Heterotic Duals Research Articles

$\mathbb{P}^1$-fibrations in F-theory and string dualities

In this work we study F-theory compactifications on elliptically fibered Calabi-Yau n-folds which have $\mathbb{P}^1$-fibered base manifolds. Such geometries, which we study in both 4- and 6-dimensions, are both ubiquitous within the set of Calabi-Yau manifolds and play a crucial role in heterotic/F-theory duality. We discuss the most general formulation of $\mathbb{P}^1$-bundles of this type, as well as fibrations which degenerate at higher codimension loci. In the course of this study, we find a number of new phenomena. For example, in both 4- and 6-dimensions we find transitions whereby the base of a $\mathbb{P}^1$-bundle can change nature, or jump, at certain loci in complex structure moduli space. We discuss the implications of this jumping for the associated heterotic duals. We argue that $\mathbb{P}^1$-bundles with only rational sections lead to heterotic duals where the Calabi-Yau manifold is elliptically fibered over the section of the $\mathbb{P}^1$- bundle, and not its base. As expected, we see that degenerations of the $\mathbb{P}^1$-fibration of the F-theory base correspond to 5-branes in the dual heterotic physics, with the exception of cases in which the fiber degenerations exhibit monodromy. Along the way, we discuss a set of useful formulae and tools for describing F-theory compactifications on this class of Calabi-Yau manifolds.

Non-perturbative heterotic duals of M-theory on G2 orbifolds

By fibering the duality between the E8 × E8 heterotic string on T3 and M-theory on K3, we study heterotic duals of M-theory compactified on G2 orbifolds of the form T7/ {mathbb{Z}}_2^3 . While the heterotic compactification space is straightforward, the description of the gauge bundle is subtle, involving the physics of point-like instantons on orbifold singularities. By comparing the gauge groups of the dual theories, we deduce behavior of a “half-G2” limit, which is the M-theory analog of the stable degeneration limit of F-theory. The heterotic backgrounds exhibit point-like instantons that are localized on pairs of orbifold loci, similar to the “gauge-locking” phenomenon seen in Hořava-Witten compactifications. In this way, the geometry of the G2 orbifold is translated to bundle data in the heterotic background. While the instanton configuration looks surprising from the perspective of the E8 × E8 heterotic string, it may be understood as T-dual Spin(32)/ℤ2 instantons along with winding shifts originating in a dual Type I compactification.

Unbroken $E7 \times E7$ nongeometric heterotic strings, stable degenerations and enhanced gauge groups in F-theory duals

Eight-dimensional non-geometric heterotic strings with gauge algebra $\mathfrak{e}_8\mathfrak{e}_7$ were constructed by Malmendier and Morrison as heterotic duals of F-theory on K3 surfaces with $\Lambda^{1,1}\oplus E_8\oplus E_7$ lattice polarization. Clingher, Malmendier and Shaska extended these constructions to eight-dimensional non-geometric heterotic strings with gauge algebra $\mathfrak{e}_7\mathfrak{e}_7$ as heterotic duals of F-theory on $\Lambda^{1,1}\oplus E_7\oplus E_7$ lattice polarized K3 surfaces. In this study, we analyze the points in the moduli of non-geometric heterotic strings with gauge algebra $\mathfrak{e}_7\mathfrak{e}_7$, at which the non-Abelian gauge groups on the F-theory side are maximally enhanced. The gauge groups on the heterotic side do not allow for the perturbative interpretation at these points. We show that these theories can be described as deformations of the stable degenerations, as a result of coincident 7-branes on the F-theory side. From the heterotic viewpoint, this effect corresponds to the insertion of 5-branes. These effects can be used to understand nonperturbative aspects of nongeometric heterotic strings. Additionally, we build a family of elliptic Calabi-Yau 3-folds by fibering elliptic K3 surfaces, which belong to the F-theory side of the moduli of non-geometric heterotic strings with gauge algebra $\mathfrak{e}_7\mathfrak{e}_7$, over $\mathbb{P}^1$. We find that highly enhanced gauge symmetries arise on F-theory on the built elliptic Calabi-Yau 3-folds.

$\frac{1}{2}$Calabi–Yau $4$-folds and four-dimensional F-theory on Calabi–Yau $4$-folds with $\mathrm{U}(1)$ factors

In this study, four-dimensional $N=1$ F-theory models with multiple U(1) gauge group factors are constructed. A class of rational elliptic 4-folds, which we call as "$\frac{1}{2}$Calabi-Yau 4-folds," is introduced, and we construct the elliptically fibered 4-folds by utilizing them. This yields a novel approach for building families of elliptically fibered Calabi-Yau 4-folds with positive Mordell-Weil ranks. The introduced $\frac{1}{2}$Calabi-Yau 4-folds possess the characteristic property wherein the sum of the ranks of the singularity type and the Mordell-Weil group is always equal to six. This interesting property enables us to construct the elliptically fibered Calabi-Yau 4-folds of various positive Mordell-Weil ranks. From one to six U(1) factors form in four-dimensional F-theory on the resulting Calabi-Yau 4-folds. We also propose the geometric condition on the base 3-fold of the built Calabi-Yau 4-folds that allows four-dimensional F-theory models that have heterotic duals to be distinguished from those that do not.

Stability and vacuum energy in open string models with broken supersymmetry

We construct type I string models with supersymmetry broken by compactifi- cation that are non-tachyonic and have exponentially small effective potential at one-loop. All open string moduli can be stabilized, while the closed string moduli remain massless at one-loop. The backgrounds of interest have rigid Wilson lines by the use of stacked branes, and some models should have heterotic duals. We also present non-tachyonic backgrounds with positive potentials of runaway type at one-loop. This class of models could be used to test various swampland conjectures.

Heterotic-type IIA duality and degenerations of K3 surfaces

We study the duality between four-dimensional N=2 compactifications of heterotic and type IIA string theories. Via adiabatic fibration of the duality in six dimensions, type IIA string theory compactified on a K3-fibred Calabi-Yau threefold has a potential heterotic dual compactification. This adiabatic picture fails whenever the K3 fibre degenerates into multiple components over points in the base of the fibration. Guided by monodromy, we identify such degenerate K3 fibres as solitons generalizing the NS5-brane in heterotic string theory. The theory of degenerations of K3 surfaces can then be used to find which solitons can be present on the heterotic side. Similar to small instanton transitions, these solitons escort singular transitions between different Calabi-Yau threefolds. Starting from well-known examples of heterotic--type IIA duality, such transitions can take us to type IIA compactifications with unknown heterotic duals.

Origin of Abelian gauge symmetries in heterotic/F-theory duality

We study aspects of heterotic/F-theory duality for compactifications with Abelian gauge symmetries. We consider F-theory on general Calabi-Yau manifolds with a rank one Mordell-Weil group of rational sections. By rigorously performing the stable degeneration limit in a class of toric models, we derive both the Calabi-Yau geometry as well as the spectral cover describing the vector bundle in the heterotic dual theory. We carefully investigate the spectral cover employing the group law on the elliptic curve in the heterotic theory. We find in explicit examples that there are three different classes of heterotic duals that have U(1) factors in their low energy effective theories: split spectral covers describing bundles with S(U(m) x U(1)) structure group, spectral covers containing torsional sections that seem to give rise to bundles with SU(m) x Z_k structure group and bundles with purely non-Abelian structure groups having a centralizer in E_8 containing a U(1) factor. In the former two cases, it is required that the elliptic fibration on the heterotic side has a non-trivial Mordell-Weil group. While the number of geometrically massless U(1)'s is determined entirely by geometry on the F-theory side, on the heterotic side the correct number of U(1)'s is found by taking into account a Stuckelberg mechanism in the lower-dimensional effective theory. In geometry, this corresponds to the condition that sections in the two half K3 surfaces that arise in the stable degeneration limit of F-theory can be glued together globally.

On E 8 and F-theory GUTs

We study correlations between the massless field spectra of F-theory fibrations supporting an SU(5) gauge symmetry extended by Abelian symmetries and the spectra that arise from the group E 8. The adjoint representation of E 8 leads to six different classes of matter spectra upon Higgsing E 8 to SU(5)×U(1) n . Of 27 different smooth F-theory elliptic fibrations constructed in the literature, satisfying certain genericness and flatness criteria, the matter spectrum of only one can be embedded in these six theories, thereby apparently ruling out any connection. We define an extension of the spectrum arising from the adjoint of E 8 by introducing new SU(5)-singlet fields with Abelian charges such that there exists a cubic gauge invariant coupling between any three representations. Higgsing by these new singlets leads to a further 20 classes of spectra, and we find that all the F-theory fibrations can then be embedded in this extended set. These results show that E 8, when extended in this specific way, may still have a role to play in controlling the possible matter spectra in F-theory. We give an explicit geometric example of the presence of the extending singlets and their Higgsing. We discuss some phenomenological applications of the new set of theories, in particular due to the existence of a ℤ2 matter parity. Finally we make some comments on the Heterotic duals of the F-theory fibrations which extend E 8 in this way.

Heterotic flux backgrounds and their IIA duals

We study four-dimensional heterotic flux vacua with N = 2 spacetime super- symmetry. A worldsheet perspective is used to clarify quantization conditions associated to the fluxes and the constraints these place on the moduli spaces of resulting compactifica- tions. We propose that these vacua fit naturally in the context of heterotic/IIA duality as heterotic duals to compactifications on K3-fibered but not elliptically fibered Calabi-Yau three-folds. We present some examples of such potential dual pairs.

T-branes and geometry

T-branes are a non-abelian generalization of intersecting branes in which the matrix of normal deformations is nilpotent along some subspace. In this paper we study the geometric remnant of this open string data for six-dimensional F-theory vacua. We show that in the dual M-theory / IIA compactification on a smooth Calabi-Yau threefold X, the geometric remnant of T-brane data translates to periods of the three-form potential valued in the intermediate Jacobian of X. Starting from a smoothing of a singular Calabi-Yau, we show how to track this data in singular limits using the theory of limiting mixed Hodge structures, which in turn directly points to an emergent Hitchin-like system coupled to defects. We argue that the physical data of an F-theory compactification on a singular threefold involves specifying both a geometry as well as the remnant of three-form potential moduli and flux which is localized on the discriminant. We give examples of T-branes in compact F-theory models with heterotic duals, and comment on the extension of our results to four-dimensional vacua.

Higgs Bundles and UV Completion in F-Theory

F-theory admits 7-branes with exceptional gauge symmetries, which can be compactified to give phenomenological four-dimensional GUT models. Here we study general supersymmetric compactifications of eight-dimensional Yang–Mills theory. They are mathematically described by meromorphic Higgs bundles, and therefore admit a spectral cover description. This allows us to give a rigorous and intrinsic construction of local models in F-theory. We use our results to prove a no-go theorem showing that local SU(5) models with three generations do not exist for generic moduli. However we show that three-generation models do exist on the Noether–Lefschetz locus. We explain how F-theory models can be mapped to non-perturbative orientifold models using a scaling limit proposed by Sen. Further we address the construction of global models that do not have heterotic duals, considering models with base CP 3 or a blow-up thereof as examples. We show how one may obtain a contractible worldvolume with a two-cycle not inherited from the bulk, a necessary condition for implementing GUT breaking using fluxes.

Model building with $F$-theory

Despite much recent progress in model building with $D$-branes, it has been problematic to find a completely convincing explanation of gauge coupling unification. We extend the class of models by considering $F$-theory compactifications, which may incorporate unification more naturally. We explain how to derive the charged chiral spectrum and Yukawa couplings in $N = 1$ compactifications of $F$-theory with $G$-flux. In a class of models which admit perturbative heterotic duals, we show that the $F$-theory and heterotic computations match.

Geometric transitions, flops and non-Kähler manifolds: II

We continue our study of geometric transitions in type II and heterotic theories. In type IIB theory we discuss an F-theory setup which clarifies many of our earlier assumptions and allows us to study gravity duals of N = 1 gauge theories with arbitrary global symmetry group G. We also point out the subtle differences between global and local metrics, and show that in many cases the global descriptions are far more complicated than discussed earlier. We determine the full global description in type I/heterotic theory. In type IIA, our analysis gives rise to a local non-Kähler metric whose global description involves a particular orientifold action with gauge fluxes localised on branes. We are also able to identify the three form fields that allow for a smooth flop in the M-theory lift. We briefly discuss the issues of generalized complex structures in type IIB theory and possible half-twisted models in the heterotic duals of our type II models. In a companion paper we will present details on the topological aspects of these models.

Topological string amplitudes, complete intersection Calabi-Yau spaces and threshold corrections

We present the most complete list of mirror pairs of Calabi-Yau complete intersections in toric ambient varieties and develop the methods to solve the topological string and to calculate higher genus amplitudes on these compact Calabi-Yau spaces. These symplectic invariants are used to remove redundancies in examples. The construction of the B-model propagators leads to compatibility conditions, which constrain multi-parameter mirror maps. For K3 fibered Calabi-Yau spaces without reducible fibers we find closed formulas for all genus contributions in the fiber direction from the geometry of the fibration. If the heterotic dual to this geometry is known, the higher genus invariants can be identified with the degeneracies of BPS states contributing to gravitational threshold corrections and all genus checks on string duality in the perturbative regime are accomplished. We find, however, that the BPS degeneracies do not uniquely fix the non-perturbative completion of the heterotic string. For these geometries we can write the topological partition function in terms of the Donaldson-Thomas invariants and we perform a non-trivial check of S-duality in topological strings. We further investigate transitions via collapsing D5 del Pezzo surfaces and the occurrence of free Z2 quotients that lead to a new class of heterotic duals.

Compactification with flux on K3 and tori

We study compactifications of Type IIB string theory on a K3 \times T^2/Z_2 orientifold in the presence of RR and NS flux. We find the most general supersymmetry preserving, Poincare invariant, vacua in this model. All the complex structure moduli and some of the Kahler moduli are stabilised in these vacua. We obtain in an explicit fashion the restrictions imposed by supersymmetry on the flux, and the values of the fixed moduli. Some T-duals and Heterotic duals are also discussed, these are non-Calabi-Yau spaces. A superpotential is constructed describing these duals.

Tadpole Analysis of Orientifolded Plane-Waves

We study orientifolds of type IIB string theory in the plane-wave background supported by null RR 3-form flux F^{(3)}. We describe how to extract the RR tadpoles in the Green-Schwarz formalism in a general setting. Two models with orientifold groups {1, \Omega} and {1,\Omega I_4}, which are T-dual to each other, are considered. Consistency of these backgrounds requires 32 D9 branes for the first model and 32 D5 branes for the second one. We study the spectra and comment on the heterotic duals of our models.

D = 4, N = 1, type IIB orientifolds

We study different aspects of the construction of D = 4, N = 1 type IIB orientifolds based on toroidal Z N and Z M × Z N, D = 4 ortifolds . We find that tadpole cancellation conditions are in general more constraining than in six dimensions and that the standard Gimon-Polchinski orientifold projection leads to the impossibility of tadpole cancellations in a number of Z N orientifolds with even N including Z 4, Z 8, Z 8′ and Z 12′ . We construct D = 4, Z N and Z N × Z M orientifolds with different configurations of 9-branes, 5-branes and 7-branes, most of them chiral. Models including the analogue of discrete torsion are constructed and shown to have features previously conjectured on the basis of F-theory compactified on four-folds. Different properties of the D = 4, N = 1 models obtained are discussed including their possible heterotic duals and effective low-energy action. These models have in general more than one anomalous U(1) and the anomalies are cancelled by a D = 4 generalized Green-Schwarz mechanism involving dilaton and moduli fields.

Four-dimensional chiral N = 1 type I vacua with and without D5-branes

In this paper we consider compactifications of type I strings on Abelian orbifolds. We discuss the tadpole cancellation conditions for the general case with D9-branes only. Such compactifications have (perturbative) heterotic duals which are also realized as orbifolds (with non-standard embedding of the gauge connection). The latter have extra twisted states that become massive once orbifold singularities are blown up. This is due to the presence of a perturbative heterotic superpotential with couplings between the extra twisted states, the orbifold blow-up modes, and (sometimes) untwisted matter fields. Anomalous U(1) (generically present in such models) also plays an important role in type 1-heterotic (tree-level) duality matching. We illustrate these issues on a particular example of Z 3 ⊗ Z 3 orbifold type I model (and its heterotic dual). The model has N = 1 supersymmetry, U(4) 3 ⊗ SO(8) gauge group, and chiral matter. We also consider compactifications of type I strings on Abelian orbifolds with both D9- and D5-branes. We discuss tadpole cancellation conditions for a certain class of such models. We illustrate the model building by considering a particular example of type I theory compactified on Z 6 orbifold. The model has N = 1 supersymmetry, [ U(6) ⊗ U(6) ⊗ U(4)] 2 gauge group, and chiral matter. This would correspond to a non-perturbative chiral vacuum from the heterotic point of view.

Non-perturbative heterotic D = 6, 4, N = 1 orbifold vacua

We consider D = 6, N = 1, Z M orbifold compactifications of heterotic strings in which the usual modular invariance constraints are violated. It is argued that in the presence of non-perturbative effects many of these vacua are nevertheless consistent. The perturbative massless sector can be computed explicitly from the perturbative mass formula subject to an extra shift in the vacuum energy. This shift is associated to a non-trivial antisymmetric B-field flux at the orbifold fixed points. The non-perturbative piece is given by five-branes either moving in the bulk or stuck at the fixed points, giving rise to Coulomb phases with tensor multiplets. The heterotic duals of some type IIB orientifolds belong to this class of orbifold models. We also discuss how to carry out this type of construction to the D = 4, N = 1 case and specific Z M × Z M examples are presented in which non-perturbative transitions changing the number of chiral generations do occur.

New branches of string compactifications and their F-theory duals

We study heterotic E 8 × E 8 models that are dual to compactifications of F-theory and type IIA string on certain classes of elliptically fibered Calabi-Yau manifolds. Different choices for the specific torus in the fibration have heterotic duals that are most easily understood in terms of E 8 × E 8 models with gauge backgrounds of type H × U(1) 8− d , where H is a non-Abelian factor. The case with d = 8 corresponds to the well-known E 8 × E 8 compactifications with non-Abelian instanton backgrounds ( k 1, k 2) whose F-theory duals are built through compactifications on fibrations of the torus P 2 (1,2,3) over F n . The new cases with d < 8 correspond to other choices for the elliptic fiber over the same base and yield unbroken U(1)′ s, some of which are anomalous and acquire a mass by swallowing zero-modes of the antisymmetric B MN field. We also study transitions to models with no tensor multiplets in D = 6 and find evidence of E d instanton dynamics. We also consider the possibility of conifold transitions among spaces with different realization of the elliptic fiber.