Abstract:We compute the Witten index of one-dimensional gauged linear sigma models with at least N = 2 supersymmetry. In the phase where the gauge group is broken to a finite group, the index is expressed as a certain residue integral. It is subject to a change as the Fayet-Iliopoulos parameter is varied through the phase boundaries. The wall crossing formula is expressed as an integral at infinity of the Coulomb branch. The result is applied to many examples, including quiver quantum mechanics that is relevant for BPS states in d = 4 N = 2 theories.

Motivated by the recent D-brane constructions of world-volume monopoles and instantons, we study the supersymmetric SU (N ) Yang-Mills theory on S 1 × R 3+1 , spontaneously broken by a Wilson loop. In addition to the usual N −1 fundamental monopoles, the N -th BPS monopole appears from the Kaluza-Klein sector. When all N monopoles are present, net magnetic charge vanishes and the solution can be reinterpreted as a Wilson-loop instanton of unit Pontryagin number. The instanton/multi-monopole moduli space is explicitly constructed, and seen to be identical to a Coulomb phase moduli space of a U (1) N gauge theory in 2 + 1 dimensions related to Kronheimer's gauge theory of SU (N ) type. This extends the results by Intriligator and Seiberg to the finite couplings that, in the infrared limit of Kronheimer's theory, the Coulomb phase parameterizes a centered SU (N ) instanton. We also elaborate on the case of restored SU (N ) symmetry.

We study baryons in an AdS/CFT model of QCD by Sakai and Sugimoto, realized as small instantons with fundamental string hairs. We introduce an effective field theory of the baryons in the five-dimensional setting, and show that the instanton interpretation implies a particular magnetic coupling. Dimensional reduction to four dimensions reproduces the usual chiral effective action, and in particular we estimate the axial coupling gA between baryons and pions and the magnetic dipole moments, both of which are proportional to Nc. We extrapolate to finite Nc and discuss subleading corrections.

We argue that M2 brane is realized as a topological soliton on a coincident pair of M5 and anti-M5 branes, as the two five-branes annihilate each other. Topology and quantum numbers of this world-volume soliton are discussed in some detail, and its formation is explained qualitatively. It follows from a compactification that a D4-anti-D4 pair annihilate and produce type II fundamental strings. The phenomenon is best described as the confinement of a world-volume U (1) gauge field on D4-anti-D4, where the confined electric flux string is identified as the fundamental string. This generalizes to other Dp-anti-Dp systems, and solves a puzzle recently pointed out by Witten.