E7 groups from octonionic magic square

In this paper we continue our program, started in [2], of building up explicit generalized Euler angle parameterizations for all exceptional compact Lie groups. Here we solve the problem for E7, by first providing explicit matrix realizations of the Tits construction of a Magic Square product between the exceptional octonionic algebra J and the quaternionic algebra H, both in the adjoint and the 56 dimensional representations. Then, we provide the Euler parametrization of E7 starting from its maximal subgroup U=(E6 x U(1))/Z3. Next, we give the constructions for all the other maximal compact subgroups.Comment: 23 pages, added sections with new construction

Plane quartics: the universal matrix of bitangents

Aronhold's classical result states that a plane quartic can be recovered by the configuration of any Aronhold systems of bitangents, i.e. special $7$-tuples of bitangents such that the six points at which any sub-triple of bitangents touches the quartic do not lie on the same conic in the projective plane. Lehavi (cf. \cite{lh}) proved that a smooth plane quartic can be explicitly reconstructed from its $28$ bitangents; this result improved Aronhold's method of recovering the curve. In a 2011 paper \cite{PSV} Plaumann, Sturmfels and Vinzant introduced an $8 \times 8$ symmetric matrix that parametrizes the bitangents of a nonsingular plane quartic. The starting point of their construction is Hesse's result for which every smooth quartic curve has exactly $36$ equivalence classes of linear symmetric determinantal representations. In this paper we tackle the inverse problem, i.e. the construction of the bitangent matrix starting from the 28 bitangents of the plane quartic, and we provide a Sage script intended for computing the bitangent matrix of a given curve

Multiloop amplitudes in superstring theory.

String theory is a candidate for a unified theory of all forces in nature. String theory is a quantum theory, and, because it includes a state that can be identified with the graviton, it is a quantum theory of gravity. So it could be a framework for describing the four fundamental interactions. The main idea is to replace the description of particles as zero dimensional object with one dimensional object, a string. Mimicking the procedure followed in QFT one can formulate the theory using the Feynmann path integral. In this context one would be able to compute the amplitudes for a process of string scattering, as, for example, in QED. It was shown that correct definition of the amplitudes is strictly connected with the construction of a suitable measure on the moduli space of certain Riemann surfaces of genus g. Thus an essential element in the computation of string amplitudes is the definition of this chiral measure. During the last fifteen years the correct expression for the superstring measure was found just up to genus two. In this thesis I will present a general method to compute the measure up to five loop which as a byproduct also improves the known results for genus two and shows that the previous formulated ansatze for the genus three amplitude was wrong. In the first part I present the well known construction of the partition function in the bosonic theory. Actually, this is the starting point for the the generalization to the supersymmetric case. Then, I introduce the mathematical background we need to obtain the superstring measure. Finally, I discuss in great details the construction of the measure up to genus five and some open problems are emphasized. Moreover, I will also point out that the path integral procedure leads to some ambiguities related to the infinite dimension of the functional measure appearing in the path integral and I will explain the main difficulties related to its mathematically rigorous definition

One-Dimensional Super Calabi-Yau Manifolds and their Mirrors

We apply a definition of generalised super Calabi-Yau variety (SCY) to supermanifolds of complex dimension one. One of our results is that there are two SCY's having reduced manifold equal to $\mathbb{P}^1$, namely the projective super space $\mathbb{P}^{1|2}$ and the weighted projective super space $\mathbb{WP}^{1|1}_{(2)}$. Then we compute the corresponding sheaf cohomology of superforms, showing that the cohomology with picture number one is infinite dimensional, while the de Rham cohomology, which is what matters from a physical point of view, remains finite dimensional. Moreover, we provide the complete real and holomorphic de Rham cohomology for generic projective super spaces $\mathbb P^{n|m}$. We also determine the automorphism groups: these always match the dimension of the projective super group with the only exception of $\mathbb{P}^{1|2}$, whose automorphism group turns out to be larger than the projective general linear supergroup. By considering the cohomology of the super tangent sheaf, we compute the deformations of $\mathbb{P}^{1|m}$, discovering that the presence of a fermionic structure allows for deformations even if the reduced manifold is rigid. Finally, we show that $\mathbb{P}^{1|2}$ is self-mirror, whereas $\mathbb{WP} ^{1|1}_{(2)}$ has a zero dimensional mirror. Also, the mirror map for $\mathbb{P}^{1|2}$ naturally endows it with a structure of $N=2$ super Riemann surface.Comment: 50 pages. Accepted for publication in JHE

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 determined by 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.Comment: 7 pages. To appear in Letters in Mathematical Physic