Dmytro Volin

Review of AdS/CFT Integrability: An Overview

This is the introductory chapter of a review collection on integrability in the context of the AdS/CFT correspondence. In the collection, we present an overview of the achievements and the status of this subject as of the year 2010.

Quantum spectral curve for planar N=4 super-Yang-Mills theory.

We present a new formalism, alternative to the old TBA-like approach, for solution of the spectral problem of planar N = 4 SYM. It takes a concise form of a non-linear matrix Riemann-Hilbert problem in terms of a few Q-functions. We demonstrate the formalism for two types of observables - local operators at weak coupling and cusped Wilson lines in a near BPS limit.

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

A bstractWe give a derivation of quantum spectral curve (QSC) — a finite set of Riemann-Hilbert equations for exact spectrum of planar N=4

Quantum folded string and integrability: From finite size effects to Konishi dimension

\mathcal{N}=4

Review of AdS/CFT Integrability, Chapter III.3: The Dressing Factor

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.

Functional BES equation

We present a new formalism, alternative to the old TBA-like approach, for solution of the spectral problem of planar N = 4 SYM. It takes a concise form of a non-linear matrix Riemann-Hilbert problem in terms of a few Q-functions. We demonstrate the formalism for two types of observables - local operators at weak coupling and cusped Wilson lines in a near BPS limit.

Quantum spectral curve as a tool for a perturbative quantum field theory

Using the algebraic curve approach we one-loop quantize the folded string solution for the type IIB superstring in AdS5 × S5. We obtain an explicit result valid for arbitrary values of its Lorentz spin S and R-charge J in terms of integrals of elliptic functions. Then we consider the limit S ~ J ~ 1 and derive the leading three coefficients of strong coupling expansion of short operators. Notably, our result evaluated for the anomalous dimension of the Konishi state gives 2λ1/4 − 4 + 2/λ1/4. This reproduces correctly the values predicted numerically in arXiv:0906.4240. Furthermore we compare our result using some new numerical data from the Y-system for another similar state. We also revisited some of the large S computations using our methods. In particular, we derive finite-size corrections to the anomalous dimension of operators with small J in this limit.

Minimal solution of the AdS/CFT crossing equation

A bstractUsing integrability and analyticity properties of the AdS5/CFT4 Y-system we reduce it to a finite set of nonlinear integral equations. The