Algebraic Bethe ansatz for integrable Kondo impurities in the one-dimensional supersymmetric t-J model

An integrable Kondo problem in the one-dimensional supersymmetric t-J model is studied by means of the boundary supersymmetric quantum inverse scattering method. the boundary K matrices depending on the local moments of the impurities are presented as a nontrivial realization of the graded reflection equation algebras in a two-dimensional impurity Hilbert space. Further, the model is solved by using the algebraic Bethe ansatz method and the Bethe ansatz equations are obtained.     

Supersymmetry in boundary integrable models

Quantum integrable models that possess N = 2 supersymmetry are investigated on the half-space. Conformal perturbation theory is used to identify some N = 2 supersymmetric boundary integrable models, and the effective boundary Landau-Ginzburg formulations are constructed. It is found that N = 2 supersymmetry largely determines the boundary action in terms of the bulk, and in particular, the boundary bosonic potential is |W|², where W is the bulk superpotential. Supersymmetry is also investigated using the affine quantum group symmetry of exact scattering matrices, and the affine quantum group symmetry of boundary reflection matrices is analyzed both for supersymmetric and more general models. Some N = 2 supersymmetry preserving boundary reflection matrices are given, and their connection with the boundary Landau-Ginzburg actions is discussed.     

Spectrum of boundary states in N=1 SUSY sine-Gordon theory

We consider N=1 supersymmetric sine-Gordon theory (SSG) with supersymmetric integrable boundary conditions (boundary SSG=BSSG). We find two possible ways to close the boundary bootstrap for this model, corresponding to two different choices for the boundary supercharge. We argue that these two bootstrap solutions should correspond to the two integrable Lagrangian boundary theories considered recently by Nepomechie.     

Row transfer matrix functional relations for Baxter's eight-vertex and six-vertex models with open boundaries via more general reflection matrices

The functional relations of the transfer matrices of fusion hierarchies for six- and eight-vertex models with open boundary conditions have been presented in this paper. We have shown the su(2) fusion rule for the models with more general reflection boundary conditions, which are represented by off-diagonal reflection matrices. Also we have discussed some physics properties, which are determined by the functional relations. Finally, the intertwining relation between the reflection K-matrices for the vertex and SOS models is discussed.     

On a2(1) reflection matrices and affine Toda theories

We construct new non-diagonal solutions to the boundary Yang-Baxter equation corresponding to a two-dimensional field theory with U_q(a₂⁽¹⁾) quantum affine symmetry on a half-line. The requirements of boundary unitarity and boundary crossing symmetry are then used to find overall scalar factors which lead to consistent reflection matrices. Using the boundary bootstrap equations we also compute the reflection factors for scalar bound states (breathers). These breathers are expected to be identified with the fundamental quantum particles in a₂⁽¹⁾ affine Toda field theory and we therefore obtain a conjecture for the affine Toda reflection factors. We compare these factors with known classical results and discuss their duality properties and their connections with particular boundary conditions.     

N=2 boundary supersymmetry in integrable models and perturbed boundary conformal field theory

Boundary integrable models with N=2 supersymmetry are considered. For the simplest boundary N=2 superconformal minimal model with a Chebyshev bulk perturbation we show explicitly how fermionic boundary degrees of freedom arise naturally in the boundary perturbation in order to maintain integrability and N=2 supersymmetry. A new boundary reflection matrix is obtained for this model and N=2 boundary superalgebra is studied. A factorized scattering theory is proposed for a N=2 supersymmetric extension of the boundary sine-Gordon model with either (i) fermionic or (ii) bosonic and fermionic boundary degrees of freedom. Exact results are obtained for some quantum impurity problems: the boundary scaling Lee–Yang model, a massive deformation of the anisotropic Kondo model at the filling values g=2/(2n+3) and the boundary Ashkin–Teller model.     

Boundary renormalisation group flows of the supersymmetric Lee–Yang model and its extensions

In this paper we examine the supersymmetric Lee–Yang model in the presence of boundaries. We determine the reflection factors for the Neveu–Schwarz type boundary conditions from the reduction of the supersymmetric sine-Gordon model and check them by using boundary truncated conformal space approach in the massless case. We explore the boundary renormalisation groups flows using boundary TBA and TCSA.     

Finite size effects in the spin-1 XXZ and supersymmetric sine-Gordon models with Dirichlet boundary conditions

Starting from the Bethe Ansatz solution of the open integrable spin-1 XXZ quantum spin chain with diagonal boundary terms, we derive a set of nonlinear integral equations (NLIEs), which we propose to describe the boundary supersymmetric sine-Gordon model BSSG⁺ with Dirichlet boundary conditions on a finite interval. We compute the corresponding boundary S matrix, and find that it coincides with the one proposed by Bajnok, Palla and Takács for the Dirichlet BSSG⁺ model. We derive a relation between the (UV) parameters in the boundary conditions and the (IR) parameters in the boundary S matrix. By computing the boundary vacuum energy, we determine a previously unknown parameter in the scattering theory. We solve the NLIEs numerically for intermediate values of the interval length, and find agreement with our analytical result for the effective central charge in the UV limit and with boundary conformal perturbation theory.     

Notes on integrable boundary interactions of open <Emphasis Type="Italic">SU</Emphasis>(4) alternating spin chains

Ref. [J. High Energy Phys. 1708, 001 (2017)] showed that the planar flavored Ahanory-Bergman-Jafferis-Maldacena (ABJM) theory is integrable in the scalar sector at two-loop order using coordinate Bethe ansatz. A salient feature of this case is that the boundary reflection matrices are anti-diagonal with respect to the chosen basis. In this paper, we relax the coefficients of the boundary terms to be general constants to search for integrable systems among this class. We found that the only integrable boundary interaction at each end of the spin chain aside from the one in ref. [J. High Energy Phys. 1708, 001 (2017)] is the one with vanishing boundary interactions leading to diagonal reflection matrices. We also construct non-supersymmetric planar flavored ABJM theory which leads to trivial boundary interactions at both ends of the open chain from the two-loop anomalous dimension matrix in the scalar sector.     

Supersymmetric Ising field theory

We construct the simplest solvable non-trivial (S ≠ 1) supersymmetric model: the supersymmetric Ising field theory. The supersymmetric generalization of the duality order/disorder algebra is discussed. Finally we attempt to “double” the model and comment on the connection between this doubled model and the unsolved problem of the supersymmetric sine-Gordon kink structure.     

Notes on integrable boundary interactions of open SU(4) alternating spin chains

Ref. [J. High Energy Phys. 1708, 001(2017)] showed that the planar flavored Ahanory-Bergman-Jafferis-Maldacena(ABJM)theory is integrable in the scalar sector at two-loop order using coordinate Bethe ansatz. A salient feature of this case is that the boundary reflection matrices are anti-diagonal with respect to the chosen basis. In this paper, we relax the coefficients of the boundary terms to be general constants to search for integrable systems among this class. We found that the only integrable boundary interaction at each end of the spin chain aside from the one in ref. [J. High Energy Phys. 1708, 001(2017)] is the one with vanishing boundary interactions leading to diagonal reflection matrices. We also construct non-supersymmetric planar flavored ABJM theory which leads to trivial boundary interactions at both ends of the open chain from the two-loop anomalous dimension matrix in the scalar sector.     

Supersymmetric Yang-Mills,octonionic instantons and triholomorphic curves

In four-dimensional gauge theory there exists a well-known correspondence between instantons and holomorphic curves, and a similar correspondence exists between certain octonionic instantons and triholomorphic curves. We prove that this latter correspondence stems from the dynamics of various dimensional reductions of ten-dimensional supersymmetric Yang-Mills theory. More precisely we show that the dimensional reduction of the (5+1)-dimensional supersymmetric sigma model with hyper-Kähler (but otherwise arbitrary) target X to a four-dimensional hyper-Kähler manifold M is a topological sigma model localising on the space of triholomorphic maps M -+ X (or hyperinstantons). When X is the moduli space M_k of instantons on a four-dimensional hyper-Kdhler manifold K, this theory has an interpretation in terms of supersymmetric gauge theory. In this case, the topological sigma model can be understood as an adiabatic limit of the dimensional reduction of ten-dimensional supersymmetric Yang-Mills on the eight-dimensional manifold M × K of holonomy Sp(1) × Sp(1) ⊂ Spin(7), which is a cohomological theory localising on the moduli space of octonionic instantons.     

The two-boundary sine-Gordon model

We study in this paper the ground state energy of a free bosonic theory on a finite interval of length R with either a pair of sine-Gordon type or a pair of Kondo type interactions at each boundary. This problem has potential applications in condensed matter (current through superconductor–Luttinger liquid–superconductor junctions) as well as in open string theory (tachyon condensation). While the application of Bethe ansatz techniques to this problem is in principle well known, considerable technical difficulties are encountered. These difficulties arise mainly from the way the bare couplings are encoded in the reflection matrices, and require complex analytic continuations, which we carry out in detail in a few cases.     

Integrable open-boundary conditions for the q-deformed supersymmetric U model of strongly correlated electrons

A general graded reflection equation algebra is proposed and the corresponding boundary quantum inverse scattering method is formulated. The formalism is applicable to all boundary lattice systems where an invertible R-matrix exists. As an application, the integrable open-boundary conditions for the q-deformed supersymmetric U model of strongly correlated electrons are investigated. The diagonal boundary K-matrices are found and a class of integrable boundary terms are determined. The boundary system is solved by means of the coordinate space Bethe ansatz technique and the Bethe ansatz equations are derived. As a sideline, it is shown that all R-matrices associated with a quantum affine superalgebra enjoy the crossing-unitarity property.     

Boundary bootstrap principle in two-dimensional integrable quantum field theories

We study the reflection amplitudes of affine Toda field theories with boundary, following the ideas developed by Fring and Koberle [A. Fring, R. Koberle, Nucl. Phys. B 421 (1994) 159; A. Fring, R. Koberle, Nucl. Phys. B 419 (1994) 647; A. Fring, R. Koberle, Int. J. Mod. Phys. A 10 (1995) 739] and focusing our attention on the E_n series elements, because of their interesting structure of higher order poles. We also investigate the corresponding minimal reflection matrices, finding, with respect to the bulk case, a more complicated relation between the spectra of bound states associated to the minimal and to the “dressed” amplitudes.     

Integrability of a family of quantum field theories related to sigma models

A method is introduced for constructing lattice discretizations of large classes of integrable quantum field theories. The method proceeds in two steps: The quantum algebraic structure underlying the integrability of the model is determined from the algebra of the interaction terms in the light-cone representation. The representation theory of the relevant quantum algebra is then used to construct the basic ingredients of the quantum inverse scattering method, the lattice Lax matrices and R-matrices. This method is illustrated with four examples: The sinh-Gordon model, the affine sl(3) Toda model, a model called the fermionic sl(2|1) Toda theory, and the N=2 supersymmetric sine-Gordon model. These models are all related to sigma models in various ways. The N=2 supersymmetric sine-Gordon model, in particular, describes the Pohlmeyer reduction of string theory on AdS₂×S², and is dual to a supersymmetric non-linear sigma model with a sausage-shaped target space.     

Classification of reflection matrices related to (super-)Yangians and application to open spin chain models

We present a classification of diagonal, antidiagonal and mixed reflection matrices related to Yangian and super-Yangian R matrices associated to the infinite series so(m), sp(n) and osp(m|n). We formulate the analytical Bethe ansatz resolution for the so(m) and sp(n) open spin chains with boundary conditions described by the diagonal solutions.     

Superfield approach to the construction of effective action in quantum field theory with extended supersymmetry

We review the current state of research on the construction of effective actions in supersymmetric quantum field theory. Special attention is paid to gauge models with extended supersymmetry in the superfield approach. The advantages of formulation of such models in harmonic superspace for the calculation of effective action are emphasized. Manifestly supersymmetric and manifestly gauge-invariant methods for constructing the low-energy effective actions and deriving the corrections to them are considered and the possibilities to obtain the exact solutions are discussed. The calculations of one-loop effective actions in N = 2 supersymmetric Yang–Mills theory with hypermultiplets and in N = 4 supersymmetric Yang–Mills theory are analyzed in detail. The relationship between the effective action in supersymmetric quantum field theory and the low-energy limit in superstring theory is discussed.