Gravity, strings, modular and quasimodular forms

Modular and quasimodular forms have played an important role in gravity and string theory. Eisenstein series have appeared systematically in the determination of spectrums and partition functions, in the description of non-perturbative effects, in higher-order corrections of scalar-field spaces, ... The latter often appear as gravitational instantons i.e. as special solutions of Einstein's equations. In the present lecture notes we present a class of such solutions in four dimensions, obtained by requiring (conformal) self-duality and Bianchi IX homogeneity. In this case, a vast range of configurations exist, which exhibit interesting modular properties. Examples of other Einstein spaces, without Bianchi IX symmetry, but with similar features are also given. Finally we discuss the emergence and the role of Eisenstein series in the framework of field and string theory perturbative expansions, and motivate the need for unravelling novel modular structures.Comment: 45 pages. To appear in the proceedings of the Besse Summer School on Quasimodular Forms - 201

Darboux Integrability of a Generalized 3D Chaotic Sprott ET9 System

في هذا البحث تم دراسة التكامل الاول من نوع داربوكس لتعميم النظام الفوضوي الثلاثي الابعاد Sprott ET9 . حيث وضحنا ان النظام لايمتلك متعددة حدود . دالة كسرية, تحليلية والداربوكس للتكامل الاول لاي قيمتين a و b. كما استطعنا ابجاد متعددة  حدود داربوكس لهذا النظام بقرب المفكوك الاسي. باستخدام وزن متعددة الحدود المتجانسة التي ساعدتنا في برهان الطريقة.In this paper, the first integrals of Darboux type of the generalized Sprott ET9 chaotic system will be studied. This study showed that the system has no polynomial, rational, analytic and Darboux first integrals for any value of . All the Darboux polynomials for this system were derived together with its exponential factors. Using the weight homogenous polynomials helped us prove the process

Hwang-Oguiso invariants and frozen singularities in Special Geometry

In Special Geometry there are two inequivalent notions of "Kodaira type" for a singular fiber: one associated with its local monodromy and one with its Hwang-Oguiso characteristic cycle. When the two Kodaira types are not equal the geometry is subtler and its deformation space gets smaller ("partially frozen" singularities). The paper analyzes the physical interpretation of the Hwang-Oguiso invariant in the context of 4d N=2 QFT and describes the surprising phenomena which appear when it does not coincide with the monodromy type. The Hwang-Oguiso multiple fibers are in one-to-one correspondence with the partially frozen singularities in M-theory compactified on a local elliptic K3: a chain of string dualities relates the two geometric set-ups. Paying attention to a few subtleties, this correspondence explains in purely geometric terms how the "same" Kodaira elliptic fiber may have different deformations spaces. The geometric computation of the number of deformations agrees with the physical expectations. At the end we briefly outline the implications of the Hwang-Oguiso invariants for the classification program of 4d N=2 SCFTs.Comment: 54 page

The Schwarzschild-Black String AdS Soliton: Instability and Holographic Heat Transport

We present a calculation of two-point correlation functions of the stress-energy tensor in the strongly-coupled, confining gauge theory which is holographically dual to the AdS soliton geometry. The fact that the AdS soliton smoothly caps off at a certain point along the holographic direction, ensures that these correlators are dominated by quasinormal mode contributions and thus show an exponential decay in position space. In order to study such a field theory on a curved spacetime, we foliate the six-dimensional AdS soliton with a Schwarzschild black hole. Via gauge/gravity duality, this new geometry describes a confining field theory with supersymmetry breaking boundary conditions on a non-dynamical Schwarzschild black hole background. We also calculate stress-energy correlators for this setting, thus demonstrating exponentially damped heat transport. This analysis is valid in the confined phase. We model a deconfinement transition by explicitly demonstrating a classical instability of Gregory-Laflamme-type of this bulk spacetime.Comment: 26 pages, 3 figure

Yang-Mills and Dirac fields in the Minkowski space-time

An existence and uniqueness theorem for the Cauchy problem for the evolution component of the coupled Yang-Mills and Dirac equations in the Minkowski space is proved in a Sobolev space for the temporal gauge condition. The constraint set C is shown to be a smooth submanifold of P preserved by the evolution. The Lie algebra gs(P) of infinitesimal gauge symmetries of P is identified. Its topology is of Beppo Levi type. The group GS(P) of gauge symmetries is of Lie type; its topology is induced by the topology of its Lie algebra. The constraint equations define a closed ideal gs(P)o of gs(P). It generates a closed connected subgroup GS(P)o of GS(P), which is showh to act properly in P. The reduced phase space is the space of GS(P)o orbits in the constraint set C. It is a smooth quotient manifold of C endowed with an exact symplectic form. The quotient group GS(P)/GS(P)o is isomorphic to the structure group G of the theory. Its action in the reduced phase space is Hamiltonian. The associated conserved quantities are colour charges. Only the charges corresponding to the centre of the Lie algebra gs(P)/ gs(P)o admit well defined local charge densities

Graphs and networks theory

This chapter discusses graphs and networks theory