String Theory  

All Roads Lead to String Theory (Polchinski)  
Prior to the First Superstring Revolution
 
Early History  SMatrix Theory Regge Trajectory  
Bosonic String Theory  Worldsheet String Bosonic String Theory String Perturbation Theory Tachyon Condensation  
Supersymmetric Revolution  Supersymmetry RNS Formalism GS Formalism BPS  
Superstring Revolutions


First Superstring Revolution  GSO Projection Type II String Theory Type IIB String Theory Type IIA String Theory Type I String Theory Type H String Theory Type HO String Theory Type HE String Theory 

Second Superstring Revolution  TDuality DBrane SDuality HoravaWitten String Theory MTheory Holographic Principle N=4 SuperYangMills Theory AdS CFT BFSS Matrix Theory Matrix String Theory (2,0) Theory Twistor String Theory FTheory String Field Theory Pure Spinor Formalism 

After the Revolutions
 
Phenomenology  String Theory Landscape Minimal Supersymmetric Standard Model String Phenomenology  
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The Type HO String Theory also known as the Spin(32)/Z_2 Heterotic String Theory, or the SO(32) Heterotic String Theory, or the , is a 10dimensional Heterotic String Theory.^{[1]} This means that the State is formed by tensoring the bosonic state (but with added MajaronaWeyl Fermions), with the Type II state. However, this raises the issue that the leftmoving state is in 26 dimensional spacetime, but the rightmoving state is in 10dimensional spacetime. To fix these mismatched dimensions, we compactify the 16 mismatched dimensions on an even unimodular lattice, which means that the lattice has to have a Cartan Determinant of 1, and the vectors must have even magnitudes. For the Type HO String Theory, it is .
Table of Contents
Action Principle
The Action for the Type HO String Theory is of course the Heterotic String Action. It is given by the following action across the Worldsheet:^{[1]} ^{[2]}
The corresponding Lagrangian Density is obviously given by:^{[1]} ^{[2]}
Where we removed the square root of the negative of the determinant of the Worldsheet metric because the Lagrangian Density is local at all points on the Worldsheet.
Compactification of mismatched dimensions
With this new action for the Type H String Theory being different from the ordinary RNS Action, we have solved the issue of the imbalance between bosons and fermions. Now, we need to tackle the problem of the leftmovers existing in 26 dimensions and the rightmovers existing in 10 dimensions, which seems like a serious inconsiswtencsy in the theory.. To do that, we will compactify the mismatched dimensions between the leftmovers and the rightmovers, on a lattice, on some lattice.. Naturally, to preserve the symmetries of the Heterotic string action above, we need to make this lattice be Unimodular, or SelfDual.
The leftmovers are from the Bosonic String State, so we have to consider compactifying 16 dimensions of the Bosonic String. So, we will do the following considerations:^{[2]}
Compactification on this lattice with "dimensionless momenta" and would lead to the following condition:
But since only the leftmovers are Bosonic Strings and need to be compactified ,.
I.e. the normsquared is even. A lattice made of such vectors is an even lattice, and thus, the lattice also needs to be somewhat even.
The only suitable even, unimodular, 16 dimensional lattices are and . The String Theory is the latter.
Note that is also the gauge group of this String Theory.
Unsuitability as a Theory of Everything
The necessity for Heterotic Strings arose when it was found that Type IIB string theory was not suitable for the Theory of Everything and neither was Type IIA.^{[3]} The Type HO String Theory, however, is not suitable for the Theory of Everything either, because its gauge group cannot contain the Standard Model Gauge group as a Subgroup. However, the gauge group of Type HE is suitable for a Theory of Everything as can easily include the Standard Model Gauge group as a subgroup.^{[3]} ...
TDuality with Type HE String Theory
Until the Second Superstring Revolution, it was thought that the two Heterotic String Theories were only connected due to their mismatched forms (i.e. with the 16 mismatched dimensions uncompactified.). However, this is useless, as this is neither a duality nor an equivalence, so one may not derive one String Theory from the other this way. During the Second Superstring Revolution, it was discovered that these two are actually related by TDuality.
The root lattice of is , whereas the root lattice of is . Since:
,
The two types of Heterotic String Theory are TDual to each other.
References
 ↑ ^{1.0} ^{1.1} ^{1.2} McMohan, David (2008). String theory demistified. Chicago: McGraw Hill. p. 207. ISBN 9780071498708. http://www.nucleares.unam.mx/~alberto/apuntes/mcmahon.pdf.
 ↑ ^{2.0} ^{2.1} ^{2.2} Polchinski, Joseph. (1998). String Theory: Volume 2, p. = 45.
 ↑ ^{3.0} ^{3.1} Mohaupt, Thomas. Introduction to String theory. http://arxiv.org/pdf/hepth/0207249v1.pdf.