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If we start from certain functional relations as definition of a quantum integrable theory, then we can derive from them a linear integral equation. It can be extended, by introducing dynamical variables, to become an equation with the form of Marchenko's. Then, we derive from the latter a classical (differential) Lax pair. We exemplify our method by focusing on the massive version of the ODE/IM (Ordinary Differential Equations/Integrable Models) correspondence from Quantum sine-Gordon (sG) with many moduli/masses to the classical sinh-Gordon (shG) equation, so describing, in a particular case, some super-symmetric gauge theories and the $AdS_3$ strong coupling scattering amplitudes/Wilson loops. Yet, we present it in a way which reveals its generality of application. In fact, we give some hints on how it works for spin chains., Comment: More comprehensive and explicit version (v2 is not to be considered as appeared wrongly preceded by v1)

We propose the Augmented Grouping Approach (AugGA) and its deployment in the Augmented Grouping GO (AugGGO) scheme, for an efficient exploration of the chemical ordering (or compositional structure) of multi-component (alloyed) nanoparticles. The approach is based on a 'grouping' strategy (previously proposed for high-symmetry structures) by which the number of compositional degrees of freedom of the system is decreased by defining sets of atoms (groups, or orbits, or shells) that are constrained to be populated by the same element. Three fundamental advances are here included with respect to previous proposals: (i) groups are defined on the basis of descriptors (no point-group symmetry is assumed), (ii) bulk groups can exploit general chemical ordering patterns taken from databases, and (iii) sub-grouping is realized

We shall interpret the null hexagonal Wilson loop (or, equivalently, six gluon scattering amplitude) in 4D ${\cal N}=4$ Super Yang-Mills, or, precisely, an integral representation of its matrix part, via an ADHM-like instanton construction. In this way, we can apply localisation techniques to obtain combinatorial expressions in terms of Young diagrams. Then, we use our general formula to obtain explicit expressions in several explicit cases. In particular, we discuss those already available in the literature and find exact agreement. Moreover, we are capable to determine explicitly the denominator (poles) of the matrix part, and find some interesting recursion properties for the residues, as well., Comment: 40 pages, 7 figures

We provide the two fundamental sets of functional relations which describe the strong coupling limit of scattering amplitudes in $\mathcal{N} = 4$ SYM dual to Wilson loops in $AdS_3$: the basic $QQ$-system and the derived $TQ$-system. We use the $TQ$ relations and the knowledge of the main properties of the $Q$-function (eigenvalue of some $Q$-operator) to write the Bethe Ansatz equations, {\it viz.} a set of ('complex') non-linear-integral equations, whose solutions give exact values to the strong coupling amplitudes/Wilson loops. Moreover, they have some advantages with respect to the ('real') non-linear-integral equations of Thermodynamic Bethe Ansatz and still reproduce, both analytically and numerically, the findings coming from the latter. In any case, these new functional and integral equations give a larger perspective on the topic., Comment: Latex, 22 pages

We consider the third order differential equation derived from the deformed Seiberg-Witten differential for pure $$ \mathcal{N} $$ N = 2 SYM with gauge group SU(3) in Nekrasov- Shatashvili limit of Ω-background. We show that this is the same differential equation that emerges in the context of Ordinary Differential Equation/Integrable Models (ODE/IM) correspondence for 2d A2 Toda CFT with central charge c = 98. We derive the corresponding QQ and related T Q functional relations and establish the asymptotic behaviour of Q and T functions at small instanton parameter q → 0. Moreover, numerical integration of the Floquet monodromy matrix of the differential equation leads to evaluation of the A-cycles a1,2,3 at any point of the moduli space of vacua parametrized by the vector multiplet scalar VEVs (tr 𝜙2) and (tr 𝜙3) even for large values of q which are well beyond the reach of instanton calculus. The numerical results at small q are in excellent agreement with instanton calculation. We conjecture a very simple relation between Baxter’s T -function and A-cycle periods a1,2,3, which is an extension of Alexei Zamolodchikov’s conjecture about Mathieu equation.

We disentangle the contribution of scalars to the OPE series of null hexagonal Wilson loops/MHV gluon scattering amplitudes in multicolour N = 4 SYM. In specific, we develop a systematic computation of the S U ( 4 ) matrix part of the Wilson loop by means of Young tableaux (with several examples). Then, we use a peculiar factorisation property (when a group of rapidities becomes large) to deduce an explicit polar form. Furthermore, we emphasise the advantages of expanding the logarithm of the Wilson loop in terms of ‘connected functions’ as we apply this procedure to find an explicit strong coupling expansion (definitively proving that the leading order can prevail on the classical A d S 5 string contribution).

The Mayer cluster expansion technique is applied to the Nekrasov instanton partition function of $\mathcal{N}=2$ $SU(N_c)$ super Yang-Mills. The subleading small $\epsilon_2$-correction to the Nekrasov-Shatashvili limiting value of the prepotential is determined by a detailed analysis of all the one-loop diagrams. Indeed, several types of contributions can be distinguished according to their origin: long range interaction or potential expansion, clusters self-energy, internal structure, one-loop cyclic diagrams, etc.. The field theory result derived more efficiently in [1], under some minor technical assumptions, receives here definite confirmation thanks to several remarkable cancellations: in this way, we may infer the validity of these assumptions for further computations in the field theoretical approach., Comment: 29 pages, 9 figures

We derive the first e 2 -correction of the instanton partition functions in 4D N = 2 Super Yang–Mills (SYM) to the Nekrasov–Shatashvili limit e 2 → 0 . In the latter we recall the emergence of the famous Thermodynamic Bethe Ansatz-like equation which has been found by Mayer expansion techniques. Here we combine efficiently these to field theory arguments. In a nutshell, we find natural and resolutive the introduction of a new operator ∇ that distinguishes the singularities within and outside the integration contour of the partition function.

We provide a description of the quantum integrable structure behind the Thermodynamic Bethe Ansatz (TBA)-like equation derived by Nekrasov and Shatashvili (NS) for N=2 $$ \mathcal{N}=2 $$ 4d Super Yang-Mills (SYM) theories. In this regime of the background, — we shall show —, the instanton partition function is characterised by the solution of a TQ-equation. Exploiting a symmetry of the contour integrals expressing the partition function, we derive a ‘dual’ TQ-equation, sharing the same T-polynomial with the former. This fact allows us to evaluate to 1 the quantum Wronskian of two dual solutions (for Q) and, then, to reproduce the NS TBA-like equation. The latter acquires interestingly the deep meaning of a known object in integrability theory, as its two second determinations give the usual non-linear integral equations (nlies) derived from the ‘dual’ Bethe Ansatz equations.

Omega-deformation of the Seiberg-Witten curve is known to be written in terms of the qq-character, namely the trace of a specific operator acting in a Hilbert space spanned by certain Young diagrams. We define a differential form acting on this space and establish two discretised versions of the Seiberg-Witten expressions for the periods and related relations for the prepotential., 19 pages, 1 figure

The contributions of scalars and fermions to the null polygonal bosonic Wilson loops/gluon MHV scattering amplitudes in \(\mathcal {N} = 4\) SYM are considered. We first examine the re-summation of scalars at strong coupling. Then, we disentangle the form of the fermion contribution and show its strong coupling expansion. In particular, we derive the leading order with the appearance of a fermion-anti-fermion bound state first and then effective multiple bound states thereof. This reproduces the string minimal area result and also applies to the Nekrasov instanton partition function \(\mathcal {Z}\) of the \(\mathcal {N}=2\) theories. Especially, in the latter case the method appears to be suitable for a systematic expansion.

Using an extension of the concept of twist field in QFT to space-time (external) symmetries, we study conical twist fields in two-dimensional integrable QFT. These create conical singularities of arbitrary excess angle. We show that, upon appropriate identification between the excess angle and the number of sheets, they have the same conformal dimension as branch-point twist fields commonly used to represent partition functions on Riemann surfaces, and that both fields have closely related form factors. However, we show that conical twist fields are truly different from branch-point twist fields. They generate different operator product expansions (short distance expansions) and form factor expansions (large distance expansions). In fact, we verify in free field theories, by re-summing form factors, that the conical twist fields operator product expansions are correctly reproduced. We propose that conical twist fields are the correct fields in order to understand null polygonal Wilson loops/gluon scattering amplitudes of planar maximally supersymmetric Yang-Mills theory., 35 pages, 16 figures. Small typos corrected in v2

We analyze the contribution of $2n$ scalars to the operator product expansion series for Maximum Helicity Violating (MHV) gluon scattering amplitudes/polygonal Wilson loops in planar $\mathcal{N}=4$ supersymmetric Yang-Mills (SYM). Hence, we sum up on $2n$ for large coupling $\ensuremath{\lambda}$: the logarithm of the amplitude is proportional to $\sqrt{\ensuremath{\lambda}}$, thus competing, unexpectedly, with the famous classical contribution. We also give explicit expressions for the first terms at large $\ensuremath{\lambda}$. For this analysis, we find an explicit computation of the $2n$-scalar term at any coupling by means of Young tableaux, similar to Nekrasov-Okounkov expressions in $\mathcal{N}=2$ SYM.

We provide a description of the quantum integrable structure behind the Thermodynamic Bethe Ansatz (TBA)-like equation derived by Nekrasov and Shatashvili (NS) for $\mathcal{N}=2$ 4d Super Yang-Mills (SYM) theories. In this regime of the background, -- we shall show --, the instanton partition function is characterised by the solution of a TQ-equation. Exploiting a symmetry of the contour integrals expressing the partition function, we derive a 'dual' TQ-equation, sharing the same T-polynomial with the former. This fact allows us to evaluate to $1$ the quantum Wronskian of two dual solutions (for $Q$) and, then, to reproduce the NS TBA-like equation. The latter acquires interestingly the deep meaning of a known object in integrability theory, as its two second determinations give the usual non-linear integral equations (nlies) derived from the 'dual' Bethe Ansatz equations., Comment: 21 pages

We study the null polygonal Wilson loops/gluon scattering amplitudes in planar N = 4 SYM by means of the pentagon Operator Product Expansion (OPE) series and its integrability features, with particular attention on the strong coupling expansion. Actually, at all couplings and for the hexagon we disentangle the S U ( 4 ) matrix structure of the squared form factors for fermions, organising them in Young diagrams similar to those previously used for scalars. Then, we concentrate on the strong coupling regime and show the appearance of a new (effective) particle in the series: the fermion-antifermion bound state, the so-called ‘meson’. Of course, we identify its interaction with itself in the OPE series, with formation of (effective) bound states, and with the gluons and bound states of them. All together these extract from the OPE series the A d S 5 minimal area result for the Wilson loop, described by a set of Thermodynamic Bethe Ansatz (TBA)-like equations, and highlight also all the one-loop contributions. Moreover, the same strategy applies to N = 2 Nekrasov partition function via the parallel meson/instanton. Eventually, to complete the strong coupling analysis, we consider the scalar sector for any polygon, confirming the emergence of a non-perturbative contribution from the string in S 5 , which eludes and corrects the minimal area argument.

Motivated by recent work on quantum integrable models without U(1) symmetry, we show that the sl(2) Hirota equation admits a Lax representation with inhomogeneous terms. The compatibility of the auxiliary linear problem leads to a new consistent family of Hirota-like equations., 15 pages; v2: minor changes

We consider scalar Wilson operators of ${\cal N}=4$ SYM at high spin, $s$, and generic twist in the multi-color limit. We show that the corresponding (non)linear integral equations (originating from the asymptotic Bethe Ansatz equations) respect certain 'reciprocity' and functional 'self-tuning' relations up to all terms $\frac{1}{s(\ln s)^n}$ (inclusive) at any fixed 't Hooft coupling $\lambda$. Of course, this relation entails straightforwardly the well-known (homonymous) relations for the anomalous dimension at the same order in $s$. On this basis we give some evidence that wrapping corrections should enter the non-linear integral equation and anomalous dimension expansions at the next order $\frac{(\ln s)^{2}}{s^2}$, at fixed 't Hooft coupling, in such a way to re-establish the aforementioned relation (which fails otherwise)., Comment: 33 pages

We derive a generalization of the Destri - De Vega equation governing the scaling functions of some excited states in the Sine-Gordon theory. In particular configurations with an even number of holes and no strings are analyzed and their UV limits found to match some of the conformal dimensions of the corresponding compactified massless free boson. Quantum group reduction allows to interpret some of our results as scaling functions of excited states of Restricted Sine-Gordon theory, i.e. minimal models perturbed by phi_13 in their massive regime. In particular we are able to reconstruct the scaling functions of the off-critical deformations of all the scalar primary states on the diagonal of the Kac-table., Comment: Latex, 12 pages

We conjecture the set of asymptotic Bethe Ansatz equations for the mirror model of the AdS 4 × CP 3 string theory, corresponding to the planar 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 direct AdS 4 / CFT 3 string theory.

In the high spin limit the minimal anomalous dimension of (fixed) twist operators in the sl ( 2 ) sector of planar N = 4 Super Yang–Mills theory expands as γ ( g , s , L ) = f ( g ) ln s + f sl ( g , L ) + ∑ n = 1 ∞ 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 ′ ⩽ 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 n ( r ) ( g ) , appearing in the limit of large twist L ∼ ln s . Finally, we provide the first orders of weak and strong coupling for the first γ ( n ) ( g , L ) (and hence f n ( r ) ( g ) ).

We study the form of the high spin expansion of the minimal anomalous dimension for long operators belonging to the sl ( 2 ) sector of N = 4 SYM. Keeping fixed the ratio j between the twist and the logarithm of the spin, the minimal anomalous dimension expands as γ ( g , j , s ) = f ( g , j ) ln s + f ( 0 ) ( g , j ) + O ( 1 / ln s ) . This particular double scaling limit is efficiently described, including the desired accuracy O ( ( ln s ) 0 ) , in terms of a linear integral equation. By its use, we are able to evaluate both at weak and strong coupling the subleading scaling function f ( 0 ) ( g , j ) as a 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.

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 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 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 → ∞ . 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 derive the Thermodynamic Bethe Ansatz equations for the relativistic sigma model describing the AdS 4 × $ \mathbb{C}\mathbb{P} $ 3 string II A theory at strong coupling (i.e. in the Alday-Maldacena decoupling limit). The corresponding Y -system involves an infinite number of Y functions and is of a new type, although it shares a peculiar feature with the Y -system for AdS 4 × $ \mathbb{C}\mathbb{P} $ 3. A truncation of the equations at level p and a further generalisation to generic rank N allow us an alternative description of the theory as the N =4, p = ∞ representative in an infinite family of models corresponding to the conformal cosets ( $ \mathbb{C}\mathbb{P} $ N −1) p × U(1), perturbed by a relevant composite field ϕ (N,p) = ϕ[( $ \mathbb{C}\mathbb{P} $ N −1) p ] × ϕ [U(1)] that couples the two independent conformal field theories. The calculation of the ultraviolet central charge confirms the conjecture by Basso and Rej and the conformal dimension of the perturbing operator, at every N and p, is obtained using the Y-system periodicity. The conformal dimension of ϕ[( $ \mathbb{C}\mathbb{P} $ N −1) p ] matches that of the field identified by Fendley while discussing integrability issues for the purely bosonic $ \mathbb{C}\mathbb{P} $ N −1 sigma model.

Moving from Beisert-Staudacher equations, the complete set of Asymptotic Bethe Ansatz equations and $S$-matrix for the excitations over the GKP vacuum is found. The resulting model on this new vacuum is an integrable spin chain of length $R=2\ln s$ ($s=$ spin) with particle rapidities as inhomogeneities, two (purely transmitting) defects and $SU(4)$ (residual R-)symmetry. The non-trivial dynamics of ${\cal N}=4$ SYM appears in elaborated dressing factors of the 2D two-particle scattering factors, all depending on the 'fundamental' one between two scalar excitations. From scattering factors we determine bound states. In particular, we study the strong coupling limit, in the non-perturbative, perturbative and giant hole regimes. Eventually, from these scattering data we construct the $4D$ pentagon transition amplitudes (perturbative regime). In this manner, we detail the multi-particle contributions (flux tube) to the MHV gluon scattering amplitudes/Wilson loops (OPE or BSV series) and re-sum them to the Thermodynamic Bubble Ansatz., 103 pages; typos corrected, references added: journal version

A generalization of the Yang-Baxter algebra is found in quantizing the monodromy matrix of two (m)KdV equations discretized on a space lattice. This braided Yang-Baxter equation still ensures that the transfer matrix generates operators in involution which form the Cartan sub-algebra of the braided quantum group. Representations diagonalizing these operators are described through relying on an easy generalization of Algebraic Bethe Ansatz techniques. The conjecture that this monodromy matrix algebra leads, {\it in the cylinder continuum limit}, to a Perturbed Minimal Conformal Field Theory description is analysed and supported., Comment: Latex file, 46 pages

In presence of a static pair of sources, the spectrum of low-lying states of any confining gauge theory in D space-time dimensions is described, at large source separations, by an effective string theory. Recently two important advances improved our understanding of this effective theory. First, it was realized that the form of the effective action is strongly constrained by the requirement of the Lorentz invariance of the gauge theory, which is spontaneously broken by the formation of a long confining flux tube in the vacuum. This constraint is strong enough to fix uniquely the first few subleading terms of the action. Second, it has been realized that the first of these allowed terms - a quartic polynomial in the field derivatives - is exactly the composite field $T\bar{T}$, built with the chiral components, $T$ and $\bar{T}$, of the energy-momentum tensor of the 2d QFT describing the infrared limit of the effective string. This irrelevant perturbation is quantum integrable and yields, through the thermodynamic Bethe Ansatz (TBA), the energy levels of the string which exactly coincide with the Nambu-Goto spectrum. In this talk we first review the general implications of these two results and then, as a test of the power of these methods, use them to construct the first few boundary corrections to the effective string action., Presented at the 31st International Symposium on Lattice Field Theory (Lattice 2013), 29 July - 3 August 2013, Mainz, Germany

Recently, it was shown that the spectrum of anomalous dimensions and other important observables in planar $\mathcal{N}=4$ supersymmetric Yang-Mills theory are encoded into a simple nonlinear Riemann-Hilbert problem: the $\mathbf{P}\ensuremath{\mu}$ system or quantum spectral curve. In this Letter, we extend this formulation to the $\mathcal{N}=6$ supersymmetric Chern-Simons theory introduced by Aharony, Bergman, Jafferis, and Maldacena. This may be an important step towards the exact determination of the interpolating function $h(\ensuremath{\lambda})$ characterizing the integrability of this model. We also discuss a surprising relation between the quantum spectral curves for the $\mathcal{N}=4$ supersymmetric Yang-Mills theory and the $\mathcal{N}=6$ supersymmetric Chern-Simons theory considered here.

We present a construction of a Virasoro symmetry of the sine-Gordon (SG) theory. It is a dynamical one and has nothing to do with the space-time Virasoro symmetry of 2D CFT. Although it is clear how it can be realized dyrectly in the SG field theory, we are rather concerned here with the corresponding N-soliton solutions. We present explicit expressions for their infinithesimal transformations and show that they are local in this case. Some preliminary stages about the quantization of the classical results presented in this paper are also given., 17 pages, corrected some typos, two references added

The knowledge of {\it non usual} and sometimes {\it hidden} symmetries of (classical) integrable systems provides a very powerful setting-out of solutions of these models. Primarily, the understanding and possibly the quantisation of intriguing symmetries could give rise to deeper insight into the nature of field spectrum and correlation functions in quantum integrable models. With this perspective in mind we will propose a general framework for discovery and investigation of local, quasi-local and non-local symmetries in classical integrable systems. We will pay particular attention to the structure of symmetry algebra and to the r\^ole of conserved quantities. We will also stress a nice unifying point of view about KdV hierarchies and Toda field theories with the result of obtaining a Virasoro algebra as exact symmetry of Sine-Gordon Model., Comment: 32 pages, Latex file

We propose an alternative description of the spectrum of local fields in the classical limit of the integrable quantum field theories. It is close to similar constructions used in the geometrical treatment of W-gravities. Our approach provides a systematic way of deriving the null-vectors that appear in this construction. We present explicit results for the case of the A_1^{1}-(m)KdV and the A_2^{2}-(m)KdV hierarchies, different classical limits of 2D CFT's. In the former case our results coincide with the classical limit of the construction of Babelon, Bernard and Smirnov.Some hints about quantization and off-critical treatment are also given., Comment: 15 pages, LATEX file, to appear in Phys.Lett.B