Yosuke Imamura

M2 to D2 revisited

We present two derivations of the multiple D2 action from the multiple M2-brane 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 S1 which defines the reduction of M theory to IIA superstring.

Index for three dimensional superconformal field theories with general R-charge assignments

We derive a general formula of an index for three dimensional

M5-brane in three-form flux and multiple M2-branes

\mathcal{N} = 2

On the Moduli Space of Elliptic Maxwell-Chern-Simons Theories

super-conformal field theories with general R-charge assignments to chiral multiplets by using the localization method in S2 × S1 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.

N = 2 supersymmetric theories on squashed three-sphere

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 three-dimensional 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 non-commutativity 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.

Relation between the 4d superconformal index and the S^3 partition function

We analyze the moduli space of the low-energy limit of 3-dimensional N = 3 MaxwellChern-Simons theories described by circular quiver diagrams, as for 4-dimensional elliptic models. We define the theories by using D3-NS5-(k,1)5-brane systems with an arbitrary number of fivebranes. The supersymmetry is expected to be enhanced to N =4i n the low-energy limit. We show that the Higgs branch, in which all bifundamental scalar fields develop vacuum expectation values, is an abelian orbifold of C 4 . We confirm that the same geometry is obtained as an M-theory dual of the brane system. We also consider theories realized by introducing more than two kinds of fivebranes, and obtain nontoric fourfolds as moduli spaces. Recently, there has been great interest in 3-dimensional superconformal field theories as theories for describing multiple M2-branes in various backgrounds. This was triggered by the proposal of a new class of 3-dimensional theories by Bagger and Lambert, 1)–3) and Gusstavson. 4),5) The model (BLG model) possesses N(d=3) =8 superconformal symmetry and is based on Lie 3-algebra. The action of the BLG model includes the structure constant f abc d of a Lie 3-algebra, which determines the form of the interactions, and a metric h ab , which appears in the coefficients of the kinetic terms. These tensors must satisfy certain conditions required by the supersymmetry invariance of the action. If these tensors satisfy the conditions, we can write down the action of a BLG model. The constraint imposed on the structure constant is called a fundamental identity. It was soon realized that the identity is very restrictive, 6) and it was proved that if we assume that the metric is positive definite and the algebra is finite dimensional, there is only one nontrivial Lie 3-algebra, 7),8) which is called an A4 algebra. The BLG model based on the A4 algebra is a SU(2)×SU(2) Chern-Simons theory with levels k and −k for each SU(2) factor. Analysis of this model showed that it describes a pair of M2-branes in certain orbifold backgrounds. 9)–11) As a theory for an arbitrary number of M2-branes, a model based on an algebra with a Lorenzian metric was proposed in Refs. 12)– 14). Because of the indefinite metric, the model includes unwanted ghost modes. Although the ghost modes can be removed by treating them as background fields satisfying classical equations of motion, 14),15) or by gauging certain symmetries and fixing them, 16),17) this procedure breaks the conformal invariance, and the theory becomes D2-brane theory 14),16),18) by the mechanism proposed in Ref. 19) unless the

= 4 Chern-Simons theories with auxiliary vector multiplets

We investigate a squashing deformation of 3d N = 2 supersymmetric theories on three-sphere, which have four supercharges. The deformation preserves SU(2)L×U(1)r isometry and all four supersymmetries. We compute the partition function and find non-trivial dependence on the squashing parameter. We also consider the large N limit of a certain class of quiver gauge theories which have free energy of order N 3/2 , and show that the free energy on the squashed sphere differs from that on round sphere by a certain factor depending only on the squashing parameter.

Perturbative partition function for a squashed S5

A relation between the 4d superconformal index and the S3 partition function is studied with focus on the 4d and 3d actions used in localization. In the case of vanishing Chern-Simons levels and round S3 we explicitly show that the 3d action is obtained from the4d action by dimensional reduction up to terms which do not affect the exact results. By combining this fact and a recent proposal concerning a squashing of S3 and SU(2) Wilson line, we obtain a formula which gives the partition function depending on the Weyl weight of chiral multiplets, real mass parameters, FI parameters, and a squashing parameter as a limit of the index of a parent 4d theory.

Chern-Simons Theories and Wrapped M-Branes in Their Gravity Duals

We investigate a class of quiver-type Chern-Simons gauge theories with some Chern-Simons couplings vanishing. The vanishing of the couplings means that the corresponding vector fields are auxiliary fields. We show that these theories possess = 4 supersymmetry by writing down the actions and the supersymmetry transformation in terms of component fields in manifestly Spin(4) covariant form.

Quiver Chern-Simons theories and crystals

We compute the index of 6d N = (1,0) theories on S 5 × R containing vector and hypermultiplets. We only consider the perturbative sector without instantons. By compactifying R to S 1 with a twisted boundary condition and taking the small radius limit, we derive the perturbative partition function on a squashed S 5 . The 1-loop partition function is represented in a simple form with the triple sine function.