Topology in Lattice Gauge Theory

Abstract

We have evidence that the vacuum of (pure) SU(2) and SU(3) gauge theories possesses an underlying instanton structure. Starting from Monte Carlo generated equilibrium gauge field configurations representing the physical vacuum, we obtain by systematically freezing the quantum fluctuations by successive relaxation (approximate) solutions of the classical equations of motion, which turn out to have discrete values of the action

$$ S \approx \beta {\text{2}}{\pi ^2}N, N = 0,1 ,2, \ldots $$

in close agreement with the continuum (multi-) instanton solutions. We show that these lattice (multi-) instantons are localized in space time, that they carry a topological charge /Q/ =N and that they give rise to a number of fermion zero modes in accordance with the Atiyah-Singer index theorem.

In order to study the role played by topology in the physics of the vacuum of QCD and SU(N) gauge theories quantitatively, we need a fast algorithm for reliably computing the topological charge Q on large lattices. Two such algorithms exist now for gauge group SU(2). Using recently derived explicit formulae for the 2- and 3-cochains, we are able to integrate the Chern-Simons density analytically and arrive at a local algebraic expression for the topological charge which is relatively easy to implement and, since it does not resort to numerical integration, fast to compute on the lattice. The other algorithm is due to Philips and Stone which is based on simplicial lattices and a geometrical interpolation of the transition functions.