Weak associativity and deformation quantization

Abstract

Non-commutativity and non-associativity are quite natural in string theory. For open strings it appear due to the presence of non-vanishing background two-form in the world volume of Dirichlet brane, while in closed string theory the flux compactifications with non-vanishing three-form also lead to non-geometric backgrounds. In this paper, working in the framework of deformation quantization, we study the violation of associativity imposing the condition that the associator of three elements should vanish whenever each two of them are equal. The corresponding star products are called alternative and satisfy an important for physical applications properties like the Moufang identities, alternative identities, Artin's theorem, etc. The condition of alternativity is invariant under the gauge transformations, just like it happens in the associative case. The price to pay is the restriction on the non-associative algebra which can be represented by the alternative star product, it should satisfy the Malcev identity. The example of nontrivial Malcev algebra is the algebra of imaginary octonions. For this case we construct an explicit expression of the non-associative and alternative star product. We also discuss the quantization of Malcev-Poisson algebras of general form, study its properties and provide the lower order expression for the alternative star product. To conclude we define the integration on the algebra of the alternative star products and show that the integrated associator vanishes.

24 pages, V2, examples corrected, discussion extended, refferences added

References in corpus (3)

• T-duality and closed string non-commutative (doubled) geometry
• Nonassociative geometry in quasi-Hopf representation categories I: Bimodules and their internal homomorphisms
• Star products made (somewhat) easier

Cited by in corpus (7)

• Non-geometric backgrounds in string theory
• Symplectic embeddings, homotopy algebras and almost Poisson gauge symmetry
• $G_2$-structures and quantization of non-geometric M-theory backgrounds
• $L_{\infty }$-Bootstrap Approach to Non-Commutative Gauge Theories
• Monopole star products are non-alternative
• Nonassociative geometry of nonholonomic phase spaces with star R-flux string deformations and (non) symmetric metrics
• Nearly associative deformation quantization