Integrable Hamiltonian systems related to the Hilbert–Schmidt ideal

By the application of the coinduction method as well as the Magri method to the ideal of real Hilbert–Schmidt operators we construct the hierarchies of integrable Hamiltonian systems on Banach Lie–Poisson spaces which consist of such types of operators. We also discuss their algebraic and analytic properties and solve them in dimensions, N=2,3,4$N=2,3,4$.     

Cusp anomalous dimension in maximally supersymmetric Yang-Mills theory at strong coupling

We construct an analytical solution to the integral equation which is believed to describe logarithmic growth of the anomalous dimensions of high-spin operators in planar N=4 super Yang-Mills theory and use it to determine the strong coupling expansion of the cusp anomalous dimension.     

Time evolution of infinite one-dimensional Coulomb system

We consider one-dimensional Coulomb systems and their time evolution given by the Newton law. We give existence and uniqueness theorems for the solutions of the equations governing the systems in the thermodynamic limits.Partially supported by the Italian C.N.R. and by the I.H.E.S.     

High-order quadratures for the solution of scattering problems in two dimensions

We construct an iterative algorithm for the solution of forward scattering problems in two dimensions. The scheme is based on the combination of high-order quadrature formulae, fast application of integral operators in Lippmann–Schwinger equations, and the stabilized bi-conjugate gradient method (BI-CGSTAB). While the FFT-based fast application of integral operators and the BI-CGSTAB for the solution of linear systems are fairly standard, a large part of this paper is devoted to constructing a class of high-order quadrature formulae applicable to a wide range of singular functions in two and three dimensions; these are used to obtain rapidly convergent discretizations of Lippmann–Schwinger equations. The performance of the algorithm is illustrated with several numerical examples.     

The property of maximal transcendentality in the N \mathcal{N} = 4 SYM

We show results for the universal anomalous dimension γ_uni(j) of Wilson twist-2 operators in the $ \mathcal{N} $ \mathcal{N} = 4 Supersymmetric Yang-Mills theory in the first three orders of perturbation theory. These expressions are obtained by extracting the most complicated contributions from the corresponding anomalous dimensions in QCD.     

Logarithmic operators in AdS3/CFT2

We discuss the relation between singletons in AdS₃ and logarithmic operators in the CFT on the boundary. In 2 dimensions there can be more logarithmic operators apart from those which correspond to singletons in AdS, because logarithmic operators can occur when the dimensions of primary fields differ by an integer instead of being equal. These operators may be needed to account for the greybody factor for gauge bosons in the bulk.     

Models of quadratic quantum algebras and their relation to classical superintegrable systems

We show how to construct realizations (models) of quadratic algebras for 2D second order superintegrable systems in terms of differential or difference operators in one variable. We demonstrate how various models of the quantum algebras arise naturally from models of the Poisson algebras for the corresponding classical superintegrable system. These techniques extend to quadratic algebras related to superintegrable systems in n dimensions and are intimately related to multivariable orthogonal polynomials. The text was submitted by the authors in English.     

Critical exponents of the random-field O(N) model

The critical behavior of the random-field Ising model has long been a puzzle. Different methods predict that its critical exponents in D dimensions are the same as in the pure (D-2)-dimensional ferromagnet with the same number of the magnetization components contrary to the experiments and simulations. We calculate the exponents of the random-field O(N) model with the (4+epsilon)-expansion and obtain values different from the exponents of the pure ferromagnet in 2+epsilon dimensions. An infinite set of relevant operators missed in previous studies leads to a breakdown of the (6-epsilon)-expansion.     

Index of a family of lattice Dirac operators and its relation to the non-Abelian anomaly on the lattice

In the continuum, a topological obstruction to the vanishing of the non-Abelian anomaly in 2n dimensions is given by the index of a certain Dirac operator in 2n+2 dimensions, or equivalently, the index of a 2-parameter family of Dirac operators in 2n dimensions. In this paper an analogous result is derived for chiral fermions on the lattice in the overlap formulation. This involves deriving an index theorem for a family of lattice Dirac operators satisfying the Ginsparg-Wilson relation. The index density is proportional to Lüscher's topological field in 2n+2 dimensions.     

Anomalous dimensions of nonlocal baryon operators

The background-field method is used to evaluate the logarithmic contributions to a nonlocal three-quark operator on the light-cone. The operator is composed of nonlocal operators of different spins, whose anomalous dimensions are obtained by choosing the Weyl representation for the Dirac matrices in the nonlocal results. The anomalous dimensions of the corresponding local three-quark operators calculated by conventional approaches are obtained by local expansion of the nonlocal results.     

Anomalous dimensions of gauge-invariant three fermion local operators of twist three

The matrices of anomalous dimensions of gauge-invariant three-fermion local operators of twist three are calculated in QCD with massless quarks. Furthermore we explicitly give the eigenvalues for operators containing up to eight derivatives, and the eigenvectors for operators containing up to three derivatives. The correspondence between eigenvectors of the anomalous dimension matrices and decuplet local baryon operators is derived by means of Fierz transformations.     

Low-temperature phase diagrams of quantum lattice systems. I. Stability for quantum perturbations of classical systems with finitely-many ground states

Starting from classical lattice systems ind2 dimensions with a regular zerotemperature phase diagram, involving a finite number of periodic ground states, we prove that adding a small quantum perturbation and/or increasing the temperature produce only smooth deformations of their phase diagrams. The quantum perturbations can involve bosons or fermions and can be of infinite range but decaying exponentially fast with the size of the bonds. For fermions, the interactions must be given by monomials of even degree in creation and annihilation operators. Our methods can be applied to some anyonic systems as well. Our analysis is based on an extension of Pirogov-Sinai theory to contour expansions ind+1 dimensions obtained by iteration of the Duhamel formula.     

QCD analysis of twist-4 contributions to the g i structure functions

We analyze the twist-4 contributions to Bjorken and Ellis-Jaffe sum rules for spin-dependent structure function g ₁(x, Q ²). We investigate the anomalous dimensions of the twist-4 operators which determine the logarithmic correction to the 1/Q ² behavior of the twist-4 contribution by evaluating off-shell Green’s functions in both flavor non-singlet and singlet case. It is shown that the operators which are proportional to the equation of motion play an important role to extract the anomalous dimensions of physical operators. The calculations to solve the operator mixing of higher-twist operators are given in detail     

Higher twist effects in asymptotically free gauge theories

The anomalous dimensions of the flavor octet, three-body operators of twist four are evaluated at the one-loop level. The calculation is complicated due to the appearance of gauge non-invariant operators which are needed to renormalize gauge invariant operators. The mixing problem of these operators is resolved by using the Ward identities derived through the Becchi-Rouet-Stora transformations. Some comments on the evaluation of the coefficient functions are also made.     

Foundations of electrodynamics: S.R. de Groot and L.G. Suttorp, (North-Holland Publishing Company,Amsterdam, 1972. xii-535 pages. Fl. 140.-)

We show that the anomalous dimension of the fundamental field is connected to the anomalous dimensions of the high spin bilinear operators. The dimensions of these operators can be determined by examining the violations of the Bjorken scaling law in deep-inelastic electron-proton scattering. The structure function at field ω near to one has a power dependence of q², the exponent being the anomalous dimension of the “parton” field.     

The Ultra-Commutation Relations

We discuss briefly the basic (integrable)representation of the ucr, comprising the operators A ,its adjoint A⁺, and N (which is equal toN⁺), satisfying AN – NA = A. There areno additional relations between the operators, in general. The ucrinclude the ccr, car, deformed bosons and fermions, andmany other systems as special cases. The principalstructure theorem asserts that every integrablerepresentation of the ucr is determined by a sequencegeneralizing the [n]-sequence of deformationtheory.     

On the absence of spontaneous breakdown of continuous symmetry for equilibrium states in two dimensions

Using the Bogoliubov inequality, we extend previously known results concerning the absence of continuous symmetry breakdown for equilibrium states of certain quantum and classical lattice, and continuum systems in two space dimensions.Partially supported by the N.S.F. under grant MCS 7801433.Partially supported by the N.S.F. under grant MCS 7906633.     

Flavored surface defects in 4d $$\mathcal{N}=1$$ N = 1 SCFTs

We discuss supersymmetric surface defects in compactifications of six-dimensional minimal conformal matter of types SU(3) and SO(8) to four dimensions. The relevant field theories in four dimensions are \(\mathcal{N}=1\) quiver gauge theories with SU(3) and SU(4) gauge groups, respectively. The defects are engineered by giving space-time-dependent vacuum expectation values to baryonic operators. We find evidence that in the case of SU(3) minimal conformal matter, the defects carry SU(2) flavor symmetry which is not a symmetry of the four-dimensional model. The simplest case of a model in this class is SU(3) SQCD with nine flavors, and thus the results suggest that this admits natural surface defects with SU(2) flavor symmetry. We analyze the defects using the superconformal index and derive analytic difference operators introducing the defects into the index computation. The duality properties of the four-dimensional theories imply that the index of the models is a kernel function for such difference operators. In turn, checking the kernel property constitutes an independent check of the dualities and the dictionary between six- dimensional compactifications and four-dimensional models.

     

On-shell methods for form factors in N = 4 SYM and their applications

Form factors are quantities that involve both asymptotic on-shell states and gauge invariant operators. They provide a natural bridge between on-shell amplitudes and off-shell correlation functions of operators, thus allowing us to use modern on-shell amplitude techniques to probe into the off-shell side of quantum field theory. In particular, form factors have been successfully used in computing the cusp(soft) anomalous dimensions and anomalous dimensions of general local operators. This review is intended to provide a pedagogical introduction to some of these developments. We will first review some amplitudes background using four-point amplitudes as main examples. Then we generalize these techniques to form factors, including(1) tree-level form factors,(2) Sudakov form factor and infrared singularities, and(3) form factors of general operators and their anomalous dimensions. Although most examples we consider are inN= 4 super-Yang-Mill theory, the on-shell methods are universal and are expected to be applicable to general gauge theories.