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Register a new userThe kernel and the injectivity of the EPRL map
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Added by
Marcin Kisielowski M.Sc.
Publication date
2011
fields
Physics
and research's language is
English
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In this paper we prove injectivity of the EPRL map for |gamma|<1, filling the gap of our previous paper.
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Do the SU(2) intertwiners parametrize the space of the EPRL solutions to the simplicity constraint? What is a complete form of the partition function written in terms of this parametrization? We prove that the EPRL map is injective for n-valent vertex in case when it is a map from SO(3) into SO(3)xSO(3) representations. We find, however, that the EPRL map is not isometric. In the consequence, in order to be written in a SU(2) amplitude form, the formula for the partition function has to be rederived. We do it and obtain a new, complete formula for the partition function. The result goes beyond the SU(2) spin-foam models framework.
In this paper we consider the Prym map for double coverings of curves of genus $g$ ramified at $r>0$ points. That is, the map associating to a double ramified covering its Prym variety. The generic Torelli theorem states that the Prym map is generically injective as soon as the dimension of the space of coverings is less or equal to the dimension of the space of polarized abelian varieties. We prove the generic injectivity of the Prym map in the cases of double coverings of curves with: (a) $g=2$, $r=6$, and (b) $g= 5$, $r=2$. In the first case the proof is constructive and can be extended to the range $rge max {6,frac 23(g+2) }$. For (b) we study the fibre along the locus of the intermediate Jacobians of cubic threefolds to conclude the generic injectivity. This completes the work of Marcucci and Pirola who proved this theorem for all the other cases, except for the bielliptic case $g=1$ (solved later by Marcucci and the first author), and the case $g=3, r=4$ considered previously by Nagaraj and Ramanan, and also by Bardelli, Ciliberto and Verra where the degree of the map is $3$. The paper closes with an appendix by Alessandro Verra with an independent result, the rationality of the moduli space of coverings with $g=2,r=6$, whose proof is self-contained.
We study the injectivity and surjectivity of the Borel map in three instances: in Roumieu-Carleman ultraholomorphic classes in unbounded sectors of the Riemann surface of the logarithm, and in classes of functions admitting, uniform or nonuniform, asymptotic expansion at the corresponding vertex. These classes are defined in terms of a log-convex sequence $mathbb{M}$ of positive real numbers. Injectivity had been solved in two of these cases by S. Mandelbrojt and B. Rodriguez-Salinas, respectively, and we completely solve the third one by means of the theory of proximate orders. A growth index $omega(mathbb{M})$ turns out to put apart the values of the opening of the sector for which injectivity holds or not. In the case of surjectivity, only some partial results were available by J. Schmets and M. Valdivia and by V. Thilliez, and this last author introduced an index $gamma(mathbb{M})$ (generally different from $omega(mathbb{M})$) for this problem, whose optimality was not established except for the Gevrey case. We considerably extend here their results, proving that $gamma(mathbb{M})$ is indeed optimal in some standard situations (for example, as far as $mathbb{M}$ is strongly regular) and puts apart the values of the opening of the sector for which surjectivity holds or not.
Timelike geodesics on a hyperplane orthogonal to the symmetry axis of the Godel spacetime appear to be elliptic-like if standard coordinates naturally adapted to the cylindrical symmetry are used. The orbit can then be suitably described through an eccentricity-semi-latus rectum parametrization, familiar from the Newtonian dynamics of a two-body system. However, changing coordinates such planar geodesics all become explicitly circular, as exhibited by Kundts form of the Godel metric. We derive here a one-to-one correspondence between the constants of the motion along these geodesics as well as between the parameter spaces of elliptic-like versus circular geodesics. We also show how to connect the two equivalent descriptions of particle motion by introducing a pair of complex coordinates in the 2-planes orthogonal to the symmetry axis, which brings the metric into a form which is invariant under Mobius transformations preserving the symmetries of the orbit, i.e., taking circles to circles.
We review recent work on the local geometry and optimal regularity of Lorentzian manifolds with bounded curvature. Our main results provide an estimate of the injectivity radius of an observer, and a local canonical foliations by CMC (Constant Mean Curvature) hypersurfaces, together with spatially harmonic coordinates. In contrast with earlier results based on a global bound for derivatives of the curvature, our method requires only a sup-norm bound on the curvature near the given observer.
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