We find a sixteen supersymmetric mass-deformed Bagger-Lambert theory with $SO(4)\times SO(4)$ global R-symmetry. The R-charge plays the `non-central' term in the superalgebra. This theory has one symmetric vacuum and two in-equivalent broken sectors of vacua. Each sector of the broken symmetry has the SO(4) geometry. We find the 1/2 BPS domain walls connecting the symmetric phase and any broken phase, and 1/4 BPS supertube-like objects, which may appear as anyonic q-balls in the symmetric phase or vortices in the broken phase. We also discuss mass deformations which reduces the number of supersymmetries.
It is known that the M2-brane worldvolume superalgebra includes two p-form central charges that encode the M-theory intersections involving M2-branes. In this paper we show by explicit computation that the Bagger-Lambert Lagrangian realizes the M2-brane superalgebra, including also the central extensions. Solitons of the Bagger-Lambert theory, that are interpreted as worldvolume realizations of intersecting branes, are shown to saturate a BPS-bound given in terms of the corresponding central charge.
In this note, we discuss membrane scattering from the three dimensional N=8 superconformal theory with SO(8) global symmetry constructed by Bagger-Lambert and Gustavsson. We discuss whether the one loop effective potential consistently reproduces the Newton potential of membranes moving in an eleven dimensional orbifold space.
Among newly discovered M2, M5 objects in the Bagger-Lambert-Gustavsson theory, our interest is about half BPS vortices which are covariantly holomorphic curves in transverse coordinates. We restrict ourselves to the case where the global symmetry is broken to so(2) x so(2)x so(4) for the mass deformed Bagger-Lambert theory. A localized object with finite energy exists in this theory where the mass parameter supports regularity. It is time independent but carries angular momentum coming solely from the gauge potential by which the energy is bounded below.
We rewrite the Bagger-Lambert action for any Lie 3-algebra as a standard Chern- Simons action coupled to matter. We use this action to compute self-energies and vertex corrections at one-loop order. Non-renormalization of the coupling constant comes out as a direct consequence of the Lie 3-algebra structure underlying the Lie algebra.
In this paper we analyse the Bagger-Lambert-Gustavsson (BLG) theory in $\mathcal{N} =1$ superspace. Furthermore, we will construct the BRST transformations for this theory. These BRST transformations will be integrated out to obtain the finite field dependent version of BRST (FFBRST) transformations. We will also analyse the effect of the FFBRST transformations on the effective action. We will thus show that the FFBRST transformations can be used to relate generating functionals of the BLG theory in two different gauges.
We reformulate the Bagger-Lambert-Gustavsson model using an N=8 superspace, thus making the full supersymmetry manifest. The formulation is based on appropriate "pure spinor wave functions" for the Chern-Simons and matter multiplets. The Lagrangian has an extremely simple structure, essentially containing a Chern-Simons-like term for the gauge field wave function and a minimally coupled matter wave function. No higher order interactions than cubic are present. The consistency of the setup relies on an interplay between the algebraic structures of the 3-algebra and of the pure spinors.
We continue our study of BPS equations and supersymmetric configurations in the Bagger-Lambert theory. The superalgebra allows three different types of central extensions which correspond to compounds of various M-theory objects: M2-branes, M5-branes, gravity waves and Kaluza-Klein monopoles which intersect or have overlaps with the M2-branes whose dynamics is given by the Bagger-Lambert action. As elementary objects they are all 1/2-BPS, and multiple intersections of $n$-branes generically break the supersymmetry into $1/2^n$, as it is well known. But a particular composite of M-branes can pr…
We study the structure of the gauge sector of the Bagger-Lambert-Gustavsson (BLG) theory in the form proposed by van Raamsdonk, adapted to 3D, N=1 superspace. By using the novel Higgs mechanism proposed by Mukhi and Papageorgakis, we derive the manifestly N=1 supersymmetric higher-order terms (beyond the supersymmetric Yang-Mills action) that follow from the BLG theory in its expansion with respect to the inverse gauge coupling constant squared. We find that all those terms have at least one anti-commutator of the super-YM field strength superfields as a factor, and thus are reducible to the S…
By compactifying the Bagger-Lambert-Gustavsson model on $\mathbb{R}^{1,1} \times S^1$, we obtain a new two dimensional massless field theory with dynamical fields valued in the Lie 3-algebra $\mathcal{A}$ coupled with an SO(1,1) scalar and vector field which are valued in the set $End(\mathcal{A})$ of the endomorphisms of the Lie 3-algebra. In the limit $g_{BLG} \to \infty$ the theory reduces to a supersymmetric Lie 3-valued generalization of the Green-Schwarz superstring in the light-cone gauge.
We classify, in a group theoretical manner, the BPS configurations in the multiple M2-brane theory recently proposed by Bagger and Lambert. We present three types of BPS equations preserving various fractions of supersymmetries: in the first type we have constant fields and the interactions are purely algebraic in nature; in the second type the equations are invariant under spatial rotation SO(2), and the fields can be time-dependent; in the third class the equations are invariant under boost SO(1,1) and provide the eleven-dimensional generalizations of the Nahm equations. The BPS equations fo…
We present two alternative field contents for Bagger-Lambert theory, based on the triality of SO(8). The first content is (φ_{A a}, χ_{\dot A a} ; A_\m{}^{a b}), where the bosonic field φis in the 8_S of SO(8) instead of the 8_V as in the original Bagger-Lambert formulation. The second field content is (φ_{\dot A a}, χ^I{}_a ; A_\m{}^{a b}), where the bosonic field φand the fermionic field χare respectively in the 8_C and 8_V of SO(8). In both of these field contents, the bosonic potentials are positive definite, as desired. Moreover, these bosonic potentials can be unified by the triality of …
We consider how to take an orbifold reduction for the multiple M2-brane theory recently proposed by Bagger and Lambert, and discuss its relation to Chern-Simons theories. Starting from the infinite dimensional 3-algebra realized as the Nambu bracket on a 3-torus, we first suggest an orbifolding prescription for various fields. Then we introduce a second truncation, which effectively reduces the internal space to a 2-torus. Eventually one obtains a large-N limit of Chern-Simons gauge theories coupled to matter fields. We consider an abelian orbifold C^4/Z_n, and illustrate how one can arrive at…
We study the SO(8) superconformal theory proposed recently by Bagger and Lambert as a possible worldvolume theory for multiple M2-branes. For their explicit example with gauge group SO(4), we rewrite the theory (originally formulated in terms of a three-algebra) as an ordinary SU(2) \times SU(2) gauge theory with bifundamental matter. In this description, the parity invariance of the theory, required for a proper description of M2-branes, is clarified. We describe the subspace of scalar field configurations on which the potential vanishes, and note that this does not coincide with the moduli s…
In this paper we will discuss the effect of a having a boundary on the supersymmetric invariance and gauge invariance of the Bagger-Lambert-Gustavsson (BLG) Theory. We will show that even though the supersymmetry and gauge invariance of the original BLG theory is broken due to the presence of a boundary, it restored by the addition of suitable boundary terms. In fact, to achieve the gauge invariance of this theory, we will have to introduce new boundary degrees of freedom. The boundary theory obeyed by these new boundary degrees of freedom will be shown to be a generalization of the gauged Wes…
In arXiv:0806.0363 the worldvolume superalgebra of the $\mathcal{N}=8$ Bagger-Lambert theory was calculated. In this paper we derive the general form for the worldvolume superalgebra of the $\mathcal{N}=6$ Bagger-Lambert theory. For a particular choice of three-algebra we derive the superalgebra of the ABJM theory. We interpret the associated central charges in terms of BPS brane configurations. In particular we find the central charge corresponding to the energy bound of the BPS fuzzy-funnel configuration of the ABJM theory. We also derive general expressions for the BPS equations of the $\ma…
We show that the N=8 superconformal Bagger-Lambert theory based on the Lorentzian 3-algebra can be derived by taking a certain scaling limit of the recently proposed N=6 superconformal U(N)xU(N) Chern-Simons-matter theories at level (k, -k). The scaling limit (and Inonu-Wigner contraction) is to scale the trace part of the bifundamental fields as X_0 -> λ^{-1} X_0 and an axial combination of the two gauge fields as B_μ -> λB_μ. Simultaneously we scale the level as k -> λ^{-1} k and then take λ-> 0 limit. Interestingly the same constraint equation \partial^2 X_0=0 is derived by imposing finiten…
We construct the totally antisymmetric structure constants f^{ABCD} of a 3-algebra with a Lorentzian bi-invariant metric starting from an arbitrary semi-simple Lie algebra. The structure constants f^{ABCD} can be used to write down a maximally superconformal 3d theory that incorporates the expected degrees of freedom of multiple M2 branes, including the "center-of-mass" mode described by free scalar and fermion fields. The gauge field sector reduces to a three dimensional BF term, which underlies the gauge symmetry of the theory. We comment on the issue of unitarity of the quantum theory, whic…
We classify Lie 3-algebras possessing an invariant lorentzian inner product. The indecomposable objects are in one-to-one correspondence with compact real forms of metric semisimple Lie algebras. We analyse the moduli space of classical vacua of the Bagger-Lambert theory corresponding to these Lie 3-algebras. We establish a one-to-one correspondence between one branch of the moduli space and compact riemannian symmetric spaces. We analyse the asymptotic behaviour of the moduli space and identify a large class of models with moduli branches exhibiting the desired N^{3/2} behaviour.
We prove a structure theorem for finite-dimensional indefinite-signature metric 3-Lie algebras admitting a maximally isotropic centre. This algebraic condition indicates that all the negative-norm states in the associated Bagger-Lambert theory can be consistently decoupled from the physical Hilbert space. As an immediate application of the theorem, new examples beyond index 2 are constructed. The lagrangian for the Bagger-Lambert theory based on a general physically admissible 3-Lie algebra of this kind is obtained. Following an expansion around a suitable vacuum, the precise relationship betw…
We recast physical properties of the Bagger-Lambert theory, such as shift-symmetry and decoupling of ghosts, the absence of scale and parity invariance, in Lie 3-algebraic terms, thus motivating the study of metric Lie 3-algebras and their Lie algebras of derivations. We prove a structure theorem for metric Lie 3-algebras in arbitrary signature showing that they can be constructed out of the simple and one-dimensional Lie 3-algebras iterating two constructions: orthogonal direct sum and a new construction called a double extension, by analogy with the similar construction for Lie algebras. We …
We review various generalizations of the notion of Lie algebras, in particular those appearing in the recently proposed Bagger-Lambert-Gustavsson model, and study their interrelations. We find that Filippov's n-Lie algebras are a special case of strong homotopy Lie algebras. Furthermore, we define a class of homotopy Maurer-Cartan equations, which contains both the Nahm and the Basu-Harvey equations as special cases. Finally, we show how the super Yang-Mills equations describing a Dp-brane and the Bagger-Lambert-Gustavsson equations supposedly describing M2-branes can be rewritten as homotopy …
We show that the Bagger-Lambert theory of multiple M2-branes fits into the general construction of maximally supersymmetric gauge theories using the embedding tensor technique. We apply the embedding tensor technique in order to systematically obtain the consistent gaugings of N=8 superconformal theories in 2+1 dimensions. This leads to the Bagger-Lambert theory, with the embedding tensor playing the role of the four-index antisymmetric tensor defining a ``3-algebra''. We present an alternative formulation of the theory in which the embedding tensor is replaced by a set of unrestricted scalar …
We elaborate on the suggestion made in arXiv:0806.3498 that the 3d \N=8 superconformal SU(N) Chern-Simons-matter theory of Lorentzian Bagger-Lambert-Gustavson type (L-BLG) can be obtained by a scaling limit (involving sending the level k to infinity and redefining the fields) from the \N=6 superconformal U(N)xU(N) Chern-Simons-matter theory of Aharony, Bergman, Jafferis and Maldacena (ABJM). We show that to implement such a limit in a consistent way one is to extend the ABJM theory by an abelian "ghost" multiplet. The corresponding limit at the 3-algebra level also requires extending the non-a…
Bagger-Lambert-Gustavsson theory with infinite dimensional gauge group has been suggested to describe M5-brane as a condensation of multiple M2-branes. Here we perform a topological twisting of the Bagger-Lambert-Gustavsson theory. The original SO(8) R-symmetry is broken to SO(3)XSO(5), where the former may be interpreted as a diagonal subgroup of the Euclidean M5-brane world-volume symmetry SO(6), while the latter is the isometry of the transverse five directions. Accordingly the resulting action contains an one-form and five scalars as for the bosonic dynamical fields. We further lift the ac…
The recently formulated Bagger-Lambert-Gustavsson (BLG) theory in three dimensions is described in terms of a constrained chiral superfield in light-cone superspace. We discuss the use of Superconformal symmetry to determine the form of its interactions, in complete analogy with N=4 SuperYang-Mills in four dimensions.
This paper examines various aspects of the recently proposed theory of coincident membranes by Bagger and Lambert. These include the properties of open membranes and the resulting boundary theory with an interpretation in terms of the fivebrane and marginal supersymmetric deformations of the interactions with the relation to the holographic dual.
The light-cone superspace version of the d=3, N=8 superconformal theory of Bagger, Lambert and Gustavsson (BLG) is obtained as a solution to constraints imposed by OSp(2,2|8) superalgebra. The Hamiltonian of the theory is shown to be a quadratic form of the dynamical supersymmetry transformation.
As we re-examine the known holographic N=1 supersymmetric renormalization group flow in four dimensions, we describe the mass-deformed Bagger-Lambert theory or equivalently the mass-deformed U(2) x U(2) Chern-Simons gauge theory with level k=1 or 2, that has G_2 symmetry, by adding a mass term for one of the eight adjoint superfields. We obtain a detailed correspondence between the fields of AdS_4 supergravity and composite operators of the infrared field theory in three dimensions. The geometric superpotential from an eleven dimensional viewpoint is obtained for M2-brane probe analysis.
In this paper, we extend the Gribov-Zwanziger framework accounting for the existence of Gribov copies to the mass deformed Bagger--Lambert--Gustavsson (BLG) theory in $\mathcal{N} =1 $ superspace. The restriction of the domain of integration in the Euclidean functional integral to the first Gribov horizon is implemented, by adding a non-local horizon term to the effective action. Furthermore, the soft breaking of the BRST symmetry due to horizon term is restored with the help of external sources. We compute the various Ward identities for this theory relying on the Lie 3-algebras.
We propose a generalization of the Bagger-Lambert-Gustavsson action as a candidate for the description of an arbitrary number of M2-branes. The action is formulated in terms of N=2 superfields in three dimensions and corresponds to an extension of the usual superfield formulation of Chern-Simons matter theories. Demanding gauge invariance of the resulting theory does not imply the total antisymmetry of the underlying 3-Lie algebra structure constants. We relax this condition and propose a class of examples for these generalized 3-Lie algebras. We also discuss how to associate various ordinary …
Maximally supersymmetric mass deformation of the Bagger-Lambert-Gustavsson (BLG) theory corresponds to a {non-central} extension of the d=3 N=8 Poincare superalgebra (allowed in three dimensions). We obtain its light-cone superspace formulation which has a novel feature of the dynamical supersymmetry generators being {cubic} in the kinematical ones. The mass deformation picks a quaternionic direction, which breaks the SO(8) R-symmetry down to SO(4)xSO(4). The Hamiltonian of the theory is shown to be a quadratic form of the dynamical supersymmetry transformations, to all orders in the mass para…
The Bagger--Lambert construction of N = 8 superconformal field theories (SCFT) in three dimensions is based on 3-algebras. Three groups of researchers recently realized that an arbitrary semisimple Lie algebra can be incorporated by using a suitable Lorentzian signature 3-algebra. The SU(N) case is a candidate for the SCFT describing coincident M2-branes. However, these theories contain ghost degrees of freedom, which is unsatisfactory. We modify them by gauging certain global symmetries. This eliminates the ghosts from these theories while preserving all of their desirable properties. The res…
A Lagrangian description of a maximally supersymmetric conformal field theory in three dimensions was constructed recently by Bagger and Lambert (BL). The BL theory has SO(4) gauge symmetry and contains scalar and spinor fields that transform as 4-vectors. We verify that this theory has OSp(8|4) superconformal symmetry and that it is parity conserving despite the fact that it contains a Chern--Simons term. We describe several unsuccessful attempts to construct theories of this type for other gauge groups and representations. This experience leads us to conjecture the uniqueness of the BL theor…
