Trace Identity Research Articles

Chiral Trace Relations in $$\mathcal{N}=2^*$$ Supersymmetric Gauge Theories

We analyze the chiral ring in Ω-deformed $$\mathcal{N}=2^*$$ supersymmetric gauge theories. Applying localization techniques, we derive closed identities for the vacuum expectation values of chiral trace operators. In the SU(2) case, we provide an AGT framework to identify chiral trace operators and the system of local integrals of motion in the related two-dimensional conformal field theory. In this setup, we predict some universal terms appearing in chiral trace identities.

Hierarchies and Hamiltonian structures of the Nonlinear Schrödinger family using geometric and spectral techniques

This paper explores the class of equations of the Non-linear Schrodinger (NLS) type by employing both geometrical and spectral analysis methods. The work is developed in three stages. First, the geometrical method (AKS theorem) is used to derive different equations of the Non-linear Schrodinger (NLS) and Derivative Non-linear Schrodinger (DNLS) families. Second, the spectral technique (Tu method) is applied to obtain the hierarchies of equations belonging to these types. Third, the trace identity along with other techniques is used to obtain the corresponding Hamiltonian structures. It is found that the spectral method provides a simple algorithmic procedure to obtain the hierarchy as well as the Hamiltonian structure. Finally, the connection between the two formalisms is discussed and it is pointed out how application of these two techniques in unison can facilitate the understanding of integrable systems. In concurrence with Tu's method, Gesztesy and Holden also formulated a method of derivation of the trace formulas for integrable nonlinear evolution equations, this method is based on a contour-integration technique.

Constructing variable coefficient nonlinear integrable coupling super AKNS hierarchy and its self‐consistent sources

Constructing variable coefficient super integrable equation hierarchy is an important problem in soliton theory. In this letter, new Lax pairs with some arbitrary functions are proposed and a variable coefficient integrable coupling of super AKNS hierarchy is generated. The super Hamiltonian structures of variable coefficient coupling equation hierarchy is derived with the aid of the super trace identity. Furthermore, the self‐consistent sources of variable coefficient super integrable coupling hierarchy are established. It is indicated that this method is an efficient way to construct the variable coefficient super integrable equation hierarchy.

Minimal J-trace identities for M(n,n) and queer trace identities for Mn(E)

It is known, see [2], that the algebra M(n,n) has a J-trace and satisfies J-trace identities and the algebra Mn(E) has a queer trace and satisfies queer trace identities, and that the degree of the minimal identities is 12(n+2)(n+1) for each of them. In this paper we construct all minimal degree identities in one variable. In the case of Mn(E) there is only one, up to constant multiple: qtr(x)qtr(x2)⋯qtr(xn+1)=0.

A super Sasa–Satsuma hierarchy and bi-Hamiltonian structures

A super Sasa–Satsuma hierarchy associated with a 3 × 3 matrix spectral problem is proposed with the aid of the zero-curvature equation and Lenard recursion equations. A typical member in the hierarchy is the super Sasa–Satsuma equation. The super bi-Hamiltonian structures of the super Sasa–Satsuma hierarchy are constructed by utilizing the super trace identity. The infinite conservation laws of the super Sasa–Satsuma equation are presented by resorting to the spectral parameter expansion.

An extension of the coupled derivative nonlinear Schrödinger hierarchy

In this paper, a 3 × 3 matrix spectral problem with six potentials is considered. With the help of the compatibility condition, a hierarchy of new nonlinear evolution equations which can be reduced to the coupled derivative nonlinear Schrödinger (CDNLS) equations is obtained. By use of the trace identity, it is proved that all the members in this new hierarchy have generalized bi-Hamiltonian structures. Moreover, infinitely many conservation laws of this hierarchy are constructed.

Two New Integrable Hierarchies and Their Nonlinear Integrable Couplings

By introducing an invertible linear transform, a new Lie algebra G is obtained from the Lie algebra H. Making use of the compatibility conditions of the respective isospectral problems, a generalized NLS-MKdV hierarchy and a new integrable soliton hierarchy are achieved by using the trace identity or the variational identity. Then, two special non-semisimple Lie algebras and are explicitly conducted. As an application, the nonlinear continuous integrable couplings of the obtained integrable systems as well as their bi-Hamiltonian structures are established, respectively.

An integrable generalization of the D-Kaup–Newell soliton hierarchy and its bi-Hamiltonian reduced hierarchy

We present a new spectral problem, a generalization of the D-Kaup–Newell spectral problem, associated with the Lie algebra sl(2,R). Zero curvature equations furnish the soliton hierarchy. The trace identity produces the Hamiltonian structure for the hierarchy and shows its Liouville integrability. Lastly, a reduction of the spectral problem is shown to have a different soliton hierarchy with a bi-Hamiltonian structure. The major motivation of this paper is to present spectral problems that generate two soliton hierarchies with infinitely many conservation laws and high-order symmetries.

Scalar, vector and tensor harmonics on the three-sphere

Scalar, vector and tensor harmonics on the three-sphere were introduced originally to facilitate the study of various problems in gravitational physics. These harmonics are defined as eigenfunctions of the covariant Laplace operator which satisfy certain divergence and trace identities, and ortho-normality conditions. This paper provides a summary of these properties, along with a new notation that simplifies and clarifies some of the key expressions. Practical methods are described for accurately and efficiently computing these harmonics numerically, and test results are given that illustrate how well the analytical identities are satisfied by the harmonics computed numerically in this way.

Two symmetry constraints for a generalized Dirac integrable hierarchy

By a two-by-two matrix spectral problem, a generalized Dirac integrable hierarchy is presented. A Hamiltonian structure of the obtained hierarchy is established by trace identity, and its Liouville integrability is proved. Then, through Bargmann symmetry constraint, spatial part of the Lax pairs and adjoint Lax pairs is nonlinearized as a completely integrable finite-dimensional Hamiltonian system. Next, under an implicit symmetry constraint, both spatial part and temporal parts of the Lax pairs and adjoint Lax pairs are all nonlinearized as completely integrable finite-dimensional Hamiltonian systems. Ultimately, the involutive representation of solution of the generalized Dirac integrable hierarchy is given.

Multi-Component Generalizations of Two Hierarchies Related to the Yajima-Oikawa Hierarchy

AbstractWe present a multi-component modified Yajima-Oikawa hierarchy and a multi-component Novikov hierarchy. We get generalized Hamiltonian structures for the two hierarchies with the aid of trace identity.

Two integrable Hamiltonian hierarchies in sl(2,R) and so(3,R) with three potentials

By introducing two specific matrix spectral problems associated with sl(2,R) and so(3,R) matrix Lie algebras, we generate two integrable Hamiltonian hierarchies with three potentials. The computation and analysis on their Hamiltonian structures by means of the trace identity show that the resulting hierarchies are Liouville integrable, namely, that each hierarchy consists of commuting Hamiltonian soliton equations.

Three-Component Generalisation of Burgers Equation and Its Bi-Hamiltonian Structures

Abstract A hierarchy of three-component generalisation of Burgers equation, which is associated with a 3×3 matrix eigenvalue problem, is generated by using the zero-curvature equation. By means of the trace identity, the bi-Hamiltonian structures of this hierarchy are constructed. Moreover, the infinite conservation laws for the hierarchy are obtained with the aid of spectral parameter expansion.

Trace identities for the topological vertex

The topological vertex is a universal series which can be regarded as an object in combinatorics, representation theory, geometry, or physics. It encodes the combinatorics of 3D partitions, the action of vertex operators on Fock space, the Donaldson-Thomas theory of toric Calabi-Yau threefolds, or the open string partition function of $\mathbb{C}^3$. We prove several identities in which a sum over terms involving the topological vertex is expressed as a closed formula, often a product of simple terms, closely related to Fourier expansions of Jacobi forms. We use purely combinatorial and representation theoretic methods to prove our formulas, but we discuss applications to the Donaldson-Thomas invariants of elliptically fibered Calabi-Yau threefolds at the end of the paper.

Complete integrability of nonlocal nonlinear Schrödinger equation

Based on the completeness relation for the squared solutions of the Lax operator L, we show that a subset of nonlocal equations from the hierarchy of nonlocal nonlinear Schrödinger equations (NLS) is a completely integrable system. The spectral properties of the Lax operator indicate that there are two types of soliton solutions. The relevant action-angle variables are parametrized by the scattering data of the Lax operator. The notion of the symplectic basis, which directly maps the variations of the potential of L to the variations of the action-angle variables has been generalized to the nonlocal case. We also show that the inverse scattering method can be viewed as a generalized Fourier transform. Using the trace identities and the symplectic basis, we construct the hierarchy Hamiltonian structures for the nonlocal NLS equations.

Nonlinear Super Integrable Couplings of Super Yang Hierarchy and Its Super Hamiltonian Structures

Nonlinear super integrable couplings of the super Yang hierarchy based upon an enlarged matrix Lie super algebra were constructed. Then its super Hamiltonian structures were established by using super trace identity. As its reduction, nonlinear integrable couplings of Yang hierarchy were obtained.

Generating integrable lattice hierarchies by some matrix and operator Lie algebras

Two types of matrix Lie algebras are presented. We make use of the first loop algebra to obtain a new \((1+1)\)-dimensional integrable discrete hierarchy, which generalizes a result given by Gordoa et al., whose reduction is a discrete modified KdV system. Then we produce another new \((2+1)\)-dimensional integrable discrete hierarchy with three fields under a \((2+1)\)-dimensional non-isospectral linear problem. We again generalize the \((1+1)\)- and \((2+1)\)-dimensional discrete hierarchies to obtain a positive and negative integrable discrete hierarchy. In addition, we obtain a discrete integrable coupling system of the \((1+1)\)-dimensional discrete hierarchy presented in the paper by enlarging such the loop algebras. Next, we apply the second matrix loop algebra to introduce an isospectral problem and deduce a new integrable discrete hierarchy, whose quasi-Hamiltonian structure is derived from the trace identity proposed by Tu Guizhang, which can be reduced to some modified Toda lattice equations. A type of Darboux transformation of a reduced discrete system of the latter integrable discrete hierarchy is obtained as well. We introduce two types of operator-Lie algebras according to a given spectral problem by a matrix Lie algebra and apply the r-matrix theory to obtain a few lattice integrable systems, including two \((2+1)\)-dimensional lattice systems.

On the quantization of continuous non-ultralocal integrable systems

We discuss the quantization of non-ultralocal integrable models directly in the continuous case, using the example of the Alday–Arutyunov–Frolov model. We show that by treating fields as distributions and regularizing the operator product, it is possible to avoid all the singularities, and allow to obtain results consistent with perturbative calculations. We illustrate these results by considering the reduction to the massive free fermion model and extracting the quantum Hamiltonian as well as other conserved charges directly from the regularized trace identities. Moreover, we show that our regularization recovers Maillet's prescription in the classical limit.

Traces in <i>SL</i>(3,C) and <i>SU</i>(2,1) Groups

In this paper we prove some trace identities in SL(3,C) and SU(2,1) groups. We also present the merits on how to parametrise pair of pants via traces and cross-ratio. Finally, we compute traces of matrices that are generated by complex reflections in complex triangle groups.

Mateos, P. (2014) Names, Ethnicity and Populations. Tracing Identity in Space

Mateos, P. (2014) Names, Ethnicity and Populations. Tracing Identity in Space