Statistics on the Heterotic Landscape: Gauge Groups and Cosmological Constants of Four-Dimensional Heterotic Strings

TL;DR

This work delivers the first large-scale statistical survey of four-dimensional heterotic string vacua, focusing on gauge groups and one-loop cosmological constants within tachyon-free, non-supersymmetric models generated by free-fermionic constructions. It uncovers strong correlations between gauge-group structure and the cosmological constant, showing that highly shattered gauge groups tend to yield smaller, often positive, $\Lambda$, and that Standard-Model–like gauge groups are preferentially realized at small $\Lambda$. A striking feature is the enormous degeneracy in the cosmological-constant values: thousands of distinct models share the same $\Lambda$ due to modular-invariance constraints and Atkin-Lehner-type symmetries, suggesting nontrivial landscape structure even at one loop. The findings highlight how worldsheet consistency conditions sculpt the heterotic landscape and motivate further exploration of higher-loop amplitudes, supersymmetric sectors, and broader model classes to assess the generality and phenomenological implications of these correlations.

Abstract

Recent developments in string theory have reinforced the notion that the space of stable supersymmetric and non-supersymmetric string vacua fills out a ``landscape'' whose features are largely unknown. It is then hoped that progress in extracting phenomenological predictions from string theory -- such as correlations between gauge groups, matter representations, potential values of the cosmological constant, and so forth -- can be achieved through statistical studies of these vacua. To date, most of the efforts in these directions have focused on Type I vacua. In this note, we present the first results of a statistical study of the heterotic landscape, focusing on more than 10^5 explicit non-supersymmetric tachyon-free heterotic string vacua and their associated gauge groups and one-loop cosmological constants. Although this study has several important limitations, we find a number of intriguing features which may be relevant for the heterotic landscape as a whole. These features include different probabilities and correlations for different possible gauge groups as functions of the number of orbifold twists. We also find a vast degeneracy amongst non-supersymmetric string models, leading to a severe reduction in the number of realizable values of the cosmological constant as compared with naive expectations. Finally, we also find strong correlations between cosmological constants and gauge groups which suggest that heterotic string models with extremely small cosmological constants are overwhelmingly more likely to exhibit the Standard-Model gauge group at the string scale than any of its grand-unified extensions. In all cases, heterotic worldsheet symmetries such as modular invariance provide important constraints that do not appear in corresponding studies of Type I vacua.

Figures (17)

• Figure 1: The absolute probabilities of obtaining distinct four-dimensional heterotic string models as a function of the degree to which their gauge groups are "shattered" into separate irreducible factors, stretching from a unique model with the irreducible rank-22 gauge group $SO(44)$ to models with only rank-one $U(1)$ and $SU(2)$ gauge-group factors. The total value of the points (the "area under the curve") is 1. As the number of gauge-group factors increases, the behavior of the probability distribution bifurcates according to whether this number is even or odd. Indeed, as this number approaches its upper limit $22$, models with even numbers of gauge-group factors become approximately ten times more numerous than those with odd numbers of gauge-group factors.
• Figure 2: The number of distinct gauge groups realized from heterotic string models with $f$ gauge-group factors, plotted as a function of $f$. Only $1301$ distinct gauge groups are realized from $\sim 10^5$ distinct heterotic string models.
• Figure 3: Average gauge-group multiplicity (defined as the number of distinct heterotic string models divided by the number of distinct gauge groups), plotted as a function of $f$, the number of gauge-group factors in the total gauge group. As the number of factors increases, we see that there are indeed more ways of producing a distinct string model with a given gauge group. Note that the greatest multiplicities occur for models with relatively large, even numbers of gauge-group factors.
• Figure 4: The composition of heterotic gauge groups, showing the average contributions to the total allowed rank from $SO(2n\geq 6)$ factors (denoted 'SO'), $SU(n\geq 3)$ factors (denoted 'SU'), exceptional group factors $E_{6,7,8}$ (denoted 'E'), and rank-one factors $U(1)$ and $SU(2)$ (denoted 'I'). In each case, these contributions are plotted as functions of the number of gauge-group factors in the string model and averaged over all string models found with that number of factors. In the case of $SU(4)\sim SO(6)$ factors, the corresponding rank contribution is apportioned equally between the 'SO' and 'SU' categories. The total of all four lines is 22, as required.
• Figure 5: The probability that a given $SO(2n)$ or $SU(n+1)$ gauge-group factor appears at least once in the gauge group of a randomly chosen heterotic string model, plotted as a function of the rank $n$ of the factor. While the 'SU' curve (solid line) is plotted for all ranks $\geq 1$, the 'SO' curve (dashed line) is only plotted for ranks $\geq 3$ since $SO(2)\sim U(1)$ and $SO(4)\sim SU(2)^2$. These curves necessarily share a common point for rank 3, where $SU(4)\sim SO(6)$.