Quantum anisotropic sigma and lambda models as spin chains

Appadu, Calan; Hollowood, Timothy J.; Price, Dafydd; Thompson, Daniel C.

doi:10.1088/1751-8121/aadc6d

Cited by 13 publications

(21 citation statements)

References 182 publications

(317 reference statements)

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“…This η-deformation algorithm, combined also with the coset construction of Refs. [16] and with the alternative λ-deformation one [72], gave rise to various constructions of the deformed integrable σ-models [10,15,17,18,20,30,31,41,73,75,81], many of them exploitable in quantum field theory [2,4,5,8,21,24,25,56,57] and in string theory via the AdS/CFT correspondence [1,7,13,16,19,22,27,32,33,36,37,60,61,64,65,68,69,80].…”

Section: Introductionmentioning

confidence: 99%

Dressing cosets and multi-parametric integrable deformations

Klimčı́k

2019

J. High Energ. Phys.

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We provide a new construction of the dressing cosets σ-models which is based on an isotropic gauging of the E-models. As an application of this new approach, we show that the recently constructed multiparametric integrable deformations of the principal chiral model are the dressing cosets, they are therefore automatically renormalizable and their dynamics can be completely characterised in terms of current algebras. Keywords: integrable sigma models, renormalization group flowRecently, Delduc, Hoare, Kameyama and Magro have found the multi-parametric integrable σ-model (1.1) living on an arbitrary simple group manifold K [15]. In their approach, they succeeded to merge consistently several deformation procedures studied previously in a separate way, like the (bi)-Yang-Baxter deformations [44], the addition of the WZW term [17] or the introduction of the so-called TsT matrix [29,59,62,66,80]. For the special case of the group SU (2), their result fits into the framework of the Lukyanov model [58].We show in Section 4 of the present paper that there exists an E-model description of the DHKM σ-model (for the emancipated parameter α), however, there is a novel element in the game comparing with the cases of the low number of deformation parameters treated in [45,46]. Namely, the E-model 1 The reason for this terminology is the fact that in the case α = 0 we recover from (1.9) the standard WZW model. In what follows, we shall say "the DHKM model" whenever the TsT matrix is switched on. However, for the case of the vanishing TsT matrix we reserve the terminology "the bi-YB-WZ model".underlying the DHKM σ-model turns out to be degenerate, that is, it is the so called dressing coset in the sense of Ref. [51].Actually, we introduce in the present work a new method of constructing the dressing cosets which is based on an appropriate isotropic gauging of the non-degenerate E-models. This new approach is technically very friendly and it plays the key role in the understanding of the structure of the DHKM model. We describe it in Section 3.3, just after reviewing the theory of the non-degenerate E-models in Section 3.1 as well as the old theory of the dressing cosets in Section 3.2.What is it good for to know that the first order Hamiltonian dynamics of a nonlinear σ-model can be described in terms of a particular (degenerate) E-model? Well, the immediate benefit of this knowledge is the fact that the σ-model underlied by the E-model is automatically renormalizable [74,76,79]. This means, in particular, that the ultraviolet corrections just let flow the parameters of the model without spoiling the form of the Lagrangian. Moreover, the E-model formalism permits to determine the renormalization group flow by a simple method introduced in Ref. [70,76]. Actually, we employ this method in Section 6 to establish the renormalizability of the bi-YB-WZ model, after proving in Section 5 its integrability. Finally, we devote Section 6.4 to a detailed analysis of the case of K = SU (2) where our results for the bi-YB-WZ RG flow are shown...

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Section: Introductionmentioning

confidence: 99%

Dressing cosets and multi-parametric integrable deformations

Klimčı́k

2019

J. High Energ. Phys.

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“…choosing E = (1 − ηR) −1 , does indeed give rise to a two-parameter anisotropic λ deformation of the WZW model for which an explicit Lax can be obtained. This kind of anisotropic, or XXZ, λ -model has been studied in [195,49,196,197]. Other more general types of λ -models including products of interacting WZW factors have been developed in a sequence of works [198,199,200,201].…”

Section: Multi-parameter λ -Modelsmentioning

confidence: 99%

“…However, the non-ultralocality (the Schwinger term / derivative of the delta function) prevents this from being done in any straightforward way. An alternative philosophy, adopted in the λ -models in [207,197], is to start directly with a bare theory on a lattice defined by an integrable Hamiltonian. One can exploit e.g.…”

Section: λ -Models Quantum Inverse Scattering and S-matricesmentioning

confidence: 99%

“…The free energy calculated from this at high temperatures (driving the system to the UV) matches that of the central charge of the WZW model, and at T = 0 but at finite chemical potential a precise matching can be made with perturbation theory including a full recovery of the β -function of the coupling λ . In [197] a similar analyse was conducted for two-parameter anisotropic λ -deformations in the case of SU(N) making a link to an XXZ spin k/2 spin chain.…”

Section: λ -Models Quantum Inverse Scattering and S-matricesmentioning

confidence: 99%

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An Introduction to Generalised Dualities and their Applications to Holography and Integrability

Thompson¹

2019

Proceedings of Corfu Summer Institute 2018 "School and Workshops on Elementary Particle Physics and Gravity" — PoS(CORFU2018)

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These pedagogical lectures given at the Corfu Summer Institute 2018 review two generalised notions of T-duality, non-Abelian T-duality and Poisson-Lie duality, and their applications. We explain how each of these has seen recent application in the context of holography. Non-Abelian T-duality has been used to construct new holographic dual geometries. Poisson-Lie duality has been used to construct new integrable string sigma-models including the η-and λ -deformations of the AdS 5 × S 5 superstring thought to encode quantum group deformations of holography. We also comment on the doubled worldsheet description that makes such dualities manifest. duality serves to interchange a conservation law, that associated to the U(1) global symmetry of the compact boson θ of radius R, with the Bianchi identity for the dual bosonθ :( 1.3) Just as with bosonisation the connection between the two variables involves derivatives, i.e. the map between variables is a non-local one. Also akin to the Abelian bosonisation, the dual theory can be established through an analogous path integral manipulation 1 known as the Buscher procedure [22,23]. Given the close analogy between T-duality and Abelian bosonisation one might wonder if there is a T-duality equivalent to non-Abelian bosonisation. More specifically, in string theory we are immediately prompted to ask if there is an extension of T-duality to the case of a non-Abelian set of isometries, or even if there are still more exotic notions of T-duality? A secondary question is then, if such dualities exist, what are they useful for? The remainder of these lectures seek to outline some answers to these two questions.Our journey will involve exploring a hierarchy of possible dualities in the following string theory scenarios:

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“…It consists in discretising and quantisingà la Faddeev-Reshetikhin. This was first developed for the Principal Chiral Model [9] (see [10][11][12] for other recent applications of this approach). This way of treating non-ultralocality relies however on an ultralocal Lax matrix which is associated with a modified canonical structure.…”

Section: Introductionmentioning

confidence: 99%

Ultralocal Lax connection for para-complex ZT-cosets

Delduc

Kameyama²,

Lacroix

et al. 2019

Nuclear Physics B

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We consider σ-models on para-complex Z T -cosets, which are analogues of those on complex homogeneous target spaces considered recently by D. Bykov. For these models, we show the existence of a gauge-invariant Lax connection whose Poisson brackets are ultralocal. Furthermore, its light-cone components commute with one another in the sense of Poisson brackets. This extends a result of O. Brodbeck and M. Zagermann obtained twenty years ago for hermitian symmetric spaces.

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Quantum anisotropic sigma and lambda models as spin chains

Cited by 13 publications

References 182 publications

Dressing cosets and multi-parametric integrable deformations

Dressing cosets and multi-parametric integrable deformations

An Introduction to Generalised Dualities and their Applications to Holography and Integrability

Ultralocal Lax connection for para-complex ZT-cosets

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