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N. J. MacKay

Publications and source records attributed to N. J. MacKay.

At least 19 recordsLinked to original sources
arXiv2007

Rational R-matrices, centralizer algebras and tensor identities for e_6 and e_7 exceptional families of Lie algebras

We use Cvitanovic's diagrammatic techniques to construct the rational solutions of the Yang-Baxter equation associated with the $e_6$ and $e_7$ families of Lie algebras, and thus explain Westbury's observations about their uniform spectral decompositions. In doing so we explore the extensions of the Brauer and symmetric group algebras to the centralizer algebras of $e_7$ and $e_6$ on their lowest-dimensional representations and (up to three-fold) tensor products thereof, giving bases for them and a number of identities satisfied by the algebras' defining invariant tensors.

math.QA↗
arXiv2006

Affine quantum groups

Affine quantum groups are certain pseudo-quasitriangular Hopf algebras that arise in mathematical physics in the context of integrable quantum field theory, integrable quantum spin chains, and solvable lattice models. They provide the algebraic framework behind the spectral parameter dependent Yang-Baxter equation. One can distinguish three classes of affine quantum groups, each leading to a different dependence of the R-matrices on the spectral parameter: Yangians lead to rational R-matrices, quantum affine algebras lead to trigonometric R-matrices and elliptic quantum groups lead to elliptic R-matrices. We will mostly concentrate on the quantum affine algebras but many results hold similarly for the other classes. After giving mathematical details about quantum affine algebras and Yangians in the first two section, we describe how these algebras arise in different areas of mathematical physics in the three following sections. We end with a description of boundary quantum groups which extend the formalism to the boundary Yang-Baxter (reflection) equation.

math.QA↗
arXiv2006

Lanchester combat models

An overview of Lanchester combat models, emphasising their pedagogical possibilities. After a description of the aimed-fire model and comments on the literature, we introduce briefly a range of further topics: a discrete equivalent, the unaimed-fire model, mixed forces, the meaning of a 'unit', support troops, Bracken's generalization and an asymmetric model.

math.HO↗
arXiv2005

Quantum, higher-spin, local charges in symmetric space sigma models

Potential anomalies are analysed for the local spin-3 and spin-4 classically conserved currents in any two-dimensional sigma model on a compact symmetric space $G/H$, with $G$ and $H$ classical groups. Quantum local conserved charges are shown to exist in exactly those models which also possess quantum non-local (Yangian) charges. The possibility of larger sets of quantum local charges is discussed and shown to be consistent with known S-matrix results and the behaviour of the corresponding Yangian representations.

hep-th↗
arXiv2004

Classically integrable boundary conditions for symmetric-space sigma models

We investigate boundary conditions for the nonlinear sigma model on the compact symmetric space $G/H$, where $H \subset G$ is the subgroup fixed by an involution $σ$ of $G$. The Poisson brackets and the classical local conserved charges necessary for integrability are preserved by boundary conditions in correspondence with involutions which commute with $σ$. Applied to $SO(3)/SO(2)$, the nonlinear sigma model on $S^2$, these yield the great circles as boundary submanifolds. Applied to $G \times G/G$, they reproduce known results for the principal chiral model.

hep-th↗
arXiv2003

Boundary scattering, symmetric spaces and the principal chiral model on the half-line

We investigate integrable boundary conditions (BCs) for the principal chiral model on the half-line, and rational solutions of the boundary Yang-Baxter equation (BYBE). In each case we find a connection with (type I, Riemannian, globally) symmetric spaces G/H: there is a class of integrable BCs in which the boundary field is restricted to lie in a coset of H; these BCs are parametrized by G/H x G/H; there are rational solutions of the BYBE in the defining representations of all classical G parametrized by G/H; and using these we propose boundary S-matrices for the principal chiral model, parametrized by G/H x G/H, which correspond to our boundary conditions.

hep-th↗
arXiv2003

Lattice quantization of Yangian charges

By placing theories with Yangian charges on the lattice in the analogue of the St Petersburg school's approach to the sine-Gordon system, we exhibit the Yangian structure of the auxiliary algebra, and explain how the two Yangians are related.

hep-th↗
arXiv2003

Twisted Yangians and symmetric pairs

We describe recent work on the twisted Yangians Y(g,h) which arise as boundary remnants of Yangians Y(g) in 1+1D integrable field theories, bringing out the special role played by the requirement that (g,h) be a symmetric pair.

math.QA↗
arXiv2003

Rational K-matrices and representations of twisted Yangians

We describe the twisted Yangians Y(g,h) which arise as boundary remnants of Yangians Y(g) in 1+1D integrable field theories. We describe and extend our recent construction of the intertwiners of their representations (the rational boundary S- or 'K'-matrices) and perform a case-by-case analysis for all pairs (g,h), giving the h-decomposition of Y(g,h)-representations where possible.

math.QA↗
arXiv2001

Boundary remnant of Yangian symmetry and the structure of rational reflection matrices

For the classical principal chiral model with boundary, we give the subset of the Yangian charges which remains conserved under certain integrable boundary conditions, and extract them from the monodromy matrix. Quantized versions of these charges are used to deduce the structure of rational solutions of the reflection equation, analogous to the 'tensor product graph' for solutions of the Yang-Baxter equation. We give a variety of such solutions, including some for reflection from non-trivial boundary states, for the SU(N) case, and confirm these by constructing them by fusion from the basic solutions.

hep-th↗
arXiv2000

Conserved charges and supersymmetry in principal chiral and WZW models

Conserved and commuting charges are investigated in both bosonic and supersymmetric classical chiral models, with and without Wess-Zumino terms. In the bosonic theories, there are conserved currents based on symmetric invariant tensors of the underlying algebra, and the construction of infinitely many commuting charges, with spins equal to the exponents of the algebra modulo its Coxeter number, can be carried out irrespective of the coefficient of the Wess-Zumino term. In the supersymmetric models, a different pattern of conserved quantities emerges, based on antisymmetric invariant tensors. The current algebra is much more complicated than in the bosonic case, and it is analysed in some detail. Two families of commuting charges can be constructed, each with finitely many members whose spins are exactly the exponents of the algebra (with no repetition modulo the Coxeter number). The conserved quantities in the bosonic and supersymmetric theories are only indirectly related, except for the special case of the WZW model and its supersymmetric extension.

hep-th↗
arXiv2000

Local conserved charges in principal chiral models

Local conserved charges in principal chiral models in 1+1 dimensions are investigated. There is a classically conserved local charge for each totally symmetric invariant tensor of the underlying group. These local charges are shown to be in involution with the non-local Yangian charges. The Poisson bracket algebra of the local charges is then studied. For each classical algebra, an infinite set of local charges with spins equal to the exponents modulo the Coxeter number is constructed, and it is shown that these commute with one another. Brief comments are made on the evidence for, and implications of, survival of these charges in the quantum theory.

hep-th↗
arXiv1999

The SO(N) principal chiral field on a half-line

We investigate the integrability of the SO(N) principal chiral model on a half-line, and find that mixed Dirichlet/Neumann boundary conditions (as well as pure Dirichlet or Neumann) lead to infinitely many conserved charges classically in involution. We use an anomaly-counting method to show that at least one non-trivial example survives quantization, compare our results with the proposed reflection matrices, and, based on these, make some preliminary remarks about expected boundary bound-states.

hep-th↗
arXiv1999

Conserved Charges and Supersymmetry in Principal Chiral Models

We report on investigations of local (and non-local) charges in bosonic and supersymmetric principal chiral models in 1+1 dimensions. In the bosonic PCM there is a classically conserved local charge for each symmetric invariant tensor of the underlying group. These all commute with the non-local Yangian charges. The algebra of the local charges amongst themselves is rather more subtle. We give a universal formula for infinite sets of mutually commuting local charges with spins equal to the exponents of the underlying classical algebra modulo its Coxeter number. Many of these results extend to the supersymmetric PCM, but with local conserved charges associated with antisymmetric invariants in the Lie algebra. We comment briefly on the quantum conservation of local charges in both the bosonic and super PCMs.

hep-th↗
arXiv1996

Remarks on excited states of affine Toda solitons

The identification in affine Toda field theory of the quantum particle with the lowest breather allows us to re-interpret discrete modes of excitation of solitons as breathers bound to solitons, and thus to investigate them through the proposed soliton-breather S-matrices. There are implications for the physical spectrum and for the semiclassical soliton mass corrections.

hep-th↗
arXiv1995

Exact S-matrices for d_{n+1}^{(2)} affine Toda solitons and their bound states

We conjecture an exact S-matrix for the scattering of solitons in $d_{n+1}^{(2)}$ affine Toda field theory in terms of the R-matrix of the quantum group $U_q(c_n^{(1)})$. From this we construct the scattering amplitudes for all scalar bound states (breathers) of the theory. This S-matrix conjecture is justified by detailed examination of its pole structure. We show that a breather-particle identification holds by comparing the S-matrix elements for the lowest breathers with the S-matrix for the quantum particles in real affine Toda field theory, and discuss the implications for various forms of duality.

hep-th↗
arXiv1995

Fusion of SO(N) reflection matrices

We examine the reflection matrix acting on the $SO(N)$ vector multiplet and fuse it to obtain that acting on the rank two particle multiplet, and give its decomposition.

hep-th↗