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TOPIC: Lie group E8
Blobrana


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standardlie
Credit Garrett Lisi

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Blobrana


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GrandUnifiedTheories(1)

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Blobrana


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Lie group F4
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Title: Mapping the geometry of the F4 group
Authors: Fabio Bernardoni, Sergio L. Cacciatori, Bianca L. Cerchiai, Antonio Scotti
(Version v2)

In this paper we present a construction of the compact form of the exceptional Lie group F4 by exponentiating the corresponding Lie algebra f4. We realize F4 as the automorphisms group of the exceptional Jordan algebra, whose elements are 3 x 3 hermitian matrices with octonionic entries. We use a parametrization which generalizes the Euler angles for SU(2) and is based on the fibration of F4 via a Spin(9) subgroup as a fibre. This technique allows us to determine an explicit expression for the Haar invariant measure on the F4 group manifold. Apart from shedding light on the structure of F4 and its coset manifold OP2=F4/Spin(9), the octonionic projective plane, these results are a prerequisite for the study of E6, of which F4 is a (maximal) subgroup.

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-- Edited by Blobrana at 16:12, 2007-11-16

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Surfer dude stuns physicists with theory of everything
An impoverished surfer has drawn up a new theory of the universe, seen by some as the Holy Grail of physics, which has received rave reviews from scientists.
Garrett Lisi, 39, has a doctorate but no university affiliation and spends most of the year surfing in Hawaii, where he has also been a hiking guide and bridge builder (when he slept in a jungle yurt).
Physicists have long puzzled over why elementary particles appear to belong to families, but this arises naturally from the geometry of E8, he says. So far, all the interactions predicted by the complex geometrical relationships inside E8 match with observations in the real world

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Title: An Exceptionally Simple Theory of Everything
Authors: A. Garrett Lisi

All fields of the standard model and gravity are unified as an E8 principal bundle connection. A non-compact real form of the E8 Lie algebra has G2 and F4 subalgebras which break down to strong su(3), electroweak su(2) x u(1), gravitational so(3,1), the frame-Higgs, and three generations of fermions related by triality. The interactions and dynamics of these 1-form and Grassmann valued parts of an E8 superconnection are described by the curvature and action over a four dimensional base manifold.

E8_Part4
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The E8 root system, with each root assigned to an elementary particle field.

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Blobrana


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A map of one of the strangest and most complex entities in mathematics should be a powerful new tool for both mathematicians and physicists pursuing a unified theory of space, time and matter.
The strange 'thing' that has been mapped is a 'Lie group' called E8 — a set of maths that describes the symmetry of an (unimaginable to most) 57-dimensional object.

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The Standard Model is the currently best accepted theory of elementary particles and their interactions, taking quantum theory and special relativity into account, but not general relativity -- i.e., not gravity. There is a huge amount of experimental evidence for this theory, but its mathematical structure remains complicated and mysterious. Nobody really knows `why' nature likes to work this way! However, the Standard Model displays many tantalising patterns, which are probably important clues.

In the Standard Model, the internal symmetry group is
G = SU(3) × SU(2) × U(1).






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Blobrana


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The E8 root system consists of 240 vectors in an eight-dimensional space.
Those vectors are the vertices (corners) of an eight-dimensional object called the Gosset polytope 421. In the 1960s, Peter McMullen drew (by hand) a 2-dimensional representation of the Gosset polytope 421. The image shown below was computer-generated by John Stembridge, based on McMullen's drawing.

e8plane2a
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Blobrana


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"This is an exciting breakthrough. Understanding and classifying the representations of E8 and Lie groups has been critical to understanding phenomena in many different areas of mathematics and science including algebra, geometry, number theory, physics and chemistry. This project will be invaluable for future mathematicians and scientists" - Peter Sarnak, a professor of mathematics at Princeton University and chair of the scientific board at the American Institute of Mathematics (AIM).

The result of the E8 calculation, which contains all the information about E8 and its representations, is 60 gigabytes. This is enough to store 45 days of continuous music in MP3-format.

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An international team of mathematicians has detailed a vast complex numerical "structure" which was invented more than a century ago.
Mapping the 248-dimensional structure, called E8, took four years of work and produced more data than the Human Genome Project, researchers said.
E8 is a member of the "Lie group" that describe symmetrical objects.
The team said their findings may assist fields of physics which use more than four dimensions, such as string theory.
Lie groups were invented by the 19th Century Norwegian mathematician Sophus Lie (pronounced "Lee").

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Mathematicians have mapped the inner workings of one of the most complicated structures ever studied: the object known as  the exceptional Lie group E8. This achievement is significant both as an advance in basic knowledge and because of the many connections between E8 and other areas, including string theory and geometry. The magnitude of the calculation is staggering: the answer, if written out in tiny print, would cover an area the size of Manhattan. Mathematicians are known for their solitary work style, but the assault on E8 is part of a large project bringing together 18 mathematicians from the U.S. and Europe for an intensive four-year collaboration.

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