Bonn 2010 – scientific programme

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MP: Fachverband Theoretische und Mathematische Grundlagen der Physik

MP 11: Noncommutative Geometry

MP 11.2: Talk

Thursday, March 18, 2010, 09:55–10:20, JUR H

Instantons in Noncommutative Gauge Theory in Four Dimensions on the Lattice — Arifa Ali Khan¹ and •Harald Markum² — ¹University of Taiz, Yemen — ²Vienna University of Technology, Austria

Theories with noncommutative space-time coordinates represent alternative candidates of grand unified theories. We discuss U(1) gauge theory in 2 and 4 dimensions on a lattice with N sites. The mapping to a U(N) plaquette model in the sense of Eguchi and Kawai can be used for computer simulations. In 2D it turned out that the value of the topological charge is decreasing during a Monte Carlo history. This shows that the topological charge is in general supressed. The situation is similar to lattice QCD where gauge field configurations are topologically trivial and one needs to apply some cooling procedure on the gluons to unhide the integer number of the instantons. In 4D the definition of a monopole observable seems to be difficult. The analogy to commutative U(1) theory of summing up the phases of an elementary cube might need a projection on the abelian part of the U(N) theory in the matrix model. Concerning the topological charge it seems straightforward. One can transcribe the plaquette and hypercube formulation to the matrix theory. There are several possible choices of noncommutativity amoung the six planes in 4D. The simplest is to use two noncommutative coordinates. One has to modify the Monte Carlo update correspondingly. It will be interesing to measure the topological charge in the non/commutative plane and in the hypercube.