Bethe Ansatz in Stringy Sigma Models

We compute the exact S-matrix and give the Bethe ansatz solution for three sigma-models which arise as subsectors of string theory in AdS(5)xS(5): Landau-Lifshitz model (non-relativistic sigma-model on S(2)), Alday-Arutyunov-Frolov model (fermionic sigma-model with su(1|1) symmetry), and Faddeev-Reshetikhin model (string sigma-model on S(3)xR).Comment: 37 pages, 11 figure

Asymptotic Bethe Ansatz S-matrix and Landau-Lifshitz type effective 2-d actions

Motivated by the desire to relate Bethe ansatz equations for anomalous dimensions found on the gauge theory side of the AdS/CFT correspondence to superstring theory on AdS_5 x S5 we explore a connection between the asymptotic S-matrix that enters the Bethe ansatz and an effective two-dimensional quantum field theory. The latter generalizes the standard ``non-relativistic'' Landau-Lifshitz (LL) model describing low-energy modes of ferromagnetic Heisenberg spin chain and should be related to a limit of superstring effective action. We find the exact form of the quartic interaction terms in the generalized LL type action whose quantum S-matrix matches the low-energy limit of the asymptotic S-matrix of the spin chain of Beisert, Dippel and Staudacher (BDS). This generalises to all orders in the `t Hooft coupling an earlier computation of Klose and Zarembo of the S-matrix of the standard LL model. We also consider a generalization to the case when the spin chain S-matrix contains an extra ``string'' phase and determine the exact form of the LL 4-vertex corresponding to the low-energy limit of the ansatz of Arutyunov, Frolov and Staudacher (AFS). We explain the relation between the resulting ``non-relativistic'' non-local action and the second-derivative string sigma model. We comment on modifications introduced by strong-coupling corrections to the AFS phase. We mostly discuss the SU(2) sector but also present generalizations to the SL(2) and SU(1|1) sectors, confirming universality of the dressing phase contribution by matching the low-energy limit of the AFS-type spin chain S-matrix with tree-level string-theory S-matrix.Comment: 52 pages, 4 figures, Imperial-TP-AT-6-2; v2: new sections 7.3 and 7.4 computing string tree-level S-matrix in SL(2) and SU(1|1) sectors, references adde

Strong coupling from the Hubbard model

It was recently observed that the one dimensional half-filled Hubbard model reproduces the known part of the perturbative spectrum of planar N=4 super Yang-Mills in the SU(2) sector. Assuming that this identification is valid beyond perturbation theory, we investigate the behavior of this spectrum as the 't Hooft parameter \lambda becomes large. We show that the full dimension \Delta of the Konishi superpartner is the solution of a sixth order polynomial while \Delta for a bare dimension 5 operator is the solution of a cubic. In both cases the equations can be solved easily as a series expansion for both small and large \lambda and the equations can be inverted to express \lambda as an explicit function of \Delta. We then consider more general operators and show how \Delta depends on \lambda in the strong coupling limit. We are also able to distinguish those states in the Hubbard model which correspond to the gauge invariant operators for all values of \lambda. Finally, we compare our results with known results for strings on AdS_5\times S^5, where we find agreement for a range of R-charges.Comment: 14 pages; v2: 17 pages, 2 figures, appendix and references added; typos fixed, minor changes; v3 fixed figures; v4 more references added, minor correctio

TBA-like equations and Casimir effect in (non-)perturbative AdS/CFT

We consider high spin, $s$, long twist, $L$, planar operators (asymptotic Bethe Ansatz) of strong ${\cal N}=4$ SYM. Precisely, we compute the minimal anomalous dimensions for large 't Hooft coupling $\lambda$ to the lowest order of the (string) scaling variable $\ell \sim L/ (\ln \mathcal{S} \sqrt{\lambda})$ with GKP string size $\sim\ln \mathcal{S}\equiv 2 \ln (s/\sqrt{\lambda})$. At the leading order $(\ln \mathcal{S}) \cdot \ell ^2$, we can confirm the O(6) non-linear sigma model description for this bulk term, without boundary term $(\ln \mathcal{S})^0$. Going further, we derive, extending the O(6) regime, the exact effect of the size finiteness. In particular, we compute, at all loops, the first Casimir correction $\ell ^0/\ln \mathcal{S}$ (in terms of the infinite size O(6) NLSM), which reveals only one massless mode (out of five), as predictable once the O(6) description has been extended. Consequently, upon comparing with string theory expansion, at one loop our findings agree for large twist, while reveal for negligible twist, already at this order, the appearance of wrapping. At two loops, as well as for next loops and orders, we can produce predictions, which may guide future string computations.Comment: Version 2 with: new exact expression for the Casimir energy derived (beyond the first two loops of the previous version); UV theory formulated and analysed extensively in the Appendix C; origin of the O(6) NLSM scattering clarified; typos correct and references adde

T-systems and Y-systems in integrable systems

The T and Y-systems are ubiquitous structures in classical and quantum integrable systems. They are difference equations having a variety of aspects related to commuting transfer matrices in solvable lattice models, q-characters of Kirillov-Reshetikhin modules of quantum affine algebras, cluster algebras with coefficients, periodicity conjectures of Zamolodchikov and others, dilogarithm identities in conformal field theory, difference analogue of L-operators in KP hierarchy, Stokes phenomena in 1d Schr\"odinger problem, AdS/CFT correspondence, Toda field equations on discrete space-time, Laplace sequence in discrete geometry, Fermionic character formulas and combinatorial completeness of Bethe ansatz, Q-system and ideal gas with exclusion statistics, analytic and thermodynamic Bethe ans\"atze, quantum transfer matrix method and so forth. This review article is a collection of short reviews on these topics which can be read more or less independently.Comment: 156 pages. Minor corrections including the last paragraph of sec.3.5, eqs.(4.1), (5.28), (9.37) and (13.54). The published version (JPA topical review) also needs these correction