Glueballs on S

It is our goal to study the influence of the structure of the configuration space on the SU(2) glueball spectrum. The method we use consists of putting the theory in a finite spatial volume and reducing the full dynamics to an effective theory of a finite number of degrees of freedom [1,2]. Asymptotic freedom implies that a small volume corresponds to a small coupling constant. Hence changing the volume from small to intermediate values allows us to monitor the onset of nonperturbative phenomena. The Yang-Mills configuration space is the space of gauge orbits A/G (A is the collection of gauge fields or connections, G the group of local gauge transformations). We know from Singer [3] that the topology of this configuration space is highly non-trivial when G is non-abelian. At increasing coupling, the wave functional will start to spread out and will become sensitive to the non-trivial topology, like non-contractable closed loops, of the configuration space. This spreading out will occur first in those directions in configuration space, where the potential rises the slowest, like in the direction of the low-energy modes of the gauge field, or in the direction of the sphalerons associated to tunnelling from one Gribov copy of the vacuum to another. As long as the global features of the configuration space only affects this finite number of modes, one can capture these non-perturbative phenomena in an effective model of the low-energy modes. One uses the Born-Oppenheimer approximation to account for the influence of all the other (highenergy) modes, that is, they are assumed to behave perturbatively and are integrated out from the path integral. We are interested in the influence of the multiple vacuum structure of the theory on the glueball spectrum; in particular we would like to see the dependence of the energies on the θ angle. The θ angle shows up when one implements gauge invariance under large gauge transformations. If n[g] is the winding number of the gauge transformation g, gauge invariance in the hamiltonian formulation is implemented by