Type HE String Theory

String Theory
All Roads Lead to String Theory (Polchinski)
Prior to the First Superstring Revolution
Early History	S-Matrix 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	T-Duality D-Brane S-Duality Horava-Witten String Theory M-Theory Holographic Principle N=4 Super-Yang-Mills Theory AdS CFT BFSS Matrix Theory Matrix String Theory (2,0) Theory Twistor String Theory F-Theory String Field Theory Pure Spinor Formalism
After the Revolutions
Phenomenology	String Theory Landscape Minimal Supersymmetric Standard Model String Phenomenology

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Type HE String Theory, also known as the $E(8) \times E(8)$ Heterotic String Theory, is a 10-dimensional Heterotic String Theory.

Action Principle

The Action is the Heterotic String Action. It is given by the following Action across the Worldsheet:

$S_H=\iint \frac{T}{2}{{h}^{ab}}\left( {{\partial }_{a}}{{X}^{\mu }}{{\partial }_{b}}{{X}^{\nu }}- \sum\limits_{k=1}^{32}{\left( i\hbar c_0 \bar{\lambda_{\mu}} \not\partial \lambda ^\mu \right)}-\left( i\hbar c_0\bar{\psi_{\mu }} \not\partial\psi^\mu \right) \right){{g}_{\mu \nu }}\sqrt{-\det h_{\alpha\beta}}\mbox{ d}\sigma\mbox{ d}\tau$

The corresponding Lagrangian Density is given by:

${{\mathsf{\mathcal{L}}}_{\mathsf{\mathcal{H}}}}=\frac{T}{2}{{h}^{ab}}\left( {{\partial }_{a}}{{X}^{\mu }}{{\partial }_{b}}{{X}^{\nu }}- \sum\limits_{k=1}^{32}{\left( i\hbar c_0 \bar{\lambda_{\mu}} \not\partial \lambda ^\mu \right)}-\left( i\hbar c_0\bar{\psi_{\mu }} \not\partial\psi^\mu \right) \right){{g}_{\mu \nu }}$

Compactification of mismatched dimensions

With this action for the Type H String Theory being different from the ordinary RNS Action, we have the issue of imbalance between bosons and fermions out of the way. Now, we need to tackle the problem of the left-movers being in 26 dimensions and the right-movers being in 10 dimensions. To do that, we will compactify the $26-10=16$ mismatched dimensions on a lattice. Naturally, to preserve the symmetries of the Heterotic String Action above, we need to make this lattice be unimodular.

The left-movers are the Bosonic String, so we have to consider compactifying 16 dimensions of the Bosonic String. So, we will do the following considerations:

Compactification on this lattice with "dimensionless momenta" $\vec v_L$ and $\vec v_R$ would lead to the following condition:

$\|\vec v_L\|^2 - \|\vec v_R\|^2 + 2 (N - \tilde N) = 0$

But since only the left-movers are Bosonic Strings and need to be compactified $\vec v_R=0$,.

$\|\vec v_L\|^2 = 0 - 2 (N - \tilde N)=-2\left(N-\tilde N\right)$

I.e. the norm-squared 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 $E(8) \times E(8)$ and $\frac{\operatorname{Spin}\left(32\right)}{\mathbb Z _2}$. The Type HE String Theory is the former.

Note that $E(8) \times E(8)$ is also the gauge group of the Type HE string theory.

Suitability 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. However, the gauge group $E(8) \times E(8)$ is suitable for a Theory of Everything as it can easily include the Standard Model gauge group as a Subgroup.

T-Duality with Type HO String Theory

Until the Second Superstring Revolution, it was thought that the Type HE String Theory and Type HO String Theory 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 T-Duality.

The root lattice of $E(8) \times E(8)$ is $\Gamma^8\oplus\Gamma^8$, whereas the root lattice of $\mathrm{Spin}(32)$ is $\Gamma^{16}$. Since:

$\Gamma^8\oplus\Gamma^8\oplus \Gamma^{1,1}=\Gamma^{16}\oplus \Gamma^{1,1}$,

The two types of Type H String Theory are T-Duality to each other.