Mass generation from Lie algebra extensions

Applied to the electroweak interactions, the theory of Lie algebra extensions suggests a mechanism by which the boson masses are generated without resource to spontaneous symmetry breaking. It starts from a gauge theory without any additional scalar field. All the couplings predicted by the Weinberg-Salam theory are present, and a few others which are nevertheless consistent within the model.

So(2,1) Lie algebra and the Green's functions for the conditionally exactly solvable potentials

The Green's functions of the recently discovered conditionally exactly solvable potentials are computed. This is done through the use of a second-order differential realization of the so(2,1) Lie algebra. So we present the dynamical symmetry underlying the solvability of such potentials and show that they belong to a general class of solvable and partially solvable potentials. © 1994 The American Physical Society.

Closed expressions for Lie algebra invariants and finite transformations

A simple procedure to obtain complete, closed expressions for Lie algebra invariants is presented. The invariants are ultimately polynomials in the group parameters. The construction of finite group elements requires the use of projectors, whose coefficients are invariant polynomials. The detailed general forms of these projectors are given. Closed expressions for finite Lorentz transformations, both homogeneous and inhomogeneous, as well as for Galilei transformations, are found as examples.

Pure Spinors in AdS and Lie Algebra Cohomology

We show that the BRST cohomology of the massless sector of the Type IIB superstring on AdS(5) x S (5) can be described as the relative cohomology of an infinite-dimensional Lie superalgebra. We explain how the vertex operators of ghost number 1, which correspond to conserved currents, are described in this language. We also give some algebraic description of the ghost number 2 vertices, which appears to be new. We use this algebraic description to clarify the structure of the zero mode sector of the ghost number two states in flat space, and initiate the study of the vertices of the higher ghost number.

The Schwinger model on the null-plane

We study the Schwinger Model on the null-plane using the Dirac method for constrained systems. The fermion field is analyzed using the natural null-plane projections coming from the γ-algebra and it is shown that the fermionic sector of the Schwinger Model has only second class constraints. However, the first class constraints are exclusively of the bosonic sector. Finally, we establish the graded Lie algebra between the dynamical variables, via generalized Dirac bracket in the null-plane gauge, which is consistent with every constraint of the theory. © World Scientific Publishing Company.

Riccati-type equations, generalised WZNW equations, and multidimensional Toda systems

We associate to an arbitrary Z-gradation of the Lie algebra of a Lie group a system of Riccati-type first order differential equations. The particular cases under consideration are the ordinary Riccati and the matrix Riccati equations. The multidimensional extension of these equations is given. The generalisation of the associated Redheffer-Reid differential systems appears in a natural way. The connection between the Toda systems and the Riccati-type equations in lower and higher dimensions is established. Within this context the integrability problem for those equations is studied. As an illustration, some examples of the integrable multidimensional Riccati-type equations related to the maximally nonabelian Toda systems are given.

The structures underlying soliton solutions in integrable hierarchies

We point out that a common feature of integrable hierarchies presenting soliton solutions is the existence of some special ''vacuum solutions'' such that the Lax operators evaluated on them, lie in some abelian subalgebra of the associated Kac-Moody algebra. The soliton solutions are constructed out of those ''vacuum solitons'' by the dressing transformation procedure.

The sl(2) affine Toda model coupled to the matter: solitons and confinement

The so-called conformal affine Toda theory coupled to the matter fields (CATM), associated to the (s) over capl(2) affine Lie algebra, is studied. The conformal symmetry is fixed by setting a connection to zero, then one defines an off-critical model, the affine Toda model coupled to the matter (ATM). Using the dressing transformation method we construct the explicit forms of the two-soliton classical solutions, and show that a physical bound soliton-antisoliton pair (breather) does not exist. Moreover, we verify that these solutions share some features of the sine-Gordon (massive Thirring) solitons, and satisfy the classical equivalence of topological and Noether currents in the ATM model. We show, using bosonization techniques that the ATM theory decouples into a sine-Gordon model and a free scalar. Imposing the Noether and topological currents equivalence as a constraint, one can show that the ATM model leads to a bag model like mechanism for the confinement of the color charge inside the sine-Gordon solitons (baryons).

Negative even grade mKdV hierarchy and its soliton solutions

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)

Curves in homogeneous spaces and their contact with 1-dimensional orbits

Let alpha be a C(infinity) curve in a homogeneous space G/H. For each point x on the curve, we consider the subspace S(k)(alpha) of the Lie algebra G of G consisting of the vectors generating a one parameter subgroup whose orbit through x has contact of order k with alpha. In this paper, we give various important properties of the sequence of subspaces G superset of S(1)(alpha) superset of S(2)(alpha) superset of S(3)(alpha) superset of ... In particular, we give a stabilization property for certain well-behaved curves. We also describe its relationship to the isotropy subgroup with respect to the contact element of order k associated with alpha.

CURRENT-ALGEBRA OF SUPER WZNW MODELS

We derive the current algebra of supersymmetric principal chiral models with a Wess-Zumino term. At the critical point one obtains two commuting super-affine Lie algebras as expected, but, in general, them are intertwining fields connecting both right and left sectors, analogously to the bosonic case. Moreover, in the present supersymmetric extension we have a quadratic algebra, rather than an affine Lie algebra, due to the mixing between bosonic and fermionic fields; the purely fermionic sector displays an affine Lie algebra as well.

Non-Abelian Sugawara construction and the q-deformed N=2 superconformal algebra

The construction of a q-deformed N = 2 superconformal algebra is proposed in terms of level-1 currents of the U-q(<(su)over cap>(2)) quantum affine Lie algebra and a single real Fermi field. In particular, it suggests the expression for the q-deformed energy-momentum tensor in the Sugawara form. Its constituents generate two isomorphic quadratic algebraic structures. The generalization to U-q(<(su)over cap>(N + 1)) is also proposed.

Higher grading conformal affine Toda theory and (generalized) sine-Gordon/massive Thirring duality

Some properties of the higher grading integrable generalizations of the conformal affine Toda systems are studied. The fields associated to the non-zero grade generators are Dirac spinors. The effective action is written in terms of the Wess-Zumino-Novikov-Witten (WZNW) action associated to an affine Lie algebra, and an off-critical theory is obtained as the result of the spontaneous breakdown of the conformal symmetry. Moreover, the off-critical theory presents a remarkable equivalence between the Noether and topological currents of the model. Related to the off-critical model we define a real and local lagrangian provided some reality conditions are imposed on the fields of the model. This real action model is expected to describe the soliton sector of the original model, and turns out to be the master action from which we uncover the weak-strong phases described by (generalized) massive Thirring and sine-Gordon type models, respectively. The case of any (untwisted) affine Lie algebra furnished with the principal gradation is studied in some detail. The example of s^l(n) (n = 2, 3) is presented explicitly. © SISSA/ISAS 2003.

Álgebras de Lie e aplicações à sistemas alternantes

Pós-graduação em Matemática - IBILCE