International Journal of Modern Physics A

A deformed q-calculus is developed on the basis of an algebraic structure involving graded brackets. A number operator and left and right shift operators are constructed for this algebra, and the whole structure is related to the algebra of a q-deformed boson. The limit of this algebra when q is an nth root of unity is also studied in detail. By means of a chain rule expansion, the left and right derivatives are identified with the charge Q and covariant derivative D encountered in ordinary/fractional supersymmetry, and this leads to new results for these operators. A generalized Berezin integral and fractional superspace measure arise as a natural part of our formalism. When q is a root of unity the algebra is found to have a nontrivial Hopf structure, extending that associated with the anyonic line. One-dimensional ordinary/fractional superspace is identified with the braided line when q is a root of unity, so that one-dimensional ordinary/fractional supersymmetry can be viewed as invariance under translation along this line. In our construction of fractional supersymmetry the q-deformed bosons play a role exactly analogous to that of the fermions in the familiar supersymmetric case.

We show how to perform systematically improvable variational calculations in the O(2N) Gross–Neveu model for generic N, in such a way that all infinities usually plaguing such calculations are accounted for in a way compatible with the perturbative renormalization group. The final point is a general framework for the calculation of nonperturbative quantities like condensates, masses, etc., in an asymptotically free field theory. For the Gross–Neveu model, the numerical results obtained from a "two-loop" variational calculation are in a very good agreement with exact quantities down to low values of N.

The symmetry algebra of the two-dimensional anisotropic quantum harmonic oscillator with rational ratio of frequencies is identified as a deformation of the u(2) algebra. The finite dimensional representation modules of this algebra are studied and the energy eigenvalues are determined using algebraic methods of general applicability to quantum superintegrable systems. For labelling the degenerate states an "angular momentum" operator is introduced, the eigenvalues of which are roots of appropriate generalized Hermite polynomials. The cases with frequency ratios 1:n correspond to generalized parafermionic oscillators, while in the special case with frequency ratio 2:1 the resulting algebra corresponds to the finite W algebra .

We discuss how to describe U(N) monopoles on the Schwarzschild and Reissner–Nordström black holes by the parameters of the moduli space of holomorphic vector bundles over S². For N = 2,3 we obtain such a description in an explicit form as well as the expressions for the corresponding monopole masses. This gives a possibility of adducing some reasonings in favor of existence of both a fine structure for black holes and the statistical ensemble tied with it which might generate the black hole entropy. Also, there arises an analogy with the famous K-theory in topology.

The presence of a heavy fermion doublet in the electroweak instanton probably leads to unsuppressed fermion number nonconservation. By calculating numerically the effective potential barrier of electroweak instanton with heavy fermion in Minkowski space by using semiclassical approximation, it is shown that, if the mass of heavy fermion exceeds a critical value  TeV (for M_H = M_W), the tunneling by electroweak instanton should be unsuppressed and the heavy fermion decays with fermion number violation. The dependence of critical fermion mass on the Higgs mass is also presented.

We construct the form factors of the trace of energy–momentum tensor for the massless model described by SU(2) principal chiral field model with WZNW term on level 1. We explain how this construction can be generalized to a class of integrable massless models including the flow from tricritical to critical Ising model.

From F. Smirnov. During several months I worked with Pierre Mejean. After his premature decease which deeply affected everybody who knew him I decided to collect and to publish the results which we obtained together.

Some aspects of chiral p-forms, in particular the obstruction that makes it hard to define covariant Green functions, are discussed. It is shown that a proposed resolution involving an infinite set of gauge fields can be made mathematically rigorous in the classical case. We also give a brief demonstration of species doubling for chiral bosons.

We derive a simple formula for the periods associated with the low energy effective action of N = 2 supersymmetric SU(2) Yang–Mills theory with massive N_f ≤ 3 hypermultiplets. This is given by evaluating explicitly the integral associated to the elliptic curve using various identities of hypergeometric functions. Following this formalism, we can calculate the prepotential with massive hypermultiplets both in the weak coupling region and in the strong coupling region. In particular, we show how the Higgs field and its dual field are expressed as generalized hypergeometric functions when the theory has a conformal point.