On Solitons in Noncommutative Gauge Theories

Recently there has been a revival of interest in noncommutative gauge theories. They are interesting examples of nonlocal field theories which in the certain limit (of large noncommutativity) become essentially equivalent to the large N ordinary gauge theories ; certain supersymmetric versions of noncommutative gauge theories arise as α′ → 0 limit of theories on Dp-branes in the presence of background B-field ; the related theories arise in Matrix compactifications with C-field turned on ; finally, noncommutativity is in some sense an intrinsic feature of the open string field theory . A lot of progress has been recently achieved in the analysis of the classical solutions of the noncommutative gauge theory on the noncommutative versions of Minkowski or Euclidean spaces. The first explicit solutions and their moduli where analyzed in 74 where instantons in the four dimensional noncommutative gauge theory (with self-dual noncommutativity) were constructed. These instantons play an important role in the construction of the discrete light cone quantization of the M-theory fivebrane , and they also gave a hope of giving an interpretation in the physical gauge theory language of the torsion free sheaves which appear in various interpretations of D-brane states , in particular those responsible for the enthropy of black holes realized via D5-D1 systems , and also enter the S-duality invariant partition functions of N = 4 super-Yang-Mills theory . One can also relax the selfduality assumption on the noncommutativity. The construction of 74 easily generalizes to the general case θ 6= 0 (see also ). In addition to the instantons (which are particles in 4+1 dimensional theory), which represent the D0-D4 system, the monopole-like solutions were found in U(1) gauge theory in 3+1 dimensions. The latter turn out to have a string attached to them. The string with the monopole at its end are the noncommutative field theory