Orbifold Theories Research Articles

Type II string theory on AdS3×S3×T4 and symmetric orbifolds

We discuss in detail the 1+1-dimensional superconformal field theory dual to type II string theory on AdS3×S3×T4, emphasizing the string theoretic aspects of this duality. For one unit of Neveu-Schwarz (NS-NS) 5-brane flux (Q5=1), this string theory has been suggested to be dual to a grand-canonical ensemble of T4N/SN free symmetric orbifold conformal field theories (CFTs). We show how the string genus expansion emerges to all orders for the free orbifold grand-canonical correlation functions. We also discuss how the strong coupling limit of the NS-NS string theory arises (even at large N) in the free orbifold description, and argue why this limit does not have a weakly coupled RR description. The dual conformal field theory (CFT) includes (for all values of Q5) an extra T4 factor that is decoupled from perturbative string theory. We discuss the exactly marginal deformations that relate the different values of Q5, including the precise JJ¯ deformations mixing this extra T4 with the symmetric orbifold. Published by the American Physical Society 2024

The lion, the witch, and the wormhole: ensemble averaging the symmetric product orbifold

We consider the ensemble average of two dimensional symmetric product orbifold CFTs SymN(\U0001d54bD) over the Narain moduli space. We argue for a bulk dual given by N copies of an abelian Chern-Simons theory coupled to topological gravity, endowed with a discrete gauge symmetry exchanging the N copies. As a check of this proposal, we calculate the ensemble average of various partition and correlation functions of the symmetric product orbifold theory and compare the resulting expressions to gauge theory quantities in the bulk. We comment on the ensemble average of the tensionless string partition function on AdS3 × S3 × \U0001d54b4 by considering the specific case of D = 4 with the addition of supersymmetry.

Spectral flow and localisation in AdS3 string theory

We study string theory in three-dimensional Anti-de Sitter spacetime in the path integral formalism. We derive expressions for generic spectrally-flowed near-boundary vertex operators in the Wakimoto representation, and relate their correlation functions to covering maps from the worldsheet to the target space boundary. We show that the path integral structurally reproduces correlation functions of the dual symmetric orbifold theory. By rephrasing spectral flow as the introduction of a background gauge field, we provide a path integral derivation of the localisation property of the near boundary theory. We then focus on the case of IIB string theory on AdS3 × S3 × \U0001d54b4 with k = 1 units of NS-NS flux, where the relationship between correlation functions and covering maps can be made sharp. We also comment on the relation of the k = 1 theory and twistor theory.

Tensionless strings on AdS3 orbifolds

The bound state of one NS5 brane (wrapped on a \U0001d54b4) and N NS1-branes has two dual descriptions: its low-energy dynamics is described by the symmetric orbifold of \U0001d54b4, while the near horizon geometry is captured by string theory on AdS3 × S3 × \U0001d54b4 with one unit of NS flux. The latter theory is exactly solvable in the hybrid formalism, and this allows one to prove the equivalence of the two descriptions. In this paper we extend this duality to ℤk orbifolds of this AdS3 × S3 background. In particular, we show that the corresponding worldsheet spectrum reproduces exactly the perturbative excitations on top of a certain non-perturbative state in the dual symmetric orbifold theory. Since the AdS/CFT duality map is exact for these models, we obtain an interesting picture of how the duality relates boundary and bulk descriptions.

The operator rings of topological symmetric orbifolds and their large N limit

We compute the structure constants of topological symmetric orbifold theories up to third order in the large N expansion. The leading order structure constants are dominated by topological metric contractions. The first order interactions are single cycles joining while at second order we can have double joining as well as splitting. At third order, single cycle joining obtains genus one contributions. We also compute illustrative small N structure constants. Our analysis applies to all second quantized Frobenius algebras, a large class of algebras that includes the cohomology ring of the Hilbert scheme of points on K3 among many others. We point out interesting open questions that our results raise.

Twisted Sectors for Lagrangian Floer Theory on Symplectic Orbifolds

The notion of twisted sectors play a crucial role in orbifold Gromov-Witten theory. We introduce the notion of dihedral twisted sectors in order to construct Lagrangian Floer theory on symplectic orbifolds and discuss related issues.

Self-duality under gauging a non-invertible symmetry

We discuss two-dimensional conformal field theories (CFTs) which are invariant under gauging a non-invertible global symmetry. At every point on the orbifold branch of c = 1 CFTs, it is known that the theory is self-dual under gauging a ℤ2 × ℤ2 symmetry, and has Rep(H8) and Rep(D8) fusion category symmetries as a result. We find that gauging the entire Rep(H8) fusion category symmetry maps the orbifold theory at radius R to that at radius 2/R. At R = 2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\sqrt{2} $$\\end{document}, which corresponds to two decoupled Ising CFTs (Ising2 in short), the theory is self-dual under gauging the Rep(H8) symmetry. This implies the existence of a topological defect line in the Ising2 CFT obtained from half-space gauging of the Rep(H8) symmetry, which commutes with the c = 1 Virasoro algebra but does not preserve the fully extended chiral algebra. We bootstrap its action on the c = 1 Virasoro primary operators, and find that there are no relevant or marginal operators preserving it. Mathematically, the new topological line combines with the Rep(H8) symmetry to form a bigger fusion category which is a ℤ2-extension of Rep(H8). We solve the pentagon equations including the additional topological line and find 8 solutions, where two of them are realized in the Ising2 CFT. Finally, we show that the torus partition functions of the Monster2 CFT and Ising×Monster CFT are also invariant under gauging the Rep(H8) symmetry.

Orbifold theory of the affine vertex operator superalgebra [formula omitted

This paper is about the orbifold theory of affine vertex operator superalgebras. Among the main results, we classify the irreducible modules and determine the fusion rules for the orbifold of the simple affine vertex operator superalgebra Losp(1|2)ˆ(k,0).

Simple-Sum Giant Graviton Expansions for Orbifolds and Orientifolds

Abstract We study giant graviton expansions of the superconformal index of 4D orbifold/orientifold theories. In general, a giant graviton expansion is given as a multiple sum over wrapping numbers. It is known that the expansion can be reduced to a simple sum for the ${\cal N}=4$ U(N) supersymmetric Yang–Mills (SYM) by choosing appropriate expansion variables. We find such a reduction occurs for a few examples of orbifold and orientifold theories: the $\mathbb {Z}_k$ orbifold and orientifolds with O3 and O7. We also argue that for a quiver gauge theory associated with a toric Calabi–Yau 3-fold the simple-sum expansion works only if the toric diagram is a triangle, i.e. the Calabi–Yau is an orbifold of $\mathbb {C}^3$.

States, symmetries and correlators of $T\bar{T}$ and $ J\bar{T} $ symmetric orbifolds

We derive various properties of symmetric product orbifolds of T\bar TTT‾ and J\bar TJT‾ - deformed CFTs from a field-theoretical perspective. First, we generalise the known formula for the torus partition function of a symmetric orbifold theory in terms of the one of the seed to non-conformal two-dimensional QFTs; specialising this to seed T\bar TTT‾ and J\bar TJT‾ - deformed CFTs reproduces previous results in the literature. Second, we show that the single-trace T\bar TTT‾ and J\bar TJT‾ deformations preserve the Virasoro and Kac-Moody symmetries of the undeformed symmetric product orbifold CFT, including their fractional counterparts, as well as the KdV charges. Finally, we discuss correlation functions in these theories. By extending a previously-proposed basis of operators for J\bar TJT‾ - deformed CFTs to the single-trace case, we explicitly compute the correlation functions of both untwisted and twisted-sector operators and compare them to an appropriate set of holographic correlators. Our derivations are based mainly on Hilbert space techniques and completely avoid the use of conformal invariance, which is not present in these models.

Transport across interfaces in symmetric orbifolds

We examine how conformal boundaries encode energy transport coefficients — namely transmission and reflection probabilities — of corresponding conformal interfaces in symmetric orbifold theories. These constitute a large class of irrational theories and are closely related to holographic setups. Our central goal is to compare such coefficients at the orbifold point (a field theory calculation) against their values when the orbifold is highly deformed (a gravity calculation) — an approach akin to past AdS/CFT-guided comparisons of physical quantities at strong versus weak coupling. At the orbifold point, we find that the (weighted-average) transport coefficients are simply averages of coefficients in the underlying seed theory. We then focus on the symmetric orbifold of the \U0001d54b4 sigma model interface CFT dual to type IIB supergravity on the 3d Janus solution. We compare the holographic transmission coefficient, which was found by [1], to that of the orbifold point. We find that the profile of the transmission coefficient substantially increases with the coupling, in contrast to boundary entropy. We also present some related ideas about twisted-sector data encoded by boundary states.

Revamping Kaluza-Klein dark matter in an orbifold theory of flavor

Revamping Kaluza-Klein dark matter in an orbifold theory of flavor

The chiral ring of a symmetric orbifold and its large N limit

We analyze the chiral operator ring of the symmetric orbifold conformal field theory on the complex two-plane ℂ2. We compute the large N limit of the ring and exhibit its factorized leading order behaviour. We moreover calculate all structure constants at the subleading and sub-subleading order. These features are coded as properties of the symmetric group and we review the relevant mathematical theorems on the product of conjugacy classes in the center of the group algebra. We illustrate the efficiency of the formalism by iteratively computing broad classes of higher point extremal correlators. We point out generalizations of our simplest of models and argue that our combinatorial analysis is relevant to the organization of the large N perturbation theory of generic symmetric orbifolds.

A fresh view on string orbifolds

In quantum field theory, an orbifold is a way to obtain a new theory from an old one by gauging a finite global symmetry. This definition of orbifold does not make sense for quantum gravity theories, that admit (conjecturally) no global symmetries. In string theory, the orbifold procedure involves the gauging of a global symmetry on the world-sheet theory describing the fundamental string. Alternatively, it is a way to obtain a new string background from an old one by quotienting some isometry.We propose a new formulation of string orbifolds in terms of the group of gauge symmetries of a given string model. In such a formulation, the ‘parent’ and the ‘child’ theories correspond to different ways of breaking or gauging all potential global symmetries of their common subsector. Through a couple of simple examples, we describe how the higher group structure of the gauge group in the parent theory plays a crucial role in determining the gauge group and the twisted sector of the orbifold theory. We also discuss the dependence of this orbifold procedure on the duality frame.

The stranger things of symmetric product orbifold CFTs

Symmetric product orbifold theories are valuable due to their universal features at large N. Here we will demonstrate that they have features that are not as pervasive: we provide evidence of strange behaviour under deformations within their moduli space. To this end, we consider the symmetric product orbifold of tensor products of mathcal{N} = 2 super-Virasoro minimal models, and classify them according to two criteria. The first criterion is the existence of a single-trace twisted exactly marginal operator that triggers the deformation. The second criterion is a sparseness condition on the growth of light states in the elliptic genera. In this context we encounter a strange variety: theories that obey the first criterion but the second criterion falls into a Hagedorn-like growth. We explain why this may be counter-intuitive and discuss how it might be accounted for in conformal perturbation theory. We also find a new infinite class of theories that obey both criteria, which are necessary conditions for each moduli space to contain a supergravity point.

Convergence in Conformal Field Theory

Convergence and analytic extension are of fundamental importance in the mathematical construction and study of conformal field theory. The author reviews some main convergence results, conjectures and problems in the construction and study of conformal field theories using the representation theory of vertex operator algebras. He also reviews the related analytic extension results, conjectures and problems. He discusses the convergence and analytic extensions of products of intertwining operators (chiral conformal fields) and of q-traces and pseudo-q-traces of products of intertwining operators. He also discusses the convergence results related to the sewing operation and the determinant line bundle and a higher-genus convergence result. He then explains conjectures and problems on the convergence and analytic extensions in orbifold conformal field theory and in the cohomology theory of vertex operator algebras.

Super orbifold theory

Let V be a simple vertex operator superalgebra and G a finite automorphism group of V containing the canonical automorphism σ such that VG is regular. It is proved that every irreducible VG-module occurs in an irreducible g-twisted V-module for some g∈G. Moreover, the quantum dimensions of irreducible VG-modules are determined, a global dimension formula for V in terms of twisted modules is obtained and a super quantum Galois theory is established. In addition the S-matrix of VG is computed.

G-twisted associative algebras of vertex operator superalgebras

Let V be a vertex operator superalgebra and G a finite automorphism group of V. Let σ be the order 2 automorphism of V associated with the superstructure of V. A sequence of associative algebras AG,n(V) are constructed to study the twisted representations of V for nonnegative n∈1T′Z where T′ is the order of group generated by G and σ. This result which generalizes many previous results on Zhu's algebras is then used to investigate the super orbifold theory. If V is a simple vertex operator superalgebra and S is a finite set of inequivalent irreducible twisted V-modules which is closed under the action of G, a duality theorem of Schur-Weyl type is obtained for the actions of certain finite dimensional semisimple associated algebra Aα(G,S) and VG on the direct sum of twisted V-modules in S. In particular, for any g∈G every irreducible g-twisted V-module is completely reducible VG-module.

Scotogenic Majorana neutrino masses in a predictive orbifold theory of flavor

The use of extra space-time dimensions provides a promising approach to the flavour problem. The chosen compactification of a 6-dimensional orbifold implies a remnant family symmetry $A_4$. This makes interesting predictions for quark and lepton masses, for neutrino oscillations and neutrinoless double beta decay, providing also a very good global description of all flavour observables. Due to an auxiliary $\mathbb{Z}_4$ symmetry, we implement a scotogenic Majorana neutrino mass generation mechanism with a viable WIMP dark matter candidate.

Symmetry defects and orbifolds of two-dimensional Yang–Mills theory

We describe discrete symmetries of two-dimensional Yang–Mills theory with gauge group G associated with outer automorphisms of G, and their corresponding defects. We show that the gauge theory partition function with defects can be computed as a path integral over the space of twisted G-bundles and calculate it exactly. We argue that its weak-coupling limit computes the symplectic volume of the moduli space of flat twisted G-bundles on a surface. Using the defect network approach to generalised orbifolds, we gauge the discrete symmetry and construct the corresponding orbifold theory, which is again two-dimensional Yang–Mills theory but with gauge group given by an extension of G by outer automorphisms. With the help of the orbifold completion of the topological defect bicategory of two-dimensional Yang–Mills theory, we describe the reverse orbifold using a Wilson line defect for the discrete gauge symmetry. We present our results using two complementary approaches: in the lattice regularisation of the path integral, and in the functorial approach to area-dependent quantum field theories with defects via regularised Frobenius algebras.