Fluctuations and energy shifts in the Bethe ansatz

Beisert, Niklas; Freyhult, Lisa

doi:10.1016/j.physletb.2005.07.015

Cited by 37 publications

(52 citation statements)

References 24 publications

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“…if corrections to thermodynamic limit on the 1-loop spin chain side are the same as the leading 1-loop corrections on the string theory side to the same linear order inλ. This matching was found on several explicit examples: near BPS (BMN) states [37,38] and circular strings [28,29,25] (see also [30,42,50]). A simple way to understand why this happens was suggested in [25] by computing the leading quantum correction to the energy of circular string state directly at the level of the effective LL model.…”

Section: Ll Action From String Theorymentioning

confidence: 52%

corrections to semiclassical AdS/CFT states from quantum Landau–Lifshitz model

Minahan

Tirziu²,

Tseytlin³

2006

Nuclear Physics B

151

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One way to relate semiclassical string states and dual gauge theory states is to show the equivalence between their low-energy effective 2d actions. The gauge theory effective action, which is represented by an effective Landau-Lifshitz (LL) model, was previously found to match the string theory world-sheet action up to the first two orders in the effective parameterλ = λ/J 2 , where λ is the 't Hooft coupling and J is the total R-charge. Here we address the question if quantizing the effective LL action reproduces the subleading 1/J corrections to the spin chain energies as well as the quantum corrections to the string energies. We demonstrate that this is indeed the case provided one chooses an appropriate regularization of the effective LL theory. Expanding near the BPS vacuum, we show that the quantum LL action gives the same 1/J corrections to energies of BMN states as found previously on the gauge theory and string theory sides. We also compute the subleading 1/J 2 corrections and show that these too match with corrections computed from the Bethe ansatz. We also compare the results from the LL action with a more direct computation from the spin chain. We repeat the same computation for the β-deformed LL action and find that the quantum LL result is again equal to the 1/J correction computed from the β-deformed Bethe ansatz equations. We also quantize the LL action near the rotating circular and folded string solutions, generalizing the known gauge/string results for 1/J corrections to the classical energies. We emphasize the simplicity of this effective field theory approach as compared to the full quantum string computations.

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Section: Ll Action From String Theorymentioning

confidence: 52%

corrections to semiclassical AdS/CFT states from quantum Landau–Lifshitz model

Minahan

Tirziu²,

Tseytlin³

2006

Nuclear Physics B

151

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“…Since the all-order Bethe Ansatz construction is based on a "discretization" [32] of the classical (integral) Bethe equations [34] and since their solution requires no regularization, it follows that a special choice of regularization is required in the algebraic curve calculation to reproduce the results of the Bethe Ansatz calculation. This is the case for strings in AdS 5 ×S 5 , where the Bethe equations and the worldsheet calculation yield the same result which is matched by the algebraic curve calculation [51,14] provided one chooses a natural regularization which accounts for certain constant shifts in the space-like momenta of fluctuations and essentially amounts to introducing different cutoffs for various partial frequency sums (cf. [14,52,35,30]).…”

Section: Comments On Comparison To the Bethe Ansatz Proposalmentioning

confidence: 68%

“…If this two-dimensional theory is also integrable, then its semiclassical states can be described using the algebraic curve techniques [34], which also determines the fluctuation frequencies [13,51,14] near the solitonic solutions and thus, effectively, the 1-loop corrections to their charges. Furthermore, there may exist a set of (discrete) Bethe equations that should provide the exact description of quantum corrections to all loop orders.…”

Section: Comments On Comparison To the Bethe Ansatz Proposalmentioning

confidence: 99%

Quantum spinning strings in AdS₄× ℂℙ³: testing the Bethe Ansatz proposal

McLoughlin

Roiban

Tseytlin

2008

J. High Energy Phys.

112

154

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Recently, an asymptotic Bethe Ansatz that is claimed to describe anomalous dimensions of "long" operators in the planar N = 6 supersymmetric three-dimensional Chern-Simons-matter theory dual to quantum superstrings in AdS 4 × CP 3 was proposed. It initially passed a few consistency checks but subsequent direct comparison to one-loop string-theory computations created some controversy. Here we suggest a resolution by pointing out that, contrary to the initial assumption based on the algebraic curve considerations, the central interpolating function h(λ) entering the BMN or magnon dispersion relation receives a non-zero one-loop correction in the natural string-theory computational scheme. We consider a basic example which has already played a key role in the AdS 5 × S 5 case: a rigid circular string stretched in both AdS 4 and along an S 1 of CP 3 and carrying two spins. Computing the leading oneloop quantum correction to its energy allows us to fix the constant one-loop term in h(λ) and also to suggest how one may establish a correspondence with the Bethe Ansatz proposal, including the non-trivial one-loop phase factor. We discuss some problems which remain in trying to match a part of world-sheet contributions (sensitive to compactness of the worldsheet space-like direction) and their Bethe Ansatz counterparts. * Also at Lebedev Institute Moscow

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“…A notion of stability exists in the semiclassical theory [22]; it demands the absence of tachyonic fluctuations. This condition does not coincide with |ρ| 1, it turns out that the latter is a stronger requirement in the case of a single cut [23].…”

Section: Introductionmentioning

confidence: 98%

Quantum stability for the Heisenberg ferromagnet

2008

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Highly spinning classical strings on R×S 3 are described by the Landau-Lifshitz model or equivalently by the Heisenberg ferromagnet in the thermodynamic limit. The spectrum of this model can be given in terms of spectral curves. However, it is a priori not clear whether any given admissible spectral curve can actually be realised as a solution to the discrete Bethe equations, a property which can be referred to as stability. In order to study the issue of stability, we find and explore the general two-cut solution or elliptic curve. It turns out that the moduli space of this elliptic curve shows a surprisingly rich structure. We present the various cases with illustrations and thus gain some insight into the features of multi-cut solutions. It appears that all admissible spectral curves are indeed stable if the branch cuts are positioned in a suitable, non-trivial fashion.

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Fluctuations and energy shifts in the Bethe ansatz

Cited by 37 publications

References 24 publications

corrections to semiclassical AdS/CFT states from quantum Landau–Lifshitz model

corrections to semiclassical AdS/CFT states from quantum Landau–Lifshitz model

Quantum spinning strings in AdS₄× ℂℙ³: testing the Bethe Ansatz proposal

Quantum stability for the Heisenberg ferromagnet

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Fluctuations and energy shifts in the Bethe ansatz

Cited by 37 publications

References 24 publications

corrections to semiclassical AdS/CFT states from quantum Landau–Lifshitz model

corrections to semiclassical AdS/CFT states from quantum Landau–Lifshitz model

Quantum spinning strings in AdS4× ℂℙ3: testing the Bethe Ansatz proposal

Quantum stability for the Heisenberg ferromagnet

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Quantum spinning strings in AdS₄× ℂℙ³: testing the Bethe Ansatz proposal