We study Euclidean 3D N = 2 supersymmetric gauge theories on squashed three-spheres preserving isometries SU (2) × U (1) or U (1) × U (1). We show that, when a suitable background U (1) gauge field is turned on, these squashed spheres support charged Killing spinors and therefore N = 2 supersymmetric gauge theories. We present the Lagrangian and supersymmetry rules for general gauge theories. The partition functions are computed using localization principle, and are expressed as integrals over Coulomb branch. For the squashed sphere with U (1) × U (1) isometry, its measure and integrand are identified with the building blocks of structure constants in Liouville or Toda conformal field theories with b = 1.

We extend the formula for partition functions of N = 2 superconformal gauge theories on S 3 obtained recently by Kapustin, Willett and Yaakov, to incorporate matter fields with arbitrary R-charge assignments. We use the result to check that the self-mirror property of N = 4 SQED with two electron hypermultiplets is preserved under a certain mass deformation which breaks the supersymmetry to N = 2.

Abstract:We compute exactly the partition function of two dimensional N = (2, 2) gauge theories on S 2 and show that it admits two dual descriptions: either as an integral over the Coulomb branch or as a sum over vortex and anti-vortex excitations on the Higgs branches of the theory. We further demonstrate that correlation functions in two dimensional Liouville/Toda CFT compute the S 2 partition function for a class of N = (2, 2) gauge theories, thereby uncovering novel modular properties in two dimensional gauge theories. Some of these gauge theories flow in the infrared to Calabi-Yau sigma modelssuch as the conifold -and the topology changing flop transition is realized as crossing symmetry in Liouville/Toda CFT. Evidence for Seiberg duality in two dimensions is exhibited by demonstrating that the partition function of conjectured Seiberg dual pairs are the same.

Abstract:We explore further our recent generalization of the N = 4 superconformal Chern-Simons theories of Gaiotto and Witten. We find and construct explicitly theories of enhanced N = 5 or 6 supersymmetry, especially N = 5, Sp(2M ) × O(N ) and N = 6, Sp(2M ) × O(2) theories. The U (M ) × U (N ) theory coincides with the N = 6 theory of Aharony, Bergman, Jafferis and Maldacena (ABJM). We argue that the N = 5 theory with Sp(2N ) × O(2N ) gauge group can be understood as an orientifolding of the ABJM model with U (2N ) × U (2N ) gauge group. We briefly discuss the Type IIB brane construction of the N = 5 theory and the geometry of the M-theory orbifold.

We extend the N = 4 superconformal Chern-Simons theories of Gaiotto and Witten to those with additional twisted hyper-multiplets. The new theories are generically linear quiver gauge theories with the two types of hyper-multiplets alternating between gauge groups. Our construction includes the Bagger-Lambert model of SO(4) gauge group. A family of abelian theories are identified with those proposed earlier in the context of the M-crystal model for M2-branes probing (C 2 /Z n ) 2 orbifolds. Possible extension with nonabelian BF couplings and string/M-theory realization are briefly discussed.In this work we generalize the Gaiotto-Witten's work to include twisted hyper-multiplets. Quiver theories appear naturally with two types of hyper-multiplets alternating between gauge groups where the quiver diagram is linear or circular with multiple nodes. The Bagger-Lambert theory with SO(4) gauge group appears naturally as a simplest kind of the quiver theory. Our work is partially motivated by attempt to understand the Bagger-Lambert theory with SO(4) gauge group in the context of the Gaiotto-Witten theory.The number of supersymmetries of three-dimensional superconformal Chern-Simons theories has a natural division with N = 3 [7]. It is rather straightforward to have the theories with N ≤ 3, and there has been some recent work on N = 2, 3 superconformal theories [8]. For the conformal theory of M2 branes, one needs more supersymmetry [9] and the recent works related to the BL theory and the GW theory can be regarded as concrete steps toward this direction.