string theory In physics, string theory is a theoretical framework in which the point-like particles of particle physics are replaced by one-dimensional objects called strings. String theory describes how these strings propagate through space and intera ...

, a heterotic string is a closed string (or loop) which is a hybrid ('heterotic') of a

superstring Superstring theory is an theory of everything, attempt to explain all of the Elementary particle, particles and fundamental forces of nature in one theory by modeling them as vibrations of tiny supersymmetry, supersymmetric String (physics), st ...

and a bosonic string. There are two kinds of heterotic superstring theories, the heterotic SO(32) and the heterotic E₈ × E₈, abbreviated to HO and HE. Apart from that there exist seven more heterotic string theories which are not supersymmetric and hence are only of secondary importance in most applications. Heterotic string theory was first developed in 1985 by

David Gross David Jonathan Gross (; born February 19, 1941) is an American theoretical physicist and string theorist. Along with Frank Wilczek and David Politzer, he was awarded the 2004 Nobel Prize in Physics for their discovery of asymptotic freedom. ...

, Jeffrey Harvey,

Emil Martinec Emil John Martinec (born 1958) is an American string theorist, a physics professor at the Enrico Fermi Institute at the University of Chicago, and director of the Kadanoff Center for Theoretical Physics. He was part of a group at Princeton Universi ...

, and Ryan Rohm (the so-called "Princeton string quartet"), in one of the key papers that fueled the

first superstring revolution The history of string theory spans several decades of intense research including two superstring revolutions. Through the combined efforts of many researchers, string theory has developed into a broad and varied subject with connections to quantum ...

Overview

, the left-moving and the right-moving excitations of strings are completely decoupled for a closed string, and it is possible to construct a string theory whose left-moving (counter-clockwise) excitations are treated as a bosonic string propagating in ''D'' = 26 dimensions, while the right-moving (clockwise) excitations are treated as a superstring in ''D'' = 10 dimensions. The mismatched 16 dimensions must be compactified on an even, self-dual lattice (a discrete subgroup of a linear space). There are two possible even self-dual lattices in 16 dimensions, and it leads to two types of the heterotic string. They differ by the

gauge group A gauge group is a group of gauge symmetries of the Yang–Mills gauge theory of principal connections on a principal bundle. Given a principal bundle P\to X with a structure Lie group G, a gauge group is defined to be a group of its vertical ...

in 10 dimensions. One gauge group is SO(32) (the HO string) while the other is E₈ × E₈ (the HE string). These two gauge groups also turned out to be the only two anomaly-free gauge groups that can be coupled to the ''N'' = 1 supergravity in 10 dimensions. (Although not realized for quite some time, U(1)⁴⁹⁶ and E₈ × U(1)²⁴⁸ are anomalous.) Every heterotic string must be a closed string, not an open string; it is not possible to define any

boundary conditions In the study of differential equations, a boundary-value problem is a differential equation subjected to constraints called boundary conditions. A solution to a boundary value problem is a solution to the differential equation which also satis ...

that would relate the left-moving and the right-moving excitations because they have a different character.

String duality

String duality String duality is a class of symmetries in physics that link different string theories, theories which assume that the fundamental building blocks of the universe are strings instead of point particles. Overview Before the so-called "duality r ...

is a class of symmetries in physics that link different string theories. In the 1990s, it was realized that the strong coupling limit of the HO theory is

type I string theory In theoretical physics, type I string theory is one of five consistent supersymmetric string theories in ten dimensions. It is the only one whose strings are unoriented (both orientations of a string are equivalent) and the only one which perturba ...

— a theory that also contains open strings; this relation is called

S-duality In theoretical physics, S-duality (short for strong–weak duality, or Sen duality) is an equivalence of two physical theories, which may be either quantum field theories or string theories. S-duality is useful for doing calculations in theore ...

. The HO and HE theories are also related by

T-duality T-duality (short for target-space duality) in theoretical physics is an equivalence of two physical theories, which may be either quantum field theories or string theories. In the simplest example of this relationship, one of the theories descr ...

. Because the various superstring theories were shown to be related by dualities, it was proposed that each type of string was a different limit of a single underlying theory called

M-theory In physics, M-theory is a theory that unifies all Consistency, consistent versions of superstring theory. Edward Witten first conjectured the existence of such a theory at a string theory conference at the University of Southern California in 1 ...

References

{{DEFAULTSORT:Heterotic String String theory E8 (mathematics)