International Journal of Modern Physics A

The problem of bosonization of standard local four-quark operators, by involving the penguin operator, with ΔI = 1/2 and ΔS = 1 selection rules, is studied. We show that the well-known Cronin's operator determined within the effective chiral Lagrangian approach with nonlinear realization of chiral SU(3) × SU(3) symmetry (so-called chiral perturbation theory at the hadronic level) can realize the hadronic level description of local four-quark operators with (V − A) × (V − A) quark structure only. Also, it is shown that any attempts to apply Cronin's operator for bosonization of the penguin operator, possessing (V − A) × (V + A) quark structure, are doomed to failure. The latter is explained by the inability of Cronin's operator to describe correctly the s-wave final-state interaction, which is very important for transitions governed by the penguin operator. The asymmetry of quark condensate is calculated. The sign of this asymmetry is opposite to that one assumed in the framework of QCD sum rules.

We show directly in the Lax operator approach how the Virasoro and W-constraints on the τ-function arise in the p-reduced KP hierarchy or generalized KdV hierarchy. In particular, we consider the KdV and the Boussinesq hierarchy to show that the Virasoro and the W-constraints follow from the string equation by expanding the "additional symmetry" operator in terms of the Lax operator. We also mention how this method could be generalized for higher KdV hierarchies.

A new two-parameter quadratic deformation of the quantum oscillator algebra and its one-parameter deformed Heisenberg subalgebra are considered. An infinite-dimensional Fock module representation is presented, which at roots of unity contains singular vectors and so is reducible to a finite-dimensional representation. The semicyclic, nilpotent and unitary representations are discussed. Witten's deformation of sl₂ and some deformed infinite-dimensional algebras are constructed from the 1d Heisenberg algebra generators. The deformation of the centerless Virasoro algebra at roots of unity is mentioned. Finally the SL_q(2) symmetry of the deformed Heisenberg algebra is explicitly constructed.

Using the representation theory of the subgroups SL₂(ℤ_p) of the modular group we investigate the induced fusion algebras in some simple examples. Only some of these representations lead to "good" fusion algebras. Furthermore, the conformal dimensions and the central charge of the corresponding rational conformal field theories are calculated. Two series of representations which can be realized by unitary theories are presented. We show that most of the fusion algebras induced by admissible representations are realized in well-known rational models.

The renormalization group equations for a class of nonrelativistic quantum σ-models targeted on flag manifolds are given. These models emerge in a continuum limit of generalized Heisenberg antiferromagnets. The case of the manifold is studied in greater detail. We show that at zero temperature there is a fixed point of the RG transformations in (2 + ε) dimensions where the theory becomes relativistic. We study the linearized RG transformations in the vicinity of this fixed point and show that half of the couplings are irrelevant. We also show that at this fixed point there is an enlargement of the global isometries of the target manifold. We construct a discrete non-Abelian enlargement of this kind.

Critical values of Wilson lines and general background fields for toroidal compactifications of heterotic string theories are constructed systematically using Dynkin diagrams.

We construct a "shifted" Z₄ × Z₄ orbifold model equivalent to a twisted Z₄ × Z₄ orbifold model at an enhancement point. It is proved that the N = 2 algebra of the 2⁶ model is coincident with that of the "shifted" orbifold model. The correspondence between the Z₄ × Z₄ orbifold model and the 2⁶ model is investigated with respect to massless spectra and Yukawa couplings. Also studied are Yukawa couplings and some other aspects of the models away from the enhancement point.

The inverse problem of Lagrangian supermechanics is investigated. A set of necessary and sufficient conditions for a system of second order differential equations in superspace to derive from a (possibly nonregular) super-Lagrangian function are given. The harmonic superoscillator is revisited and a family of even and odd alternative super-Lagrangians are constructed for it. Finally, we comment on the existence of recursion operators.

The classical limit of quantum q-oscillators suggests an interpretation of the deformation as a way to introduce nonlinearity. Guided by this idea, we considered q fields, the partition function, and compute a consequence on the specific heat and second order correlation function of the q-oscillator which may serve for experimental checks for the nonlinearity.

We propose a random matrix model as a representation for D = 1 open strings. We show that the model with one flavor of boundary fields is equivalent to N fermions with spin in a central potential that also includes a long-range ferromagnetic interaction between the fermions that falls off as 1/(r_ij)². We also generalize this theory to contain an arbitrary number of flavors. For an appropriate choice of the matrix model potential the ground state of the system can be found. Using this potential, we calculate the free energy in the double scaling limit and show that the free energy expansion has the expected form for a theory of open and closed strings if the boundary field mass and couplings have a logarithmic divergence. We then examine the critical properties of this theory and show that the length of the boundary around a hole remains finite, even near the critical point. We also argue that unlike critical string theory or a D = 0 theory, the open string coupling constant is a free parameter.

I construct classical superextensions of the Virasoro algebra by employing the Ward identities of a linearly realized subalgebra. For the N = 4 superconformal algebra, this subalgebra is generated by the N = 2 U(1) supercurrent and a spin 0 N = 2 superfield. I show that this structure can be extended to an N = 4 super W₃ algebra, and give the complete form of this algebra.