We compute the supersymmetric partition function of an N = 1 chiral multiplet coupled to an external Abelian gauge field on complex manifolds with T 2 ×S 2 topology. The result is locally holomorphic in the complex structure moduli of T 2 × S 2 . This computation illustrates in a simple example some recently obtained constraints on the parameter dependence of supersymmetric partition functions.We also devise a simple method to compute the chiral multiplet partition function on any four-manifold M 4 preserving two supercharges of opposite chiralities, via supersymmetric localization. In the case of M 4 = S 3 × S 1 , we provide a path integral derivation of the previously known result, the elliptic gamma function, which emphasizes its dependence on the S 3 × S 1 complex structure moduli.

We construct the type IIB supergravity solutions describing D3-branes ending on 5-branes, in the near-horizon limit of the D3-branes. Our solutions are holographically dual to the 4d N = 4 SU (N ) super-Yang-Mills (SYM) theory on a half-line, at large N and large 't Hooft coupling, with various boundary conditions that preserve half of the supersymmetry. The solutions are limiting cases of the general solutions with the same symmetries constructed in 2007 by D'Hoker, Estes and Gutperle. The classification of our solutions matches exactly with the general classification of boundary conditions for D3-branes ending on 5-branes by Gaiotto and Witten. We use the gravity duals to compute the one-point functions of some chiral operators in the N = 4 SYM theory on a half-line at strong coupling, and find that they do not match with the expectation values of the same operators with the same boundary conditions at small 't Hooft coupling. Our solutions may also be interpreted as the gravity duals of 4d N = 4 SYM on AdS 4 , with various boundary conditions.

We study Quantum Electrodynamics in d = 3 (QED3) coupled to N f flavors of fermions. The theory flows to an IR fixed point for N f larger than some critical number N c f . For N f ≤ N c f , chiralsymmetry breaking is believed to take place. In analogy with the Wilson-Fisher description of the critical O(N ) models in d = 3, we make use of the existence of a fixed point in d = 4 − 2 to study the three-dimensional conformal theory. We compute in perturbation theory the IR dimensions of fermion bilinear and quadrilinear operators. For small N f , a quadrilinear operator can become relevant in the IR and destabilize the fixed point. Therefore, the -expansion can be used to estimate N c f . An interesting novelty compared to the O(N ) models is that the theory in d = 3 has an enhanced symmetry due to the structure of 3d spinors. We identify the operators in d = 4−2 that correspond to the additional conserved currents at d = 3 and compute their infrared dimensions.

We study three dimensional N = 2 supersymmetric QCD theories with O(N c ) gauge groups and with N f chiral multiplets in the vector representation. We argue that for N f < N c − 2 there is a runaway potential on the moduli space and no vacuum. For N f ≥ N c − 2 there is a moduli space also in the quantum theory, and for N f ≥ N c − 1 there is a superconformal fixed point at the origin of this moduli space that has a dual description as the low-energy fixed point of an O(N f − N c + 2) gauge theory. We test this duality in various ways; in some cases the duality for an O(2) gauge theory may be related to the known duality for U (1) gauge theories. We also discuss real mass deformations, which allow to connect theories with a different Chern-Simons level. This allows us to connect our duality with the known duality in O(N c ) theories with a Chern-Simons term of level k, where the dual gauge group is given by O(N f + |k| − N c + 2).

The presence of a boundary (or defect) in a conformal field theory allows one to generalize the notion of an exactly marginal deformation. Without a boundary, one must find an operator of protected scaling dimension ∆ equal to the space-time dimension d of the conformal field theory, while with a boundary, as long as the operator dimension is protected, one can make up for the difference d − ∆ by including a factor z ∆−d in the deformation where z is the distance from the boundary. This coordinate dependence does not lead to a reduction in the underlying SO(d, 1) global conformal symmetry group of the boundary conformal field theory. We show that such terms can arise from boundary flows in interacting field theories. Ultimately, we would like to be able to characterize what types of boundary conformal field theories live on the orbits of such deformations. As a first step, we consider a free scalar with a conformally invariant mass term z −2 φ 2 , and a fermion with a similar mass. We find a connection to double trace deformations in the AdS/CFT literature.