Compactifications of the heterotic string on T^d are the simplest, yet rich enough playgrounds to uncover swampland ideas: the U(1)^{d+16} left-moving gauge symmetry gets enhanced at special points in moduli space only to certain groups. We state criteria, based on lattice embedding techniques, to establish whether a gauge group is realized or not. For generic d, we further show how to obtain the moduli that lead to a given gauge group by modifying the method of deleting nodes in the extended Dynkin diagram of the Narain lattice II_{1,17}. More general algorithms to explore the moduli space are also developed. For d=1 and 2 we list all the maximally enhanced gauge groups, moduli, and other relevant information about the embedding in II_{d,d+16}. In agreement with the duality between heterotic on T^2 and F-theory on K3, all possible gauge groups on T^2 match all possible ADE types of singular fibers of elliptic K3 surfaces. We also present a simple method to transform the moduli under the duality group, and we build the map that relates the charge lattices and moduli of the compactification of the E_8 x E_8 and Spin(32)/Z_2 heterotic theories.
We use deep autoencoder neural networks to draw a chart of the heterotic $\mathbb{Z}_6$-II orbifold landscape. Even though the autoencoder is trained without knowing the phenomenological properties of the $\mathbb{Z}_6$-II orbifold models, we are able to identify fertile islands in this chart where phenomenologically promising models cluster. Then, we apply a decision tree to our chart in order to extract the defining properties of the fertile islands. Based on this information we propose a new search strategy for phenomenologically promising string models.
We investigate the impact of sterile neutrinos on the decay rate of extra $Z'$s with mass in the TeV range in heterotic string derived models. We explore the impact of sterile neutrinos on the current $Z'$ mass exclusion limits at the LHC, and how these bounds change when the parameter space of this specific class of models is modified.
The immensity of the string landscape and the difficulty of identifying solutions that match the observed features of particle physics have raised serious questions about the predictive power of string theory. Modern methods of optimisation and search can, however, significantly improve the prospects of constructing the standard model in string theory. In this paper we scrutinise a corner of the heterotic string landscape consisting of compactifications on Calabi-Yau three-folds with monad bundles and show that genetic algorithms can be successfully used to generate anomaly-free supersymmetric…
Discovering a selection principle and the origin of flavor symmetries from an ultraviolet completion of particle physics is an interesting open task. As a step in this direction, we classify all possible flavor symmetries of 4D massless spectra emerging from supersymmetric Abelian orbifold compactifications, including roto-translations and non-factorizable compact spaces, for generic moduli values. Although these symmetries are valid in all string theories, we focus on the E8 x E8 heterotic string. We perform the widest known search of E8 x E8 Abelian orbifold compactifications, yielding over …
We investigate M-theory and heterotic compactifications to 7 and 3 dimensions. In 7 dimensions we discuss a class of massive supergravities that arise from M-theory on K3 and point out obstructions to realizing these theories in a dual heterotic framework with a geometric description. Taking M-theory further down to 3 dimensions on K3xK3 with a choice of flux leads to a rich landscape of theories with various amounts of supersymmetry, including those preserving 6 supercharges. We explore possible heterotic realizations of these vacua and prove a no--go theorem: every heterotic geometry that pr…
The single-valued projection (sv) is a relation between scattering amplitudes of gauge bosons in heterotic and open superstring theories. Recently we have studied sv from the aspect of nonlinear sigma models [1], where the gauge physics of open string sigma model is under the Wilson loop representation but the gauge physics of heterotic string sigma model is under the fermionic representation since the Wilson loop representation is absent in the heterotic case. There we showed that the sv comes from a sum of six radial orderings of heterotic vertices on the complex plane. In this paper, we pro…
We review here some aspects of our recent works about the geometric engineering of heterotic little string theories using F-theory. Building on the seminal work by Aspinwall and Morrison as well as Intrilligator and Blum, we solve some longstanding open questions thanks to recent progress in our understanding of 6D (1,0) theories and their generalized symmetries. On the geometry side, these systems correspond to non-compact elliptically fibered Calabi-Yau varieties that must admit the structure of an elliptic K3 fibration. From fiberwise F-theory/Heterotic duality the K3 plays a central role -…
We apply deep-learning techniques to the string landscape, in particular, $SO(32)$ heterotic string theory on simply-connected Calabi-Yau threefolds with line bundles. It turns out that three-generation models cluster in particular islands specified by deep autoencoder networks and k-means++ clustering. Especially, we explore mutual relations between model parameters and the cluster with densest three-generation models (called "3-generation island"). We find that the 3-generation island has a strong correlation with the topological data of Calabi-Yau threefolds, in particular, second Chern cla…
Geometric transitions between Calabi-Yau manifolds have proven to be a powerful tool in exploring the intricate and interconnected vacuum structure of string compactifications. However, their role in N=1, 4-dimensional string compactifications remains relatively unexplored. In this work we present a novel proposal for transitioning the background geometry (including NS5-branes and holomorphic, slope-stable vector bundles) of 4-dimensional, N=1 heterotic string compactifications through a conifold transition connecting Calabi-Yau threefolds. Our proposal is geometric in nature but informed by t…
Compactifications of the heterotic string on special T^d/Z_2 orbifolds realize a landscape of string models with 16 supercharges and a gauge group on the left-moving sector of reduced rank d+8. The momenta of untwisted and twisted states span a lattice known as the Mikhailov lattice II_{(d)}, which is not self-dual for d > 1. By using computer algorithms which exploit the properties of lattice embeddings, we perform a systematic exploration of the moduli space for d=1 and 2, and give a list of maximally enhanced points where the U(1)^{d+8} enhances to a rank d+8 non-Abelian gauge group. For d …
In this thesis we provide new tools to determine and explore the Landscape of four-dimensional effective field theories originating from string and M-theory. The main aim is to introduce, within four-dimensional effective descriptions, elements that are predicted from string theory. To this end, a hierarchy of forms is introduced within the four-dimensional $\mathcal{N}=1$ supergravity theories. The inclusion of gauge three-forms delivers a dynamical way to generate flux-induced superpotentials. Instead, gauge two-forms, dual descriptions of axions, may be eventually gauged by three-forms to g…
The $\mathbb{Z}_2\times \mathbb{Z}_2$ heterotic string orbifold yielded a large space of phenomenological three generation models and serves as a testing ground to explore how the Standard Model of particle physics may be incorporated in a theory of quantum gravity. In this paper we explore the existence of type 0 models in this class of string compactifications. We demonstrate the existence of type 0 $\mathbb{Z}_2\times \mathbb{Z}_2$ heterotic string orbifolds, and show that there exist a large degree of redundancy in the space of GGSO projection coefficients when the type 0 restrictions are …
We propose an organizing principle for string theory moduli spaces in six dimensions with $\mathcal{N} = (1,1)$, based on a rank reduction map, into which all known constructions fit. In the case of cyclic orbifolds, which are the main focus of the paper, we make an explicit connection with meromorphic 2D (s)CFTs with $c = 24$ ($c = 12$) and show how these encode every possible gauge symmetry enhancement in their associated 6D theories. These results generalize naturally to non-cyclic orbifolds, into which the only known string construction (to our awareness) also fits. This framework suggests…
University of Oxford（牛津大学）；Wolfson College（沃尔夫森学院）
The goal of identifying the Standard Model of particle physics and its extensions within string theory has been one of the principal driving forces in string phenomenology. Recently, the incorporation of artificial intelligence in string theory and certain theoretical advancements have brought to light unexpected solutions to mathematical hurdles that have so far hindered progress in this direction. In this review we focus on model building efforts in the context of the $E_8\times E_8$ heterotic string compactified on smooth Calabi-Yau threefolds and discuss several areas in which machine lear…
Exploring the Potential Energy Landscape Over a Large Parameter-Space
City University, London（伦敦城市大学）；NanKai University（南开大学）；University of Oxford（牛津大学）；Syracuse University（雪城大学）；Colorado State University（科罗拉多州立大学）；Universität Hamburg（汉堡大学）；Trinity College, University of Oxford（牛津大学三一学院）
We apply techniques from data mining to the heterotic orbifold landscape in order to identify new MSSM-like string models. To do so, so-called contrast patterns are uncovered that help to distinguish between areas in the landscape that contain MSSM-like models and the rest of the landscape. First, we develop these patterns in the well-known $\mathbb{Z}_6$-II orbifold geometry and then we generalize them to all other $\mathbb{Z}_N$ orbifold geometries. Our contrast patterns have a clear physical interpretation and are easy to check for a given string model. Hence, they can be used to scale down…
We classified in arXiv:2201.04152 certain 10d supergravity solutions with a 4d de Sitter, Minkowski or anti-de Sitter spacetime. We then found new solutions in previously unexplored classes. In this paper we study their properties, compare them to swampland conjectures, and make new observations. Using new numerical tools, we first identify all Lie algebras underlying the 6d group manifolds, allowing us to discuss their compactness. We then investigate scale separation, and prove related no-go theorems. Last but not least, we automatize and analyze the stability of all solutions. This leads us…
We generalize soft theorems of the nonlinear sigma model beyond the $\mathcal{O} (p^2)$ amplitudes and the coset of $\text{SU} (N) \times \text{SU} (N) / \text{SU} (N) $. We first discuss the universal flavor ordering of the amplitudes for the Nambu-Goldstone bosons, so that we can reinterpret the known $\mathcal{O} (p^2)$ single soft theorem for $\text{SU} (N) \times \text{SU} (N) / \text{SU} (N) $ in the context of a general symmetry group representation. We then investigate the special case of the fundamental representation of $\text{SO} (N)$, where a special flavor ordering of the "pair ba…
