SEARCH · Searcharxiv
Search indexed arXiv papers on artificial intelligence, large language models, computer vision and robotics. Read source abstracts and follow links to arXiv.
Quote a phrase for an exact phrase match. Source license links do not imply unrestricted reuse.
The nested algebraic Bethe ansatz is presented for the anisotropic supersymmetric $U$ model maintaining quantum supersymmetry. The Bethe ansatz equations of the model are obtained on a one-dimensional closed lattice and an expression for the energy is given.
As is known, tetrahedron equations lead to the commuting family of transfer-matrices and provide the integrability of corresponding three-dimensional lattice models. We present the modified version of these equations which give the commuting family of more complicated two-layer transfer-matrices. In the static limit we have succeeded in constructing the solution of these equations in terms of elliptic functions.
We derive a canonical form for smooth vector fields on $\Re^{n+1}$. We use this to demonstrate the local multi-Hamiltonian nature of the corresponding flows. Associated with the canonical form is an inhomogenious linear PDE whose solutions provide conserved measures. These can be used to construct the local Hamiltonians.
We construct infinite sets of local conserved charges for the conformal affine Toda model. The technique involves the abelianization of the two-dimensional gauge potentials satisfying the zero-curvature form of the equations of motion. We find two infinite sets of chiral charges and apart from two lowest spin charges all the remaining ones do not possess chiral densities. Charges of different chiralities Poisson commute among themselves. We discuss the algebraic properties of these charges and use the fundamental Poisson bracket relation to show that the charges conserved in time are in involution. Connections to other Toda models are established by taking particular limits.
A lattice regularized Lax operator for the nonultralocal modified Korteweg de Vries (mKdV) equation is proposed at the quantum level with the basic operators satisfying a $q$-deformed braided algebra. Finding further the associated quantum $R$ and $Z$-matrices the exact integrability of the model is proved through the braided quantum Yang--Baxter equation, a suitably generalized equation for the nonultralocal models. Using the algebraic Bethe ansatz the eigenvalue problem of the quantum mKdV model is exactly solved and its connection with the spin-$\ha$ XXZ chain is established, facilitating the investigation of the corresponding conformal properties.
Reflection equation for the scattering of lines moving in half-plane is obtained. The corresponding geometric picture is related with configurations of half-planes touching the boundary plane in 2+1 dimensions. This equation can be obtained as an additional to the tetrahedron equation consistency condition for a modified Zamolodchikov algebra.
We show that the set of transfer matrices of an arbitrary fusion type for an integrable quantum model obey these bilinear functional relations, which are identified with an integrable dynamical system on a Grassmann manifold (higher Hirota equation). The bilinear relations were previously known for a particular class of transfer matrices corresponding to rectangular Young diagrams. We extend this result for general Young diagrams. A general solution of the bilinear equations is presented.
A manifestly N=2 supersymmetric coset formalism is applied to analyse the "fermionic" extensions of N=2 $a=4$ and $a=-2$ KdV hierarchies. Both these hierarchies can be obtained from a manifest N=2 coset construction. This coset is defined as the quotient of some local but non-linear superalgebra by a $\hat{U(1)}$ subalgebra. Three superextensions of N=2 KdV hierarchy are proposed, among which one seems to be entirely new.
We discuss the symmetric homogeneous polynomial solutions of the generalized Laplace's equation which arises in the context of the Calogero-Sutherland model on a line. The solutions are expressed as linear combinations of Jack polynomials and the constraints on the coefficients of expansion are derived. These constraints involve generalized binomial coefficients defined through Jack polynomials. Generalized binomial coefficients for partitions of $k$ upto $k=6$ are tabulated.
We present the Lax operator for the N=3 KdV hierarchy and consider its extensions. We also construct a new infinite family of N=2 supersymmetric hierarchies by exhibiting the corresponding super Lax operators. The new realization of N=4 supersymmetry on the two general N=2 superfields, bosonic spin 1 and fermionic spin 1/2, is discussed.
It is shown that all 3-body quantal integrable systems that emerge in the Hamiltonian reduction method possess the same hidden algebraic structure. All of them are given by a second degree polynomial in generators of an infinite-dimensional Lie algebra of differential operators. It leads to new families of the orthogonal polynomials in two variables.
The local Sugawara constructions of the "small" N=4 SCA in terms of supercurrents of N=2 extensions of the affinization of the sl(2|1) and sl(3) algebras are investigated. The associated super mKdV type hierarchies induced by N=4 SKdV ones are defined. In the sl(3) case the existence of two inequivalent Sugawara constructions is found. The long one involves all the affine sl(3)-valued currents, while the "short" one deals only with those from the subalgebra sl(2)\oplus u(1). As a consequence, the sl(3)-valued affine superfields carry two inequivalent mKdV type super hierarchies induced by the correspondence between "small" N=4 SCA and N=4 SKdV hierarchy. However, only the first hierarchy posseses genuine global N=4 supersymmetry. We discuss peculiarities of the realization of this N=4 supersymmetry on the affine supercurrents.
Presented is a free boson representation of the type II vertex operators for the $A_{n-1}^{(1)}$ face model. Using the bosonization, we derive some properties of the type II vertex operators, such as commutation, inversion and duality relations.
The Equivalence Principle (EP), stating that all physical systems are connected by a coordinate transformation to the free one with vanishing energy, univocally leads to the Quantum Stationary HJ Equation (QSHJE). Trajectories depend on the Planck length through hidden variables which arise as initial conditions. The formulation has manifest p-q duality, a consequence of the involutive nature of the Legendre transform and of its recently observed relation with second-order linear differential equations. This reflects in an intrinsic psi^D-psi duality between linearly independent solutions of the Schroedinger equation. Unlike Bohm's theory, there is a non-trivial action even for bound states. No use of any axiomatic interpretation of the wave-function is made. Tunnelling is a direct consequence of the quantum potential which differs from the usual one and plays the role of particle's self-energy. The QSHJE is defined only if the ratio psi^D/psi is a local self-homeomorphism of the extended real line. This is an important feature as the L^2 condition, which in the usual formulation is a consequence of the axiomatic interpretation of the wave-function, directly follows as a basic theorem which only uses the geometrical gluing conditions of psi^D/psi at q=\pm\infty as implied by the EP. As a result, the EP itself implies a dynamical equation that does not require any further assumption and reproduces both tunnelling and energy quantization. Several features of the formulation show how the Copenhagen interpretation hides the underlying nature of QM. Finally, the non-stationary higher dimensional quantum HJ equation and the relativistic extension are derived.
The thermodynamic Bethe ansatz (TBA) and the excited state TBA equations for an integrable spin chain related to the Lie superalgebra osp(1|2) are proposed by the quantum transfer matrix (QTM) method. We introduce the fusion hierarchy of the QTM and derive the functional relations among them (T-system) and their certain combinations (Y-system). Their analytical property leads to the non-linear integral equations which describe the free energy and the correlation length at any finite temperatures. With regard to the free energy, they coincide with the TBA equation (math-ph/9911010, Mod. Phys. Lett. A, 14, 2427 (1999)) based on the string hypothesis.
The SU(2) invariant massive Thirring model with a boundary is considered on the basis of the vertex operator approach. The bosonic formulae are presented for the vacuum vector and its dual in the presence of the boundary. The integral representations are also given for form factors of the present model.
Gorsky et al. presented an explicit construction of Whitham deformations of the Seiberg-Witten curve for the $SU(N+1)$ $\calN = 2$ SUSY Yang-Mills theory. We extend their result to all classical gauge groups and some other cases such as the spectral curve of the $A^{(2)}_{2N}$ affine Toda Toda system. Our construction, too, uses fractional powers of the superpotential $W(x)$ that characterizes the curve. We also consider the $u$-plane integral of topologically twisted theories on four-dimensional manifolds $X$ with $b_2^{+}(X) = 1$ in the language of these explicitly constructed Whitham deformations and an integrable hierarchy of the KdV type hidden behind.
The bihamiltonian structure of the N=2 Supersymmetric Boussinesq equation is found. It is not reduced to the corresponding classical structure and hence it describes the pure supersymmetric effect. For the supersymmetric Boussinesq equation which contains the classical partner the Lax pair is given explicitly. Thus we prove the integrability of the equation.