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Fiona K. Seibold

Publications and source records attributed to Fiona K. Seibold.

At least 19 recordsLinked to original sources
arXiv2025

Elliptic deformations of the $\mathsf{AdS}_3 \times \mathsf{S}^3 \times \mathsf{T}^4$ string

With the aim of investigating the existence of an integrable elliptic deformation of strings on $\mathsf{AdS}_3 \times \mathsf{S}^3 \times \mathsf{T}^4$, we compute the tree-level worldsheet S-matrix of the elliptically-deformed bosonic sigma model on $\mathsf{AdS}_3 \times \mathsf{S}^3$ in uniform light-cone gauge. The resulting tree-level S-matrix is compatible with the integrability of the model and has interesting features, including a hidden $\mathsf{U}(1)$ symmetry not manifest in the Lagrangian. We find that it cannot be embedded in the known exact integrable R-matrices describing deformations of the undeformed $\mathsf{AdS}_3 \times \mathsf{S}^3 \times \mathsf{T}^4$ light-cone gauge S-matrix including fermions. Therefore, we construct embeddings of the deformed 6-d metric in type II supergravity with constant dilaton and homogeneous fluxes. The simplicity of these solutions suggests they are promising candidates to lead to an integrable string sigma model including fermions.

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arXiv2025

Supersymmetry and integrability of the elliptic $\mathrm{AdS}_3 \times \mathrm{S}^3 \times \mathrm{T}^4$ superstring

We construct a 1-parameter family of Ramond-Ramond fluxes supporting the elliptic $\mathrm{AdS}_3 \times \mathrm{S}^3 \times \mathrm{T}^4$ metric with constant dilaton and preserving 8 of the 16 supercharges of the undeformed background. On the supersymmetric locus, we compute the tree-level worldsheet S-matrix in uniform light-cone gauge up to quadratic order in fermions and find that it non-trivially satisfies the classical Yang-Baxter equation. Moreover, imposing classical integrability and symmetries, we conjecture compatible processes quartic in fermions. We also investigate different limits of interest, including trigonometric deformations and the limit to the $\mathrm{AdS}_2 \times \mathrm{S}^2 \times \mathrm{T}^6$ superstring. Our results provide strong evidence for a supersymmetric and integrable elliptic deformation of the $\mathrm{AdS}_3 \times \mathrm{S}^3 \times \mathrm{T}^4$ superstring supported by Ramond-Ramond flux and a constant dilaton.

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arXiv2025

AdS3 Integrability, Tensionless Limits, and Deformations: A Review

Motivated by the recent advances in the understanding of integrability for $AdS_3$ backgrounds, we present a lightning review of this approach, with particular attention to the "tensionless" limits (with zero and one unit of NSNS flux), and to the many integrable deformations of the supergravity backgrounds. Our aim is to concisely but comprehensively take stock of the state of the art in the field, in a way accessible to non-experts, and to highlight outstanding challenges. Along the way we reference where the various derivation of these results, which we mostly omit, can be found in full detail.

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arXiv2025

Interpolating families of integrable AdS3 backgrounds

We construct families of integrable deformations that interpolate between $AdS_3\times S^3\times S^3\times S^1$ and either $AdS_3\times S^3\times S^2\times T^2$ or $AdS_3\times S^2\times S^2\times T^3$. They preserve half of the supersymmetry of the original background, namely one copy of the $\mathfrak{d}(2,1;α)$ algebra. From this it follows a similar integrable interpolation between $AdS_3\times S^3\times T^4$ and $AdS_3\times S^2\times T^5$, which also preserves half of the supersymmetry, namely a copy of the $\mathfrak{psu}(1,1|2)$ algebra. In all cases, the interpolating backgrounds are constructed by using TsT transformations, which makes it easy to implement them in the integrability formalism in the full quantum theory. To illustrate this point, we discuss the lightcone gauge fixing of the models and compute their pp-wave Hamiltonian.

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arXiv2025

Scattering on the supermembrane

We compute the one-loop $2 \rightarrow 2$ scattering amplitude of massless scalars on the world volume of an infinite $D = 11$ supermembrane quantized in the static gauge. The resulting expression is manifestly finite and turns out to be much simpler than in the bosonic membrane case in arXiv:2308.12189 being simply proportional to the tree-level scattering amplitude. We also consider the case of $\mathbb{R}^{1,1} \times S^1$ membrane with one dimension compactified on a circle of radius R and demonstrate how the supermembrane scattering amplitude reduces to the one on an infinite $D = 10$ Green-Schwarz superstring in the limit of R $\rightarrow 0$.

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arXiv2025

S-matrix on effective string and compactified membrane

Expanding Nambu-Goto action near infinitely long string vacuum one can compute scattering amplitudes of 2d massless fields representing transverse string coordinates. As was shown in arXiv:1203.1054, the resulting S-matrix is integrable, in agreement with the known free string spectrum and also with an interpretation of the static-gauge NG action as a $T\bar T$ deformation of a free massless theory. We consider a generalization of this computation to the case of a membrane, expanding its 3d action near an infinite membrane vacuum that has cylindrical $\mathbb R \times S^1$ shape (we refer to such membrane as "compactified"). Representing 3d fields as Fourier series in $S^1$ coordinate we get an effective 2d model in which the massless string modes are coupled to an infinite KK tower of massive 2d modes. We find that the resulting 2d S-matrix is not integrable already at the tree level. We also compute 1-loop scattering amplitude of massless string modes with all compactified membrane modes propagating in the loop. The result is UV finite and is a non-trivial function of the kinematic variables. In the large momentum limit or when the radius of $S^1$ is taken to infinity we recover the expression for the 1-loop scattering amplitude of the uncompactified $\mathbb R^2$ membrane. We also consider a 2d model which is the $T\bar T$ deformation to the free theory with the same massless plus infinite massive tower of modes. The corresponding 2d S-matrix is found, as expected, to be integrable.

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arXiv2024

Inequivalent light-cone gauge-fixings of strings on $AdS_n \times S^n$ backgrounds

Light-cone gauge-fixed sigma-models on $AdS_n\times S^n$ backgrounds play an important role in the integrability formulation of the AdS/CFT correspondence. The string spectrum of the sigma-model is gauge-independent, however the Hamiltonian and scattering matrix of the transverse worldsheet fields are not. We study how these change for a large family of inequivalent light-cone gauges, which are interpreted as $T\bar{T}$, $\tilde{J}T_τ$, $JT_σ$ and $J^τ$ deformations. We investigate the moduli space of equivalent light-cone gauges and, specialising to $AdS_5 \times S^5$, compute the different light-cone gauge symmetry algebras, well-known to be $\mathfrak{psu}(2|2)^{\oplus 2} \oplus \mathfrak{u}(1)^{\oplus 2}$ for the standard gauge-fixing. Many integrable deformations require a non-standard light-cone gauge, hence our classification and analysis of inequivalent gauges will be important for analysing such models.

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arXiv2024

Exact approaches on the string worldsheet

We review different exact approaches to string theory. In the context of the Green-Schwarz superstring, we discuss the action in curved backgrounds and its supercoset formulation, with particular attention to superstring backgrounds of the $AdS_3$ type supported by both Ramond-Ramond and Neveu-Schwarz-Neveu-Schwarz fluxes. This is the basis for the discussion of classical integrability, of worldsheet-scattering factorisation in the uniform lightcone gauge, and eventually of the string spectrum through the mirror thermodynamic Bethe ansatz, which for $AdS_3$ backgrounds was only derived and analysed very recently. We then illustrate some aspects of the Ramond-Neveu-Schwarz string, and introduce the formalism of Berkovits-Vafa-Witten, which has seen very recent applications to $AdS_3$ physics, which we also briefly review. Finally, we present the relation between M-theory in the discrete lightcone quantisation and decoupling limits of string theory that exhibit non-relativistic behaviours, highlighting the connection with integrable $T\bar{T}$ deformations, as well as the relation between spin-matrix theory and Landau-Lifshitz models. This review is based on lectures given at the Young Researchers Integrability School and Workshop 2022 "Taming the string worldsheet" at NORDITA, Stockholm.

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arXiv2023

Bi-$η$ and bi-$λ$ deformations of $\mathbb{Z}_4$ permutation supercosets

Integrable string sigma models on AdS$_3$ backgrounds with 16 supersymmetries have the distinguishing feature that their superisometry group is a direct product. As a result the deformation theory of these models is particularly rich since the two supergroups in the product can be deformed independently. We construct bi-$η$ and bi-$λ$ deformations of two classes of $\mathbb{Z}_4$ permutation supercoset sigma models, which describe sectors of the Green-Schwarz and pure-spinor string worldsheet theories on type II AdS$_3$ backgrounds with pure R-R flux. We discuss an important limit of these models when one supergroup is undeformed. The associated deformed supergravity background should preserve 8 supersymmetries and is expected to have better properties than the full bi-deformation. As a step towards investigating the quantum properties of these models, we study the two-loop RG flow of the bosonic truncation of the bi-$λ$ deformation.

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arXiv2022

Integrable supersymmetric deformations of $\rm AdS_3 \times S^3 \times T^4$

We construct a family of type IIB string backgrounds that are deformations of $\rm AdS_3 \times S^3 \times T^4$ with a "squashed" $\rm AdS_3 \times S^3$ metric supported by a combination of NSNS and RR fluxes. They have global $\rm SU(1,1) \times SU(2)$ symmetry, regular curvature, constant dilaton and preserve 8 supercharges. Upon compactification to 4 dimensions they reduce to $\mathcal N=2$ supersymmetric $\rm AdS_2 \times S^2$ solutions with electric and magnetic Maxwell fluxes. These type IIB supergravity solutions can be found from the undeformed $\rm AdS_3 \times S^3 \times T^4$ background by a combination of T-dualities and S-duality. In contrast to T-duality, S-duality transformations of a type IIB supergravity background do not generally preserve the classical integrability of the corresponding Green-Schwarz superstring sigma model. Nevertheless, we show that integrability is preserved in the present case. Indeed, we find that these backgrounds can be obtained, up to T-dualities, from an integrable inhomogeneous Yang-Baxter deformation (with unimodular Drinfel'd-Jimbo R-matrix) of the original $\rm AdS_3 \times S^3$ supercoset model.

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arXiv2022

Transfer matrices for AdS3/CFT2

We work out the algebraic Bethe ansatz for the worldsheet theory of the $AdS_3\times S^3\times T^4$ superstring, and use it to derive the transfer matrices for fundamental particles and bound states of the string and mirror model. We also show how the Bethe equations and transfer matrices are modified in the presence of an Abelian twist. These will be an important ingredient in the exploration of the mirror thermodynamic Bethe ansatz equations recently proposed by Frolov and Sfondrini, and their generalisation to twisted and deformed models.

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arXiv2021

Bethe Ansatz for Quantum-Deformed Strings

Two distinct $η$-deformations of strings on AdS$_5\times$S$^5$ can be defined; both amount to integrable quantum deformations of the string non-linear sigma model, but only one is itself a superstring background. In this paper we compare their conjectured all-loop worldsheet S matrices and derive the corresponding Bethe equations. We find that, while the S matrices are apparently different, they lead to the same Bethe equations. Moreover, in either case the eigenvalues of the transfer matrix, which encode the conserved charges of each system, also coincide. We conclude that the integrable structure underlying the two constructions is essentially the same. Finally, we write down the full Bethe-Yang equations describing the asymptotic spectrum of the superstring background.

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arXiv2021

On quantum deformations of $AdS_3 \times S^3 \times T^4$ and mirror duality

We consider various integrable two-parameter deformations of the $AdS_3 \times S^3 \times T^4$ superstring with quantum group symmetry. Working on the string worldsheet in light-cone gauge and to quadratic order in fermions, we obtain their common massive tree-level two-body S matrix, which matches the expansion of the conjectured exact $q$-deformed S matrix. We then analyze the behavior of the exact S matrix under mirror transformation -- a double Wick rotation on the worldsheet -- and find that it satisfies a mirror duality relation analogous to the distinguished $q$-deformed $AdS_5 \times S^5$ S matrix in the one parameter deformation limit. Finally, we show that the fermionic $q$-deformed $AdS_5 \times S^5$ S matrix also satisfies such a relation.

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arXiv2021

The twisted story of worldsheet scattering in $η$-deformed $AdS_5 \times S^5$

We study the worldsheet scattering theory of the $η$ deformation of the AdS$_\mathsf{5} \times $S$^\mathsf{5}$ superstring corresponding to the purely fermionic Dynkin diagram. This theory is a Weyl-invariant integrable deformation of the AdS$_\mathsf{5} \times $S$^\mathsf{5}$ superstring, with trigonometric quantum-deformed symmetry. We compute the two-body worldsheet S matrix of this string in the light-cone gauge at tree level to quadratic order in fermions. The result factorizes into two elementary blocks, and solves the classical Yang-Baxter equation. We also determine the corresponding exact factorized S matrix, and show that its perturbative expansion matches our tree-level results, once we correctly identify the deformed light-cone symmetry algebra of the string. Finally, we briefly revisit the computation of the corresponding S matrix for the $η$ deformation based on the distinguished Dynkin diagram, finding a tree-level S matrix that factorizes and solves the classical Yang-Baxter equation, in contrast to previous results.

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arXiv2019

Two-parameter integrable deformations of the $AdS_3 \times S^3 \times T^4$ superstring

For supercosets with isometry group of the form $\hat{G} \times \hat{G}$, the eta-deformation can be generalised to a two-parameter integrable deformation with independent $q$-deformations of the two copies. We study its kappa-symmetry and write down a formula for the Ramond-Ramond fluxes. We then focus on $\hat{G}= PSU(1,1|2)$ and construct two supergravity backgrounds for the two-parameter integrable deformation of the $AdS_3 \times S^3 \times T^4$ superstring, as well as explore their limits. We also construct backgrounds that are solutions of the weaker generalised supergravity equations of motion and compare them to the literature.

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arXiv2019

Poisson-Lie duals of the $η$-deformed $\mathrm{AdS}_2 \times \mathrm{S}^2 \times \mathrm{T}^6$ superstring

We investigate Poisson-Lie duals of the $η$-deformed $\mathrm{AdS}_2 \times \mathrm{S}^2 \times \mathrm{T}^6$ superstring. The $η$-deformed background satisfies a generalisation of the type II supergravity equations. We discuss three Poisson-Lie duals, with respect to (i) the full $\mathfrak{psu}(1,1|2)$ superalgebra, (ii) the full bosonic subalgebra and (iii) the Cartan subalgebra, for which the corresponding backgrounds are expected to satisfy the standard type II supergravity equations. The metrics and B-fields for the first two cases are the same and given by an analytic continuation of the $λ$-deformed model on $\mathrm{AdS}_2 \times \mathrm{S}^2 \times \mathrm{T}^6$ with the torus undeformed. However, the RR fluxes and dilaton will differ. Focusing on the second case we explicitly derive the background and show agreement with an analytic continuation of a known embedding of the $λ$-deformed model on $\mathrm{AdS}_2 \times \mathrm{S}^2$ in type II supergravity.

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arXiv2019

Supergravity backgrounds of the eta-deformed AdS2 x S2 x T6 and AdS5 x S5 superstrings

We construct supergravity backgrounds for the integrable eta-deformations of the AdS2 x S2 x T6 and AdS5 x S5 superstring sigma models. The eta-deformation is governed by an R-matrix that solves the non-split modified classical Yang-Baxter equation on the superisometry algebra of the model. Such R-matrices include those of Drinfel'd-Jimbo type, which are constructed from a Dynkin diagram and the associated Cartan-Weyl basis. Drinfel'd-Jimbo R-matrices associated with inequivalent bases will typically lead to different deformed backgrounds. For the two models under consideration we find that the unimodularity condition, implying that there is no Weyl anomaly, is satisfied if and only if all the simple roots are fermionic. For AdS2 x S2 x T6 we construct backgrounds corresponding to the three Dynkin diagrams. When all the simple roots are fermionic we find a supergravity background previously obtained by directly solving the supergravity equations. For AdS5 x S5 we construct a supergravity background corresponding to the Dynkin diagram with all fermionic simple roots.

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