We give the full set of $S$ matrices for extensions of $D(n)_1$ permutation orbifolds, extending our previous work to the yet unknown case of integer spin spinor currents. The main tool is triality of SO(8). We also provide fixed point resolution matrices for spinor currents of $D(n)_1$ permutation orbifolds with $n$ even and not multiple of four, where the spinor currents have half-integer spin.
I consider the $\mathbb{Z}_λ,$ $λ$ prime free-bosonic permutation orbifolds as interacting physical string systems at $\hat{c} = 26λ$. As a first step, I introduce twisted tree diagrams which confirm at the interacting level that the physical spectrum of each twisted sector is equivalent to that of an ordinary $c=26$ closed string. The untwisted sectors are surprisingly more difficult to understand, and there are subtleties in the sewing of the loops, but I am able to propose provisional forms for the full modular-invariant cosmological constants and one-loop diagrams with insertions.
We summarize the results obtained in the last few years about permutation orbifolds in two-dimensional conformal field theories, their application to string theory and their use in the construction of four-dimensional heterotic string models.
We find a formula for the resolution of fixed points in extensions of permutation orbifold conformal field theories by its (half-)integer spin simple currents. We show that the formula gives a unitary and modular invariant S matrix.
This is the first in a series of papers in which I investigate the orbifolds of permutation-type as candidates for {\it new physical string systems} at multiples of critical closed-string central charges. Examples include the bosonic orientation orbifolds, the bosonic permutation orbifolds and others, as well as superstring extensions. In this paper I use the extended (twisted) Virasoro algebras of these orbifolds to construct the corresponding extended action formulations of the twisted open- and closed-string sectors of all the bosonic orbifolds at $\hatc=26K$. The extended actions exhibit a…
In this paper we apply the previously derived formalism of permutation orbifold conformal field theories to N=2 supersymmetric minimal models. By interchanging extensions and permutations of the factors we find a very interesting structure relating various conformal field theories that seems not to be known in literature. Moreover, unexpected exceptional simple currents arise in the extended permuted models, coming from off-diagonal fields. In a few situations they admit fixed points that must be resolved. We determine the complete CFT data with all fixed point resolution matrices for all simp…
We review extensions by integer spin simple currents in two-dimensional conformal field theories and their applications in string theory. In particular, we study the problem of resolving the fixed points of a simple current and apply the formalism to the permutation orbifold. In terms of string compactifications, we construct permutations of N=2 minimal models and use them as building blocks in heterotic Gepner models.
We study the spectrum of permutation orbifolds of 2d CFTs. We find examples where the light spectrum grows faster than Hagedorn, which is different from known cases such as symmetric orbifolds. We also describe how to compute their partition functions using a generalization of Hecke operators.
We study orbifolds by permutations of two identical N=2 minimal models within the Gepner construction of four dimensional heterotic strings. This is done using the new N=2 supersymmetric permutation orbifold building blocks we have recently developed. We compare our results with the old method of modding out the full string partition function. The overlap between these two approaches is surprisingly small, but whenever a comparison can be made we find complete agreement. The use of permutation building blocks allows us to use the complete arsenal of simple current techniques that is available …
In this fourth paper of the series, I clarify the somewhat mysterious relation between the large class of {\it orientation orbifolds} (with twisted open-string CFT's at $\hat c=52$) and {\it orientifolds} (with untwisted open strings at $c=26$), both of which have been associated to division by world-sheet orientation-reversing automorphisms. In particular -- following a spectral clue in the previous paper -- I show that, even as an {\it interacting string system}, a certain half-integer-moded orientation orbifold-string system is in fact equivalent to the archetypal orientifold. The subtitle …
We determine the simple currents and fixed points of the orbifold theory $CFT\otimes CFT/\mathbb{Z}_2$, given the simple currents and fixed point of the original $CFT$. We see in detail how this works for the $SU(2)_k$ WZW model, focusing on the field content (i.e. $h$-spectrum of the primary fields) of the theory. We also look at the fixed point resolution of the simple current extended orbifold theory and determine the $S^J$ matrices associated to each simple current for $SU(2)_2$ and for the $B(n)_1$ and $D(n)_1$ series.
This is the second in a series of papers which consider the orbifolds of permutation-type as candidates for new physical string systems at higher central charge. In the first paper, I worked out the extended actions of the twisted sectors of these orbifolds -- which exhibit new permutation-twisted world-sheet gravities and correspondingly extended diffeomorphism groups. In this paper I begin the study of these systems as operator string theories, limiting the discussion for simplicity to the strings with ${\hat c} = 52$ matter (which are those governed by ${\mathbb Z}_2$-twisted permutation gr…
We consider properties of solitons in general orbifolds in the algebraic quantum field theory framework and constructions of solitons in affine and permutation orbifolds. Under general conditions we show that our construction gives all the twisted representations of the fixed point subnet. This allows us to prove a number of conjectures: in the affine orbifold case we clarify the issue of ``fixed point resolutions''; in the permutation orbifold case we determine all irreducible representations of the orbifold, and we also determine the fusion rules in a nontrivial case, which imply an integral…
Recently the operator algebra, including the twisted affine primary fields, and a set of twisted KZ equations were given for the WZW permutation orbifolds. In the first part of this paper we extend this operator algebra to include the so-called orbifold Virasoro algebra of each WZW permutation orbifold. These algebras generalize the orbifold Virasoro algebras (twisted Virasoro operators) found some years ago in the cyclic permutation orbifolds. In the second part, we discuss the reducibility of the twisted affine primary fields of the WZW permutation orbifolds, obtaining a simpler set of singl…
Symmetric product orbifolds, i.e. permutation orbifolds of the full symmetric group S_{n} are considered by applying the general techniques of permutation orbifolds. Generating functions for various quantities, e.g. the torus partition functions and the Klein-bottle amplitudes are presented, as well as a simple expression for the discrete torsion coefficients.
Explicit formulae describing the genus one characters and modular transformation properties of permutation orbifolds of arbitrary Rational Conformal Field Theories are presented, and their relation to the theory of covering surfaces is investigated.
The theory of permutation orbifolds is reviewed and applied to the study of symmetric product orbifolds and the congruence subgroup problem. The issue of discrete torsion, the combinatorics of symmetric products, the Galois action and questions related to the classification of RCFTs are also discussed.
The space of permutation orbifolds is a simple landscape of two dimensional CFTs, generalizing the well-known symmetric orbifolds. We consider constraints which a permutation orbifold with large central charge must obey in order to be holographically dual to a weakly coupled (but possibly stringy) theory of gravity in AdS. We then construct explicit examples of permutation orbifolds which obey these constraints. In our constructions the spectrum remains finite at large N, but differs qualitatively from that of symmetric orbifolds. We also discuss under what conditions the correlation functions…
We study out-of-time-ordered correlation functions in permutation orbifolds at large central charge. We show that they do not decay at late times for arbitrary choices of low-dimension operators, indicating that permutation orbifolds are non-chaotic theories. This is in agreement with the fact they are free discrete gauge theories and should be integrable rather than chaotic. We comment on the early-time behaviour of the correlators as well as the deformation to strong coupling.
The torus and the Klein bottle amplitude coefficients are computed in permutation orbifolds of RCFT-s in terms of the same quantities in the original theory and the twist group. An explicit expression is presented for the number of self conjugate primaries in the orbifold as a polynomial of the total number of primaries and the number of self conjugate ones in the parent theory. The formulae in the $Z_2$ orbifold illustrate the general results.
In the second paper of this series, I obtained the twisted BRST systems and extended physical-state conditions of all twisted open and closed $\hat{c} = 52$ strings. In this paper, I supplement the extended physical-state conditions with the explicit form of the extended (twisted) Virasoro generators of all $\hat{c} = 52$ strings, which allows us to discuss the physical spectra of these systems. Surprisingly, all the $\hat{c}=52$ spectra admit an equivalent description in terms of generically-unconventional Virasoro generators at $c=26$. This description strongly supports our prior conjecture …
We consider the application of permutation orbifold constructions towards a new possible understanding of the genus zero property in Monstrous and Generalized Moonshine. We describe a theory of twisted Hecke operators in this setting and conjecture on the form of Generalized Moonshine replication formulas.
A general theory of permutation orbifolds is developed for arbitrary twist groups. Explicit expressions for the number of primaries, the partition function, the genus one characters, the matrix elements of modular transformations and for fusion rule coefficients are presented, together with the relevant mathematical concepts, such as Lambda-matrices and twisted dimensions. The arithmetic restrictions implied by the theory for the allowed modular representations in CFT are discussed. The simplest nonabelian example with twist group S_3 is described to illustrate the general theory.
The hypothesis that every theory of quantum gravity in AdS_3 is a dimensional reduction of string/M-theory leads to a natural conjecture for the density of states of two dimensional CFTs with a large central charge limit. We prove this conjecture for 2D CFTs which are orbifolds by permutation groups. In particular, we characterize those permutation groups which give CFTs with well-defined large N limits and can thus serve as holographic duals to bulk gravity theories in AdS_3. We then show that the holographic dual of a permutation orbifold will have a Hagedorn spectrum in the large N limit. T…
To illustrate the general results of the previous paper, we discuss here a large concrete example of the orbifold-string theories of permutation-type. For each of the many subexamples, we focus on evaluation of the \emph{target space-time dimension} $\hat{D}_j(σ)$, the \emph{target space-time signature} and the \emph{target space-time symmetry} of each cycle $j$ in each twisted sector $σ$. We find in particular a gratifying \emph{space-time symmetry enhancement} which naturally matches the space-time symmetry of each cycle to its space-time dimension. Although the orbifolds of $\Z_{2}$-permuta…
Two dimensional conformal field theories with large central charge and a sparse low-lying spectrum are expected to admit a classical string holographic dual. We construct a large class of such theories employing permutation orbifold technology. In particular, we describe the group theoretic constraints on permutation groups to ensure a (stringy) holographic CFT. The primary result we uncover is that in order for the degeneracy of states to be finite in the large central charge limit, the groups of interest are the so-called oligomorphic permutation groups. Further requiring that the low-lying …
We carry out a systematic study of primary operators in the conformal field theory of a free Weyl fermion. Using SO(4,2) characters we develop counting formulas for primaries constructed using a fixed number of fermion fields. By specializing to particular classes of primaries, we derive very explicit formulas for the generating functions for the number of primaries in these classes. We present a duality map between primary operators in the fermion field theory and polynomial functions. This allows us to construct the primaries that were counted. Next we show that these classes of primary fiel…
The connection between the theory of permutation orbifolds, covering surfaces and uniformization is investigated, and the higher genus partition functions of an arbitrary permutation orbifold are expressed in terms of those of the original theory.
To illustrate our recent discussions of the target space-times in general orbifold-string theories of permutation-type, we return here to a detailed analysis of some simple examples of this type, namely an explicit set of orientation-orbifold string systems. These orientation-orbifold string systems provide twisted, multisector generalizations of ordinary critical open-closed bosonic string systems -- each such system exhibiting a unique graviton. Furthermore, each sector $σ$ of these string systems shows the following properties: a) 26 effective degrees of freedom, b) a Lorentzian space-time …
We continue our discussion of the general bosonic prototype of the new orbifold-string theories of permutation type. Supplementing the extended physical-state conditions of the previous paper, we construct here the extended Virasoro generators with cycle central charge $\hat{c}_j(σ)=26f_j(σ)$, where $f_j(σ)$ is the length of cycle $j$ in twisted sector $σ$. We also find an equivalent, reduced formulation of each physical-state problem at reduced cycle central charge $c_j(σ)=26$. These tools are used to begin the study of the target space-time dimension $\hat{D}_j(σ)$ of cycle $j$ in sector $σ$…
We investigate some aspects of the relationship between matrix string theory and light-cone string field theory by analysing the correspondence between the two-loop thermal partition function of DLCQ strings in flat space and the integrated two-point correlator of twist fields in a symmetric product orbifold conformal field theory at one-loop order. This is carried out by deriving combinatorial expressions for generic twist field correlation functions in permutation orbifolds using the covering surface method, by deriving the one-loop modification of the twist field interaction vertex, and by …
We resume our discussion of the new orbifold-string theories of permutation-type, focusing in the present series on the algebraic formulation of the general bosonic prototype and especially the target space-times of the theories. In this first paper of the series, we construct one twisted BRST system for each cycle $j$ in each twisted sector $σ$ of the general case, verifying in particular the previously-conjectured algebra $[Q_{i}(σ),Q_{j}(σ)]_{+} =0$ of the BRST charges. The BRST systems then imply a set of extended physical-state conditions for the matter of each cycle at cycle central char…
We investigate the large $N$ limit of permutation orbifolds of vertex operator algebras. To this end, we introduce the notion of nested oligomorphic permutation orbifolds and discuss under which conditions their fixed point VOAs converge. We show that if this limit exists, then it has the structure of a vertex algebra. Finally, we give an example based on $\mathrm{GL}(N,q)$ for which the fixed point VOA limit is also the limit of the full permutation orbifold VOA.
We prove the first nontrivial reconstruction theorem for modular tensor categories: the category associated to any twisted Drinfeld double of any finite group, can be realised as the representation category of a completely rational conformal net. We also show that any twisted double of a solvable group is the category of modules of a completely rational vertex operator algebra. In the process of doing this, we identify the 3-cocycle twist for permutation orbifolds of holomorphic conformal nets: unexpectedly, it can be nontrivial, and depends on the value of the central charge modulo 24. In add…
The characteristic property of the $Z_2\times Z_2$ orbifold is its cyclic permutation symmetry. It is argued that this property may be instrumental in explaining simultaneously the fermion mass hierarchy and the squark mass degeneracy. Detailed studies in free fermionic models that preserve the cyclic permutation symmetry of anomalous U(1) charges of the three generations are discussed.
We find new duality transformations which allow us to construct the stress tensors of all the twisted sectors of any orbifold A(H)/H, where A(H) is the set of all current-algebraic conformal field theories with a finite symmetry group H \subset Aut(g). The permutation orbifolds with H = Z_λand H = S_3 are worked out in full as illustrations but the general formalism includes both simple and semisimple g. The motivation for this development is the recently-discovered orbifold Virasoro master equation, whose solutions are identified by the duality transformations as sectors of the permutation or…
Using the Polya Enumeration Theorem, we count with particular attention to C^3/Gamma up to C^6/Gamma, abelian orbifolds in various dimensions which are invariant under cycles of the permutation group S_D. This produces a collection of multiplicative sequences, one for each cycle in the Cycle Index of the permutation group. A multiplicative sequence is controlled by its values on prime numbers and their pure powers. Therefore, we pay particular attention to orbifolds of the form C^D/Gamma where the order of Gamma is p^alpha. We propose a generalization of these sequences for any D and any p.
Recently a construction was given for the stress tensors of all sectors of the general current-algebraic orbifold A(H)/H, where A(H) is any current-algebraic conformal field theory with a finite symmetry group H. Here we extend and further analyze this construction to obtain the mode formulation of each sector of each orbifold A(H)/H, including the twisted current algebra, the Virasoro generators, the orbifold adjoint operation and the commutator of the Virasoro generators with the modes of the twisted currents. As applications, general expressions are obtained for the twisted current-current …
We investigate gauge anomalies in the context of orbifold conformal field theories. Such anomalies manifest as failures of modular invariance in the constituents of the orbifold partition function. We review how this irregularity is classified by cohomology and how extending the orbifold group can remove it. Working with such extensions requires an understanding of the consistent ways in which extending groups can act on the twisted states of the original symmetry, which leads us to a discrete-torsion like choice that exists in orbifolds with trivially-acting subgroups. We review a general met…
We obtain the operator algebra of each twisted sector of all WZW orbifolds, including the general twisted current algebra and the algebra of the twisted currents with the twisted affine primary fields. Surprisingly, the twisted right and left mover current algebras are not a priori copies of each other. Using the operator algebra we also derive world-sheet differential equations for the twisted affine primary fields of all WZW orbifolds. Finally we include ground state properties to obtain the twisted Knizhnik-Zamolodchikov equations of the WZW permutation orbifolds and the inner-automorphic W…
Chiral orbifold models are defined as gauge field theories with a finite gauge group $Γ$. We start with a conformal current algebra A associated with a connected compact Lie group G and a negative definite integral invariant bilinear form on its Lie algebra. Any finite group $Γ$ of inner automorphisms or A (in particular, any finite subgroup of G) gives rise to a gauge theory with a chiral subalgebra $A^Γ\subset A$ of local observables invariant under $Γ$. A set of positive energy $A^Γ$ modules is constructed whose characters span, under some assumptions on $Γ$, a finite dimensional unitary re…
We consider a $A_{N-1}$ type of spin dependent Calogero-Sutherland model, containing an arbitrary representation of the permutation operators on the combined internal space of all particles, and find that such a model can be solved as easily as its standard $su(M)$ invariant counterpart through the diagonalisation of Dunkl operators. A class of novel representations of the permutation operator $P_{ij}$, which pick up nontrivial phase factors along with interchanging the spins of $i$-th and $j$-th particles, are subsequently constructed. These `anyon like' representations interestingly lead to …
A possible extension of the Standard Model of elementary particles is Gauge-Higgs unification, where the Higgs field is identified with (some of) the extra dimensional components of a five-dimensional gauge field. In this scenario there is evidence for the potential and the mass of the Higgs field to be finite. Here we show the behavior of the static potential of a five-dimensional SU(2) lattice gauge theory with orbifold boundary conditions. The potentials are computed within the mean-field approximation including first order corrections.
It is natural to believe that the free symmetric product orbifold CFT is dual to the tensionless limit of string theory on AdS3 x S3 x T4. At this point in moduli space, string theory is expected to contain a Vasiliev higher spin theory as a subsector. We confirm this picture explicitly by showing that the large level limit of the N=4 cosets of arXiv:1305.4181, that are dual to a higher spin theory on AdS3, indeed describe a closed subsector of the symmetric product orbifold. Furthermore, we reorganise the full partition function of the symmetric product orbifold in terms of representations of…
The possibility of appearance of GUT precursors near the TeV scale (suggested by Dienes-Dudas-Gherghetta) is addressed within 5D GUTs compactified on an $S^{(1)}/Z_2\tm Z_2'$ orbifold. For a low compactification scale (large radius), there is a significant non universal logarithmic contribution in the relative running of gauge couplings. This within 5D SU(5), with the minimal field content, gives wrong prediction for $\al_3(M_Z)$ unless one goes beyond the minimal setting. The realization of the light precursors' idea thus requires some specific extensions. As a scenario alternative to SU(5) w…
