We associate some (old) convergent series related to definite integrals with the cyclotomic equation x m − 1 = 0, for several natural numbers m ; for example, for m = 3, x 3 − 1 = ( x − 1)(1 + x + x 2 ) leads to . In some cases, we express the results in terms of the Dirichlet characters. Generalizations for arbitrary m are well defined but do imply integrals and/or series summations rather involved.

An attempt is made to approach the irreducible representations of the exceptional Lie groups G2 and F4 by symmetrization of some defining representations by means of Young tableaux, procedure that works rather well for most of the classical groups. ForG2 the program is completely successful, while it is not quite so for F4. As the five exceptional Lie groups are related to octonions, we also comment on the “octonion” character of these two groups, in particular the relation of F4 to some other connected Spin groups.

Regular polytopes, the generalization of the five Platonic solids in 3 space dimensions, exist in arbitrary dimension n >= -1; now in dim. 2, 3 and 4 there are extra polytopes, while in general dimensions only the hyper-tetrahedron, the hyper-cube and its dual hyper-octahedron exist. We attribute these peculiarites and exceptions to special properties of the orthogonal groups in these dimensions: the SO(2) = U(1) group being (abelian and) divisible, is related to the existence of arbitrarily-sided plane regular polygons, and the splitting of the Lie algebra of the 0(4) group will be seen responsible for the Schlafli special polytopes in 4-dim., two of which percolate down to three. In spite of dim. 8 being also special (Cartan's triality), we argue why there are no extra polytopes, while it has other consequences: in particular the existence of the three division algebras over the reals R, complex C, quatemions H and octonions O is seen also as another feature of the special properties of corresponding orthogonal groups, and of the spheres of dimension 0, 1, 3 and 7.

We analyze the geometrical background under which many Lie groups relevant to particle physics are endowed with a (possibly multiple) hexagonal structure. There are several groups appearing, either as special holonomy groups on the compactification process from higher dimensions, or as dynamical string gauge groups; this includes groups like SU(2), SU(3), G 2, Spin(7), O(8) as well as E 8 and O(32). We emphasize also the relation of these hexagonal structures with the octonion division algebra, as we expect as well eventually some role for octonions in the interpretation of symmetries in High Energy Physics.

Este ano de 2013 es el centenario del atomo de Bohr, que inauguro la aplicacion de la teoria cuantica al atomo. En este trabajo se recuerda la construccion de Bohr en su “Trilogia” (junio de 1913), se comentan algunos exitos de la teoria cuantica antigua (1913-1925), asi como algunos de sus fallos.

We describe the collection of finite simple groups, with a view to physical applications. We recall first the prime cyclic groups Zp and the alternating groups Altn > 4. After a quick revision of finite fields Fq?>, q = pf, with p prime, we consider the 16 families of finite simple groups of Lie type. There are also 26 extra ‘sporadic’ groups, which gather in three interconnected ‘generations’ (with 5+7+8 groups) plus the pariah groups (6). We point out a couple of physical applications, including constructing the biggest sporadic group, the ‘Monster’ group, with close to 1054 elements from arguments of physics, and also the relation of some Mathieu groups with compactification in string and M-theory.

This is an introduction to finite simple groups, in particular sporadic groups, intended for physicists. After a short review of group theory, we enumerate the 1 + 1 + 16 = 18 families of finite simple groups, as an introduction to the sporadic groups. These are described next, in three levels of increasing complexity, plus the six isolated "pariah" groups. The (old) five Mathieu groups make up the first, smallest order level. The seven groups related to the Leech lattice, including the three Conway groups, constitute the second level. The third and highest level contains the Monster group M, plus seven other related groups. Next a brief mention is made of the remaining six pariah groups, thus completing the 5 + 7 + 8 + 6 = 26 sporadic groups. The review ends up with a brief discussion of a few of physical applications of finite groups in physics, including a couple of recent examples which use sporadic groups.

We compare the extra degeneracy of both the Oscillator and the Kepler problems in d dimensions as superintegrable systems. Addition of a centrifugal term like 1/r 2 does not substantially change the solutions. We argue that both systems just describe a kind of rotation, and that the centrifugal potential is ineffective because its scale invariance.

We make an attempt to describe the spectrum of masses of elementary particles, as it comes out empirically in six distinct scales. We argue for some rather well defined mass scales, like the electron mass: it seems to us that there is a minimum mass associated to any electric charge, so we elaborate on this assumption; indeed, some scales of masses will cover also masses of composite particles or mass differences. We extend some plausibility arguments for other scales, as binding or self-energy effects of the microscopic forces, plus some speculative uses, here and there, of gravitation. We also consider briefly exotics like supersymmetry and extra dimensions in relation to the mass scale problem, including some mathematical arguments (e.g. triality), which might throw light on the three-generation problem. The paper is rather tentative and speculative and does not make many predictions, but it seems to explain some features of the particle spectrum.

We make an attempt to describe the spectrum of masses of elementary particles, as it comes out empirically in six distinct scales. We argue for some rather well defined mass scales, like the electron mass; we elaborate on the assumption that there is a minimum mass associated to any electric charge. Another natural mass scale is Λ = Λ QCD coming arbitrarily at quantizing a classically conformal SU (3) c theory. Indeed, some scales of masses will cover also masses of composite particles or mass differences. We extend some plausible arguments for other scales, as binding or self-energy effects of the microscopic forces, plus some speculative uses, here and there, of gravitation. We also consider briefly exotics like supersymmetry and extra dimensions in relation to the mass scale problem, including some mathematical arguments (e.g. triality), which might throw light on the three-generation problem. We also address briefly the issues of dark matter and dark energy. The paper is rather tentative and speculative and does not make many predictions, but it aims to explain some features of the particle spectrum.

We discuss quiver gauge models with bi-fundamental and fundamental matter obtained from F-theory compactified on ALE spaces over a four-dimensional base space. We focus on the base geometry which consists of intersecting F 0 = CP 1 × CP 1 Hirzebruch complex surfaces arranged as Dynkin graphs classified by three kinds of Kac–Moody (KM) algebras: ordinary, i.e. finite-dimensional, affine and indefinite, in particular hyperbolic. We interpret the equations defining these three classes of generalized Lie algebras as the anomaly cancelation condition of the corresponding N = 1 F-theory quivers in four dimensions. We analyze in some detail hyperbolic geometries obtained from the affine [Formula: see text] base geometry by adding a node, and we find that it can be used to incorporate fundamental fields to a product of SU-type gauge groups and fields.

We study holonomy groups coming from F-theory compactifications. We focus mainly on SO(8) as 12−4=8 and subgroups SU(4), Spin(7), G 2 and SU(3) suitable for descent from F-theory, M-theory and Superstring theories. We consider the relation of these groups with the octonions, which is striking and reinforces their role in higher dimensions and dualities. These holonomy groups are related in various mathematical forms, which we exhibit.

We consider composition and division algebras over the real numbers: We note two roles for the group G(2): as automorphism group of the octonions and as the isotropy group of a generic three-form in seven dimensions. We show why they are equivalent, by means of a regular metric. We express in some diagrams the relation between some pertinent groups, most of them related to the octonions. Some applications to physics are also discussed.

We discuss local F-theory geometries and their gauge theory dualities in terms of intersecting D7-branes wrapped four-cycles in Type IIB superstring. The manifolds are built as elliptic K3 surface fibrations over intersecting F 0 = CP 1 × CP 1 base geometry according to ADE Dynkin Diagrams. The base is obtained by blowing up the extended ADE hyper-Kähler singularities of eight-dimensional manifolds considered as sigma model target spaces with eight supercharges. The resulting gauge theory of such local F-theory models are given in terms of Type IIB D7-branes wrapped intersecting F 0 . The four-dimensional N = 1 anomaly cancelation requirement translates into a condition on the associated affine Lie algebras.

We propose to substitute Newton’s constant G N for another constant G 2, as if the gravitational force would fall off with the 1/r law, instead of the 1/r 2; so we describe a system of natural units with G 2, c and ℏ. We adjust the value of G 2 so that the fundamental length L = L Pl is still the Planck’s length and so G N = L × G 2. We argue for this system as (1) it would express longitude, time and mass without square roots; (2) G 2 is in principle disentangled from gravitation, as in (2 + 1) dimensions there is no field outside the sources. So G 2 would be truly universal; (3) modern physics is not necessarily tied up to (3 + 1)-dim. scenarios and (4) extended objects with p = 2 (membranes) play an important role both in M-theory and in F-theory, which distinguishes three (2, 1) dimensions.