We explicitly derive, following a Noether-like approach, the criteria for preserving Poincare invariance in noncommutative gauge theories. Using these criteria we discuss the various spacetime symmetries in such theories. It is shown that, interpreted appropriately, Poincare invariance holds. The analysis is performed in both the commutative as well as noncommutative descriptions and a compatibility between the two is also established.Comment: LaTeX, 20 pages, slightly improved version, one reference added, version appearing in Phys. Rev.

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In today’s world, technology has become an inevitable part of human life. In fact, during the Covid-19 pandemic, everything from the corporate world to educational institutes has shifted from offline to online. It leads to exponential increase in intrusions and attacks over the Internet-based technologies. One of the lethal threat surfacing is the Distributed Denial of Service (DDoS) attack that can cripple down Internet-based services and applications in no time. The attackers are updating their skill strategies continuously and hence elude the existing detection mechanisms. Since the volume of data generated and stored has increased manifolds, the traditional detection mechanisms are not appropriate for detecting novel DDoS attacks. This paper systematically reviews the prominent literature specifically in deep learning to detect DDoS. The authors have explored four extensively used digital libraries (IEEE, ACM, ScienceDirect, Springer) and one scholarly search engine (Google scholar) for searching the recent literature. We have analyzed the relevant studies and the results of the SLR are categorized into five main research areas: (i) the different types of DDoS attack detection deep learning approaches, (ii) the methodologies, strengths, and weaknesses of existing deep learning approaches for DDoS attacks detection (iii) benchmarked datasets and classes of attacks in datasets used in the existing literature, and (iv) the preprocessing strategies, hyperparameter values, experimental setups, and performance metrics used in the existing literature (v) the research gaps, and future directions.

We derive maps relating currents and their divergences in non-Abelian U(N ) noncommutative gauge theory with the corresponding expressions in the ordinary (commutative) description. For the U(1) theory, in the slowly-varying-field approximation, these maps are also seen to connect the star-gauge-covariant anomaly in the noncommutative theory with the standard Adler-Bell-Jackiw anomaly in the commutative version. For arbitrary fields, derivative corrections to the maps are explicitly computed up to O(θ 2 ).

We exploit the Seiberg-Witten maps for fields and currents in a U(1) gauge theory relating the noncommutative and commutative (usual) descriptions to obtain the O(θ) structure of the commutator anomalies in noncommutative electrodynamics. These commutators involve the (covariant) current-current algebra and the (covariant) current-field algebra. We also establish the compatibility of the anomalous commutators with the noncommutative covariant anomaly through the use of certain consistency conditions derived here.PACS: 11.10.Nx, 11.15.-q; Appearing in Phys. Rev. D The stability of the O(θ) map among the currents J µ and J µ under gauge transformations is easily attained by mimicking the map (2) among the field tensors:where (· · · ) indicates the freedom of adding more O(θ) terms that are invariant under ordinary gauge transformations. It is clear that the most general structure is given by where c 1 , c 2 and c 3 are undetermined coefficients. Demanding the simultaneous conservation D µ ⋆ J µ = ∂ µ J µ = 0 immediately fixes c 1 = 2c 2 = 1 and c 3 = 0, so thatThis is the O(θ) map among the currents obtained in an algebraic approach. The map can be generalised to higher orders in θ in a dynamical approach [9]. Using the maps (1) and (18), the covariant divergence of J µ ,