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} (2), the mathfrak{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.
Highlights
The only exact nontrivial1 models, which have been constructed so far along the lines of [17], and exhibiting the flat commutative limit, are the three-dimensional U(1) theory with su(2) noncommutativity [17] and the two-dimensional U(1) model with kappa-Minkowski2 noncommutativity [19]
We review the connection of the formalism with the L∞ bootstrap and with symplectic embeddings, and derive the L∞-algebra, which underlies our model
In this article we constructed a family of four-dimensional noncommutative deformations of the U(1) gauge theory, implementing a class of noncommutative spaces (1.13) in the general framework of [17]
Summary
According to [17] the infinitesimal deformed gauge transformations, which close the algebra (1.12), and reproduce the correct undeformed limit (1.9), can be constructed by allowing for a field-dependent deformation as follows: δf Aμ = γμν (A)∂ν f + {Aμ, f }. Any symplectic embedding, related to the deformed gauge transformation (2.1), the L∞ algebra is there, and can be constructed as follows. The proposition, mentioned above, asserts that L∞-algebras which correspond to different choices of γ (i.e. different symplectic embeddings), associated with the same Poisson bivector Θ via eq (2.3), are necessarily connected by L∞-quasi-isomorphisms From this point of view the L∞ structure, which underlies a given deformed gauge transformation of the form (2.1) is “unique”. This result is a direct generalisation of the L∞ algebra, presented in the Example 6.4 of [6] for the three-dimensional su(2)-case
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