Gepner Models Research Articles

Conformal perturbation theory on K3: the quartic Gepner point

The Gepner model (2)4 describes the sigma model of the Fermat quartic K3 surface. Moving through the nearby moduli space using conformal perturbation theory, we investigate how the conformal weights of its fields change at first and second order and find approximate minima. This serves as a toy model for a mechanism that could produce new chiral fields and possibly new nearby rational CFTs.

On quantum Poisson-Lie T-duality of WZNW models

We study Poisson-Lie T-duality of the Wess-Zumino-Novikov-Witten (WZNW) models which are obtained from a class of Drinfel’d doubles and its generalization. In this case, the resultant WZNW models are known to be classically self-dual under Poisson-Lie T-duality. We describe an explicit construction of the associated currents, and discuss the conformal invariance under this duality. In a concrete example of the SU(2) WZNW model, we find that the self-duality is represented as a chiral automorphism of the sû\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\hat{\\mathfrak{su}} $$\\end{document}(2) affine Lie algebra, though the transformation of the currents is non-local and non-linear. This classical automorphism can be promoted to the quantum one through the parafermionic formulation of sû\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\hat{\\mathfrak{su}} $$\\end{document}(2), which in turn induces an isomorphism of the WZNW model. We thus find a full quantum equivalence of the dual pair under Poisson-Lie T-duality. The isomorphism is represented by a sign-change of a chiral boson or the order-disorder duality of the parafermionic conformal field theory as in Abelian T-duality on tori or in the mirror symmetry of the Gepner model.

Mirror symmetry and new approach to constructing orbifolds of Gepner models

Motivated by the principles of the conformal bootstrap, primarily the principle of Locality, simultaneously with the requirement of space-time supersymmetry, we reconsider constructions of compactified superstring models. Starting from requirements of space-time supersymmetry and mutual locality, we construct a complete set of physical fields of orbifolds of Gepner models. To technically implement this, we use spectral flow generators to construct all physical fields from the chiral primary fields. The set of these spectral flow operators forms a so-called admissible group Gadm, which defines a given orbifold. The action of these operators produces a collection of physical fields consistent with the action of supersymmetry generators. The selection of mutually local fields from this collection is carried out using the mirror group Gadm⁎. The permutation of Gadm and Gadm⁎ replaces the original orbifold with a mirror one that satisfies the same conditions as the original one. This also implies that the resulting model is modular invariant.

Non-SUSY heterotic string vacua of Gepner models with vanishing cosmological constant

Abstract We study a natural generalization of the results given in K. Aoyama and Y. Sugawara, Prog. Theor. Exp. Phys. 2020, 103B01 (2020) to heterotic strings. Namely, starting from the generic Gepner models for Calabi –Yau three-folds, we construct non-SUSY heterotic string vacua with vanishing cosmological constant at the one-loop level. We especially focus on asymmetric orbifolding based on some discrete subgroup of the chiral $U(1)$ action which acts on both the Gepner model and the $SO(32)$ or $E_8\times E_8$ sector. We present a classification of the relevant orbifold models leading to the string vacua with the properties mentioned above. In some cases, the desired vacua can be constructed in a manner quite similar to those given in the previous paper for the type II string, in which the orbifold groups contain two generators with discrete torsions. On the other hand, we also have simpler models that are just realized as asymmetric orbifolds of cyclic groups with only one generator.

Non-SUSY Gepner models with vanishing cosmological constant

Abstract In this article we discuss a construction of non-SUSY type II string vacua with the vanishing cosmological constant at the one-loop level based on the generic Gepner models for Calabi–Yau 3-folds. We make an orbifolding of the Gepner models by $\Bbb Z_2 \times \Bbb Z_4$, which asymmetrically acts with some discrete torsions incorporated. We demonstrate that the obtained type II string vacua indeed lead to the vanishing cosmological constant at the one-loop level, whereas any space-time supercharges cannot be constructed as long as we assume chiral forms such as “$ \displaystyle {\cal Q}^{\alpha}_L \equiv \oint dz\, {\cal J}_L^{\alpha}(z) $”. We further discuss possible generalizations of the models described above.

Supersymmetric indices on I × T2, elliptic genera and dualities with boundaries

We study three dimensional N=2 supersymmetric theories on I×M2 with 2d N=(0,2) boundary conditions at the boundaries ∂(I×M2)=M2⊔M2, where M2=C or T2. We introduce supersymmetric indices of three dimensional N=2 theories on I×T2 that couple to elliptic genera of 2d N=(0,2) theories at the two boundaries. We evaluate the I×T2 indices in terms of supersymmetric localization and study dualities on the I×M2. We consider the dimensional reduction of I×T2 to I×S1 and obtain the localization formula of 2d N=(2,2) supersymmetric indices on I×S1. We illustrate computations of open string Witten indices based on gauged linear sigma models. Correlation functions of Wilson loops on I×S1 agree with Euler pairings in the geometric phase and also agree with cylinder amplitudes for B-type boundary states of Gepner models in the Landau-Ginzburg phase.

Heterotic/type II duality and non-geometric compactifications

We present a new class of dualities relating non-geometric Calabi-Yau com- pactifications of type II string theory to T-fold compactifications of the heterotic string, both preserving four-dimensional mathcal{N} = 2 supersymmetry. The non-geometric Calabi-Yau space is a K 3 fibration over T2 with non-geometric monodromies in the duality group O(Γ4,20); this is dual to a heterotic reduction on a T4 fibration over T2 with the O(Γ4,20) monodromies now viewed as heterotic T-dualities. At a point in moduli space which is a minimum of the scalar potential, the type II compactification becomes an asymmetric Gepner model and the monodromies become automorphisms involving mirror symmetries, while the heterotic dual is an asymmetric toroidal orbifold. We generalise previous constructions to ones in which the automorphisms are not of prime order. The type II construction is perturbatively consistent, but the naive heterotic dual is not modular invariant. Modular invariance on the heterotic side is achieved by including twists in the circles dual to the winding numbers round the T2, and this in turn introduces non-perturbative phases depending on NS5-brane charge in the type II construction.

Calabi-Yau manifolds and sporadic groups

A few years ago a connection between the elliptic genus of the K3 manifold and the largest Mathieu group M24 was proposed. We study the elliptic genera for Calabi-Yau manifolds of larger dimensions and discuss potential connections between the expansion coefficients of these elliptic genera and sporadic groups. While the Calabi-Yau 3-fold case is rather uninteresting, the elliptic genera of certain Calabi-Yau d-folds for d > 3 have expansions that could potentially arise from underlying sporadic symmetry groups. We explore such potential connections by calculating twined elliptic genera for a large number of Calabi-Yau 5-folds that are hypersurfaces in weighted projected spaces, for a toroidal orbifold and two Gepner models.

Trisecting non-Lagrangian theories

We propose a way to define and compute invariants of general smooth 4-manifolds based on topological twists of non-Lagrangian 4d mathcal{N}=2kern0.5em mathrm{and}kern0.5em mathcal{N}=3 theories in which the problem is reduced to a fairly standard computation in topological A-model, albeit with rather unusual targets, such as compact and non-compact Gepner models, asymmetric orbifolds, mathcal{N}=left(2,2right) linear dilaton theories, “self-mirror” geometries, varieties with complex multiplication, etc.

Non-geometric Calabi-Yau backgrounds and K3 automorphisms

We consider compactifications of type IIA superstring theory on mirror-folds obtained as K3 fibrations over two-tori with non-geometric monodromies involving mirror symmetries. At special points in the moduli space these are asymmetric Gepner models. The compactifications are constructed from non-geometric automorphisms that arise from the diagonal action of an automorphism of the K3 surface and of an automorphism of the mirror surface. We identify the corresponding gaugings of mathcal{N} = 4 supergravity in four dimensions, and show that the minima of the potential describe the same four-dimensional low-energy physics as the worldsheet formulation in terms of asymmetric Gepner models. In this way, we obtain a class of Minkowski vacua of type II string theory which preserve mathcal{N} = 2 supersymmetry. The massless sector consists of mathcal{N} = 2 supergravity coupled to 3 vector multiplets, giving the STU model. In some cases there are additional massless hypermultiplets.

The asymmetric CFT landscape in with extended supersymmetry

We study asymmetric simple‐current extensions of Gepner models in dimensions with at least eight supercharges in the right‐moving sector. The models obtained in an extensive stochastic computer search belong to a small number of different classes. These classes can be categorized as dimensional reductions, asymmetric orbifolds with , extra gauge enhancement and as coming from the super Higgs‐effect. Models in the latter class are particularly interesting, as they may correspond to non‐geometric flux compactifications.

Partial SUSY breaking for asymmetric Gepner models and non-geometric flux vacua

Using the method of simple current extensions, asymmetric Gepner models of Type IIB with mathcal{N} = 1 space-time supersymmetry are constructed. The combinatorics of the massless vector fields suggests that these classical Minkowski string vacua provide fully backreacted solutions corresponding to mathcal{N} = 1 minima of mathcal{N} = 2 gauged supergravity. The latter contain abelian gaugings along the axionic isometries in the hypermultiplet moduli space, and can be considered as Type IIB flux compactifications on Calabi-Yau manifolds equipped with (non-)geometric fluxes. For a particular class of asymmetric Gepner models, we are able to explicitly specify the underlying CICYs and to check necessary conditions for a GSUGRA interpretation. If this conjecture is correct, there exists a large class of exactly solvable non-geometric flux compactifications on CY threefolds.

Tracing symmetries and their breakdown through phases of heterotic (2,2) compactifications

We are considering the class of heterotic $\mathcal{N}=(2,2)$ Landau-Ginzburg orbifolds with 9 fields corresponding to $A_1^9$ Gepner models. We classify all of its Abelian discrete quotients and obtain 152 inequivalent models closed under mirror symmetry with $\mathcal{N}=1,2$ and $4$ supersymmetry in 4D. We compute the full massless matter spectrum at the Fermat locus and find a universal relation satisfied by all models. In addition we give prescriptions of how to compute all quantum numbers of the 4D states including their discrete R-symmetries. Using mirror symmetry of rigid geometries we describe orbifold and smooth Calabi-Yau phases as deformations away from the Landau-Ginzburg Fermat locus in two explicit examples. We match the non-Fermat deformations to the 4D Higgs mechanism and study the conservation of R-symmetries. The first example is a $\mathbb{Z}_3$ orbifold on an E$_6$ lattice where the R-symmetry is preserved. Due to a permutation symmetry of blow-up and torus K\"{a}hler parameters the R-symmetry stays conserved also smooth Calabi-Yau phase. In the second example the R-symmetry gets broken once we deform to the geometric $\mathbb{Z}_3 \times \mathbb{Z}_{3,\text{free}}$ orbifold regime.

Asymmetric Gepner models in type II

We describe new four-dimensional type II compactifications with N=2 supersymmetry, based on asymmetric Gepner models for K3 x T^2. In more than half of these models, all the K3 moduli are lifted, giving at low energies N=2 supergravity with the STU vector multiplets and no hypermultiplets.

MOTIVIC L-FUNCTION IDENTITIES FROM CFT AND ARITHMETIC MIRROR SYMMETRY

Exactly solvable mirror pairs of Calabi–Yau threefolds of hypersurface type exist in the class of Gepner models that include nondiagonal affine invariants. Motivated by the string theoretic automorphy established previously for models in this class, it is natural to ask whether the arithmetic structure of mirror pairs varieties reflects the fact that as conformal field theories, they are isomorphic. Mirror symmetry in particular predicts that the L-functions of the Ω-motives of such pairs are identical. In this paper this prediction is confirmed by showing that the Ω-motives of exactly solvable mirror pairs are isomorphic. This follows as a corollary of the proof of a more general result establishing an isomorphism between nondiagonally and diagonally induced motives in this class of varieties. The motivic approach formulated here circumvents the difficulty that no mirror construction of the Hasse–Weil zeta function is known.

Discrete gauge symmetries in discrete MSSM-like orientifolds

Motivated by the necessity of discrete ZN symmetries in the MSSM to insure baryon stability, we study the origin of discrete gauge symmetries from open string sector U(1)ʼs in orientifolds based on rational conformal field theory. By means of an explicit construction, we find an integral basis for the couplings of axions and U(1) factors for all simple current MIPFs and orientifolds of all 168 Gepner models, a total of 32 990 distinct cases. We discuss how the presence of discrete symmetries surviving as a subgroup of broken U(1)ʼs can be derived using this basis. We apply this procedure to models with MSSM chiral spectrum, concretely to all known U(3)×U(2)×U(1)×U(1) and U(3)×Sp(2)×U(1)×U(1) configurations with chiral bi-fundamentals, but no chiral tensors, as well as some SU(5) GUT models. We find examples of models with Z2 (R-parity) and Z3 symmetries that forbid certain B and/or L violating MSSM couplings. Their presence is however relatively rare, at the level of a few percent of all cases.

Elliptic genera of non-compact Gepner models and mirror symmetry

Abstract We consider tensor products of N = 2 minimal models and non-compact conformal field theories with N = 2 superconformal symmetry, and their orbifolds. The elliptic genera of these models give rise to a large and interesting class of real Jacobi forms. The tensor product of conformal field theories leads to a natural product on the space of completed mock modular forms. We exhibit families of non-compact mirror pairs of orbifold models with c = 9 and show explicitly the equality of elliptic genera, including contributions from the long multiplet sector. The Liouville and cigar deformed elliptic genera transform into each other under the mirror transformation.

Simple current extensions and the permutation orbifold

We review extensions by integer spin simple currents in two-dimensional conformal field theories and their applications in string theory. In particular, we study the problem of resolving the fixed points of a simple current and apply the formalism to the permutation orbifold. In terms of string compactifications, we construct permutations of N=2 minimal models and use them as building blocks in heterotic Gepner models.

Permutation orbifolds of heterotic Gepner models

We study orbifolds by permutations of two identical N = 2 minimal models within the Gepner construction of four-dimensional heterotic strings. This is done using the new N = 2 supersymmetric permutation orbifold building blocks we have recently developed. We compare our results with the old method of modding out the full string partition function. The overlap between these two approaches is surprisingly small, but whenever a comparison can be made we find complete agreement. The use of permutation building blocks allows us to use the complete arsenal of simple current techniques that is available for standard Gepner models, vastly extending what could previously be done for permutation orbifolds. In particular, we consider ( 0 , 2 ) models, breaking of SO ( 10 ) to subgroups, weight-lifting for the minimal models and B-L lifting. Some previously observed phenomena, for example concerning family number quantization, extend to this new class as well, and in the lifted models three-family models occur with abundance comparable to two or four.

Asymmetric Gepner models III. B-L lifting

In the same spirit as heterotic weight lifting, B-L lifting is a way of replacing the superfluous and ubiquitous U ( 1 ) B – L with something else with the same modular properties, but different conformal weights and ground state dimensions. This method works in principle for all variants of ( 2 , 2 ) constructions, such as orbifolds, Calabi–Yau manifolds, free bosons and fermions and Gepner models, since it only modifies the universal SO ( 10 ) × E 8 part of the CFT. However, it can only yield chiral spectra if the “internal” sector of the theory provides a simple current of order 5. Here we apply this new method to Gepner models. Including exceptional invariants, 86 of them have the required order 5 simple current, and 69 of these yield chiral spectra. Three family spectra occur abundantly.