Symmetric Orbifold Research Articles

Operator mixing for string states in the D1-D5 CFT near the orbifold point

In the context of the fuzzball programme, we investigate deforming the microscopic string description of the D1-D5 system on T^4xS^1 away from the orbifold point. Using conformal perturbation theory and a generalization of Lunin-Mathur symmetric orbifold technology for computing twist-nontwist correlators developed in a companion work, we initiate a program to compute the anomalous dimensions of low-lying string states in the D1-D5 superconformal field theory. Our method entails finding four-point functions involving a string operator O of interest and the deformation operator, taking coincidence limits to identify which other operators mix with O, subtracting the identified conformal family to isolate other contributions to the four-point function, finding the mixing coefficients, and iterating. For the lowest-lying string modes, this procedure should truncate in a finite number of steps. We check our method by showing how the operator dual to the dilaton does not participate in mixing that would change its conformal dimension, as expected. Next we complete the first stage of the iteration procedure for a low-lying string state of the form \partial X \partial X \bar\partial X \bar\partial X and find its mixing coefficient. Our main qualitative result is evidence of operator mixing at first order in the deformation parameter, which means that the string state acquires an anomalous dimension. After diagonalization this will mean that anomalous dimensions of some string states in the D1-D5 SCFT must decrease away from the orbifold point while others increase.

Classification of symmetric toroidal orbifolds

Abstract We provide a complete classification of six-dimensional symmetric toroidal orbifolds which yield $ \mathcal{N}\geq 1 $ supersymmetry in 4D for the heterotic string. Our strategy is based on a classification of crystallographic space groups in six dimensions. We find in total 520 inequivalent toroidal orbifolds, 162 of them with Abelian point groups such as $ {{\mathbb{Z}}_3},{{\mathbb{Z}}_4},{{\mathbb{Z}}_6}\hbox{-}\mathrm{I} $ etc. and 358 with non-Abelian point groups such as S 3, D 4, A 4 etc. We also briefly explore the properties of some orbifolds with Abelian point groups and $ \mathcal{N}=1 $ , i.e. specify the Hodge numbers and comment on the possible mechanisms (local or non-local) of gauge symmetry breaking.

Phase transitions in symmetric orbifold CFTs and universality

Since many thermodynamic properties of black holes are universal, the thermodynamics of their holographic duals should be universal too. We show how this universality is exhibited in the example of symmetric orbifolds of general two dimensional CFTs. We discuss the free energies and phase diagrams of such theories and show that they are indeed universal in the large N limit. We also comment on the implications of our results for the classification of CFTs that can have an interpretation as holographic duals to gravity theories on AdS(3).

A spin chain for the symmetric product CFT2

We consider “gauge invariant” operators in SymN T 4, the symmetric product orbifold of N copies of the 2d supersymmetric sigma model with T 4 target. We discuss a spin chain representation for single-cycle operators and study their two point functions at large N. We perform systematic calculations at the orbifold point (“tree level”), where non-trivial mixing is already present, and some sample calculations to first order in the blow-up mode of the orbifold (“one loop”).

Extremal correlators and Hurwitz numbers in symmetric product orbifolds

We study correlation functions of single-cycle chiral operators in ${\mathrm{Sym}}^{N}{T}^{4}$, the symmetric product orbifold of $N$ supersymmetric four-tori. Correlators of twist operators are evaluated on covering surfaces, generally of different genera, where fields are single valued. We compute some simple four-point functions and study how the sum over inequivalent branched covering maps splits under operator product expansions. We then discuss extremal $n$-point correlators, i.e. correlators of $n\ensuremath{-}1$ chiral and one antichiral operators. They obey simple recursion relations involving numbers obtained from counting branched covering maps with particular properties. In most cases we are able to solve explicitly the recursion relations. Remarkably, extremal correlators turn out to be equal to Hurwitz numbers.

Diagrams for symmetric product orbifolds

We develop a diagrammatic language for symmetric product orbifolds of two-dimensional conformal field theories. Correlation functions of twist operators are written as sums of diagrams: each diagram corresponds to a branched covering map from a surface where the fields are single-valued to the base sphere where twist operators are inserted. This diagrammatic language facilitates the study of the large N limit and makes more transparent the analogy between symmetric product orbifolds and free non-abelian gauge theories. We give a general algorithm to calculate the leading large N contribution to four-point correlators of twist fields.

Three-point functions in superstring theory on AdS3× S3× T4

We consider R and NS spectral flow sectors of type IIB superstring theory on AdS(3)xS(3)xT(4) in the context of the AdS(3)/CFT(2) correspondence. We present a derivation of the vertex operators creating spectral flow images of chiral primary states previously proposed in the literature. We compute spectral flow conserving three-point functions involving these operators on the sphere. Using the bulk-to-boundary dictionary, we compare the results with the corresponding correlators in the dual conformal field theory, the symmetric product orbifold of T(4). In the limit of small string coupling, agreement is found in all the cases considered.

Infinite product decomposition of orbifold mapping spaces

Physicists showed that the generating function of orbifold elliptic genera of symmetric orbifolds can be written as an infinite product. We show that there exists a geometric factorization on space level behind this infinite product formula in much more general framework, where factors in the infinite product correspond to isomorphism classes of connected finite covering spaces of manifolds involved. From this formula, a concept of geometric Hecke operators for functors emerges. We show that these Hecke operators indeed satisfy the usual identity of Hecke operators for the case of 2-dimensional tori.

Exact [formula omitted] correlators of [formula omitted

We extend to chiral N=4 operators the holographic agreement recently found between correlators of the symmetric orbifold of M4 at large N and type IIB strings propagating in AdS3×S3×M4, where M4=T4 or K3. We also present expressions for some bulk correlators not yet computed in the boundary.

The loop orbifold of the symmetric product

Using the loop orbifold of the symmetric product, we give a formula for the Poincaré polynomial of the free loop space of the Borel construction of the symmetric product. We also show that the Chas–Sullivan orbifold product structure in the homology of the free loop space of the Borel construction of the symmetric product induces a ring structure in the homology of the inertia orbifold of the symmetric product. For a general almost complex orbifold, we define a new ring structure on the cohomology of its inertia orbifold which we call the virtual intersection ring. Finally we show that under Poincaré duality in the case of the symmetric product orbifold, both ring structures are isomorphic.

LACK OF TRINIFICATION IN Z3 ORBIFOLDS OF THE SO(32) HETEROTIC STRING

We report results relating to the trinification scenario in some explicit string constructions that contain SU (3)3 as a gauge symmetry. These models are obtained from symmetric Z3 orbifolds of the SO (32) heterotic string with one discrete Wilson line. We highlight the obstacles that were encountered: the absence of the usual Higgs sector that would break SU (3)3→ SU (3)c× SU (2)L× U (1)Y; the presence of exotics that would generically befoul gauge coupling unification and lead to fractionally-charged states in the low energy spectrum.

Searching for realistic 4d string models with a Pati–Salam symmetry. Orbifold grand unified theories from heterotic string compactification on a [formula omitted] orbifold

Motivated by orbifold grand unified theories, we construct a class of three-family Pati–Salam models in a Z 6 Abelian symmetric orbifold with two discrete Wilson lines. These models have marked differences from previously-constructed three-family models in prime-order orbifolds. In the limit where one of the six compactified dimensions (which lies in a Z 2 sub-orbifold) is large compared to the string length scale, our models reproduce the supersymmetry and gauge symmetry breaking pattern of 5d orbifold grand unified theories on an S 1 / Z 2 orbicircle. We find a horizontal 2 + 1 splitting in the chiral matter spectra—2 families of matter are localized on the Z 2 orbifold fixed points, and 1 family propagates in the 5d bulk—and identify them as the first-two and third families. Remarkably, the first two families enjoy a non-Abelian dihedral D 4 family symmetry, due to the geometric setup of the compactified space. In all our models there are always some color triplets, i.e., ( 6 , 1 , 1 ) representations of the Pati–Salam group, survive orbifold projections. They could be utilized to spontaneously break the Pati–Salam symmetry to that of the Standard Model. One model, with a 5d E 6 symmetry, may give rise to interesting low energy phenomenology. We study gauge coupling unification, allowed Yukawa couplings and some of their phenomenological consequences. The E 6 model has a renormalizable Yukawa coupling only for the third family. It predicts a gauge-Yukawa unification relation at the 5d compactification scale, and is capable of generating reasonable quark/lepton masses and mixings. Potential problems are also addressed, they may point to the direction for refining our models.

On the construction of asymmetric orbifold models

Various asymmetric orbifold models based on chiral shifts and chiral reflections are investigated. Special attention is devoted to the consistency of the models with two fundamental principles for asymmetric orbifolds: modular invariance and the existence of a proper Hilbert space formulation for states and operators. The interplay between these two principles is non-trivial. It is shown, for example, that their simultaneous requirement forces the order of a chiral reflection to be 4, instead of the naive 2. A careful explicit construction is given of the associated one-loop partition functions. At higher loops, the partition functions of asymmetric orbifolds are built from the chiral blocks of associated symmetric orbifolds, whose pairings are determined by degenerations to one-loop.

Constructing 5d orbifold grand unified theories from heterotic strings

A three-generation Pati–Salam model is constructed by compactifying the heterotic string on a particular T6/Z6 Abelian symmetric orbifold with two discrete Wilson lines. The compactified space is taken to be the Lie algebra lattice G2⊕SU(3)⊕SO(4). When one dimension of the SO(4) lattice is large compared to the string scale, this model reproduces many features of a 5d SO(10) grand unified theory compactified on an S1/Z2 orbifold. (Of course, with two large extra dimensions we can obtain a 6d SO(10) grand unified theory.) We identify the orbifold parities and other ingredients of the orbifold grand unified theories in the string model. Our construction provides a UV completion of orbifold grand unified theories, and gives new insights into both field theoretical and string theoretical constructions.

Generalized orbifold Euler characteristics of symmetric orbifolds and covering spaces

Let G be a finite group and let M be a G-manifold. We introduce the concept of generalized orbifold invariants of M/G associated to an arbitrary group Gamma, an arbitrary Gamma-set, and an arbitrary covering space of a connected manifold Sigma whose fundamental group is Gamma. Our orbifold invariants have a natural and simple geometric origin in the context of locally constant G-equivariant maps from G-principal bundles over covering spaces of Sigma to the G-manifold M. We calculate generating functions of orbifold Euler characteristic of symmetric products of orbifolds associated to arbitrary surface groups (orientable or non-orientable, compact or non-compact), in both an exponential form and in an infinite product form. Geometrically, each factor of this infinite product corresponds to an isomorphism class of a connected covering space of a manifold Sigma. The essential ingredient for the calculation is a structure theorem of the centralizer of homomorphisms into wreath products described in terms of automorphism groups of Gamma-equivariant G-principal bundles over finite Gamma-sets. As corollaries, we obtain many identities in combinatorial group theory. As a byproduct, we prove a simple formula which calculates the number of conjugacy classes of subgroups of given index in any group. Our investigation is motivated by orbifold conformal field theory.

Symmetric products, permutation orbifolds and discrete torsion

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.

Renormalization of the Planck mass for type II superstrings on symmetric orbifolds

We compute the one-loop renormalization of the Planck mass for type II string theories compactified to four dimensions on symmetric orbifolds that preserve ${\cal N}=2$ supersymmetry. Depending on the orbifold, the effect can be as large as to compete with the standard tree-level value.

Superstrings on PP-Wave Backgrounds and Symmetric Orbifolds

We study the superstring theory on pp-wave background with NSNS-flux that is realized as the Penrose limit of AdS_3 x S^3 x M^4, where M^4 is T^4 or T^4/Z_2(~ K3). Quantizing this system in the covariant gauge, we explicitly construct the space-time supersymmetry algebra and the complete set of DDF operators. We analyse the spectrum of physical states by using the spectrally flowed representations of current algebra. This spectrum is classified by the ``short string sectors'' and the ``long string sectors'' as in AdS_3 string theory. The states of the latter propagate freely along the transverse plane of pp-wave background, but the states of the former do not. We compare the short string spectrum with the BPS and almost BPS states which have large R-charges in the symmetric orbifold conformal theory, which is known as the candidate of dual theory of superstrings on AdS_3 x S^3 x M^4. We show that every short string states can be embedded successfully in the single particle Hilbert space of symmetric orbifold conformal theory.

Superstrings on PP-Wave Backgrounds and Symmetric Orbifolds

Superstrings on PP-Wave Backgrounds and Symmetric Orbifolds

Multi-strings on AdS3 × S3 from matrix string theory

We analyze the Coulomb branch of Matrix string theory in the presence of NS5-branes. If we regard the components of U(1) gauge fields as the dualized longitudinal coordinates, we obtain the symmetric product of AdS_3 x S^3 x R^4 as the geometry of Coulomb branch. We observe that the absence or presence of the nonzero electric flux determines whether the string propagates in bulk as an ordinary closed string or is forced to live near the boundary. We further discuss the issues of the physical spectrum from the viewpoint of Matrix string theory. We show that the twisted sectors of CFT on the symmetric orbifold, which correspond to glued strings, turn out to yield many chiral primaries that were hitherto considered to be missing. We also comment on the threshold energy in Liouville sector where continuous spectrum begins.