Editing Heterotic string theory

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In [[string theory]], a '''heterotic string''' is a closed string (or loop) which is a hybrid ('heterotic') of a [[superstring]] and a [[bosonic string]]. There are two kinds of heterotic superstring theories, the heterotic SO(32) and the heterotic E<sub>8</sub>&nbsp;×&nbsp;E<sub>8</sub>, 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.<ref>{{Cite book |last=Polchinski |first=Joseph |title=String Theory: Superstring Theory and Beyond |publisher=Cambridge University Press |year=1998 |isbn=9780521633048 |volume=2 |pages=55-59 |language=English}}</ref> Heterotic string theory was first developed in 1985 by [[David Gross]], [[Jeffrey A. Harvey|Jeffrey Harvey]], [[Emil Martinec]], and [[Ryan Rohm]]<ref>{{cite journal | last1=Gross | first1=David J. | last2=Harvey | first2=Jeffrey A. | last3=Martinec | first3=Emil | last4=Rohm | first4=Ryan | title=Heterotic String | journal=Physical Review Letters | publisher=American Physical Society (APS) | volume=54 | issue=6 | date=1985-02-11 | issn=0031-9007 | doi=10.1103/physrevlett.54.502 | pages=502–505| pmid=10031535 | bibcode=1985PhRvL..54..502G }}</ref> (the so-called "Princeton string quartet"<ref>{{cite web|author=Dennis Overbye|author-link=Dennis Overbye|url=https://www.nytimes.com/2004/12/07/science/07stri.html?pagewanted=2&fta=y|title=String theory, at 20, explains it all (or not)|work=The New York Times|date=2004-12-07|access-date=2020-03-15}}</ref>), in one of the key papers that fueled the [[first superstring revolution]].

==Overview==
In [[string theory]], the left-moving and the right-moving excitations of strings are completely decoupled for a closed string,<ref>{{cite book | last1=Becker | first1=Katrin | last2=Becker|first2=M.|last3=Schwarz|first3=J. H.|title=String theory and M-theory : a modern introduction | url=https://archive.org/details/stringtheorymthe00beck_649 | url-access=limited | publisher=Cambridge University Press | publication-place=Cambridge New York | year=2007 | isbn=978-0-521-86069-7 | oclc=607562796 | page=[https://archive.org/details/stringtheorymthe00beck_649/page/n270 253]}}</ref> and it is possible to construct a string theory whose left-moving (counter-clockwise) excitations are treated as a bosonic string propagating in ''D''&nbsp;=&nbsp;26 dimensions, while the right-moving (clockwise) excitations are treated as a superstring in ''D''&nbsp;=&nbsp;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]] in 10 dimensions. One gauge group is [[special orthogonal group|SO(32)]] (the HO string) while the other is [[E8 (mathematics)|E<sub>8</sub>&nbsp;×&nbsp;E<sub>8</sub>]] (the HE string).<ref>[[Joseph Polchinski]] (1998). ''String Theory: Volume 2'', p. 45.</ref>

These two gauge groups also turned out to be the only two [[anomaly (physics)|anomaly]]-free gauge groups that can be coupled to the [[type I supergravity|''N''&nbsp;=&nbsp;1 supergravity]] in 10 dimensions. (Although not realized for quite some time, U(1)<sup>496</sup> and E<sub>8</sub>&nbsp;×&nbsp;U(1)<sup>248</sup> are anomalous.<ref>{{cite journal | last1=Adams | first1=Allan | last2=Taylor | first2=Washington | last3=DeWolfe | first3=Oliver | title=String Universality in Ten Dimensions | journal=Physical Review Letters | volume=105 | issue=7 | date=2010-08-10 | issn=0031-9007 | doi=10.1103/physrevlett.105.071601 | page=071601| pmid=20868028 |arxiv=1006.1352| bibcode=2010PhRvL.105g1601A | s2cid=13916249 }}</ref>)

Every heterotic string must be a [[closed string]], not an [[String (physics)|open string]]; it is not possible to define any [[boundary conditions]] that would relate the left-moving and the right-moving excitations because they have a different character.

==String duality==
[[String duality]] 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]] &mdash; a theory that also contains [[Open string (physics)|open strings]]; this relation is called [[S-duality]]. The HO and HE theories are also related by [[T-duality]].

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]].

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==References==
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{{String theory topics |state=collapsed}}

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[[Category:String theory]]
[[Category:E8 (mathematics)]]