Preserving levels of projective determinacy by tree forcings

We prove that various classical tree forcings -- for instance Sacks forcing, Mathias forcing, Laver forcing, Miller forcing and Silver forcing -- preserve the statement that every real has a sharp and hence analytic determinacy. We then lift this result via methods of inner model theory to obtain level-by-level preservation of projective determinacy (PD). Assuming PD, we further prove that projective generic absoluteness holds and no new equivalence classes classes are added to thin projective transitive relations by these forcings.Comment: 3 figure

How to get a conservative well-posed linear system out of thin air. Part II. Controllability and stability

Published versio

Global Fukaya category and quantum Novikov conjecture I

Conceptually, the goal here is a construction which functorially translates a Hamiltonian fibre bundle to a certain ``derived vector bundle'' over the same space, with fiber an $A _{\infty}$ category. This ``derived vector bundle'' must remember the continuity of the original bundle. Concretely, using Floer-Fukaya theory for a monotone $(M, \omega)$ we construct a natural continuous map \begin{equation*} BHam (M, \omega) \to (\mathcal{S}, NFuk (M)), \end{equation*} with $(\mathcal{S}, NFuk (M))$ denoting the $NFuk (M)$ component of the ``space'' of $\infty$-categories, where $NFuk (M)$ is the $A _{\infty}$-nerve of the Fukaya category $Fuk (M)$. This construction is very closely related to the theory of the Seidel homomorphism and the quantum Chern classes of the author, and this map is intended to be the deepest expression of their underlying geometric theory. In part II the above map is shown to be non trivial by an explicit calculation. In particular we arrive at a new non-trivial ``quantum'' invariant of any smooth manifold and a ``quantum'' Novikov conjecture.Comment: v5, 41 pages. This adds significant detail and fixes some language issue

Regularised Kalb-Ramond Magnetic Monopole with Finite Energy

In a previous work we suggested a self-gravitating electromagnetic monopole solution in a string-inspired model involving global spontaneous breaking of a $SO(3)$ internal symmetry and Kalb-Ramond (KR) axions, stemming from an antisymmetric tensor field in the massless string multiplet. These axions carry a charge, which, in our model, also plays the r\^ole of the magnetic charge. The resulting geometry is close to that of a Reissner-Nordstr\"om (RN) black hole with charge proportional to the KR-axion charge. We proposed the existence of a thin shell structure inside a (large) core radius as the dominant mass contribution to the energy functional. The resulting energy was finite, and proportional to the KR-axion charge; however, the size of the shell was not determined and left as a phenomenological parameter. In the current article, we can calculate the mass-shell size, on proposing a regularisation of the black hole singularity via a matching procedure between the RN metric in the outer region and, in the inner region, a de Sitter space with a (positive) cosmological constant proportional to the scale of the spontaneous symmetry breaking of $SO(3)$ . The matching, which involves the Israel junction conditions for the metric and its first derivatives at the inner surface of the shell, determines the inner mass-shell radius. The axion charge plays an important r\^ole in guaranteeing positivity of the "mass coefficient" of the gravitational potential term appearing in the metric component; so the KR electromagnetic monopole shows normal attractive gravitational effects. This is to be contrasted with the global monopole case (in the absence of KR axions) where such a matching is known to yield a negative "mass coefficient" (and, hence, repulsive gravitational effects). The total energy of the monopole within the shell is calculated.Comment: 9 pages revtex, 1 pdf figure incorporated; added clarifying discussion in sections II and III, better motivating the use of de Sitter regularisation of the core region of the self gravitating monopole solution from string theory considerations. No effect on conclusions. Version to be published in Physical Review