We extend earlier ideas about the appearance of noncommutative geometry in string theory with a nonzero B-field. We identify a limit in which the entire string dynamics is described by a minimally coupled (supersymmetric) gauge theory on a noncommutative space, and discuss the corrections away from this limit. Our analysis leads us to an equivalence between ordinary gauge fields and noncommutative gauge fields, which is realized by a change of variables that can be described explicitly. This change of variables is checked by comparing the ordinary Dirac-Born-Infeld theory with its noncommutative counterpart.We obtain a new perspective on noncommutative gauge theory on a torus, its T -duality, and Morita equivalence. We also discuss the D0/D4 system, the relation to M -theory in DLCQ, and a possible noncommutative version of the six-dimensional (2, 0) theory. 8/99to the ADHM equations. This constant had been argued, following [36], to arise in the description of instantons on D-branes upon turning on a constant B-field [37], 1 so putting the two facts together it was proposed that instantons on branes with a B-field should be described by noncommutative Yang-Mills theory [35,38].Another very cogent argument for this is as follows. Consider N parallel threebranes of Type IIB. They can support supersymmetric configurations in the form of U (N ) instantons.If the instantons are large, they can be described by the classical self-dual Yang-Mills equations. If the instantons are small, the classical description of the instantons is no longer good. However, it can be shown that, at B = 0, the instanton moduli space M in string theory coincides precisely with the classical instanton moduli space. The argument for this is presented in section 2.3. In particular, M has the small instanton singularities that are familiar from classical Yang-Mills theory. The significance of these singularities in string theory is well known: they arise because an instanton can shrink to a point and escape as a −1-brane [39,40]. Now if one turns on a B-field, the argument that the stringy instanton moduli space coincides with the classical instanton moduli space fails, as we will also see in section 2.3. Indeed, the instanton moduli space must be corrected for nonzero B.The reason is that, at nonzero B (unless B is anti-self-dual) a configuration of a threebrane and a separated −1-brane is not BPS, 2 so an instanton on the threebrane cannot shrink to a point and escape. The instanton moduli space must therefore be modified, for nonzero B, to eliminate the small instanton singularity. Adding a constant to the ADHM equations resolves the small instanton singularity [41], and since going to noncommutative R 4 does add this constant [35], this strongly encourages us to believe that instantons with the B-field should be described as instantons on a noncommutative space.This line of thought leads to an apparent paradox, however. Instantons come in all sizes, and however else they can be described, big instantons can surely be described by conventio...

We study the vacuum structure and dyon spectrum of N = 2 supersymmetric gauge theory in four dimensions, with gauge group SU (2). The theory turns out to have remarkably rich and physical properties which can nonetheless be described precisely; exact formulas can be obtained, for instance, for electron and dyon masses and the metric on the moduli space of vacua. The description involves a version of Olive-Montonen electric-magnetic duality.The "strongly coupled" vacuum turns out to be a weakly coupled theory of monopoles, and with a suitable perturbation confinement is described by monopole condensation. 6/94

We study four dimensional N = 2 supersymmetric gauge theories with matter multiplets.For all such models for which the gauge group is SU (2), we derive the exact metric on the moduli space of quantum vacua and the exact spectrum of the stable massive states.A number of new physical phenomena occur, such as chiral symmetry breaking that is driven by the condensation of magnetic monopoles that carry global quantum numbers.For those cases in which conformal invariance is broken only by mass terms, the formalism automatically gives results that are invariant under electric-magnetic duality. In one instance, this duality is mixed in an interesting way with SO(8) triality. 8/94