There are two main problems in finding the higher genus superstring measure. The first one is tha... more There are two main problems in finding the higher genus superstring measure. The first one is that for g ≥ 5 the super moduli space is not projected. Furthermore, the supermeasure is regular for g ≤ 11, a bound related to the source of singularities due to the divisor in the moduli space of Riemann surfaces with even spin structure having holomorphic sections, such a divisor is called the θ-null divisor. A result of this paper is the characterization of such a divisor. This is done by first extending the Dirac propagator, that is the Szegö kernel, to the case of an arbitrary number of zero modes, that leads to a modification of the Fay trisecant identity, where the determinant of the Dirac propagators is replaced by the product of two determinants of the Dirac zero modes. By taking suitable limits of points on the Riemann surface, this holomorphic Fay trisecant identity leads to identities that include points dependent rank 3 quadrics in P g−1. Furthermore, integrating over the homological cycles gives relations for the Riemann period matrix which are satisfied in the presence of Dirac zero modes. Such identities characterize the θ-null divisor. Finally, we provide the geometrical interpretation of the above points dependent quadrics and shows, via a new θ-identity, its relation with the Andreotti-Mayer quadric.
We formulate Friedmann's equations as second-order linear differential equations. This is done us... more We formulate Friedmann's equations as second-order linear differential equations. This is done using techniques related to the Schwarzian derivative that selects the β-times t β := t a −2β , where a is the scale factor. In particular, it turns out that Friedmann's equations are equivalent to the eigenvalue problems
The geometrical formulation of the quantum Hamilton-Jacobi theory shows that the quantum potentia... more The geometrical formulation of the quantum Hamilton-Jacobi theory shows that the quantum potential is never vanishing, so that it plays the rôle of intrinsic energy. Such a key property selects the Wheeler-DeWitt (WDW) quantum potential Q[g jk ] as the natural candidate for the dark energy. This leads to the WDW Hamilton-Jacobi equation with a vanishing kinetic term, and with the identification Λ = − κ 2 √ḡ Q[g jk ]. This shows that the cosmological constant is a quantum correction of the Ein-stein tensor, reminiscent of the von Weizsäcker correction to the kinetic term of the Thomas-Fermi theory. The quantum potential also defines the Madelung pressure tensor. The geometrical origin of the vacuum energy density, a strictly non-perturbative phenomenon, provides strong evidence that it is due to a graviton condensate. Time independence of the regularizied WDW equation suggests that the ratio between the Planck length and the Hubble radius may be a time constant , providing an infrared/ultraviolet duality. Such a duality is related to the local to global geometry theorems for constant curvatures, suggesting a key rôle of Thurston geometry and higher dimensional uniformization theory. This shows that understanding Universe's geometry is crucial for a formulation of Quantum Gravity.
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Papers by marco matone