Sigma model from spin chain

Bellucci, Stefano; Casteill, Pierre-Yves

doi:10.1016/j.nuclphysb.2006.02.021

Cited by 20 publications

(25 citation statements)

References 57 publications

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“…In the limit in which the spin chain is long, one can furthermore find a long wavelength limit. This procedure results in a sigma-model description in a semi-classical limit of the spin chain [14] (see also [15][16][17][18][19] and [4]). The sigma-model is based on the coset related to the representation of the spin group in Table 1.…”

Section: Spin Matrix Theorymentioning

confidence: 99%

“…When applied to strings on AdS 5 × S 5 this scaling limit is realized by the SMT limits of [2,4]. The SMT limit is in turn closely related to limits of spin chains [14] (see also [15][16][17][18][19]) that have been studied in connection to integrability in AdS/CFT. In particular, the simplest example of the non-relativistic world-sheet theory describes a covariant version of the Landau-Lifshitz sigma-model which is the continuum limit of the ferromagnetic XXX 1/2 Heisenberg spin chain.…”

Section: Introductionmentioning

confidence: 95%

See 1 more Smart Citation

Strings with non-relativistic conformal symmetry and limits of the AdS/CFT correspondence

Harmark

Hartong

Menculini

et al. 2018

J. High Energ. Phys.

137

308

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We find a Polyakov-type action for strings moving in a torsional Newton-Cartan geometry. This is obtained by starting with the relativistic Polyakov action and fixing the momentum of the string along a non-compact null isometry. For a flat target space, we show that the world-sheet theory becomes the Gomis-Ooguri action. From a target space perspective these strings are non-relativistic but their world-sheet theories are still relativistic. We show that one can take a scaling limit in which also the world-sheet theory becomes non-relativistic with an infinite-dimensional symmetry algebra given by the Galilean conformal algebra. This scaling limit can be taken in the context of the AdS/CFT correspondence and we show that it is realized by the 'Spin Matrix Theory' limits of strings on AdS 5 × S 5 . Spin Matrix theory arises as non-relativistic limits of the AdS/CFT correspondence close to BPS bounds. The duality between non-relativistic strings and Spin Matrix theory provides a holographic duality of its own and points towards a framework for more tractable holographic dualities whereby non-relativistic strings are dual to near BPS limits of the dual field theory.2 As will be clear in Sec. 2.1 we find in this paper that the TNC geometry is extended with a periodic target space direction.3 This Nambu-Goto form was also obtained in [20]. 4 The GCA was also observed in earlier work on non-relativistic limits of AdS/CFT [21]. See also Ref. [22] for useful work on representations of the GCA and aspects of non-relativistic conformal two-dimensional field theories.

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Section: Spin Matrix Theorymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 95%

Strings with non-relativistic conformal symmetry and limits of the AdS/CFT correspondence

Harmark

Hartong

Menculini

et al. 2018

J. High Energ. Phys.

137

308

View full text Add to dashboard Cite

show abstract

“…An appealing answer in this respect comes from the application of the Landau-Lifshitz limit to the spin chain. This method proved the equivalence between various sectors of four-dimensional N = 4 Yang-Mills theory and type IIB strings in the AdS 5 × S 5 background [6][7][8][9][10][11][12][13][14][15][16][17][18][19][20]. As reviewed in [21], the procedure starts with the expectation value of the Hamiltonian of the spin chain in a general coherent state.…”

mentioning

confidence: 94%

The SU(2) Wess-Zumino-Witten spin chain sigma model

Hernández

Nieto

Ruiz

2019

J. High Energ. Phys.

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Classical strings propagating in AdS 3 × S 3 × T 4 supported with Neveu-Schwarz-Neveu-Schwarz flux are described by a Wess-Zumino-Witten model. In this note, we study the emergence of their semiclassical SU(2) spectrally flowed sectors as the Landau-Lifshitz limit of the underlying quantum spin chain. We consider the propagator in the coherent state picture, and find that the time interval is discretized proportionally to the lattice spacing. In the Landau-Lifshitz limit, where both time and space become continuous, we derive a path integral representation of the propagator for each spectrally flowed sector. We prove that the arbitrariness of the global phase of coherent states is mapped to the gauge freedom of the B-field in the classical action. We show that higher order corrections in the Landau-Lifshitz limit are suppressed by inverse powers of the 't Hooft coupling.

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“…Thus, the theory is in a P T unbroken phase which guarantees the reality of the energy spectrum as well as the unitarity of the S-matrix. simpler (FR) model proposed by Faddeev and Reshetikhin is quite important from this point of view in connection with quantization of strings on AdS 5 × S 5 [2,28,31,[33][34][35][36][37][38][39][40].Recently, Klose and Zarembo (KZ) [41] calculated the S-matrix for a number of 1 + 1 dimensional integrable models (see also [42][43][44][45][46]) using standard field theoretic methods [47][48][49][50] in a simple manner. The simplicity of their method arises from the fact that the calculations are carried out in the wrong vacuum [51,52].…”

mentioning

confidence: 99%

mentioning

confidence: 99%

The S-matrix of the Faddeev-Reshetikhin model, diagonalizability andPTsymmetry

Das¹,

Melikyan²,

Rivelles³

2007

J. High Energy Phys.

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We study the question of diagonalizability of the Hamiltonian for the Faddeev-Reshetikhin (FR) model in the two particle sector. Although the two particle S-matrix element for the FR model, which may be relevant for the quantization of strings on AdS 5 ×S 5 , has been calculated recently using field theoretic methods, we find that the Hamiltonian for the system in this sector is not diagonalizable. We trace the difficulty to the fact that the interaction term in the Hamiltonian violating Lorentz invariance leads to discontinuity conditions (matching conditions) that cannot be satisfied. We determine the most general quartic interaction Hamiltonian that can be diagonalized. This includes the bosonic Thirring model as well as the bosonic chiral Gross-Neveu model which we find share the same S-matrix. We explain this by showing, through a Fierz transformation, that these two models are in fact equivalent. In addition, we find a general quartic interaction Hamiltonian, violating Lorentz invariance, that can be diagonalized with the same two particle S-matrix element as calculated by Klose and Zarembo for the FR model. This family of generalized interaction Hamiltonians is not Hermitian, but is P T symmetric. We show that the wave functions for this system are also P T symmetric. Thus, the theory is in a P T unbroken phase which guarantees the reality of the energy spectrum as well as the unitarity of the S-matrix. simpler (FR) model proposed by Faddeev and Reshetikhin is quite important from this point of view in connection with quantization of strings on AdS 5 × S 5 [2,28,31,[33][34][35][36][37][38][39][40].Recently, Klose and Zarembo (KZ) [41] calculated the S-matrix for a number of 1 + 1 dimensional integrable models (see also [42][43][44][45][46]) using standard field theoretic methods [47][48][49][50] in a simple manner. The simplicity of their method arises from the fact that the calculations are carried out in the wrong vacuum [51,52]. In this case, it is well known, for example, that in the two particle sector, the contribution to the S-matrix comes only from the bubble diagrams and if the system is integrable, all other scattering elements can be related to the two particle S-matrix [53][54][55][56]. One of the models studied by KZ is indeed the FR model, whose S-matrix element for the positive energy two particle states has a simple form that reflects the violation of Lorentz invariance present in the interaction Hamiltonian. This is interesting and, in fact, is relevant as a first step in understanding the quantization of the string itself. However, since the S-matrix element for the FR model is calculated in the wrong vacuum, it is necessary, as a next step, to go to the true vacuum to extract physical results [57]. This can be carried out by diagonalizing the (quartic) Hamiltonian of the theory in this sector.In this paper, we study the question of diagonalization of the two particle Hamiltonian for the FR model systematically. Surprisingly, we find that the quartic Hamiltonian for the FR model cannot be...

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Sigma model from spin chain

Cited by 20 publications

References 57 publications

Strings with non-relativistic conformal symmetry and limits of the AdS/CFT correspondence

Strings with non-relativistic conformal symmetry and limits of the AdS/CFT correspondence

The SU(2) Wess-Zumino-Witten spin chain sigma model

The S-matrix of the Faddeev-Reshetikhin model, diagonalizability andPTsymmetry

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