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Register a new userA Note on Gauge Invariant Operators in Noncommutative Gauge Theories and the Matrix Model
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In this note we discuss local gauge-invariant operators in noncommutative gauge theories. Inspired by the connection of these theories with the Matrix model, we give a simple construction of a complete set of gauge-invariant operators. We make connection with the recent discussions of candidate operators which are dual to closed strings modes. We also discuss large Wilson loops which in the limit of vanishing noncommutativity, reduce to the closed Wilson loops of the ordinary gauge theory.
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I review results from recent investigations of anomalies in fermion--Yang Mills systems in which basic notions from noncommutative geometry (NCG) where found to naturally appear. The general theme is that derivations of anomalies from quantum field theory lead to objects which have a natural interpretation as generalization of de Rham forms to NCG, and that this allows a geometric interpretation of anomaly derivations which is useful e.g. for making these calculations efficient. This paper is intended as selfcontained introduction to this line of ideas, including a review of some basic facts about anomalies. I first explain the notions from NCG needed and then discuss several different anomaly calculations: Schwinger terms in 1+1 and 3+1 dimensional current algebras, Chern--Simons terms from effective fermion actions in arbitrary odd dimensions. I also discuss the descent equations which summarize much of the geometric structure of anomalies, and I describe that these have a natural generalization to NCG which summarize the corresponding structures on the level of quantum field theory. Contribution to Proceedings of workshop `New Ideas in the Theory of Fundamental Interactions, Szczyrk, Poland 1995; to appear in Acta Physica Polonica B.
We calculate conformal anomalies in noncommutative gauge theories by using the path integral method (Fujikawas method). Along with the axial anomalies and chiral gauge anomalies, conformal anomalies take the form of the straightforward Moyal deformation in the corresponding conformal anomalies in ordinary gauge theories. However, the Moyal star product leads to the difference in the coefficient of the conformal anomalies between noncommutative gauge theories and ordinary gauge theories. The $beta$ (Callan-Symanzik) functions which are evaluated from the coefficient of the conformal anomalies coincide with the result of perturbative analysis.
The Connes and Lott reformulation of the strong and electroweak model represents a promising application of noncommutative geometry. In this scheme the Higgs field naturally appears in the theory as a particular `gauge boson, connected to the discrete internal space, and its quartic potential, fixed by the model, is not vanishing only when more than one fermion generation is present. Moreover, the exact hypercharge assignments and relations among the masses of particles have been obtained. This paper analyzes the possibility of extensions of this model to larger unified gauge groups. Noncommutative geometry imposes very stringent constraints on the possible theories, and remarkably, the analysis seems to suggest that no larger gauge groups are compatible with the noncommutative structure, unless one enlarges the fermionic degrees of freedom, namely the number of particles.
Using path integral method (Fujikawas method) we calculate anomalies in noncommutative gauge theories with fermions in the bi-fundamental and adjoint representations. We find that axial and chiral gauge anomalies coming from non-planar contributions are derived in the low noncommutative momentum limit $widetilde{p}^{mu}(equiv theta^{mu u}p_{ u}) to 0$. The adjoint chiral fermion carries no anomaly in the non-planar sector in $D=4k (k=1,2,...,)$ dimensions. It is naturally shown from the path integral method that anomalies in non-planar sector originate in UV/IR mixing.
We refine a previous proposal for obtaining the multi-instanton partition function from the supersymmetric index of the 1d supersymmetric gauge theory on the worldline of D0-branes. We provide examples where the refinements are crucial for obtaining the correct result.
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