arXiv subjects

Matthias R. Gaberdiel

Publications and source records attributed to Matthias R. Gaberdiel.

At least 55 records · Page 3Linked to original sources

arXiv2018

The supersymmetric affine Yangian

The affine Yangian of $\mathfrak{gl}_1$ is known to be isomorphic to ${\cal W}_{1+\infty}$, the $W$-algebra that characterizes the bosonic higher spin -- CFT duality. In this paper we propose defining relations of the Yangian that are relevant for the ${\cal N}=2$ superconformal version of ${\cal W}_{1+\infty}$. Our construction is based on the observation that the ${\cal N}=2$ superconformal ${\cal W}_{1+\infty}$ algebra contains two commuting bosonic ${\cal W}_{1+\infty}$ algebras, and that the additional generators transform in bi-minimal representations with respect to these two algebras. The corresponding affine Yangian can therefore be built up from two affine Yangians of $\mathfrak{gl}_1$ by adding in generators that transform appropriately.

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arXiv2018

The plane-wave limit of ${\rm AdS}_3 \times {\rm S}^3 \times {\rm S}^3 \times {\rm S}^1$

The plane-wave limit of ${\rm AdS}_3 \times {\rm S}^3 \times {\rm S}^3 \times {\rm S}^1$ is analysed for generic null-geodesics that are not necessarily BPS. For the case of pure NS-NS flux it is shown how the resulting spectrum can be reproduced as a suitable limit of the world-sheet description in terms of WZW models. Since supersymmetry is broken, most of the degeneracies are lifted, and thus the identification of states is quite unambiguous.

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arXiv2018

Tensionless String Spectra on ${\rm AdS}_3$

The spectrum of superstrings on ${\rm AdS}_3 \times {\rm S}^3 \times \mathbb{M}_4$ with pure NS-NS flux is analysed for the background where the radius of the AdS space takes the minimal value $(k=1)$. Both for $\mathbb{M}_4={\rm S}^3 \times {\rm S}^1$ and $\mathbb{M}_4 = \mathbb{T}^4$ we show that there is a special set of physical states, coming from the bottom of the spectrally flowed continuous representations, which agree in precise detail with the single particle spectrum of a free symmetric product orbifold. For the case of ${\rm AdS}_3 \times {\rm S}^3 \times \mathbb{T}^4$ this relies on making sense of the world-sheet theory at $k=1$, for which we make a concrete proposal. We also comment on the implications of this striking result.

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arXiv2018

Mathieu Moonshine and Symmetry Surfing

Mathieu Moonshine, the observation that the Fourier coefficients of the elliptic genus on K3 can be interpreted as dimensions of representations of the Mathieu group M24, has been proven abstractly, but a conceptual understanding in terms of a representation of the Mathieu group on the BPS states, is missing. Some time ago, Taormina and Wendland showed that such an action can be naturally defined on the lowest non-trivial BPS states, using the idea of `symmetry surfing', i.e., by combining the symmetries of different K3 sigma models. In this paper we find non-trivial evidence that this construction can be generalized to all BPS states.

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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.

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arXiv2017

BPS spectrum on AdS$_3\times $S$^3 \times $S$^3 \times $S$^1$

The BPS spectrum of string theory on AdS$_3\times {\rm S}^3 \times {\rm S}^3 \times {\rm S}^1$ is determined using a world-sheet description in terms of WZW models. It is found that the theory only has BPS states with $j^+ = j^-$ where $j^{\pm}$ refer to the spins of the $\mathfrak{su}(2)$ algebras of the large ${\cal N}=4$ superconformal algebra. We then re-examine the BPS spectrum of the corresponding supergravity and find that, in contradistinction to previous claims in the literature, also in supergravity only the states with $j^+=j^-$ are BPS. This resolves a number of long-standing puzzles regarding the BPS spectrum of string theory and supergravity in this background.

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arXiv2017

A holographic dual for string theory on $\mathrm{AdS}_3 \times \mathrm{S}^3 \times \mathrm{S}^3 \times \mathrm{S}^1$

The CFT dual of string theory on $\mathrm{AdS}_3 \times \mathrm{S}^3 \times \mathrm{S}^3 \times \mathrm{S}^1$ is conjectured to be the symmetric orbifold of the $\mathcal{S}_κ$ theory, provided that one of the two $Q_5^\pm$ quantum numbers is a multiple of the other. We determine the BPS spectrum of the symmetric orbifold in detail, and show that it reproduces precisely the BPS spectrum that was recently calculated in supergravity. We also determine the BPS spectrum of the world-sheet theory that is formulated in terms of WZW models, and show that, apart from some gaps (which are reminiscent of those that appear in the corresponding $\mathbb{T}^4$ calculation), it also reproduces this BPS spectrum. In fact, the matching seems to work as well as for the familiar $\mathbb{T}^4$ case, and thus our results give strong support for this proposal.

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arXiv2017

Stringy AdS3 from the Worldsheet

We investigate the behaviour of the bosonic string on AdS3 with H-flux at stringy scales, looking in particular for a `tensionless' limit in which there are massless higher spin gauge fields. We do this by revisiting the physical spectrum of the sl(2,R)$_k$ WZW model and considering the limit in which k becomes small. At k=3 we find that there is an infinite stringy tower of massless higher spin fields which are part of a continuum of light states. This can be viewed as a novel tensionless limit, which appears to be distinct from that inferred from the symmetric orbifold description of superstring AdS3 vacua.

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arXiv2017

Higher spins on AdS$_{3}$ from the worldsheet

It was recently shown that the CFT dual of string theory on ${\rm AdS}_3 \times {\rm S}^3 \times T^4$, the symmetric orbifold of $T^4$, contains a closed higher spin subsector. Via holography, this makes precise the sense in which tensionless string theory on this background contains a Vasiliev higher spin theory. In this paper we study this phenomenon directly from the worldsheet. Using the WZW description of the background with pure NS-NS flux, we identify the states that make up the leading Regge trajectory and show that they fit into the even spin ${\cal N}=4$ Vasiliev higher spin theory. We also show that these higher spin states do not become massless, except for the somewhat singular case of level $k=1$ where the theory contains a stringy tower of massless higher spin fields coming from the long string sector.

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arXiv2017

Higher Spins and Yangian Symmetries

The relation between the bosonic higher spin ${\cal W}_\infty[λ]$ algebra, the affine Yangian of $\mathfrak{gl}_{1}$, and the SH$^c$ algebra is established in detail. For generic $λ$ we find explicit expressions for the low-lying ${\cal W}_\infty[λ]$ modes in terms of the affine Yangian generators, and deduce from this the precise identification between $λ$ and the parameters of the affine Yangian. Furthermore, for the free field cases corresponding to $λ=0$ and $λ=1$ we give closed-form expressions for the affine Yangian generators in terms of the free fields. Interestingly, the relation between the ${\cal W}_\infty$ modes and those of the affine Yangian is a non-local one, in general. We also establish the explicit dictionary between the affine Yangian and the SH$^c$ generators. Given that Yangian algebras are the hallmark of integrability, these identifications should pave the way towards uncovering the relation between the integrable and the higher spin symmetries.

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arXiv2017

The large $\mathcal{N}=4$ superconformal $\mathcal{W}_\infty$ algebra

The most general large ${\cal N}=4$ superconformal ${\cal W}_{\infty}$ algebra, containing in addition to the superconformal algebra one supermultiplet for each integer spin, is analysed in detail. It is found that the ${\cal W}_{\infty}$ algebra is uniquely determined by the levels of the two $\mathfrak{su}(2)$ algebras, a conclusion that holds both for the linear and the non-linear case. We also perform various cross-checks of our analysis, and exhibit two different types of truncations in some detail.

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arXiv2016

Twisted sectors from plane partitions

Twisted sectors arise naturally in the bosonic higher spin CFTs at their free points, as well as in the associated symmetric orbifolds. We identify the coset representations of the twisted sector states using the description of W_\infty representations in terms of plane partitions. We confirm these proposals by a microscopic null-vector analysis, and by matching the excitation spectrum of these representations with the orbifold prediction.

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arXiv2016

Higgsing the stringy higher spin symmetry

It has recently been argued that the symmetric orbifold theory of T4 is dual to string theory on AdS3 x S3 x T4 at the tensionless point. At this point in moduli space, the theory possesses a very large symmetry algebra that includes, in particular, a $W_\infty$ algebra capturing the gauge fields of a dual higher spin theory. Using conformal perturbation theory, we study the behaviour of the symmetry generators of the symmetric orbifold theory under the deformation that corresponds to switching on the string tension. We show that the generators fall nicely into Regge trajectories, with the higher spin fields corresponding to the leading Regge trajectory. We also estimate the form of the Regge trajectories for large spin, and find evidence for the familiar logarithmic behaviour, thereby suggesting that the symmetric orbifold theory is dual to an AdS background with pure RR flux.

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arXiv2016

The symmetric orbifold of N=2 minimal models

The large level limit of the N=2 minimal models that appear in the duality with the N=2 supersymmetric higher spin theory on AdS_3 is shown to be a natural subsector of a certain symmetric orbifold theory. We study the relevant decompositions in both the untwisted and the twisted sector, and analyse the structure of the higher spin representations in the twisted sector in some detail. These results should help to identify the string background of which the higher spin theory is expected to describe the leading Regge trajectory in the tensionless limit.

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arXiv2016

Cosets of Meromorphic CFTs and Modular Differential Equations

Some relations between families of two-character CFTs are explained using a slightly generalised coset construction, and the underlying theories (whose existence was only conjectured based on the modular differential equation) are constructed. The same method also gives rise to interesting new examples of CFTs with three and four characters.

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arXiv2015

Higher spins in the symmetric orbifold of K3

The symmetric orbifold of K3 is believed to be the CFT dual of string theory on AdS3 x S3 x K3 at the tensionless point. For the case when the K3 is described by the orbifold T4/Z2, we identify a subsector of the symmetric orbifold theory that is dual to a higher spin theory on AdS3. We analyse how the BPS spectrum of string theory can be described from the higher spin perspective, and determine which single-particle BPS states are accounted for by the perturbative higher spin theory.

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arXiv2014

The $\mathfrak{so}$-Kazama-Suzuki Models at Large Level

The large level limit of the $\mathcal{N}=2$ ${\rm SO}(2N)$ Kazama-Suzuki coset models is argued to be equivalent to the orbifold of $4N$ free fermions and bosons by the Lie group ${\rm SO}(2N) \times {\rm SO}(2)$. In particular, it is shown that the untwisted sector of the continuous orbifold accounts for a certain closed subsector of the coset theory. Furthermore, the ground states of the twisted sectors are identified with specific coset representations, and this identification is checked by various independent arguments.

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arXiv2014

Genus two partition functions of chiral conformal field theories

A systematic analysis of the genus two vacuum amplitudes of chiral self-dual conformal field theories is performed. It is explained that the existence of a modular invariant genus two partition function implies infinitely many relations among the structure constants of the theory. All of these relations are shown to be a consequence of the associativity of the OPE, as well as the modular covariance properties of the torus one-point functions. Using these techniques we prove that for the proposed extremal conformal field theories at c=24k a consistent genus two vacuum amplitude exists for all k, but that this does not actually check the consistency of these theories beyond what is already testable at genus one.

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