Abstract:

In this thesis, we study gauge theories with N = 2 supersymmetry in four dimensions. The low energy effective action of these theories on their Coulomb branch is described by a holomorphic function called the prepotential. In the first half, we study linear conformal quiver theories with gauge group SU(2). These theories have an SU(2) gauge group at each node of the quiver, and matter arranged in the fundamental and the bifundamental representations, such that at each node the betafunction vanishes. To compute the prepotential for these theories, we follow three different approaches. These are (i) the classic SeibergWitten approach, in which we consider an Mtheory construction of the SeibergWitten curve and the associated differential, (ii) equivariant localization as developed by Nekrasov, and (iii) the 2d/4d correspondence of the four dimensional gauge theory with the two dimensional Liouville conformal field theory, as put forward by Alday, Gaiotto, and Tachikawa. Matching the prepotential, we find out the precise map between the various parameters that appear in the three descriptions.
In the latter half of the thesis, we study surface operators in the context of N=2* theories with gauge group SU(N). These theories describe the dynamics of a vector multiplet, and a massive hypermultiplet in the adjoint representation of the gauge group. Surface operators are nonlocal operators that have support on a two dimensional submanifold of the four dimensional spacetime. They are defined by the singularities they induce in the fourdimensional gauge fields, or can be characterized by the twodimensional theory they support on their worldvolume. The infrared dynamics on the worldvolume of the twodimensional surface operator is described by a holomorphic function called the twisted superpotential. Using localization techniques, we obtain the instanton partition function, and thereby the twisted superpotential of these theories. This involves taking a suitable orbifold of the original action without the surface operator. Imposing constraints from Sduality, we obtain a modular anomaly equation for the coefficients that appear in the mass expansion of the twisted superpotential. Solving the modular anomaly equation at each order, and comparing with the results obtained from localization, we resum the twisted superpotential in a mass series, whose coefficient functions depend on (quasi) modular forms and elliptic functions of the bare coupling constant and the continuous (complex) parameters that describe the surface operator. We further show that our results for monodromy defects in the fourdimensional theory, match the effective twisted superpotential that describes the infrared properties of certain two dimensional sigma models couples to N=2* gauge theories. This provides strong evidence for the proposed duality between the two descriptions. 