Heterotic String Compactifications Research Articles

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.

Looijenga’s weighted projective space, Tate’s algorithm and Mordell-Weil lattice in F-theory and heterotic string theory

It is now well known that the moduli space of a vector bundle for heterotic string compactifications to four dimensions is parameterized by a set of sections of a weighted projective space bundle of a particular kind, known as Looijenga’s weighted projective space bundle. We show that the requisite weighted projective spaces and the Weierstrass equations describing the spectral covers for gauge groups E N (N = 4, · · · , 8) and SU(n + 1) (n = 1, 2, 3) can be obtained systematically by a series of blowing-up procedures according to Tate’s algorithm, thereby the sections of correct line bundles claimed to arise by Looijenga’s theorem can be automatically obtained. They are nothing but the four-dimensional analogue of the set of independent polynomials in the six-dimensional F-theory parameterizing the complex structure, which is further confirmed in the constructions of D 4, A 5, D 6, E 3 and SU(2) × SU(2) bundles. We also explain why we can obtain them in this way by using the structure theorem of the Mordell-Weil lattice, which is also useful for understanding the relation between the singularity and the occurrence of chiral matter in F-theory.

Infinitesimal moduli of G2 holonomy manifolds with instanton bundles

We describe the infinitesimal moduli space of pairs $(Y, V)$ where $Y$ is a manifold with $G_2$ holonomy, and $V$ is a vector bundle on $Y$ with an instanton connection. These structures arise in connection to the moduli space of heterotic string compactifications on compact and non-compact seven dimensional spaces, e.g. domain walls. Employing the canonical $G_2$ cohomology developed by Reyes-Carri\'on and Fern\'andez and Ugarte, we show that the moduli space decomposes into the sum of the bundle moduli $H^1_{\check{d}_A}(Y,\mathrm{End}(V))$ plus the moduli of the $G_2$ structure preserving the instanton condition. The latter piece is contained in $H^1_{\check{d}_\theta}(Y,TY)$, and is given by the kernel of a map ${\cal\check F}$ which generalises the concept of the Atiyah map for holomorphic bundles on complex manifolds to the case at hand. In fact, the map ${\cal\check F}$ is given in terms of the curvature of the bundle and maps $H^1_{\check{d}_\theta}(Y,TY)$ into $H^2_{\check{d}_A}(Y,\mathrm{End}(V))$, and moreover can be used to define a cohomology on an extension bundle of $TY$ by $\mathrm{End}(V)$. We comment further on the resemblance with the holomorphic Atiyah algebroid and connect the story to physics, in particular to heterotic compactifications on $(Y,V)$ when $\alpha'=0$.

Heterotic String Compactification and New Vector Bundles

We propose a construction of Kähler and non-Kähler Calabi–Yau manifolds by branched double covers of twistor spaces. In this construction we use the twistor spaces of four-manifolds with self-dual conformal structures, with the examples of connected sum of n $${\mathbb{P}^{2}}$$ s. We also construct K3-fibered Calabi–Yau manifolds from the branched double covers of the blow-ups of the twistor spaces. These manifolds can be used in heterotic string compactifications to four dimensions. We also construct stable and polystable vector bundles. Some classes of these vector bundles can give rise to supersymmetric grand unified models with three generations of quarks and leptons in four dimensions.

K3 Surfaces, Modular Forms, and Non-Geometric Heterotic Compactifications

We construct non-geometric compactifications by using the F-theory dual of the heterotic string compactified on a two-torus, together with a close connection between Siegel modular forms of genus two and the equations of certain K3 surfaces. The modular group mixes together the K\"ahler, complex structure, and Wilson line moduli of the torus yielding weakly coupled heterotic string compactifications which have no large radius interpretation.

General aspects of heterotic string compactifications on stacks and gerbes

In this paper we work out some basic results concerning heterotic string compactifications on stacks and, in particular, gerbes. A heterotic string compactification on a gerbe can be understood as, simultaneously, both a compactification on a space with a restriction on nonperturbative sectors, and also, a gauge theory in which a subgroup of the gauge group acts trivially on the massless matter. Gerbes admit more bundles than corresponding spaces, which suggests they are potentially a rich playground for heterotic string compactifications. After we give a general characterization of heterotic strings on stacks, we specialize to gerbes, and consider three different classes of `building blocks' of gerbe compactifications. We argue that heterotic string compactifications on one class is equivalent to compactification of the same heterotic string on a disjoint union of spaces, compactification on another class is dual to compactifications of other heterotic strings on spaces, and compactification on the third class is not perturbatively consistent, so that we do not in fact recover a broad array of new heterotic compactifications, just combinations of existing ones. In appendices we explain how to compute massless spectra of heterotic strings on stacks, derive some new necessary conditions for a heterotic string on a stack or orbifold to be well-defined, and also review some basic properties of bundles on gerbes.

Geometric constraints in dual F-theory and heterotic string compactifications

We systematically analyze a broad class of dual heterotic and F-theory models that give four-dimensional supergravity theories, and compare the geometric constraints on the two sides of the duality. Specifically, we give a complete classification of models where the heterotic theory is compactified on a smooth Calabi-Yau threefold that is elliptically fibered with a single section and carries smooth irreducible vector bundles, and the dual F-theory model has a corresponding threefold base that has the form of a ℙ1 bundle. We formulate simple conditions for the geometry on the F-theory side to support an elliptically fibered Calabi-Yau fourfold. We match these conditions with conditions for the existence of stable vector bundles on the heterotic side, and show that F-theory gives new insight into the conditions under which such bundles can be constructed. In particular, we find that many allowed F-theory models correspond to vector bundles on the heterotic side with exceptional structure groups, and determine a topological condition that is only satisfied for bundles of this type. We show that in many cases the F-theory geometry imposes a constraint on the extent to which the gauge group can be enhanced, corresponding to limits on the way in which the heterotic bundle can decompose. We explicitly construct all (4962) F-theory threefold bases for dual F-theory/heterotic constructions in the subset of models where the common twofold base surface is toric, and give both toric and non-toric examples of the general results.

N = 2 heterotic-type II duality and bundle moduli

Heterotic string compactifications on a $K3$ surface $\mathfrak{S}$ depend on a choice of hyperk\"ahler metric, anti-self-dual gauge connection and Kalb-Ramond flux, parametrized by hypermultiplet scalars. The metric on hypermultiplet moduli space is in principle computable within the $(0,2)$ superconformal field theory on the heterotic string worldsheet, although little is known about it in practice. Using duality with type II strings compactified on a Calabi-Yau threefold, we predict the form of the quaternion-K\"ahler metric on hypermultiplet moduli space when $\mathfrak{S}$ is elliptically fibered, in the limit of a large fiber and even larger base. The result is in general agreement with expectations from Kaluza-Klein reduction, in particular the metric has a two-stage fibration structure, where the $B$-field moduli are fibered over bundle and metric moduli, while bundle moduli are themselves fibered over metric moduli. A more precise match must await a detailed analysis of $R^2$-corrected ten-dimensional supergravity.

Jacobi forms of higher index and paramodular groups in $ \mathcal{N} $ = 2, D = 4 compactifications of string theory

We associate a Jacobi form over a rank s lattice to N=2, D=4 heterotic string compactifications which have s Wilson lines at a generic point in the vector multiplet moduli space. Jacobi forms of index m=1 and m=2 have appeared earlier in the context of threshold corrections to heterotic string couplings. We emphasize that higher index Jacobi forms as well as Jacobi forms of several variables over more generic even lattices also appear and construct models in which they arise. In particular, we construct an orbifold model which can be connected to models that give index m=3, 4 or 5 Jacobi forms through the Higgsing process. Constraints from being a Jacobi form are then employed to get threshold corrections using only partial information on the spectrum. We apply this procedure for index m=3, 4 or 5 Jacobi form examples and also for Jacobi forms over A_2 and A_3 root lattices. Examples with a single Wilson line are examined in detail and we display the relation of Siegel forms over a paramodular group \Gamma_m to these models, where \Gamma_m is associated with the T-duality group of the models we study. Finally, results on the heterotic string side are used to clarify the linear mapping of vector multiplet moduli to Type IIA duals without using the one-loop cubic part of the prepotential on the Type II side, and also to give predictions for the geometry of the dual Calabi-Yau manifolds.

Matter from Toric Geometry and its Search at the LHC

Toric geometry is applied for construction the enhanced gauge groups in F-theory compactified on elliptic Calabi-Yau fourfolds. The Hodge numbers calculated from the polyhedra for the chain H = SU (1), ... ,SU (5), SO(10), E6, E7 determine the number of tensor multiplets, vector multiplets and hypermultiplets of soli- tonic states that appear from singularities of elliptic fibration. Due to duality between the compactification of E8timesE8 heterotic string and the type IIA string compactification on a Calabi-Yau manifold there is a natural sequence of E-group embeddings which gives the matter content of Minimal Supersymmetric Standard Model and the possibility of searching for supersymmetry at the LHC.

Heterotic non-abelian orbifolds

Abstract We perform the first systematic analysis of particle spectra obtained from heterotic string compactifications on non-Abelian toroidal orbifolds. After developing a new technique to compute the particle spectrum in the case of standard embedding based on higher dimensional supersymmetry, we compute the Hodge numbers for all recently classified 331 non-Abelian orbifold geometries which yield $ \mathcal{N}=1 $ supersymmetry for heterotic compactifications. Surprisingly, most Hodge numbers follow the empiric pattern h (1,1) − h (2,1) = 0 mod 6, which might be related to the number of three standard model generations. Furthermore, we study the fundamental groups in order to identify the possibilities for non-local gauge symmetry breaking. Three examples are discussed in detail: the simplest non-Abelian orbifold S 3 and two more elaborate examples, T 7 and Δ(27), which have only one untwisted Kähler and no untwisted complex structure modulus. Such models might be especially interesting in the context of no-scale supergravity. Finally, we briefly discuss the case of orbifolds with vanishing Euler numbers in the context of enhanced (spontaneously broken) supersymmetry.

Vacuum varieties, holomorphic bundles and complex structure stabilization in heterotic theories

We discuss the use of gauge fields to stabilize complex structure moduli in Calabi-Yau three-fold compactifications of heterotic string and M-theory. The requirement that the gauge fields in such models preserve supersymmetry leads to a complicated landscape of vacua in complex structure moduli space. We develop methods to systematically map out this multi-branched vacuum space, in a computable and explicit manner. In analysing the resulting vacua, it is found that the associated Calabi-Yau three-folds are sometimes stabilized at a value of complex structure resulting in a singular compactification manifold. We describe how it is possible to resolve these singularities, in some cases, while maintaining computational control over the moduli stabilization mechanism. The discussion is illustrated throughout the paper with explicit worked examples.

Schoen manifold with line bundles as resolved magnetized orbifolds

Abstract We give an alternative description of the Schoen manifold as the blow-up of a $ {{\mathbb{Z}}_2}\times {{\mathbb{Z}}_2} $ orbifold in which one $ {{\mathbb{Z}}_2} $ factor acts as a roto-translation. Since for this orbifold the fixed tori are only identified in pairs but not orbifolded, four-dimensional chirality can never be obtained in heterotic string compactifications using standard techniques alone. However, chirality is recovered when its tori become magnetized. To exemplify this, we construct an E8 × E8′ heterotic SU(5) GUT on the Schoen manifold with Abelian gauge fluxes, which becomes an MSSM with three generations after an appropriate Wilson line is associated to its freely acting involution. We reproduce this model as a standard heterotic orbifold CFT of the (partially) blown down Schoen manifold with a magnetic flux. Finally, in analogy to a proposal for non-perturbative heterotic models by Aldazabal et al. we suggest modifications to the heterotic orbifold spectrum formulae in the presence of magnetized tori.

Moduli stabilising in heterotic nearly Kähler compactifications

We study heterotic string compactifications on nearly K\"ahler homogeneous spaces, including the gauge field effects which arise at order \alpha'. Using Abelian gauge fields, we are able to solve the Bianchi identity and supersymmetry conditions to this order. The four-dimensional external space-time consists of a domain wall solution with moduli fields varying along the transverse direction. We find that the inclusion of \alpha' corrections improves the moduli stabilization features of this solution. In this case, one of the dilaton and the volume modulus asymptotes to a constant value away from the domain wall. It is further shown that the inclusion of non-perturbative effects can stabilize the remaining modulus and "lift" the domain wall to an AdS vacuum. The coset SU(3)/U(1)^2 is used as an explicit example to demonstrate the validity of this AdS vacuum. Our results show that heterotic nearly K\"ahler compactifications can lead to maximally symmetric four-dimensional space-times at the non-perturbative level.

Five-dimensional heterotic black holes and its dual IR-CFT

We analyze the possible dynamical emergence of IR conformal field theory describing the low- energy excitations of near-extremal black holes in five-dimensional compactification of heterotic strings. We find that, by tuning the mass and charges in such a way that the extremal black holes have a classically vanishing horizon area, the near-horizon develops an AdS3 throat and when we combine the low-energy limit with vanishing Newton coupling constant, the system has a dual conformal field theory description. We compare our results with c-extremization and the Kerr/CFT predictions.

One-loop BPS amplitudes as BPS-state sums

AbstractRecently, we introduced a new procedure for computing a class of one-loop BPS-saturated amplitudes in String Theory, which expresses them as a sum of one-loop contributions of all perturbative BPS states in a manifestly T-duality invariant fashion. In this paper, we extend this procedure toallBPS-saturated amplitudes of the form ∫FΓd+k,dΦ, withΦbeing a weak (almost) holomorphic modular form of weight − k/2. We use the fact that any suchΦcan be expressed as a linear combination of certain absolutely convergent Poincaré series, against which the fundamental domain F can be unfolded. The resulting BPS-state sum neatly exhibits the singularities of the amplitude at points of gauge symmetry enhancement, in a chamber-independent fashion. We illustrate our method with concrete examples of interest in heterotic string compactifications.

U(n) spectral covers from decomposition

We construct decomposed spectral covers for bundles on elliptically fibered Calabi-Yau threefolds whose structure groups are S(U(1) x U(4)), S(U(2) x U(3)) and S(U(1) x U(1) x U(3)) in heterotic string compactifications. The decomposition requires not only the tuning of the SU(5) spectral covers but also the tuning of the complex structure moduli of the Calabi-Yau threefolds. This configuration is translated to geometric data on F-theory side. We find that the monodromy locus for two-cycles in K3 fibered Calabi-Yau fourfolds in a stable degeneration limit is globally factorized with squared factors under the decomposition conditions. This signals that the monodromy group is reduced and there is a U(1) symmetry in a low energy effective field theory. To support that, we explicitly check the reduction of a monodromy group in an appreciable region of the moduli space for an $E_6$ gauge theory with (1+2) decomposition. This may provide a systematic way for constructing F-theory models with U(1) symmetries.

Old issues and linear sigma models

Using mirror symmetry, we resolve an old puzzle in the linear sigma model description of the spacetime Higgs mechanism in a heterotic string compactification with (2,2) worldsheet supersymmetry. The resolution has a nice spacetime interpretation via the normalization of physical fields and suggests that with a little care deformations of the linear sigma model can describe heterotic Higgs branches.

Heterotic bundles on Calabi-Yau manifolds with small Picard number

We undertake a systematic scan of vector bundles over spaces from the largest database of known Calabi-Yau three-folds, in the context of heterotic string compactification. Specifically, we construct positive rank five monad bundles over Calabi-Yau hypersurfaces in toric varieties, with the number of Kahler moduli equal to one, two, and three and extract physically interesting models. We select models which can lead to three families of matter after dividing by a freely-acting discrete symmetry and including Wilson lines. About 2000 such models on two manifolds are found.

On marginal deformations of [formula omitted] non-linear sigma models

An N=1, d=4 supersymmetric compactification of the perturbative heterotic string is described by a d=2 (0,2) superconformal field theory. The first order marginal deformations of the internal (0,2) SCFT are in 1 to 1 correspondence with massless gauge-neutral scalars in the spacetime theory. Working at tree-level in the α′ expansion, we describe these first order deformations for SCFTs with a (0,2) non-linear sigma model description. Our results clarify the structure of deformations of heterotic Calabi–Yau compactifications and more general heterotic flux vacua.