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.

We extend the localization calculation of the 3D Chern-Simons partition function over Seifert manifolds to an analogous calculation in five dimensions. We construct a twisted version of N=1 supersymmetric Yang-Mills theory defined on a circle bundle over a four dimensional symplectic manifold. The notion of contact geometry plays a crucial role in the construction and we suggest a generalization of the instanton equations to five dimensional contact manifolds. Our main result is a calculation of the full perturbative partition function on a five sphere for the twisted supersymmetric Yang-Mills theory with different Chern-Simons couplings. The final answer is given in terms of a matrix model. Our construction admits generalizations to higher dimensional contact manifolds. This work is inspired by the work of Baulieu-Losev-Nekrasov from the mid 90's, and in a way it is covariantization of their ideas for a contact manifold.Comment: 28 pages; v2: minor mistake corrected; v3: matches published versio

We solve the long standing problem of finding an off-shell supersymmetric formulation for a general N = (2, 2) nonlinear two dimensional sigma model. Geometrically the problem is equivalent to proving the existence of special coordinates; these correspond to particular superfields that allow for a superspace description. We construct and explain the geometric significance of the generalized Kähler potential for any generalized Kähler manifold; this potential is the superspace Lagrangian.Recently general N = (2, 2) supersymmetric sigma models have attracted considerable attention; the renewed interest comes both from physics and mathematics. The physics is related to compactifications with NS-NS fluxes, whereas the mathematics is associated with generalized complex geometry, in particular, generalized Kähler geometry, which is precisely the geometry of the target space of N = (2, 2) supersymmetric sigma models.The general N = (2, 2) sigma model originally described in [1] has been studied extensively in the physics literature. However, until now an N = (2, 2) off-shell supersymmetric formulation has not been known in the general case. At the physicist's level of rigor, a description in terms N = (2, 2) superfields would imply the existence of a single function that encodes the local geometry-a generalized Kähler potential. Geometrically the problem of N = (2, 2) off-shell supersymmetry amounts to the proper understanding of certain natural local coordinates and the generalized Kähler potential.In the present work we resolve the issue of what constitutes a complete description of the target space geometry of a general N = (2, 2) sigma model. We show that the full set of fields consists of chiral, twisted chiral and semichiral fields. This was was a natural guess after semichiral superfields were discovered in [2], and was explicitly conjectured by Sevrin and Troost [3]; however, in [4], which contains many useful and interesting results, the erroneous conclusion that this is not the case was reached.The bulk of the paper is devoted to the proof that certain local coordinates for generalized Kähler geometry exist. From the point of view of N = (2, 2) supersymmetry these coordinates are natural and correspond to the basic superfield ingredients of the model. The paper is organized as follows. In Section 2 we review the general N = (2, 2) sigma model and describe the generalized Kähler geometry. Section 3 states the problem of off-shell supersymmetry and explains what should be done to solve it. In Section 4 we describe three relevant Poisson structures and their symplectic foliations, and identify coordinates adapted to these foliations. For the sake of clarity in Section 5 we start with a special case when ker[J + , J − ] = ∅. In this case we show that the correct coordinates exist and we explain the existence of the generalized Kähler potential. Next, in Section 6 we extend our results to the general case. Finally, in Section 7 we summarize our results and explain some open problems.Warning to mathematicians: Due t...

Based on the construction by Hosomichi, Seong and Terashima we consider N = 1 supersymmetric 5D Yang-Mills theory with matter on a five-sphere with radius r. This theory can be thought of as a deformation of the theory in flat space with deformation parameter r and this deformation preserves 8 supercharges. We calculate the full perturbative partition function as a function of r/g 2 Y M , where g Y M is the Yang-Mills coupling, and the answer is given in terms of a matrix model. We perform the calculation using localization techniques. We also argue that in the large N -limit of this deformed 5D Yang-Mills theory this matrix model provides the leading contribution to the partition function and the rest is exponentially suppressed.