Four-dimensional noncommutative deformations of U(1) gauge theory and L∞ bootstrap.

Abstract

We construct a family of four-dimensional noncommutative deformations of U(1) gauge theory following a general scheme, recently proposed in JHEP 08 (2020) 041 for a class of coordinate-dependent noncommutative algebras. This class includes the $$ \mathfrak{su} $$ su (2), the $$ \mathfrak{su} $$ su (1, 1) and the angular (or λ-Minkowski) noncommutative structures. We find that the presence of a fourth, commutative coordinate x0 leads to substantial novelties in the expression for the deformed field strength with respect to the corresponding three-dimensional case. The constructed field theoretical models are Poisson gauge theories, which correspond to the semi-classical limit of fully noncommutative gauge theories. Our expressions for the deformed gauge transformations, the deformed field strength and the deformed classical action exhibit flat commutative limits and they are exact in the sense that all orders in the deformation parameter are present. We review the connection of the formalism with the L∞ bootstrap and with symplectic embeddings, and derive the L∞-algebra, which underlies our model.

On the L$_\infty$ structure of Poisson gauge theory

“…which in addition to the brackets ℓ 2 (f, g) and ℓ n+1 (f, A ⊗n ) constructed in [10, Section 6.1] define the L f ull ∞ algebra corresponding to the su(2)-like Poisson gauge theory. The results of this section can be generalized to the four-dimensional case (with a commutative fourth coordinate) and/or to the angular non-commutativity [16] using the explicit expressions for F , presented in [17].…”

On the L∞ structure of Poisson gauge theory

“…which in addition to the brackets 2 ( f , g) and n+1 ( f , A ⊗n ) constructed in [11, section 6.1] define the L full ∞ algebra corresponding to the su(2)-like Poisson gauge theory. The results of this section can be generalized to the four-dimensional case (with a commutative fourth coordinate) and/or to the angular non-commutativity [21] using the explicit expressions for F , presented in [22].…”

Localization and observers in $\varrho$-Minkowski spacetime

“…In [13] a field theory on this space has been built; in the same paper a different physical identification of the non-commuting variables has been also considered, with time a commuting coordinate. The latter has been studied in [14] in the context of Poisson gauge models and in [15] in the context of double quantization 2 .…”