Quantum spectral curve for arbitrary state/operator in AdS5/CFT4

Abstract

We give a derivation of quantum spectral curve (QSC) -a finite set of Riemann-Hilbert equations for exact spectrum of planar N = 4 SYM theory proposed in our recent paper Phys. Rev. Lett. 112 (2014). We also generalize this construction to all local single trace operators of the theory, in contrast to the TBA-like approaches worked out only for a limited class of states. We reveal a rich algebraic and analytic structure of the QSC in terms of a so called Q-system -a finite set of Baxter-like Q-functions. This new point of view on the finite size spectral problem is shown to be completely compatible, though in a far from trivial way, with already known exact equations (analytic Y-system/TBA, or FiNLIE). We use the knowledge of this underlying Q-system to demonstrate how the classical finite gap solutions and the asymptotic Bethe ansatz emerge from our formalism in appropriate limits.

The full Quantum Spectral Curve for AdS4/CFT3

“…While we do not treat in full generality the representation theory aspects, we construct explicitly an enlarged set of Q functions, and prove that they satisfy exact Bethe equations reflecting the full JHEP09(2017)140 supergroup structure. Generalizing arguments of [17], we will show that, in the limit of large volume, some of these exact Bethe equations reduce to the Asymptotic Bethe Ansatz.…”

“…p4 ,j , mod(2π), (3.16) where 17) and {u 4,j } K 4 j=1 , u4 ,j K4 j=1 denote the momentum-carrying Bethe roots, see [35]. We will show that the phase P agrees with (3.16) up to the first two orders at weak coupling,…”

“…We shall now show how to construct an equivalent version of the QSC which is more appropriate to the description of AdS 4 degrees of freedom, and contains, in the classical limit, the four quasi-momenta parametrizing the motion of a classical string solution in AdS 4 . As in the case of AdS 5 /CF T 4 considered in [17], this entails a swap between the physical and the mirror section of the Riemann surface. In addition, we will see that this alternative system naturally encodes the relevant symmetry group SO(3, 2), which was not explicitly visible in the previous formulation.…”

“…As discussed in [44], in this form the equations are, from a purely algebraic point of view, exactly the same as the Pµ-system of N = 4 SYM [16,17], with the redefinitions…”

“…In particular, the problem of computing the conformal spectrum of the theory was tackled by tailoring integrable QFT techniques to this new setting, in particular the Bethe Ansatz [5,7,8], the TBA, the Y and T-systems [9][10][11][12][13][14][15], leading to the discovery of the very effective Quantum Spectral Curve (QSC) formulation [16,17]. The latter is a very satisfactory simplification and probably the most elementary formulation of the problem.…”