The String Landscape: On Formulas for Counting Vacua

TL;DR

This paper develops formulas to count specific vacua in the string/M theory landscape, focusing on M-theory compactifications with G_2 holonomy, and explores their gauge theory dualities and symmetry breaking patterns.

Contribution

It introduces new formulas for counting vacua in M-theory compactifications and analyzes their gauge dualities and symmetry breaking differences.

Findings

Formulas for counting vacua in G_2 holonomy compactifications.

Identification of dual gauge theories with equal U(1) factors.

Differences between Wilson line and Higgs symmetry breaking patterns.

Abstract

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 G_2 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.