Abstract:We study Ω-deformation of B-twisted gauge theories in two dimensions. As an application, we construct an Ω-deformed, topologically twisted five-dimensional maximally supersymmetric Yang-Mills theory on the product of a Riemann surface Σ and a three-manifold M , and show that when Σ is a disk, this theory is equivalent to analytically continued Chern-Simons theory on M . Based on these results, we establish a correspondence between three-dimensional N = 2 superconformal theories and analytically continued Chern-Simons theory. Furthermore, we argue that there is a mirror symmetry between Ω-deformed two-dimensional theories.

We perform a series of dimensional reductions of the 6d, N = (2, 0) SCFT on S 2 × Σ × I × S 1 down to 2d on Σ. The reductions are performed in three steps: (i) a reduction on S 1 (accompanied by a topological twist along Σ) leading to a supersymmetric Yang-Mills theory on S 2 × Σ × I, (ii) a further reduction on S 2 resulting in a complex Chern-Simons theory defined on Σ × I, with the real part of the complex Chern-Simons level being zero, and the imaginary part being proportional to the ratio of the radii of S 2 and S 1 , and (iii) a final reduction to the boundary modes of complex Chern-Simons theory with the Nahm pole boundary condition at both ends of the interval I, which gives rise to a complex Toda CFT on the Riemann surface Σ. As the reduction of the 6d theory on Σ would give rise to an N = 2 supersymmetric theory on S 2 × I × S 1 , our results imply a 4d-2d duality between four-dimensional N = 2 supersymmetric theory with boundary and two-dimensional complex Toda theory.

In this paper, we study the restoration of gauge symmetry and up to half the supersymmetry (N = (2, 0) or N = (1, 1) in two dimensions) for N = 2 non-Abelian Chern-Simons theories in the presence of a boundary. We describe the boundary action which is a supersymmetric WZW model coupled to the bulk Chern-Simons theory. Unlike the N = 1 case, higher supersymmetry (N = (2, 0)) will endow the group manifold of the WZW model with a complex structure. Therefore, the N = (2, 0) WZW model in our paper is constructed via a coset space G c /G, where G is the same as the gauge group in the Chern-Simons action.

We study N = 2 supersymmetric gauge theories on the product of a twosphere and a cylinder. We show that the low-energy dynamics of a BPS sector of such a theory is described by a quantum integrable system, with the Planck constant set by the inverse of the radius of the sphere. If the sphere is replaced with a hemisphere, then our system reduces to an integrable system of the type studied by Nekrasov and Shatashvili. In this case we establish a correspondence between the effective prepotential of the gauge theory and the Yang-Yang function of the integrable system.

We study the ground states and left-excited states of the A k−1 N = (2, 0) little string theory. Via a theorem by Atiyah [1], these sectors can be captured by a supersymmetric nonlinear sigma model on CP 1 with target space the based loop group of SU (k). The ground states, described by L 2 -cohomology classes, form modules over an affine Lie algebra, while the left-excited states, described by chiral differential operators, form modules over a toroidal Lie algebra. We also apply our results to analyze the 1/2 and 1/4 BPS sectors of the M5-brane worldvolume theory.In loving memory of See-Hong Tan.