Embeddings of Sz(32) in E_8(5)

We show that the Suzuki group Sz(32) is a subgroup of E_8(5), and so is its automorphism group. Both are unique up to conjugacy in E_8(F) for any field F of characteristic 5, and the automorphism group Sz(32):5 is maximal in E_8(5)

Modular Invariants for Lattice Polarized K3 Surfaces

We study the class of complex algebraic K3 surfaces admitting an embedding of H+E8+E8 inside the Neron-Severi lattice. These special K3 surfaces are classified by a pair of modular invariants, in the same manner that elliptic curves over the field of complex numbers are classified by the J-invariant. Via the canonical Shioda-Inose structure we construct a geometric correspondence relating K3 surfaces of the above type with abelian surfaces realized as cartesian products of two elliptic curves. We then use this correspondence to determine explicit formulas for the modular invariants.Comment: 29 pages, LaTe

The Master Equation for the Prepotential-Pub

The perturbative prepotential and the K\"ahler metric of the vector multiplets of the N=2 effective low-energy heterotic strings is calculated directly in N=1 six-dimensional toroidal compactifications of the heterotic string vacua. This method provides the solution for the one loop correction to the N=2 vector multiplet prepotential for compactifications of the heterotic string for any rank three and four models, as well for compactifications on $K_3 \times T^2$. In addition, we complete previous calculations, derived from string amplitudes, by deriving the differential equation for the third derivative of the prepotential with respect of the usual complex structure U moduli of the $T^2$ torus. Moreover, we calculate the one loop prepotential, using its modular properties, for N=2 compactifications of the heterotic string exhibiting modular groups similar with those appearing in N=2 sectors of N=1 orbifolds based on non-decomposable torus lattices and on N=2 supersymmetric Yang-Mills.Comment: Shorter version of hep-th/9802099 to appear in Nuclear Physics

Conjugacy of Embeddings of Alternating Groups in Exceptional Lie Groups

We discuss conjugacy classes of embeddings of Alternating groups in Exceptional Lie groups. We settle the count of classes of embeddings in E8 of a subgroup Alt10 and its double cover. This involves computation and the reduction of the problems to relative eigenvector problems. We update previously published tables of embeddings. We comment on the improvements present in our table and on the remaining unsettled conjugacy questions

Tinkertoys for the Twisted D-Series

We study 4D N=2 superconformal field theories that arise from the compactification of 6D N=(2,0) theories of type D_N on a Riemann surface, in the presence of punctures twisted by a Z_2 outer automorphism. Unlike the untwisted case, the family of SCFTs is in general parametrized, not by M_{g,n}, but by a branched cover thereof. The classification of these SCFTs is carried out explicitly in the case of the D_4 theory, in terms of three-punctured spheres and cylinders, and we provide tables of properties of twisted punctures for the D_5 and D_6 theories. We find realizations of Spin(8) and Spin(7) gauge theories with matter in all combinations of vector and spinor representations with vanishing beta-function, as well as Sp(3) gauge theories with matter in the 3-index traceless antisymmetric representation.Comment: 75 pages, 270 figure

Modular Symmetries of Threshold Corrections for Abelian Orbifolds with Discrete Wilson Lines

The modular symmetries of string loop threshold corrections for gauge coupling constants are studied in the presence of discrete Wilson lines for all examples of abelian orbifolds, where the point group is realised by the action of Coxeter elements or generalised Coxeter elements on the root lattices of the Lie groups.Comment: 36 pages, Late

Mathematics Underlying the F-Theory/Heterotic String Duality in Eight Dimensions

One of the dualities in string theory, the F-theory/heterotic string duality in eight dimensions, predicts an interesting correspondence between two seemingly disparate geometrical objects. On one side of the duality there are elliptically fibered K3 surfaces with section. On the other side, one finds elliptic curves endowed with certain flat connections and complexified Kahler classes. This paper is part of a project aimed at establishing the rigorous mathematical results describing the geometry underlying the classical aspects of this duality. The task involves understanding and comparing the classical moduli spaces on the two sides.Comment: 43 pages, LaTe

Centralizers and Fixed Points of Automorphisms in Finite and Locally Finite Groups

This Habilitation thesis is a collection of former research of K. Ersoy. In Chapter 1, Introduction, an outline of his research program in general is given. • Chapter 2 is the paper of Ersoy, “Infinite Groups with an Anticentral Element”, [Ers12], appeared in Comm. Alg. 2012. • Chapter 3 is the paper of Ersoy, “Finite Groups with a Splitting Automor- phism of Odd Order”, [Ers16], appeared in Arch. Math. 2016. • Chapter 4 is the paper of Ersoy together with C.K. Gupta, with name “Locally Finite Groups with Centralizers of Finite Rank”, [EG], appeared in Comm. Alg. 2016. • Chapter 5 is the paper of Ersoy together with M. Kuzucuoğlu and P. Shumy- atsky, with name “Locally Finite Groups and Their Subgroups with Small Centralizers”, [EKS], appeared in J. Algebra, 2017. • Chapter 6 is the paper of Ersoy, “Centralizers of p-subgroups in Simple Locally Finite Groups”, [Ers19], that appeared in Glasgow Mathematical Journal, 2020. • Chapter 7 is the paper of Ersoy, together with A. Tortora and M. Tota, with name “On groups with all subgroups subnormal or soluble of bounded derived length”, [ETT], appeared in Glasgow Math. J., 201