Heterotic String Compactifications Research Articles

Back to heterotic strings on ALE spaces. Part II. Geometry of T-dual little strings

This work is the second of a series of papers devoted to revisiting the properties of Heterotic string compactifications on ALE spaces. In this project we study the geometric counterpart in F-theory of the T-dualities between Heterotic ALE instantonic Little String Theories (LSTs) extending and generalising previous results on the subject by Aspinwall and Morrison. Since the T-dualities arise from a circle reduction one can exploit the duality between F-theory and M-theory to explore a larger moduli space, where T-dualities are realised as inequivalent elliptic fibrations of the same geometry. As expected from the Heterotic/F-theory duality the elliptic F-theory Calabi-Yau we consider admit a nested elliptic K3 fibration structure. This is central for our construction: the K3 fibrations determine the flavor groups and their global forms, and are the key to identify various T-dualities. We remark that this method works also more generally for LSTs arising from non-geometric Heterotic backgrounds. We study a first example in detail: a particularly exotic class of LSTs which are built from extremal K3 surfaces that admit flavor groups with maximal rank 18. We find all models are related by a so-called T-hexality (i.e. a 6-fold family of T-dualities) which we predict from the inequivalent elliptic fibrations of the extremal K3.

Quantum gravity bounds on mathcal{N} = 1 effective theories in four dimensions

We propose quantum gravitational constraints on effective four-dimensional theories with mathcal{N} = 1 supersymmetry. These Swampland constraints arise by demanding consistency of the worldsheet theory of a class of axionic, or EFT, strings whose existence follows from the Completeness Conjecture of quantum gravity. Modulo certain assumptions, we derive positivity bounds and quantization conditions for the axionic couplings to the gauge and gravitational sector at the two- and four-derivative level, respectively. We furthermore obtain general bounds on the rank of the gauge sector in terms of the gravitational couplings to the axions. We exemplify how these bounds rule out otherwise consistent effective supergravity theories as theories of quantum gravity. Our derivations of the quantum gravity bounds are tested and further motivated in concrete string theoretic settings. In particular, this leads to a sharper version of the bound on the gauge group rank in F-theory on elliptic four-folds with a smooth base, which improves the known geometrical Kodaira bounds. We furthermore provide a detailed derivation of the EFT string constraints in heterotic string compactifications including higher derivative corrections to the effective action and apply the bounds to M-theory compactifications on G2 manifolds.

Three-generation super no-scale models in heterotic superstrings

We derive the conditions for the one-loop contribution to the cosmological constant to be exponentially suppressed in a class of heterotic string compactifications with three generations of chiral matter, where supersymmetry is spontaneously broken à la Scherk-Schwarz. Using techniques of partial unfolding based on the Hecke congruence subgroup Γ0(2) of the modular group to extract the leading asymptotics, we show that the super no-scale condition nB=nF between the degeneracies of massless bosons nB and fermions nF in the full string spectrum is necessary but not sufficient, and needs to be supplemented by additional conditions which we identify. We use these results to construct three-generation Pati–Salam models with interesting phenomenological characteristics.

Gauge group topology of 8D Chaudhuri-Hockney-Lykken vacua

Compactifications of the Chaudhuri-Hockney-Lykken (CHL) string to eight dimensions can be characterized by embeddings of root lattices into the rank 12 momentum lattice ${\mathrm{\ensuremath{\Lambda}}}_{M}$, the so-called Mikhailov lattice. Based on these data, we devise a method to determine the global gauge group structure including all $U(1)$ factors. The key observation is that, while the physical states correspond to vectors in the momentum lattice, the gauge group topology is encoded in its dual. Interpreting a nontrivial ${\ensuremath{\pi}}_{1}(G)\ensuremath{\equiv}\mathcal{Z}$ for the non-Abelian gauge group $G$ as having gauged a $\mathcal{Z}$ 1-form symmetry, we also prove that all CHL gauge groups are free of a certain anomaly [1] that would obstruct this gauging. We verify this by explicitly computing $\mathcal{Z}$ for all 8D CHL vacua with $\mathrm{rank}(G)=10$. Since our method applies also to ${T}^{2}$ compactifications of heterotic strings, we further establish a map that determines any CHL gauge group topology from that of a ``parent'' heterotic model.

Chern-Simons invariants and heterotic superpotentials

The superpotential in four-dimensional heterotic effective theories contains terms arising from holomorphic Chern-Simons invariants associated to the gauge and tangent bundles of the compactification geometry. These effects are crucial for a number of key features of the theory, including vacuum stability and moduli stabilization. Despite their importance, few tools exist in the literature to compute such effects in a given heterotic vacuum. In this work we present new techniques to explicitly determine holomorphic Chern-Simons invariants in heterotic string compactifications. The key technical ingredient in our computations are real bundle morphisms between the gauge and tangent bundles. We find that there are large classes of examples, beyond the standard embedding, where the Chern-Simons superpotential vanishes. We also provide explicit examples for non-flat bundles where it is non-vanishing and non-integer quantized, generalizing previous results for Wilson lines.

Witten anomaly in 4d heterotic compactificaitons with mathcal{N} = 2 supersymmetry

We showed that there is no SU(2) Witten anomaly in a large class of 4d mathcal{N} = 2 supersymmetric Heterotic string compactifications. The consistency conditions we consider are the modularity of the new supersymmetric index, the integrality of BPS indices, and the discrete Peccei-Quinn shift symmetries. We also found an example where these conditions are not sufficient to show that the theory is anomaly free. This suggests that there are more conditions in the worldsheet SCFT essential for consistent string compactifications.

Exponentially suppressed cosmological constant with enhanced gauge symmetry in heterotic interpolating models

Abstract A few 9D interpolating models with two parameters are constructed and the massless spectra are studied by considering compactification of heterotic strings on a twisted circle with a Wilson line. It is found that there are some conditions between radius $R$ and Wilson line $A$ under which the gauge symmetry is enhanced. In particular, when the gauge symmetry is enhanced to $SO(18)\times SO(14)$, the cosmological constant is exponentially suppressed. We also construct a non-supersymmetric string model that is tachyon-free in all regions of moduli space and whose gauge symmetry involves $E_8$.

Rparity from string compactification

In this paper, we embed the $\Z_{4R}$ parity as a discrete subgroup of a global symmetry \UoR\,obtained from $\Z_{12-I}$ compactification of heterotic string $\EE8$. A part of \UoR\,transformation is the shift of the anticommuting variable $\vartheta$ to $e^{i\alpha}\vartheta$ which necessarily incoorporate the transformation of internal space coordinate. Out of six internal spaces, we identify three U(1)'s whose charges are denoted as $Q_{18},Q_{20}$, and $Q_{22}$. The \UoR~is defined as \UEE$\times$\UKK~where \UEE~is the part from $\EE8$ and \UKK~is the part generated by $Q_{18},Q_{20}$, and $Q_{22}$. We propose a method to define a \UoR~direction. The needed vacuum expectation values for breaking gauge U(1)'s except U(1)$_Y$ of the standard model carry \UoR~charge 4 modulo 4 such that \UoR~is broken down to $\Z_{4R}$ at the grand unification scale. $\Z_{4R}$ is broken to $\Z_{2R}$ between the intermediate ($\sim 10^{11\,}\gev$) and the electroweak scales ($100\,\gev\sim 1\,\tev$). The conditions we impose are proton longevity, a large top quark mass, and acceptable magnitudes for the $\mu$ term and neutrino masses.

Non-Cartan Mordell-Weil lattices of rational elliptic surfaces and heterotic/F-theory compactifications

The Mordell-Weil lattices (MW lattices) associated to rational elliptic surfaces are classified into 74 types. Among them, there are cases in which the MW lattice is none of the weight lattices of simple Lie algebras or direct sums thereof. We study how such “non-Cartan MW lattices” are realized in the six-dimensional heterotic/F-theory compactifications. In this paper, we focus on non-Cartan MW lattices that are torsion free and whose associated singularity lattices are sublattices of A7. For the heterotic string compactification, a non-Cartan MW lattice yields an instanton gauge group H with one or more U(1) group(s). We give a method for computing massless spectra via the index theorem and show that the U(1) instanton number is limited to be a multiple of some particular non-one integer. On the F-theory side, we examine whether we can construct the corresponding threefold geometries, i.e., rational elliptic surface fibrations over ℙ1. Except for some cases, we obtain such geometries for specific distributions of instantons. All the spectrum derived from those geometries completely match with the heterotic results.

Six Line Configurations and String Dualities

We study the family of K3 surfaces of Picard rank sixteen associated with the double cover of the projective plane branched along the union of six lines, and the family of its Van Geemen-Sarti partners, i.e., K3 surfaces with special Nikulin involutions, such that quotienting by the involution and blowing up recovers the former. We prove that the family of Van Geemen-Sarti partners is a four-parameter family of K3 surfaces with $H \oplus E_7(-1) \oplus E_7(-1)$ lattice polarization. We describe explicit Weierstrass models on both families using even modular forms on the bounded symmetric domain of type $IV$. We also show that our construction provides a geometric interpretation, called geometric two-isogeny, for the F-theory/heterotic string duality in eight dimensions. As a result, we obtain novel F-theory models, dual to non-geometric heterotic string compactifications in eight dimensions with two non-vanishing Wilson line parameters.

A stringy test of the Scalar Weak Gravity Conjecture

We prove a version of the Weak Gravity Conjecture for 6d F-theory or heterotic string compactifications with 8 supercharges. This sharpens our previous analysis by including massless scalar fields. The latter are known to modify the Weak Gravity Conjecture bound in two a priori independent ways: First, the extremality condition of a charged black hole is modified, and second, the test particles required to satisfy the Weak Gravity Conjecture are subject to additional Yukawa type interactions. We argue on general grounds that at weak coupling, the two types of effects are equivalent for a tower of asymptotically massless charged test particles predicted by the Swampland Distance Conjecture.We then specialize to F-theory compactified on elliptic Calabi–Yau three-folds and prove that the precise numerical bound on the charge-to-mass ratio is satisfied at weak coupling. This amounts to an intriguing coincidence of two a priori different notions of extremality, namely one based on the balance of gauge, gravitational and scalar forces for extremal (non-BPS) black holes, and the other encoded in the modular properties of certain Jacobi forms. In the presence of multiple abelian gauge group factors, the elliptic genus counting these states is a lattice quasi-Jacobi form of higher rank, and we exemplify this in a model with two abelian gauge group factors.

(0,2) hybrid models

We introduce a class of (0,2) superconformal field theories based on hybrid geometries, generalizing various known constructions. We develop techniques for the computation of the complete massless spectrum when the theory can be interpreted as determining a perturbative heterotic string compactification. We provide evidence for surprising properties regarding RG flows and IR accidental symmetries in (0,2) hybrid CFTs. We also study the conditions for embedding a hybrid theory in a particular class of gauged linear sigma models. This perspective suggests that our construction generates models which cannot be realized or analyzed by previously known methods.

Conway subgroup symmetric compactifications of heterotic string

We investigate special compactifications of the heterotic string for which the space of half-BPS states is, in a natural way, a representation of various subgroups of the Conway group. These compactifications provide a useful framework for analyzing the action of some of the large symmetry groups appearing in discussions of moonshine in the physics literature. We investigate toroidal compactifications of heterotic string with 16 supersymmetries as well as asymmetric toroidal orbifolds with N = 2 supersymmetry in four dimensions that arise as compactifications. The latter Conway subgroup symmetric compactifications of the heterotic string might have some interesting implications for D-brane bound states on Calabi–Yau manifolds.

Physics of superheavy dark matter in supergravity

New trends in inflationary model building and dark matter production in supergravity are considered. Starobinsky inflation is embedded into [Formula: see text] supergravity, avoiding instability problems, when the inflaton belongs to a vector superfield associated with a [Formula: see text] gauge symmetry, instead of a chiral superfield. This gauge symmetry can be spontaneously broken by the super-Higgs mechanism resulting in a massive vector supermultiplet including the (real scalar) inflaton field. Both supersymmetry (SUSY) and the R-symmetry can also be spontaneously broken by the Polonyi mechanism at high scales close to the inflationary scale. In this case, Polonyi particles and gravitinos become superheavy, and can be copiously produced during inflation by the Schwinger mechanism sourced by the universe expansion. The Polonyi mass slightly exceeds twice the gravitino mass, so that Polonyi particles are unstable and decay into gravitinos. Considering the mechanisms of superheavy gravitino production, we find that the right amount of cold dark matter composed of gravitinos can be achieved. In our scenario, the parameter space of the inflaton potential is directly related to the dark matter one, providing a new unifying framework of inflation and dark matter genesis. A multi-superfield extension of the supergravity framework with a single (inflaton) superfield can result in a formation of primordial nonlinear structures like mini- and stellar-mass black holes, primordial nongaussianity, and the running spectral index of density fluctuations. This framework can be embedded into the SUSY GUTs inspired by heterotic string compactifications on Calabi–Yau three-folds, thus unifying particle physics with quantum gravity.

ASK/PSK-correspondence and the r-map

We formulate a correspondence between affine and projective special K\"ahler manifolds of the same dimension. As an application, we show that, under this correspondence, the affine special K\"ahler manifolds in the image of the rigid r-map are mapped to one-parameter deformations of projective special K\"ahler manifolds in the image of the supergravity r-map. The above one-parameter deformations are interpreted as perturbative $\alpha'$-corrections in heterotic and type-II string compactifications with $N=2$ supersymmetry. Also affine special K\"ahler manifolds with quadratic prepotential are mapped to one-parameter families of projective special K\"ahler manifolds with quadratic prepotential. We show that the completeness of the deformed supergravity r-map metric depends solely on the (well-understood) completeness of the undeformed metric and the sign of the deformation parameter.

The Infinitesimal Moduli Space of Heterotic G2 Systems

Heterotic string compactifications on integrable G2 structure manifolds Y with instanton bundles {(V,A), (TY,tilde{theta})} yield supersymmetric three-dimensional vacua that are of interest in physics. In this paper, we define a covariant exterior derivative {mathcal{D}} and show that it is equivalent to a heterotic G2 system encoding the geometry of the heterotic string compactifications. This operator {mathcal{D}} acts on a bundle {mathcal{Q}=T^*Y oplus {rm End}(V) oplus {rm End}(TY)} and satisfies a nilpotency condition {check{{mathcal{D}}}^2=0} , for an appropriate projection of {mathcal D}. Furthermore, we determine the infinitesimal moduli space of these systems and show that it corresponds to the finite-dimensional cohomology group {check H^1_{check{{mathcal{D}}}}(mathcal{Q})}. We comment on the similarities and differences of our result with Atiyah’s well-known analysis of deformations of holomorphic vector bundles over complex manifolds. Our analysis leads to results that are of relevance to all orders in the {alpha'} expansion.

BPS algebras, genus zero and the heterotic Monster

In this note, we expand on some technical issues raised in (Paquette et al 2016 Commun. Number Theory Phys. 10 433–526) by the authors, as well as providing a friendly introduction to and summary of our previous work. We construct a set of heterotic string compactifications to 0 + 1 dimensions intimately related to the Monstrous moonshine module of Frenkel, Lepowsky, and Meurman (and orbifolds thereof). Using this model, we review our physical interpretation of the genus zero property of Monstrous moonshine. Furthermore, we show that the space of (second-quantized) BPS-states forms a module over the Monstrous Lie algebras —some of the first and most prominent examples of Generalized Kac–Moody algebras—constructed by Borcherds and Carnahan. In particular, we clarify the structure of the module present in the second-quantized string theory. We also sketch a proof of our methods in the language of vertex operator algebras, for the interested mathematician.

New evidence for (0,2) target space duality

In the context of (0, 2) gauged linear sigma models, we explore chains of perturbatively dual heterotic string compactifications. The notion of target space duality originates in non-geometric phases and can be used to generate distinct GLSMs with shared geometric phases leading to apparently identical target space theories. To date, this duality has largely been studied at the level of counting states in the effective theories. We extend this analysis to the effective potential and loci of enhanced symmetry in dual theories. By engineering vector bundles with non-trivial constraints arising from slope-stability (i.e. D-terms) and holomorphy (i.e. F-terms) the detailed structure of the vacuum space of the dual theories can be explored. Our results give new evidence that GLSM target space duality may provide important hints towards a more complete understanding of (0, 2) string dualities.

Niemeier Lattices in the Free Fermionic Heterotic–String Formulation

The spinor–vector duality was discovered in free fermionic constructions of the heterotic string in four dimensions. It played a key role in the construction of heterotic–string models with an anomaly-free extra Z′ symmetry that may remain unbroken down to low energy scales. A generic signature of the low scale string derived Z′ model is via diphoton excess that may be within reach of the LHC. A fascinating possibility is that the spinor–vector duality symmetry is rooted in the structure of the heterotic–string compactifications to two dimensions. The two-dimensional heterotic–string theories are in turn related to the so-called moonshine symmetries that underlie the two-dimensional compactifications. In this paper, we embark on exploration of this connection by the free fermionic formulation to classify the symmetries of the two-dimensional heterotic–string theories. We use two complementary approaches in our classification. The first utilises a construction which is akin to the one used in the spinor–vector duality. Underlying this method is the triality property of SO(8) representations. In the second approach, we use the free fermionic tools to classify the twenty-four-dimensional Niemeier lattices.

From Type II string theory toward BSM/dark sector physics

Four-dimensional compactifications of string theory provide a controlled set of possible gauge representations accounting for BSM particles and dark sector components. In this review, constraints from perturbative Type II string compactifications in the geometric regime are discussed in detail and then compared to results from heterotic string compactifications and nonperturbative/nongeometric corners. As a prominent example, an open string realization of the QCD axion is presented. The status of deriving the associated low-energy effective action in four dimensions is discussed and open avenues of major phenomenological importance are highlighted. As examples, a mechanism of closed string moduli stabilization by D-brane backreaction as well as one-loop threshold corrections to the gauge couplings and balancing a low string scale [Formula: see text] with unisotropic compact dimensions are discussed together with implications on potential future new physics observations. For illustrative purposes, an explicit example of a globally consistent D6-brane model with MSSM-like spectrum on [Formula: see text] is presented.