ABJM quantum spectral curve at twist 1: algorithmic perturbative solution

Abstract

We present an algorithmic perturbative solution of ABJM quantum spectral curve at twist 1 in sl(2) sector for arbitrary spin values, which can be applied to, in principle, arbitrary order of perturbation theory. We determined the class of functions -products of rational functions in spectral parameter with sums of Baxter polynomials and Hurwitz functions -closed under elementary operations, such as shifts and partial fractions, as well as differentiation. It turns out, that this class of functions is also sufficient for finding solutions of inhomogeneous Baxter equations involved. For the latter purpose we present recursive construction of the dictionary for the solutions of Baxter equations for given inhomogeneous parts. As an application of the proposed method we present the computation of anomalous dimensions of twist 1 operators at six loop order. There is still a room for improvements of the proposed algorithm related to the simplifications of arising sums. The advanced techniques for their reduction to the basis of generalized harmonic sums will be the subject of subsequent paper. We expect this method to be generalizable to higher twists as well as to other theories, such as N = 4 SYM.

A review of the AdS/CFT Quantum Spectral Curve

“…• The QSC has been also formulated for the Hubbard model [60,61] and for the ABJM theory [62,63]. For ABJM it provided high-order perturbative results and very precise numerics [64,65,66,67,68], the latter confirming and extending a much older TBA calculation [69]. For ABJM one can again compute the small S expansion analytically which led to a conjecture for the exact form of the function h(λ) appearing in all integrability-based results for ABJM [70].…”

Solving the Hagedorn temperature of AdS5/CFT4 via the Quantum Spectral Curve: Chemical potentials and deformations

“…[29,75]. Further examples of integrable theories to which our approach should be applicable are the ABJM and ABJ theory in the context of AdS 4 /CFT 3 [76,77], for which a QSC has equally been studied [78][79][80][81][82][83]. Additional examples occur 15 The TBA for the conformal fishnet theory is currently known only for a subclass of operators [73,74].…”

Anomalous dimensions of twist 2 operators and $\mathcal{N}=4$ SYM quantum spectral curve

“…Note, that the normalization of our Baxter polynomial is different from [22] and is chosen in such a way as to have simple elementary operations for the latter. Now, we claim that the solutions of all other difference equations can be done using techniques similar to [27,39]. In particular all calculations can be done algebraically within the special class of functions: products of rational functions in spectral parameter with sums of above Baxter polynomials and Hurwitz functions 5 .…”

“…The approach we are going to present here on the other hand does all calculations directly in spectral parameter space and gives algorithmic perturbative solution at any loop order. It is further development of the technique originally developed for twist 1 operators in ABJM model in [27,39]. Similar to [27,39], the presented solution is based on the existence of a class of functions -products of rational functions in spectral parameter with sums of Baxter polynomials and Hurwitz functions, which is closed under elementary operations, such as shifts and partial fractions, as well as differentiation.…”

“…It is further development of the technique originally developed for twist 1 operators in ABJM model in [27,39]. Similar to [27,39], the presented solution is based on the existence of a class of functions -products of rational functions in spectral parameter with sums of Baxter polynomials and Hurwitz functions, which is closed under elementary operations, such as shifts and partial fractions, as well as differentiation. The introduced class of function is sufficient for finding solutions of the involved inhomogeneous Baxter and first order difference equations using recursive construction of the dictionary of particular solutions for canonical types of inhomogeneities.…”