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.

We prove a recent conjecture that the partition function of N = (2, 2) gauge theories on the two-sphere which flow to Calabi-Yau sigma models in the infrared computes the exact Kähler potential on the quantum Kähler moduli space of the corresponding Calabi-Yau. This establishes the two-sphere partition function as a new method of computation of worldsheet instantons and Gromov-Witten invariants. We also calculate the exact two-sphere partition function for N = (2, 2) Landau-Ginzburg models with an arbitrary twisted su-

Instantons and W-bosons in 5d maximally supersymmetric Yang-Mills theory arise from a circle compactification of the 6d (2,0) theory as Kaluza-Klein modes and winding self-dual strings, respectively. We study an index which counts BPS instantons with electric charges in Coulomb and symmetric phases. We first prove the existence of unique threshold bound state of (noncommutative) U (1) instantons for any instanton number, and also show that charged instantons in the Coulomb phase correctly give the degeneracy of SU (2) self-dual strings. By studying SU (N ) self-dual strings in the Coulomb phase, we find novel momentum-carrying degrees on the worldsheet. The total number of these degrees equals the anomaly coefficient of SU (N ) (2,0) theory. We finally show that our index can be used to study the symmetric phase of this theory, and provide an interpretation as the superconformal index of the sigma model on instanton moduli space.

We present a new duality between the F-terms of supersymmetric field theories defined in two-and four-dimensions respectively. The duality relates N = 2 supersymmetric gauge theories in four dimensions, deformed by an Ω-background in one plane, to N = (2, 2) gauged linear σ-models in two dimensions. On the four dimensional side, our main example is N = 2 SQCD with gauge group G = SU(L) and N F = 2L fundamental flavours. Using ideas of Nekrasov and Shatashvili, we argue that the Coulomb branch of this theory provides a quantization of the classical Heisenberg SL(2) spin chain. Agreement with the standard quantization via the Algebraic Bethe Ansatz implies the existence of an isomorphism between the chiral ring of the 4d theory and that of a certain two-dimensional theory. The latter can be understood as the worldvolume theory on a surface operator/vortex string probing the Higgs branch of the same 4d theory. We check the proposed duality by explicit calculation at low orders in the instanton expansion. One striking consequence is that the SeibergWitten solution of the 4d theory is captured by a one-loop computation in two dimensions.The duality also has interesting connections with the AGT conjecture, matrix models and topological string theory where it corresponds to a refined version of the geometric transition.