as the largest and most difficult of the exceptional Lie groups. It has 248 dimensions, and its smallest non-trivial realization requires a space of 57 dimensions1,2 (see ref. 4 for a physicist's description of this object). In short, E8 is as intricate as symmetry can get. Pictured here is a two-dimensional projection of E8's 'root system' — a lattice-like system in eight dimensions that embodies its full complexity.

Like other Lie groups, E8 comes in different versions, called real forms. Roughly speaking, these differ according to whether 'rotations' are performed with a real-number angle or an imaginary-number angle. More specifically, if it is possible to return to the starting point after a finite rotation, one speaks of a compact realization. A simple example is rotation in space by 360°, which can be represented mathematically through multiplication by e with the (real) angle α = 2π. A simple non-compact transformation would be translation along a line, which is realized mathematically as multiplication by eα. This is equivalent to rotation by an imaginary angle −: because i2 = −1, then ei(−iα) = eα. E8 admits three real forms, one compact and two non-compact. Quite generally, the non-compact forms are much more tricky to deal with. This makes the main advance just reported1 so impressive: it concerns the most subtle of all non-compact forms in Lie-group theory, the 'split-real form' of E8, sometimes denoted E8(8).

Aside from pure mathematics, what is the wider significance of this achievement? One answer lies in fundamental physics. Symmetry concepts played a central role in the establishment of the two most successful theories of modern physics: general relativity, and quantum-field theory as embodied in particle physics' standard model. In general relativity, symmetry enters through the principle of general covariance: that the laws of physics should not depend on the coordinate system in which they are formulated. This principle enabled Albert Einstein to formulate in one stroke the equations of the gravitational field governing the evolution of the Universe, as well as many other phenomena that would otherwise be intractable (the interaction of light with gravity, for instance).

In the standard model, symmetry is embodied by the principle of gauge invariance, which determines the way in which elementary particles can interact. Given this principle, and the apparatus of modern quantum-field theory, all that is needed to properly formulate the standard model is the specification of the symmetry group, the matter-particle content, and the transformation properties of these matter fields (quarks and leptons) under the chosen symmetry group. Gauge invariance automatically ensures the mathematical consistency ('renormalizability') of the theory, allowing us to extract definite predictions from seemingly infinite expressions, and thus making the standard model one of the best-tested theories of physics.

Yet in spite of their success, neither general relativity nor the standard model can be final theories of physics5. This is first of all because of a basic incompatibility between the two theories, reflected in the appearance of 'non-renormalizable' infinities when Einstein's theory is quantized following the standard rules of quantum mechanics. Equally importantly, neither theory is able to answer some obvious questions. For instance, what sets the pattern of elementary particles found in nature apart from other possible such patterns? Similarly, what is so special about the standard model's symmetry group, denoted SU(3) × SU(2) × U(1), which seems mathematically undistinguished? And, connected to those questions, how did the Universe, and with it space-time and matter, come into being at the moment of the Big Bang?

To avoid the existing mathematical discrepancies, the yet-to-be-constructed unified theory (sometimes dubbed 'M theory') must be tightly constrained, and possibly even uniquely determined, by symmetry principles. One important difference between Einstein's theory of gravity and the standard model concerns the way in which symmetries are realized. In general relativity, symmetries act in physical space and time, whereas the gauge transformations of particle physics act in an abstract internal space (in which one can, for example, 'rotate' a proton into a neutron and vice versa).

An important step on the long road to a unified theory was the development of supersymmetry, a new kind of symmetry relating the particle groups known as bosons and fermions6. This led to supergravity, an extension of Einstein's theory, and superstring theory7, which is considered by many to be the leading contender to unify physics. Surprisingly, it turned out that the 'most supersymmetric' extension of Einstein's theory — supergravity in 11 space-time dimensions8,9 — has the split-real forms of E6, E7 and E8 automatically built into it10, albeit in a rather hidden form.

This seminal discovery was all the more remarkable because it revealed completely unsuspected connections. Who could have anticipated what is, in effect, a link between the esoterics of exceptional Lie groups and the absence of long-range (tensor) forces other than gravity in nature? More recent studies of gauged maximal supergravity theories in three dimensions11 have confirmed the intimate links between supergravity and the split-real form E8(8).

As yet, we have no idea what the true extent of E8's involvement in the scheme of things will be. If proponents of superstring theory are right, the compact form of E8 could be realized as a gauge symmetry in the framework of 'grand unification'. But it is equally possible that E8 will be realized in a different and more subtle way, intertwining space-time and matter, and possibly involving the split-real form, rather than the compact form.

The ambitious search for a fundamental symmetry of nature might even force us to venture into the unknown territory of infinite-dimensional exceptional symmetry groups, of which the finite-dimensional E8 is just a subset. The prime candidate is E10, about which we know next to nothing, other than that it exists. Physicists should not let themselves get carried away by these intriguing possibilities, as experiment remains the final arbiter. But they would be well advised to take note of the exciting developments1 in deciphering the E8 group.

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Nicolai, H. A beauty and a beast. Nature 447, 41–42 (2007). https://doi.org/10.1038/447041a

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