43 research outputs found
The role of the Beltrami parametrization of complex structures in 2-d Free Conformal Field Theory
This talk gives a review on how complex geometry and a Lagrangian formulation
of 2-d conformal field theory are deeply related. In particular, how the use of
the Beltrami parametrization of complex structures on a compact Riemann surface
fits perfectly with the celebrated locality principle of field theory, the
latter requiring the use infinite dimensional spaces. It also allows a direct
application of the local index theorem for families of elliptic operators due
to J.-M. Bismut, H. Gillet and C. Soul\'{e}. The link between determinant line
bundles equipped with the Quillen\'s metric and the so-called holomorphic
factorization property will be addressed in the case of free spin b-c
systems or more generally of free fields with values sections of a holomorphic
vector bundles over a compact Riemann surface.Comment: Actes du Colloque "Complex Geometry '98
Nucleon spin decomposition and differential geometry
In the last few years, the so-called Chen et al. approach of the nucleon spin
decomposition has been widely discussed and elaborated on. In this letter we
propose a genuine differential geometric understanding of this approach. We
mainly highligth its relation to the "dressing field method" we advocated in
[C. Fournel, J. Fran\c{c}ois, S. Lazzarini, T. Masson, Int. J. Geom. Methods
Mod. Phys. 11, 1450016 (2014)]. We are led to the conclusion that the claimed
gauge-invariance of the Chen et al. decomposition is actually unreal.Comment: 9 pages. v3: minor corrections in the text, addition of a new
referenc
Local description of generalized forms on transitive Lie algebroids and applications
In this paper we study the local description of spaces of forms on transitive
Lie algebroids. We use this local description to introduce global structures
like metrics, -Hodge operation and integration along the algebraic part
of the transitive Lie algebroid (its kernel). We construct a \v{C}ech-de Rham
bicomplex with a Leray-Serre spectral sequence. We apply the general theory to
Atiyah Lie algebroids and to derivations on a vector bundle
Gauge field theories: various mathematical approaches
This paper presents relevant modern mathematical formulations for (classical)
gauge field theories, namely, ordinary differential geometry, noncommutative
geometry, and transitive Lie algebroids. They provide rigorous frameworks to
describe Yang-Mills-Higgs theories or gravitation theories, and each of them
improves the paradigm of gauge field theories. A brief comparison between them
is carried out, essentially due to the various notions of connection. However
they reveal a compelling common mathematical pattern on which the paper
concludes.Comment: 33 pages. To be published in the book: Mathematical Structures of the
Universe (Copernicus Center Press, Krak\'ow, Poland, 2014
Kodaira-Spencer deformation of complex structures and Lagrangian field theory
In complete analogy with the Beltrami parametrization of complex structures
on a (compact) Riemann surface, we use in this paper the Kodaira-Spencer
deformation theory of complex structures on a (compact) complex manifold of
higher dimension. According to the Newlander-Nirenberg theorem, a smooth change
of local complex coordinates can be implemented with respect to an integrable
complex structure parametrized by a Beltrami differential. The question of
constructing a local field theory on a complex compact manifold is addressed
and the action of smooth diffeomorphisms is studied in the BRS algebraic
approach. The BRS cohomology for the diffeomorphisms gives rise to generalized
Gel'fand-Fuchs cocycles provided that the Kodaira-Spencer integrability
condition is satisfied. The diffeomorphism anomaly is computed and turns out to
be holomorphically split as in the bidimensional Lagrangian conformal models.
Moreover, its algebraic structure is much more complicated than the one
proposed in a quite recent paper hep-th/9606082 (Nucl. Phys. B484 (1997) 196).Comment: LaTeX, 30 pages, no figure. Submitted to Journ. Math. Phy
Formulation of gauge theories on transitive Lie algebroids
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 non-commutative gauge theories.Comment: 30 pages. Final version. arXiv admin note: substantial text overlap
with arXiv:1109.428
Cartan Connections and Atiyah Lie Algebroids
This work extends previous developments carried out by some of the authors on
Ehresmann connections on Atiyah Lie algebroids. In this paper, we study Cartan
connections in a framework relying on two Atiyah Lie algebroids based on a
-principal fiber bundle and its associated -principal fiber
bundle , where defines the
model for a Cartan geometry. The first main result of this study is a
commutative and exact diagram relating these two Atiyah Lie algebroids, which
allows to completely characterize Cartan connections on .
Furthermore, in the context of gravity and mixed anomalies, our construction
answers a long standing mathematical question about the correct
geometrico-algebraic setting in which to combine inner gauge transformations
and infinitesimal diffeomorphisms.Comment: 27 pages. Published versio