Sebastien Leurent

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 AdS_5/CFT_4

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.

Solving the AdS/CFT Y-system

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

Wronskian solution for AdS/CFT Y-system

{{\mathbb{Z}}_4}

Multiple zeta functions and double wrapping in planar

symmetry of the underlying coset sigma model, in its quantum version, allows for a deeper insight into the analyticity structure of the corresponding Y-functions and T-functions, as well as for their analyticity friendly parameterization in terms of Wronskian determinants of Q-functions. As a check for the new equations, we reproduce the numerical results for the Konishi operator previously obtained from the original infinite Y-system.

N=4

Using the discrete Hirota integrability we find the general solution of the full quantum Y-system for the spectrum of anomalous dimensions of operators in the planar AdS5/CFT4 correspondence in terms of Wronskian-like determinants parameterized by a finite number of Baxter’s Q-functions. We consider it as a useful step towards the construction of a finite system of non-linear integral equations (FiNLIE) for the full spectrum. The explicit asymptotic form of all the Q-functions for the large size operators is presented. We establish the symmetries and the analyticity properties of the asymptotic Q-functions and discuss their possible generalization to any finite size operators.

SYM

Using the FiNLIE solution of the AdS/CFT Y-system, we compute the anomalous dimension of the Konishi operator in planar N = 4 SYM up to eight loops, i.e. up to the leading double wrapping order. At this order a non-reducible Euler Zagier sum, zeta(1,2.8), appears for the first time. We find that at all orders in perturbation, every spectral-dependent quantity of the Y-system is expressed through multiple Hurwitz zeta functions, hence we provide a Mathematica package to manipulate these functions, including the particular case of Euler-Zagier sums. Furthermore, we conjecture that only Euler Zagier sums can appear in the answer for the anomalous dimension at any order in perturbation theory. We also resum the leading transcendentality terms of the anomalous dimension at all orders, obtaining a simple result in terms of Bessel functions. Finally, we demonstrate that exact Bethe equations should be related to an absence of poles condition that becomes especially non-trivial at double wrapping.

Classical tau-function for quantum spin chains

A bstractFor an arbitrary generalized quantum integrable spin chain we introduce a “master T -operator” which represents a generating function for commuting quantum transfer matrices constructed by means of the fusion procedure in the auxiliary space. We show that the functional relations for the transfer matrices are equivalent to an infinite set of model-independent bilinear equations of the Hirota form for the master T -operator, which allows one to identify it with τ -function of an integrable hierarchy of classical soliton equations. In this paper we consider spin chains with rational GL(N)-invariant R-matrices but the result is independent of a particular functional form of the transfer matrices and directly applies to quantum integrable models with more general (trigonometric and elliptic) R-matrices and to supersymmetric spin chains.

Baxter's Q-operators and operatorial Backlund flow for quantum (super)-spin chains

We propose the operatorial Baxter’s TQ-relations in a general form of the operatorial Bäcklund flow describing the nesting process for the inhomogeneous rational gl(K|M) quantum (super)spin chains with twisted periodic boundary conditions. The full set of Q-operators and T-operators on all levels of nesting is explicitly defined. The results are based on a generalization of the identities among the group characters and their group co-derivatives with respect to the twist matrix, found by one of the authors Kazakov and Vieira (JHEP 0810:050, 2008). Our formalism, based on this new “master” identity, allows a systematic and rather straightforward derivation of the whole set of nested Bethe ansatz equations for the spectrum of quantum integrable spin chains, starting from the R-matrix.

Finite size spectrum of SU(N) principal chiral field from discrete Hirota dynamics

Using recently proposed method of discrete Hirota dynamics for integrable (1 + 1)D quantum field theories on a finite space circle of length L we derive and test numerically a finite system of nonlinear integral equations for the exact spectrum of energies of SU(N) x SU(N) principal chiral field model as functions of mL, where m is the mass scale. We propose a determinant solution of the underlying Y-system, or Hirota equation, in terms of Wronskian determinants of N x N matrices parameterized by N - 1 functions of the spectral parameter theta with the known analytic properties at finite L. Although the method works in principle for any state, the explicit equations are written for states in the U(1) sector only. For N > 2, we encounter and clarify a few subtleties in these equations related to the presence of bound states, absent in the previously considered N = 2 case. As a demonstration of efficiency of our method, we solve these equations numerically for a few low-lying states at N = 3 in a wide range of mL