Large scale and large period limits of symmetric calorons
- Derek Harland
PhD thesis Pure Yang-Mills instantons are considered on S 1 × R 3 – so-called calorons. The holonomy – or Polyakov loop around the thermal S 1 at spatial infinity – is assumed to be a non-centre element of the gauge group SU(n) as most appropriate for QCD applications in the confined phase. It is shown that a charge k caloron can be seen as a collection of nk massive magnetic monopoles – coming in n types – each carrying fractional topological charge. This interpretation offers a physically appealing way of introducing monopole degrees of freedom into pure gluodynamics: as constituents of finite temperature instantons. Using the Nahm transform an elaborate treatment is given for arbitrary topo-logical charge and new exact and explicit solutions are found for SU(2) and charge 2. The k zero-modes of the Dirac operator in the background of a charge k caloron are computed, and are shown to 'hop' between the n types of monopoles as a function of the temporal boundary condition. The abelian limit – where it is assumed that the massive field components can be dropped – is analysed in great detail and is shown that the abelian charge distribution of each monopole type coincides with the corresponding fermion zero-mode density. The 4nk dimensional hyperkähler moduli space is identified as an algebraic variety, defined by four matrices obeying a constraint, modulo a natural adjoint action. This moduli space is proved to be the same as the moduli space of stable holomorphic bundles over the complex projective plane which are trivial on two complex lines. A description is given for its twistor space which allows for the computation of the exact hyperkähler metric – at least in principle. Finally, lattice gauge theoretic applications are mentioned and is explicitly demonstrated how to obtain calorons on the lattice using the method of cooling.