arXiv subjects

Lorenz Eberhardt

Publications and source records attributed to Lorenz Eberhardt.

At least 55 records · Page 3Linked to original sources

arXiv2021

Summing over Geometries in String Theory

We examine the question how string theory achieves a sum over bulk geometries with fixed asymptotic boundary conditions. We discuss this problem with the help of the tensionless string on $\mathcal{M}_3 \times \mathrm{S}^3 \times \mathbb{T}^4$ (with one unit of NS-NS flux) that was recently understood to be dual to the symmetric orbifold $\text{Sym}^N(\mathbb{T}^4)$. We strengthen the analysis of arXiv:2008.07533 and show that the perturbative string partition function around a fixed bulk background already includes a sum over semi-classical geometries and large stringy corrections can be interpreted as various semi-classical geometries. We argue in particular that the string partition function on a Euclidean wormhole geometry factorizes completely into factors associated to the two boundaries of spacetime. Central to this is the remarkable property of the moduli space integral of string theory to localize on covering spaces of the conformal boundary of $\mathcal{M}_3$. We also emphasize the fact that string perturbation theory computes the grand canonical partition function of the family of theories $\bigoplus_N\text{Sym}^N(\mathbb{T}^4)$. The boundary partition function is naturally expressed as a sum over winding worldsheets, each of which we interpret as a `stringy geometry'. We argue that the semi-classical bulk geometry can be understood as a condensate of such stringy geometries. We also briefly discuss the effect of ensemble averaging over the Narain moduli space of $\mathbb{T}^4$ and of deforming away from the orbifold by the marginal deformation.

hep-th↗

arXiv2020

Scattering Equations in AdS: Scalar Correlators in Arbitrary Dimensions

We introduce a bosonic ambitwistor string theory in AdS space. Even though the theory is anomalous at the quantum level, one can nevertheless use it in the classical limit to derive a novel formula for correlation functions of boundary CFT operators in arbitrary space-time dimensions. The resulting construction can be treated as a natural extension of the CHY formalism for the flat-space S-matrix, as it similarly expresses tree-level amplitudes in AdS as integrals over the moduli space of Riemann spheres with punctures. These integrals localize on an operator-valued version of scattering equations, which we derive directly from the ambitwistor string action on a coset manifold. As a testing ground for this formalism we focus on the simplest case of ambitwistor string coupled to two current algebras, which gives bi-adjoint scalar correlators in AdS. In order to evaluate them directly, we make use of a series of contour deformations on the moduli space of punctured Riemann spheres and check that the result agrees with tree level Witten diagram computations to all multiplicity. We also initiate the study of eigenfunctions of scattering equations in AdS, which interpolate between conformal partial waves in different OPE channels, and point out a connection to an elliptic deformation of the Calogero-Sutherland model.

hep-th↗

arXiv2020

Superconformal symmetry and representations

We give an introduction to conformal and superconformal algebras and their representations in various dimensions. Special emphasis is put on 4d $\mathcal{N}=2$ superconformal symmetry. This is the writeup of the lectures given at the Winter School "YRISW 2020" to appear in a special issue of JPhysA.

hep-th↗

arXiv2020

The Grassmannian VOA

We study the 3-parametric family of vertex operator algebras based on the unitary Grassmannian coset CFT $\mathfrak{u}(M+N)_k/(\mathfrak{u}(M)_k \times \mathfrak{u}(N)_k)$. This VOA serves as a basic building block for a large class of cosets and generalizes the $\mathcal{W}_\infty$ algebra. We analyze representations and their characters in detail and find surprisingly simple character formulas for the representations in the generic parameter regime that admit an elegant combinatorial formulation. We also discuss truncations of the algebra and give a conjectural formula for the complete set of truncation curves. We develop a theory of gluing for these algebras in order to build more complicated coset and non-coset algebras. We demonstrate the power of this technology with some examples and show in particular that the $\mathcal{N}=2$ supersymmetric Grassmannian can be obtained by gluing three bosonic Grassmannian algebras in a loop. We finally speculate about the tantalizing possibility that this algebra is a specialization of an even larger 4-parametric family of algebras exhibiting pentality symmetry. Specialization of this conjectural family should include both the unitary Grassmannian family as well as the Lagrangian Grassmannian family of VOAs which interpolates between the unitary and the orthosymplectic cosets.

hep-th↗

arXiv2020

Deriving the $\text{AdS}_{3}/\text{CFT}_{2}$ Correspondence

It was recently argued that string theory on ${\rm AdS}_3\times {\rm S}^3\times \mathbb{T}^4$ with one unit ($k=1$) of NS-NS flux is exactly dual to the symmetric orbifold CFT ${\rm Sym}^N(\mathbb{T}^4)$. In this paper we show how to directly relate the $n$-point correlators of the two sides to one another. In particular, we argue that the correlators of the world-sheet theory are delta-function-localised in string moduli space to those configurations that allow for a holomorphic covering map of the $\text{S}^2$-boundary of $\text{AdS}_3$ by the world-sheet. This striking feature can be seen both from a careful Ward identity analysis, as well as from semi-classically exact AdS$_3$ solutions that are pinned to the boundary. The world-sheet correlators therefore have exactly the same structure as in the Lunin-Mathur construction of symmetric orbifold CFT correlators in terms of a covering surface -- which now gets identified with the world-sheet. Together with the results of arXiv:1803.04423 and arXiv:1812.01007 this essentially demonstrates how the $k=1$ $\text{AdS}_3$ string theory becomes equivalent to the spacetime orbifold CFT in the genus expansion.

hep-th↗

arXiv2020

$\text{AdS}_3/\text{CFT}_2$ at higher genus

We continue our study of the worldsheet theory of superstrings on $\mathrm{AdS}_3 \times \mathrm{S}^3 \times \mathbb{T}^4$ in the tensionless limit arXiv:1911.00378. We consider the theory on higher genus surfaces. We give evidence that the worldsheet correlators localise on certain worldsheets that cover the boundary of $\mathrm{AdS}_3$ holomorphically. This simplifies the string moduli space integral dramatically to a finite sum. This property shows that the higher genus corrections of the string worldsheet reproduce the structure of the $1/N$ corrections in the dual symmetric orbifold CFT $\mathrm{Sym}^N(\mathbb{T}^4)$.

hep-th↗

arXiv2020

Correlators of the symmetric product orbifold

We exploit null vectors of the fractional Virasoro algebra of the symmetric product orbifold to compute correlation functions of twist fields in the large $N$ limit. This yields a new method to derive correlation functions in these orbifold CFTs that is purely based on the symmetry algebra. We explore various generalisations, such as subleading (torus) contributions or correlation functions of other fields than the bare twist fields. We comment on the consequences of our computation for the $\text{AdS}_3/\text{CFT}_2$ correspondence.

hep-th↗

arXiv2019

The matrix-extended $W_{1+\infty}$ algebra

We construct a quadratic basis of generators of matrix-extended $\mathcal{W}_{1+\infty}$ using a generalization of the Miura transformation. This makes it possible to conjecture a closed-form formula for the operator product expansions defining the algebra. We study truncations of the algebra. An explicit calculation at low levels shows that these are parametrized in a way consistent with the gluing description of the algebra. It is perhaps surprising that in spite of the fact that the algebras are rather complicated and non-linear, the structure of their truncations follows very simple gluing rules.

hep-th↗

arXiv2019

Three-point functions in AdS$_3$/CFT$_2$ holography

Recently, string theory on $\text{AdS}_3 \times \text{S}^3 \times \mathbb{T}^4$ with one unit of NS-NS flux ($k=1$) was argued to be exactly dual to the symmetric orbifold of $ \mathbb{T}^4$, and in particular, the full (unprotected) spectrum was matched between the two descriptions. This duality was later extended to the case with higher NS-NS background flux for which the long string sector was argued to be described by the symmetric product orbifold of $(\mathcal{N}=4$ $\text{Liouville}) \times \mathbb{T}^4$. In this paper we study correlation functions for the bosonic analogue of this duality, relating bosonic string theory on $\text{AdS}_3 \times X$ to the symmetric orbifold of $\text{Liouville} \times X$. More specifically, we show that the low-lying null vectors of Liouville theory correspond to BRST exact states from the worldsheet perspective, and we demonstrate that they give rise to the expected BPZ differential equations for the dual CFT correlators. Since the structure constants of Liouville theory are uniquely fixed by these constraints, this shows that the seed theory of the dual CFT contains indeed the Liouville factor.

hep-th↗

arXiv2019

Stringy $\mathcal{N}=(2,2)$ holography for AdS${_3}$

We propose a class of ${\rm AdS}_3/{\rm CFT}_2$ dualities with $\mathcal{N}=(2,2)$ supersymmetry. These dualities relate string theory on ${\rm AdS}_3 \times ({\rm S}^3\times \mathbb{T}^4)/{\rm G}$ to marginal deformations of the symmetric product orbifold of $\mathbb{T}^4/{\rm G}$, where ${\rm G}$ is a dihedral group. We demonstrate that the BPS spectrum calculated from supergravity and string theory agrees with that of the dual CFT. Moreover, the supergravity elliptic genus is shown to reproduce the CFT answer, thus providing further non-trivial evidence in favour of the proposal.

hep-th↗

arXiv2019

Strings on $\text{AdS}_3 \times \text{S}^3 \times \text{S}^3 \times \text{S}^1$

String theory on ${\rm AdS}_3 \times {\rm S}^3 \times {\rm S}^3 \times {\rm S}^1$ with pure NS-NS flux and minimal flux through one of the two ${\rm S}^3$'s is studied from a world-sheet perspective. It is shown that the spacetime spectrum, as well as the algebra of spectrum generating operators, matches precisely that of the symmetric orbifold of ${\rm S}^3\times \mathrm{S}^1$ in the large $N$ limit. This gives strong support for the proposal that these two descriptions are exactly dual to one another.

hep-th↗

arXiv2019

String theory on $\boldsymbol{\text{AdS}_{\mathbf{3}}}$ and the symmetric orbifold of Liouville theory

For string theory on AdS$_3$ with pure NS-NS flux a complete set of DDF operators is constructed, from which one can read off the symmetry algebra of the spacetime CFT. Together with an analysis of the spacetime spectrum, this allows us to show that the CFT dual of superstring theory on ${\rm AdS}_3 \times {\rm S}^3 \times \mathbb{T}^4$ for generic NS-NS flux is the symmetric orbifold of $({\cal N}=4$ Liouville theory$)\times \mathbb{T}^4$. For the case of minimal flux ($k=1$), the Liouville factor disappears, and we just obtain the symmetric orbifold of $\mathbb{T}^4$, thereby giving further support to a previous claim. We also show that a similar analysis can be done for bosonic string theory on ${\rm AdS}_3 \times X$.

hep-th↗

arXiv2018

The Worldsheet Dual of the Symmetric Product CFT

Superstring theory on ${\rm AdS}_3\times {\rm S}^3\times \mathbb{T}^4$ with the smallest amount of NS-NS flux (`$k=1$') is shown to be dual to the spacetime CFT given by the large $N$ limit of the free symmetric product orbifold $\mathrm{Sym}^N(\mathbb{T}^4)$. To define the worldsheet theory at $k=1$, we employ the hybrid formalism in which the ${\rm AdS}_3\times {\rm S}^3$ part is described by the $\mathfrak{psu}(1,1|2)_1$ WZW model (which is well defined). Unlike the case for $k\geq2$, it turns out that the string spectrum at $k=1$ does {\it not} exhibit the long string continuum, and perfectly matches with the large $N$ limit of the symmetric product. We also demonstrate that the fusion rules of the symmetric orbifold are reproduced from the worldsheet perspective. Our proposal therefore affords a tractable worldsheet description of a tensionless limit in string theory, for which the dual CFT is also explicitly known.

hep-th↗

arXiv2018

Long strings and chiral primaries in the hybrid formalism

We revisit two related phenomena in $\mathrm{AdS}_3$ string theory backgrounds. At pure NS-NS flux, the spectrum contains a continuum of long strings which can escape to the boundary of $\mathrm{AdS}_3$ at a finite cost of energy. Related to this are certain gaps in the BPS spectrum one computes from the RNS worldsheet description. One expects that both these effects disappear when perturbing slightly away from the pure NS-NS flux background. We employ the hybrid formalism for mixed flux backgrounds to demonstrate directly from the worldsheet that this is indeed the case.

hep-th↗

arXiv2018

The plane-wave spectrum from the worldsheet

We study string theory on $\mathrm{AdS}_3$ backgrounds with mixed flux using the hybrid formalism of Berkovits, Vafa and Witten. We solve the worldsheet description of the theory completely in the plane-wave limit. This constitutes a direct derivation of the plane-wave spectrum from the worldsheet with mixed flux.

hep-th↗

arXiv2018

$\mathcal{N}=(3,3)$ holography on ${\rm AdS}_3 \times ({\rm S}^3 \times {\rm S}^3 \times {\rm S}^1)/\mathbb Z_2$

We consider string theory on AdS$_3$ $\times$ (S$^3$ $\times$ S$^3$ $\times$ S$^1)/\mathbb Z_2$, a background supporting $\mathcal N=(3,3)$ spacetime supersymmetry. We propose that string theory on this background is dual to the symmetric product orbifold of $\mathcal S_0/\mathbb Z_2$ where $\mathcal S_0$ is a theory of four free fermions and one free boson. We show that the BPS spectra of the two sides of the duality match precisely. Furthermore, we compute the elliptic genus of the dual CFT and that of the supergravity limit of string theory and demonstrate that they match, hence providing non-trivial support for the holographic proposal.

hep-th↗

arXiv2018

Supersymmetric $\mathrm{AdS}_3$ supergravity backgrounds and holography

We analyse the conditions for $\mathrm{AdS}_3 \times \mathcal{M}_7$ backgrounds with pure NS-NS flux to be supersymmetric. We find that a necessary condition is that $\mathcal{M}_7$ is a $\mathrm{U}(1)$-fibration over a balanced manifold. We subsequently classify all $\mathcal{N}=(2,2)$ solutions where $\mathcal{M}_7$ satisfies the stronger condition of being a $\mathrm{U}(1)$-fibration over a Kahler manifold. We compute the BPS spectrum of all the backgrounds in this classification. We assign a natural dual CFT to the backgrounds and confirm that the BPS spectra agree, thus providing evidence in favour of the proposal.

hep-th↗

arXiv2018

Higher spin algebras and large $\mathcal{N}=4$ holography

A new family of higher spin algebras that arises upon restricting matrix extensions of $\mathfrak{shs}[λ]$ is found. We identify coset CFTs realising these symmetry algebras, and thus propose new higher spin-CFT dual pairs. These higher spin theories arise naturally as a subsector of string theory on ${\rm AdS}_3\times {\rm S}^3 \times {\rm S}^3 \times {\rm S}^1$ for specific ratios of the radii of the two spheres.

hep-th↗