29,761 research outputs found
From Kirchberg's inequality to the Goldberg conjecture
The main result of this note is that a compact Kaehler manifold whose Ricci tensor has two distinct constant non-negative eigenvalues is locally the product of two Kaehler-Einstein manifolds. The problem of existence of Kaehler metrics whose Ricci tensor has two distinct constant eigenvalues is related to the limiting case of Kirchberg's inequality for the first eigenvalue of the Dirac operator on compact Kaehler manifolds, as well as to the celebrated (still open) conjecture of Goldberg
On Kirchberg's Inequality for Compact Kähler Manifolds of Even Complex Dimension
International audienceIn 1986 Kirchberg showed that each eigenvalue of the Dirac operator on a compact Kähler manifold of even complex dimension satisfies some inequality involving the scalar curvature. It is conjectured that the manifolds for the limiting case of this inequality are products T^2×N, where T^2 is a flat torus and N is the twistor space of a quaternionic Kähler manifold of positive scalar curvature. In 1990 Lichnerowicz announced an affirmative answer for this conjecture, but his proof seems to work only when assuming that the Ricci tensor is parallel. The aim of this note is to prove several results about manifolds satisfying the limiting case of Kirchberg''s inequality and to prove the above conjecture in some particular cases
Dirac cohomology, elliptic representations and endoscopy
The first part (Sections 1-6) of this paper is a survey of some of the recent
developments in the theory of Dirac cohomology, especially the relationship of
Dirac cohomology with (g,K)-cohomology and nilpotent Lie algebra cohomology;
the second part (Sections 7-12) is devoted to understanding the unitary
elliptic representations and endoscopic transfer by using the techniques in
Dirac cohomology. A few problems and conjectures are proposed for further
investigations.Comment: This paper will appear in `Representations of Reductive Groups, in
Honor of 60th Birthday of David Vogan', edited by M. Nervins and P. Trapa,
published by Springe
Chiral Symmetry Breaking and Chiral Polarization: Tests for Finite Temperature and Many Flavors
It was recently conjectured that, in SU(3) gauge theories with fundamental
quarks, valence spontaneous chiral symmetry breaking is equivalent to
condensation of local dynamical chirality and appearance of chiral polarization
scale . Here we consider more general association involving the
low-energy layer of chirally polarized modes which, in addition to its width
(), is also characterized by volume density of participating
modes () and the volume density of total chirality (). Few
possible forms of the correspondence are discussed, paying particular attention
to singular cases where emerges as the most versatile characteristic.
The notion of finite-volume "order parameter", capturing the nature of these
connections, is proposed. We study the effects of temperature (in N=0 QCD)
and light quarks (in N=12), both in the regime of possible symmetry
restoration, and find agreement with these ideas. In N=0 QCD, results from
several volumes indicate that, at the lattice cutoff studied, the deconfinement
temperature is strictly smaller than the overlap-valence chiral
transition temperature in real Polyakov line vacuum. Somewhat similar
intermediate phase (in quark mass) is also seen in N=12. It is suggested
that deconfinement in N=0 is related to indefinite convexity of absolute
X-distributions.Comment: 45 pages, 20 figures; v2: reduced the size of submission and fixed
references to appendices; v3: minor changes - published for
On the validity of Strong Cosmic Censorship Conjecture in presence of Dirac fields
A well posed theory of nature is expected to determine the future of an
observer uniquely from a given set of appropriate initial data. In the context
of general relativity, this is ensured by Penrose's strong cosmic censorship
conjecture. But in recent years, several examples are found which suggest
breakdown of the deterministic nature of the theory in Reissner-Nordstrom-de
Sitter black holes under the influence of different fundamental fields.
Nevertheless, the situation has been reassuring for the case of astrophysically
meaningful Kerr-de Sitter black hole solutions which seems to respect the
conjecture. However, the previous analyses were done considering only the
effect of scalar fields. In this paper, we extend the study by considering
Dirac fields in Kerr-de Sitter background and show that there exist a parameter
space which does not respect the conjecture.Comment: 13 pages, 2 figures, Accepted in European Physical Journal
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