Symmetric Orbifold Research Articles

Spacetime dilaton in AdS3 × X holography

Abstract We study the spectrum generating operators (DDF operators) for bosonic strings and superstrings in AdS3 × X with pure, generic NSNS flux. The dual CFT is conjectured to be a deformed symmetric orbifold of $$ {\mathbb{R}}_{\overset{\sim }{Q}}\times X $$ ℝ Q ~ × X , where $$ {\mathbb{R}}_{\overset{\sim }{Q}} $$ ℝ Q ~ denotes a linear dilaton theory with slope $$ \overset{\sim }{Q} $$ Q ~ . In this paper, we construct DDF operators corresponding to the spacetime linear dilaton field in the limit where the worldsheet theory becomes free. The DDF operator gives another evidence supporting the proposed dual CFT. Furthermore, the DDF operator is constructed purely from the AdS3 fields and thus, strongly implies the existence of $$ {\mathbb{R}}_{\overset{\sim }{Q}} $$ ℝ Q ~ for generic X that defines a consistent background.

The light we can see: extracting black holes from weak Jacobi forms

We quantify how constraints on light states affect the asymptotic growth of heavy states in weak Jacobi forms. The constraints we consider are sparseness conditions on the Fourier coefficients of these forms, which are necessary to interpret them as gravitational path integrals. Using crossing kernels, we extract the leading and subleading behavior of these coefficients and show that the leading Cardy-like growth is robust in a wide regime of validity. On the other hand, we find that subleading corrections are sensitive to the constraints placed on the light states, and we quantify their imprint on the asymptotic growth of states. Our approach is tested against the generating function of symmetric product orbifolds, where we provide new insights into the factors contributing to the asymptotic growth of their Fourier coefficients. Finally, we use our methods to revisit the UV/IR connection that relates black hole microstate counting to modular forms. We provide a microscopic interpretation of the logarithmic corrections to the entropy of BPS black holes in N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 2, 4 ungauged supergravity in four and five dimensions, and tie it to consistency conditions in AdS3/CFT2.

Solving AdS3 string theory at minimal tension: tree-level correlators

We revisit the minimal tension (k = 1) string theory on AdS3 × S3 × \U0001d54b4. We propose a new free-field description of the worldsheet theory and show how localization of string amplitudes emerges from the path integral. We exemplify our proposal by reproducing the worldsheet partition function of the psu\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathfrak{psu} $$\\end{document}(1, 1|2)1 WZW model and providing explicit expressions for spectrally-flowed vertex operators and DDF operators. We compute string correlators in the path integral formalism and obtain a precise tree-level match with correlation functions of the boundary symmetric orbifold.

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

Comments on integrability in the symmetric orbifold

We present a map between the excitation of the symmetric-product orbifold CFT of T4, and of the worldsheet-integrability description of AdS3 × S3 × T4 of Lloyd, Ohlsson Sax, Sfondrini, and Stefański at k = 1. We discuss the map in the absence of RR fluxes, when the theory is free, and at small RR flux, h ≪ 1, where the symmetric-orbifold CFT is deformed by a marginal operator from the twist-two sector. We discuss the recent results of Gaberdiel, Gopakumar, and Nairz, who computed from the perturbed symmetric-product orbifold the central extension to the symmetry algebra of the theory and its coproduct. We show that it coincides with the h ≪ 1 expansion of the lightcone symmetry algebra known from worldsheet integrability, and that hence the S matrix found by Gaberdiel, Gopakumar, and Nairz maps to the one bootstrapped by the worldsheet integrability approach.

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.

Engineering perturbative string duals for symmetric product orbifold CFTs

Constructing a holographic string theory dual for a CFT in the perturbative, weakly coupled regime is a holy grail for gauge/string dualities that would not only open the door for proofs of the AdS/CFT correspondence but could also provide novel examples of string duals with and without supersymmetry. In this work we consider some marginal perturbation of a family of symmetric product orbifolds in two dimensions. From their correlation functions we engineer a bosonic string theory whose amplitudes are shown to reproduce the CFT correlation function order-by-order both in the coupling and in 1/N. Our derivation does not require to compute and compare correlation functions explicitly but rather relies on a sequence of identities that can be derived using path integral methods. The bosonic string theory we engineer is based on the field content of the Kac-Wakimoto representation of strings in AdS3 with k units of pure NSNS flux, but the interaction terms we obtain are different. They include current algebra preserving interaction terms with one unit of spectral flow.

Beyond the tensionless limit: integrability in the symmetric orbifold

The symmetric orbifold of \U0001d54b4 is exactly dual to string theory on AdS3 × S3 × \U0001d54b4 with minimal (k = 1) NS-NS flux. In this paper we study the perturbation of the symmetric orbifold that is dual to switching on R-R flux, and hence to deforming the theory away from the tensionless point. More specifically, we determine systematically the action of a centrally extended supersymmetry algebra on the CFT states, and deduce from it the anomalous conformal dimensions. In the w-twisted sector with large w the structure is similar to what was found for N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 4 SYM: the basic excitations are multi-magnons whose individual dispersion relation is fixed by symmetry, and the comparison with the BMN answer suggests that the result is true to all orders in perturbation theory. Finally we show that the multi-magnon states interact via an integrable S-matrix and possess a natural family of bound states.

On full asymptotics of real analytic torsions for compact locally symmetric orbifolds

On full asymptotics of real analytic torsions for compact locally symmetric orbifolds

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.

Modified supersymmetric indices in AdS3/CFT2

We consider the \U0001d4a9 = (2, 2) AdS3/CFT2 dualities proposed by Eberhardt, where the bulk geometry is AdS3 × (S3 × T4)/ℤk, and the CFT is a deformation of the symmetric orbifold of the supersymmetric sigma model T4/ℤk (with k = 2, 3, 4, 6). The elliptic genera of the two sides vanish due to fermionic zero modes, so for microstate counting applications one must consider modified supersymmetric indices. In an analysis similar to that of Maldacena, Moore, and Strominger for the standard \U0001d4a9 = (4, 4) case of T4, we study the appropriate helicity-trace index of the boundary CFTs. We encounter a strange phenomenon where a saddle-point analysis of our indices reproduces only a fraction (respectively 12\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\frac{1}{2} $$\\end{document}, 23\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\frac{2}{3} $$\\end{document}, 34\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\frac{3}{4} $$\\end{document}, 56\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\frac{5}{6} $$\\end{document}) of the Bekenstein-Hawking entropy of the associated macroscopic black branes.

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.

Infinite $\mathrm{T\bar T}$-like symmetries of compactified LST

We show that the three-dimensional asymptotically linear dilaton background that arises in the near-horizon decoupling region of NS5-branes compactified on T^4T4 admits boundary conditions that lead to an infinite set of symmetries. The associated conserved charges, which implement field-dependent coordinate transformations, are found to be identical to the corresponding generators in a symmetric product orbifold of T\bar TTT‾ - deformed CFTs. Their algebra is a non-linear modification of the Virasoro × Virasoro algebra, which precisely coincides with the algebra of the “unrescaled” symmetry generators in T\bar TTT‾-deformed CFTs. This further strengthens a previously proposed link between the single-trace T\bar TTT‾ deformation and compactified little string theory.

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.

Lagrangian configurations and Hamiltonian maps

We study configurations of disjoint Lagrangian submanifolds in certain low-dimensional symplectic manifolds from the perspective of the geometry of Hamiltonian maps. We detect infinite-dimensional flats in the Hamiltonian group of the two-sphere equipped with Hofer's metric, prove constraints on Lagrangian packing, find instances of Lagrangian Poincaré recurrence, and present a new hierarchy of normal subgroups of area-preserving homeomorphisms of the two-sphere. The technology involves Lagrangian spectral invariants with Hamiltonian term in symmetric product orbifolds.

Freely acting orbifolds of type IIB string theory on T5

We study freely acting orbifolds of type IIB string theory on T5 that spontaneously break supersymmetry from mathcal{N} = 8 to mathcal{N} = 6, 4, 2 or 0 in five dimensions. We focus on orbifolds that are a ℤp quotient by a T-duality acting on T4 and a shift on the remaining S1. Modular invariant partition functions are constructed and detailed examples of both symmetric and asymmetric orbifolds are presented, including new examples of five-dimensional non-supersymmetric string theories with no tachyons. The orbifolds we consider arise at special points in the moduli space of string theory compactifications with a duality twist. The supergravity limit of these are Scherk-Schwarz reductions which generate gauged supergravities with positive definite classical potentials on the moduli space in five dimensions. Both symmetric and asymmetric freely acting orbifolds give a landscape of Minkowski vacua. For gauged supergravities to belong to this landscape, we find a number of constraints and conditions. Firstly, the scalar potential should lead to a massive spectrum with masses that obey quantization conditions arising from a string theory orbifold, which we discuss in detail. Secondly, we find constraints on the massless sector, e.g. in the examples of orbifolds preserving sixteen supersymmetries in five dimensions that we consider, only an odd number of vector multiplets arises. Lastly, we present new examples of candidate asymmetric orbifolds with modular invariant partition functions, but with non-integral coefficients in the qoverline{q} -expansion in the twisted sectors.

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.

Perturbing the symmetric orbifold from the worldsheet

The symmetric orbifold of \U0001d54b4 is the analogue of free SYM in four dimensions, and its dual is described by a tensionless string propagating in AdS3 × S3 × \U0001d54b4. In this paper we study the deformation of this exact AdS/CFT duality away from the free point. On the symmetric orbifold side this amounts to perturbing the theory by the exactly marginal operator from the 2-cycle twisted sector. We identify the corresponding perturbation in the dual worldsheet description, and show that the anomalous conformal dimensions of a number of symmetric orbifold currents are correctly reproduced from this worldsheet perspective.

On the universal behavior of Toverline{T} -deformed CFTs: single and double-trace partition functions at large c

We study universal properties of the torus partition function of Toverline{T} -deformed CFTs under the assumption of modular invariance, for both the original version, referred to as the double-trace version in this paper, and the single-trace version defined as the symmetric product orbifold of double-trace Toverline{T} -deformed CFTs. In the double-trace case, we specify sparseness conditions for the light states for which the partition function at low temperatures is dominated by the vacuum when the central charge of the undeformed CFT is large. Using modular invariance, this implies a universal density of high energy states, in analogy with the behavior of holographic CFTs. For the single-trace Toverline{T} deformation, we show that modular invariance implies that the torus partition function can be written in terms of the untwisted partition function and its modular images, the latter of which can be obtained from the action of a generalized Hecke operator. The partition function and the energy of twisted states match holographic calculations in previous literature, thus providing further evidence for the conjectured holographic correspondence. In addition, we show that the single-trace partition function is universal when the central charge of the undeformed CFT is large, without needing to assume a sparse density of light states. Instead, the density of light states is shown to always saturate the sparseness condition.