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

Abstract

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.

Quantum Landau-Lifshitz model at four loops:1/Jand1/J2corrections to Berenstein-Maldacena-Nastase energies

“…As in [13,8] here dC = ǫ ijk n i dn j ∧ dn k , i.e. C is a monopole potential on S 2 ,λ = λ/J 2 , where…”

“…As in [13,8], let us now rewrite the LL Lagrangian (2.2) in terms of two independent fields. Solving the constraint | n| 2 = 1 as n 3 = 1 − n 2 1 − n 2 2 we get the following SO(2) invariant expression for the Lagrangian in terms of n 1 and n 2 (a, b = 1, 2; n 2 = n a n a )…”

“…The 1/J corrections can be found from the Bethe ansatz on the spin chain [20,21] or from a direct superstring quantization [22,6]. As explained in [8], the derivation from the LL action turns out to be much simpler than the string-theory derivation. Here we will extend the method of [8,13] tõ λ 4 /J andλ 4 /J 2 orders.…”

“…The only effect of these modes in the effective LL theory in the SU(2) sector is the proper regularization, which one needs to consider. As shown in [8] this can be accomplished with a combination of normal ordering and ζ-function regularization. Let us point out that this discussion is valid only for analytic terms inλ.…”

“…In Section 3 after reviewing the quantization of LL action developed in [8], we compute the 1/J and 1/J 2 corrections in Section 3.1 and 3.2, respectively. In section 4 we collect the results and find completely both the gauge and string LL actions.…”