We derive a general formula of an index for three dimensional N = 2 superconformal field theories with general R-charge assignments to chiral multiplets by using the localization method in S 2 ×S 1 background. As examples we compute the index for theories in a few mirror pairs, and confirm the agreement of the indices in each mirror pair. *

Abstract:We investigate the Bagger-Lambert-Gustavsson model associated with the Nambu-Poisson algebra as a theory describing a single M5-brane. We argue that the model is a gauge theory associated with the volume-preserving diffeomorphism in the threedimensional internal space. We derive gauge transformations, actions, supersymmetry transformations, and equations of motions in terms of six-dimensional fields. The equations of motions are written in gauge-covariant form, and the equations for tensor fields have manifest self-dual structure. We demonstrate that the double dimensional reduction of the model reproduces the non-commutative U(1) gauge theory on a D4-brane with a small noncommutativity parameter. We establish relations between parameters in the BLG model and those in M-theory. This shows that the model describes an M5-brane in a large C-field background.

We argue a squashing deformation of 3D N = 2 supersymmetric theories on round S 3 , which preserves SU (2) L × U (1)r isometry and the supersymmetries. We stress that the choice of a supersymmetry for the localization affects whether the partition function calculated using the supersymmetry depends on the squashing parameter or not.

We present two derivations of the multiple D2 action from the multiple M2brane model proposed by Bagger-Lambert and Gustavsson. The first one is to start from Lie 3-algebra associated with given (arbitrary) Lie algebra. The Lie 3-algebra metric is not positive definite but the zero-norm generators merely correspond to Lagrange multipliers. Following the work of Mukhi and Papageorgakis, we derive D2-brane action from the model by giving a variable a vacuum expectation value. The second derivation is based on the correspondence between M2 and M5. We compactify one dimension and wind M5-brane along this direction. This leads to a noncommutative D4 action. Multiple D2 action is then obtained by suitably choosing the non-commutative parameter on the two-torus. It also implies a natural interpretation to the extra generator in Lie 3-algebra, namely the winding of M5 world volume around S 1 which defines the reduction of M theory to II A superstring.