On the Lie-algebraic origin of metric 3-algebras

Abstract

Since the pioneering work of Bagger-Lambert and Gustavsson, there has been a proliferation of three-dimensional superconformal Chern-Simons theories whose main ingredient is a metric 3-algebra. On the other hand, many of these theories have been shown to allow for a reformulation in terms of standard gauge theory coupled to matter, where the 3-algebra does not appear explicitly. In this paper we reconcile these two sets of results by pointing out the Lie-algebraic origin of some metric 3-algebras, including those which have already appeared in three-dimensional superconformal Chern-Simons theories. More precisely, we show that the real 3-algebras of Cherkis-Saemann, which include the metric Lie 3-algebras as a special case, and the hermitian 3-algebras of Bagger-Lambert can be constructed from pairs consisting of a metric real Lie algebra and a faithful (real or complex, respectively) unitary representation. This construction generalises and we will see how to construct many kinds of metric 3-algebras from pairs consisting of a real metric Lie algebra and a faithful (real, complex or quaternionic) unitary representation. In the real case, these 3-algebras are precisely the Cherkis-Saemann algebras, which are then completely characterised in terms of this data. In the complex and quaternionic cases, they constitute generalisations of the Bagger-Lambert hermitian 3-algebras and anti-Lie triple systems, respectively, which underlie N=6 and N=5 superconformal Chern-Simons theories, respectively. In the process we rederive the relation between certain types of complex 3-algebras and metric Lie superalgebras.

arXiv comment29 pages (v4: really final version to appear in CMP. Example 7 has been improved.)

References in corpus (33)

• N=6 superconformal Chern-Simons-matter theories, M2-branes and their gravity duals
• Gauge Symmetry and Supersymmetry of Multiple M2-Branes
• Algebraic structures on parallel M2-branes
• Modeling Multiple M2's
• Fractional M2-branes
• Janus Configurations, Chern-Simons Couplings, And The Theta-Angle in N=4 Super Yang-Mills Theory
• Superconformal Chern-Simons Theories and AdS_4/CFT_3 Correspondence
• N=5,6 Superconformal Chern-Simons Theories and M2-branes on Orbifolds
• Notes on Superconformal Chern-Simons-Matter Theories
• Three-Algebras and N=6 Chern-Simons Gauge Theories
• Superconformal Chern-Simons Theories
• Comments on the Bagger-Lambert theory and multiple M2-branes
• N=4 Superconformal Chern-Simons Theories with Hyper and Twisted Hyper Multiplets
• Constraining Maximally Supersymmetric Membrane Actions
• M2 to D2 revisited
• M2-branes, 3-Lie Algebras and Plucker relations
• A simple class of N=3 gauge/gravity duals
• Classification of N=6 superconformal theories of ABJM type
• Multiple M2-branes and Generalized 3-Lie algebras
• One-loop corrections to Bagger-Lambert theory
• Bagger-Lambert Theory for General Lie Algebras
• Supersymmetry and homogeneity of M-theory backgrounds
• Pluecker-type relations for orthogonal planes
• An N=1 Superfield Action for M2 branes
• Multiple M2-branes and the Embedding Tensor
• Multiple Membranes from Gauged Supergravity
• Superconformal Chern-Simons Theories and the Squashed Seven Sphere
• Metric Lie 3-algebras in Bagger-Lambert theory
• Prolongations of Lie algebras and applications
• On Superspace Actions for Multiple M2-Branes, Metric 3-Algebras and their Classification
• Lorentzian Lie 3-algebras and their Bagger-Lambert moduli space
• Octonions, G_2 and generalized Lie 3-algebras
• Construction of Lie Superalgebras from Triple Product Systems

Cited by in corpus (61)

• n-ary algebras: a review with applications
• Classification of N=6 superconformal theories of ABJM type
• Multiple Membranes in M-theory
• On 3d N=8 Lorentzian BLG theory as a scaling limit of 3d superconformal N=6 ABJM theory
• Infrared Stability of N=2 Chern-Simons Matter Theories
• Lorentzian Lie (3-)algebra and toroidal compactification of M/string theory
• Marginal Deformations and 3-Algebra Structures
• Six-Dimensional (1,0) Superconformal Models and Higher Gauge Theory
• Three-algebras, triple systems and 3-graded Lie superalgebras
• On Superspace Actions for Multiple M2-Branes, Metric 3-Algebras and their Classification
• Deformations of 3-algebras
• Symplectic, product and complex structures on 3-Lie algebras
• M-brane Models from Non-Abelian Gerbes
• Differential Cohomotopy implies intersecting brane observables via configuration spaces and chord diagrams
• Three-Algebras in N = 5, 6 Superconformal Chern-Simons Theories: Representations and Relations
• Symplectic Three-Algebra Unifying N=5,6 Superconformal Chern-Simons-Matter Theories
• Monopoles, three-algebras and ABJM theories with $\mathbb{N}=5,6,8$ supersymmetry
• Quantized Nambu-Poisson Manifolds in a 3-Lie Algebra Reduced Model
• Freudenthal Gauge Theory
• Octonions, G_2 and generalized Lie 3-algebras
• Superpotentials for superconformal Chern-Simons theories from representation theory
• Lie 2-algebra models
• Nambu Bracket for M Theory
• PT symmetry, Cartan decompositions, Lie triple systems and Krein space related Clifford algebras
• Model of M-theory with Eleven Matrices
• The ABJM Model is a Higher Gauge Theory
• Topological M-Strings and Supergroup WZW Models
• Lessons from M2's and Hopes for M5's
• Three lectures on 3-algebras
• Quadratic color Hom-Lie algebras
• Simplicity in the Faulkner construction
• Constructing Generalized Self-Dual Strings
• Supersymmetric solutions from N=5 gauged supergravity
• Supersymmetry and DLCQ Limit of Lie 3-algebra Model of M-theory
• $\delta$-derivations of n-ary algebras
• Superspace Formulation in a Three-Algebra Approach to D=3, N=4,5 Superconformal Chern-Simons Matter Theories
• Unifying N=5 and N=6
• $3$-post-Lie algebras and relative Rota-Baxter operators of nonzero weight on $3$-Lie algebras
• Supersymmetric solutions from N=6 gauged supergravity
• A New Three-Algebra Representation for the {\cal N}=6 su(N)Xsu(N) Superconformal Chern-Simons Theory
• Squashing Gravity Dual of N=6 Superconformal Chern-Simons Gauge Theory
• Aspects of U-duality in BLG models with Lorentzian metric 3-algebras
• N=5 three-algebras and 5-graded Lie superalgebras
• Three-Algebra BFSS Matrix Theory
• Remark About Scaling Limit of ABJ Theory
• Extension of IIB Matrix Model by Three-Algebra
• The Background Field Method for N = 2, d3 Super Chern-Simons-Matter Theories
• Mock Modular Index of M2-M5 Brane System
• Matrix Quantization of Classical Nambu Brackets and Super $p$-Branes
• Four-algebraic extension of the IIB matrix model
• Deformations and cohomologies of embedding tensors on 3-Lie algebras
• Superconformal Chern-Simons theories beyond leading order
• Lie n-algebras and cohomologies of relative Rota-Baxter operators on n-Lie algebras
• Hypermatrix factors for string and membrane junctions
• k-Leibniz algebras from lower order ones: from Lie triple to Lie l-ple systems
• Reduced Chern-Simons Quiver Theories and Cohomological 3-Algebra Models
• A class of hermitian generalized Jordan triple systems and Chern-Simons gauge theory
• Stability and rigidity of 3-Lie algebra morphisms
• 3-Leibniz bialgebra in $N=6$ Chern-Simons gauge theories, multiple M2 to D2 branes and vice versa
• Cohomologies of 3-Lie algebras with derivations
• Metric 3-Leibniz algebras and M2-branes