Quantum Spectral Curve of theN=6Supersymmetric Chern-Simons Theory

Abstract

Recently, it was shown that the spectrum of anomalous dimensions and other important observables in planar N=4 supersymmetric Yang-Mills theory are encoded into a simple nonlinear Riemann-Hilbert problem: the Pμ system or quantum spectral curve. In this Letter, we extend this formulation to the N=6 supersymmetric Chern-Simons theory introduced by Aharony, Bergman, Jafferis, and Maldacena. This may be an important step towards the exact determination of the interpolating function h(λ) characterizing the integrability of this model. We also discuss a surprising relation between the quantum spectral curves for the N=4 supersymmetric Yang-Mills theory and the N=6 supersymmetric Chern-Simons theory considered here.

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

“…Suppose that we do not know about connection to representation theory but try to answer the question what generic restrictions can be imposed on the asymptotics of Q-functions solely from the analytic structure of QSC. We will answer this question below, derive in this way (C. 19), (C.20), (C.21) and hence demonstrate that analyticity of QSC naturally encodes the unitarity constraints. While the quantization conditions (C. 19), (C.20) will be demonstrated from scratch, for (C.21) we will need certain bits of information from the large volume approximation and then use the continuity argument.…”

“…The method was proven to be very powerful for exact analytic nonperturbative calculations for the generalized cusp anomalous dimension [15,49,50], for slope and curvature function [18,20]. Recently, following our methods, the QSC was also build in the ABJM theory [19,[51][52][53]. Exact slope function in ABJM theory computed using the QSC methods has lead to a well justified conjecture for the interpolation function h(λ), entering into all integrability based calculations in this theory [20].…”

“…Some of these functions, like P 1 , P 2 and ω ab , were already familiar from our previous, FiNLIE construction of [12], whereas the others, like P 3 , P 4 , Q i and µ ab , seem to be new and somewhat mysterious. In this section, we will reveal a nice mathematical structure emerging behind the Riemann-Hilbert equations allowing to interpret 19 This restriction becomes too strong in some cases. For example for analytic continuation in the Lorentz spin S for twist two operators one could also have exponential factors.…”

“…Our main assumption is that they have power-like asymptotic at infinity. 19 First, ω on the sheet with short cuts must be also periodic at large u. The only power-like periodic function is a constant.…”

“…[16,17] as well as at strong coupling [15], for the slope and curvature functions for twist-2 operators at any coupling and pomeron intercept at strong coupling [18]. Recently, the QSC was also found for the ABJM model in [19], which was used [20] to make a well-grounded conjecture for the interpolation function h(λ) entering numerous physically relevant quantities such as cusp anomalous dimension and magnon dispersion relation in this theory.…”