Serge Lazzarini

W‐gauge structures and their anomalies: An algebraic approach

Starting from flat two‐dimensional gauge potentials we propose the notion of W‐gauge structure in terms of a nilpotent Becchi–Rouet–Stora (BRS) differential algebra. The decomposition of the underlying Lie algebra with respect to an SL(2) subalgebra is crucial for the discussion of conformal covariance, in particular the appearance of a projective connection. Different SL(2) embeddings lead to different W‐gauge structures. We present a general soldering procedure which allows one to express zero curvature conditions for the W‐currents in terms of conformally covariant differential operators acting on the W‐gauge fields and to obtain, at the same time, the complete nilpotent BRS differential algebra generated by W‐currents, gauge fields, and the ghost fields corresponding to W‐diffeomorphisms. As illustrations we treat the cases of SL(2) itself and the two different SL(2) embeddings in SL(3), viz., the W (1)3‐ and W (2)3‐gauge structures, in some detail. In these cases we determine algebraically W‐anomalie...

Improved Epstein–Glaser renormalization. II. Lorentz invariant framework

The Epstein–Glaser type T-subtraction introduced by one of the authors in a previous paper is extended to the Lorentz invariant framework. The advantage of using our subtraction instead of the Epstein and Glaser standard W-subtraction method is especially important when working in Minkowski space, as then the counterterms necessary to keep Lorentz invariance are simplified. We show how T-renormalization of primitive diagrams in the Lorentz invariant framework directly relates to causal Riesz distributions. A covariant subtraction rule in momentum space is found, sharply improving upon the BPHZL method for massless theories.

Gauge invariant composite fields out of connections, with examples

In this paper, we put forward a systematic and unifying approach to construct gauge invariant composite fields out of connections. It relies on the existence in the theory of a group-valued field with a prescribed gauge transformation. As an illustration, we detail some examples. Two of them are based on known results: the first one provides a reinterpretation of the symmetry breaking mechanism of the electroweak part of the Standard Model of particle physics; the second one is an application to Einsteins theory of gravity described as a gauge theory in terms of Cartan connections. The last example depicts a new situation: starting with a gauge field theory on Atiyah Lie algebroids, the gauge invariant composite fields describe massive vector fields. Some mathematical and physical discussions illustrate and highlight the relevance and the generality of this approach.

Formulation of gauge theories on transitive Lie algebroids

a b s t r a c t In this paper we introduce and study some mathematical structures on top of transitive Lie algebroids in order to formulate gauge theories in terms of generalized connections and their curvature: metrics, Hodge star operator and integration along the algebraic part of the transitive Lie algebroid (its kernel). Explicit action functionals are given in terms of global objects and in terms of their local description as well. We investigate applications of these constructions to Atiyah–Lie algebroids and to derivations on a vector bundle. The obtained gauge theories are discussed with respect to ordinary and to similar noncommutative gauge theories.

Connections on Lie algebroids and on derivation-based noncommutative geometry

Abstract In this paper, we show how connections and their generalizations on transitive Lie algebroids are related to the notion of connections in the framework of the derivation-based noncommutative geometry. In order to compare the two constructions, we emphasize the algebraic approach of connections on Lie algebroids, using a suitable differential calculus. Two examples allow this comparison: on the one hand, the Atiyah Lie algebroid of a principal fiber bundle and, on the other hand, the space of derivations of the algebra of endomorphisms of an S L ( n , C ) -vector bundle. Gauge transformations are also considered in this comparison.

Flat Complex Vector Bundles, the Beltrami Differential and W-Algebras

Since the appearance of the paper by Bilal et al. in 1991, it has been widely assumed that W-algebras originating from the Hamiltonian reduction of an SL(n,C)-bundle over a Riemann surface give rise to a flat connection, in which the Beltrami differential may be identified. In this Letter, it is shown that the use of the Beltrami parametrization of complex structures on a compact Riemann surface over which flat complex vector bundles are considered, allows the construction of the above mentioned flat connection. It is stressed that the modulus of the Beltrami differential is of necessity less than one, and that solutions of the so-called Beltrami equation give rise to an orientation-preserving smooth change of local complex coordinates. In particular, the latter yields a smooth equivalence between flat complex vector bundles. The role of smooth diffeomorphisms which induce equivalent complex structures is specially emphasized. Furthermore, it is shown that, while the construction given here applies to the special case of the Virasoro algebra, the extension to flat complex vector bundles of arbitrary rank does not provide ‘generalizations’ of the Beltrami differential usually considered as central objects for such non-linear symmetries.

Polyakov Soldering and Second-Order Frames: The Role of the Cartan Connection

The so-called ‘soldering’ procedure performed by Polyakov (Int J Math Phys A5, 833–842, 1990) for a

A farewell to unimodularity

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