G 2-structures and quantization of non-geometric M-theory backgrounds

Abstract

Abstract:We describe the quantization of a four-dimensional locally non-geometric Mtheory background dual to a twisted three-torus by deriving a phase space star product for deformation quantization of quasi-Poisson brackets related to the nonassociative algebra of octonions. The construction is based on a choice of G 2 -structure which defines a nonassociative deformation of the addition law on the seven-dimensional vector space of Fourier momenta. We demonstrate explicitly that this star product reduces to that of the three-dimensional parabolic constant R-flux model in the contraction of M-theory to string theory, and use it to derive quantum phase space uncertainty relations as well as triproducts for the nonassociative geometry of the four-dimensional configuration space. By extending the G 2 -structure to a Spin(7)-structure, we propose a 3-algebra structure on the full eight-dimensional M2-brane phase space which reduces to the quasi-Poisson algebra after imposing a particular gauge constraint, and whose deformation quantisation simultaneously encompasses both the phase space star products and the configuration space triproducts. We demonstrate how these structures naturally fit in with previous occurences of 3-algebras in M-theory.

Nonassociative Snyder ϕ4 quantum field theory

“…A further generalization of Snyder spacetime deformations was recently introduced in [36][37][38]. Also several nonassociative star/cross product geometries and related quantum field theories have been discussed recently in [39].…”

UV-IR mixing in nonassociative Snyder ϕ4 theory

“…First, from the underlying mathematics, like L ∞ algebras (see [45] and references within [46]), new structures arise, for example the star-product algebra of functions, which were studied through nongeometric strings, probing noncommutative and nonassociative deformations of closed string background geometries [47][48][49], see also the celebrated paper by Kontsevich [50]. Second, the quantization of these backgrounds through explicit constructions of phase space star products were provided in [51,52], and subsequently applied to construct nonassociative theories [53]. In this article, we show the active role originating from the nonassociativity of the star product.…”

Locally non-geometric fluxes and missing momenta in M-theory

“…This fact that there is a missing momentum mode was crucial in the conjecture of [18] that the phase space of M2-branes in a locally non-geometric M-theory background is governed by the non-associative algebra of the imaginary octonions, which was subsequently quantised in [19]. The reduction to type IIA string theory can then be understood as a contraction of the algebra of imaginary octonions which yields (1.2).…”

“…This played an important role in the conjecture of [18] that the phase space algebra of M2-branes in such spaces is isomorphic to the non-associative algebra of imaginary octonions. Furthermore, include the missing momentum mode leads to an eight-dimensional phase space governed by a 3-algebra [19] with Spin(8) structure. Implementing the constraint (5.6) reduces this 3-algebra to the non-associative M-theory R-flux algebra isomorphic to the imaginary octonions.…”

“…This algebra was also be deformation quantised in [14,16] using a membrane model. Recently, four-dimensional non-geometric backgrounds of M-theory were studied in [17][18][19][20], particularly focussing on local non-geometry. The lift of the string theory R-flux R ijk to a four-dimensional M-theory background was constructed in [17], using the duality manifest formulation of exceptional field theory [21].…”