𝒩 = 4 Superconformal Chern-Simons theories with hyper and twisted hyper multiplets

Abstract

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.

Three-dimensionalN=6superconformal field theories and their membrane dynamics

“…Instead, we argue that they are diagonal if all the matrices φ A φ B commute with each other. 7 The coordinates of C 4 /Z k can then be understood as eigenvalues of these matrices, similarly to what occurs for D-branes [34,35].…”

“…For a membrane theory, the rescaled limit 7 We denote with a bar the transposed complex conjugate matrix, (φ A ) i i ′ ≡ (φ * A ) i ′ i , reserving the dagger for the harmonic oscillator's creation operators that we shall introduce later. See the appendix for more details on our conventions.…”

“…A flurry of activity has followed this observation. In particular, the Penrose limit of the AdS 4 × CP 3 background was considered in [21,22,23] 1 Superconformal extensions of Chern-Simons theory with various amounts of supersymmetry have been also considered recently in [4,5,6,7]. 2 The supersymmetries of this action have been checked carefully in [8,9], while its relation to the construction in terms of three-algebras has been described in [10,11].…”

Instanton effects in rank deformed superconformal Chern-Simons theories from topological strings

“…In the subsequent works [20][21][22][23][24], the investigation started with a special class of N = 3 theories: the circular quiver superconformal Chern-Simons theories with unitary gauge groups. 2 It was shown in [30][31][32][33][34] that, for the circular quiver, the theory enjoys the supersymmetry N = 4 when the Chern-Simons level k a associated to each vertex is expressed as k a = k(s a − s a−1 )/2 with s a = ±1. Namely, the full information of the N = 4 theories is encoded in the list of s a = ±1, and we may refer to the theory with {s a } = {+1, +1, · · · , +1 as (p 1 , q 1 , p 2 , q 2 , · · · ) model [21].…”

Deconstructing supersymmetricSmatrices inD≤2+1

“…Further details and additional properties of the ρ tensors relevant to proving off-shell SUSY invariance are provided in appendix A. Focusing now on on-shell quantities 4 , it is convenient to introduce a three dimensional polarization vector…”