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

We demonstrate electric-magnetic duality in N=1 supersymmetric non-Abelian gauge the-

A q-form global symmetry is a global symmetry for which the charged operators are of space-time dimension q; e.g. Wilson lines, surface defects, etc., and the charged excitations have q spatial dimensions; e.g. strings, membranes, etc. Many of the properties of ordinary global symmetries (q = 0) apply here. They lead to Ward identities and hence to selection rules on amplitudes. Such global symmetries can be coupled to classical background fields and they can be gauged by summing over these classical fields. These generalized global symmetries can be spontaneously broken (either completely or to a subgroup). They can also have 't Hooft anomalies, which prevent us from gauging them, but lead to 't Hooft anomaly matching conditions. Such anomalies can also lead to anomaly inflow on various defects and exotic Symmetry Protected Topological phases. Our analysis of these symmetries gives a new unified perspective of many known phenomena and uncovers new results.

We study the perturbative dynamics of noncommutative field theories on R d , and find an intriguing mixing of the UV and the IR. High energies of virtual particles in loops produce non-analyticity at low momentum. Consequently, the low energy effective action is singular at zero momentum even when the original noncommutative field theory is massive. Some of the nonplanar diagrams of these theories are divergent, but we interpret these divergences as IR divergences and deal with them accordingly. We explain how this UV/IR mixing arises from the underlying noncommutativity. This phenomenon is reminiscent of the channel duality of the double twist diagram in open string theory.

Dynamical supersymmetry breaking in a long-lived meta-stable vacuum is a phenomenologically viable possibility. This relatively unexplored avenue leads to many new models of dynamical supersymmetry breaking. Here, we present a surprisingly simple class of models with meta-stable dynamical supersymmetry breaking: N = 1 supersymmetric QCD, with massive flavors. Though these theories are strongly coupled, we definitively demonstrate the existence of meta-stable vacua by using the free-magnetic dual. Model building challenges, such as large flavor symmetries and the absence of an R-symmetry, are easily accommodated in these theories. Their simplicity also suggests that broken supersymmetry is generic in supersymmetric field theory and in the landscape of string vacua.

We study (non-renormalizable) five dimensional supersymmetric field theories. The theories are parametrized by quark masses and a gauge coupling. We derive the metric on the Coulomb branch exactly. We use stringy considerations to learn about new non-trivial interacting field theories with exceptional global symmetry E n (E 8 , E 7 , E 6 , E 5 = Spin(10), E 4 = SU (5), E 3 = SU (3) × SU (2), E 2 = SU (2) × U (1) and E 1 = SU (2)). Their Coulomb branch is R + and their Higgs branch is isomorphic to the moduli space of E n instantons. One of the relevant operators of these theories leads to a flow to SU (2) gauge theories with N f = n − 1 flavors. In terms of these SU (2) IR theories this relevant parameter is the inverse gauge coupling constant. Other relevant operators (which become quark masses after flowing to the SU (2) theories) lead to flows between them. Upon further compactifications to four and three dimensions we find new fixed points with exceptional symmetries.