A regularization algorithm for matrices of bilinear and sesquilinear forms

We give an algorithm that uses only unitary transformations and for each square complex matrix constructs a *congruent matrix that is a direct sum of a nonsingular matrix and singular Jordan blocks.Comment: 18 page

Extremal varieties 3-rationally connected by cubics, quadro-quadric Cremona transformations and rank 3 Jordan algebras

For any $n\geq 3$, we prove that there exist equivalences between these apparently unrelated objects: irreducible $n$-dimensional non degenerate projective varieties $X\subset \mathbb P^{2n+1}$ different from rational normal scrolls and 3-covered by twisted cubic curves, up to projective equivalence; quadro-quadric Cremona transformations of $\mathbb P^{n-1}$, up to linear equivalence; $n$-dimensional complex Jordan algebras of rank three, up to isotopy. We also provide some applications to the classification of particular classes of varieties in the class defined above and of quadro-quadric Cremona transformations, proving also a structure theorem for these birational maps and for varieties 3-covered by twisted cubics by reinterpreting for these objects the solvability of the radical of a Jordan algebra.Comment: 30 pages, 1 figure. Corrected typo

Orbits of Exceptional Groups, Duality and BPS States in String Theory

We give an invariant classification of orbits of the fundamental representations of exceptional groups $E_{7(7)}$ and $E_{6(6)}$ which classify BPS states in string and M theories toroidally compactified to d=4 and d=5. The exceptional Jordan algebra and the exceptional Freudenthal triple system and their cubic and quartic invariants play a major role in this classification. The cubic and quartic invariants correspond to the black hole entropy in d=5 and d=4, respectively. The classification of BPS states preserving different numbers of supersymmetries is in close parallel to the classification of the little groups and the orbits of timelike, lightlike and space-like vectors in Minkowski space. The orbits of BPS black holes in N=2 Maxwell-Einstein supergravity theories in d=4 and d=5 with symmetric space geometries are also classified including the exceptional N=2 theory that has $E_{7(-25)}$ and $E_{6(-26)}$ as its symmety in the respective dimensions.Comment: New references and two tables added, a new section on the orbits of N=2 Maxwell-Einstein supergravity theories in d=4 and d=5 included and some minor changes were made in other sections. 17 pages. Latex fil

Spectral projections and resolvent bounds for partially elliptic quadratic differential operators

We study resolvents and spectral projections for quadratic differential operators under an assumption of partial ellipticity. We establish exponential-type resolvent bounds for these operators, including Kramers-Fokker-Planck operators with quadratic potentials. For the norms of spectral projections for these operators, we obtain complete asymptotic expansions in dimension one, and for arbitrary dimension, we obtain exponential upper bounds and the rate of exponential growth in a generic situation. We furthermore obtain a complete characterization of those operators with orthogonal spectral projections onto the ground state.Comment: 60 pages, 3 figures. J. Pseudo-Differ. Oper. Appl., to appear. Revised according to referee report, including minor changes to Corollary 1.8. The final publication will be available at link.springer.co

Unusual square roots in the ghost-free theory of massive gravity

A crucial building block of the ghost free massive gravity is the square root function of a matrix. This is a problematic entity from the viewpoint of existence and uniqueness properties. We accurately describe the freedom of choosing a square root of a (non-degenerate) matrix. It has discrete and (in special cases) continuous parts. When continuous freedom is present, the usual perturbation theory in terms of matrices can be critically ill defined for some choices of the square root. We consider the new formulation of massive and bimetric gravity which deals directly with eigenvalues (in disguise of elementary symmetric polynomials) instead of matrices. It allows for a meaningful discussion of perturbation theory in such cases, even though certain non-analytic features arise.Comment: 24 pages; minor changes, final versio