Modular forms and three-loop superstring amplitudes

We study a proposal of D'Hoker and Phong for the chiral superstring measure for genus three. A minor modification of the constraints they impose on certain Siegel modular forms leads to a unique solution. We reduce the problem of finding these modular forms, which depend on an even spin structure, to finding a modular form of weight 8 on a certain subgroup of the modular group. An explicit formula for this form, as a polynomial in the even theta constants, is given. We checked that our result is consistent with the vanishing of the cosmological constant. We also verified a conjecture of D'Hoker and Phong on modular forms in genus 3 and 4 using results of Igusa.     

The elemental composition of Stradivari's musical instruments: new results through non‐invasive EDXRF analysis

During recent decades, many researchers have tried to understand the main influences on the extraordinary sound and beauty of the masterpieces made by the ancient violin makers. This is still a challenge for many others today. Mainly because of a lack of written historical documents, the rediscovery of some of the ancient violin‐making processes was made possible thanks to scientific analyses performed on their materials by means of diagnostic techniques. However, understanding which substances were adopted is a very hard task, because the analyses are influenced by many factors: for example, alterations, wear, retouches, and the heterogeneity of materials. This paper presents some new EDXRF results collected on eight‐stringed musical instruments made by Antonio Stradivari between 1669 and 1734 (“Clisbee” 1669, “Hellier” 1679, “Ford‐Rougemont” 1703, “Joachim‐Ma” 1714, “Russian Federation” viola 1715, “Cremonese” 1715, “Vesuvius” 1727, and “Scotland” 1734) and now preserved at the Museo del Violino in Cremona. A brief comparison with a modern violin made by Simone Ferdinando Sacconi (“Hellier copy” 1941), one of the most eminent violin makers of the 20th century and one of the greatest experts on Stradivari's work, is also provided. This represents the first comparative analysis of a wide number of ancient musical instruments made by the same violin maker over an extended period. A non‐destructive and non‐invasive approach was followed to (a) understand the elemental composition differences between the best conserved and most worn‐out surfaces; (b) check if there are elemental similarities among the finishing materials of violins made in different years by the same violin maker; (c) give new suggestions about the materials used. To distinguish the best conserved areas from the worn‐out ones, a preliminary investigation by UV‐induced fluorescence photography was performed. In addition, stereomicroscopic observations and Fourier transform infrared spectroscopy (FTIR) analyses were performed on selected areas to validate the hypotheses. The results, in some cases comparable with previous research on Stradivari instruments, have increased the pool of information about materials and treatments adopted in the Stradivari workshop.     

Tunneling method for Hawking radiation in the Nariai case

We revisit the tunneling picture for the Hawking effect in light of the charged Nariai manifold, because this general relativistic solution, which displays two horizons, provides the bonus to allow the knowledge of exact solutions of the field equations. We first perform a revisitation of the tunneling ansatz in the framework of particle creation in external fields à la Nikishov, which corroborates the interpretation of the semiclassical emission rate \({\varGamma }_{emission}\) as the conditional probability rate for the creation of a couple of particles from the vacuum. Then, particle creation associated with the Hawking effect on the Nariai manifold is calculated in two ways. On the one hand, we apply the Hamilton–Jacobi formalism for tunneling, in the case of a charged scalar field on the given background. On the other hand, the knowledge of the exact solutions for the Klein–Gordon equations on Nariai manifold, and their analytic properties on the extended manifold, allow us a direct computation of the flux of particles leaving the horizon, and, as a consequence, we obtain a further corroboration of the semiclassical tunneling picture from the side of S-matrix formalism.     

Genus Four Superstring Measures

A main issue in superstring theory are the superstring measures. D’Hoker and Phong showed that for genus two these reduce to measures on the moduli space of curves which are the product of modular forms of weight eight and the bosonic measure. They also suggested a generalisation to higher genus. We showed that their approach works, with a minor modification, in genus three and we announced a positive result also in genus four. Here we give the modular form in genus four explicitly. Recently, S. Grushevsky published this result as part of a more general approach.