Self-dual monopole loops, instantons and confinement

Abstract

It is well-known that the standard instanton analysis in 4d Yang-Mills is plagued with the instanton size moduli problem, which renders the instanton contribution to vacuum energy density (or one-instanton partition function) infrared divergent. The formalism also ignores the implications of long range (magnetic dipole type) $1/r^4$ interaction between the small instantons, since it is weaker than Coulomb interaction. We show that in $U(1)$ lattice gauge theory, where finite action configurations are monopole loops, small loops at large separations also interact with the same type of $1/r^4$ interaction. If one ignores the classical interactions between monopoles, following the same idea as in Yang-Mills theory, the one-monopole partition function is also infrared divergent at strong coupling. However, $1/r^4$ interactions among small loops should be viewed as a consequence of multipole expansion, and emanate from $1/r^2$ interaction between current segments. Taking interactions into account, one can prove that the strongly coupled $U(1)$ lattice gauge theory is dual to a lattice abelian Higgs model, and more importantly, free of infrared divergences. The model exhibits mass gap and confinement by monopole condensation. We suggest that the structure of moduli space of instantons, ADHM data, and the long ranged classical interactions in pure Yang-Mills theory should be examined with this refined perspective. We conjecture that, in contradistinction to the current views on the subject, internal structure of instantons in Yang-Mills theory is responsible for confinement in $4d$ , similar to sigma model in $d=2$ dimensions.