arXiv subjects
Publications and source records attributed to Davide Fioravanti.
We conjecture the set of asymptotic Bethe Ansatz equations for the {\it mirror} model of the $\text{AdS}_4\times\mathbb{CP}^3$ string theory, corresponding to the planar $\mathcal{N}=6$ superconformal Chern-Simons gauge theory in three dimensions. Hence, we derive the (vacuum energy) thermodynamic Bethe Ansatz equations and the Y-system describing the {\it direct} $\text{AdS}_4/\text{CFT}_3$ string theory.
In the high spin limit the minimal anomalous dimension of (fixed) twist operators in the $sl(2)$ sector of planar ${\cal N}=4$ Super Yang-Mills theory expands as $γ(g,s,L)=f(g) \ln s + f_{sl}(g,L) + \sum \limits_{n=1}^\infty γ^{(n)}(g,L) (\ln s)^{-n} + ... $. We find that the sub-logarithmic contribution $γ^{(n)}(g,L) $ is governed by a linear integral equation, depending on the solution of the linear integral equations appearing at the steps $n'\leq n-3$. We work out this recursive procedure and determine explicitly $γ^{(n)}(g,L) $ (in particular $γ^{(1)}(g,L)=0$ and $γ^{(n)}(g,2)=γ^{(n)}(g,3)=0$). Furthermore, we connect the $γ^{(n)}(g,L) $ (for finite $L$) to the generalised scaling functions, $f^{(r)}_n(g)$, appearing in the limit of large twist $L\sim\ln s$. Finally, we provide the first orders of weak and strong coupling for the first $γ^{(n)}(g,L)$ (and hence $f^{(r)}_n(g)$).
Moving from the mirror theory Bethe-Yang equations proposed by Arutyunov and Frolov, we derive the thermodynamic Bethe Ansatz equations which should control the spectrum of the planar $\text{AdS}_5/\text{CFT}_4$ correspondence. The associated set of universal functional relations (Y-system) satisfied by the exponentials of the TBA pseudoenergies is deduced, confirming the structure inferred by Gromov, Kazakov and Vieira.
We compute classical and first quantum finite-size corrections to the recently found giant magnon solutions in two different subspaces of $\mathbb{CP}^3$. We use the Lüscher approach on the recently proposed exact S-matrix for $\mathcal{N}=6$ superconformal Chern-Simons theory. We compare our results with the string and algebraic curve computations and find agreement, thus providing a non-trivial test for the new $AdS_4/CFT_3$ correspondence within an integrability framework.
We study the first sub-leading correction $O((\ln s)^0)$ to the cusp anomalous dimension in the high spin expansion of finite twist operators in ${\cal N}=4$ SYM theory. Since this approximation is still governed by a linear integral equation (derived already from the Bethe Ansatz equations in a previous paper), we finalise it better in order to study the weak and strong coupling regimes. In fact, we emphasise how easily the weak coupling expansion can be obtained, confirms the known four loop result and predicts the higher orders. Eventually, we pay particular attention to the strong coupling regime showing agreement and predictions in comparison with string expansion; speculations on the 'universal' part (upon subtracting the collinear anomalous dimension) are brought forward.
We study the the high spin expansion of the anomalous dimension for long operators belonging to the $sl(2)$ sector of ${\cal N}=4$ SYM. Keeping the ratio $j$ between the twist and the logarithm of the spin fixed, the anomalous dimensions expand as $γ= f(g,j)\ln s + f^{(0)}(g,j)+O(1/\ln s)$. This particular double scaling limit is efficiently described, up to the desired accuracy $O(\ln s ^0)$, in terms of linear integral equations. By using them, we are able to evaluate both at weak and strong coupling the sub-leading scaling function $f^{(0)}(g,j)$ as series in $j$, up to the order $j^5$. Thanks to these results, the possible extension of the liaison with the O(6) non-linear sigma model may be tackled on a solid ground.
A method for determining the generalised scaling function(s) arising in the high spin behaviour of long operator anomalous dimensions in the planar $sl(2)$ sector of ${\cal N}=4$ SYM is proposed. The all-order perturbative expansion around the strong coupling is detailed for the prototypical third and fourth scaling functions, showing the emergence of the O(6) Non-Linear Sigma Model mass-gap from different SYM 'mass' functions. Remarkably, only the fourth one gains contribution from the non-BES reducible densities and also shows up, as first, NLSM interaction and specific model dependence. Finally, the computation of the $n$-th generalised function is sketched and might be easily finalised for checks versus the computations in the sigma model or the complete string theory.
We propose a scheme for determining a generalised scaling function, namely the Sudakov factor in a peculiar double scaling limit for high spin and large twist operators belonging to the $sl(2)$ sector of planar ${\cal N}=4$ SYM. In particular, we perform explicitly the all-order computation at strong 't Hooft coupling regarding the first (contribution to the) generalised scaling function. Moreover, we compare our asymptotic results with the numerical solutions finding a very good agreement and evaluate numerically the non-asymptotic contributions. Eventually, we illustrate the agreement and prediction on the string side.
Anomalous dimension and higher conserved charges in the $sl(2)$ sector of ${\cal N}=4$ SYM for generic spin $s$ and twist $L$ are described by using a novel kind of non-linear integral equation (NLIE). The latter can be derived under typical situations of the SYM sectors, i.e. when the scattering need not depend on the difference of the rapidities and these, in their turn, may also lie on a bounded range. Here the non-linear (finite range) integral terms, appearing in the NLIE and in the dimension formula, go to zero as $s\to \infty$. Therefore they can be neglected at least up to the $O(s^0)$ order, thus implying a linear integral equation (LIE) and a linear dimension/charge formula respectively, likewise the 'thermodynamic' (i.e. infinite spin) case. Importantly, these non-linear terms go faster than any inverse logarithm power $(\ln s)^{-n}$, $n>0$, thus extending the linearity validity.
We exactly compute the energy density of the integrable O(n) non-linear sigma model as a convergent series. This series is specifically analysed for the very important O(6) symmetry, since it was suggested to result as a peculiar limit of the AdS string theory by Alday and Maldacena [1]. In this respect, the O(6) model gives also refined confirmations and predictions once compared with the SYM Bethe Ansatz [2,3].
The generalized (1+1)-D non-linear Schrodinger (NLS) theory with particular integrable boundary conditions is considered. More precisely, two distinct types of boundary conditions, known as soliton preserving (SP) and soliton non-preserving (SNP), are implemented into the classical $gl_N$ NLS model. Based on this choice of boundaries the relevant conserved quantities are computed and the corresponding equations of motion are derived. A suitable quantum lattice version of the boundary generalized NLS model is also investigated. The first non-trivial local integral of motion is explicitly computed, and the spectrum and Bethe Ansatz equations are derived for the soliton non-preserving boundary conditions.
We survey and discuss the applications of the non-linear integral equation in the framework of the Bethe Ansatz type equations which are conjectured to give the eigenvalues of the dilatation operator in ${\cal {N}}=4$ SYM. Moreover, an original idea (different from that of \cite {FMQR}) to derive a non-linear integral equation is briefly depicted in Section 4.
We consider the highest anomalous dimension operator in the SU(2) sector of planar ${\cal N}=4$ SYM at all-loop, though neglecting wrapping contributions. In any case, the latter enter the loop expansion only after a precise length-depending order. In the thermodynamic limit we write both a linear integral equation for the Bethe root density and a linear system obeyed by the commuting charges. Consequently, we determine the leading strong coupling contribution to the density and from this an approximation to the leading and sub-leading terms of any charge $Q_r$: it scales as $λ^{1/4-r/2}$, which generalises the Gubser-Klebanov-Polyakov energy law. In the end, we briefly extend these considerations to finite lengths and 'excited' operators by using the idea of a non-linear integral equation.
As the Hubbard energy at half filling is believed to reproduce at strong coupling (part of) the all loop expansion of the dimensions in the SU(2) sector of the planar $ {\cal N}=4$ SYM, we compute an exact non-perturbative expression for it. For this aim, we use the effective and well-known idea in 2D statistical field theory to convert the Bethe Ansatz equations into two coupled non-linear integral equations (NLIEs). We focus our attention on the highest anomalous dimension for fixed bare dimension or length, $L$, analysing the many advantages of this method for extracting exact behaviours varying the length and the 't Hooft coupling, $λ$. For instance, we will show that the large $L$ (asymptotic) expansion is exactly reproduced by its analogue in the BDS Bethe Ansatz, though the exact expression clearly differs from the BDS one (by non-analytic terms). Performing the limits on $L$ and $λ$ in different orders is also under strict control. Eventually, the precision of numerical integration of the NLIEs is as much impressive as in other easier-looking theories.
Non-linear integral equations derived from Bethe Ansatz are used to evaluate finite size corrections to the highest (i.e. {\it anti-ferromagnetic}) and immediately lower anomalous dimensions of scalar operators in ${\cal N}=4$ SYM. In specific, multi-loop corrections are computed in the SU(2) operator subspace, whereas in the general SO(6) case only one loop calculations have been finalised. In these cases, the leading finite size corrections are given by means of explicit formulæand compared with the exact numerical evaluation. In addition, the method here proposed is quite general and especially suitable for numerical evaluations.
Bound state excitations of the spin 1/2-XYZ model are considered inside the Bethe Ansatz framework by exploiting the equivalent Non-Linear Integral Equations. Of course, these bound states go to the sine-Gordon breathers in the suitable limit and therefore the scattering factors between them are explicitly computed by inspecting the corresponding Non-Linear Integral Equations. As a consequence, abstracting from the physical model the Zamolodchikov-Faddeev algebra of two $n$-th elliptic breathers defines a tower of $n$-order Deformed Virasoro Algebras, reproducing the $n=1$ case the usual well-known algebra of Shiraishi-Kubo-Awata-Odake \cite{SKAO}.
Initially, we derive a nonlinear integral equation for the vacuum counting function of the spin 1/2-XYZ chain in the {\it disordered regime}, thus paralleling similar results by Klümper \cite{KLU}, achieved through a different technique in the {\it antiferroelectric regime}. In terms of the counting function we obtain the usual physical quantities, like the energy and the transfer matrix (eigenvalues). Then, we introduce a double scaling limit which appears to describe the sine-Gordon theory on cylindrical geometry, so generalising famous results in the plane by Luther \cite{LUT} and Johnson et al. \cite{JKM}. Furthermore, after extending the nonlinear integral equation to excitations, we derive scattering amplitudes involving solitons/antisolitons first, and bound states later. The latter case comes out as manifestly related to the Deformed Virasoro Algebra of Shiraishi et al. \cite{SKAO}. Although this nonlinear integral equations framework was contrived to deal with finite geometries, we prove it to be effective for discovering or rediscovering S-matrices. As a particular example, we prove that this unique model furnishes explicitly two S-matrices, proposed respectively by Zamolodchikov \cite{ZAMe} and Lukyanov-Mussardo-Penati \cite{LUK, MP} as plausible scattering description of unknown integrable field theories.
The g-function was introduced by Affleck and Ludwig in the context of critical quantum systems with boundaries. In the framework of the thermodynamic Bethe ansatz (TBA) method for relativistic scattering theories, all attempts to write an exact integral equation for the off-critical version of this quantity have, up to now, been unsuccesful. We tackle this problem by using an n-particle cluster expansion, close in spirit to form-factor calculations of correlators on the plane. The leading contribution already disagrees with all previous proposals, but a study of this and subsequent terms allows us to deduce an exact infrared expansion for g, written purely in terms of TBA pseudoenergies. Although we only treat the thermally-perturbed Ising and the scaling Lee-Yang models in detail, we propose a general formula for g which should be valid for any model with entirely diagonal scattering.