Supersymmetry versus Gauge Symmetry on the Heterotic Landscape
April 2007
Supersymmetry versus Gauge Symmetry on the
Heterotic Landscape
Keith R. Dienes1∗, Michael Lennek1†, David Sénéchal2‡,
Vaibhav Wasnik1§
1Department of Physics, University of Arizona, Tucson, AZ 85721 USA
2Département de Physique, Université de Sherbrooke, Sherbrooke, Québec J1K 2R1
Canada
Abstract
One of the goals of the landscape program in string theory is to extract in-
formation about the space of string vacua in the form of statistical correlations
between phenomenological features that are otherwise uncorrelated in field the-
ory. Such correlations would thus represent predictions of string theory that
hold independently of a vacuum-selection principle. In this paper, we study
statistical correlations between two features which are likely to be central to
any potential description of nature at high energy scales: gauge symmetries and
spacetime supersymmetry. We analyze correlations between these two kinds of
symmetry within the context of perturbative heterotic string vacua, and find a
number of striking features. We find, for example, that the degree of spacetime
supersymmetry is strongly correlated with the probabilities of realizing certain
gauge groups, with unbroken supersymmetry at the string scale tending to fa-
vor gauge-group factors with larger rank. We also find that nearly half of the
heterotic landscape is non-supersymmetric and yet tachyon-free at tree level;
indeed, less than a quarter of the tree-level heterotic landscape exhibits any
supersymmetry at all at the string scale.
∗ E-mail address: dienes@physics.arizona.edu
†E-mail address: mlennek@physics.arizona.edu
‡E-mail address: david.senechal@usherbrooke.ca
§E-mail address: wasnik@physics.arizona.edu
http://arxiv.org/abs/0704.1320v1
1 Introduction
Recent developments in string theory suggest that there exists a huge “landscape”
of self-consistent string vacua [1]. The existence of this landscape is of critical im-
portance for string phenomenology since the specific low-energy phenomenology that
can be expected to emerge from string theory depends critically on the particular
choice of vacuum state. Detailed quantities such as particle masses and mixings, and
even more general quantities and structures such as the choice of gauge group, num-
ber of chiral particle generations, and the magnitude of the supersymmetry-breaking
scale, can be expected to vary significantly from one vacuum solution to the next.
Thus, in the absence of some sort of vacuum-selection principle, it is natural to deter-
mine whether there might exist generic string-derived statistical correlations between
different phenomenological features that would otherwise be uncorrelated in field the-
ory [2]. In this way, one can still hope to extract phenomenological predictions from
string theory.
To date, there has been considerable work in this direction [2, 3, 4, 5, 6, 7, 8,
9, 10, 11, 12]; for recent reviews, see Ref. [13]. Collectively, this work addresses
questions ranging from the formal (such as the finiteness of the number of string
vacua and the methods by which they may be efficiently scanned and classified) to
the phenomenological (such as the value of the cosmological constant, the scale of
supersymmetry breaking, and the statistical prevalence of the Standard Model gauge
group and three chiral generations).
In this paper, we shall undertake a statistical study of the correlations between
two phenomenological features which are likely to be central to any description of na-
ture at high energy scales: spacetime supersymmetry and gauge symmetry. Indeed,
over the past twenty years, a large amount of theoretical effort has been devoted
to studying string models with N=1 spacetime supersymmetry. However, it is im-
portant to understand the implications of choosing N=1 supersymmetry over other
classes of string models (such as models with N=2 or N=4 supersymmetry, or even
non-supersymmetric string models) within the context of the landscape. Moreover,
since N=1 supersymmetry plays a huge role in current theoretical efforts to extend
the Standard Model, we shall also be interested in understanding the statistical preva-
lence of spacetime supersymmetry across the landscape and the degree to which the
presence or absence of supersymmetry affects other phenomenological features such
as the choice of gauge group and the resulting particle spectrum.
In this paper, we shall investigate such questions within the context of the het-
erotic string landscape. There are several reasons why we shall focus on the heterotic
landscape. First, heterotic strings are of tremendous phenomenological interest in
their own right; indeed, these strings the framework in which most of the original
work in string phenomenology was performed in the late 1980’s and early 1990’s.
Second, heterotic strings have internal constructions and self-consistency constraints
which are, in many ways, more constrained than those of their Type I (open) coun-
terparts. Thus, they are likely to exhibit phenomenological correlations which differ
from those that might be observed on the landscape of, say, intersecting D-brane
models or Type I flux vacua. Finally, in many cases these perturbative supersym-
metric heterotic strings are dual to other strings (e.g., Type I orientifold models)
whose statistical properties are also being analyzed in the literature. Thus, analysis of
the perturbative heterotic landscape, both supersymmetric and non-supersymmetric,
might eventually enable statistical tests of duality symmetries across the entire string
landscape.
The first statistical study of the heterotic landscape appeared in Ref. [8]. This
study, which focused exclusively on the statistical properties of non-supersymmetric
(N=0) tachyon-free heterotic string vacua, was based on a relatively small data set of
four-dimensional heterotic string models [14] which were randomly generated using
software originally developed in Ref. [15]. Since then, there have been several ad-
ditional statistical examinations of certain classes of N=1 supersymmetric heterotic
strings [10, 11]. Together, such studies can therefore be viewed as providing heterotic
analogues of the Type I statistical studies reported in Refs. [5, 6, 7].
Although the study we shall undertake here is similar in spirit to that of Ref. [8],
there are several important differences which must be highlighted. First, as discussed
above, we shall be focusing here on the effects of spacetime supersymmetry. Thus, we
shall be examining models with all levels of spacetime supersymmetry (N=0, 1, 2, 4),
not just non-supersymmetric models, and examining how the level of spacetime su-
persymmetry correlates with gauge symmetry. Second, the current study is based on
a much larger data set consisting of approximately 107 heterotic string models which
was newly generated for this purpose using an update of the software originally devel-
oped in Ref. [15]. This data set is thus approximately two orders of magnitude larger
than that used for Ref. [8], and represents literally the largest set of distinct heterotic
string models ever constructed. Indeed, for reasons we shall discuss in Sect. 3, we
believe that data sets of this approximate size are probably among the largest that
can be generated using current computer technology.
But perhaps most importantly, because our heterotic-string data set was newly
generated for the purpose of this study, we are able to quote results that take into
account certain subtleties concerning so-called “floating correlations”. As discussed in
Ref. [9], the problem of floating correlations is endemic to investigations of this type,
and reflects the fact that not all physically distinct string models are equally likely
to be sampled in any random search through the landscape. This thereby causes
statistical correlations to “float” as a function of sample size. In Ref. [9], several
methods were developed that can be used to overcome this problem, and it was shown
through explicit examples that these methods allow one to extract correlations and
statistical distributions which are not only stable as a function of sample size, but
which also differ significantly from those which would have been näıvely apparent from
a direct counting of generated models. We shall therefore employ these techniques in
the current paper, extracting each of our statistical results in such a way that they
represent stable correlations across the entire heterotic landscape we are examining.
As with most large-scale statistical studies of this type, there are several lim-
itations which must be borne in mind. First, our sample size is relatively small,
consisting of only ∼ 107 distinct models. However, although this number is miniscule
compared with the numbers of string models that are currently quoted in most land-
scape discussions, we believe that the statistical results we shall obtain are stable as a
function of sample size and would not change significantly as more models are added
to the data sample. We shall discuss this feature in more detail in Sect. 3. Indeed, as
mentioned above, data samples of the current size are likely to be the largest possible
given current computer technology.
Second, the analysis in this paper shall be limited to correlations between only
two phenomenological properties of these models: their low-energy gauge groups, and
their levels of supersymmetry. More detailed examinations of the particle spectra of
these models will be presented in Ref. [16].
Finally, the models we shall be discussing are stable only at tree level. For exam-
ple, the models with spacetime supersymmetry continue to have flat directions which
have not been lifted. Even worse, the non-supersymmetric models (even though
tachyon-free) will generally have non-zero dilaton tadpoles and thus are not stable
beyond tree level. Despite these facts, each of the string models we shall be studying
represents a valid string solution at tree level, satisfying all of the necessary string
self-consistency constraints. These include the requirements of worldsheet confor-
mal/superconformal invariance, modular-invariant one-loop and multi-loop ampli-
tudes, proper spacetime spin-statistics relations, and physically self-consistent layers
of sequential GSO projections and orbifold twists. Thus, although such models may
not represent the sorts of truly stable vacua that we would ideally like to be study-
ing, it is reasonable to hope that any statistical correlations we uncover are likely
to hold even after vacuum stabilization. Indeed, since no stable perturbative non-
supersymmetric heterotic strings have yet been constructed, this sort of analysis is
currently the state of the art for large-scale statistical studies of this type, and mirrors
the situation on the Type I side, where state-of-the-art statistical analyses [5, 6, 7]
have also focused on models which are only stable at tree level. Eventually, once
the heterotic model-building technology develops further and truly stable vacua can
be analyzed, it will be interesting to compare those results with these in order to
ascertain the degree to which vacuum stabilization might affect these other phe-
nomenological properties.
This paper is organized as follows. In Sect. 2, we describe the class of models that
we shall be examining in this paper. In Sect. 3, we summarize our method of analysis
which enables us to overcome the problem of floating correlations in order to extract
statistically meaningful correlations. In Sect. 4, we present our results concerning the
prevalence of spacetime supersymmetry across the heterotic landscape, and in Sect. 5
we present our results concerning correlations between spacetime supersymmetry and
gauge groups. Finally, our conclusions are presented in Sect. 6.
2 The models
The models we shall be examining in this paper are similar to those studied in
Ref. [8]. Specifically, each of the vacua we shall be examining in this paper represents
a weakly coupled critical heterotic string compactified to four large (flat) spacetime
dimensions. In general, such a string may be described in terms of its left- and right-
moving worldsheet conformal field theories (CFT’s). For a string in four dimensions,
these must have central charges (cR, cL) = (9, 22) in order to enforce worldsheet
conformal anomaly cancellation, and must exhibit conformal invariance for the left-
movers and superconformal invariance for the right-movers. While any such CFT’s
may be considered, in this paper we shall focus on those string models for which
these internal worldsheet CFT’s may be taken to consist of tensor products of free,
non-interacting, complex (chiral) bosonic or fermionic fields.
As discussed in Ref. [8], this is a huge class of models which has been discussed
and analyzed in many different ways in the string literature. On the one hand, taking
these worldsheet fields as fermionic leads to the so-called “free-fermionic” construc-
tion [17] which will be our primary tool throughout this paper. In the language
of this construction, different models are achieved by varying (or “twisting”) the
boundary conditions of these fermions around the two non-contractible loops of the
worldsheet torus while simultaneously varying the phases according to which the con-
tributions of each such spin-structure sector are summed in producing the one-loop
partition function. However, alternative but equivalent languages for constructing
such models exist. For example, we may bosonize these worldsheet fermions and
construct “Narain” models [18, 19] in which the resulting complex worldsheet bosons
are compactified on internal lattices of appropriate dimensionality with appropriate
self-duality properties. Furthermore, many of these models have additional geomet-
ric realizations as orbifold compactifications with appropriately chosen Wilson lines;
in general, the process of orbifolding is quite complicated in these models, involving
many sequential layers of projections and twists. All of these constructions generally
overlap to a large degree, and all are capable of producing models in which the cor-
responding gauge groups and particle contents are quite intricate. Nevertheless, in
all cases, we must ensure that all required self-consistency constraints are satisfied.
These include modular invariance, physically sensible GSO projections, proper spin-
statistics identifications, and so forth. Thus, each of these vacua represents a fully
self-consistent string solution at tree level.
In order to efficiently survey the space of such four-dimensional string-theoretic
vacua, we implemented a computer search based on the free-fermionic spin-structure
construction [17]. Details of this study are similar to those of the earlier study de-
scribed in Ref. [8], and utilize an updated version of the model-generating software
that was originally written for Ref. [15]. In our analysis, we restricted our attention to
those models for which our real worldsheet fermions can always be uniformly paired
to form complex fermions, and therefore it was possible to specify the boundary con-
ditions (or spin-structures) of these real fermions in terms of the complex fermions
directly. We also restricted our attention to cases in which the worldsheet fermions
exhibited either antiperiodic (Neveu-Schwarz) or periodic (Ramond) boundary con-
ditions around the non-contractible loops of the torus. Of course, in order to build a
self-consistent string model in this framework, these boundary conditions must sat-
isfy tight constraints. These constraints are necessary in order to ensure that the
one-loop partition function is modular invariant and that the resulting Fock space of
states can be interpreted as arising through a physically sensible projection from the
space of all worldsheet states onto the subspace of physical states with proper space-
time spin-statistics. Thus, within a given string model, it is necessary to sum over
appropriate sets of untwisted and twisted sectors with different boundary conditions
and projection phases.
Our statistical analysis consisted of an examination of over 107 distinct vacua
in this class. Essentially, each set of fermion boundary conditions and GSO projec-
tion phases was chosen randomly in each sector, subject only to the required self-
consistency constraints. However, in our statistical sampling, we placed essentially
no limits on the complexity of the orbifold twisting (i.e., in the free-fermionic lan-
guage, we allowed as many as sixteen linearly independent basis vectors). Thus, our
statistical analysis included models of arbitrary intricacy and sophistication. We also
made use of techniques developed specifically for analyzing string models generated
in random searches, allowing for the mitigation of many of the effects of bias which
are endemic to studies of this sort.
As part of our study, we generated string models with all degrees of spacetime
supersymmetry (N=0, 1, 2, 4) that can arise in four dimensions. For N=0 models,
we further demanded that supersymmetry be broken without introducing tachyons.
Thus, the N=0 vacua are all non-supersymmetric but tachyon-free, and can be con-
sidered as four-dimensional analogues of the ten-dimensional SO(16)× SO(16) het-
erotic string [20] which is also non-supersymmetric but tachyon-free. However, other
than this, we placed no requirements on other possible phenomenological properties
of these vacua such as their possible gauge groups, numbers of chiral generations,
or other aspects of the particle content. We did, however, require that our string
construction begin with a supersymmetric theory in which the supersymmetry may
or may not be broken by subsequent orbifold twists. (In the language of the free-
fermionic construction, this is tantamount to demanding that our fermionic boundary
conditions include a superpartner sector, typically denoted W1 or V1.) This is to be
distinguished from a potentially more general class of models in which supersymme-
try does not appear at any stage of the construction. This is merely a technical detail
in our construction, and we do not believe that this ultimately affects our results.
As with any string-construction method, the free-fermionic formalism contains
numerous redundancies in which different choices of worldsheet fermion boundary
conditions and/or GSO phases lead to identical string models in spacetime. Indeed,
a given unique string model can have many different representations in terms of
worldsheet constructions. For this reason, we judged string vacua to be distinct
based on their spacetime characteristics — i.e., their low-energy gauge groups and
massless particle content.
SUSY class # distinct models
N=0 (tachyon-free) 4 946 388
N=1 3 772 679
N=2 492 790
N=4 1106
Total: 9 212 963
Table 1: The data set of perturbative heterotic strings analyzed in this paper. For each
level of supersymmetry allowed in four dimensions, we list the number of corresponding
distinct models generated. As discussed in the text, models are judged to be distinct
based on their spacetime properties (e.g., gauge groups and particle content). All non-
supersymmetric models listed here are tachyon-free and thus are four-dimensional analogs
of the SO(16) × SO(16) string model in ten dimensions.
Given this, our ultimate data set of heterotic strings is as described in Table 1.
Note that all non-supersymmetric models listed in Table 1 are tachyon-free, and thus
are stable at tree level. We should mention that while generating these models, we
also generated over a million distinct non-supersymmetric tachyonic vacua which are
not even stable at tree level. We therefore did not include their properties in our
analysis, and recorded their existence only as a way of gauging the overall degree to
which the tree-level heterotic string landscape is tachyon-free. Also note that as the
level of supersymmetry increases, the number of distinct models in our sample set
decreases. This reflects the fact that relatively fewer of these models exist, so they
become more and more difficult to generate. This will be discussed further in Sects. 3
and 4.
Of course, the free-fermionic construction realizes only certain points in the full
model space of self-consistent heterotic string models. For example, since each world-
sheet fermion is nothing but a worldsheet boson compactified at a specific radius, a
larger (infinite) class of models can immediately be realized through a bosonic for-
mulation by varying these radii away from their free-fermionic values. However, this
larger class of models has predominantly only abelian gauge groups and rather lim-
ited particle representations. Indeed, the free-fermionic points typically represent
precisely those points at which additional (non-Cartan) gauge-boson states become
massless, thereby enhancing the gauge symmetries to become non-abelian. Thus, the
free-fermionic construction naturally leads to precisely the set of models which are
likely to be of direct phenomenological relevance.
We should note that it is also possible to go beyond the class of free-field string
models altogether, and consider models built from more complicated worldsheet
CFT’s (e.g., Gepner models). One could even go beyond the model space of crit-
ical string theories, and consider non-critical strings and/or strings with non-trivial
background fields. Likewise, we may consider heterotic strings beyond the usual per-
turbative limit. However, although such models may well give rise to phenomenolo-
gies very different from those that emerge in free-field constructions, their spectra
are typically very difficult to analyze and are thus not amenable to an automated
statistical investigation.
3 Method of analysis
Each string model-construction technique provides a mapping between a space of
internal parameters and a corresponding physical string model in spacetime. In the
case of closed strings, for example, such internal parameters might include compact-
ification moduli, boundary-condition phases, Wilson-line coefficients, or topological
quantities specifying Calabi-Yau manifolds; in the case of open strings, by contrast,
they might include D-brane dimensionalities and charges, wrapping numbers or in-
tersection angles, fluxes, and the vevs of moduli fields. Regardless of the construction
technique at hand, however, there is a well-defined procedure through which one can
derive the spectrum and couplings of the corresponding model in spacetime.
Given this, one generally conducts a random search through the space of models
by randomly choosing self-consistent values of these internal parameters, and then
deriving the physical properties of the corresponding string models. Questions about
statistical correlations are then addressed in terms of the relative abundances of
models that emerge with different spacetime characteristics. Indeed, if {α, β, γ, ...}
denote these different spacetime characteristics (or different combinations of these
characteristics), then we are generally interested in extracting ratios of population
abundances of the form Nα/Nβ, where Nα and Nβ are the numbers of models which
exhibit physical characteristics α and β across the landscape as a whole.
Clearly, we cannot survey the entire landscape, and thus we are forced to at-
tempt to extract such ratios with relatively limited information. In particular, let us
assume that our search has consisted of analyzing D different randomly generated
sets of internal parameters, ultimately yielding a set of different models in spacetime
exhibiting varying physical characteristics. Let Mα(D) denote the number of distinct
models which are found which exhibit characteristic α. Our natural tendency is then
to attempt to associate
Mα(D)
Mβ(D)
(3.1)
for some sufficiently large value of D. While this relation might not hold exactly
for relatively small values of D, the expectation is that we might be able to reach
sufficiently large values of D for which we might hope to extract reasonably accurate
predictions for Nα/Nβ.
Unfortunately, as has recently been discussed in Ref. [9], Eq. (3.1) does not gen-
erally hold for any reasonable value of D (short of exploring the full landscape).
Indeed, the violations of this relation are striking, even in situations in which sizable
fractions of the landscape are explored, and will ultimately doom any attempt at
extracting population fractions in this manner. In the remainder of this section, we
shall first explain why Eq. (3.1) fails. We shall then summarize the methods which
were developed in Ref. [9] for circumventing these difficulties, and which we will be
employing in the remainder of this paper.
As stated above, each string model-construction technique provides a mapping
between a space of internal parameters and a physical string model in spacetime.
However, this mapping is not one-to-one, and there generally exists a huge redun-
dancy wherein a single physical string model in spacetime can have multiple realiza-
tions or representations in terms of internal parameters. For this reason, the space of
internal parameters is usually significantly larger than the space of obtainable distinct
models.
The failure of this mapping to be one-to-one is critical because any random statis-
tical study of the string landscape must ultimately take the form of a random explo-
ration of the space of internal parameters that lead to these models. First, one must
randomly choose a self-consistent configuration of internal parameters; only then
can one derive and tabulate the spacetime properties of the corresponding model.
But then we are faced with the question of determining whether spacetime models
with multiple internal realizations should be weighted more strongly in our statistical
analysis than models with relatively few realizations. In other words, we must decide
whether our landscape measure should be based on internal parameters (wherein
each model is weighted according to its number of internal realizations) or based
on spacetime properties (wherein each physically distinct model is weighted equally
regardless of the number of its internal realizations).
If we were to base our landscape measure on internal parameters, then these
redundancies would not represent problems; they would instead become vital ingre-
dients in our numerical analysis. However, if we are to perform statistics in the space
of models in a physically significant way, it is easy to see that we are forced to count
distinct models rather than distinct combinations of internal parameters. The reason
for this is as follows. In many cases, these redundancies arise as the result of world-
sheet symmetries (e.g., mirror symmetries), and even though such symmetries may
be difficult to analyze and eliminate analytically for reasonably complicated models,
their associated redundancies are similar to the redundancies of gauge transforma-
tions and do not represent new physics. In other cases, such redundancies are simply
reflections of the failures or limitations of a particular model-construction technique;
once again, however, they do not represent new physics, but rather reflect a poor
choice of degrees of freedom for our internal parameters, or a mathematical difficulty
or inability to properly define their independent domains. Finally, such redundancies
can also emerge because entirely different model-construction techniques can often
lead to identical models in spacetime. Thus, two landscape researchers using dif-
ferent construction formalisms might independently generate random sets of models
which partially overlap, but once again this does not mean that the models which
are common to both sets should be double-counted when their statistical results are
merged. Indeed, in all of these cases, redundancies in the mapping between inter-
nal parameters and spacetime properties do not represent differences of physics, but
rather differences in the description of that physics. We thus must use spacetime
characteristics (rather than the parameters internal to a given string construction)
as our means of counting and distinguishing string models.
Many of these ideas can be illustrated by considering the E8×E8 heterotic string
in ten dimensions. As is well known, this string model can be represented in many
ways: as a ZZ2 orbifold of the SO(32) supersymmetric string, as a ZZ2 × ZZ2 orbifold of
the non-supersymmetric SO(32) heterotic string, and so forth. Likewise, this model
can be realized through an orbifold construction, through a free-fermionic construc-
tion, through a bosonic lattice construction, and through other constructions as well.
Yet, there is only a single E8 × E8 string model in ten dimensions. It is therefore
necessary to tally distinct string models, and not distinct internal formulations, when
performing landscape calculations and interpreting their results.
Unfortunately, this redundancy inherent in the mapping between internal param-
eters and their corresponding string models implies that in any random exploration
of the space of models, certain string models are likely to be sampled much more
frequently than other models. Thus, one must filter out this effect by keeping a
record of each distinct model that has already been sampled so that each time an
additional model is generated (i.e., each time there is a new “attempt”), it can be
compared against all previous models and discarded if it is not new. Although this
is a memory-intensive and time-consuming process which ultimately limits the sizes
of the resulting data sets that can be generated using current automated technology,
this filtering can successfully be employed to eliminate model redundancies.
However, there remains the converse problem: because some models strongly
dominate the random search, others effectively recede and are therefore extremely
difficult to reach. They therefore do not tend to show up during the early stages
of a random search, and tend to emerge only later in the search process after the
dominant models have been more fully tallied. Indeed, as the search proceeds into
its later stages, it is only the models with “rare” characteristics which increasingly
tend to be generated, precisely because those models with “common” characteristics
will have already been generated and tabulated. Thus, the proportion of models
with “rare” characteristics tends to evolve rather dramatically as a function of time
through the model-generation process.
This type of bias is essentially unavoidable, and has the potential to seriously
distort the values of any numerical correlations that might be extracted from a ran-
dom search through the landscape. In particular, as discussed in Ref. [9], this type of
bias generally causes statistical correlations to “float” or evolve as a function of the
sample size of models examined. Moreover, since one can ultimately explore only a
limited portion of the landscape, there is no opportunity to gather statistics at the
endpoint of the search process at which these correlations would have floated to their
true values. This, then, is the problem of floating correlations.
Fortunately, as discussed in Ref. [9], there are several statistical methods which
can be used in order to overcome this difficulty. These methods enable one to extract
statistical correlations and distributions which are stable as a function of sample
size and which, with some reasonable assumptions, represent the statistical results
that would be obtained if the full space of models could be explored. We shall now
describe the most important of these methods, since we shall be using this technique
throughout the rest of this paper.
In general, a model search proceeds as follows. One randomly generates a self-
consistent set of internal parameters, and calculates the properties of the correspond-
ing string model. One then compares this model against all models which have pre-
viously been generated: if the model is distinct, it is recorded and saved; if it is
redundant, it is discarded. One then repeats this process. Early in the process, most
attempts result in new distinct models because very few models have already been
found. However, as the search proceeds, an increasing fraction of attempts fail to
produce new models. This rise in the ratio of attempts per new model indicates that
the space of models is becoming more and more explored. Thus, attempts per model
can be used as a measure of how far into the full space of corresponding models our
search has penetrated.
Therefore, if we are interested in extracting the ratio Nα/Nβ for two physical
characteristics α and β, as discussed above Eq. (3.1), the solution is not to extract
this ratio through Eq. (3.1) because such a relation assumes that the spaces of α-
models and β-models are being penetrated at exactly the same rates during the
random search process. Rather, the solution [9] is to keep a record not only of the
models generated as the search proceeds, but also of the cumulative average attempts
per model that are needed in order to generate these models. We then extract the
desired ratio Nα/Nβ through a relation of the form
Mα(dα)
Mβ(dβ)
Mα(dα)
Mβ (dβ)
(3.2)
where dα and dβ respectively represent the numbers of attempts that resulted in α-
models and β-models, regardless of whether the models in each class were distinct.
Thus, we must essentially perform two independent search processes, one for α-
models and one for β-models, and we terminate these searches only when they have
each reached the same degree of penetration as measured through their respective
numbers of attempts per model dα/Mα. The value of Nα/Nβ obtained in this way
should then be independent of the chosen reference value of dα/Mα for sufficiently
large dα/Mα. This method of extracting Nα/Nβ is discussed more fully in Ref. [9],
where the derivation and limitations of this method are outlined in detail.
Of course, in the process of randomly generating string models, we cannot nor-
mally control whether a random new model is of the α- or β-type. Both will tend
to be generated together, as part of the same random search. Thus, our procedure
requires that we completely disregard the additional models of one type that might
be generated in the process of continuing to generate the required, additional models
of the other type. This is the critical implication of Eq. (3.2). Rather than let our
model-generating procedure continue for a certain duration, with statistics gathered
at the finish line as in Eq. (3.1), we must instead establish two separate finish lines
for our search process, one for α-models and one for β-models. Of course, these finish
lines are not completely arbitrary, and must be chosen such they correspond to the
same relative degree of penetration of the α- and β-model spaces. Indeed, these finish
lines must be balanced so that they correspond to points at which the same ratio of
attempts per model has been reached. However, these finish lines will not generally
coincide with each other, which requires that some data actually be disregarded in
order to extract meaningful statistical correlations.
As discussed in Ref. [9], Eq. (3.2) will enable us to extract a value for the ratio
Nα/Nβ which is stable as a function of sample size only when the biases within the
α-model space are the same as those within the β-model space. In such cases, we
can refer to the physical characteristics α and β as being in the same universality
class. However, for a given model-generation method (such as the free-fermionic
construction which we shall be employing in this paper), it turns out that many
physical characteristics of interest {α, β, ...} have the property that they are in the
same universality class. In the rest of this paper, correlations for physical quantities
will be quoted only when the physical characteristics being compared are in the
same universality class. The above method is then used in order to extract these
correlations.
4 Supersymmetry on the heterotic landscape
In this section, we begin our analysis of the structure of the heterotic string
landscape. In so doing, we shall also provide an explicit example of the method
described in Sect. 3. Our focus in this section is to determine the extent to which
string models with different levels of unbroken supersymmetry (N=0, 1, 2, 4) populate
the tree-level four-dimensional heterotic landscape. ForN=0 models, we shall further
distinguish between models which are tachyon-free at tree level, and those which are
tachyonic. Note that these characteristics are all mutually exclusive and together
span the entire landscape of heterotic string models in four dimensions. Thus, our
goal is to achieve nothing less than a partitioning of the full set of tree-level heterotic
string models according to their degrees of supersymmetry. (We stress that this
analysis will be the only case in which unstable tachyonic N=0 string models will
be considered in this paper.) We will then proceed in Sect. 5 to examine questions
related to correlations between the numbers of unbroken supersymmetry generators
and the corresponding gauge groups.
The landscape of four-dimensional heterotic strings is a relatively large and com-
plex structure. It may therefore be useful, as an initial step, to quickly recall the
much smaller “landscape” of ten-dimensional heterotic strings. In ten dimensions,
the maximal allowed supersymmetry is N=1, and thus our tree-level ten-dimensional
landscape may be partitioned into only three categories: N=1 models, N=0 tachyon-
free models, and N=0 tachyonic models. Note that since the N=0 tachyonic models
are not even stable at tree level, the tree-level “landscape” actually consists only
of models in the first two categories. However, for convenience, in this section we
shall use the word “landscape” to describe the full set of heterotic vacuum solutions
regardless of stability.
SUSY class % of 10D landscape % of reduced 10D landscape
N=0 (tachyonic) 66.7 62.5
N=0 (tachyon-free) 11.1 12.5
N=1 22.2 25.0
Table 2: Classification of the ten-dimensional tree-level heterotic “landscape” as a function
of the number of spacetime supersymmetries and the presence/absence of tachyons at tree
level. As always, models are judged to be distinct based on their gauge groups and parti-
cle contents. The full ten-dimensional heterotic landscape consists of nine distinct string
models, while the landscape of models accessible through our random search methods is
reduced by one model. In either case, we see that two thirds of the tachyon-free portion
of the ten-dimensional landscape is supersymmetric. Thus unbroken supersymmetry tends
to dominate the “landscape” consisting of ten-dimensional models which are stable at tree
level.
As is well known [21], the full set of D = 10 heterotic strings consists of nine
distinct string models: two are supersymmetric [these are the SO(32) and E8 × E8
models], one is non-supersymmetric but tachyon-free [this is the SO(16) × SO(16)
string model [20]], and six additional models are non-supersymmetric and tachyonic.
Expressed as proportions of a full ten-dimensional heterotic landscape, we therefore
find the results shown in the middle column of Table 2. It is important to note,
however, that not all of these models would be realizable through the methods we
shall be employing in this paper (involving a construction in which all degrees of
freedom are represented in terms of complex worldsheet fermions). Indeed, one of
the tachyonic non-supersymmetric models exhibits rank-reduction and thus would
not be realizable in a random search of the sort we shall be conducting. Statistics
for the corresponding “reduced” landscape of accessible models are therefore listed
along the third column of Table 2; these are the statistics which will form the basis
for future comparisons. Note that in either case, the tachyon-free portion of the
ten-dimensional landscape is dominated by supersymmetric models. This suggests
that breaking supersymmetry without introducing tachyons is relatively difficult in
ten dimensions.
Our goal is to understand how this picture changes after compactification to four
dimensions. Towards this end, one procedure might be to randomly generate a large
set of string models, and see how many models one obtains of each type after a
certain fixed time as elapsed. However, as discussed in Sect. 3, these percentages
will generally float or evolve as a function of the total number of models examined.
This behavior is shown in Fig. 1, and we see that while the non-supersymmetric per-
centages seem to be floating towards greater values, the supersymmetric percentages
seem to be floating towards lesser values.
Figure 1: The numbers of distinct string models exhibiting different amounts of spacetime
supersymmetry, plotted as functions of the total number of distinct string models examined.
Models exhibiting N=4 supersymmetry are too few to appear on this figure.
As discussed in Sect. 3, it is easy to understand the reason for this phenomenon.
Clearly, as we continue to generate models randomly, an ever-increasing fraction of
these models consists of models without supersymmetry. This in turn suggests that
at any given time, we have already discovered a greater fraction of the space of su-
persymmetric models than non-supersymmetric models. This would explain why it
becomes increasingly more difficult to randomly generate new, distinct supersym-
metric models as compared with non-supersymmetric models, and why their relative
percentages show the floating behavior illustrated in Fig. 1.
How then can we extract meaningful information? As discussed in Sect. 3, the
remedy involves keeping track of not only the total numbers of distinct models found
in each supersymmetric class, but also the total number of attempts which yielded
a model in each class, even though such models were not necessarily new. This
information is shown in Table 3 for our total sample of >∼ 10
7 models.
SUSY class # distinct models # attempts avg. attempts/model
N=0 (tachyonic) 1 279 484 3 810 838 2.98
N=0 (tachyon-free) 4 946 388 18 000 000 3.64
N=1 3 772 679 24 200 097 6.41
N=2 492 790 13 998 843 28.41
N=4 1106 6 523 277 5 898.08
Total: 10 492 447 66 533 055 6.34
Table 3: This table expands on Table 1 by including the numbers of attempts to gener-
ate models in each class as well as the corresponding average numbers of attempts per
distinct model. We also include information about the attempts which resulted in non-
supersymmetric models whose spectra are tachyonic at tree level. It is apparent that the
number of attempts per model increases rather dramatically as the level of supersymmetry
increases, indicating that our heterotic string sample has penetrated further into the spaces
of models with greater numbers of supersymmetries than into those with fewer.
As we see from Table 3, the number of required attempts per model increases dra-
matically with the level of supersymmetry. This in turn implies, for example, that
although we may have generated many fewer distinct N=4 models thanN=1 models,
the full space of N=4 models has already been penetrated much more fully than the
space of N=1 models. Thus, as we continue to generate more models, it should be-
come relatively easier to generate non-supersymmetric models than supersymmetric
models. If true, this would imply that the relative proportion of non-supersymmetric
models should increase as we continue to generate more models, while the relative
proportion of supersymmetric models should decrease. This is, of course, exactly
what we have already seen in Fig. 1.
In order to extract final information concerning the relative sizes of these spaces,
the procedure outlined in Sect. 3 instead requires that we do something different,
and compare the numbers of distinct models generated in each class at those points
in our model-generating process when their corresponding numbers of attempts per
model are equal . It is only in this way that we can overcome the effects of floating
correlations and extract stable relative percentages which do not continue to evolve
as functions of the total sample size.
For example, let us consider the relative numbers of N=1 and N=2 models.
Although we see from Table 3 that our full sample of >∼ 10
7 models contains approx-
imately 7.66 times as many N=1 models as N=2 models, this is not the relative size
of their corresponding model spaces because the N=2 space of models has already
been explored more fully than the N=1 model space, with 6.41 attempts per N=1
model compared with 28.41 attempts per N=2 model. However, at an earlier point
in our search, we found that it took an average of approximately 6.41 attempts to
generate a new, distinct N=2 model: this occurred when we had generated only ap-
proximately 90 255 models with N=2 supersymmetry. This suggests that the space
of N=1 models is actually 3772679/90255 ≈ 41.8 times as large as the space of N=2
models.
Moreover, we can verify that this ratio is actually stable as a function of sample
size. For example, at an even earlier point in our search when we had generated only
≈ 2.22×106 N=1 models, we found that an average of 3.64 attempts were required to
generate a new, distinct N=1 model. However, this same average number of attempts
per model occurred in our N=2 sample when we had generated only ≈ 53 000 N=2
models. Thus, once again, the N=1 and N=2 model spaces appear to have a size
ratio of ≈ 41.8 : 1.
In this way, by comparing total numbers of models examined at equal values of
attempts per model, we can extract the relative sizes of the spaces of models with
differing degrees of supersymmetry and verify that these results are stable as functions
of sample size (i.e., stable as functions of the chosen value of attempts per model).
Our results are shown in Table 4. As far as we can determine, the percentages quoted
in Table 4 represent the values to which the percentages in Fig. 1 would float if we
could analyze what is essentially the full landscape. However, short of examining
the full landscape, we see that there is no single point at which these percentages
would simultaneously appear in any finite extrapolation of Fig 1. Instead, it is only
by comparing the numbers of models obtained at different points in our analysis that
the true ratios quoted in Table 4 can be extracted.
SUSY class % of heterotic landscape
N=0 (tachyonic) 32.1
N=0 (tachyon-free) 46.5
N=1 20.9
N=2 0.5
N=4 0.003
Table 4: Classification of the four-dimensional tree-level heterotic landscape as a function of
the number of unbroken spacetime supersymmetries and the presence/absence of tachyons
at tree level. This table is thus the four-dimensional counterpart of Table 2, which quoted
analogous results for ten dimensions. Relative to the situation in ten dimensions, we see that
compactification to four dimensions tends to favor breaking all spacetime supersymmetries
without introducing tachyons at tree level.
Table 4 thus represents our final partitioning of the tree-level four-dimensional
landscape according to the amount of supersymmetry exhibited. There are several
rather striking facts which are evident from these results:
• First, we see that nearly half of the heterotic landscape is non-supersymmetric
and yet tachyon-free.
• Second, we see that the supersymmetric portion of the heterotic landscape ap-
pears to account for less than one-quarter of the full four-dimensional heterotic
landscape.
• Finally, models exhibiting extended (N ≥ 2) supersymmetries are exceedingly
rare, representing less than one percent of the full landscape.
Of course, we stress once again that these results hold only for the tree-level
landscape, i.e., models which are stable at tree level only. It is not clear whether
these results would persist after full moduli stabilization. However, assuming that
they do, these results lead to a number of interesting conclusions.
The first conclusion is that the properties of the tachyon-free heterotic landscape
as a whole are statistically dominated by the properties of string models which do
not have spacetime supersymmetry. Indeed, the N=0 string models account for
over three-quarters of this portion of the heterotic string landscape. The fact that
the N=0 string models dominate the tachyon-free portion of the landscape suggests
that breaking supersymmetry without introducing tachyons is actually favored over
preserving supersymmetry for this portion of the landscape. Indeed, we expect this
result to hold even after full moduli stabilization, unless an unbroken supersymmetry
is somehow restored by stabilization.
The second conclusion which can be drawn from these results is that the super-
symmetric portion of the landscape is almost completely comprised of N=1 string
models. Indeed, only 2% of the supersymmetric portion of the heterotic landscape
has more than N=1 supersymmetry. This suggests that the correlations present for
the supersymmetric portion of the landscape can be interpreted as the statistical
correlations within the N=1 string models, with the N=2 correlations represent-
ing a correction at the level of 2% and the N=4 correlations representing a nearly
negligible correction.
It is natural to ask what effects are responsible for this hierarchy. As was dis-
cussed in Sect. 3, two string models are considered distinct if any of their spacetime
properties are found to be different. Two models which have the same number of
unbroken spacetime supersymmetries must therefore differ in other features, such as
their gauge groups and particle representations. Thus, if there exist more models
with one level of supersymmetry than another, this must mean that there are more
string-allowed configurations of gauge groups and particle representations with one
level of supersymmetry than the other. Indeed, given the results of Table 4, our
expectation is that increasing the level of supersymmetry will have the effect of de-
creasing the number of distinct models with a given gauge group, and possibly even
the range of allowed gauge groups. We shall test both of these expectations explicitly
in Sect. 5.
5 Supersymmetry versus gauge groups
Within the heterotic string, worldsheet self-consistency conditions arising from the
requirements of conformal anomaly cancellation, one-loop and multi-loop modular in-
variance, physically sensible GSO projections, etc., impose many tight constraints on
the allowed particle spectrum. These constraints simultaneously affect not only the
spacetime Lorentz structure of the theory (such as is involved in spacetime super-
symmetry), but also the internal gauge structure of the theory. Thus, it is precisely
within the context of string theory that we expect to find correlations between super-
symmetries and gauge symmetries — features which would otherwise be uncorrelated
in theories based on point particles.
In general, these correlations can lead to certain tensions in a given string con-
struction. Models exhibiting large numbers of unbroken supersymmetries may be
expected to have relatively rigid gauge structures, and vice versa. There are two spe-
cific types of correlations which we shall study. First, we shall analyze how the degree
of supersymmetry affects the range of possible allowed gauge groups. For example, in
extreme cases it may occur that certain gauge symmetries may not even be allowed
for certain levels of spacetime supersymmetry. Second, even within the context of a
fixed gauge group, we can expect the degree of spacetime supersymmetry to affect
the range of allowed particle representations which can appear at the massless level.
In other words, the number of distinct string models with a given fixed gauge group
may be highly sensitive to the degree of spacetime supersymmetry.
Some of these features are already on display in the ten-dimensional heterotic
“landscape”. For example, no gauge group is shared between those ten-dimensional
models with supersymmetry and those without. Moreover, in each case, there is
only a single model with each allowed gauge group. Thus, in ten dimensions, the
specification of the level of supersymmetry (and/or the gauge group) is sufficient to
completely fix the corresponding particle spectrum.
Clearly, in four dimensions, things will be far more complex. In particular, we
shall study three correlations in this section:
• First, we shall focus on the number of allowed gauge groups as a function of
the degree of supersymmetry. We shall also study gauge-group multiplicities
— i.e., the probabilities that there exist distinct string models with the same
gauge group but different particle spectra. This will be the focus of Sect. 5.1.
• Second, as a function of the degree of supersymmetry, we shall investigate
“shatter” — i.e., the degree to which our total (rank-22) gauge group is “shat-
tered” into distinct irreducible factors, or equivalently the average rank of each
irreducible gauge-group factor. This will be the focus of Sect. 5.2.
• Finally, as a function of the degree of supersymmetry, we shall study the prob-
abilities of realizing specific (combinations of) gauge-group factors in a given
string model. This will be the focus of Sect. 5.3.
As we shall see, these studies will find deep correlations which ultimately reflect the
string-theoretic tension between supersymmetry and the string consistency condi-
tions.
5.1 Numbers and multiplicities of unique gauge groups
We begin by studying the total numbers of distinct gauge groups which can be
realized as a function of the number of unbroken supersymmetries in a given string
model.
To do this, one direct approach can might be to classify models according to
their numbers of unbroken spacetime supersymmetries, and tabulate the numbers of
distinct gauge groups which appear as functions of the total number of models in
each class. As we continue to generate more and more models, we then obtain the
results shown in Fig. 2.
It is evident from Fig. 2 that for a fixed sample size, models with more unbroken
supersymmetries tend to exhibit larger numbers of distinct gauge groups, or equiv-
alently smaller numbers of model multiplicities per gauge group. For example, we
see from Fig. 2 that when each class of models has reached a sample size of 500 000
models, the tachyon-free N=0 models have a greater multiplicity per gauge group
than N=1 models by an approximate factor ≈ 1.4, while the N=2 models have a
smaller multiplicity per gauge group than the N=1 models by an approximate factor
≈ 0.8. However, it is easy to understand this behavior. As the level of supersym-
metry increases, there are more constraints on the possible particle spectra that can
emerge for a given gauge group. This in turn implies that there are likely to be fewer
ways for two models with the same gauge group to be distinct, which in turn implies
that there is a greater chance that distinct models will be forced to exhibit distinct
gauge groups. Thus, models exhibiting greater amounts of supersymmetry are likely,
on average, to exhibit greater numbers of gauge groups amongst a fixed number of
models.
Of course, as also evident from Fig. 2, the multiplicity of distinct models per gauge
group exhibits a strong, floating dependence on the sample size. Therefore, in order
to extract a stable ratio of multiplicity ratios — one which presumably represents
the values of these ratios when extrapolated to the full landscape — we must employ
the methods described in Sect. 3. We then obtain the results shown in the middle
column of Table 5. Using these results in conjunction with the corresponding ratios
of landscape magnitudes in Table 4, we can also calculate the relative numbers of
Figure 2: Numbers of distinct gauge groups obtained as functions of the number of dis-
tinct string models generated. Each curve corresponds to models with a different num-
ber of unbroken spacetime supersymmetries, with N=0 signifying models which are non-
supersymmetric but tachyon-free. We see that for a fixed sample size, models with more
unbroken supersymmetries tend to exhibit a larger number of distinct gauge groups. (Note
that models with N=4 supersymmetry are too few to be shown in this plot.)
distinct gauge groups realizable within each SUSY class of models. These results
are shown in the final column of Table 5. Note that in each case, these quantities
are quoted as ratios relative to their N=1 values; this represents the most detailed
information that can be extracted using the methods of Sect. 3.
We see from Table 5 that both the average multiplicities per gauge group and the
total numbers of realizable gauge groups are monotonically decreasing functions of
the number of unbroken supersymmetries. While this is to be expected on the basis
of the arguments described above, we must realize that our class of N=0 models does
not consist of all non-supersymmetric models, but merely those which are tachyon-
free. Thus, the requirement of avoiding tachyons could have turned out to be more
stringent than the requirement of maintaining an unbroken supersymmetry, at least as
far as generating a variety of gauge groups is concerned. This is indeed what happens
in the ten-dimensional landscape, where there are fewer realizable gauge groups for
non-supersymmetric tachyon-free models than for models with N=1 supersymmetry.
avg. multiplicity # of realizable
SUSY class per gauge group gauge groups
N=0 (tachyon-free) 1.65 1.35
N=1 1.00 1.00
N=2 0.89 0.03
Table 5: The average relative multiplicities (distinct models per gauge group) and total
numbers of realizable gauge groups, evaluated for heterotic string models with N = 0, 1, 2
unbroken spacetime supersymmetries. In each case, these quantities are normalized to their
N=1 values.
However, the results in Table 5 indicate that the opposite is true in D = 4.
Note that in Table 5, we do not quote results for the N=4 portion of the heterotic
landscape because the absolute numbers of models in this class are so small that no
stable numerical results can be extracted relative to the other levels of supersymmetry.
However, it is worth noting that literally each N=4 model in our sample has a unique
gauge group, so the absolute (rather than relative) gauge-group multiplicity in the
N=4 case is exactly 1.000. This only reinforces our general observation that increased
levels of supersymmetry reduce the gauge-group multiplicity; indeed, we now see that
the case of maximal supersymmetry appears to result in the minimal allowed gauge-
group multiplicity. It is likely that this result can be proven analytically for the N=4
landscape as a whole.
5.2 Shatter/average rank
Having studied the numbers of different possible gauge groups, we now turn our
attention to the gauge groups themselves. Once again, our goal is to study how these
gauge groups depend on the presence or absence of spacetime supersymmetry.
To begin the discussion, our focus in this section will be on what we call “shat-
ter” [8]. Recall that the heterotic string models we are considering all have gauge
groups with total rank 22. This stretches from models with gauge group SO(44)
all the way down to models with gauge groups of the form U(1)n × SU(2)22−n with
potentially all values of n in the range 0 ≤ n ≤ 22. Following Ref. [8], we shall define
the “shatter” for a given string model as the number of distinct irreducible gauge-
group factors into which its total rank-22 gauge group has been shattered. Note that
for this purpose, factors of SO(4) ∼ SU(2)× SU(2) contribute two units to shatter.
Since the total rank of the gauge group is fixed at 22 for such models, this means that
shatter is also a measure of the average rank of the individual group group factors,
with 〈rank〉 = 22/shatter. Roughly speaking, shatter can also be taken as a measure
of the degree of complexity needed for the construction of a given string model, with
increasingly smaller individual gauge-group factors tending to require increasingly
many non-overlapping sequences of orbifold twists and Wilson lines.
Given this definition of shatter, we may then calculate the distribution of shatter
across the landscape of heterotic strings. We may calculate, for example, the relative
probabilities that models with certain levels of shatter emerge across the landscape,
and ask how these probability distributions vary with the amount of spacetime su-
persymmetry present in the model.
Our results are shown in Fig. 3. Once again, we stress that our raw data tends
to evolve significantly as a function of the sample size of models considered. It is
therefore necessary to employ the techniques described in Sect. 3 in order to extract
stable results which should apply across the landscape as a whole. In practice, this
requires a difficult and time-consuming process in which each of the data points
shown in Figs. 3 for N=0, 1, 2 has individually been extracted through the limiting
procedure described in Sect. 3. Only then is an entire “curve” constructed for each
level of supersymmetry, as shown.
For theN=4 case, by contrast, our sample size is too small to permit stable results
to be extracted. However, the fact that the attempts per model count in Table 3 is
so large for the N=4 models suggests that our N=4 sample has already explored
a significant fraction (and perhaps even most) of the corresponding landscape. The
N=4 curve in Fig. 3 thus represents a direct tally of our N=4 sample set.
As evident from Fig. 3, certain features of these plots are independent of the level
of spacetime supersymmetry. These therefore represent general trends which hold
across the entire tachyon-free heterotic string landscape. For example, one general
trend is a strong preference for models with relatively high degrees of shatter and
correspondingly small average ranks for individual gauge-group factors — models
exhibiting shatters near or in the teens clearly dominate. On the other hand, this
preference for highly shattered gauge groups does not appear to extend to the limit
of completely shattered models with shatter=22; indeed, the set of models with
only rank-one gauge-group factors seems to represent a fairly negligible portion of
the landscape regardless of the degree of supersymmetry. This indicates that most
models in this class have gauge groups which contain at least one factor of rank
greater than one.∗
Another universal trend implied by (though not explicitly shown in) Fig. 3 is
that string models with shatters of less than four accrue relatively little measurable
amount of probability. Even in the N=4 case, these models are thus actually quite
rare across the landscape as a whole. In some sense, this too is to be expected, since
∗ Of course, we stress that this conclusion applies only for models in the free-fermionic class. In
general, it is always possible to deform away from the free-fermionic limit by adjusting the internal
radii of the worldsheet fields away from their free-fermionic values; in such cases, we expect all gauge
symmetries to be broken down to U(1)22. However, as noted earlier, the free-fermionic points typi-
cally represent precisely those points at which additional (non-Cartan) gauge-boson states become
massless, thereby enhancing the gauge symmetries to become non-abelian. Thus, as discussed more
fully in Sect. 2, the free-fermionic construction naturally leads to precisely the set of models which
are likely to be of direct phenomenological relevance.
Figure 3: The absolute probabilities of obtaining distinct four-dimensional heterotic string
models with different numbers of unbroken supersymmetries, plotted as functions of the
degree to which their gauge groups are “shattered” into separate irreducible factors. The
total value of the points (the “area under the curve”) in each case is 1. Here N=0 refers
to models which are non-supersymmetric but tachyon-free.
there are many more ways of breaking a large gauge symmetry through orbifolds and
non-trivial Wilson lines than of preserving it.
Despite these universal features, we see that spacetime supersymmetry neverthe-
less does have a significant effect on the shapes of these curves. In this regard, there
are two features to note.
First, we observe that as the degree of unbroken supersymmetry increases, the
range of probable shatter values also tends to increase, with probability shifting from
models with high shatters to models with lower shatters. This is especially noticeable
when comparing the distribution of theN=2 andN=4 models with those of theN=0
and N=1 models. These results indicate that models exhibiting smaller amounts
of shatter (i.e., models whose gauge-group factors have larger individual average
ranks) become somewhat more probable as the level of supersymmetry increases.
Ultimately, this correlation between unbroken supersymmetry and unbroken gauge
symmetry emerges since both have their underlying origins in how our orbifold twists
and Wilson lines are chosen.
Second, and perhaps more unexpectedly, we see that the degree of supersymmetry
also affects the overall profiles of these curves. While the N=0 curve is relatively
smooth, exhibiting a single peak at shatter=20, these curves begin to experience
even/odd oscillations as the degree of supersymmetry increases, with odd values of
shatter significantly favored over even values when supersymmetry is present. The
origins of this phenomenon are less apparent, and perhaps lie in the modular invari-
ance and anomaly cancellation constraints which correlate the orders of the allowed
twists leading to self-consistent string models. Interestingly, this even/odd behav-
ior continues into the N=4 case, although these oscillations are significantly less
pronounced and flip sign, with evens now dominating over odds.
One notable feature of the N=4 curve is its approximate reflection symmetry
around shatter=10. It is unclear whether this is an exact symmetry which holds in
situations with maximal supersymmetry, or whether this is merely an accident.
5.3 Specific gauge-group factors
Finally, we turn to an analysis of the probabilities of realizing individual gauge-
group factors. Just how likely is it, say, that a randomly chosen heterotic string
model will exhibit an SU(3) factor in its gauge group, and how does this probability
correlate with the spacetime supersymmetry of the model?
Just as with previous questions, addressing this issue requires a detailed analysis
along the lines discussed in Sect. 3. This is because the probabilities of realizing
different gauge-group factors also float quite strongly as a function of sample size.
As dramatic illustration of this fact, let us restrict our attention to models with
N=1 spacetime supersymmetry and calculate the probability that a given model
will exhibit an SU(3) gauge-group factor as a function of the number of models we
have examined. We then obtain the result shown in Fig. 4, and it is clear that the
Figure 4: The percentage of distinct four-dimensionalN=1 supersymmetric heterotic string
models exhibiting at least one SU(3) gauge-group factor, plotted as a function of the
number of models examined for the first 1.25 million models. We see that as we generate
further models, SU(3) gauge-group factors become somewhat more ubiquitous — i.e., the
fraction of models with this property floats. One must therefore account for this floating
behavior using the methods described in Sect. 3 in order to extract meaningful information
concerning the relative probabilities of specific gauge-group factors.
percentage of models with SU(3) gauge-group factors floats rather significantly as a
function of the sample size. Indeed, on the basis of this information alone, it would be
quite impossible to determine the final value to which this curve might float. Just as
with previous examples, this floating behavior ultimately occurs because models with
SU(3) gauge-group factors are relatively difficult to generate using the construction
methods we are employing; thus, they tend to emerge in increasing numbers only
after other models are exhausted. As discussed more fully in Ref. [9], this does not
imply that there are fewer of these models or that our construction method cannot
ultimately reach them— all we can conclude is that they are less likely to be generated
in a random search than other models, and thus they tend to emerge only later in
the search process. Indeed, as we shall shortly see, models with SU(3) gauge-group
factors actually tend to dominate the landscape.
Therefore, in order to extract meaningful results, we again employ the methods
discussed in Sect. 3. We then obtain the final percentages quoted in Table 6. We
observe, in particular, that the probability of models with N=1 supersymmetry ex-
hibiting at least one SU(3) gauge-group factor has actually risen all the way to 98%.
The fact that this probability has floated from nearly 55% to 98% only reinforces
the importance of the analysis method presented in Sect. 3, and illustrates the need
to properly account for floating correlations when quoting statistical results for such
studies.
gauge group N=0 N=1 N=2 N=4
U1 99.9 94.5 68.4 89.6
SU2 62.46 97.4 64.3 60.9
SU3 99.3 98.0 93.0 45.1
SU4 14.46 30.0 39.0 53.5
SU5 16.78 43.5 66.3 33.8
SU>5 0.185 1.7 10.6 73.0
SO8 0.482 1.6 6.2 21.1
SO10 0.084 0.2 1.6 18.7
SO>10 0.005 0.038 0.77 7.5
E6,7,8 0.0003 0.03 0.16 11.5
Table 6: Percentages of heterotic string models exhibiting specific gauge-group factors as
functions of their spacetime supersymmetry. Here SU>5 and SO>10 collectively indicate
gauge groups SU(n) and SO(2n) for any n > 5, while N refers to the number of unbroken
supersymmetries at the string scale. Note that the N=0 models are all tachyon-free.
As we see from Table 6, supersymmetry can have quite sizable effects upon the
probability of realizing specific groups. However, there are some general trends that
hold for the full heterotic landscape. These trends include:
• A preference for SU(n + 1) over SO(2n) groups for each rank n. Even though
these two groups have the same rank, it seems that SU groups are more common
than the SO groups for all levels of supersymmetry.
• Groups with smaller rank are much more common than groups with larger rank.
Once again, this also appears to hold for all levels of supersymmetry.
• Finally, the gauge-group factors comprising Standard-Model gauge group
GSM ≡ SU3 × SU2 × U1 are particularly common, much more so than those
of any of its grand-unified extensions.
As we found in Sect. 4, the N=0 string models dominate the tachyon-free portion
of the heterotic landscape. Similarly, the N=1 string models are the dominant part
of the supersymmetric portion of the landscape. Nevertheless, it is interesting to
gauge entire SUSY
group landscape subset
U1 98.00 93.89
SU2 73.22 96.62
SU3 98.85 97.88
SU4 19.42 30.21
SU5 25.37 44.03
SU>5 0.73 1.92
SO8 0.87 1.71
SO10 0.13 0.23
SO>10 0.02 0.06
E6,7,8 0.01 0.03
Table 7: Percentage of heterotic string models exhibiting specific gauge-group factors,
quoted across the entire landscape of tachyon-free models (both supersymmetric and non-
supersymmetric) as well as across only that subset of models with at least N≥1 spacetime
supersymmetry. These results are derived from those of Table 6 using the landscape weight-
ings in Table 4.
examine the gauge-group probabilities across both of these portions of the landscape.
These probabilities are easy to calculate by combining the results in Tables 4 and 6,
leading to the results shown in Table 7.
Several features are immediately apparent from Table 7. First, gauge groups
with larger ranks appear to be favored more strongly across the supersymmetric
subset of the landscape than across the tachyon-free landscape as a whole. Since
each of our heterotic string models in this class has a gauge group of fixed total
rank, this preference for higher-rank gauge groups necessarily comes at the price of
sacrificing smaller-rank gauge groups. Indeed, it often happens that this preference
for larger-rank gauge groups actually precludes the appearance of any small-rank
gauge groups whatsoever. Interestingly, the supersymmetric portion of the landscape
seems to sacrifice U(1) primarily and SU(3) to a lesser extent. This is in contrast to
SU(2), which is actually more strongly favored in the supersymmetric portion of the
landscape than in the general tachyon-free landscape as a whole.
Second, the level of supersymmetry also appears to affect the probability dis-
tributions across the different possible gauge-group factors. The supersymmetric
portion of the landscape has a much greater representation of the large rank groups.
This suggests that the constraints placed on the string spectrum in order to preserve
spacetime supersymmetry also have the effect of favoring larger gauge symmetries,
a fact already noted in Sect. 5.2. In other words, there tends to be a decrease in
the gauge-group multiplicity for highly shattered gauge groups which consist of only
very small gauge-group factors, and thus the larger-rank gauge groups make up a
larger proportion of the whole landscape. Indeed, this effect is particularly acute
for that subset of the landscape exhibiting maximal N=4 supersymmetry, where the
larger-rank SU gauge groups are particularly well represented.
6 Discussion
In this paper, we have examined both the prevalence of spacetime supersymmetry
across the heterotic string landscape and the statistical correlations between the
appearance of spacetime supersymmetry and the gauge structure of the corresponding
string models. Somewhat surprisingly, we found that nearly half of the heterotic
landscape is non-supersymmetric and yet tachyon-free at tree level; indeed, less than
a quarter of the tree-level heterotic landscape exhibits any supersymmetry at all at
the string scale. Moreover, we found that the degree of spacetime supersymmetry
is strongly correlated with the probabilities of realizing certain gauge groups, with
unbroken supersymmetry at the string scale tending to favor gauge-group factors
with larger rank.
There are several extensions to these results which are currently under investiga-
tion. For example, we would like to understand how the presence of supersymmetry
affects the statistical appearance of the entire composite Standard-Model gauge group
GSM ≡ SU3 ×SU2 ×U1, and not merely the appearance of its individual factors. We
would also like to understand how the presence or absence of supersymmetry affects
other features which are equally important for the overall architecture of the Stan-
dard Model: these include the appearance of three chiral generations of quarks and
leptons, along with a potentially correct set of gauge couplings and Yukawa couplings.
This work has already been completed, and will be reported shortly [16].
Despite this progress, such studies have a number of intrinsic limitations which
must continually be borne in mind. A number of these have been emphasized by us
in recent articles (see, e.g., the concluding sections of Refs. [8, 9]) and will not be
repeated here. However, other limitations are particularly relevant for the results we
have quoted here and thus deserve emphasis.
First, we must continually bear in mind that our study has been limited to models
in which rank-cutting is absent. Thus, all of the four-dimensional heterotic string
models we have examined exhibit a fixed maximal rank=22. This has the potential
to skew the statistics of the different gauge-group factors. For example, it is possible
that gauge-group factors with smaller ranks might be over-represented in this sample
simply because the appearance of such groups is often the only way in which a given
model can precisely saturate the total rank bound. By contrast, for models which can
exhibit rank-cutting, this saturation would not be needed and it is therefore possible
that lower-rank groups are consequently less abundant.
A second limitation of this study stems from the nature of performing random
search studies in general. In Sect. 3, we summarized several methods by which
the problematic issue of floating correlations can be transcended, and this paper has
provided several examples of not only the need for such methods but also of the means
by which they are implemented. As more fully discussed in Ref. [9], such problems are
going to arise — and such methods are going to be necessary — whenever we attempt
to extract statistical correlations from a large data set to which our computational
access is necessarily limited. However, despite the apparent success of such methods,
it is always a logical possibility that there exists a huge pool of string models with
non-standard physical characteristics remaining just beyond our computational power
to observe. The existence of such a pool of models would completely change the
nature of our statistical results, to an extent which is essentially unbounded, yet we
may miss this completely because of limited computational power. Although this
becomes increasingly unlikely as our search through the landscape becomes larger
and increasingly sophisticated, this nevertheless always remains a logical possibility
which cannot be discounted.
But finally, perhaps the most serious limitation of our study is the fact that we are
analyzing the statistical properties of string models which are not necessarily stable
beyond tree level. Indeed, since none of our non-supersymmetric string models has a
vanishing one-loop cosmological constant, these models in particular necessarily have
non-zero dilaton tadpoles at one-loop order and thus become unstable. Even our su-
persymmetric models have flat directions which have not been lifted. Thus, as we have
stressed throughout this paper, the “landscape” we have examined in this paper is at
best a tree-level one. Despite this fact, however, it is important to realize that these
models do represent self-consistent string solutions at tree level. Specifically, these
models satisfy all of the constraints needed for worldsheet conformal/superconformal
invariance, modular-invariant one-loop and multi-loop amplitudes, proper spacetime
spin-statistics relations, and physically self-consistent layers of sequential GSO pro-
jections and orbifold twists. Indeed, since no completely stable perturbative heterotic
strings have yet been constructed, this sort of analysis is currently the state of the
art for large-scale statistical studies of this type. This mirrors the situation on the
Type I side, where state-of-the-art statistical analyses have also focused on models
which are only stable at tree level.
Nevertheless, we are then left with the single over-arching question: to what
extent can we believe that the results we have found for this “tree-level” landscape
actually apply to the true landscape that would emerge after all moduli are stabilized?
The answer to this question clearly depends on the extent to which the statistical
correlations we have uncovered here are likely to hold even after vacuum stabilization.
A priori , this is completely unknown. However, one surprising result of this paper
is the observation that the string self-consistency requirements themselves — even
merely at tree-level — do not preferentially give rise to supersymmetric solutions at
the string scale. Indeed, as we discussed in Sect. 4, less than a quarter of the tree-
level heterotic landscape appears to exhibit any supersymmetry at all at the string
scale. Thus, breaking supersymmetry without introducing tachyons is actually sta-
tistically favored over preserving supersymmetry, even at the string scale and even
when the requirements of avoiding tachyons are implemented. Observations such
as these tend to shift the burden of proof onto the SUSY enthusiasts, and perhaps
reframe the question to one in which we might ask whether an unbroken supersym-
metry is somehow restored by modulus stabilization. This seems unlikely, especially
since most modern methods of modulus stabilization rely on breaking rather than
introducing supersymmetry. In either case, however, this shows how the results of
such studies — even though limited to only the tree-level landscape — can have the
power to dramatically reframe the relevant questions. Indeed, once the technology
for building heterotic string models develops further and truly stable vacua can be
statistically analyzed in large quantities, it will be interesting to compare the statis-
tical properties of those vacua with these in order to ascertain the degree to which
vacuum stabilization might affect these other phenomenological properties.
Thus, it is our belief that such statistical landscape studies of this sort have their
place, particularly when the results of such studies are interpreted correctly and in
the proper context. As such, we hope that this study of the perturbative heterotic
landscape may represent one small step in this direction.
Acknowledgments
The work of KRD, ML, and VW is supported in part by the U.S. National Science
Foundation under Grant PHY/0301998, by the U.S. Department of Energy under
Grant DE-FG02-04ER-41298, and by a Research Innovation Award from Research
Corporation.
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	Introduction
	The models
	Method of analysis
	Supersymmetry on the heterotic landscape
	Supersymmetry versus gauge groups
	Numbers and multiplicities of unique gauge groups
	Shatter/average rank
	Specific gauge-group factors
	Discussion
