Orbifold Construction Research Articles

Commutativity of automorphisms of subfactors modulo inner automorphisms

We introduce a new algebraic invariant χ a ( M , N ) \chi _{a}(M,N) of a subfactor N ⊂ M N \subset M . We show that this is an abelian group and that if the subfactor is strongly amenable, then the group coincides with the relative Connes invariant χ ( M , N ) \chi (M,N) introduced by Y. Kawahigashi. We also show that this group is contained in the center of Out ⁡ ( M , N ) \operatorname {Out}(M,N) in many interesting examples such as quantum S U ( n ) k SU(n)_{k} subfactors with level k k ( k ≥ n + 1 ) (k \geq n+1) , but not always contained in the center. We also discuss its relation to the most general setting of the orbifold construction for subfactors.

On the uniqueness of the twisted representation in the [formula omitted]2 orbifold construction of a conformal field theory from a lattice

Following on from recent work describing the representation content of a meromorphic bosonic conformal field theory in terms of a certain state inside the theory corresponding to a fixed state in the representation, and using work of Zhu on a correspondence between the representations of the conformal field theory and representations of a particular associative algebra constructed from it, we construct a general solution for the state defining the representation and identify the further restrictions on it necessary for it to correspond to a ground state in the representation space. We then use this general theory to analyze the representations of the Heisenberg algebra and its Z 2-projection. The conjectured uniqueness of the twisted representation is shown explicitly, and we extend our considerations to the reflection-twisted FKS construction of a conformal field theory from a lattice.

On representations of conformal field theories and the construction of orbifolds

We consider representations of meromorphic bosonic chiral conformal field theories, and demonstrate that such a representation is completely specified by a state within the theory. The necessary and sufficient conditions upon this state are derived, and, because of their form, we show that we may extend the representation to a representation of a suitable larger conformal field theory. In particular, we apply this procedure to the lattice (FKS) conformal field theories, and deduce that Dong's proof of the uniqueness of the twisted representation for the reflection-twisted projection of the Leech lattice conformal field theory generalises to an arbitrary even (self-dual) lattice. As a consequence, we see that the reflection-twisted lattice theories of Dolan et al are truly self-dual, extending the analogies with the theories of lattices and codes which were being pursued. Some comments are also made on the general concept of the definition of an orbifold of a conformal field theory in relation to this point of view.

On Flatness of Ocneanu′s Connections on the Dynkin Diagrams and Classification of Subfactors

We will give a proof of Ocneanu′s announced classification of subfactors of the AFD type II 1 factor with the principal graphs A n , D n , E 7 , the Dynkin diagrams, and give a single explicit of exp π √ -1/24 and exp π √-1/60 for each of E 6 and E 8 such that its validity is equivalent to the existence of two (and only two) subfactors for these principal graphs. Our main tool is the flatness of connections on finite graphs, which is the key notion of Ocneanu′s paragroup theory. We give the difference between the diagrams D 2 n and D 2 n+1 a meaning as a Z/2 Z-obstruction for flatness arising in orbifold construction, which is an analogue of orbifold models in solvable lattice models.

ORBIFOLD CONSTRUCTION FOR NON-AFD SUBFACTORS

We extend the orbifold construction to arbitrary (not necessarily AFD) subfactors. That is, we construct subfactors with principal graphs D2n from those with principal graphs A4n−3 by taking simultaneous Z2-crossed products with non-strongly outer actions. Our result can be applied to Popa's universal subfactors with principal graphs A4n−3, for example.

Simple currents versus orbifolds with discrete torsion — a complete classification

We give a complete classification of all simple current modular invariants, extending previous results for (Z p ) k to arbitrary centers. We obtain a simple explicit formula for the most general case. Using orbifold techniques to this end, we find a one-to-one correspondence between simple current invariants and subgroups of the center with discrete torsions. As a by-product, we prove the conjectured monodromy independence of the total number of such invariants. The orbifold approach works in a straightforward way for symmetries of odd order, but some modifications are required to deal with symmetries of even order. With these modifications the orbifold construction with descrete torsion is complete within the class of simple current invariants. Surprisingly, there are cases where discrete torsion is a necessity rather than a possibility.

On the relationship between Monstrous Moonshine and the uniqueness of the Moonshine Module

We consider the relationship between the conjectured uniqueness of the Moonshine Module, , and Monstrous Moonshine, the genus zero property of the modular invariance group for each Monster group Thompson series. We first discuss a family of possibleZ n meromorphic orbifold constructions of based on automorphisms of the Leech lattice compactified bosonic string. We reproduce the Thompson series for all 51 non-Fricke classes of the Monster groupM together with a new relationship between the centralisers of these classes and 51 corresponding Conway group centralisers (generalising a well-known relationship for 5 such classes). Assuming that is unique, we consider meromorphic orbifoldings of and show that Monstrous Moonshine holds if and onlyZ r if the only meromorphic orbifoldings of are itself or the Leech theory. This constraint on the meromorphic orbifoldings of therefore relates Monstrous Moonshine to the uniqueness of in a new way.

Anomaly cancellation in six dimensions

It is shown that anomaly cancellation conditions are sufficient to determine the two most important topological numbers relevant for Calabi–Yau (CY) compactification to six dimensions. This reflects the fact that K3 is the only nontrivial CY manifold in two complex dimensions. The Green–Schwarz counterterms are explicitly constructed and sum rules for charges of additional enhanced U(1) factors are derived and the results with all possible Abelian orbifold constructions of K3 are compared. This includes asymmetric orbifolds as well, showing that it is possible to regain a geometrical interpretation for this class of models. Finally, some models with a broken E7 gauge group which will be useful for more phenomenological applications are discussed herein.

An orbifold compactification with three families from twisted sectors

We obtain a three generational SU(3) c×SU(3) w×U(1) 4×[SO(12)×U(1) 2]′ model from an orbifold construction with the requirement that three generations arise from twisted sectors. There exist supersymmetric vacua realizing the standard model. In one example the anomalous U(1) breaks the gauge symmetry down to SU(3) c×SU(3) w×U(1) y×SO(12)′.

COMPLETE STRUCTURE OF Zn YUKAWA COUPLINGS

We give the complete twisted Yukawa couplings for all the Zn orbifold constructions in the most general case, i.e. when orbifold deformations are considered. This includes a certain number of tasks. Namely, determination of the allowed couplings, calculation of the explicit dependence of the Yukawa couplings values on the moduli expectation values (i.e. the parameters determining the size and shape of the compactified space), etc. The final expressions are completely explicit, which allows a counting of the different Yukawa couplings for each orbifold (with and without deformations). This knowledge is crucial to determine the phenomenological viability of the different schemes, since it is directly related to the fermion mass hierarchy. Other facts concerning the phenomenological profile of Zn orbifolds are also discussed, e.g. the existence of nondiagonal entries in the fermion mass matrices, which is related to a nontrivial structure of the Kobayashi-Maskawa matrix. Some theoretical results are also given, e.g. the non-participation of (1,2) moduli in twisted Yukawa couplings. Likewise, (1,1) moduli associated with fixed tori which are involved in the Yukawa coupling, do not affect the value of the coupling. We discuss the relevance of these facts for the supersymmetry breaking process.

Soft SUSY breaking terms in stringy scenarios. Computation and phenomenological viability

We calculate the soft SUSY breaking terms arising from a large class of string scenarios, namely symmetric orbifold constructions, and study its phenomenological viability. They exhibit a certain lack of universality, unlike the usual assumptions of the minimal supersymmetric standard model. Assuming gaugino condensation in the hidden sector as the source of SUSY breaking, it turns out that squark and slepton masses tend to be much larger than gaugino masses. Furthermore, we show that these soft breaking terms can be perfectly consistent with both experimental and naturalness constraints (the latter comes from the absence of fine tuning in the SU(2) × U(1) Y → U(1) em breaking process). This is certainly non-trivial and in fact imposes interesting constraints on measurable quantities. More precisely, we find that the gluino mass ( M 3) and the chargino mass ( M χ ±) cannot be much higher than their present experimental lower bounds ( M 3⪅285 GeV; M χ ± ⪅80 GeV), while squark and slepton masses must be much larger ( ⪆1 TeV ). This can be considered as an observational signature of this kind of stringy scenarius. Besides, the top mass is constrained to be within a range (80⪅ m t ⪅165 GeV) remarkably consistent with its present experimental bounds.

Fitting the quark and lepton masses in string theories

The capability of string theories to reproduce at low energy the observed pattern of quark and lepton masses and mixing angles is examined, focusing the attention on orbifold constructions, where the magnitude of Yukawa couplings depends on the values of the deformation parameters which describe the size and shape of the compactified space. A systematic exploration shows that for Z 3, Z 4, Z 6-I and possibly Z 7 orbifolds a correct fit of the physical fermion masses is feasible. In this way the experimental masses, which are low-energy quantities, select a particular size and shape of the compactified space, which turns out to be very reasonable (in particular the modulus T defining the former is T=O(1)). The rest of the Z N orbifolds are rather hopeless and should be discarded on the assumption of a minimal SU(3)×SU(2)×U(1) Y scenario. On the other hand, due to stringy selection rules, there is no possibility of fitting the Kobayashi-Maskawa parameters at the renormalizable level, although it is remarked that this job might well be done by non-renormalizable couplings.

Monstrous Moonshine from orbifolds

We consider the Monster Module of Frenkel, Lepowsky, and Meurman as aZ 2 orbifold of a bosonic string compactified by the Leech lattice. We show that the main Conway and Norton Monstrous Moonshine properties, stating that the Thompson series for each Monster group conjugacy class has a modular invariance group of genus zero, follow from an orbifold construction based on an orbifold group composed of Monster group elements. it is shown that a conjectured vacuum structure for the orbifold twisted sectors is sufficient to specify the modular group and the genus zero property for each Thompson series. It is also shown that the Power Map formula of Conway and Norton follows from the same vacuum structure. Finally, we demonstrate the validity of the vacuum conjectures for sectors twisted by Leech lattice automorphisms in many cases.

SU(2)k MODULAR-INVARIANT PARTITION FUNCTIONS FROM ORBIFOLDS

We present a method which generates the modular-invariant partition functions of the ADE series of SU(2)k. Dividing the diagonal theory by discrete subgroups of the conformal group, we construct all the modular-invariant partition functions, thus proving that orbifold construction generates all the partition functions of SU(2)k.

Asymmetric orbifolds: Path integral and operator formulations

We discuss some of the details involved in asymmetric orbifold construction. In particular, the path-integral formalism is developed and shown to be consistent with operator interpretation. Conditions of one-loop modular invariance are discussed in detail. We also describe how one can construct correlation functions involving twisted states.

Duality in orbifolds and Ginzburg-Landau models

The modular transformation properties of all fields in the orbifold construction are given. The same is done for the modal deformations in Ginzburg-Landau potentials, via a parametrization of these in terms of correlators of twist fields.

TYPE II SUPERSTRINGS ON TWISTED GROUP MANIFOLDS AND THEIR HETEROTIC COUNTERPARTS

We construct Type II superstrings in four space-time dimensions compactified on a twisted Wess-Zumino-Witten model based on the group SU(2)3. It is shown that within this framework, it is possible to obtain models with N=6, 5 and 3 space-time supersymmetries, in addition to the usual models with N=8, 4 and 2. The map of these models into the corresponding heterotic superstrings with N=4, 2 and 1 space-time supersymmetries is also derived: in complete analogy to the compactifications on six-dimensional manifolds also in this case this map from Type II to heterotic superstrings corresponds geometrically to the embedding of the 9-dimensional compact manifold spin-connection into the gauge connection. The superstring compactifications we discussed are equivalent to asymmetric orbifold constructions in six-dimensions with no necessity, however, of introducing chiral bosons.

Strings on world-sheet orbifolds

A new orbifold construction is proposed in which orbifold groups can also act on the ghost fields. The crucial role of BRST invariance in the construction is analyzed. Type-I superstrings (on ordinary orbifolds) fit naturally into this scheme as world-sheet orbifolds of the type-II superstring theory. More exotic examples are presented as well. Finally, some geometrical aspects of the construction are discussed.

A unified formalism for strings in four dimensions

We develop a simple and unified formalism to construct four-dimensional string models consistent with modular invariance. The boundary conditions are allowed to be twisted for fermions while bosons are either twisted or shifted. We show that the generalized GSO projection conditions follow from our solution to the modular invariance of the total partition function. The Green-Schwarz strings and the Neveu-Schwarz-Ramond strings are treated in a unified way. We give the conditions needed to preserve world-sheet supersymmetry and to characterize N = 1, 2, 4 space-time supersymmetry. Our formalism is general enough to contain the fermionic, lattice, and orbifold constructions of four-dimensional strings and we provide its connection to some of them.

Non-critical orbifolds

We generalize the notion of orbifold to statistical mechanical models away from their critical points. In the simplest models, our non-critical orbifold construction reduces to Kramers-Wannier duality. More generally, it gives a new model whose toroidal partition function can be expressed in terms of that of the original model. The construction also preserves the star-triangle relations, so that the “orbifold” of an exactly solvable model remains exactly solvable. At a critical point, our construction reduces to the conventional orbifold construction of conformal field theory.