Nonabelian noncommutative gauge fields and Seiberg-Witten map

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, 2010

There are strong restrictions on the possible representations and in general on the matter content of gauge theories formulated on noncommutative Moyal spaces, termed as noncommutative gauge theory no-go theorem. According to the no-go theorem [1], matter fields in the noncommutative U(1) gauge theory can only have ±1 or zero charges and for a generic noncommutative n i=1 U(N i) gauge theory matter fields can be charged under at most two of the U(N i) gauge group factors. On the other hand, it has been argued in the literature that, since a noncommutative U(N) gauge theory can be mapped to an ordinary U(N) gauge theory via the Seiberg-Witten map, seemingly it can bypass the no-go theorem. In this note we show that the Seiberg-Witten map [2] can only be consistently defined and used for the gauge theories which respect the no-go theorem. We discuss the implications of these arguments for the particle physics model building on noncommutative space.

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.

Physics Letters B, 2010

It is known that gauge transformation of the Kalb-Ramond field B μν with vector parameter Λ μ is symmetry of the closed string world-sheet action. It fails at the end points of the open string and can be restored by introducing Maxwell field A μ. We show that the same conclusion valid for space-time equations of motion, because they are conditions for conformal invariance of the world-sheet action. This is example how the symmetries of space-time theory can be investigated using properties of σ-model energy-momentum tensor. We also show that the reducible part of the closed string symmetry transformation, with Λ μ = ∂ μ λ, turns to irreducible part of the open string one. As well as Maxwell field A μ , the parameter λ is nontrivial only on the string endpoints. We show that after quantization the symmetry transformations of the background fields B μν and A μ turn to the modified symmetries defined in terms of Moyal star product. The modified Λ μ-transformation related different regularizations as well as Seiberg-Witten map.

The European Physical Journal C - Particles and Fields, 2000

We introduce a formulation of gauge theory on noncommutative spaces based on the concept of covariant coordinates. Some important examples are discussed in detail. A Seiberg-Witten map is established in all cases.

Physics Letters B, 2004

Massive vector fields can be described in a gauge invariant way with the introduction of compensating fields. In the unitary gauge one recovers the original formulation. Although this gauging mechanism can be extended to noncommutative spaces in a straightforward way, non trivial aspects show up when we consider the Seiberg-Witten map. As we show here, only a particular class of its solutions leads to an action that admits the unitary gauge fixing.

Physics Letters B, 2001

Following the formalism of enveloping algebras and star product calculus we formulate and analyze a model of gauge gravity on noncommutative spaces and examine the conditions of its equivalence to the general relativity theory. The corresponding Seiberg-Witten maps are established which allow the definition of respective dynamics for a finite number of gravitational gauge field components on noncommutative spaces.

2003

We analyze in detail the recursive construction of the Seiberg-Witten map and give an exhaustive description of its ambiguities. The local BRST cohomology for noncommutative Yang-Mills theory is investigated in the framework of the effective commutative Yang-Mills type theory. In particular, we analyze how some of the conformal symmetries get obstructed by the noncommutative deformation.

Nuclear Physics B, 2000

We study the generalization of noncommutative gauge theories to the case of orthogonal and symplectic groups. We find out that this is possible, since we are allowed to define orthogonal and symplectic subgroups of noncommutative unitary gauge transformations even though the gauge potentials and gauge transformations are not valued in the orthogonal and symplectic subalgebras of the Lie algebra of antihermitean matrices. Our construction relies on an antiautomorphism of the basic noncommutative algebra of functions which generalizes the charge conjugation operator of ordinary field theory. We show that the corresponding noncommutative picture from low energy string theory is obtained via orientifold projection in the presence of a non-trivial NSNS B-field.

2005

Canonically deformed spacetime, where the commutator of two coordinates is a constant, is the most commonly studied noncommutative space. Noncommutative gauge theories that have ordinary gauge theory as their commutative limit have been constructed there. But these theories have their drawbacks: First of all, constant noncommutativity can only be an approximation of a realistic theory, and therefore it is necessary to study more complicated space-dependent structures as well. Secondly, in the canonical case, the noncommutativity didn't fulfill the initial hope of curing the divergencies of quantum field theory. Therefore it is very desirable to understand noncommutative spaces that really admit finite QFTs. These two aspects of going beyond the canonical case will be the main focus of this thesis. They will be addressed within two different formalisms, each of which is especially suited for the purpose. In the first part noncommutative spaces created by star-products are studied...