Nonassociative field theory on non‐geometric spaces

Abstract

We describe quasi-Hopf twist deformations of flat closed string compactifications with non-geometric Rflux using a suitable cochain twist, and construct nonassociative deformations of fields and differential calculus. We report on our new findings in using this formalism to construct perturbative nonassociative field theories on these backgrounds. We describe the modifications to the usual classification of Feynman diagrams into planar and non-planar graphs. The example of ϕ 4 theory is studied in detail and the one-loop contributions to the two-point function are calculated. Theory and Gravity, Corfu, Greece, September 8-15, 2013. Non-geometric backgrounds arise as consistent string vacua in p-form flux compactifications via Tduality transformations. Consider the standard example of closed strings propagating in a three-torus endowed with non-vanishing constant 3-form flux H = dB. Employing T-duality along all three directions takes H to its T-dual 3-vector flux R and results in a purely non-geometric background where transition functions between patches cannot even be defined locally [6]. This "R-space" exhibits an intriguing nonassociative deformation of geometry which is consistent with the original nonassociative deformations of spacetime discovered in [3], where standard conformal field theory approaches were used to study closed strings propagating in a constant H-flux background; we refer to the lecture notes [2] of these proceedings for further details of these approaches and for a more exhaustive list of references. Based on talk given by D.M. at the Workshop on Noncommutative FieldA geometrization of R-space is provided by the phase space of its T-dual background. This geometry is induced by regarding the fundamental degrees of freedom in the non-geometric background as membranes in a Courant σ-model whose boundary dynamics are described by a closed string quasi-Poisson σ-model with target space the cotangent bundle of the original membrane spacetime, which can be quantized using Kontsevich's deformation quantization [8]. This yields a nonassociative star product of fields on R-space that reproduces the nonassociative geometry discovered in [3,6], and leads to Seiberg-Witten maps which untwist the nonassociative product to a family of associative noncommutative star products; we refer to the lectures of P. Schupp from these proceedings for further details of this approach. It is also equivalent to a strict deformation quantization approach that is based on integrating a pertinent Lie 2-algebra to a Lie 2-group, which leads to the formulation of the nonassociative star products discussed by D. Lüst in these proceedings.These deformations can also be acquired by twisting the Hopf algebra of symmetries of R-space to a quasi-Hopf algebra using a suitable cochain twist [9]; see also [10] for a review and for further references. The advantage of this technique is that it is algorithmic in the sense that once a twist has been found, it can be used to deform all geometric structures on R-space. In this co...

Twist for Snyder space

“…Another example of non-associative deformations arising in quantum mechanics and gravity in three dimensions is given by considering the dynamics of electrons in uniform distributions of magnetic charge. All of the above lead to non-associativity becoming an interesting area of research in physics in recent years [45,46,47,48,49,50,61]; for example, in connection to non-geometric Rflux backgrounds, non-associative star products have been investigated [46] and perturbative non-associative field theories were constructed [47]; the construction of non-associative Weyl star products was undertaken [48]; physical consequences of non-associativity were examined [50], finding that momentum space is quantized and implying a coarse-graining of momentum space with a uniform monopole background. Regarding different directions of the research on non-associativity in quantum physics, the twist we have constructed in this paper could be helpful in studying the properties of the non-associative star product corresponding to the Snyder spacetime.…”

“…In [44], a truncated form of the nonassociative and noncommutative Snyder φ 4 field theory was defined and quantized using the functional method in momentum space. Different nonassociative star/cross product geometries, as well as the related field theories, have also been considered in [45,46,47,48,49,50].…”

Magnetic monopoles and nonassociative deformations of quantum theory

“…and Mac Lane's coherence theorem asserts that this uniquely defines the insertion of suitable associator factors into higher order iterated star products. This principle enables the extension of our considerations here to the construction of some nonassociative quantum field theories [31,32].…”

Perspectives on Nonassociative Geometry

“…Physically consistent models with novel properties in the context of quantum mechanics were constructed in [25] using this formalism, and of Euclidean scalar quantum field theory in [23]. To extend these considerations to more complicated field theories, a general systematic formalism was developed in [6,7] for differential geometry on noncommutative and nonassociative spaces internal to the representation category of any quasi-Hopf algebra, generalizing and extending earlier work [15,8,2].…”