Roberto Catenacci

On the body of supermanifolds

The problem of constructing the body of a G∞ manifold is considered. It is shown that any such manifold is foliated, and the body is defined to be the space of the leaves of this foliation. Under certain regularity conditions on the foliation, the body is a smooth finite‐dimensional real manifold.

Group actions and anomalies in gauge theories

Abstract The transformation properties are studied of the vacuum functional W ( A ) for chiral fermions in a gauge potential A under the group A ×U(1)× R + of gauge, chiral and scale transformations. The vacuum functional W is identified with a section of a G ×U(1)× R + line bundle over the space A of all gauge potentials. Known results on bundles carrying group actions give a simple and unifying clue to non-abelian, abelian chiral anomalies, as well as to trace anomalies. While the first are due to the twisting of a line bundle on A / G , the abelian chiral and trace anomalies are related to characters of U(1) and R + respectively. Characters of U(1) are basically controlled by “winding numbers”, i.e. again by topology. Opposite to these, trace anomalies seem to have little to do with topology, with the exception of two-dimensional theories.

Almost‐structures and structures in Lorentzian manifolds. I. Almost‐Hermite‐ and almost‐product‐ (2×2) ‐structures

We present a list of the most important almost‐structures which have been found of interest in general relativity, in the null‐bivectors formalism. We discusss some of the relevant properties of such almost‐structures and various new or more or less known results. We also present theorems on the relations between almost‐product‐structures and almost‐Hermitian‐structures.

TOPOLOGICAL EFFECTIVE FIELD THEORIES FOR DIRAC FERMIONS FROM INDEX THEOREM

Dirac fermions have a central role in high energy physics but it is well known that they emerge also as quasiparticles in several condensed matter systems supporting topological order. We present a general method for deriving the topological effective actions of (3+1) massless Dirac fermions living on general backgrounds and coupled with vector and axial-vector gauge fields. The first step of our strategy is standard (in the Hermitian case) and consists in connecting the determinants of Dirac operators with the corresponding analytical indices through the zeta-function regularization. Then, we introduce a suitable splitting of the heat kernel that naturally selects the purely topological part of the determinant (i.e. the topological effective action). This topological effective action is expressed in terms of gauge fields using the Atiyah-Singer index theorem which computes the analytical index in topological terms. The main new result of this paper is to provide a consistent extension of this method to the non Hermitian case where a well-defined determinant does not exist. Quantum systems supporting relativistic fermions can thus be topologically classified on the basis of their response to the presence of (external or emergent) gauge fields through the corresponding topological effective field theories.

A note on Killing vectors in algebraically special vacuum space-times

A study is made on the connection between the existence of a Killing vector whose associated bivector (Ka;b) is null and the Petrov type. A theorem of Debney is refined. Also considered is a particular case (omitted in Debneys papers) in which the bivector is nonnull.

Effect of gravitational radiation upon electromagnetic waves in a dispersive medium

We show that linearized gravitational radiation produces fluctuations in intensity and position of a distant source if the ray travels in a dispersive medium. The effect, however, depends upon the nongeodesic character of the ray and does not occur in an electrostatic plasma. When the index of refraction n is greater than unity a Cerenkov type resonance produces scintillation proportional to DO3/2 (DO being the distance of the source) and a ‘dancing’ proportional to DO1/2 if, instead, n<1 the scintillation behaves like DO and the dancing does not diverge as DO→ ∞. The calculation is performed in detail for a random and isotropic spectrum of gravitational waves W(ω). This effect allows one to set an upper limit to W(ω) at the frequency at which the fluctuations are observed, but for the rarified interstellar and intergalactic plasmas these limits are not very interesting.

Generalized Killing-Yano tensors

We consider the Killing-Yano equation by dropping the antisymmetry hypothesis and studying some consequences of the integrability conditions. We find that the cases of interest are more general, and we discuss the relevance of solutions to the construction of Killing tensors.

Irreducible mass for the Tomimatsu-Sato space-times

A global definition of irreducible mass for the odd-δ T-S metrics is investigated. We find that its expression in terms of the source parameters is the same for all the members of the family and reduces to the formula that holds in the Kerr case (δ=1). As a consequence, we show that processes withmir = const no longer imply zero variations of the horizons area for δ>1.

Invariance of the electromagnetic field in static space-times

Abstract Using the Debever formalism and classical geometrodynamics it is shown that in a static space-time, solution of Einstein-Maxwell equations, the electromagnetic field is also static.