arXiv subjects

Luis J. Boya

Publications and source records attributed to Luis J. Boya.

At least 19 recordsLinked to original sources

arXiv2015

Relation of the cyclotomic equation with the harmonic and derived series

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 $\int_0^1dx\frac{1}{(1+x+x^2)} = \fracπ{(3\sqrt{3})} = (1-\frac{1}{2}) + (\frac{1}{4}-\frac{1}{5}) + (\frac{1}{7}-\frac{1}{8}) + \ldots$ . 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.

math.NT↗

arXiv2013

Introduction to Sporadic Groups for physicists

We describe the collection of finite simple groups, with a view on physical applications. We recall first the prime cyclic groups $Z_p$, and the alternating groups $Alt_{n>4}$. After a quick revision of finite fields $\mathbb{F}_q$, $q = p^f$, with $p$ prime, we consider the 16 families of finite simple groups of Lie type. There are also 26 \emph{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 $10^{54}$ elements from arguments of physics, and also the relation of some Mathieu groups with compactification in string and M-theory.

math-ph↗

arXiv2012

On regular polytopes

Regular polytopes, the generalization of the five Platonic solids in 3 space dimensions, exist in arbitrary dimension $n\geq-1$; now in {\rm dim}. 2, 3 and 4 there are \emph{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 $\mathrm{SO}(2)=\mathrm{U}(1)$ group being (abelian and) \emph{divisible}, is related to the existence of arbitrarily-sided plane regular polygons, and the \emph{splitting} of the Lie algebra of the $\mathrm{O}(4)$ group will be seen responsible for the Schläfli special polytopes in 4-dim., two of which percolate down to three. In spite of {\rm dim}. 8 being also special (Cartan's \emph{triality}), we argue why there are no \emph{extra} polytopes, while it has other consequences: in particular the existence of the three \emph{division algebras} over the reals $\mathbb{R}$: complex $\mathbb{C}$, quaternions $\mathbb{H}$ and octonions $\mathbb{O}$ 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.

math-ph↗

arXiv2012

On Hexagonal Structures in Higher Dimensional Theories

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), SO(8) as well as E_8 and SO(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.

hep-th↗

arXiv2011

On the Scales of Masses in Elementary Particles

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.

hep-ph↗

arXiv2011

Introduction to Sporadic Groups

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 $\mathbb M$, 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.

math-ph↗

arXiv2009

Composition algebras and the two faces of $G_{2}$

We consider composition and division algebras over the real numbers: We note two rôles for the group $G_{2}$: as automorphism group of the octonions and as the isotropy group of a generic 3-form in 7 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.

math-ph↗

arXiv2009

On Local F-theory Geometries and Intersecting D7-branes

We discuss local F-theory geometries and theirs 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 \times CP^1 base geometry according to ADE Dynkin Diagrams. The base is obtained by blowing up the extended ADE hyper-Kahler 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.

hep-th↗

arXiv2009

Holonomy Groups Coming From F-Theory Compactification

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), G2 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.

hep-th↗

arXiv2009

On F-theory Quiver Models and Kac-Moody Algebras

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 F0=CP1xCP1 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 A 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.

hep-th↗

arXiv2009

Commutativity of missing label operators in terms of Berezin brackets

We obtain a criterion on the commutativity of polynomials in the enveloping algebra of a Lie algebra in terms of an involution condition with respect to the Berezin bracket. As an application, it is shown that the commutativity requirement of missing label operators for reduction chains in the missing label problem can be solved analytically.

math-ph↗

arXiv2008

Symmetric Spaces of Exceptional Groups

We adress the problem of the reasons for the existence of 12 symmetric spaces with the exceptional Lie groups. The 1+2 cases for $G_2$ and $F_4$ respectively are easily explained from the octonionic nature of these groups. The 4+3+2 cases on the $E_{6,7,8}$ series require the magic square of Freudenthal and, for the split case, an appeal to the supergravity chain in $5, 4$ and 3 spacetime dimensions.

math-ph↗

arXiv2008

Geometry of density sates

We reconsider the geometry of pure and mixed states in a finite quantum system. The rangesof eigenvalues of the density matrices delimit a regular simplex (Hypertetrahedron TN) in any dimension N; the polytope isometry group is the symmetric group SN+1, and splits TN in chambers, the orbits of the states under the projective group PU(N + 1). The type of states correlates with the vertices, edges, faces, etc. of the polytope, with the vertices making up a base of orthogonal pure states. The entropy function as a measure of the purity of these states is also easily calculable; we draw and consider some isentropic surfaces. The Casimir invariants acquire then also a more transparent interpretation.

quant-ph↗

arXiv2008

Supersymmetry and Polytopes

We make an imaginative comparison between the Minimal Supersymmetric Standard Model and the 24-cell polytope in four dimensions, the Octacube.

hep-th↗

arXiv2008

Relation Between Holonomy Groups in Superstrings, M and F-theories

We consider manifolds with special holonomy groups SU(3), G2 and Spin(7) as suitable for compactification of superstrings, M-theory and F-theory (with only one time) respectively. The relations of these groups with the octonions are discussed, reinforcing their role in the physics of string theory and duality. We also exhibit three triple exact sequences explaining the connections between the mentioned special holonomy groups.

hep-th↗

arXiv2007

The Spin-Statistics Theorem in Arbitrary Dimensions

We investigate the spin-statistics connection in arbitrary dimensions for hermitian spinor or tensor quantum fields with a rotationally invariant bilinear Lagrangian density. We use essentially the same simple method as for space dimension D = 3. We find the usual connection (tensors as bosons and spinors as fermions) for D = 8n + 3; 8n + 4; 8n + 5, but only bosons for spinors and tensors in dimensions 8n +/- 1 and 8n. In dimensions 4n + 2 the spinors may be chosen as bosons or fermions. The argument hinges on finding the identity representation of the rotation group either on the symmetric or the antisymmetric part of the square of the field representation.

hep-th↗

arXiv2006

Special Holonomy Manifolds in Physics

This is a pedagogical exposition of holonomy groups intended for physicists. After some pertinent definitions, we focus on special holonomy manifolds, two per division algebras, and comment upon several cases of interest in physics, associated with compactification from F-, M- and string theory, on manifolds of 8, 7 and 6 dimensions respectively.

math-ph↗

arXiv2005

The Uniqueness of the World

We follow some (wild) speculations on trying to understand the uniqueness of our physical world, from the field concept to F-Theory.

physics.pop-ph↗