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

superstring Superstring theory is an attempt to explain all of the particles and fundamental forces of nature in one theory by modeling them as vibrations of tiny supersymmetric strings. 'Superstring theory' is a shorthand for supersymmetric string theor ...

and a bosonic string. There are two kinds of heterotic string, the heterotic SO(32) and the heterotic E₈ × E₈, abbreviated to HO and HE. 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 Univer ...

, 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 quantu ...

.

Overview

In string theory, the left-moving and the right-moving excitations are completely decoupled, 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 In mathematics, a topological group ''G'' is called a discrete group if there is no limit point in it (i.e., for each element in ''G'', there is a neighborhood which only contains that element). Equivalently, the group ''G'' is discrete if and ...

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 In physics, a gauge theory is a type of field theory in which the Lagrangian (and hence the dynamics of the system itself) does not change (is invariant) under local transformations according to certain smooth families of operations (Lie groups ...

in 10 dimensions. One gauge group is

SO(32) In mathematics, the orthogonal group in dimension , denoted , is the group of distance-preserving transformations of a Euclidean space of dimension that preserve a fixed point, where the group operation is given by composing transformations. ...

(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 Anomaly may refer to: Science Natural *Anomaly (natural sciences) ** Atmospheric anomaly ** Geophysical anomaly Medical * Congenital anomaly (birth defect), a disorder present at birth ** Physical anomaly, a deformation of an anatomical struct ...

-free gauge groups that can be coupled to the ''N'' = 1

supergravity In theoretical physics, supergravity (supergravity theory; SUGRA for short) is a modern field theory that combines the principles of supersymmetry and general relativity; this is in contrast to non-gravitational supersymmetric theories such as ...

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 In physics, a string is a physical entity postulated in string theory and related subjects. Unlike elementary particles, which are zero-dimensional or point-like by definition, strings are one-dimensional extended entities. Researchers often h ...

, not an open string; it is not possible to define any

boundary conditions In mathematics, in the field of differential equations, a boundary value problem is a differential equation together with a set of additional constraints, called the boundary conditions. A solution to a boundary value problem is a solution to t ...

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

String duality

String duality String or strings may refer to: * String (structure), a long flexible structure made from threads twisted together, which is used to tie, bind, or hang other objects Arts, entertainment, and media Films * ''Strings'' (1991 film), a Canadian ani ...

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 contains ...

— a theory that also contains

open strings ''Open Strings'' is an album by French jazz fusion artist Jean-Luc Ponty, released in 1971 on vinyl by the MPS label. Track listing All songs written by Jean-Luc Ponty, except where noted. Side one #"Flipping, Pt.1" – 4:40 #"Flipping, Pt.2 ...

; 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 In theoretical physics, T-duality (short for target-space duality) 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 desc ...

. 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 M-theory is a theory in physics that unifies all 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 1995. Witte ...

.

References

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