# Type H String Theory

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In String Theory, a Type H String is a Closed String which is a hybrid ("heterotic") of a Type II String and a Bosonic String. There are two kinds of Heterotic String, the Type HO String and the Type HE String. Heterotic String Theory was first developed in 1985 by David Gross, Jeffrey Harvey, Emil Martinec, and Ryan Rohm ("Princeton String Quartet" [2]), in one of the key papers that fueled the first superstring revolution .[3][4] .

In String Theory, the left-moving and the right-moving excitations almost do not interact with each other, 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 (clock-wise) excitations are treated as a superstring in $D = 10$ dimensions.[3] [4] 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 $\mathrm{Spin}(32)/\mathbb{Z}_2$ while the other is $E(8)\times E(8)$.[3] .

These two gauge groups also turned out to be the only two anomaly-free gauge groups that can be coupled to the $\mathcal N = 1$ supergravity in 10 dimensions other than $U(1)^{496}$ and $E(8)\times U(1)^{248}$ , which is suspected to lie in the swampland.[4]

Every heterotic string must be a Closed String, not an 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. Reason being that the right-moving state belongs to the Type II String Theory, and therefore, only closed Bosonic Strings can be taken for the left-movers, and the entire string is closed.

## Action principleEdit this section

Here, we will derive the Lagrangian Density for the Type H String Theory. Firstly, by the premise, the left-moving component is from the Bosonic String Theory whereas the right-moving component is from Type II String Theory (both Type IIB String Theory and the Type IIA String Theory are fine since the left- and right- moving excitations don't interact). However, there are two apparent contradictions here.

1. How is it possible that the left-moving component has only bosons?
2. How is it possible that the left-moving component lives in 26 dimensional spacetime whereas the right-moving component lives in 10-dimensional spacetime?

We will first address (1). Instead of simply tensoring the Bosonic left-mover and the Type II right-mover here, we will add some fermions, to the left-mover. At first glance, one is tempted to incorporate supersymmetry to the left-mover, but that would be stupid, as then one would obtain the Type II String Theory in the left-movers too, and then the entire state, and thus the entire theory would become a Type II state and the Type II String Theory respectively.

So, we will add some Majorana-Weyl Fermions to the left-moving state without adding Supersymmetry (i.e. somewhat artificially).

Now, since the right-moving state is Type II String Theory, this means there are 32 Supercharges in the right-moving sector, so the left-moving sector needs 32 Majorana-Weyl Fermions to counter these 32 supercharges.

The Lagrangian Density for 1 Majorana-Weyl Fermion is [4]:

${{\mathsf{\mathcal{L}}}_{\mathsf{\mathcal{MW}}}}=\frac{T}{2}{{h}^{ab}}\left( - {\left( i\hbar c_0 \bar{\lambda_{\mu}} \not\partial \lambda ^\mu \right)} \right){{g}_{\mu \nu }}$

Where the use of $\lambda$ as opposed to $\psi$ indicates that we are dealing with a Majorana-Weyl Fermion here, as opposed to a Dirac Fermion. So, the Lagrangian Density of 32 Majorana-Weyl Fermions would be:[4]

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

The indices $k$ are hidden in the summation to avoid overcrowding of indices. Now, we need to add this to RNS Lagrangian Density $\mathcal{L_{RNS}}$ which describes the rest of the Type H String Theory, Type II String Theory and all. So, the RNS Lagrangian Density is:

$\mathcal{L_{RNS}} = {\frac{T }{2}}h^{ab} \left(\partial_a X^\mu \partial_b X^\nu - i\hbar c_0\bar{\psi_\mu }\not\partial\psi^\mu \right)g_{\mu \nu}$

Adding up the Lagrangian Densityies, we obtain: [4] [5]

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

This is the [[Lagrangian Density] for the Heterotic Strings. Notice the elegant symmetries:

(1) The Supersymmetry between $X$ and $\psi$ and
(2) The symmetry between $\psi$ and $\lambda$ with appropriate summations.

The Action is then clearly:

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

This Heterotic String Theory has thus a Supersymmetry of $\mathcal{ N } = 1$.

## The Type HO String TheoryEdit this section

The Type HO String Theory, also known as the $\mathrm{Spin}(32)/\mathbb{Z}_2$ String Theory or occassionally the $SO(32)$ String Theory or the Spin(32) String Theory, is the Heterotic String Theory with the mismatched dimensions compactified on the lattice $\frac{\mathrm{Spin}\left( 32 \right) }{\mathbb{Z}_2 }$.

## The E(8) X E(8) string theoryEdit this section

The Type HE String Theory, also known as the $E(8) X E(8)$ String Theory is the Heterotic String Theory with the mismatched dimensions compactified on the lattice $E(8) \times E(8)$.

## T-Duality and Type H String TheoryEdit this section

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 $\frac{\operatorname{Spin}\left(32\right)}{\mathbb Z_2}$ is $\Gamma^{16}$. Since:

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

The Type HE String Theory and the Type HO String Theory are thus T-Dual to each other.

## ReferencesEdit this section

1. The Heterotic String. David J. Gross, Jeffrey A. Harvey, Emil J. Martinec, Ryan Rohm, Published in Phys. Rev. Lett. 54:502–505, 1985.
2. Overbye, Dennis (7/12/2004). "String Theory, at 20, explains it all (or not)". NY Times. Retrieved 29 July 2013.
3. 3.0 3.1 3.2 Polchinski, Joseph. (1998). String Theory: Volume 2, p. 45.
4. 4.0 4.1 4.2 4.3 4.4 4.5 McMohan, David (2009). String theory DeMystified. Chicago: McGrawHill. ISBN 978-0071498708.
5. Becker, Becker, Schwarz, Katrin, Melanie, John H. (2007). String theory and M-theory. United Kingdom: Cambridge University Press. p. 120. ISBN 978-0521860697.