1 Introduction
Abstract

The observation of a scalar resonance at the LHC, compatible with perturbative electroweak symmetry breaking, reinforces the Standard Model parameterisation of all subatomic data. The logarithmic evolution of the SM gauge and matter parameters suggests that this parameterisation remains viable up to the Planck scale, where gravitational effects are of comparable strength. String theory provides a perturbatively consistent scheme to explore how the parameters of the Standard Model may be determined from a theory of quantum gravity. The free fermionic heterotic string models provide concrete examples of exact string solutions that reproduce the spectrum of the Minimal Supersymmetric Standard Model. Contemporary studies entail the development of methods to classify large classes of models. This led to the discovery of exophobic heterotic–string vacua and the observation of spinor–vector duality, which provides an insight to the global structure of the space of (2,0) heterotic–string vacua. Future directions entail the study of the role of the massive string states in these models and their incorporation in cosmological scenarios. A complementary direction is the formulation of quantum gravity from the principle of manifest phase space duality and the equivalence postulate of quantum mechanics, which suggest that space is compact. The compactness of space, which implies intrinsic regularisation, may be tightly related to the intrinsic finite length scale, implied by string phenomenology.

unification; string phenomenology; particle physics; quantum gravity
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x \doinum10.3390/—— \pubvolume2 2014 \historyReceived: 28 February 2014; in revised form: 10 April 2014 / Accepted: 15 April 2014 /
Published: 2014 \TitleString Phenomenology: Past, Present and Future Perspectives \AuthorAlon E. Faraggi* \corresalon.faraggi@liv.ac.uk.

1 Introduction

The experimental observation of a scalar resonance by the ATLAS atlas and CMS cms experiments of the Large Hadron Collider at CERN, compatible with the scalar particle of the Standard Electroweak Model weinberg , is a pivotal moment in the quest for the unification of the fundamental theories of matter and interactions. Indeed, nearly thirty years have elapsed since the experimental discovery of the and –vector bosons wpmzdiscovery ; wpmzdiscovery2 , and forty years since the demonstration of renormalizability of spontaneously broken non–Abelian gauge symmetries thooft , which were the earlier milestones on this journey. The discovery of the Higgs boson solidifies the Standard Model parameterisation of all subatomic experimental observations to date. The observation of a Higgs boson at 125GeV suggests that the electroweak symmetry breaking mechanism is perturbative, rather than nonperturbative. This reinforces the view that the Standard Model provides a viable perturbative parameterisation of the subatomic interactions up to an energy scale, which is separated by orders of magnitude from the scale within reach of contemporary accelerator experiments. If this is indeed the scenario selected by nature, it entails that alternative experimental tests will be required to establish its validity. These tests will unavoidably look for astrophysical and cosmological imprints, that can probe the much higher energy scales.

The possibility that the Standard Model provides a viable effective parameterisation, up to a much higher scale, has been entertained in the context of Grand Unified Theories (GUTs) and string theories ibanezuranga . The gauge charges of the Standard Model matter states are strongly suggestive of the embedding of the Standard Model states in representations of larger gauge groups. This is most striking in the context of GUT, in which each of the Standard Model chiral generations fits into a spinorial representation of . The gauge charges of the Standard Model matter states are experimental observables. The Standard Model contains three generations, which are split into six multiplets that are charged under its three gauge sectors. Therefore, in the framework of the Standard Model one needs fifty–four parameters to account for these gauge charges. Embedding the Standard Model in reduces this number of parameters to one parameter, which is the number of spinorial representations of needed to accommodate the Standard Model spectrum. Additional evidence for the high scale unification stems from:

  • the logarithmic running of the Standard Model parameters, which is compatible with observations in the gauge sectors cwogcu and the heavy generation Yukawa couplings cwohms . Logarithmic running in the scalar sector is spoiled by radiative corrections from the Standard Model cutoff scale. Restoration of the logarithmic running mandates the existence of a new symmetry. Supersymmetry is a concrete example that fulfils the task. The observation of a scalar resonance at 125GeV, and the fact that no other particles have been observed up to the multi–TeV energy scale, indicates that the resonance is a fundamental scalar rather than a composite state nussinov . This outcome agrees with the Higgs states in heterotic–string vacua.

  • Further evidence for the validity of the renormalizable Standard Model up to a very high energy scale stems from the suppression of proton decay mediating operators. The Standard Model should be regarded as providing a viable effective parametrisation, but not as a fundamental accounting of the observable phenomena. The reason being in that it does not provide a complete description. Obviously, gravitational effects are not accounted for. Moreover, the Standard Model itself is not mathematically self–consistent. It gives rise to singularities in the ultraviolet limit. For these reasons the Standard Model can only be regarded as an effective theory below some cutoff. A plausible cutoff is the Planck scale, at which the gravitational coupling is of comparable strength to the gauge couplings. The renormalizability of the Standard Model is not valid beyond its cutoff scale. Nonrenormalizable operators are induced by whatever theory extends the Standard Model at and beyond the cutoff scale. We should therefore take into account all the nonrenormalizable operators that are allowed by the Standard Model gauge symmetries, and that are suppressed by powers of the cutoff scale. Such dimension six operators, which are invariant under the Standard Model gauge symmetries, lead to proton decay. They indicate that the cutoff scale must be above GeV, unless they are forbidden by some new symmetries. As global symmetries are, in general, expected to be violated by quantum gravity effects, the new symmetries should be either gauge symmetries or local discrete symmetries ps .

  • Suppression of left–handed neutrino masses is compatible with the generation of heavy mass to the right–handed neutrinos by the seesaw mechanism seesaw .

The Standard Model multiplet structure, and the additional evidence provided by logarithmic running, proton longevity and neutrino masses indicates that the primary guides in the search of a realistic string vacuum are the existence of three chiral generations and their embedding in representations. This embedding does not entail the existence of an gauge symmetry in the effective low energy field theory. Rather, the symmetry is broken at the string level to a maximal subgroup, and preferably directly to the Standard Model gauge group.

The Standard Model of particle physics is founded on a causal and renormalizable quantum field theory with local phase invariance under a product of Abelian and non–Abelian gauge symmetries. These symmetry principles encode all the subatomic experimental observations to date. Alas, the effects of the gravitational interactions are not included in this picture. Moreover, there is a fundamental dichotomy between the principles underlying quantum mechanics and gravitational observations. In particular, with regard to the treatment of the vacuum. While quantum field theories give rise to energy sources that contribute to the vacuum energy with scale of the order of the QCD scale and above, observations show that the vacuum energy is smaller by orders of magnitude. Another point of contention is with regard to the nature of space. In general relativity, the contemporary theory of gravity, space is a dynamical field satisfying Einstein’s equations of motion. In quantum field theories, on the other hand, space provides background parameters and does not correspond to the fundamental degrees of freedom, which are encoded in the particle wave–functions and their conjugate momenta. Furthermore, gravity as a quantum field theory is not renormalizable, which is therefore plagued with infinities and is inconsistent at a fundamental level.

The conundrum may be seen to arise from the fact that quantum field theories may, in principle, probe space distances that are infinitely small, provided that the corresponding momenta is infinitely large. We may envision that this outcome is fundamentally unphysical and what we need is a fundamental description of matter and interactions, which excludes the possibility of probing infinitely small distances. String theory provides such a theory. Moreover, the equivalence postulate formulation of quantum mechanics implies that space is compact and the existence of a fundamental length in quantum mechanics tqc . The fundamental cutoff may therefore be intrinsically built into quantum mechanics, provided that its full set of symmetries are incorporated.

As a finite theory string theory provides a consistent framework for perturbative quantum gravity stringreviews . The consistency of string theory at the quantum level dictates that it must accommodate a specific number of worldsheet degrees of freedom to produce an anomaly free and finite theory. Some of degrees of freedom give rise to the gauge symmetries that we may identify with the subatomic interactions. Moreover, similar consistency constraints at the quantum level in the case of the superstring and heterotic–string give rise to matter states that are charged under the gauge degrees of freedom, and may be identified with the Standard Model matter states. Hence, string theory provides a viable framework for the consistent unification of gravity with the subatomic matter and interactions. In turn, this feature of string theory allows for the development of a phenomenological approach to quantum gravity.

String theory is therefore a mundane extension of the idealisation of point particles with internal attributes. Furthermore, the rank of the gauge group accounting for the internal attributes is fixed by the consistency conditions of the theory. The string action is parameterised by two worldsheet degrees of freedom, corresponding to the proper time and the string internal dimension. The equation of motion of the worldsheet degrees of freedom is a two dimensional wave equation. The solutions are separated into left– and right–moving solutions. The physical states of the quantised string give rise to a tachyonic state, which is eliminated from the spectrum if the bosonic worldsheet fields are augmented with fermionic fields. This is achieved provided that the theory possesses supersymmetry on the worldsheet, which guarantees the existence of spacetime supersymmetry. Since the tachyonic state does not have a corresponding fermionic superpartner, the existence of spacetime supersymmetry guarantees that the tachyonic state is excluded from the physical spectrum. Additionally, the fermionic string gives rise to spacetime fermions that transforms in representations of the internal gauge symmetry.

String theory is formulated as a perturbative scattering expansion. Using the conformal symmetry on the worldsheet, the lowest order amplitudes can be mapped to the sphere with vertex operator insertions corresponding to the external string states. Higher order amplitudes are mapped to higher genus tori, with the genus one torus being the lowest order quantum correction. The vacuum to vacuum amplitude is the first order correction when there are no external states and all the physical states can propagate in the closed time–like loop. The conformal worldsheet symmetry is translated to invariance of the torus amplitude under modular transformations of the complex worldsheet parameter . The worldsheet fermionic fields can pick up non trivial phases when parallel transported around the non–contractible loops of the worldsheet torus. The possible transformations for all the worldsheet fermions are encoded in the so called spin structures and are mixed non–trivially by the modular transformations. Requiring invariance under modular transformations leads to a set of non–trivial constraints on the allowed spin structures stringreviews .

Different string theories may be formulated depending on the existence, or not, of worldsheet fermionic fields in the left– and right–moving sectors of the string. Type IIA and type IIB superstring arise if worldsheet fermions are added in both the left– and right–moving sectors. Adding worldsheet fermions only to the left–moving sector produces the heterotic–string with , or gauge symmetry in ten dimensions. In the low energy point particle approximation, we expect a string theory to correspond to an effective field theory approximation. That is when the energy involved is not sufficiently high to reveal the internal structure of the string, we expect that it should be described effectively as some point particle field theory. In the case of the fermionic strings these are type IIA or IIB supergravities, or an effective ten dimensional supergravity with or gauge symmetry. Additionally, the non–perturbative effective field theory limits of the ten dimensional string are related to compactifications of eleven dimensional supergravity. For example, the type IIA superstring is related to compactification of eleven dimensional supergravity on a circle, whereas the ten dimensional heterotic string corresponds to compactification on a circle moded by a reflection symmetry. The full set of relations at the quantum level is yet to be unravelled, and is traditionally dubbed as M–theory or F–theory stringreviews .

The lesson is that our understanding of the synthesis of gravity and the gauge interactions is still very rudimentary. String theory is clearly a step in the right direction. It provides a framework to ask questions about the gauge and gravity unification and to seek consistent answers within that framework. By giving rise to all the basic fields that are used to parameterise the subatomic and gravitational experimental data, it enables the development of a phenomenological approach to quantum gravity. However, its is clear that string theory is not the final answer. The contemporary string theories are believed to be effective limits of a more fundamental theory. From that perspective each of the string theories can be used to probe some properties of the vacuum of the fundamental theory, but not to fully characterise it. The heterotic string is the effective limit that gives rise to spinorial representation in the perturbative spectrum. The heterotic–string therefore is the effective limit that should be used if the properties that we would like to preserve are the existence of three chiral generations and their embedding in spinorial representations.

2 Past

Realistic string models are obtained by compactifying the heterotic–string heterotic from ten to four dimensions chsw . Alternatively, we can construct realistic string models directly in four dimensions by representing the compactified dimensions in terms of internal conformal field theories propagating on the string worldsheet. The simplest such theories are in terms of free worldsheet field theories, i.e. in terms of free bosons orbifolds or free fermions fff , with the main simplification being the implementation of the modular invariance constraints. Nevertheless, constructions using interacting worldsheet conformal field theories exist as well gepner , and can be used to construct phenomenological vacua grs . It should be remarked that the representations of the four dimensional string vacua as compactifications on internal manifold or in terms of internal conformal field theories are not necessarily distinct. For example, theories that utilise two dimensional worldsheet free bosons or free fermions are mathematically equivalent. Similarly, it was demonstrated in some cases that string models with interacting internal CFT correspond to string compactification on a Calabi–Yau manifold at specific points in the moduli space gepner . This is an important point for the following reason. While the space of distinct string vacua in the effective field theory limit may seem to be huge, many of these vacua are related by various perturbative and nonperturbative dualities at the string level. The reason is that at the string level massless and massive physical states can be exchanged. Thus, vacua that are topologically and physically distinct in the effective field theory level are in fact connected at the string level. This feature is particularly important if we envision the existence of a dynamical vacuum selection mechanism in string theory.

The simplest phenomenological string models can therefore be constructed by using a free internal conformal field theory. String theories in which the internal CFT is written in terms of free fermions corresponds to compactifications on a flat six dimensional torus at a special point in the moduli space n4n2n1 . Exactly marginal deformations from the free fermionic point are obtained by adding Thirring interactions among the worldsheet fermions chang . The number of allowed deformations correspond exactly to the number of allowed deformations in compactifications of the corresponding string theory on a flat torus. Compactifications of the heterotic–string on a flat six dimensional torus produce spacetime supersymmetry, which is reduced to by modding out the internal six dimensional torus by an internal symmetry. This produces the so called orbifold compactifications. The simplest such orbifolds correspond to modding out the internal six dimensional torus by symmetries. Modding out by a single reduces the number of spacetime supersymmetries from to . Therefore, to reduce the number of supersymmetries to necessitates modding out by two independent symmetries, i.e. by .

2.1 NAHE–based models

In the free fermionic formulation fff of toroidal compactifications narain ; nsw all the internal degrees of freedom needed to cancel the worldsheet conformal anomaly are represented in terms of free fermions propagating on the string worldsheet. In the usual notation the 64 worldsheet fermions in the light–cone gauge are denoted as:

   :                             

   

In this notation the are the fermionic superpartners of the left–moving bosonic coordinates. The are the worldsheet real fermions corresponding to the six compactified dimensions of the internal manifold. The remaining sixteen complex fermions generate the Cartan subalgebra of the ten dimensional gauge group, with being those that generate the symmetry, and are those that generate the hidden sector gauge group. The complex worldsheet fermions generate three symmetries.

Under parallel transport around the noncontractible loops of the torus amplitude the worldsheet fermionic fields can pick up a phase. The 64 phases are encoded in boundary condition basis vectors, which generate the one loop partition function,

where and denote all possible combinations of the basis vectors. The requirement of modular invariance leads to a set of constraints on the allowed basis vectors and one loop phases. The basis vectors generate a finite additive group and each sector in the additive group, , produces a Fock space by acting on the vacuum with fermionic and bosonic oscillators. Worldsheet fermionic fields that are periodic under parallel transport produce a doubly degenerate vacuum that generate the spinorial charges. The physical states in the Hilbert space are obtained by applying the Generalised GSO projections, which arise due to the modular invariance requirement. The cubic level and higher order terms in the superpotential are obtained by calculating scattering amplitudes between vertex operators superpoterm . Finally, string vacua often give rise to an pseudo–anomalous symmetry, which is cancelled by the Green–Schwarz mechanism gs ; dsw . The anomalous gives rise to a Fayet–Iliopoulos –term dsw ; ads , which breaks spacetime supersymmetry at the string scale. Restoration of supersymmetry is obtained by assigning non–trivial Vacuum Expectation Value (VEV) to a set of fields in the physical spectrum, and imposing that all the supersymmetry breaking – and –terms vanish.

In this manner a large set of string vacua can be obtained. The early quasi–realistic free fermionic models were constructed in the late 1980s – early 1990s, and consist of the so called NAHE–based models. The NAHE–set is a set of five boundary condition basis vectors, , which is common to a large class of the early models nahe . The two basis vectors correspond to a toroidally compactified model with spacetime supersymmetry and an gauge group. The sectors , and correspond to the three twisted sectors of a orbifold compactification. They reduce the number of supersymmetries to and the gauge symmetry to . Additionally, they produce 48 multiplets in the spinorial representation of . The number of these multiplets is reduced to three by adding three additional basis vectors to the NAHE set, typically denoted by , which also reduce the gauge symmetry. The symmetry is reduced to a maximal subgroup and the flavour symmetries are reduced to , with . Using this construction three generation models with

were obtained, whereas models with su421 did not yield three generations. It is noted that in all these models the Standard Model weak hypercharge possess the embedding, and yield the canonical GUT normalisation , where is the string unification scale. This is an important feature of these models because it facilitates agreement with the measured gauge coupling parameters at the electroweak scale df . It should also be contrasted with other possible embedding of the weak hypercharge that do not yield the canonical GUT embedding. Such is the case, for instance, in many orientifold models. However, in orientifold models the string scale may be lowered relative to the gravitational scale. Hence, in orientifold models smaller values of may be accommodated. Heterotic–string models may also yield smaller values of , by modifying the identification of the weak hypercharge in the string models. We recall that arises as a result of the relative normalisation of the weak hypercharge relative to the non–Abelian generators at dfm . In heterotic string models this normalisation is affected by the number of Cartan generators in the weak hypercharge combination relative to the number of Cartan subgenerators of the non–Abelian group factors. However, in the perturbative heterotic string the unification scale is fixed and therefore lower values of are disfavoured. This constraint may be relaxed in the non–perturbative heterotic–string witten . Another point to note in regard to the definition of the weak hypercharge is the existence of string states that carry fractional electric charge. This is a general feature of string models. The reason being the breaking of the non–Abelian gauge symmetries by Wilson lines. A general observation by Wen and Witten wenwitten , and a theorem by Schellekens schellekens , notes that breaking a non–Abelian gauge group by a Wilson line in string theory, with a left over unbroken symmetry, produces states that do not satisfy the charge quantisation of the unbroken non–Abelian symmetry. This outcome further depends on the identification of the weak–hypercharge. That is if we relax the canonical GUT embedding of the weak hypercharge, we modify the GUT quantisation of the charges and may therefore obtain integrally charged states. The important point to note is that these are phenomenological properties of string constructions and it is yet to be determined how they play out in fully realistic string constructions.

2.2 Phenomenology of string unification

Subsequent to the construction of the string models and analysis of their spectra, we calculate the cubic level and higher order terms in the superpotential, up to a desired order for a specific phenomenological problem. The next step entails the analysis of supersymmetric and –flat directions. Requiring that the vacuum at the string scale is supersymmetric necessitates the assignment of non–vanishing VEVs to a set of Standard Model singlets in the string models. In this process some of the higher order terms in the superpotential become effective renormalizable operators, which are suppressed relative to the leading order cubic terms, i.e.

(1)

where are fermionic and bosonic vertex operators, respectively; is the order of the nonrenormalisable operator; and is the string cutoff scale. Using this methodology many of the issues pertaining to the phenomenology of the Standard Model and string unification have been studied in the framework of the quasi–realistic free fermionic heterotic–string models. A partial list includes:

  • The analysis of fermion masses entails the calculation of cubic and higher order terms in the superpotential that are reduced to dimension four terms in eq. (1). The Standard Model fermion mass terms arise from couplings to the electroweak Higgs with an assumed VEV of the order of the electroweak scale. Other fermion mass terms arise from coupling to other scalar fields, and their mass scales may therefore be higher than the electroweak scale. Analysis of Standard Model fermion masses yielded a viable prediction for top quark mass prior to its experimental observation top . The calculation proceeds as follows. First the top quark Yukawa coupling is calculated at the cubic level of the superpotential, giving , where is the gauge coupling at the unification scale. Subsequently, the Yukawa couplings for the bottom quark and tau lepton are obtained from quartic order terms. The magnitude of the quartic order coefficients are calculated using standard CFT techniques, and the VEV of the Standard Model singlet field in the relevant terms is extracted from analysis of the – and –flat directions. This analysis yields effective Yukawa couplings for the bottom quark and tau lepton in terms of the unified gauge coupling given by top . This result for the top quark Yukawa coupling is common in a large class of free fermionic models, whereas those for the bottom quark and tau lepton differ between models. Similarly, the Yukawa coupling for the two lighter generations differ between models and depend on the flat direction VEVs. Subsequent to extracting the Yukawa couplings at the string scale, they are run to the electroweak scale using the Minimal Supersymmetric Standard Model (MSSM) Renormalisation Group Equations (RGEs). It is further assumed that the unified gauge coupling at the string scale is compatible with the value required by the gauge coupling data at the electroweak scale. The bottom Yukawa is further run to the bottom mass scale, which is used to extract a value for , with and being the VEVs of the two MSSM electroweak Higgs doublets. The top quark mass is then given by

    with , yielding . It is noted that, up to the assumptions stated above, the top Yukawa coupling is found near a fixed point. Namely, varying the top Yukawa between at the unification scale yields at the electroweak scale. This calculation demonstrates the important advantage of string theory over other attempts of developing a viable framework for quantum gravity. It unifies the gauge and Yukawa couplings and enables the calculation of the Standard Model Yukawa couplings in terms of the unified string coupling. While the calculation of the top Yukawa is robust and shared between a large class of models, the calculation of the corresponding couplings for the lighter quarks and leptons involve a large degree of model dependence. Before investing substantial efforts to calculate the Yukawa couplings of the lighter quarks and leptons in a given model, we should enhance the prospect that a given model is the right model. This line of reasoning underlies the contemporary approach that is outlined below.

  • The analysis of the effective Yukawa couplings for the lighter two generations proceeds by analysing higher order terms in the superpotential and extracting the effective dimension four operators superpoterm . The analysis should be regarded as demonstrating in principle the potential of string models to explain the detailed features of the Standard Model flavour parameters. It is still marred by too many uncertainties and built in assumptions to be regarded as a predictive framework. Nevertheless, once an appealing model is constructed the methodology is in place to attempt a more predictive analysis. The explorations to date included, for example, demonstration of the generation mass hierarchy gmh ; Cabibbo–Kobayashi–Maskawa (CKM) mixing fh1 ; fh3 ; light generation masses lgm ; and neutrino masses fh2 ; cfneutrinomasses .

  • An important issue in heterotic–string models is compatibility with the experimental gauge coupling data at the electroweak scale. The perturbative heterotic–string predicts that the gauge couplings unify at the string scale, which is of the order of . On the other hand extrapolation of the gauge couplings, assuming MSSM spectra, from the –boson mass scale to the GUT scale shows that the couplings converge at a scale of the order of . Thus, the two scales differ by a factor of about 20. This extrapolation should be taken with caution as the the parameters are extrapolated over 14 orders of magnitude, with rather strong assumptions on the physics in the region of extrapolation. Indeed, in view of the more recent results from the LHC the analysis needs to be revised as the assumption of MSSM spectrum at the –boson scale has been invalidated. Nevertheless, the issue can be studied in detail in perturbative heterotic–string models and a variety of possible effects have been examined, including heavy string threshold corrections, light SUSY thresholds, additional gauge structures and additional intermediate matter states df . Within the context of the free fermionic models only the existence of additional matter states may resolve the discrepancy and such states indeed exist in the spectrum of concrete string models gcu . This result may be relaxed in the nonperturbative heterotic–string witten or if the moduli are away from the free fermionic point nillesstein .

  • Proton longevity is an important problem in quantum gravity, in general, and in string models in particular. The reason being that we expect only gauge symmetries, or local discrete symmetries that arise as remnants of broken gauge symmetries, to be respected in quantum gravity. Within the Standard Model itself baryon and lepton are accidental global symmetries at the renormalizable level. Thus, we expect, in general, all operators that are compatible with the local gauge and discrete symmetries in given string models to be generated from nonrenormalizable terms. Such terms can then give rise to dimension four, five and six baryon and lepton number violating operators that may lead to rapid proton decay. Possible resolutions have been studied in specific free fermionic models and include the existence of an additional light symmetry extrazprime and local discrete symmetries ps .

  • String models may, in general, lead to non-degenerate squark masses, depending on the specific SUSY breaking mechanism. For example, SUSY breaking mechanism which is dominated by the moduli –term will lead to non–degenerate squark masses, because of the moduli dependence of the flavour parameters. Similarly, –term SUSY breaking depends on the charges of the Standard Model fields under the gauge symmetry in the SUSY breaking sector, and those are in general family non–universal. Free fermionic models can give rise to a family universal anomalous auone . If the SUSY breaking mechanism is dominated by the anomalous –term it may produce family universal squark masses of order squarkdg .

  • Three generation semi–realistic string models produce, in general, additional massless vector–like states that are charged under the Standard Model gauge symmetries. Some of these additional vector–like states arise from the Wilson line breaking of the GUT symmetry and therefore carry fractional charge with respect to the remnant unbroken symmetries. In particular, they may carry fractional electric charge, which is highly constrained by observations. These fractionally charged states must therefore be sufficiently massive or diluted to evade the experimental limits. Mass terms for the vector–like states may arise from cubic and higher level terms in the superpotential. In the model of ref. fny its has been demonstrated in fractional that all the exotic fractionally charged states couple to a set of singlets. In ref. cfn – and –flat solutions that incorporate this set of fields have been found. Additionally, all the extra standard–like fields in the model, beyond the MSSM, receive mass terms by the same set of VEVs. These solutions therefore give rise to the first known string solutions that produce in the low energy effective theory of the observable sector solely the states of the MSSM, and are dubbed Minimal Standard Heterotic–String Model (MSHSM). Three generation Pati–Salam free fermionic models in which fractionally charged exotic states arise only in the massive spectrum were found in ref. exophobic . Flat directions that lead to MSHSM with one leading Yukawa coupling were found in an exemplary model in this class cfr .

  • An important issue in string models is that of moduli stabilisation. The free fermionic models are formulated near the self–dual point in the moduli space. However, the geometrical moduli that allow deformations from that point exist in the spectrum and can be incorporated in the form of Thirring worldsheet interactions chang . The correspondence of the free fermionic models with orbifolds implies that the geometrical moduli correspond to three complex and three Kähler structure moduli. String theory as a theory of quantum geometry, rather than classical geometry, allows for assignment of asymmetric boundary conditions with respect to the worldsheet fermions that correspond to the internal dimensions. These correspond to the asymmetric bosonic identifications under . In the free fermionic models, and consequently in orbifolds, it is possible to assign asymmetric boundary conditions with respect to six circles of the six dimensional compactified torus. In such a model all the complex and Kähler moduli of the untwisted moduli are projected out modulifixing . Additionally, the breaking of the worldsheet supersymmetry in the bosonic sector of the heterotic–string results in projection of the would–be twisted moduli modulifixing . Thus, all the fields that are naively identified as moduli in models with worldsheet supersymmetry can be projected out in concrete models. However, the identification of the moduli in models with worldsheet supersymmetry is not well understood and there may exist other fields in the spectrum of such models that may be identified as moduli fields. Furthermore, as long as supersymmetry remains unbroken in the vacuum there exist moduli fields associated with the supersymmetric flat directions. However, it has been proposed that there exit quasi–realistic free fermionic models which do not admit supersymmetric flat directions minimalhiggs . This is obtained when both symmetric and asymmetric twistings of the internal dimensions are implemented, resulting in reduction of the number of moduli fields. In the relevant models supersymmetry is broken due to the existence of a Fayet–Iliopoulos term, which is generated by an anomalous symmetry. It was argued in minimalhiggs that the relevant models do not admit exact flat directions and therefore supersymmetry is broken at some level. In such models all the moduli are fixed. It should be noted that this possibility arises only in very particular string models, rather than in a generic string vacua baylorgang .

3 Present

Most of the studies discussed so far were done by studying concrete examples of NAHE–based models, i.e. models that contain the common set plus the three (or four) additional basis vectors that extend the NAHE–set and differ between models, with the most studied models being those of ref. fny and eu . More recent studies involve the exploration of large number of models. This provides an insight into the general properties of the space of vacua, as well as the development of a “fishing algorithm” to fish models with specific phenomenological properties. This method led to discovery of spinor–vector duality spinvecdual and of exophobic vacua exophobic ; asseltwo ; cfr ; bfgrs . More recently the method has been applied for the classification of flipped free fermionic models frs , as well as the classification with respect to the top quark Yukawa coupling johnstopclass .

3.1 Classification of fermionic orbifolds

Over the past decade a systematic method is being developed that allows the explorations of large number of string vacua and analysis of their spectra. In this method the set of basis vectors is fixed. The Pati–Salam class of models is generated by a set of thirteen basis vectors

where

(2)

In the notation employed in Eq. (2) the worldsheet fields appearing in a given basis vector have periodic boundary conditions, whereas all other fields have anti–periodic boundary conditions. The first twelve vectors in this set are identical to those used in fknr ; fkr for the classification of fermionic orbifolds with GUT symmetry. The thirteenth basis vector, , breaks the symmetry and generates the Pati–Salam class of models. The set generate an supersymmetric model, with gauge symmetry. The vectors give rise to all possible symmetric shifts of the six internal fermionized coordinates (). Their addition breaks the gauge group, but preserves supersymmetry. The vectors and define the gauge symmetry and the orbifold twists, which break to supersymmetry. The and basis vectors reduce the untwisted gauge group generators from to . Finally is the additional new vector that breaks the GUT symmetry to , and the hidden symmetry to .

The second ingredient that is needed to define the string vacuum are the GGSO projection coefficients that appear in the one–loop partition function, , spanning a matrix. Only the elements with are independent while the others are fixed by modular invariance. A priori there are therefore 78 independent coefficients corresponding to string vacua. Eleven coefficients are fixed by requiring that the models possess supersymmetry. Additionally, the phase only affects the overall chirality. Without loss of generality the associated GGSO projection coefficients are fixed, leaving 66 independent coefficients. Each of the 66 independent coefficients can take two discrete values and thus a simple counting gives (that is approximately ) models in the class of superstring vacua under consideration.

The utility of the classification method is that it provides the means to span all the massless producing sectors in the models. For example, the twisted matter states arise from the sectors

where , and is given by the vector . These sectors give rise to and representations of decomposed under . The important feature of this classification method is that each of the sectors for given gives rise to one spinorial, or one anti–spinorial, or neither, i.e the states arising at each fixed point of the corresponding are controlled individually. Similarly, the states from the sectors produce states in the vectorial representation of decomposed under the Pati–Salam gauge group.

The power of the free fermionic classification method is that it enables translation of the GGSO projections into generic algebraic forms. The general expression for the GSO projections on the states from a given sector is given by fff

From this expression we note that, whenever the overlap of periodic fermions between the basis vector and the sector is empty, the operator on the left of this expression is fixed. Hence, depending on the choice of the GGSO phase on the right, the given state is either in or out of the physical spectrum. For any given state from specific sectors there are several basis vectors that act as projectors. Introducing the notation with , we can collect these projectors into algebraic system of equations of the form  where the unknowns are the fixed point labels  The and are given in terms of the GGSO projection coefficients for each of the three planes. For example, on the first plane for the spinorial or states we have

(3)

and  with similar expressions for the second and third planes. The number of solutions per plane is determined by the relative rank of the matrix and the rank of the augmented matrix . For a given choice of GGSO projection coefficients, the number of states surviving in the spectrum, is therefore readily obtained. Similar, algebraic expressions can be obtained for all the sectors that produce massless states in the given basis, as well as for the chirality of the fermions with periodic boundary conditions.

The methodology outlined above enables the classification of a large number of fermionic orbifolds. Compared to the earlier construction it enables a scan of a large number of models and extraction of some of the desired phenomenological properties. We can develop a fishing algorithm to extract models with specific characteristics. For example, a class of Pati–Salam models in which exotic fractionally charged states appear as massive states but not in the massless spectrum was found using these tools. The systematic classification methods were developed to date only for models that admit symmetric boundary conditions with respect to the set of internal worldsheet fermions . On the other hand, NAHE–based models were constructed using symmetric and asymmetric boundary conditions, with the assignment of asymmetric boundary conditions having distinct phenomenological implications dtsm ; udysr .

3.1.1 Spinor–vector duality

Another example of the utility of the fermionic classification method is given by the spinor–vector duality, which was discovered by using these methods and elucidates the global structure of the free fermionic models, in particular, and that of the larger string landscape, in general. The spinor–vector duality is a duality in the space of string vacua generated by the basis set with , and unbroken symmetry. The duality entails an invariance under the exchange of the total number of representations and the total number of representations of . That is, for a given vacuum with a number of and representations, there exist another vacuum in which the two numbers are interchanged. The origin of this duality is revealed when the symmetry is enhanced to . Under the decomposition of the and representations decompose as and . Therefore, in the case of vacua with symmetry the total number of representations is equal to the total of representations. Hence, models with enhanced symmetry are self–dual under the spinor–vector duality map.

The spinor–vector duality therefore arises from the breaking of the symmetry to . This breaking is generated in the orbifold language by Wilson–lines, or in the free fermionic construction, by choices of the GGSO projection coefficients. It is important to recognise that these two descriptions are not distinct, but are mathematically identical. That is we can translate the GGSO projection coefficients to Wilson line and visa versa n4n2n1 . Thus, when the symmetry is broken to , there exist a choice of GGSO projection coefficients, or of Wilson lines, that keeps a number of spinorial and a number of vectorial representations of , and another choice for which the two numbers are interchanged. It is important to note that this is an exact duality symmetry operating in the entire space of string vacua in which the symmetry is not enhanced to spinvecdual ; tristan ; aft ; ffmt . It is further noted that the spinor–vector duality can be interpreted in terms of a spectral flow operator ffmt . In this context the spectral flow operator in the twisted sector may be seen as a deformed version of the operator inducing the Massive Spectral boson–fermion Degeneracy Symmetry (MSDS) msds . Therefore, the spinor–vector duality extends to the massive sectors ffmt , albeit in a fashion that still needs to be determined in the general case. Similarly, we note that the generalisation of the spinor–vector duality to the case of interacting internal CFTs can be studied by adopting the following methodology e.g. in the case of minimal models. The starting point is an heterotic–string compactified to four dimensions with worldsheet supersymmetry and an internal interacting CFT representing the compact space. The next step is to break the worldsheet supersymmetry in the bosonic sector of the heterotic–string. The spectral flow operator then induces a map between distinct vacua afg .

We can also understand the spinor–vector duality operationally in terms of the free phases in the fermionic language tristan or as discrete torsion in the orbifold picture aft ; ffmt . For that purpose we recall the level one characters manno

where

and is the partition function of a single worldsheet complex fermion, given in terms of theta functions manno . The partition function of the heterotic–string compactified on a six dimensional torus is given by

(4)

where as usual, for each circle,

Next, a projection is applied, where the first is a freely acting Scherk–Schwarz like projection, which couples a fermion number in the observable and hidden sectors with a –shift in a compactified coordinate, and is given by  where the fermion numbers act on the spinorial representations of the observable and hidden groups as  and identifies points shifted by a shift in the direction, i.e. The effect of the shift is to insert a factor of into the lattice sum in eq. (4), i.e.  The second acts as a twist on the internal coordinates given by  The effect of the first is to reduce the gauge symmetry from to . The twist reduces the number of spacetime supersymmetries from to , and reduces the gauge symmetry arising from to . Additionally, it produces a twisted sector that gives rise to massless states in the spinorial and , and vectorial , representations of . In this vacuum the spinor–vector duality operates in terms of the representations of rather than in terms of representations of , as the enhanced symmetry point possess an symmetry rather than . The spinor–vector duality operates identically in the two cases and the case of the single non–freely acting twist elucidates more readily the underlying structure of the spinor–vector duality. The orbifold partition function is given by

The partition function contains an untwisted sector and three twisted sectors. The winding modes in the sectors twisted by and are shifted by , and therefore these sectors only produce massive states. The sector twisted by gives rise to the massless twisted matter states. The partition function has two modular orbits and one discrete torsion . Massless states are obtained for vanishing lattice modes. The terms in the sector contributing to the massless spectrum take the form

(5)

where

(6)

Depending on the sign of the discrete torsion we note from eq. (6) that either the spinorial states, or the vectorial states, are massless. In the case with we see from eq. (7) that in this case massless momentum modes from the shifted lattice arise in whereas produces only massive modes. Therefore, in his case the vectorial character in eq. (6) produces massless states, whereas the spinorial character generates massive states. In the case with we note from eq. (8) that exactly the opposite occurs.

(7)
(8)

Another observation from the term appearing in eq. (5) is the matching of the number of massless degrees of freedom in the two cases. In the case with the number of degrees of freedom in the spinorial representation of is 32. In the case with the number of degrees of freedom in the vectorial representation of is 12. As seen from the first line in eq. (5) the term in the partition function producing the vectorial states also transforms as a spinor under the symmetry. Hence the total number of states is 24, i.e. there is still a mismatch of 8 states between the two cases. However, we note from the second line in eq. (5) that in the case with eight additional states are obtained from the first excited states of the internal lattice. We note therefore that the total number of degrees of freedom is preserved under the duality map, i.e.

Given the relation of free fermionic models to toroidal orbifolds, we can anticipate that the spinor–vector duality can be realised in terms of the moduli of the toroidal lattices. Those are the six dimensional metric, the antisymmetric tensor field and the Wilson lines narain . Indeed, the discrete torsion appearing in eq. (5) can be translated to a map between two Wilson lines ffmt . We note that in the case of (5) the map between the Wilson lines is continuous. The reason is the fact that we employed a single twist on the internal coordinates. The moduli associated with the Wilson line mapping are not projected out in this case and therefore the interpolation between the two Wilson lines is continuous. In the more general case with a twist these moduli are projected out and the mapping between the two Wilson lines is discrete ffmt .

Additionally, we can understand the spinor–vector duality in terms of a spectral flow operator ffmt , which may be generalised to other cases. We recall that vacua with extended gauge symmetry are self–dual under the spinor–vector duality, and that they correspond to vacua with worldsheet supersymmetry. Just like the case of the worldsheet supersymmetry in the supersymmetric sector of the heterotic–string, there is a spectral flow operator that acts as a generator of in the vacua with enhanced symmetry. On the supersymmetric side the spectral flow operator mixes states with different spacetime spin, whereas on the non–supersymmetric side it mixes states that differ by their charge in the decomposition , i.e. it mixes the states that transform as spinors and vectors of . When the symmetry is broken, i.e. when the worldsheet supersymmetry is broken from to , the spectral flow operator induces the spinor–vector duality map between the two distinct vacua ffmt .

The spinor–vector duality is a novel symmetry that operates in the global space of and heterotic–string orbifolds and provides valuable insight and interesting questions for future research. First, we note that the spinor–vector duality is a map between vacua that are completely unrelated in the effective field theory limit. For example, we may envision a map between a model with 3 spinorial representations, and one vectorial representation, to a model with 3 vectorial representations and one spinorial representation. In terms of the low energy physics the two cases are fundamentally different. On the other hand, from the point of view of string theory they are identical. Namely, there is an exact map from one to the other. The distinction between the string representation versus the effective field theory limit is that the string can access its massive modes, which are not seen in the effective field theory limit. Therefore, vacua that seem distinct in the effective field theory limit are in fact related in the full string theory. We may further envision that at some early stage in the evolution of the universe, when the heavy string modes are excited that the two vacua can in fact mix. This possibility has implications on the counting of distinct string vacua and therefore on the string landscape. It is evident that our contemporary understanding of the string landscape is still very rudimentary and we should proceed with caution before overstating our case. The spinor–vector duality may also have interesting implications from a purely mathematical point of view. Namely, in the effective field theory limit there should exist a description of the massless degrees of freedom in terms of a smooth effective field theory i.e. in terms of a supergravity theory with a classical geometry (i.e. some Calabi–Yau six dimensional manifold) with a vector bundle accounting for the gauge degrees of freedom. The existence of the spinor–vector duality map implies that there should be a similar map between the two effective theory limits of the two vacua. This is particularly interesting in terms of the counting of the additional states that are needed to compensate for the mismatch in the number of states between the two vacua. How do they arise in the effective field theory limit? In the very least, the spinor–vector duality provides a valuable tool to study the moduli spaces of heterotic–string compactifications.

4 other approaches

The free fermionic models represents one of the approaches to string phenomenology. Several other approaches are being pursued, leading to overlapping and complementary results, in the perturbative and nonperturbative domains. The literature on these subjects is vast and include several monographs, including, for example, ibanezuranga . A partial and incomplete list of some of these studies include: geometrical studies gkmr ; dopw ; bmrw ; hv ; aglp ; orbifolds ino ; blt ; krz ; lnrrrvw ; bgrtv ; interacting CFTs g ; sy ; gatos ; orientifolds csu ; imr ; kst . It should be emphasised that the present article does not aim to review these important contributions, but merely those of the author. A comprehensive review is provided in reference ibanezuranga , as well as in stringreviews .

5 Future

With the observation that the agent of electroweak symmetry breaking is compatible with an elementary scalar, particle physics and string phenomenology are set for a bright future. In the particle physics realm the main questions are experimental. Are there additional states associated with the electroweak symmetry breaking mechanism? e.g. Is spacetime supersymmetry realised in nature, and within reach of contemporary colliders? Can we improve on the contemporary measurements of the Standard Model parameters and by how much? Can we build accelerators to probe energy scales in the deca–TeV region and above? These are rather general questions and experiments should target more specific questions, e.g. can we cool the muon phase space in a muon storage ring or a muon collider? The construction of a muon based facility will advance the accelerator based technology to a new era, and may be used as a Higgs factory in one of its initial missions muonhiggs .

Particle physics and string phenomenology are two sides of the same coin, and should not be regarded as distinct entities. Particle physics shows that experimental data can be parameterised by a model, which is based on the principles of point quantum field theories, i.e. locality, causality and renormalizability. This led to the development of the Standard Model, which is a quantum field theory with internal symmetries. A point quantum gravity theory fails to satisfy these criteria. String theory resolves the problem with the third property by relaxing the first. String models provide consistent approaches to quantum gravity, in which the internal symmetries are dictated by the consistency of the theory. As a common setting for the gauge and gravitational interactions string theory facilitates the calculation of the Standard Particle Model parameters in a reduced framework.

5.1 Toward string predictions

String theory leads to distinct signatures beyond the Standard Model. In the first instance all the known stable string vacua at the Planck scale are supersymmetric bachas . Whether supersymmetry is manifested within reach of contemporary experiments is a wild speculation. Nevertheless, this hypothesis is motivated on the ground that it facilitates extrapolation of the Standard Model parameters from the unification scale to the electroweak scale. Furthermore, electroweak symmetry breaking at the low scale is generated in the supersymmetric scheme by the interplay of the top quark Yukawa coupling and the gauge coupling of the strong interaction ewsb .

Low scale supersymmetry is therefore not a necessary outcome of string theory, but certainly its observation will provide further evidence that the different structures of string constructions are realised in nature. Specific SUSY breaking scenarios in string models give rise to distinct supersymmetric spectra and that in turn will be used to constrain further the phenomenological string vacua dedes . It is further noted that –parity is generically broken in string vacua rparity and that the LSP is not expected therefore to provide a viable dark matter candidate.

A generic prediction of string theory is the existence of additional gauge degrees of freedom, beyond those of the Standard Model, and is dictated by the consistency conditions of string theory. However, construction of viable string models that allow for extra gauge symmetries within reach of contemporary experiments is highly non–trivial. On the other hand, an extra symmetry may be instrumental to understand some phenomenological features of the Supersymmetric Standard Model, like the suppression of proton decay mediating operators and the –parameter.

Another generic outcome of string models is the existence of exotic matter states. This feature of string constructions arises as a result of the breaking of the non–Abelian GUT symmetries by Wilson lines, which results in exotic states that do not obey the quantisation rules of the original GUT group. Thus, one can get, for example, states that carry fractional electric charge. The lightest of the fractionally charged states is necessarily stable by electric charge conservations. The experimental restrictions on states that carry fractional electric charge are severe and they must be either sufficiently heavy, and/or sufficiently diluted to evade detection. Nevertheless, given that the bulk of the matter in the universe is dark, i.e. does not interact electromagnetically, stable string relics with a variety of properties can be contemplated ssr . This includes for example the possibility that the string relics come as fractionally charged hadrons and leptons, with charge . Such states will continue to scatter in the early universe until they form a bound hydrogen–like state with another fractionally charged companion. Provided that they are sufficiently heavy and sufficiently rare they could have evaded detection by searches for rare isotopes. Another possibility of exotic stable string relics arises when the GUT symmetry is broken to . This case gives rise to states that carry the regular Standard Model charges, but carry fractional charges with respect to the extra . This case, depending on the Higgs representations that break the , can result in discrete symmetries that forbid the decay of the exotic states to the Standard Model states. It can therefore give rise to meta–stable heavy string relics that are Standard Model singlets. Depending on the cosmological evolution in the early universe they could have been diluted and reproduced as super–heavy states after reheating ssr . Such states can produce viable dark matter ssr candidates as well as candidates for Ultra High Energy Cosmic Rays (UHECR) cfp .

5.2 Cosmological evolution

The early studies in string phenomenology, articulated in section 2, entailed the in depth exploration of exemplary models and the study of phenomenological properties. These studies focussed on the properties of the massless spectra of these exemplary models and led to the construction of the first known Minimal Standard Heterotic String Models (MSHSM) fny ; cfn .

The more recent studies, articulated in section 3, involve the classification of large classes of models and the relations between them. The string vacua in this investigation are fermionic orbifolds and are therefore related to the exemplary models in section 2. More importantly, the contemporary studies involve the analysis of the partition functions associated with this class of string vacua. In that context they aim to explore how the massive string spectrum may play a role in the determination of the phenomenological and mathematical properties of string models. This led to the discovery of spinor–vector duality in heterotic–string models spinvecdual .

One direction therefore in future string phenomenology studies will involve the investigation of the associated string partition function, and in particular away from the free fermionic point. The most general form of the partition function affiliated with the orbifolds, and hence with the phenomenological free fermionic models is given by

where the internal lattice is for one compact dimension is given by

and is a modular invariant phase. The properties of the string vacua, away from the free fermionic point, can be explored by studying this partition function and the role of the massive states. Furthermore, while the current understanding of string theory is primarily limited to static solutions, exploration of dynamical scenarios can be pursued by compactifying the time coordinate on a circle and using the Scherk–Schwarz mechanism ss in the compactified time–like coordinate. One then obtains a finite temperature–like partition function that can be used to explore cosmological scenarios. Indeed, this is the string cosmology program pursued by the Paris group over the past few years pariscos . In a similar spirit partition functions of string compactifications to two dimensions have been explored revealing rich mathematical structures and the so–called massive supersymmetry, in which the massive spectrum exhibits Fermi–Bose degeneracy, whereas the massless spectrum does not msds . One can envision interpolations of the two dimensional partition functions, associated with the cosmological and massive supersymmetry scenarios to the four dimensional partition functions associated with the phenomenological free fermionic models. The ultimate aim of this program will be to explore possible mechanisms for dynamical vacuum selection in string theory.

5.3 Dualities and fundamental principles

Physics is first and foremost an experimental science. There is no absolute truth. There is only perception of reality as registered in an experimental apparatus111including astrophysical observations.. Be that as it may, the language that is used to interpret the experimental signals is mathematics. The scientific methodology then entails:

  • the existence of some initial conditions, which are either preset or set up in an experiment;

  • the construction of a mathematical model that predict (or postdict) the outcome of the experiment;

  • the confrontation of the predictions of the mathematical model with the outcome of the experimental observations.

A successful mathematical model is the one that is able to account for a wider range of experimental observations. This scientific methodology has been developed over the past five hundred years or so.

To construct a mathematical model one needs to define a set of variables that are to be measured experimentally. The set of variables is key to the interpretation of the experimental outcome. Over the years modern physics has undergone a process of evolution in terms of these basic set of variables. In the Galilean–Newtonian system the basic set of variables are the position and velocities. In modern experiments the relevant measured variables are typically the initial and final energy and momenta. In the Lagrangian formalism the set of variable is generalised to any set of configuration coordinates and their derivatives with respect to time. In the Hamiltonian formalism the set of variables are the generalised configuration coordinates and their conjugate momenta, which constitute the phase space. This represents a nontrivial conceptual evolution from the Galilean–Newtonian system of position and velocities.

String theory provides a consistent framework for the perturbative unification of the gauge and gravitational interactions. The string characterisation of the basic constituents of matter reproduces the picture of elementary particles with internal attributes. String theory unifies the spacetime and internal properties of elementary particles. In the modern description of matter and interactions, the three subatomic interactions are in a sense already unified. They are based on the gauge principle. By giving rise to the mediators of the subatomic interactions that satisfy the gauge principle, and at the same time giving rise to the mediator of the gravitational interactions, that satisfy the gravitational gauge principle, string theory also unifies the principles underlying these theories.

Can string theory be the final chapter in the unification of the gauge and gravitational interactions. Unlike general relativity and quantum mechanics, string theory is not formulated by starting from a fundamental principle and deriving the physical consequences. Ultimately this is what we would like to have.

Perturbative and nonperturbative dualities have played a key role in trying to obtain a rigorous understanding of string theory. –duality is an important perturbative property of string theory tduality . We may interpret –duality and phase space duality in compact space. An additional important property of –duality in string theory is the existence of self–dual states under –duality.

We may envision promoting phase–space duality to a level of a fundamental principle. This is the program that was undertaken in ref. fm2 . The key is the relation between the phase space variables via a generating function , . To obtain a dual structure we define a dual generating function , with . The two generating functions are related by the dual Legendre transformations,

(9)

and

(10)

Furthermore, one can show that transforms as a scalar function under the –transformations

(11)

where . We can associate the two Legendre transformations (9) and (10) with a second order differential equations whose solutions are and , respectively. A special class of solutions are those which satisfy the two sets of differential equations, i.e. , with . These are the self–dual solutions under the Legendre duality of (9) and (10).

Given that the Legendre transformations are not defined for linear functions we have that the phase–space duality is not consistent for physical systems with , i.e. precisely for the self–dual states. It is further noted that the second order differential equations are covariant under coordinate transformations, but that their potential functions are only invariant under the Möbius transformations (11). This suggests the fundamental equivalence postulate fm2 ; fmijmp :

Given two physical systems labelled by potential functions and , where denotes the space of all possible ’s, there always exists a coordinate transformations such that .

This postulate implies that there should always exist a coordinate transformation connecting any state to the state . Inversely, this means that any nontrivial state can be obtained from the states by a coordinate transformation.

The classical Hamilton–Jacobi (HJ) formalism provides a natural setting to apply this postulate. In the HJ formalism a mechanical problem is solved by using canonical transformations to map the Hamiltonian of a nontrivial physical system, with nonvanishing kinetic and potential energies, to a trivial Hamiltonian. The solution is given by the Classical Hamilton–Jacobi Equation (CHJE) and the functional relation between the phase space variables is extracted by the relation , with being the solution of the HJ equation. We can pose a similar question, but imposing the functional relations on the trivialising transformation and