Editing Heterotic string theory (section)

==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.