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Groenewold-Moyal Product, α^\star-Cohomology, and Classification of Translation-Invariant Non-Commutative Structures  [PDF]
Amir Abbass Varshovi
Physics , 2012,
Abstract: The theory of alpha_star-cohomology is studied thoroughly and it is shown that in each cohomology class there exists a unique 2-cocycle, the harmonic form, which generates a particular Groenewold-Moyal star product. This leads to an algebraic classification of translation-invariant non-commutative structures and shows that any general translationinvariant non-commutative quantum field theory is physically equivalent to a Groenewold- Moyal non-commutative quantum field theory.
Translation-invariant models for non-commutative gauge fields  [PDF]
Daniel N. Blaschke,Francois Gieres,Erwin Kronberger,Manfred Schweda,Michael Wohlgenannt
Mathematics , 2008, DOI: 10.1088/1751-8113/41/25/252002
Abstract: Motivated by the recent construction of a translation-invariant renormalizable non-commutative model for a scalar field (see arXiv:0802.0791 [math-ph]), we introduce models for non-commutative U(1) gauge fields along the same lines. More precisely, we include some extra terms into the action with the aim of getting rid of the UV/IR mixing.
Translation-Invariant Noncommutative Gauge Theories, Matrix Modeling and Noncommutative Geometry  [PDF]
Amir Abbass Varshovi
Physics , 2011,
Abstract: A matrix modeling formulation for translation-invariant noncommutative gauge theories is given in the setting of differential graded algebras and quantum groups. Translation-invariant products are discussed in the setting of {\alpha}-cohomology and it is shown that loop calculations are entirely determined by {\alpha}-cohomology class of star product in all orders. Noncommutative version of geometric quantization and (anti-) BRST transformations is worked out which leads to a noncommutative description of consistent anomalies and Schwinger terms.
A translation-invariant renormalizable non-commutative scalar model  [PDF]
R. Gurau,J. Magnen,V. Rivasseau,A. Tanasa
Physics , 2008, DOI: 10.1007/s00220-008-0658-3
Abstract: In this paper we propose a translation-invariant scalar model on the Moyal space. We prove that this model does not suffer from the UV/IR mixing and we establish its renormalizability to all orders in perturbation theory.
Running couplings and triviality of field theories on non-commutative spaces  [PDF]
Emil T. Akhmedov,Philip DeBoer,Gordon W. Semenoff
Physics , 2000, DOI: 10.1103/PhysRevD.64.065005
Abstract: We examine the issue of renormalizability of asymptotically free field theories on non-commutative spaces. As an example, we solve the non-commutative O(N) invariant Gross-Neveu model at large N. On commutative space this is a renormalizable model with non-trivial interactions. On the noncommutative space, if we take the translation invariant ground state, we find that the model is non-renormalizable. Removing the ultraviolet cutoff yields a trivial non-interacting theory.
Gauge invariant operators in field theories on non-commutative spaces  [PDF]
Harald Dorn
Physics , 2002, DOI: 10.1002/1521-3978(200209)50:8/9<884::AID-PROP884>3.0.CO;2-B
Abstract: We review some selected aspects of the construction of gauge invariant operators in field theories on non-commutative spaces and their relation to the energy momentum tensor as well as to the non-commutative loop equations.
Consistent Anomalies in Translation-Invariant Noncommutative Gauge Theories  [PDF]
Amir Abbass Varshovi
Physics , 2011, DOI: 10.1063/1.4704797
Abstract: Translation-invariant noncommutative gauge theories are discussed in the setting of matrix modeled gauge theories. Using the matrix model formulation the explicit form of consistent anomalies and consistent Schwinger terms for translation-invariant noncommutative gauge theories are derived.
Bivariant cyclic cohomology and Connes' bilinear pairings in Non-commutative motives  [PDF]
Goncalo Tabuada
Mathematics , 2010,
Abstract: In this article we further the study of non-commutative motives. We prove that bivariant cyclic cohomology (and its variants) becomes representable in the category of non-commutative motives. Furthermore, Connes' bilinear pairings correspond to the composition operation. As an application, we obtain a simple model, given in terms of infinite matrices, for the (de)suspension of these bivariant cohomology theories.
Scalar and gauge translation-invariant noncommutative models  [PDF]
Adrian Tanasa
Physics , 2008,
Abstract: We make here a short overview of the recent developments regarding translation-invariant models on the noncommutative Moyal space. A scalar model was first proposed and proved renormalizable. Its one-loop renormalization group flow and parametric representation were calculated. Furthermore, a mechanism to take its commutative limit was recently given. Finally, a proposition for a renormalizable, translation-invariant gauge model was made.
On the third cohomology group of commutative monoids  [PDF]
María Calvo-Cervera,Antonio M. Cegarra,Benjamín A. Heredia
Mathematics , 2014,
Abstract: We interpret Grillet's symmetric thrid cohomology classes of commutative monoids in terms of strictly symmetric monoidal abelian groupoids. We state and prove a classification result that generalizes the well-known one for strictly commutative Picard categories by Deligne and Fr\"ohlich and Wall, and Sinh.
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