abstract

In these lecture notes we give a “historical” introduction to topological gauge theories. Our main aim is to clearly explain the origin of the Hamiltonian which forms the basis of Witten’s construction of topological gauge theory. We show how this Hamiltonian arises from Witten’s formulation of Morse theory as applied by Floer to the infinite dimensional space of gauge connections, with the Chern–Simons functional as the appropriate Morse function(al). We therefore discuss the De Rham cohomology, Hodge theory, Morse theory, Floer homology, Witten’s construction of the Lagrangian for topological gauge theory, the subsequent BRST formulation of topological quantum field theory and finally Witten’s construction of the Donaldson polynomials.

abstract

New extensive computer simulations of the bosonic string carried out in collaboration with D. Boulatov and V. Kazakov are presented.

abstract

The classical electroweak theory is unstable for magnetic fields \(H\) satisfying \(eH \gt m^2_W\), \(m_W\) being the \(W\) mass. We discuss the origin of this instability and the peculiar anti-screening property of the \(W\)-condensate formed. The anti-screening is closely related to the asymptotic freedom of the SU(2)\(_W\)-fields, thereby motivating the name “Asymptotic Freeon”. For sufficiently large magnetic fields we expect symmetry restoration. This is verified by Bogomol’nyi’s method which allows us to reduce the ordinary second order electroweak equations to first order equations in the special case where the Higgs mass \(m_h\) is equal to the \(Z\)-mass \(m_Z\).

abstract

The glueball spectrum is studied for all \(J^{PC}\) values for pure gauge SU(3) colour interactions using lattice gauge theory simulation.

abstract

Finite size effects can be dramatically different for periodic and twisted boundary conditions, as long as we are in small volumes where perturbation theory prevails. In that case glueballs are constituted from gluons; whereas the glueballs feel only periodic boundary conditions, its constituent gluons experience the twist and have to wind at least as many times around the box as there are colours. We review Montecarlo results on glueball masses and string tension with periodic and twisted boundary conditions. For lattice sizes now available they seem to be in reasonable agreement. Various improvements are suggested.

abstract

In the present paper we would like to emphasize that the terms gauge invariance and independence of the gauge choke have different meaning and to discuss the natural choice of the gauge in QCD when the rising potential in included. To demonstrate the situation we consider the infrared behaviour of the gluon propagator and the running coupling constant.

abstract

We discuss the present status of experimental information on, and theoretical understanding of the production of very soft (\(p_{\rm T} \lt \) 30 MeV/\(c\) or so) real and virtual photons in hadronic interactions. We expect that the data will undergo a time evolution towards the shape required by Low theorem and Landau–Pomeranchuk mechanism. Some recently suggested proposals for intermediate stage processes leading to enhancement of very soft photon production are qualitatively analyzed.