# Vacuum Constraints for Realistic Strongly Coupled Heterotic M-Theories

## Abstract

**:**

## 1. Introduction

## 2. The Compactification Vacuum

#### 2.1. The d = 11 → d = 5 Compactification

#### 2.1.1. The Calabi-Yau Threefold

#### 2.1.2. The Observable Sector Gauge Bundle

#### 2.1.3. The Hidden Sector Gauge Bundle

**Hidden Sector $\mathit{SU}\left(\mathit{N}\right)$ Factor**

**Hidden Sector Line Bundles**

#### 2.1.4. Bulk Space Five-Branes

#### 2.1.5. Anomaly Cancellation

#### 2.2. The d = 5→ d = 4 Compactification

#### The Linearized Double Domain Wall

## 3. The d = 4 ${\mathit{E}}_{\mathbf{8}}\times {\mathit{E}}_{\mathbf{8}}$ Effective Theory

#### 3.1. The ${\kappa}_{11}^{2/3}$ Lagrangian

**Stability of the Observable Sector Vector Bundle**

**Poly-Stability of the Hidden Sector Vector Bundle**

- ${V}^{\left(2\right)}={\mathcal{V}}_{N}$:In this case, since for a single vector bundle slope-stability implies poly-stability, one need only check that ${\mathcal{V}}_{N}$ is slope-stable. For example, one could choose ${\mathcal{V}}_{N}$ to be identical to the $SU\left(4\right)$ bundle in the observable sector, ${V}^{\left(1\right)}$, presented above. Note that, since we are restricting all hidden sector non-Abelian bundles to have structure group $SU\left(N\right)$, it follows that $\mu \left({\mathcal{V}}_{N}\right)$ must vanish. As with the observable sector bundle $SU\left(4\right)$ bundle, stability of a generic non-Abelian vector bundle will only occur within a specific region of Kähler moduli space.
- ${V}^{\left(2\right)}=L$:In this case, one need only check that the line bundle L is slope-stable, which will imply poly-stability. Fortunately, every line bundle is trivially slope-stable, so any line bundle can be used. It is important to note that the slope of a line bundle which appears as a lone factor in the Whitney sum has–a priori–no further constraints. Using (65), (26) and (4), it follows that the slope of an arbitrary line bundle specified by $L={\mathcal{O}}_{X}({\ell}^{1},{\ell}^{2},{\ell}^{3})$ is given by$$\begin{array}{c}\hfill \mu \left(L\right)={d}_{ijk}{\ell}^{i}{a}^{j}{a}^{k}=\frac{1}{3}\left({a}^{2}({a}^{2}+6{a}^{3}){\ell}^{1}+{({a}^{1})}^{2}{\ell}^{2}+6{a}^{1}{a}^{3}{\ell}^{2}+2{a}^{1}{a}^{2}(2{\ell}^{1}+{\ell}^{2}+6{\ell}^{3})\right).\end{array}$$
- ${V}^{\left(2\right)}={\mathcal{V}}_{N}\oplus L:$As specified above, the non-Abelian vector bundle ${\mathcal{V}}_{N}$ must be slope-stable in a region of Kähler moduli space. Furthermore, since we are restricting the structure group in our discussion to be $SU\left(N\right)$, it follows that $\mu ({\mathcal{V}}_{N})=0$. As we just indicated, any line bundle L will be slope-stable everywhere in Kähler moduli space. However, the full Whitney sum ${V}^{\left(2\right)}={\mathcal{V}}_{N}\oplus L$ will be poly-stable–and, hence, preserve $N=1$ supersymmetry–if and only if $\mu \left(L\right)=\mu ({\mathcal{V}}_{N})=0$. That is, because of the existence of a non-Abelian $SU\left(N\right)$ factor, the line bundle L now has the additional constraint that its slope vanish identically. It is clear from (67) that this will be the case only in a restricted region of Kähler moduli space. It follows that the full Whitney sum ${V}^{\left(2\right)}={\mathcal{V}}_{N}\oplus L$ will only be a viable hidden sector bundle if the region of stability of ${\mathcal{V}}_{N}$ has a non-vanishing intersection with the region where the slope of L vanishes. This is a very non-trivial requirement. To give a concrete example, let us choose ${\mathcal{V}}_{N}={V}^{\left(1\right)}$, where ${V}^{\left(1\right)}$ is the $SU\left(4\right)$ observable sector bundle specified above. Recall that the region of slope-stability of this bundle in Kähler moduli space is delineated by the inequalities in (66) and shown in Figure 1. Plotted in 3-dimensions, this region of slope-stability over a limited region of Kähler moduli space is shown in Figure 2a. Furthermore, let us specify that $L={\mathcal{O}}_{X}(1,2,-3)$. Note that L satisfies condition (23), as it must. It follows from (67) that the region of moduli space in which $\mu \left(L\right)=0$ is given by the equation$$\frac{2}{3}{({a}^{1})}^{2}-4{a}^{1}{a}^{2}+\frac{1}{3}{({a}^{2})}^{2}+4{a}^{1}{a}^{3}+\frac{3}{2}{a}^{2}{a}^{3}=0.$$

#### 3.2. The ${\kappa}_{11}^{4/3}$ Lagrangian

#### 3.2.1. Corrections to a Fayet-Iliopoulos Term

#### 3.2.2. Gauge Threshold Corrections

#### 3.3. A Specific Class of Examples

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 2.**The region of poly-stability for the hidden sector vector bundle ${V}^{\left(2\right)}={V}^{\left(1\right)}\oplus {\mathcal{O}}_{X}(1,2,-3)$. The red volume in Figure 2a is the sub-region of Kähler moduli space where the bundle ${V}^{\left(1\right)}$ is slope stable, whereas the green volume of Figure 2b is a sub-region of where $\mu ({\mathcal{O}}_{X}(1,2,-3))=0$. They have a substantial region of overlap in Kähler moduli space, indicated by the yellow volume in Figure 2c.

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Ovrut, B.A.
Vacuum Constraints for Realistic Strongly Coupled Heterotic M-Theories. *Symmetry* **2018**, *10*, 723.
https://doi.org/10.3390/sym10120723

**AMA Style**

Ovrut BA.
Vacuum Constraints for Realistic Strongly Coupled Heterotic M-Theories. *Symmetry*. 2018; 10(12):723.
https://doi.org/10.3390/sym10120723

**Chicago/Turabian Style**

Ovrut, Burt A.
2018. "Vacuum Constraints for Realistic Strongly Coupled Heterotic M-Theories" *Symmetry* 10, no. 12: 723.
https://doi.org/10.3390/sym10120723