Codimension two defects of the (0, 2) six dimensional theory X[j] have played an important role in understanding dualities for certain N = 2 SCFTs in four dimensions. These defects are typically understood by their behaviour under various dimensional reduction schemes. In their various guises, the defects admit partial descriptions in terms of singularities of Hitchin systems, Nahm boundary conditions or Toda operators. Here, a uniform dictionary between these descriptions is given for a large class of such defects in X[j], j ∈ A, D, E.

The role played by the Euler anomaly in the dictionary relating sphere partition functions of four dimensional theories of class $\mathcal{S}$ and two dimensional nonrational CFTs is clarified. On the two dimensional side, this involves a careful treatment of scale factors in Liouville/Toda correlators. Using ideas from tinkertoy constructions for Gaiotto duality, a framework is proposed for evaluating these scale factors. The representation theory of Weyl groups plays a critical role in this framework.Comment: 55 pages, 16 figures; v2:fixed referencing & typos ; v3: argument about scale factors in Liouville/Toda now phrased in terms of stripped correlators, leading to a sharper conjecture (earlier version had some inaccurate statements). Presentation improved, typos fixed, refs added. I thank the anonymous referee for comments. Version accepted for publication in JHE

We study mass deformations of certain three dimensional N 4 Superconformal Field Theories (SCFTs) that have come to be called T ρ [G] theories. These are associated to tame defects of the six dimensional (0, 2) SCFT X[j] for j A, D, E. We describe these deformations using a refined version of the theory of sheets, a subject of interest in Geometric Representation Theory. In mathematical terms, we parameterize local mass-like deformations of the tamely ramified Hitchin integrable system and identify the subset of the deformations that do admit an interpretation as a mass deformation for the theories under consideration. We point out the existence of non-trivial Rigid SCFTs among these theories. We classify the Rigid theories within this set of SCFTs and give a description of their Higgs and Coulomb branches. We then study the implications for the endpoints of RG flows triggered by mass deformations in these 3d N 4 theories. Finally, we discuss connections with the recently proposed idea of Symplectic Duality and describe some conjectures about its action.

This note announces results on the relations between the approach of Beilinson and Drinfeld to the geometric Langlands correspondence based on conformal field theory, the approach of Kapustin and Witten based on N = 4 SYM, and the AGT-correspondence. The geometric Langlands correspondence is described as the Nekrasov-Shatashvili limit of a generalisation of the AGT-correspondence in the presence of surface operators. Following the approaches of Kapustin -Witten and Nekrasov -Witten we interpret some aspects of the resulting picture using an effective description in terms of two-dimensional sigma models having Hitchin's moduli spaces as target-manifolds.

We pursue the symplectic description of toric Kähler manifolds. There exists a general local classification of metrics on toric Kähler manifolds equipped with Hamiltonian two-forms due to Apostolov, Calderbank and Gauduchon(ACG). We derive the symplectic potential for these metrics.Using a method due to Abreu, we relate the symplectic potential to the canonical potential written by Guillemin. This enables us to recover the moment polytope associated with metrics and we thus obtain global information about the metric. We illustrate these general considerations by focusing on six-dimensional Ricci-flat metrics and obtain Ricci-flat metrics associated with real cones over L pqr and Y pq manifolds. The metrics associated with cones over Y pq manifolds turn out to be partially resolved with two blow-up parameters taking special (non-zero)values. For a fixed Y pq manifold, we find explicit metrics for several inequivalent blow-ups parametrised by a natural number k in the range 0 < k < p. We also show that all known examples of resolved metrics such as the resolved conifold and the resolution of C 3 /Z 3 also fit the ACG classification.Abreu had a simple suggestion to obtain Einstein/extremal metrics from the canonical one [6]. Adapting a method due to Calabi in the complex context [7], Abreu modified the canonical symplectic potential by adding a 'function' to it as follows:where h(P ) is non-singular in the interior as well as the boundary of the polytope.We will refer to h(P ) as the Abreu function in this paper. The Abreu function is determined by requiring that the new metric has the required property such as extremality. For instance, the differential equation for h(P ) is the analog of the Monge-Ampère equation that appears when one imposes Ricci-flatness on the Kähler potential [8]. The function h(P ) has been determined in only a small number of examples [6, 9, 10]. However, there have been a recent attempt to obtain the function numerically [11]. This paper focuses on a special sub-class of toric Kähler manifolds, those that admit a Hamiltonian 2-form. For Kähler manifolds that admit such a 2-form (and possibly non-toric), there exists a classification of these metrics due to Apostolov, Calderbank and Gauduchon(ACG) [12]. The main merit of such metrics is that it replaces a PDE in m variables that one needs to solve to obtain the symplectic potential by an ODE's in m functions of one-variable in the best of situations.We obtain the symplectic potential for these metrics and find that it can be easily written in the form given in Eqn.(1.3). Then the associated polytope is easily recovered. We find that all known examples of resolved metrics in six-dimensions admit a Hamiltonian 2-form and add a new infinite family of partially resolved spaces to the list of known examples. Another application of these methods is in the context of the AdS-CFT correspondence which relates four-dimensional conformal field theories with type IIB string theory on AdS 5 × X 5 , where X 5 is a compact five-dimensional Sasaki-Einstein...