Kac-Moody Extensions of 3-Algebras and M2-branes

Abstract

We study the 3-algebraic structure involved in the recently shown M2-branes worldvolume gauge theories. We first extend an arbitrary finite dimensional 3-algebra into an infinite dimensional 3-algebra by adding a mode number to each generator. A unique central charge in the algebra of gauge transformations appears naturally in this extension. We present an infinite dimensional extended 3-algebra with a general metric and also a different extension with a Lorentzian metric. We then study ordinary finite dimensional 3-algebras with different signatures of the metric, focusing on the cases with a negative eigenvalue and the cases with a zero eigenvalue. In the latter cases we present a new algebra, whose corresponding theory is a decoupled abelian gauge theory together with a free theory with global gauge symmetry, and there is no negative kinetic term from this algebra.

v3: Appendix A added proving an identity; minor corrections and typos fixed. v4: slight refinement in section 2.1, no other change. 18 pages, Latex

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• n-ary algebras: a review with applications
• On M5-branes in N=6 Membrane Action
• Multiple M2-branes and Generalized 3-Lie algebras
• One-loop corrections to Bagger-Lambert theory
• An N=1 Superfield Action for M2 branes
• Multiple Membranes from Gauged Supergravity
• Ternary Virasoro - Witt Algebra
• From N M2's to N D2's
• Scaling limit of N=6 superconformal Chern-Simons theories and Lorentzian Bagger-Lambert theories
• Coulomb Branch of the Lorentzian Three Algebra Theory
• Lorentzian Lie (3-)algebra and toroidal compactification of M/string theory
• Holographic Supergravity Dual to Three Dimensional N=2 Gauge Theory
• Topological branes, p-algebras and generalized Nahm equations
• Truncated Nambu-Poisson Bracket and Entropy Formula for Multiple Membranes
• M-Theory Superalgebra From Multiple Membranes
• $w_{\infty }$ 3-algebra
• Towards Holographic Gravity Dual of N=1Superconformal Chern-Simons Gauge Theory
• Classical and Quantal Ternary Algebras
• Multiple M2-Branes and Plane Waves
• A note on Quantum Aspects of Multiple Membranes
• Hydrodynamics of N=6 Superconformal Chern-Simons Theories at Strong Coupling
• D2 to M2 Procedure for D2-Brane DBI Effective Action
• N=6 Membrane Worldvolume Superalgebra
• Infinite-dimensional 3-algebra and integrable system
• M2-branes on a resolved $C_4/Z_4$
• On Nambu-Lie 3-algebra representations
• Virasoro 3-algebra from scalar densities
• Bagger-Lambert theory on an orbifold and its relation to Chern-Simons-matter theories
• Triality and Bagger-Lambert Theory
• On thermodynamics of N=6 superconformal Chern-Simons theories at strong coupling
• A note on interpreting N M2 branes
• On q-deformed infinite-dimensional n-algebra
• Quasi-Instantons and Supersymmetry Breaking
• Janus Field Theories from Non-Linear BF Theories for Multiple M2-Branes
• On $W_{1+\infty }$ 3-algebra and integrable system
• 3-Algebraic structures of the quantum Calogero-Moser model
• Generalized Virasoro algebra: left-symmetry and related algebraic and hydrodynamic properties