arXiv subjects

Matthias R. Gaberdiel

Publications and source records attributed to Matthias R. Gaberdiel.

At least 37 records · Page 2Linked to original sources

arXiv2022

The free field realisation of the BVW string

The symmetric orbifold of $\mathbb{T}^4$ was recently shown to be exactly dual to string theory on ${\rm AdS}_3\times {\rm S}^3 \times \mathbb{T}^4$ with minimal ($k=1$) NS-NS flux. The worldsheet theory is best formulated in terms of the hybrid formalism of Berkovits, Vafa & Witten (BVW), in terms of which the ${\rm AdS}_3\times {\rm S}^3$ factor is described by a $\mathfrak{psu}(1,1|2)_k$ WZW model. At level $k=1$, $\mathfrak{psu}(1,1|2)_1$ has a free field realisation that is obtained from that of $\mathfrak{u}(1,1|2)_1$ upon setting a $\mathfrak{u}(1)$ field, often called $Z$, to zero. We show that the free field version of the ${\cal N}=2$ generators of BVW (whose cohomology defines the physical states) does not give rise to an ${\cal N}=2$ algebra, but is rather contaminated by terms proportional to the $Z$-field. We also show how to overcome this problem by introducing additional ghost fields that implement the quotienting by $Z$.

hep-th↗

arXiv2021

D-branes in $\mathrm{AdS}_3\times \mathrm{S}^3\times \mathbb{T}^4$ at $k=1$ and their holographic duals

String theory on $\text{AdS}_3\times \text{S}^3\times \mathbb{T}^4$ with minimal $k=1$ NS-NS flux can be described in terms of a free field worldsheet theory in the hybrid formalism. We construct various D-branes of this string theory and calculate their associated cylinder amplitudes. We find that these amplitudes match with the cylinder correlators of certain boundary states of the dual symmetric orbifold CFT $\text{Sym}(\mathbb{T}^4)$, thus suggesting a direct correspondence between these boundary conditions. We also show that the disk amplitudes of these D-branes localise to those points in the worldsheet moduli space where the worldsheet disk holomorphically covers the spacetime disk.

hep-th↗

arXiv2021

The physical states of the Hybrid Formalism

String theory on ${\rm AdS}_3\times {\rm S}^3 \times\mathbb{T}^4$ with one unit ($k=1$) of NS-NS flux is considered in the hybrid formalism of Berkovits, Vafa & Witten (BVW). Using the free field realisation of the world-sheet theory at $k=1$, we identify explicitly the BRST cohomology classes corresponding to some of the low-lying states of the dual CFT. In particular, we do this for the ${\cal N}=4$ superconformal generators of the symmetric orbifold theory, and we confirm these identifications by showing that the worldsheet correlators reproduce the expected dual CFT answer. Along the way we note that the physical vertex operators on the worldsheet have a simpler form if one works with a different, but equivalent, choice for the BRST operators relative to BVW.

hep-th↗

arXiv2021

The String Dual to Free ${\cal N}=4$ Super Yang-Mills

We propose a worldsheet description for the ${\rm AdS}_5\times {\rm S}^5$ string theory dual to large $N$, free ${\cal N}=4$ super Yang-Mills in four dimensions. The worldsheet theory is a natural generalisation of the recently investigated tensionless string on ${\rm AdS}_3\times {\rm S}^3\times \mathbb{T}^4$. As in the case of ${\rm AdS}_3$ it has a free field description, with spectrally flowed sectors, and is closely related to an (ambi-)twistor string theory. Here, however, we view it as a critical $N=4$ (closed) string background. We argue that the corresponding worldsheet gauge constraints reduce the degrees of freedom to a finite number of oscillators (string bits) in each spectrally flowed sector. Imposing a set of residual gauge constraints on this reduced oscillator Fock space then determines the physical spectrum of the string theory. Quite remarkably, we find compelling evidence that this prescription reproduces precisely the entire planar spectrum - of single trace operators - of the free super Yang-Mills theory.

hep-th↗

arXiv2021

From Symmetric Product CFTs to ${\rm AdS}_3$

Correlators in symmetric orbifold CFTs are given by a finite sum of admissible branched covers of the 2d spacetime. We consider a Gross-Mende like limit where all operators have large twist, and show that the corresponding branched covers can be described via a Penner-like matrix model. The limiting branched covers are given in terms of the spectral curve for this matrix model, which remarkably turns out to be directly related to the Strebel quadratic differential on the covering space. Interpreting the covering space as the world-sheet of the dual string theory, the spacetime CFT correlator thus has the form of an integral over the entire world-sheet moduli space weighted with a Nambu-Goto-like action. Quite strikingly, at leading order this action can also be written as the absolute value of the Schwarzian of the covering map. Given the equivalence of the symmetric product CFT to tensionless string theory on ${\rm AdS}_3$, this provides an explicit realisation of the underlying mechanism of gauge-string duality originally proposed in arXiv:hep-th/0504229 and further refined in arXiv:0803.2681.

hep-th↗

arXiv2021

The Worldsheet Dual of Free Super Yang-Mills in 4D

The worldsheet string theory dual to free 4d ${\cal N}=4$ super Yang-Mills theory was recently proposed in arXiv:2104.08263. It is described by a free field sigma model on the twistor space of ${\rm AdS}_5\times {\rm S}^5$, and is a direct generalisation of the corresponding model for tensionless string theory on ${\rm AdS}_3\times {\rm S}^3$. As in the case of ${\rm AdS}_3$, the worldsheet theory contains spectrally flowed representations. We proposed in arXiv:2104.08263 that in each such sector only a finite set of generalised zero modes (`wedge modes') are physical. Here we show that after imposing the appropriate residual gauge conditions, this worldsheet description reproduces precisely the spectrum of the planar gauge theory. More specifically, the states in the sector with $w$ units of spectral flow match with single trace operators built out of $w$ super Yang-Mills fields (`letters'). The resulting physical picture is a covariant version of the BMN light-cone string, now with a finite number of twistorial string bit constituents of an essentially topological worldsheet.

hep-th↗

arXiv2021

Deformed Shatashvili-Vafa algebra for superstrings on AdS$_3\times {\cal M}_7$

String backgrounds of the form $\mathbb{M}_3 \times {\cal M}_7$ where $\mathbb{M}_3$ denotes $3$-dimensional Minkowski space while ${\cal M}_7$ is a $7$-dimensional G$_2$-manifold, are characterised by the property that the world-sheet theory has a Shatashvili-Vafa (SV) chiral algebra. We study the generalisation of this statement to backgrounds where the Minkowski factor $\mathbb{M}_3$ is replaced by ${\rm AdS}_3$. We argue that in this case the world-sheet theory is characterised by a certain ${\cal N}=1$ superconformal ${\cal W}$-algebra that has the same spin spectrum as the SV algebra and also contains a tricritical Ising model ${\cal N}=1$ subalgebra. We determine the allowed representations of this ${\cal W}$-algebra, and analyse to which extent the special features of the SV algebra survive this generalisation.

hep-th↗

arXiv2020

Stress-energy tensor correlators from the world-sheet

The large $N$ limit of symmetric orbifold theories was recently argued to have an AdS/CFT dual world-sheet description in terms of an $\mathfrak{sl}(2,\mathbb{R})$ WZW model. In previous work the world-sheet state corresponding to the symmetric orbifold stress-energy tensor was identified. We calculate certain 2- and 3-point functions of the corresponding vertex operator on the world-sheet, and demonstrate that these amplitudes reproduce exactly what one expects from the dual symmetric orbifold perspective.

hep-th↗

arXiv2020

Free field world-sheet correlators for ${\rm AdS}_3$

We employ the free field realisation of the $\mathfrak{psu}(1,1|2)_1$ world-sheet theory to constrain the correlators of string theory on ${\rm AdS}_3\times {\rm S}^3\times \mathbb{T}^4$ with unit NS-NS flux. In particular, we directly obtain the unusual delta function localisation of these correlators onto branched covers of the boundary ${\rm S}^2$ by the (genus zero) world-sheet -- this is the key property which makes the equivalence to the dual symmetric orbifold manifest. In our approach, this feature follows from a remarkable `incidence relation' obeyed by the correlators, which is reminiscent of a twistorial string description. We also illustrate our results with explicit computations in various special cases.

hep-th↗

arXiv2020

The Small ${\cal N}=4$ Superconformal ${\cal W}_{\infty}$ Algebra

The symmetric orbifold of $\mathbb{T}^4$ is the CFT dual of string theory on AdS$_3\times {\rm S}^3 \times \mathbb{T}^4$ with minimal NS-NS flux. We study its symmetry algebra and provide evidence that it does not have any deformation parameter. This suggests that the symmetric orbifold is (at least locally) the most symmetrical CFT in its moduli space.

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↗

arXiv2019

Stringy CFT Duals with $\mathcal{N} = (2,2)$ Supersymmetry

It was recently shown that string theory on $\mathrm{AdS}_3 \times \mathrm{S}^3 \times \mathbb{T}^4$ with minimal NS-NS flux ($k = 1$) is exactly dual to the symmetric orbifold of T4. Here we show that a similar statement also holds for the $D_n$ orbifolds of these backgrounds that have $\mathcal{N} = (2, 2)$ supersymmetry. In this case the CFT dual is the symmetric orbifold of $\mathbb{T}^4/D_n$.

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

Twin-plane-partitions and $\mathcal{N}=2$ affine Yangian

The universal enveloping algebra of ${\cal W}_{1+\infty}$ is isomorphic to the affine Yangian of $\mathfrak{gl}_1$. We study the ${\cal N}=2$ supersymmetric version of this correspondence, and identify the full set of defining relations of the supersymmetric affine Yangian. These relations can be deduced by demanding that the algebra has a representation on twin-plane-partitions, which we construct by gluing pairs of plane partitions. We define the action of the algebra on these twin-plane-partitions explicitly.

hep-th↗