The Particle Spectrum of Heterotic Compactifications

TL;DR

The paper presents explicit techniques to compute the full particle spectrum in four-dimensional heterotic vacua by evaluating the cohomology of stable holomorphic vector bundles on elliptically fibered Calabi–Yau threefolds via spectral data and Fourier–Mukai transforms. It shows that, while generic vector-bundle moduli yield a fixed spectrum, the spectrum can jump on subvarieties of moduli space, enabling transitions between different particle contents in the same vacuum. Through an explicit SU(5) GUT model with base $B= ext{F}_1$ and spectral data ${oldsymbol{ ext C}}_V i|5oldsymbol{ ext σ}+oldsymbol{ ho}^*(12S+15oldsymbol{ ext E})|$, the authors compute $h^1(X,U)$ for $U eq Voldsymbol{ extstyleigotimes}V^*$ and demonstrate moduli-driven jumps in $h^1(X,oldsymbol{ extstyleigwedge^2}V)$ and $h^1(X,oldsymbol{ extstyleigwedge^2}V^*)$. The results yield a spectrum with $n_{78}=1$, $n_{ar{10}}=0$, $n_{10}=3$, and variable $n_{ar{5}}$ and $n_{5}$ (ranging generically as $n_{ar{5}}=37$ and $n_{5}=34$, but capable of jumps up to $n_{ar{5}}=94$ and $n_{5}=91$), while the total vector-bundle moduli number is $n_1=223$. This framework generalizes to arbitrary heterotic vacua and highlights the potential for moduli-induced spectral transitions within a fixed compactification, contrasting with the standard embedding where many cohomologies are topological invariants.

Abstract

Techniques are presented for computing the cohomology of stable, holomorphic vector bundles over elliptically fibered Calabi-Yau threefolds. These cohomology groups explicitly determine the spectrum of the low energy, four-dimensional theory. Generic points in vector bundle moduli space manifest an identical spectrum. However, it is shown that on subsets of moduli space of co-dimension one or higher, the spectrum can abruptly jump to many different values. Both analytic and numerical data illustrating this phenomenon are presented. This result opens the possibility of tunneling or phase transitions between different particle spectra in the same heterotic compactification. In the course of this discussion, a classification of SU(5) GUT theories within a specific context is presented.

Figures (3)

• Figure 1: A subspace of the moduli space ${\mathcal{M}}$ of $\phi$'s spanned by $\phi^{[(4)1]}_{p=1,2}$, $\phi^{[(4)2]}_{q=1,2,3}$ and $\phi^{[(4)3]}_{r=1,2,3,4}$. Generically, in the bulk, $n_{\overline{5}} = 37$, its minimal value. As we restrict to various planes and intersections thereof, we are confining ourselves to special sub-spaces of co-dimension one or higher. In these subspaces, the value of $n_{\overline{5}}$ can increase dramatically.
• Figure 2: In 100,000 random initializations of the matrix $M_2$ of integers valued between 1 and 3, the numbers of occurrences of the various values of $\hbox{rk}(M_2)$ are plotted. We see that the generic value 85 dominates by far.
• Figure 3: A subspace of the moduli space ${\mathcal{M}}$ of $\phi$'s spanned by $\phi^{[(4)1]}_{p=1,2}$, $\phi^{[(4)2]}_{q=1,2,3}$ and $\phi^{[(4)3]}_{r=1,2,3,4}$. Generically, in the bulk, the rank of $M_2$ is 85, its maximal value. As we restrict to various planes and intersections thereof, we are confining ourselves to special sub-spaces of co-dimension one or higher. In these subspaces, the rank of $M_2$ can drop dramatically.