International Journal of Modern Physics A

We use Chen iterated line integrals to construct a topological algebra of separating functions on the group of loopsLℳ_p. has a Hopf algebra structure which allows the construction of a group structure on its spectrum. We call this topological group the group of generalized loops Then we develop a loop calculus, based on the end point and area derivative operators, providing a rigorous mathematical treatment of the early heuristic ideas of Gambini, Trias and also Mandelstam, Makeenko and Migdal. Finally, we define a natural action of the “pointed” diffeomorphism group Diff_p(ℳ) on , and consider a variational derivative which allows the construction of homotopy invariants. This formalism is useful for constructing a mathematical theory of loop representation of gauge theories and quantum gravity.

The nonlocal regularization of QED is shown to possess an axial anomaly of the same form as other regularization schemes. The Noether current is explicitly constructed and the symmetries are shown to be violated, whereas the identities constructed when one properly considers the contribution from the path integral measure are respected. We also discuss the merits and new features of the regularization scheme, as well as the barrier to quantizing the fully gauged chiral-invariant theory.

We consider a modification of the Wilson-Yukawa model to overcome the difficulty that the fermion mass is not proportional to the Higgs vacuum expectation value. In the modification scalar and fermionic regulator fields are introduced so that all the physical fermion fields possess shift symmetry when the Yukawa coupling vanishes. With the fermionic hopping parameter expansion it is shown that the fermion mass is proportional to the Higgs vacuum expectation value. We find, however, that the coupling of fermion to the external gauge field is always vectorlike in the continuum limit and that further modifications to the scalar action cannot change this undesirable conclusion.

We calculate the probabilities of slepton and photino production via a charged lepton decay induced by an arbitrary electromagnetic wave or by a constant homogeneous external field, F_μν. This allows us to estimate the domains where this reaction could be significant. Comparing the process with the synchrotron radiation we conclude that for all values of the background field parameter , with p^ν being the lepton’s momentum, the following estimate holds: .

We also study the impact of the intense field on radiative corrections to the lepton’s anomalous magnetic moment. It is found that, when χ exceeds the cube of the masses’ ratios, , , the sparticle’s contribution to the anomalous magnetic moment is of the same order as the QED correction. In this region the magnitude of the anomalous magnetic moment of the lepton is very sensitive to the masses of the slepton and the photino.

In QCD at high temperature, a naive perturbative calculation produces infrared divergences. We construct the leading term of an effective potential by summing up diagrams with leading infrared divergences. Based on this potential, we show that the temporal component of gauge field A₄ cannot develop a thermal expectation value <gA₄>≪O(T) at high temperature. We explain why the usual perturbative calculation which was used in previous literature may lead to a wrong result.

Following a recent proposal of Richard Borcherds to regard fusion as the ringlike tensor product of modules of a quantum ring, a generalization of rings and vertex algebras, we define fusion as a certain quotient of the (vector space) tensor product of representations of the symmetry algebra . We prove that this tensor product is associative and symmetric up to equivalence. We also determine explicitly the action of on it, under which the central extension is preserved.

Having defined fusion in this way, determining the fusion rules is then the algebraic problem of decomposing the tensor product into irreducible representations. We demonstrate how to solve this for the case of the WZW and the minimal models and recover thereby the well-known fusion rules.

The action of the symmetry algebra on the tensor product is given in terms of a comultiplication. We calculate the R matrix of this comultiplication and find that it is triangular. This seems to shed some new light on the possible rôle of the quantum group in conformal field theory.

We identify the untwisted moduli of heterotic orbifold compactifications for the case where the gauge twist is realized by a rotation. The Wilson lines are found to have both continuous and discrete parts. For the case of the standard Z₃ orbifold we classify all possibilities of breaking the gauge group E(6) ⊗ SU(3) by nine of the eighteen Wilson moduli and by additional discrete Wilson lines.