The Gribov problem in noncommutative gauge theory

Nuclear Physics B, 2001

The concept of covariant coordinates on noncommutative spaces leads directly to gauge theories with generalized noncommutative gauge fields of the type that arises in string theory with background B-fields. The theory is naturally expressed in terms of cochains in an appropriate cohomology; we discuss how it fits into the framework of projective modules. The equivalence of star products that arise from the background field with and without fluctuations and Kontsevich's formality theorem allow an explicitly construction of a map that relates ordinary gauge theory and noncommutative gauge theory (Seiberg-Witten map.) As application we show the exact equality of the Dirac-Born-Infeld action with B-field in the commutative setting and its semi-noncommutative cousin in the intermediate picture. Using noncommutative extra dimensions the construction is extended to noncommutative nonabelian gauge theory for arbitrary gauge groups; an explicit map between abelian and nonabelian gauge fields is given. All constructions are also valid for non-constant B-field, Poisson structure and metric.

The European Physical Journal C, 2001

We present a formalism to explicitly construct non-Abelian gauge theories on noncommutative spaces (induced via a star product with a constant Poisson tensor) from a consistency relation. This results in an expansion of the gauge parameter, the noncommutative gauge potential and fields in the fundamental representation, in powers of a parameter of the noncommutativity. This allows the explicit construction of actions for these gauge theories.

Physics Letters B, 2002

Studying the general structure of the noncommutative (NC) local groups, we prove a no-go theorem for NC gauge theories. According to this theorem, the closure condition of the gauge algebra implies that: 1) the local NC u(n) algebra only admits the irreducible n×n matrix-representation. Hence the gauge fields are in n×n matrix form, while the matter fields can only be in fundamental, adjoint or singlet states; 2) for any gauge group consisting of several simple-group factors, the matter fields can transform nontrivially under at most two NC group factors. In other words, the matter fields cannot carry more than two NC gauge group charges. This no-go theorem imposes strong restrictions on the NC version of the Standard Model and in resolving the standing problem of charge quantization in noncommutative QED.

Journal of High Energy Physics, 2013

We discuss noncommutative gauge theory from the generalized geometry point of view. We argue that the equivalence between the commutative and semiclassically noncommutative DBI actions is naturally encoded in the generalized geometry of D-branes.

Physics Letters B, 1989

A noncommutative extension of abelian gauge theory is proposed and is compared with standard nonabelian gauge theory.

Journal of High Energy Physics

We propose a field theoretical model defined on non-commutative space-time with non-constant non-commutativity parameter Θ(x), which satisfies two main requirements: it is gauge invariant and reproduces in the commutative limit, Θ → 0, the standard U(1) gauge theory. We work in the slowly varying field approximation where higher derivatives terms in the star commutator are neglected and the latter is approximated by the Poisson bracket, −i[f, g] ≈ {f, g}. We derive an explicit expression for both the NC deformation of Abelian gauge transformations which close the algebra [δ f , δ g ]A = δ {f,g} A, and the NC field strength F , covariant under these transformations, δ f F = {F , f }. NC Chern-Simons equations are equivalent to the requirement that the NC field strength, F , should vanish identically. Such equations are non-Lagrangian. The NC deformation of Yang-Mills theory is obtained from the gauge invariant action, S = F 2. As guiding example, the case of su(2)-like non-commutativity, corresponding to rotationally invariant NC space, is worked out in detail.

The European Physical Journal C, 2007

We consider an external gauge potential minimally coupled to a renormalisable scalar theory on 4-dimensional Moyal space and compute in position space the oneloop Yang-Mills-type effective theory generated from the integration over the scalar field. We find that the gauge invariant effective action involves, beyond the expected noncommutative version of the pure Yang-Mills action, additional terms that may be interpreted as the gauge theory counterpart of the harmonic oscillator term, which for the noncommutative ϕ 4-theory on Moyal space ensures renormalisability. The expression of a possible candidate for a renormalisable action for a gauge theory defined on Moyal space is conjectured and discussed.

2013

We study the space-time symmetries and transformation properties of the non-commutative U(1) gauge theory, by using Noether charges. We carry out our analysis by keeping an open view on the possible ways θ µν could transform. We conclude that θ µν cannot transform under any space-time transformation since the theory is not invariant under the conformal transformations, with the only exception of space-time translations. The same analysis applies to other gauge groups.

Arxiv preprint hep-th/0202014, 2002