A Lax equation for the non-linear sigma model

Brunelli, J. C.; Constandache, A.; Das, Ashok

doi:10.1016/s0370-2693(02)02663-1

Cited by 4 publications

(5 citation statements)

References 33 publications

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“…Because of the presence of terms involving derivatives of delta functions, the Poisson bracket algebra becomes non-ultralocal. (We remark here parenthetically that the Poisson bracket for the nonlinear sigma model is not unique [65]. However, in any form, there always exist some derivatives of delta function, making them non-ultralocal.)…”

Section: The Faddeev-reshetikhin Modelmentioning

confidence: 88%

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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Section: The Faddeev-reshetikhin Modelmentioning

confidence: 88%

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

Das¹,

Melikyan²,

Rivelles³

2007

J. High Energy Phys.

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“…Indeed, the fermionic part of our action, being at most quartic in the spinors, may be imagined to metamorphose, in certain special cases, into Gross-Neveu models [13], with various symmetry structures for the quartic interactions. The integrability of the bosonic sector is somewhat easier to establish (see for example [24]). The guiding principle here is to find a known candidate integrable system, and using its symmetries on the worldsheet, guess at the corresponding background geometry using the form of (7).…”

Section: Discussionmentioning

confidence: 99%

“…And indeed, it is when the B-field is turned on in the time direction that the dual theory is more than a field theory, but a full (non-commutative) open string theory. Yet the solutions given by ( 23) and (24) are not the only possible ones. Ours is a condition which is the analogue of the BPS condition for a closed string in flat space.…”

Section: Nonzero B (1)−+mentioning

confidence: 99%

Holography with Ramond-Ramond fluxes

Sahakian

2002

J. High Energy Phys.

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Starting from the non-linear sigma model of the IIB string in the light-cone gauge, we analyze the role of RR fluxes in Holography. We find that the worldsheet theory of states with only left or right moving modes does not see the presence of RR fields threading a geometry. We use this significant simplification to compute part of the strong coupling spectrum of the two dimensional NCOS theory. We also reproduce the action of a closed string in a PP-wave background using this general formalism; and we argue for various strategies to find new systems where the closed string theory may be exactly solvable.1 vvs@mail.lns.cornell.edu 2 Note that the compactified NCOS theory has also a sector of Newtonian gravity [9]- [11].

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“…The integrability of the bosonic sector is somewhat easier to establish (see for example [24]). The guiding principle here is to find a known candidate integrable system, and using its symmetries on the worldsheet, guess at the corresponding background geometry using the form of (1.7).…”

Section: Jhep12(2002)030mentioning

confidence: 99%