Nonassociativity, Malcev Algebras and String Theory

Abstract

Nonassociative structures have appeared in the study of D-branes in curved backgrounds. In recent work, string theory backgrounds involving three-form fluxes, where such structures show up, have been studied in more detail. We point out that under certain assumptions these nonassociative structures coincide with nonassociative Malcev algebras which had appeared in the quantum mechanics of systems with non-vanishing three-cocycles, such as a point particle moving in the field of a magnetic charge. We generalize the corresponding Malcev algebras to include electric as well as magnetic charges. These structures find their classical counterpart in the theory of Poisson-Malcev algebras and their generalizations. We also study their connection to Stueckelberg's generalized Poisson brackets that do not obey the Jacobi identity and point out that nonassociative string theory with a fundamental length corresponds to a realization of his goal to find a non-linear extension of quantum mechanics with a fundamental length. Similar nonassociative structures are also known to appear in the cubic formulation of closed string field theory in terms of open string fields, leading us to conjecture a natural string-field theoretic generalization of the AdS/CFT-like (holographic) duality.

33 pages; Latex file; References added, minor changes, typos corrected

References in corpus (9)

• Gauge Symmetry and Supersymmetry of Multiple M2-Branes
• Algebraic structures on parallel M2-branes
• Modeling Multiple M2's
• T-duality and closed string non-commutative (doubled) geometry
• The effective theory of type IIA AdS4 compactifications on nilmanifolds and cosets
• Type IIA AdS4 compactifications on cosets, interpolations and domain walls
• Non-commutative/non-associative IIA (IIB) Q- and R-branes and their intersections
• Twisted Poisson Structures and Non-commutative/non-associative Closed String Geometry
• Quantum Gravity, Dynamical Energy-Momentum Space and Vacuum Energy

Cited by in corpus (14)

• Quantum Gravity, Dynamical Phase Space and String Theory
• On the Non-commutativity of Closed String Zero Modes
• Non-associativity in non-geometric string and M-theory backgrounds, the algebra of octonions, and missing momentum modes
• $G_2$-structures and quantization of non-geometric M-theory backgrounds
• States in non-associative quantum mechanics: Uncertainty relations and semiclassical evolution
• Gravitizing the Quantum
• Triple Interference, Non-linear Talbot Effect and Gravitization of the Quantum
• Magnetic monopoles and nonassociative deformations of quantum theory
• Weak associativity and deformation quantization
• Nonlinear Field Space Cosmology
• Nearly associative deformation quantization
• Non-Associative Algebras and Quantum Physics -- A Historical Perspective
• Weyl Semimetal and Nonassociative Nambu Geometry
• Extension of Malcev algebra and applications to gravity