The String Landscape: On Formulas for Counting Vacua

We derive formulas for counting certain classes of vacua in the string/M theory landscape. We do so in the context of the moduli space of M-theory compactifications on singular manifolds with G2 holonomy. Particularly, we count the numbers of gauge theories with different gauge groups but equal numbers of U (1) factors which are dual to each other. The vacua correspond to various symmetry breaking patterns of grand unified theories. Counting these dual vacua is equivalent to counting the number of conjugacy classes of elements of finite order inside Lie groups. We also point out certain cases where the conventional expectation is that symmetry breaking patterns by Wilson lines and Higgs fields are the same, but we show they are in fact different

Differentiable stratified groupoids and a de Rham theorem for inertia spaces

We introduce the notions of a differentiable groupoid and a differentiable stratified groupoid, generalizations of Lie groupoids in which the spaces of objects and arrows have the structures of differentiable spaces, respectively differentiable stratified spaces, compatible with the groupoid structure. After studying basic properties of these groupoids including Morita equivalence, we prove a de Rham theorem for locally contractible differentiable stratified groupoids. We then focus on the study of the inertia groupoid associated to a proper Lie groupoid. We show that the loop and the inertia space of a proper Lie groupoid can be endowed with a natural Whitney B stratification, which we call the orbit Cartan type stratification. Endowed with this stratification, the inertia groupoid of a proper Lie groupoid becomes a locally contractible differentiable stratified groupoid

Counting Conjugacy Classes of Elements of Finite Order in Lie Groups

We use combinatorial techniques to derive explicit formulas for the number of conjugacy classes of elements of finite order in unitary, symplectic, and orthogonal Lie groups, as well as the number of such conjugacy classes whose elements have a specified number of distinct eigenvalues