This is the foreword to the special volume on localization techniques in quantum field theory. The summary of individual chapters is given and their interrelation is discussed.

The five-dimensional N = 1 supersymmetric gauge theory with Sp(N ) gauge group and SO(2N f ) flavor symmetry describes the physics on N D4-branes with N f D8-branes on top of a single O8 orientifold plane in Type I ′ theory. This theory is known to be superconformal at the strong coupling limit with the enhanced global symmetry E N f +1 for N f ≤ 7. In this work we calculate the superconformal index on S 1 × S 4 for the Sp(1) gauge theory by the localization method and confirm such enhancement of the global symmetry at the superconformal limit for N f ≤ 5 to a few leading orders in the chemical potential. Both perturbative and (anti)instanton contributions are present in this calculation. For N f = 6, 7 cases some issues related the pole structure of the instanton calculation could not be resolved and here we could provide only some suggestive answer for the leading contributions to the index. For the Sp(N ) case, similar issues related to the pole structure appear.

Abstract:We explore further our recent generalization of the N = 4 superconformal Chern-Simons theories of Gaiotto and Witten. We find and construct explicitly theories of enhanced N = 5 or 6 supersymmetry, especially N = 5, Sp(2M ) × O(N ) and N = 6, Sp(2M ) × O(2) theories. The U (M ) × U (N ) theory coincides with the N = 6 theory of Aharony, Bergman, Jafferis and Maldacena (ABJM). We argue that the N = 5 theory with Sp(2N ) × O(2N ) gauge group can be understood as an orientifolding of the ABJM model with U (2N ) × U (2N ) gauge group. We briefly discuss the Type IIB brane construction of the N = 5 theory and the geometry of the M-theory orbifold.

We investigate the self-dual Yang-Mills gauge configurations on R 3 × S 1 when the gauge symmetry SU(2) is broken to U(1) by the Wilson loop. We construct the explicit field configuration for a single instanton by the Nahm method and show that an instanton is composed of two self-dual monopoles of opposite magnetic charge. We normalize the moduli space metric of an instanton and study various limits of the field configuration and its moduli space metric.

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 extend the N = 4 superconformal Chern-Simons theories of Gaiotto and Witten to those with additional twisted hyper-multiplets. The new theories are generically linear quiver gauge theories with the two types of hyper-multiplets alternating between gauge groups. Our construction includes the Bagger-Lambert model of SO(4) gauge group. A family of abelian theories are identified with those proposed earlier in the context of the M-crystal model for M2-branes probing (C 2 /Z n ) 2 orbifolds. Possible extension with nonabelian BF couplings and string/M-theory realization are briefly discussed.In this work we generalize the Gaiotto-Witten's work to include twisted hyper-multiplets. Quiver theories appear naturally with two types of hyper-multiplets alternating between gauge groups where the quiver diagram is linear or circular with multiple nodes. The Bagger-Lambert theory with SO(4) gauge group appears naturally as a simplest kind of the quiver theory. Our work is partially motivated by attempt to understand the Bagger-Lambert theory with SO(4) gauge group in the context of the Gaiotto-Witten theory.The number of supersymmetries of three-dimensional superconformal Chern-Simons theories has a natural division with N = 3 [7]. It is rather straightforward to have the theories with N ≤ 3, and there has been some recent work on N = 2, 3 superconformal theories [8]. For the conformal theory of M2 branes, one needs more supersymmetry [9] and the recent works related to the BL theory and the GW theory can be regarded as concrete steps toward this direction.

We study magnetically charged classical solutions of a spontaneously broken gauge theory interacting with gravity. We show that nonsingular monopole solutions exist only if the Higgs field vacuum expectation value v is less than or equal to a critical value v cr , which is of the order of the Planck mass. In the limiting case, the monopole becomes a black hole, with the region outside the horizon described by the critical Reissner-Nordstrom solution. For v < v cr , we find additional solutions which are singular at r = 0, but which have this singularity hidden within a horizon. These have nontrivial matter fields outside the horizon, and may be interpreted as small black holes lying within a magnetic monopole. The nature of these solutions as a function of v and of the total mass M and their relation to the Reissner-Nordstrom solutions is discussed.

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