We consider an alternative derivation of the GSO Projection in the free fermionic construction of the weakly coupled heterotic string in terms of root systems, as well as the interpretation of the GSO Projection in this picture. We then present an algorithm to systematically and efficiently generate input sets (i.e. basis vectors) in order to study Landscape statistics with minimal computational cost. For example, the improvement at order 6 is approximately 10^{-13} over a traditional brute force approach, and improvement increases with order. We then consider an example of statistics on a relatively simple class of models.
A glossary overview of the phenomenological studies of quasi-realistic free fermionic heterotic string models is presented. I elaborate on the correspondence of these models with Z2XZ2 orbifolds.
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 $Z^\prime$ symmetry that may remain unbroken down to low energy scales. A generic signature of the low scale string derived $Z^\prime$ model is via di-photon 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…
Free fermionic models and symmetric heterotic toroidal orbifolds both constitute exact backgrounds that can be used effectively for phenomenological explorations within string theory. Even though it is widely believed that for Z2xZ2 orbifolds the two descriptions should be equivalent, a detailed dictionary between both formulations is still lacking. This paper aims to fill this gap: We give a detailed account of how the input data of both descriptions can be related to each other. In particular, we show that the generalized GSO phases of the free fermionic model correspond to generalized torsi…
Inspired by the tachyon-free non-supersymmetric heterotic SO(16)xSO(16) string we consider a special class of non-supersymmetric field theories: Those that can be obtained from supersymmetric field theories by supersymmetry breaking twists. We argue that such theories, like their supersymmetric counter parts, may still possess some fermionic symmetries as left-overs of the super gauge transformations and have special one-loop non-renormalization properties due to holomorphicity. In addition, we extend the supergraph techniques to these theories to calculate some explicit supersymmetry-breaking…
Using software under development at Baylor University, we explicitly construct all layer 1 gauge, weakly coupled free fermionic heterotic string models up to order 22 in four large space-time dimensions. The gauge models consist primarily of gauge content making a systematic construction process efficient. We present an overview of the model building procedure, redundancies in the process, methods used to reduce such redundancies and statistics regarding the occurrence of various combinations of gauge group factors and GUT groups. Statistics for both N=4 and N=0 models are presented.
I survey the parameter space of NAHE-based free fermionic heterotic string models. First, I discuss flat directions of the low energy effective field theories and show that D-flat directions need not be isomorphic to gauge invariant superpotential terms. Next, I review recent studies of three generation SU(3)_C x SU(2)_L x U(1)_Y, SU(3)_C x SU(2)_L x SU(2)_R x U(1)_B-L, SU(4)_C x SU(2)_L x U(1)_Y', and flipped SU(5) models, constructed by Wilson breaking of the underlying SO(10) observable sector gauge group of the NAHE Z_2 x Z_2 basis. Based on talk presented at the First International Confer…
We investigate the explicit form of the fermionic zeromodes in heterotic fivebrane backgrounds. By explicitly solving the fermionic field equations in fivebrane backgrounds, two normalizable and physical fermionic zeromodes are obtained. Each of these zeromodes has a non-vanishing gravitino component. We suggest a possible scenario of the gravitino pair condensation.
We discuss instanton fermionic zero modes in the heterotic N=(0,2) modification of the CP(1) sigma model in two dimensions. By calculating its chiral anomaly we prove that the number of fermionic zero modes is same as in the standard N=(2,2) CP(1) case, and determine their explicit form.
Free fermionic construction of four dimensional string vacua, are related to the Z2XZ2 orbifolds at special points in the moduli space, and yielded the most realistic three family string models to date. Using free fermionic construction techniques we are able to classify more than 10^10 string vacua by the net family and anti-family number. Using a montecarlo technique we find that a bell shaped distribution that peaks at vanishing net number of chiral families. We also observe that ~15% of the models have three net chiral families. We find that in addition to mirror symmetry that the distribu…
The three generation heterotic-string models in the free fermionic formulation are among the most realistic string vacua constructed to date, which motivated their detailed investigation. The classification of free fermion heterotic string vacua has revealed a duality under the exchange of spinor and vector representations of the SO(10) GUT symmetry over the space of models. We demonstrate the existence of the spinor-vector duality using orbifold techniques, and elaborate on the relation of these vacua to free fermionic models.
The free fermionic construction of the heterotic string in four dimensions produced a large space of three generation models with the underlying $SO(10)$ embedding of the Standard Model states. The $SO(10)$ symmetry is broken to a subgroup directly at the string scale. Over the past few years free fermionic models with the Pati-Salam and flipped $SU(5)$ subgroups have been classified. In this paper we extend this classification program to models in which the $SO(10)$ symmetry is broken at the string level to the $SU(4)\times SU(2)_L\times U(1)_R$ (SU421) subgroup. The subspace of free fermioni…
This thesis provides a classification of the chiral content of the heterotic $\mathbbm{Z}_2 \times \mathbbm{Z}_2$ orbifold models. We show that the chiral content of the heterotic $\mathbbm{Z}_2 \times \mathbbm{Z}_2$ orbifold models at any point in the moduli space can be described by a free fermionic model. We present a direct translation between the orbifold formulation and the free fermionic construction. We use the free fermionic description for the classification wherein we consider orbifolds with symmetric shifts. We show that perturbative three generation models are not obtained in the …
We classify N=1 orbifolds on the E_6 root lattice and investigate explicit model constructions on the Z_3xZ_3 orbifold in heterotic string theory. Interestingly some of the twisted sectors from the Z_3xZ_3 orbifold on the E_6 root lattice have just three fixed tori respectively, and generate three degenerate massless states. We also found three point functions with flavor mixing terms. We assume only non-standard gauge embeddings and find that they lead to three-family SU(5) and SO(10) GUT-like models. These models also include strongly coupled sectors in the low energy and messenger states ch…
We consider flux compactifications of heterotic string theory in the presence of fermionic condensates on M_{1,2} times X_7 with both factors carrying a Killing spinor. In other words, M_{1,2} is either de Sitter, anti-de Sitter or Minkowski, and X_7 possesses a nearly parallel G_2-structure or has G_2-holonomy. We solve the complete set of field equations and the Bianchi identity to order alpha'. The latter is satisfied via a non-standard embedding by choosing the gauge field to be a G_2-instanton. It is shown that none of the solutions to the field equations is supersymmetric.
The first particle physics observable whose origin may be sought in string theory is the triple replication of the matter generations. The class of Z2XZ2 orbifolds of six dimensional compactified tori, that have been most widely studied in the free fermionic formulation, correlate the family triplication with the existence of three twisted sectors in this class. In this work we seek an improved understanding of the geometrical origin of the three generation free fermionic models. Using fermionic and orbifold techniques we classify the Z2XZ2 orbifold with symmetric shifts on six dimensional com…
An overview of old and new results in studies of the quasi-realistic free fermionic models is presented, which include the recent discovery of exophobic string vacua and reproduction of the Higgs-matter splitting mechanism in a corresponding orbifold construction.
Classification of the N=1 space-time supersymmetric fermionic Z2XZ2 heterotic-string vacua with symmetric internal shifts, revealed a novel spinor-vector duality symmetry over the entire space of vacua, where the S_t <-> V duality interchanges the spinor plus anti-spinor representations with vector representations. In this paper we demonstrate that the spinor--vector duality exists also in fermionic Z2 heterotic string models, which preserve N=2 space-time supersymmetry. In this case the interchange is between spinorial and vectorial representations of the unbroken SO(12) GUT symmetry. We prov…
We study heterotic backgrounds with non-trivial $H$-flux and non-vanishing expectation values of fermionic bilinears, often referred to as gaugino condensates. The gaugini appear in the low energy action via the gauge-invariant three-form bilinear $Σ_{MNP}={\rm tr}\:\barχΓ_{MNP}χ$. For Calabi-Yau compactifications to four dimensions, the gaugino condensate corresponds to an internal three-form $Σ_{mnp}$ that must be a singlet of the holonomy group. This condition does not hold anymore when an internal $H$-flux is turned on and ${\cal O}(α')$ effects are included. In this paper we study flux co…
We continue the classification of the fermionic Z2XZ2 heterotic string vacua with symmetric internal shifts. The space of models is spanned by working with a fixed set of boundary condition basis vectors and by varying the sets of independent Generalized GSO (GGSO) projection coefficients (discrete torsion). This includes the Calabi-Yau like compactifications with (2,2) world-sheet superconformal symmetry, as well as more general vacua with only (2,0) superconformal symmetry. In contrast to our earlier classification that utilized a montecarlo technique to generate random sets of GGSO phases, …
The evidence for neutrino masses in atmospheric and solar neutrino experiments provides further support for the embedding of the Standard Model fermions in the chiral 16 SO(10) representation. Such an embedding is afforded by the realistic free fermionic heterotic-string models. In this paper we advance the study of these string models toward a non-perturbative analysis by generalizing the work of Donagi, Pantev, Ovrut and Waldram from the case of G=SU(2n+1) to G=SU(2n) stable holomorphic vector bundles on elliptically fibered Calabi-Yau manifolds with fundamental group Z2. We demonstrate exis…
Recently it was demonstrated that free fermionic heterotic-strings can produce models with solely the Minimal Supersymmetric Standard Model states in the low energy spectrum. This unprecedented result provides further strong evidence for the possibility that the true string vacuum shares some of the properties of the free fermionic models. Past free fermionic models have focused on several possible unbroken observable SO(10) subgroups at the string scale, which include the flipped SU(5) (FSU5), the Pati-Salam (PS) string models, and the string Standard-like Models (SLM). We extend this study t…
Classification of Left-Right Symmetric (LRS) heterotic-string vacua in the free fermionic formulation, using random generation of generalised GSO (GGSO) projection coefficients, produced phenomenologically viable models with probability $4\times 10^{-11}$. Extracting substantial number of phenomenologically viable models requires modification of the classification method. This is achieved by identifying phenomenologically amenable conditions on the Generalised GSO projection coefficients that are randomly generated at the $SO(10)$ level. Around each of these fertile cores we perform a complete…
To assist in the search for phenomenologically realistic models in the string landscape, we must develop tools for investigating all gauge charges, including U(1) charges, in string models. We introduce the process for constructing fermionic string models and present an algorithm for determining the U(1) gauge states and U(1) charges in weakly-coupled free fermionic heterotic string (WCFFHS) models given their matter and non-Abelian gauge content. We determine the computational complexity of this algorithm and discuss a particular implementation that can be used in conjunction with a framework…
Using the RNS-like fermionic vector variables introduced in arXiv:1305.0693, the pure spinor $b$ ghost in a curved heterotic superstring background is easily constructed. This construction simplifies and completes the $b$ ghost construction in a curved background of arXiv:1311.7012.
We extend the classification of fermionic $\mathbb{Z}_2\times\mathbb{Z}_2$ heterotic string orbifolds to non--supersymmetric Pati--Salam (PS) models in two classes of vacua, that we dub $\tilde S$--models and $S$--models. The first correspond to compactifications of a tachyonic ten--dimensional vacuum, whereas the second correspond to compactifications of the ten--dimensional tachyon--free $SO(16)\times SO(16)$ heterotic string. In both cases we develop a systematic method to extract tachyon--free four--dimensional models. We show that tachyon--free configurations arise with probability $\sim0…
The existence of an extra Z' inspired from heterotic-string theory at accessible energy scales attracted considerable interest in the particle physics literature. Surprisingly, however, the construction of heterotic--string derived models that allow for an extra Z' to remain unbroken down to low scales has proven to be very difficult. The main reason being that the U(1) symmetries that are typically discussed in the literature are either anomalous or have to be broken at a high scale to generate light neutrino masses. In this paper we use for that purpose the self duality property under the sp…
There is good support for the embedding of the Standard Model fermions in the chiral 16 representation of SO(10). Such an embedding is provided by the realistic free fermionic heterotic-string models. In this talk we demonstrate the existence of solutions with 3 generations and SO(10) observable gauge group, in the case of compactification on a torus-fibred Calabi-Yau space over a Hirzebruch base surface. The SO(10)symmetry is broken to SU(5)xU(1) by a Wilson line. The overlap with the realistic free fermionic heterotic-string models is discussed.
We consider the heterotic E8 X E8 string theory, which gives rise to four-dimensional Standard-like Models and allows for their SO(10) embedding. We investigate two different schemes of compactification: the free fermionic formulation and the orbifold construction. In the examples presented in the free fermionic formulation we explore the removal of the extra Higgs representations by using the free fermion boundary conditions directly at the string level, rather than in the effective low energy field theory. Moreover, by employing the standard analysis of flat directions we present a quasi-rea…
The gauging of isometries in general sigma-models which include fermionic terms which represent the interaction of strings with background Yang-Mills fields is considered. Gauging is possible only if certain obstructions are absent. The quantum gauge anomaly is discussed, and the (1,0) supersymmetric generalisation of the gauged action given.
We extend the classification of free fermionic heterotic-string models to vacua in which the SO(10) GUT symmetry is broken at the string level to the Pati-Salam subgroup. Using our classification method we recently presented the first example of a quasi--realistic heterotic-string vacuum that is free of massless exotic states. Within this method we are able to derive algebraic expressions for the generalised GSO projections for all sectors that appear in the models. This facilitates the programming of the entire spectrum analysis in a computer code. The total number of vacua in the class of mo…
The heterotic-string models in the free fermionic formulation are among the most realistic string vacua constructed to date, which motivates their detailed exploration. Classification of free fermionic heterotic-string vacua revealed a duality under exchange of spinor and vector representations of the SO(10) GUT symmetry. The spinor-vector duality was subsequently demonstrated in a Z2 X Z2 X Z2 orbifold, in which the map is realised as exchange of discrete torsions. Analysis of the orbifold partition function shows that the duality map preserves the number of massless twisted states. While the…
Three generation heterotic-string vacua in the free fermionic formulation gave rise to models with solely the MSSM states in the observable Standard Model charged sector. The relation of these models to Z_2 x Z_2 orbifold compactifications dictates that they produce three pairs of untwisted Higgs multiplets. The reduction to one pair relies on the analysis of supersymmetric flat directions, that give superheavy mass to the dispensable Higgs states. We explore the removal of the extra Higgs representations by using the free fermion boundary conditions and hence directly at the string level, rat…
The main aspects of a recently developed method for classification of heterotic superstring vacua in the Free Fermionic Formulation are presented. Using these techniques we classify a big number of approximately $10^{15}$ heterotic string vacua with Pati--Salam, SU(4)x{SU(2)}_Lx{SU(2)}_R gauge symmetry with respect to their main phenomenological features as the number of families, Pati--Salam breaking Higgs, Standard Model Higgs doublets, additional triplets and exotic charge states. We identify an interesting subclass of these vacua, approximately one to one million, whose massless spectrum i…
Using Baylor University's C++ software for construction of weakly coupled free fermionic heterotic string models, called the FF Framework, we explicitly construct the level 1 Kač-Moody ten dimensional heterotic string models with the aim of understanding the redundancies endemic to this construction method. We show that for models in any even number of large space-time dimensions with a massless left mover and an odd ordered right mover, the maximal number of space-time supersymmetries are present. We show that in order to produce all of the models for a given order, different basis vectors mu…
We discuss non-geometric supersymmetric heterotic string models in D=4, in the framework of the free fermionic construction. We perform a systematic scan of models with four a priori left-right asymmetric Z_2 projections and shifts. We analyze some 2^{20} models, identifying 18 inequivalent classes and addressing variants generated by discrete torsions. They do not contain geometrical or trivial neutral moduli, apart from the dilaton. However, we show the existence of flat directions in the form of exactly marginal deformations and identify patterns of symmetry breaking where product gauge gro…
We discuss toroidal orbifolds of the E8XE8 heterotic string, in which the free fermionic Higgs-matter splitting is implemented by a shift in the internal lattice coupled with the fermion numbers of the gauge degrees of freedom. We consider models in which some choices of the orbifold result in the projection of the graviton. In the models that we consider the projection also results in flipping the spin-statistics assignments in the massive string spectrum, whereas the massless spectrum retains the conventional spin-statistics assignments. We argue that the partition functions are mathematical…
The classification method of the free fermionic heterotic string vacua is extended to models where the $SO(10)$ GUT symmetry is broken directly at the string scale to the Left-Right Symmetric subgroup. Our method involves using a fixed set of basis vectors which are defined by the boundary conditions assigned to the free fermions before enumerating the string vacua by varying the Generalised GSO (GGSO) projection coefficients. It allows the derivation of algebraic expressions for the GGSO projections for each sector that generates massless states in the models. This enables a computerised anal…
Recently it was proposed that ten-dimensional tachyonic string vacua may serve as starting points for the construction of viable four dimensional phenomenological string models which are tachyon free. This is achieved by projecting out the tachyons in the four-dimensional models using projectors other than the projector which is utilised in the supersymmetric models and those of the $SO(16)\times SO(16)$ heterotic string. We continue the exploration of this class of models by developing systematic computerised tools for their classification, the analysis of their tachyonic and massless spectra…
The heterotic string free fermionic formulation produced a large class of three generation models, with an underlying SO(10) GUT symmetry which is broken directly at the string level by Wilson lines. A common subset of boundary condition basis vectors in these models is the NAHE set, which corresponds to Z2 X Z2 orbifold of an SO(12) Narain lattice, with (h11,h21)=(27,3). Alternatively, a manifold with the same data is obtained by starting with a Z2 X Z2 orbifold at a generic point on the lattice, with (h11,h21)=(51,3), and adding a freely acting Z2 involution. The equivalence of the two const…
Consistent heterotic free fermionic string models are classified in terms of their number of spacetime supersymmetries, N. For each of the six distinct choices of gravitino sector, we determine what number of supersymmetries can survive additional GSO projections. We prove by exhaustive search that only three of the six can yield N = 1, in addition to the N = 4, 2, or 0 that five of the six can yield. One choice of gravitino sector can only produce N = 4 or 0. Relatedly, we find that only Z_2, Z_4, and Z_8 twists of the internal fermions with worldsheet supersymmetry are consistent with N=1 in…
We extend the classification of the free fermionic heterotic-string vacua to models in which the SO(10) GUT symmetry at the string scale is broken to the flipped SU(5) subgroup. In our classification method, the set of basis vectors defined by the boundary conditions which are assigned to the free fermions is fixed and the enumeration of the string vacua is obtained in terms of the Generalised GSO (GGSO) projection coefficients entering the one-loop partition function. We derive algebraic expressions for the GGSO projections for all the physical states appearing in the sectors generated by the…
The only string models known to reproduce the Minimal Supersymmetric Standard Model in the low energy effective field theory are those constructed in the free fermionic formulation. We demonstrate the existence of quasi-realistic free fermionic heterotic-string models in which supersymmetric singlet flat directions do not exist. This raises the possibility that supersymmetry is broken perturbatively in such models by the one-loop Fayet-Iliopoulos term. We show, however, that supersymmetric flat directions that utilize VEVs of some non-Abelian fields in the massless string spectrum do exist in …
I propose that self-duality in quantum phase-space provides the criteria for the selection of the quantum gravity vacuum. The evidence for this assertion arises from two independent considerations. The first is the phenomenological success of the free fermionic heterotic-string models, which are constructed in the vicinity of the self-dual point under T-duality. The relation between the free fermionic models and the underlying Z2 X Z2 toroidal orbifolds is discussed. Recent analysis revealed that the Z2 X Z2 free fermionic orbifolds utilize an asymmetric shift in the reduction to three generat…
We show that in the absence of a Ramond-Ramond sector both the type IIA and type IIB free string gases have a thermal instability due to low temperature tachyon modes. The gas of free IIA strings undergoes a thermal duality transition into a gas of free IIB strings at the self-dual temperature. The free heterotic string gas is a tachyon-free ensemble with gauge symmetry SO(16)$\times$SO(16) in the presence of a timelike Wilson line background. It exhibits a holographic duality relation undergoing a self-dual phase transition with positive free energy and positive specific heat. The type IB ope…
We describe a method for constructing genuinely asymmetric (2,0) heterotic strings out of N=2 minimal models in the fermionic sector, whereas the bosonic sector is only partly build out of N=2 minimal models. This is achieved by replacing one minimal model plus the superfluous E_8 factor by a non-supersymmetric CFT with identical modular properties. This CFT generically lifts the weights in the bosonic sector, giving rise to a spectrum with fewer massless states. We identify more than 30 such lifts, and we expect many more to exist. This yields more than 450 different combinations. Remarkably,…
In the context of the free-fermionic formulation of the heterotic superstring, we construct a three generation N=1 supersymmetric SU(4)xSU(2)LxSU(2)R model supplemented by an SU(8) hidden gauge symmetry and five Abelian factors. The symmetry breaking to the standard model is achieved using vacuum expectation values of a Higgs pair in (4bar,2R)+(4,2R) at a high scale. One linear combination of the Abelian symmetries is anomalous and is broken by vacuum expectation values of singlet fields along the flat directions of the superpotential. All consistent string vacua of the model are completely cl…
With the apparent discovery of the Higgs boson, the Standard Model has been confirmed as the theory accounting for all sub-atomic phenomena. This observation lends further credence to the perturbative unification in Grand Unified Theories (GUTs) and string theories. The free fermionic formalism yielded fertile ground for the construction of quasi--realistic heterotic--string models, which correspond to toroidal Z2xZ2 orbifold compactifications. In this paper we study a new class of heterotic-string models in which the GUT group is SU(6)xSU(2) at the string level. We use our recently developed …
Super-BMS$_4$ algebras -- also called BMS$_4$ superalgebras -- are graded extensions of the BMS$_4$ algebra. They can be of two different types: they can contain either a finite number or an infinite number of fermionic generators. We show in this letter that, with suitable boundary conditions on the graviton and gravitino fields at spatial infinity, supergravity on asymptotically flat spaces possesses as superalgebra of asymptotic symmetries a (nonlinear) super-BMS$_4$ algebra containing an infinite number of fermionic generators, which we denote SBMS$_4$. These boundary conditions are not on…
We study the cosmology of a string derived supersymmetric flipped $SU(5)$ model in the context of free-fermionic heterotic constructions that allow full calculability of the effective supergravity in perturbation theory around the fermionic vacuum where all string moduli have fixed values. The model has 3 generations of chiral families and a Higgs sector leading to particle phenomenology consistent with low energy data, that has been extensively studied in the past. Here, we show that it can also accommodate a novel successful cosmology, based on the no-scale effective supergravity derived fro…
Imprints of spinor-vector dualities have been uncovered in various string constructions. They are typically induced by changing certain free general GSO phases in the underlying string partition functions. This paper shows that spinor-vector dualities in six dimensions are constrained by a fundamental effective field theory consistency condition, namely that any six dimensional low energy theory must be free of irreducible SO(2N) anomalies. Aspects of spinor-vector dualities are analysed in four six-dimensional free fermionic models which are distinguished by two generalised GSO phases. In add…
We construct supergravity backgrounds for the integrable eta-deformations of the AdS2 x S2 x T6 and AdS5 x S5 superstring sigma models. The eta-deformation is governed by an R-matrix that solves the non-split modified classical Yang-Baxter equation on the superisometry algebra of the model. Such R-matrices include those of Drinfel'd-Jimbo type, which are constructed from a Dynkin diagram and the associated Cartan-Weyl basis. Drinfel'd-Jimbo R-matrices associated with inequivalent bases will typically lead to different deformed backgrounds. For the two models under consideration we find that th…
We investigate the existence of string vacua with Massive Spectrum Degeneracy Symmetry (MSDS) in Heterotic and Type II orbifold constructions. We present a classification of all possible Z_2^N-orbifolds with MSDS symmetry that can be constructed in the formalism of the 2d free fermionic construction. We explicitly construct several two-dimensional models whose Reduced Massive Spectrum Degeneracy Symmetry (RMSDS) is due to a set of Z_2-orbifold projections induced naturally in the framework of the free fermionic construction. In all proposed models the massive boson and fermion degrees of freed…
We introduce a new a family of $Z(N)$ multispins quantum chains with a free-fermionic ($N=2$) or free-parafermionic ($N>2$) eigenspectrum. The models have $(p+1)$ interacting spins ($p=1,2,\dots$), being Hermitian in the $Z(2)$ (Ising) case and non-Hermitian for $N>2$. We construct a set of mutually commuting charges that allows us to derive the eigenenergies in terms of the roots of polynomials generated by a recurrence relation of order $(p+1)$. In the critical limit we identify these polynomials with certain hypergeometric polynomials ${}_{p+1}F_p$. Also in the critical regime, we calculate…
