arXiv subjects

Ben Hoare

Publications and source records attributed to Ben Hoare.

At least 19 recordsLinked to original sources

arXiv2025

Integrable models from 4d holomorphic BF theory

We show how to construct 2d field theories with holomorphic integrability from defect setups in 4d holomorphic BF. In a simple example setup, we explicitly construct the 2d theory and perform an initial classical analysis. Making use of the symmetries, we are able to write down an infinite family of solutions to the equations of motion. Comparing with more typical integrable systems, we explain how 2d holomorphic integrability sits between the standard notions of integrability in one and two dimensions. These 2d theories are designed as toy models for integrable theories in three and four dimensions, many of which can be understood as partially or totally holomorphic. We comment on the implications for higher-dimensional integrability and aspects of quantization in the concluding remarks.

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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

Twists of trigonometric sigma models

We introduce the $\mathbb{Z}_N$-twisted trigonometric sigma models, a new class of integrable deformations of the principal chiral model. Starting from 4d Chern-Simons theory on a cylinder, the models are constructed by introducing a $\mathbb{Z}_N$ branch cut running along the non-compact direction. As we pass through the branch cut we apply a $\mathbb{Z}_N$ automorphism to the algebra-valued and group-valued fields of the theory. Two instances of models in this class have appeared in the literature and we explain how these fit into our general construction. We generalise both to the case of $\mathbb{Z}_N$-twistings, and for each we construct two further deformations, leading to four doubly-deformed models. Mapping the cylinder to a sphere, a novel feature of the construction is that the branch points are at the simple poles corresponding to the ends of the cylinder. As a result, the untwisted and twisted models have the same number of degrees of freedom. While twisting does not always lead to inequivalent models, we show that if we use the $\mathbb{Z}_2$ outer automorphism of $\mathfrak{g}^{\mathbb{C}} = \mathfrak{sl}(n;\mathbb{C})$ to twist, then the untwisted and twisted models have different symmetries, hence the latter are new integrable sigma models.

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arXiv2024

Gauging The Diamond: Integrable Coset Models from Twistor Space

Recent work has shown that certain integrable and conformal field theories in two dimensions can be given a higher-dimensional origin from holomorphic Chern-Simons in six dimensions. Along with anti-self-dual Yang-Mills and four-dimensional Chern-Simons, this gives rise to a diamond correspondence of theories. In this work we extend this framework to incorporate models realised through gaugings. As well as describing a higher-dimensional origin of coset CFTs, by choosing the details of the reduction from higher dimensions, we obtain rich classes of two-dimensional integrable models including homogeneous sine-Gordon models and generalisations that are new to the literature.

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arXiv2023

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 $T\bar{T}$, $\tilde{J}T_\tau$, $JT_\sigma$ and $J^\tau$ 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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arXiv2023

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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arXiv2023

Integrable Deformations from Twistor Space

Integrable field theories in two dimensions are known to originate as defect theories of 4d Chern-Simons and as symmetry reductions of the 4d anti-self-dual Yang-Mills equations. Based on ideas of Costello, it has been proposed in work of Bittleston and Skinner that these two approaches can be unified starting from holomorphic Chern-Simons in 6 dimensions. We provide the first complete description of this diamond of integrable theories for a family of deformed sigma models, going beyond the Dirichlet boundary conditions that have been considered thus far. Starting from 6d holomorphic Chern-Simons theory on twistor space with a particular meromorphic 3-form $\Omega$, we construct the defect theory to find a novel 4d integrable field theory, whose equations of motion can be recast as the 4d anti-self-dual Yang-Mills equations. Symmetry reducing, we find a multi-parameter 2d integrable model, which specialises to the $\lambda$-deformation at a certain point in parameter space. The same model is recovered by first symmetry reducing, to give 4d Chern-Simons with generalised boundary conditions, and then constructing the defect theory.

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arXiv2023

Towards a quadratic Poisson algebra for the subtracted classical monodromy of Symmetric Space Sine-Gordon theories

Symmetric Space Sine-Gordon theories are two-dimensional massive integrable field theories, generalising the Sine-Gordon and Complex Sine-Gordon theories. To study their integrability properties on the real line, it is necessary to introduce a subtracted monodromy matrix. Moreover, since the theories are not ultralocal, a regularisation is required to compute the Poisson algebra for the subtracted monodromy. In this article, we regularise and compute this Poisson algebra for certain configurations, and show that it can both satisfy the Jacobi identity and imply the existence of an infinite number of conserved quantities in involution.

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arXiv2022

Supersphere non-linear sigma model on the lattice

Two-dimensional $O(N)$ non-linear sigma models are exactly solvable theories and have many applications, from statistical mechanics to their use as QCD toy models. We consider a supersymmetric extension, the non-linear sigma model on the supersphere~$S^{N+2m-1|2m}\equiv \frac{OSP(N+2m|2m)}{OSP(N+2m-1|2m)}$. We briefly describe its renormalization properties and lattice discretization, and present a strategy for numerical simulations together with some preliminary numerical results.

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arXiv2022

Bi-$\eta$ and bi-$\lambda$ 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-$\eta$ and bi-$\lambda$ 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-$\lambda$ 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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arXiv2021

Integrable Deformations of Sigma Models

In this pedagogical review we introduce systematic approaches to deforming integrable 2-dimensional sigma models. We use the integrable principal chiral model and the conformal Wess-Zumino-Witten model as our starting points and explore their Yang-Baxter and current-current deformations. There is an intricate web of relations between these models based on underlying algebraic structures and worldsheet dualities, which is highlighted throughout. We finish with a discussion of the generalisation to other symmetric integrable models, including some original results related to Z T cosets and their deformations, and the application to string theory. This review is based on notes written for lectures delivered at the school "Integrability, Dualities and Deformations", which ran from 23 to 27 August 2021 in Santiago de Compostela and virtually.

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arXiv2020

Dual description of $\eta$-deformed OSP sigma models

We study the dual description of the $\eta$-deformed $OSP(N|2m)$ sigma model in the asymptotically free regime ($N>2m+2$). Compared to the case of classical Lie groups, for supergroups there are inequivalent $\eta$-deformations corresponding to different choices of simple roots. For a class of such deformations we propose the system of screening charges depending on a continuous parameter $b$, which defines the $\eta$-deformed $OSP(N|2m)$ sigma model in the limit $b\rightarrow\infty$ and a certain Toda QFT as $b\rightarrow0$. In the sigma model regime we show that the leading UV asymptotic of the $\eta$-deformed model coincides with a perturbed Gaussian theory. In the perturbative regime $b\rightarrow0$ we show that the tree-level two-particle scattering matrix matches the expansion of the trigonometric $OSP(N|2m)$ $S$-matrix.

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arXiv2020

Sigma models with local couplings: a new integrability -- RG flow connection

We consider several classes of $\sigma$-models (on groups and symmetric spaces, $\eta$-models, $\lambda$-models) with local couplings that may depend on the 2d coordinates, e.g. on time $\tau$. We observe that (i) starting with a classically integrable 2d $\sigma$-model, (ii) formally promoting its couplings $h_\alpha$ to functions $h_\alpha(\tau)$ of 2d time, and (iii) demanding that the resulting time-dependent model also admits a Lax connection implies that $h_\alpha(\tau)$ must solve the 1-loop RG equations of the original theory with $\tau$ interpreted as RG time. This provides a novel example of an 'integrability - RG flow' connection. The existence of a Lax connection suggests that these time-dependent $\sigma$-models may themselves be understood as integrable. We investigate this question by studying the possibility of constructing non-local and local conserved charges. Such $\sigma$-models with $D$-dimensional target space and time-dependent couplings subject to the RG flow naturally appear in string theory upon fixing the light-cone gauge in a $(D+2)$-dimensional conformal $\sigma$-model with a metric admitting a covariantly constant null Killing vector and a dilaton linear in the null coordinate.

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arXiv2019

Integrable sigma models and 2-loop RG flow

Following arXiv:1907.04737, we continue our investigation of the relation between the renormalizability (with finitely many couplings) and integrability in 2d $\sigma$-models. We focus on the "$\lambda$-model," an integrable model associated to a group or symmetric space and containing as special limits a (gauged) WZW model and an "interpolating model" for non-abelian duality. The parameters are the WZ level $k$ and the coupling $\lambda$, and the fields are $g$, valued in a group $G$, and a 2d vector $A_\pm$ in the corresponding algebra. We formulate the $\lambda$-model as a $\sigma$-model on an extended $G \times G \times G$ configuration space $(g, h, \bar{h})$, defining $h $ and $\bar{h}$ by $A_+ = h \partial_+ h^{-1}$, $A_- = \bar{h} \partial_- \bar{h}^{-1}$. Our central observation is that the model on this extended configuration space is renormalizable without any deformation, with only $\lambda$ running. This is in contrast to the standard $\sigma$-model found by integrating out $A_\pm$, whose 2-loop renormalizability is only obtained after the addition of specific finite local counterterms, resulting in a quantum deformation of the target space geometry. We compute the 2-loop $\beta$-function of the $\lambda$-model for general group and symmetric spaces, and illustrate our results on the examples of $SU(2)/U(1)$ and $SU(2)$. Similar conclusions apply in the non-abelian dual limit implying that non-abelian duality commutes with the RG flow. We also find the 2-loop $\beta$-function of a "squashed" principal chiral model.

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arXiv2019

Integrable 2d sigma models: quantum corrections to geometry from RG flow

Classically integrable $\sigma$-models are known to be solutions of the 1-loop RG equations, or "Ricci flow", with only a few couplings running. In some of the simplest examples of integrable deformations we find that in order to preserve this property at 2 (and higher) loops the classical $\sigma$-model should be corrected by quantum counterterms. The pattern is similar to that of effective $\sigma$-models associated to gauged WZW theories. We consider in detail the examples of the $\eta$-deformation of $S^2$ ("sausage model") and $H^2$, as well as the closely related $\lambda$-deformation of the $SO(1,2)/SO(2)$ coset. We also point out that similar counterterms are required in order for non-abelian duality to commute with RG flow beyond the 1-loop order.

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arXiv2018

On the massless tree-level S-matrix in 2d sigma models

Motivated by the search for new integrable string models, we study the properties of massless tree-level S-matrices for 2d sigma models expanded near the trivial vacuum. We find that, in contrast to the standard massive case, there is no apparent link between massless S-matrices and integrability: in well-known integrable models the tree-level massless S-matrix fails to factorize and exhibits particle production. Such tree-level particle production is found in several classically integrable models: the principal chiral model, its classically equivalent "pseudo-dual" model, its non-abelian dual model and also the SO(N+1)/SO(N) coset model. The connection to integrability may, in principle, be restored if one expands near a non-trivial vacuum with massive excitations. We discuss IR ambiguities in 2d massless tree-level amplitudes and their resolution using either a small mass parameter or the i epsilon-regularization. In general, these ambiguities can lead to anomalies in the equivalence of the S-matrix under field redefinitions, and may be linked to the observed particle production in integrable models. We also comment on the transformation of massless S-matrices under sigma model T-duality, comparing the standard and the "doubled" formulations (with T-duality covariance built into the latter).

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arXiv2018

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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