Quantum inverse scattering and the lambda deformed principal chiral model

Appadu, Calan; Hollowood, Timothy J.; Price, Dafydd

doi:10.1088/1751-8121/aa7958

Cited by 25 publications

(40 citation statements)

References 113 publications

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“…Then, the AdS 5 × S 5 lambda model (and in general any lambda model on a symmetric and semi-symmetric space) is still outside the grasp of the QISM even in its simplifying λ → 0 limit. The quantization of non-ultralocal 1+1 dimensional integrable field theories has been a longstanding challenging problem and different approaches to handle this situation in diverse scenarios have been considered along the years, see for instance the references [15][16][17][18][19][20][21][22][23][24][25], but, unfortunately, finding a systematic quantization scheme for treating this kind of theories has been quite elusive.…”

Section: Introductionmentioning

confidence: 99%

Lambda models from Chern-Simons theories

Schmidtt¹

2018

J. High Energ. Phys.

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In this paper we refine and extend the results of [1], where a connection between the AdS 5 × S 5 superstring lambda model on S 1 = ∂D and a double Chern-Simons (CS) theory on D based on the Lie superalgebra psu(2, 2|4) was suggested, after introduction of the spectral parameter z. The relation between both theories mimics the well-known CS/WZW symplectic reduction equivalence but is non-chiral in nature. All the statements are now valid in the strong sense, i.e. valid on the whole phase space, making the connection between both theories precise. By constructing a z-dependent gauge field in the 2+1 Hamiltonian CS theory it is shown that: i) by performing a symplectic reduction of the CS theory the Maillet algebra satisfied by the extended Lax connection of the lambda model emerges as a boundary current algebra and ii) the Poisson algebra of the supertraces of z-dependent Wilson loops in the CS theory obey some sort of spectral parameter generalization of the Goldman bracket. The latter algebra is interpreted as the precursor of the (ambiguous) lambda model monodromy matrix Poisson algebra prior to the symplectic reduction. As a consequence, the problematic non-ultralocality of lambda models is avoided (for any value of the deformation parameter λ ⊂ [0, 1]), showing how the lambda model classical integrable structure can be understood as a byproduct of the symplectic reduction process of the z-dependent CS theory.

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Section: Introductionmentioning

confidence: 99%

Lambda models from Chern-Simons theories

Schmidtt¹

2018

J. High Energ. Phys.

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“…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%

“…One may still wish to motivate the bare lattice theory directly from the classical field theory. An approach to this is Faddeev-Reshetikhin [209] alleviation idea, which in this case [207] consists of taking a delicate limit of k → 0 and λ → 0 whilst keeping fixed…”

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

confidence: 99%

“…In [207] this analysis was similarly carried out for the SU(N) equivalent λ -deformation, providing a derivation of a previously conjectured S-matrix. 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 .…”

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 inverse scattering and the lambda deformed principal chiral model

Cited by 25 publications

References 113 publications

Lambda models from Chern-Simons theories

Lambda models from Chern-Simons theories

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

Ultralocal Lax connection for para-complex ZT-cosets

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